From b1737afc25e6d2fecdd32ad4270205e789fcb73d Mon Sep 17 00:00:00 2001
From: lovebird <cgoflyn@gmail.com>
Date: Thu, 18 Jan 2024 09:35:47 +0100
Subject: [PATCH] paper:hydrogel 3dprint

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+<A name=1></a>Rauwendaal<br>
+<b>Polymer Extrusion</b><br>
+<hr>
+<A name=2></a><hr>
+<A name=3></a>Chris Rauwendaal<br>
+<b>Polymer Extrusion </b><br>
+5th Edition<br>
+With contributions from<br>Paul J. Gramann, Bruce A. Davis, and Tim A. Osswald<br>
+Hanser Publishers, Munich                  Hanser Publications, Cincinnati<br>
+<hr>
+<A name=4></a>The Author:<br>
+Dr. Chris Rauwendaal<br>
+10556 Combie Road # 6677, Auburn, CA 95602-8908, USA<br>
+Distributed in North and South America by: <br>
+Hanser Publications <br>
+6915 Valley Avenue, Cincinnati, Ohio 45244-3029, USA <br>
+Fax: (513) 527-8801 <br>
+Phone: (513) 527-8977 <br>
+www.hanserpublications.com<br>
+Distributed in all other countries by <br>
+Carl Hanser Verlag <br>
+Postfach 86 04 20, 81631 München, Germany <br>
+Fax: +49 (89) 98 48 09 <br>
+www.hanser-fachbuch.de<br>
+The use of general descriptive names, trademarks, etc., in this publication, even if the former are not <br>
+especial y identified, is not to be taken as a sign that such names, as understood by the Trade Marks and <br>
+Merchandise Marks Act, may accordingly be used freely by anyone. While the advice and information <br>
+in this book are believed to be true and accurate at the date of going to press, neither the author nor the <br>
+editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. <br>
+The publisher makes no warranty, express or implied, with respect to the material contained herein.<br>
+ <br>
+Library of Congress Cataloging-in-Publication Data<br>
+ Rauwendaal, Chris.<br>
+  Polymer extrusion / Chris Rauwendaal. -- 5th edition.<br>
+       pages cm<br>
+  ISBN 978-1-56990-516-6 (hardcover) -- ISBN 978-1-56990-539-5 (e-book)  1.  Plastics--Extrusion.  I. <br>
+Title. <br>
+  TP1175.E9R37 2013<br>
+  668.4'13--dc23<br>
+                                                            2013037810<br>
+Bibliografische Information Der Deutschen Bibliothek <br>
+Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; <br>
+detaillierte bibliografische Daten sind im Internet über &lt;http://dnb.d-nb.de&gt; abrufbar.<br>
+ISBN  978-1-56990-516-6<br>
+E-Book ISBN  978-1-56990-539-5<br>
+All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, <br>
+electronic or mechanical, including photocopying or by any information storage and retrieval system, <br>
+without permission in writing from the publisher.<br>© Carl Hanser Verlag, Munich 2014 <br>
+Production Management: Steffen Jörg <br>
+Coverconcept: Marc Müller-Bremer, www.rebranding.de, München<br>
+Coverdesign: Stephan Rönigk<br>
+Typsetted, printed and bound by Kösel, Krugzel<br>
+Printed in Germany<br>
+<hr>
+<A name=5></a>The Author:<br>
+Dr. Chris Rauwendaal<br>
+<b>Preface to the </b><br>
+10556 Combie Road # 6677, Auburn, CA 95602-8908, USA<br>
+<b>Fifth Edition</b><br>
+Distributed in North and South America by: <br>
+Hanser Publications <br>
+6915 Valley Avenue, Cincinnati, Ohio 45244-3029, USA <br>
+Fax: (513) 527-8801 <br>
+Phone: (513) 527-8977 <br>
+It has been twelve years since the fourth edition of Polymer Extrusion was pub-<br>
+www.hanserpublications.com<br>
+lished and twenty six years since the book was first published. Extrusion technology <br>continues to advance; as a result, a fifth edition is needed to keep the Polymer Extru-<br>
+Distributed in all other countries by <br>
+Carl Hanser Verlag <br>
+sion book up to date and relevant.<br>
+Postfach 86 04 20, 81631 München, Germany <br>
+Fax: +49 (89) 98 48 09 <br>
+New material has been added throughout the book. The general literature survey <br>
+www.hanser-fachbuch.de<br>
+has been updated since several books on extrusion have been published since 2001. <br>A new theory for predicting developing melt temperatures has been incorporated <br>
+The use of general descriptive names, trademarks, etc., in this publication, even if the former are not <br>
+especial y identified, is not to be taken as a sign that such names, as understood by the Trade Marks and <br>
+into Chapter 7. This theory allows accurate prediction of changes in melt tempera-<br>
+Merchandise Marks Act, may accordingly be used freely by anyone. While the advice and information <br>
+ture along the length of the extruder; complete analytical solutions are presented <br>
+in this book are believed to be true and accurate at the date of going to press, neither the author nor the <br>
+editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. <br>
+to the relevant equations. As a result, melt temperatures can be predicted without <br>
+The publisher makes no warranty, express or implied, with respect to the material contained herein.<br>
+having to result to numerical techniques and computer simulation.<br>In Chapter 8, a new section on efficient extrusion of medical devices has been added. <br>
+ <br>
+Library of Congress Cataloging-in-Publication Data<br>
+It covers good manufacturing practices in medical extrusion and automation. The <br>
+ Rauwendaal, Chris.<br>
+  Polymer extrusion / Chris Rauwendaal. -- 5th edition.<br>
+effect of processing conditions and screw design on molecular degradation is covered <br>
+       pages cm<br>
+in detail. Screws designs that minimize molecular degradation are discussed and <br>
+  ISBN 978-1-56990-516-6 (hardcover) -- ISBN 978-1-56990-539-5 (e-book)  1.  Plastics--Extrusion.  I. <br>
+Title. <br>
+explained.<br>
+  TP1175.E9R37 2013<br>
+In Chapter 11, the section on gels in extruded products has been expanded as this <br>
+  668.4'13--dc23<br>
+                                                            2013037810<br>
+continues to be a problem experienced by many extrusion companies. There is also <br>a new section on discolored specks in extruded products. In this section expressions <br>
+Bibliografische Information Der Deutschen Bibliothek <br>
+are included that allow prediction of the incidence and frequency of specks or gels <br>
+Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; <br>
+based on their frequency in the incoming raw material. Included is a discussion on <br>
+detaillierte bibliografische Daten sind im Internet über &lt;http://dnb.d-nb.de&gt; abrufbar.<br>
+new instruments that are now available to detect defects in pellets produced at the <br>
+ISBN  978-1-56990-516-6<br>
+resin supplier with the ability to remove pellets with defects from the pellet stream.<br>
+E-Book ISBN  978-1-56990-539-5<br>
+Over the past five to ten years very high speed single screw extruders have been <br>
+All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, <br>
+developed. These extruders are now commercially available and they are used <br>
+electronic or mechanical, including photocopying or by any information storage and retrieval system, <br>
+without permission in writing from the publisher.<br>
+by dozens of companies around the world. These machines run at speeds up to <br>
+© Carl Hanser Verlag, Munich 2014 <br>
+1500 rpm; they achieve outputs that are about an order of magnitude above those of <br>
+Production Management: Steffen Jörg <br>
+conventional extruders. This high speed single screw extruder technology is one of <br>
+Coverconcept: Marc Müller-Bremer, www.rebranding.de, München<br>
+the most significant developments. Therefore, this topic has been added to the new <br>
+Coverdesign: Stephan Rönigk<br>
+Typsetted, printed and bound by Kösel, Krugzel<br>
+edition in Chapter 2.<br>
+Printed in Germany<br>
+<hr>
+<A name=6></a><b>VI</b>  <b>Preface to the Fifth Edition</b><br>
+The author would like to thank Professor Jürgen Miethlinger and Michael Aigner from <br>Johannes Kepler University in Linz, Austria for checking equations in Chapter 7 and <br>finding a few mistakes; these have been corrected.<br>The author would also like to thank Cheryl Hamilton and Nadine Warkotsch at Carl <br>Hanser Verlag for their encouragement and help in making the fifth edition a reality. <br>The author is grateful for a long and fruitful relationship with Carl Hanser Verlag. <br>Finally, I would like to extend a special thank you to my wife Sietske. She has sup-<br>ported and helped me in many ways for the past forty years. I am grateful for her <br>love and support--I feel very fortunate to have a friend and spouse who makes life <br>worth living.<br>
+<i>Chris Rauwendaal </i><br>
+<i> </i><br>
+<i>Auburn, California </i><br>
+<i> </i><br>
+<i>October 2013</i><br>
+<hr>
+<A name=7></a><b>Contents</b><br>
+<b>Preface to the Fifth Edition</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . V<br>
+<b>1  Introduction</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 1<br>1.1  Basic Process   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 1<br>1.2  Scope of the Book   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 2<br>1.3  General Literature Survey   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 3<br>1.4  History of Polymer Extrusion  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 6<br>
+<b>Part I ­ Extrusion Machinery</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 11<br><b>2  Different Types of Extruders</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 13<br>2.1  The Single Screw Extruder  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 13<br>
+2.1.1 <br>
+Basic Operation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 15<br>
+2.1.2  Vented Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 16<br>
+2.1.3  Rubber Extruders  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 17<br>
+2.1.4  High-Speed Extrusion   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 22<br>
+2.1.4.1  Melt Temperature   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 22<br>
+2.1.4.2  Extruders without Gear Reducer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23<br>
+2.1.4.3  Energy Consumption   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23<br>
+2.1.4.4  Change-over Resin Consumption  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23<br>
+2.1.4.5  Change-over Time and Residence Time  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 24<br>
+2.2  The Multiscrew Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 24<br>
+2.2.1  The Twin Screw Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 24<br>
+2.2.2  The Multiscrew Extruder With More Than Two Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 25<br>
+2.2.3  The Gear Pump Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 27<br>
+2.3  Disk Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 28<br>
+2.3.1  Viscous Drag Disk Extruders  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 28<br>
+2.3.1.1  Stepped Disk Extruder  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 28<br>
+2.3.1.2  Drum Extruder  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 30<br>
+2.3.1.3  Spiral Disk Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 31<br>
+2.3.1.4  Diskpack Extruder  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 31<br>
+2.3.2  The Elastic Melt Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 35<br>
+2.3.3  Overview of Disk Extruders  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 36<br>
+2.4  Ram Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 36<br>
+2.4.1  Single Ram Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 37<br>
+2.4.1.1  Solid State Extrusion   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 38<br>
+2.4.2  Multi Ram Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 41<br>
+<hr>
+<A name=8></a><b>VIII</b>  <b>Contents</b><br>
+2.4.3  Appendix 2.1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 42<br>
+2.4.3.1  Pumping Efficiency in Diskpack Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 42<br>
+<b>3  Extruder Hardware</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 49<br>3.1  Extruder Drive   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 49<br>
+3.1.1 <br>
+AC Motor Drive System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 49<br>
+3.1.1.1 <br>
+Mechanical Adjustable Speed Drive   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 50<br>
+3.1.1.2  Electric Friction Clutch Drive   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 50<br>
+3.1.1.3  Adjustable Frequency Drive   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 51<br>
+3.1.2  DC Motor Drive System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 53<br>
+3.1.2.1  Brushless DC Drives   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 55<br>
+3.1.3  Hydraulic Drive System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 55<br>
+3.1.4  Comparison of Various Drive Systems  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 58<br>
+3.1.5  Reducer   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 59<br>
+3.1.6  Constant Torque Characteristics   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 60<br>
+3.2  Thrust Bearing Assembly  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 61<br>3.3  Barrel and Feed Throat  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 64<br>3.4  Feed Hopper   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 68<br>3.5  Extruder Screw  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 70<br>3.6  Die Assembly   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 72<br>
+3.6.1  Screens and Screen Changers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 72<br>
+3.7  Heating and Cooling Systems   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 75<br>
+3.7.1  Electric Heating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 75<br>
+3.7.1.1 <br>
+Resistance Heating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 75<br>
+3.7.1.2  Induction Heating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 76<br>
+3.7.2  Fluid Heating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 77<br>
+3.7.3  Extruder Cooling   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 77<br>
+3.7.4  Screw Heating and Cooling   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 80<br>
+<b>4  Instrumentation and Control</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 85<br>4.1  Instrumentation Requirements   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  85<br>
+4.1.1 <br>
+Most Important Parameters   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 86<br>
+4.2  Pressure Measurement   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 87<br>
+4.2.1  The Importance of Melt Pressure  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 87<br>
+4.2.2  Different Types of Pressure Transducers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 88<br>
+4.2.3  Mechanical Considerations   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  92<br>
+4.2.4  Specifications   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 93<br>
+4.2.5  Comparisons of Different Transducers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 96<br>
+4.3  Temperature Measurement   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 96<br>
+4.3.1  Methods of Temperature Measurement  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 97<br>
+4.3.2  Barrel Temperature Measurement  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 100<br>
+4.3.3  Stock Temperature Measurement   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 102<br>
+4.3.3.1  Ultrasound Transmission Time   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 105<br>
+4.3.3.2  Infrared Melt Temperature Measurement  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 106<br>
+4.4  Other Measurements   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 107<br>
+4.4.1  Power Measurement  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 107<br>
+<hr>
+<A name=9></a> Contents <br>
+<b>IX</b><br>
+4.4.2  Rotational Speed  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 109<br>
+4.4.3  Extrudate Thickness   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 110<br>
+4.4.4  Extrudate Surface Conditions   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 114<br>
+4.5  Temperature Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 116<br>
+4.5.1  On-Off Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 117<br>
+4.5.2  Proportional Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 119<br>
+4.5.2.1  Proportional-Only Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 119<br>
+4.5.2.2  Proportional and Integral Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 123<br>
+4.5.2.3  Proportional and Integral and Derivative Control  .  .  .  .  .  .  .  .  .  .  .  .  . 124<br>
+4.5.2.4  Dual Sensor Temperature Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 125<br>
+4.5.3  Controllers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 126<br>
+4.5.3.1  Temperature Controllers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 126<br>
+4.5.3.2  Power Controllers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 126<br>
+4.5.3.3  Dual Output Controllers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 128<br>
+4.5.4  Time-Temperature Characteristics  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 128<br>
+4.5.4.1  Thermal Characteristics of the System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 128<br>
+4.5.4.2  Modeling of Response in Linear Systems   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 130<br>
+4.5.4.3  Temperature Characteristics with On-Off Control  .  .  .  .  .  .  .  .  .  .  .  .  . 133<br>
+4.5.5  Tuning of the Controller Parameters   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 135<br>
+4.5.5.1  Performance Criteria   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 135<br>
+4.5.5.2  Effect of PID Parameters   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 136<br>
+4.5.5.3  Tuning Procedure When Process Model Is Unknown  .  .  .  .  .  .  .  .  .  . 137<br>
+4.5.5.4  Tuning Procedure When Process Model Is Known   .  .  .  .  .  .  .  .  .  .  .  . 138<br>
+4.5.5.5  Pre-Tuned Temperature Controllers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 139<br>
+4.5.5.6  Self-Tuning Temperature Controllers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 140<br>
+4.6  Total Process Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 140<br>
+4.6.1  True Total Extrusion Process Control   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 141<br>
+<b>Part II ­ Process Analysis</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 147<br><b>5  Fundamental Principles</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 149<br>5.1  Balance Equations   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 149<br>
+5.1.1 <br>
+The Mass Balance Equation  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 149<br>
+5.1.2  The Momentum Balance Equation  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 150<br>
+5.1.3  The Energy Balance Equation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 151<br>
+5.2  Basic Thermodynamics   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 153<br>
+5.2.1  Rubber Elasticity   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 157<br>
+5.2.2  Strain-Induced Crystallization  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 159<br>
+5.3  Heat Transfer   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 160<br>
+5.3.1  Conductive Heat Transfer   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 160<br>
+5.3.2  Convective Heat Transfer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 161<br>
+5.3.3  Dimensionless Numbers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 161<br>
+5.3.3.1  Dimensional Analysis   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 161<br>
+5.3.3.2  Important Dimensionless Numbers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 163<br>
+5.3.4  Viscous Heat Generation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 168<br>
+5.3.5  Radiative Heat Transport   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 169<br>
+5.3.5.1  Dielectric Heating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 171<br>
+5.3.5.2  Microwave Heating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 173<br>
+<hr>
+<A name=10></a><b>X</b> <br>
+<b>Contents</b><br>
+5.4  Basics of Devolatilization   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 175<br>
+5.4.1  Devolatilization of Particulate Polymer   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 180<br>
+5.4.2  Devolatilization of Polymer Melts   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 181<br>
+Appendix 5.1   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 185<br>Example: Pipe Flow of Newtonian Fluid   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 185<br>
+<b>6  Important Polymer Properties</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 191<br>6.1  Properties of Bulk Materials   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 191<br>
+6.1.1 <br>
+Bulk Density  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 191<br>
+6.1.2  Coefficient of Friction  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 194<br>
+6.1.3  Particle Size and Shape   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 200<br>
+6.1.4  Other Properties  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 201<br>
+6.2  Melt Flow Properties   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 202<br>
+6.2.1  Basic Definitions   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 202<br>
+6.2.2  Power Law Fluid  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 208<br>
+6.2.3  Other Fluid Models   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 213<br>
+6.2.4  Effect of Temperature and Pressure  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 214<br>
+6.2.5  Viscoelastic Behavior   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 218<br>
+6.2.6  Measurement of Flow Properties  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 220<br>
+6.2.6.1  Capillary Rheometer   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 220<br>
+6.2.6.2  Melt Index Tester   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 223<br>
+6.2.6.3  Cone and Plate Rheometer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 227<br>
+6.2.6.4  Slit Die Rheometer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 228<br>
+6.2.6.5  Dynamic Analysis   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 230<br>
+6.3  Thermal Properties  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 233<br>
+6.3.1  Thermal Conductivity   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 234<br>
+6.3.2  Specific Volume and Morphology   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 236<br>
+6.3.3  Specific Heat and Heat of Fusion  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 239<br>
+6.3.4  Specific Enthalpy   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 241<br>
+6.3.5  Thermal Diffusivity   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 242<br>
+6.3.6  Melting Point   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 245<br>
+6.3.7  Induction Time  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 245<br>
+6.3.8  Thermal Characterization   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 247<br>
+6.3.8.1  DTA and DSC   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 247<br>
+6.3.8.2  TGA   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 247<br>
+6.3.8.3  TMA   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 248<br>
+6.3.8.4  Other Thermal Characterization Techniques   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 248<br>
+6.4  Polymer Property Summary  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 248<br>
+<b>7  Functional Process Analysis</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 255<br>7.1  Basic Screw Geo metry   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 255<br>7.2  Solids Conveying   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 258<br>
+7.2.1  Gravity Induced Solids Conveying  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 259<br>
+7.2.1.1 <br>
+Pressure Distribution  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 262<br>
+7.2.1.2  Flow Rate   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 265<br>
+7.2.1.3  Design Criteria  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 266<br>
+<hr>
+<A name=11></a> Contents <br>
+<b>XI</b><br>
+7.2.2  Drag Induced Solids Conveying   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 268<br>
+7.2.2.1  Frictional Heat Generation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 282<br>
+7.2.2.2  Grooved Barrel Sections   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 284<br>
+7.2.2.3  Adjustable Grooved Barrel Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 296<br>
+7.2.2.4  Starve Feeding Versus Flood Feeding   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 302<br>
+7.3  Plasticating  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 305<br>
+7.3.1  Theoretical Model of Contiguous Solids Melting  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 306<br>
+7.3.1.1 <br>
+Non-Newtonian, Non-Isothermal Case  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 316<br>
+7.3.2  Other Melting Models   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 326<br>
+7.3.3  Power Consumption in the Melting Zone   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 330<br>
+7.3.4  Computer Simulation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 332<br>
+7.3.5  Dispersed Solids Melting   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 333<br>
+7.4  Melt Conveying  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 340<br>
+7.4.1  Newtonian Fluids   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 343<br>
+7.4.1.1 <br>
+Effect of Flight Flanks   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 348<br>
+7.4.1.2  Effect of Clearance  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 350<br>
+7.4.1.3  Power Consumption in Melt Conveying   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 353<br>
+7.4.2  Power Law Fluids   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 356<br>
+7.4.2.1  One-Dimensional Flow  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 356<br>
+7.4.2.2  Two-Dimensional Flow  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 361<br>
+7.4.3  Non-Isothermal Analysis   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 367<br>
+7.4.3.1  Newtonian Fluids with Negligible Viscous Dissipation   .  .  .  .  .  .  .  . 367<br>
+7.4.3.2  Non-Isothermal Analysis of Power Law Fluids   .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 374<br>
+7.4.3.3  Developing Temperatures   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 387<br>
+7.4.3.4  Estimating Fully Developed Melt Temperatures   .  .  .  .  .  .  .  .  .  .  .  .  .  . 404<br>
+7.4.3.5  Assumption of Stationary Screw and Rotating Barrel  .  .  .  .  .  .  .  .  .  . 411<br>
+7.5  Die Forming   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 419<br>
+7.5.1  Velocity and Temperature Profiles  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 420<br>
+7.5.2  Extrudate Swell   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 429<br>
+7.5.3  Die Flow Instabilities   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 431<br>
+7.5.3.1  Shark Skin   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 431<br>
+7.5.3.2  Melt Fracture   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 432<br>
+7.5.3.3  Draw Resonance  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 434<br>
+7.6  Devolatilization  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 435<br>7.7  Mixing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 441<br>
+7.7.1 <br>
+Mixing in Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 442<br>
+7.7.1.1 <br>
+Distributive Mixing in Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 447<br>
+7.7.2  Static Mixing Devices  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 457<br>
+7.7.2.1  Geo metry of Static Mixers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 460<br>
+7.7.2.2  Functional Performance Characteristics  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 464<br>
+7.7.2.3  Miscellaneous Considerations  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 468<br>
+7.7.3  Dispersive Mixing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 469<br>
+7.7.3.1  Solid-Liquid Systems   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 469<br>
+7.7.3.2  Liquid-Liquid System  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 471<br>
+7.7.4  Backmixing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 483<br>
+7.7.4.1  Cross-Sectional Mixing and Axial Mixing  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 483<br>
+<hr>
+<A name=12></a><b>XII</b>  <b>Contents</b><br>
+7.7.4.2  Residence Time Distribution   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 485<br>
+7.7.4.3  RTD in Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 487<br>
+7.7.4.4  Methods to Improve Backmixing  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 488<br>
+7.7.4.5  Conclusions for Backmixing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 490<br>
+Appendix 7.1   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 491<br>Appendix 7.2   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 492<br>Appendix 7.3   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 493<br>
+<b>8  Extruder Screw Design</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 509<br>8.1  Mechanical Considerations   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 510<br>
+8.1.1 <br>
+Torsional Strength of the Screw Root  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 510<br>
+8.1.2  Strength of the Screw Flight   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 512<br>
+8.1.3  Lateral Deflection of the Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 514<br>
+8.2  Optimizing for Output   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 519<br>
+8.2.1  Optimizing for Melt Conveying   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 519<br>
+8.2.2  Optimizing for Plasticating   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 530<br>
+8.2.2.1  Effect of Helix Angle   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 532<br>
+8.2.2.2  Effect of Multiple Flights   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 532<br>
+8.2.2.3  Effect of Flight Clearance  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 534<br>
+8.2.2.4  Effect of Compression Ratio  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 536<br>
+8.2.3  Optimizing for Solids Conveying   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 537<br>
+8.2.3.1  Effect of Channel Depth  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 537<br>
+8.2.3.2  Effect of Helix Angle   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 538<br>
+8.2.3.3  Effect of Number of Flights   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 539<br>
+8.2.3.4  Effect of Flight Clearance  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 539<br>
+8.2.3.5  Effect of Flight Geometry  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 539<br>
+8.3  Optimizing for Power Consumption   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 540<br>
+8.3.1  Optimum Helix Angle   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 541<br>
+8.3.2  Effect of Flight Clearance  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 543<br>
+8.3.3  Effect of Flight Width  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 544<br>
+8.4  Single-Flighted Extruder Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 548<br>
+8.4.1  The Standard Extruder Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 549<br>
+8.4.2  Modifications of the Standard Extruder Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 550<br>
+8.5  Devolatilizing Extruder Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 553<br>
+8.5.1  Functional Design Considerations  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 554<br>
+8.5.2  Various Vented Extruder Screw Designs   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 558<br>
+8.5.2.1  Conventional Vented Extruder Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 558<br>
+8.5.2.2  Bypass Vented Extruder Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 559<br>
+8.5.2.3  Rearward Devolatilization   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 560<br>
+8.5.2.4  Multi-Vent Devolatilization   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 561<br>
+8.5.2.5  Cascade Devolatilization   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 562<br>
+8.5.2.6  Venting through the Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 563<br>
+8.5.2.7  Venting through a Flighted Barrel  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 564<br>
+8.5.3  Vent Port Configuration   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 565<br>
+8.6  Multi-Flighted Extruder Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 568<br>
+8.6.1  The Conventional Multi-Flighted Extruder Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 568<br>
+<hr>
+<A name=13></a> Contents <br>
+<b>XIII</b><br>
+8.6.2  Barrier Flight Extruder Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 569<br>
+8.6.2.1  The Maillefer Screw  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 572<br>
+8.6.2.2  The Barr Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 575<br>
+8.6.2.3  The Dray and Lawrence Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 578<br>
+8.6.2.4  The Kim Screw  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 579<br>
+8.6.2.5  The Ingen Housz Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 580<br>
+8.6.2.6  The CRD Barrier Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 581<br>
+8.6.2.7  Summary of Barrier Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 582<br>
+8.7  Mixing Screws   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 584<br>
+8.7.1  Dispersive Mixing Elements   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 584<br>
+8.7.1.1 <br>
+The CRD Mixer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 602<br>
+8.7.1.2  Mixers to Break Up the Solid Bed   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 616<br>
+8.7.1.3  Summary of Dispersive Mixers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 618<br>
+8.7.2  Distributive Mixing Elements   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 619<br>
+8.7.2.1  Ring or Sleeve Mixers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 621<br>
+8.7.2.2  Variable Depth Mixers  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 622<br>
+8.7.2.3  Summary of Distributive Mixers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 623<br>
+8.8  Efficient Extrusion of Medical Devices   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 624<br>
+8.8.1  Introduction   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 624<br>
+8.8.2  Good Manufacturing Practices in Medical Extrusion   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 625<br>
+8.8.3  Automation of the Medical Extrusion Process  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 625<br>
+8.8.4  Minimizing Polymer Degradation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 626<br>
+8.8.5  Melt Temperatures Inside the Extruder   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 626<br>
+8.8.6  Melt Temperatures and Screw Design   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 627<br>
+8.8.7  Molecular Degradation and Screw Design  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 630<br>
+8.8.8  Conclusions   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 634<br>
+8.9  Scale-Up   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 635<br>
+8.9.1  Common Scale-Up Factors   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 635<br>
+8.9.2  Scale-Up for Heat Transfer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 638<br>
+8.9.3  Scale-Up for Mixing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 639<br>
+8.9.4  Comparison of Various Scale-Up Methods  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 640<br>
+8.10  Rebuilding Worn Screws and Barrels   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 642<br>
+8.10.1  Application of Hardfacing Materials   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 644<br>
+8.10.1.1  Oxyacetylene Welding   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 644<br>
+8.10.1.2  Tungsten Inert Gas Welding   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 645<br>
+8.10.1.3  Plasma Transfer Arc Welding   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 645<br>
+8.10.1.4  Metal Inert Gas Welding   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 646<br>
+8.10.1.5  Laser Hardfacing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 646<br>
+8.10.2  Rebuilding of Extruder Barrels   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 646<br>
+<b>9  Die Design</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 653<br>9.1  Basic Considerations   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 654<br>
+9.1.1 <br>
+Balancing the Die by Adjusting the Land Length   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 655<br>
+9.1.2  Balancing by Channel Height   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 659<br>
+9.1.3  Other Methods of Die Balancing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 662<br>
+<hr>
+<A name=14></a><b>XIV</b>  <b>Contents</b><br>
+9.2  Film and Sheet Dies   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 663<br>
+9.2.1  Flow Adjustment in Sheet and Film Dies  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 664<br>
+9.2.2  The Horseshoe Die   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 667<br>
+9.3  Pipe and Tubing Dies   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 668<br>
+9.3.1  Tooling Design for Tubing   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 671<br>
+9.3.1.1  Definitions of Various Draw Ratios   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 672<br>
+9.3.1.2  Land Length  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 673<br>
+9.3.1.3  Taper Angles   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 674<br>
+9.3.1.4  Special Features  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 676<br>
+9.4  Blown Film Dies   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 676<br>
+9.4.1  The Spiral Mandrel Geometry   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 679<br>
+9.4.2  Effect of Die Geometry on Flow Distribution  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 680<br>
+9.4.3  Summary of Spiral Mandrel Die Design Variables   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 684<br>
+9.5  Profile Extrusion Dies   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 684<br>9.6  Coextrusion   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 686<br>
+9.6.1  Interface Distortion   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 690<br>
+9.7  Calibrators   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 692<br>
+<b>10  Twin Screw Extruders</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 697<br>10.1  Introduction   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 697<br>10.2  Twin versus Single Screw Extruder  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 699<br>10.3  Intermeshing Co-Rotating Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 701<br>
+10.3.1  Closely Intermeshing Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 701<br>
+10.3.2  Self-Wiping Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 704<br>
+10.3.2.1  Geometry of Self-Wiping Extruders  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 705<br>
+10.3.2.2  Conveying in Self-Wiping Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 713<br>
+10.4  Intermeshing Counter-Rotating Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 720<br>10.5  Non-Intermeshing Twin Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 730<br>10.6  Coaxial Twin Screw Extruders  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 743<br>10.7  Devolatilization in Twin Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 745<br>10.8  Commercial Twin Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 749<br>
+10.8.1  Screw Design Issues for Co-Rotating Twin Screw Extruders   .  .  .  .  .  .  .  .  .  .  . 753<br>
+10.8.2  Scale-Up in Co-Rotating Twin Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 756<br>
+10.9  Overview of Twin Screw Extruders   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 758<br>
+<b>11  Troubleshooting Extruders</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 763<br>11.1  Requirements for Efficient Troubleshooting   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 763<br>
+11.1.1  Instrumentation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 764<br>
+11.1.2  Understanding of the Extrusion Process   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 764<br>
+11.1.3  Collect and Analyze Historical Data (Timeline)  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 765<br>
+11.1.4  Team Building   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 766<br>
+11.1.5  Condition of the Equipment  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 766<br>
+11.1.6  Information on the Feedstock   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 767<br>
+11.2  Tools for Troubleshooting  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 768<br>
+11.2.1  Temperature Measurement Devices   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 768<br>
+<hr>
+<A name=15></a> Contents <br>
+<b>XV</b><br>
+11.2.2  Data Acquisition Systems (DAS)   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 769<br>
+11.2.2.1  Portable Data Collectors/Machine Analyzers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 769<br>
+11.2.2.2  Fixed Station Data Acquisition Systems   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 770<br>
+11.2.3  Light Microscopy   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 772<br>
+11.2.4  Thermochromic Materials   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 773<br>
+11.2.5  Thermal Analysis   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 775<br>
+11.2.6  Miscellaneous Tools   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 775<br>
+11.3  Systematic Troubleshooting  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 776<br>
+11.3.1  Upsets versus Development Problems  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 776<br>
+11.3.2  Machine-Related Problems  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 776<br>
+11.3.2.1  Drive System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 776<br>
+11.3.2.2  The Feed System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 778<br>
+11.3.2.3  Different Feeding Systems  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 778<br>
+11.3.2.4  Heating and Cooling System   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 778<br>
+11.3.2.5  Wear Problems  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 780<br>
+11.3.2.6  Screw Binding   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 797<br>
+11.3.3  Polymer Degradation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 803<br>
+11.3.3.1  Types of Degradation  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 803<br>
+11.3.3.2  Degradation in Extrusion   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 807<br>
+11.3.4  Extrusion Instabilities   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 820<br>
+11.3.4.1  Frequency of Instability  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 821<br>
+11.3.4.2  Functional Instabilities   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 828<br>
+11.3.4.3  Solving Extrusion Instabilities   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 833<br>
+11.3.5  Air Entrapment   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 834<br>
+11.3.6  Gels, Gel Content, and Gelation   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 836<br>
+11.3.6.1  Measuring Gels   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 839<br>
+11.3.6.2  Gels Created in the Extrusion Process   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 840<br>
+11.3.6.3  Removing Gels Produced in Polymerization   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 841<br>
+11.3.7  Die Flow Problems   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 843<br>
+11.3.7.1  Melt Fracture   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 843<br>
+11.3.7.2  Die Lip Build-Up (Die Drool)   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 845<br>
+11.3.7.3  V- or W-Patterns  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 846<br>
+11.3.7.4  Specks and Discoloration  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 846<br>
+11.3.7.5  Lines in Extruded Product  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 851<br>
+11.3.7.6  Optical Properties   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 853<br>
+<b>12  Modeling and Simulation of the Extrusion Process</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 861<br>12.1  Introduction   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 861<br>12.2  Background   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 862<br>
+12.2.1  Analytical Techniques   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 862<br>
+12.2.2  Numerical Methods   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 864<br>
+12.2.2.1  Finite Difference Method  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 865<br>
+12.2.2.2  Finite Element Method   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 866<br>
+12.2.2.3  Boundary Element Method   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 867<br>
+12.2.3  Remeshing Techniques in Moving Boundary Problems   .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 868<br>
+12.2.4  Rheology  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 870<br>
+<hr>
+<A name=16></a><b>XVI</b>  <b>Contents</b><br>
+12.3  Simulating 3-D Flows with 2-D Models   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 871<br>
+12.3.1  Simulating Flows in Internal Batch Mixers with 2-D Models   .  .  .  .  .  .  .  .  .  .  . 871<br>
+12.3.2  Simulating Flows in Extrusion with 2-D Models   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 877<br>
+12.3.3  Simulating Flows in Extrusion Dies with 2-D Models   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 881<br>
+12.4  Three-Dimensional Simulation  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 885<br>
+12.4.1  Simulating Flows in the Banbury Mixer with Three-Dimensional Models   885<br>
+12.4.2  Simulating Flows in Extrusion Dies with 3-Dimensional Models   .  .  .  .  .  .  .  . 887<br>
+12.4.3  Simulating Flows in Extrusion with 3-Dimensional Models   .  .  .  .  .  .  .  .  .  .  .  . 892<br>
+12.4.3.1  Regular Conveying Screw   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 892<br>
+12.4.3.2  Energy Transfer Mixer  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 893<br>
+12.4.3.3  Twin Screw Extruder  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 895<br>
+12.4.3.4  Rhomboidal Mixers and Fluted Mixers (Leroy/Maddock)  .  .  .  .  .  .  . 901<br>
+12.4.3.5  Turbo-Screw  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 906<br>
+12.4.3.6  CRD Mixer   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 908<br>
+12.4.4  Static Mixers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 910<br>
+12.5  Conclusions   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 912<br>
+<b>Conversion Constants</b>  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 919<br>Length   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 919<br>Volume   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 919<br>Mass   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 920<br>Density   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 920<br>Force   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 920<br>Stress  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 921<br>Viscosity   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 921<br>Energy/Work  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 922<br>Power  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 922<br>Specific Energy   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 922<br>Thermal Conductivity   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 923<br>Temperature   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 923<br>
+<b>Index</b>   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 925<br>
+<hr>
+<A name=17></a><b>1 Introduction</b><br>
+<b></b><br>
+ <b>1.1Basic Process</b><br>
+The extruder is indisputably the most important piece of machinery in the polymer <br>processing industry. To extrude means to push or to force out. Material is extruded <br>when it is pushed through an opening. When toothpaste is squeezed out of a tube, it <br>is extruded. The part of the machine containing the opening through which the <br>material is forced is referred to as the extruder die. As material passes through the <br>die, the material acquires the shape of the die opening. This shape generally changes <br>to some extent as the material exits from the die. The extruded product is referred to <br>as the extrudate.<br>Many different materials are formed through an extrusion process: metals, clays, <br>ceramics, foodstuffs, etc. The food industry, in particular, makes frequent use of <br>extruders to make noodles, sausages, snacks, cereal, and numerous other items. In <br>this book, the materials that will be treated are confined to polymers or plastics.<br>Polymers can be divided into three main groups: thermoplastics, thermosets, and <br>elastomers. Thermoplastic materials soften when they are heated and solidify when <br>they are cooled. If the extrudate does not meet the specifications, the material can <br>generally be reground and recycled. Thus, the basic chemical nature of a thermo-<br>plastic usually does not change significantly as a result of the extrusion process. <br>Thermosets undergo a crosslinking reaction when the temperature is raised above a <br>certain temperature. This crosslinking bonds the polymer molecules together to <br>form a three-dimensional network. This network remains intact when the tempera-<br>ture is reduced again. Crosslinking causes an irreversible change in the material. <br>Therefore, thermosetting materials cannot be recycled as can thermoplastic mate-<br>rials. Elastomers or rubbers are materials capable of very large deformations with <br>the material behaving in a largely elastic manner. This means that when the deform-<br>ing force is removed, the material completely, or almost completely, recovers. The <br>emphasis of this book will be on thermoplastics; only a relatively small amount of <br>attention will be paid to thermosets and elastomers.<br>
+<hr>
+<A name=18></a><b>2</b> <br>
+<b>1Introduction</b><br>
+Materials can be extruded in the molten state or in the solid state. Polymers are <br>generally extruded in the molten state; however, some applications involve solid-<br>state extrusion of polymers. If the polymer is fed to the extruder in the solid state <br>and the material is melted as it is conveyed by the extruder screw from the feed port <br>to the die, the process is called plasticating extrusion. In this case, the extruder <br>performs an additional function, namely melting, besides the regular extrusion <br>function. Sometimes the extruder is fed with molten polymer; this is called melt fed <br>extrusion. In melt fed extrusion, the extruder acts purely as a pump, developing the <br>pressure necessary to force the polymer melt through the die.<br>There are two basic types of extruders: continuous and discontinuous or batch type <br>extruders. Continuous extruders are capable of developing a steady, continuous flow <br>of material, whereas batch extruders operate in a cyclic fashion. Continuous extrud-<br>ers utilize a rotating member for transport of the material. Batch extruders gener-<br>ally have a reciprocating member to cause transport of the material.<br>
+<b></b><br>
+ <b>1.2Scope of the Book</b><br>
+This book will primarily deal with plasticating extrusion in a continuous fashion. <br>Chapters 2, 3, and 4 deal with a description of extrusion machinery. Chapter 5 will <br>briefly review the fundamental principles that will be used in the analysis of the <br>extrusion process. Chapter 6 deals with the polymer properties important in the <br>extrusion process. This is a very important chapter, because one cannot understand <br>the extrusion process if one does not know the special characteristics of the mate-<br>rial to be extruded. Understanding just the extrusion machinery is not enough. <br> <br>Rheological and thermal properties of a polymer determine, to a large extent, the <br>characteristics of the extrusion process. The process engineer, therefore, should be <br>a mechanical or chemical engineer on the one hand, and a rheologist on the other <br>hand. Since most engineers have little or no formal training in polymer rheology, the <br>subject is covered in Chapter 6, inasmuch as it is necessary for the analysis of the <br>extrusion process.<br>Chapter 7 covers the actual analysis of the extrusion process. The process is ana-<br>lyzed in discrete functional zones with particular emphasis on developing a quanti-<br>tative understanding of the mechanisms operational in each zone.<br>The theory developed in Chapter 7 is applied to the design of extruder screws in <br>Chapter 8 and to the design of extruder dies in Chapter 9. Chapter 10 is devoted to <br>twin screw extruders. Twin screw extruders have become an increasingly important <br>branch of the extrusion industry. It is felt, therefore, that any book on extrusion can-<br>not ignore this type of extruder. Troubleshooting extruders is covered in Chapter 11. <br>
+<hr>
+<A name=19></a> <br>
+1.3General Literature Survey <br>
+<b>3</b><br>
+This is perhaps the most critical and most important function performed in indus-<br>trial operation of extruders. Extrusion problems causing downtime or off-spec pro-<br>ducts can become very costly in a short period of time: losses can often exceed the <br>purchase price of the entire machine in a few hours or a few days. Thus, it is very <br>important that the process engineer can troubleshoot quickly and accurately. This <br>requires a solid understanding of the operational principles of extruders and the <br>basic mechanisms behind them. Therefore, the chapter on troubleshooting is a prac-<br>tical application of the functional process analysis developed in Chapter 7.<br>The final chapter of the book, Chapter 12 on modeling and computer simulation was <br>added in the fourth edition. This chapter was written by Paul Gramann, Bruce Davis, <br>and Tim Osswald; three workers who have made substantial contributions to the <br>development of this branch of polymer processing. Computer aided engineering <br>(CAE) is now an integral part of extrusion engineering and a current book on extru-<br>sion would not be complete without dealing with this important subject.<br>The book is therefore divided in four main parts. The first part deals with the hard-<br>ware and/or mechanical aspects of extruders: this is covered in Chapters 2, 3, and 4. <br>The second part deals with the process analysis: this is covered in Chapters 5, 6, <br>and 7. The third part deals with practical applications of the extrusion theory: this <br>is covered in Chapters 8, 9, 10, and 11. Part four is the chapter on modeling and <br>computer simulation. Parts I, II, and IV can be studied independently; Part III, how-<br>ever, cannot be fully appreciated without studying Part II.<br>
+<b></b><br>
+ <b>1.3General Literature Survey</b><br>
+Considering the fact that there are already several books on extrusion of polymers, <br>the question can be asked why the need for another book on extrusion. The three <br>most comprehensive books written on extrusion as of the early 1980s were the <br>books by Bernhardt [1], Schenkel [2], and Tadmor [5]. The book by Bernhardt is on <br>polymer processing, but has a very good chapter on extrusion. It is well written and <br>describes extrusion theory and its practical applications to screw and die design. <br>Because of the age of the book, however, the extrusion theory is incomplete in that <br>it does not cover plasticating--this theory was developed later by Tadmor [5]--and <br>devolatilization theory.<br>The book by Schenkel [2] is a translation of the German text "Kunststoff Extruder-<br>Technik" [3], which is an extension of the original book "Schneckenpressen fuer <br>Kunststoffe" [4]. This book is very complete: dealing with flow properties of poly-<br>mers, extrusion theory, and design of extrusion equipment. The emphasis of Schen-<br>kel's book is on the mechanical or machinery aspects of extruders. As in Bernhardt's <br>
+<hr>
+<A name=20></a><b>4</b> <br>
+<b>1Introduction</b><br>
+book, the book by Schenkel suffers from the fact that it was written a long time ago, <br>in the early 1960s. Thus, the extrusion theory is incomplete and new trends in <br>extrusion machinery are absent. The book by Tadmor [5] is probably the most com-<br>plete theoretical treatise on extrusion. From a theoretical point of view, it is an out-<br>standing book. It is almost completely devoted to a detailed engineering analysis of <br>the extrusion process. Consequently, relatively little attention is paid to extrusion <br>machinery and practical applications to screw and die design. In order to fully <br>appreciate this book, the reader should possess a significant degree of mathematical <br>dexterity. The book is ideally suited for those who want to delve into detailed math-<br>ematical analysis and computer simulation of the extrusion process. The book is <br>less suited for those who need to design extruder screws and/or extruder dies, or <br>those who need to solve practical extrusion problems.<br>Another book on extrusion is "Plastics Extrusion Technology" edited by Hensen <br>et al. [44]. This is quite a comprehensive book dealing in detail with a number of <br>extrusion processes, such as compounding, pipe extrusion, profile extrusion, etc. <br>The book gives good information on the various extrusion operations with detailed <br>information on downstream equipment. This book is an English version of the two-<br>volume German original, "Kunststoff-Extrusionstechnik I und II" [45, 46]. Volume I, <br>Grundlagen (Fundamentals), covers the fundamental aspects such as rheology, ther-<br>modynamics, fluid flow analysis, single- and twin-screw extruders, die design, heat-<br>ing and cooling, etc. Volume II, Extrusionsanlagen, covers various extrusion lines; <br>this is the volume translated into English.<br>Another important addition to the extrusion literature is the book on twin screw <br>extrusion by White [47]. This book has an excellent coverage of the historical devel-<br>opment of various twin screw extruders. It also discusses in some detail recent <br>experimental work and developments in twin screw theory. This book covers inter-<br>meshing and non-intermeshing extruders, both co- and counter-rotating extruders. <br>Another extrusion process, which has been gaining interest and is being used more <br>widely, is reactive extrusion. A good book on this subject was edited by Xanthos <br>[48]; it covers applications, review, and engineering fundamentals of reactive extru-<br>sion.<br>Statistical process control in extrusion is covered by a book by Rauwendaal [49]; an <br>updated version of this book also covering injection molding and pre-control was <br>published in 2000 [56]. Mixing in extrusion processes is covered by a book edited <br>by Rauwendaal [50]. It covers basic aspects of mixing and mixing in various extru-<br>sion machinery, such as single screw extruders, twin screw extruders, reciprocating <br>screw extruders, internal mixers, and co-rotating disk processors. Another book on <br>mixing is the book edited by Manas-Zloczower and Tadmor [51]. This book covers <br>four major sections: mixing mechanisms and theory, modeling and flow visualiza-<br>tion, material considerations, and mixing practices. The book on mixing in polymer <br>processing [54] by Rauwendaal covers the basics of mixing and a comprehensive <br>
+<hr>
+<A name=21></a> <br>
+1.3General Literature Survey <br>
+<b>5</b><br>
+description of mixing machinery. This book is written as a self-study guide with <br>questions at the end of each chapter.<br>In order not to duplicate other texts on extrusion, this book will emphasize new <br>trends and developments in extrusion machinery. The extrusion theory will be co -<br>vered as completely as possible; however, the mathematical complexity will be kept <br>to a minimum. This is done to enhance the ease of applying theory to practical cases <br>and to make the book accessible to a larger number of people. A significant amount <br>of attention will be paid to practical application of the extrusion theory to screw and <br>die design and solving extrusion problems. After all, practical applications are most <br>important to practicing polymer process engineers or chemists.<br>Various other books on extrusion have been written [6­22, 52]. These books gener-<br>ally cover only specific segments of extrusion technology: e.g., screw design, twin <br>screw extrusion, etc., or provide only introductory material. Thus, it is felt that a <br>comprehensive book on polymer extrusion with recent information on machinery, <br>theory, and application does fulfill a need. There are several books on the more gen-<br>eral subject of polymer processing [24­36], in addition to the book by Bernhardt [7] <br>already mentioned. Many of these give a good review of the field of polymer process-<br>ing, some emphasizing the general principles involved in process analysis [24­33] <br>or machine design [34], while others concentrate more on a description of the pro-<br>cess machinery and products [35, 36]. Since the field of polymer processing is tre-<br>mendously large, books on the subject inevitably cover extrusion in less detail than <br>possible in a book devoted exclusively to extrusion.<br>The book "Understanding Extrusion" [55] is a very much-simplified and abbreviated <br>version of "Polymer Extrusion" without any equations in the main body of the text. <br>Even though the mathematical level of Polymer Extrusion is at approximately the <br>bachelors engineering level, the mathematics is too much for people who are not <br>interested in the engineering details. Understanding Extrusion became an immedi-<br>ate best seller, indicating that a non-engineering level book on extrusion can attract <br>a wide readership. This book is the basis of a computer based interactive training <br>program on extrusion, ITX®.<br>The book "Troubleshooting the Extrusion Process--A Systematic Approach to Solv-<br>ing Plastic Extrusion Problems" [59] by del Pilar Noriega and Rauwendaal was the <br>first book devoted exclusively to troubleshooting extrusion problems. A second edi-<br>tion of this book was published in 2010.<br>Chung's book "Extrusion of Polymers" [57] covers some of the same material as <br>Polymer Extrusion, however, it does not cover die design, troubleshooting, and mod-<br>eling and computer simulation. As such, the book is limited in scope; it also suffers <br>from the fact that most references are from before 1980. The book "Screw Extru-<br>sion: Technology and Science" [58] was edited by White and Potente. This book has <br>a broad scope and multiple contributors; it covers fundamentals, single screw extru-<br>
+<hr>
+<A name=22></a><b>6</b> <br>
+<b>1Introduction</b><br>
+sion technology, reciprocating extruders, single screw extruder analysis and design, <br>and multiscrew extrusion.<br>A recent book on extrusion is the book "Analyzing and Troubleshooting Single-<br>Screw Extruders" [60] by Campbell and Spalding. The authors claim that this is the <br>first book that focuses on the actual physics of the process-screw rotation physics. <br>This claim is not entirely correct considering that the 4th edition of this book already <br>examined this issue in detail, see Section 7.4.3.5.<br>
+<b></b><br>
+ <b>1.4History of Polymer Extrusion</b><br>
+The first machine for extrusion of thermoplastic materials was built around 1935 by <br>Paul Troester in Germany [37]. Before this time, extruders were primarily used for <br>extrusion of rubber. These earlier rubber extruders were steam-heated ram extrud-<br>ers and screw extruders; with the latter having very short length to diameter (L/D) <br>ratios, about 3 to 5. After 1935, extruders evolved into electrically heated screw <br>extruders with increased length. Around this time, the basic principle of twin screw <br>extruders for thermoplastics was conceived in Italy by Roberto Colombo of LMP. <br>He was working with Carlo Pasquetti on mixing cellulose acetate. Colombo devel-<br>oped an intermeshing co-rotating twin screw extruder. He obtained patents in many <br>different countries and several companies acquired the rights to use these patents. <br>Pasquetti followed a different concept and developed and patented the intermeshing <br>counter-rotating twin screw extruder.<br>The first detailed analyses of the extrusion process were concerned with the melt <br>conveying or pumping process. The earliest publication was an anonymous article <br>[38], which is often erroneously credited to Rowell and Finlayson, who wrote an arti-<br>cle of the same title in the same journal six years later [39]. Around 1950, scientific <br>studies of the extrusion process started appearing with increased frequency. In the <br>mid-fifties, the first quantitative study on solids conveying was published by Darnell <br>and Mol [40]. An important conference in the development of extrusion theories <br>was the 122nd ACS meeting in 1953. At this symposium, members of the Polychemi-<br>cals Department of E.I. DuPont de Nemours &amp; Co. presented the latest developments <br>in extrusion theory [41]. These members, Carley, Strub, Mallouk, McKelvey, and <br>Jepson were honored in 1983 by the SPE Extrusion Division for original develop-<br>ment of extrusion theories. In the mid-sixties, the first quantitative study on melt-<br>ing was published by Tadmor [42], based on earlier qualitative studies by Maddock <br>[43]. Thus, it was not until about 1965 that the entire extrusion process, from the <br>feed hopper to the die, could be described quantitatively. The theoretical work since <br>this time has concentrated, to a large extent, on generalizing and extending the <br>
+<hr>
+<A name=23></a> <br>
+1.4History of Polymer Extrusion <br>
+<b>7</b><br>
+extrusion theory and the development of numerical techniques and computer meth-<br>ods to solve equations that can no longer be solved by analytical methods.<br>As a result, there has been a shift in the affiliation of the workers involved in scien-<br>tific extrusion studies. While the early work was done mostly by investigators in the <br>polymer industry, later workers have been academicians. This has created some-<br>what of a gap between extrusion theoreticians and practicing extrusion technolo-<br>gists. This is aggravated by the fact that some workers are so concerned about the <br>scientific pureness of the work, which is commendable in itself, that it becomes <br>increasingly unappealing to the industrial process engineer who wants to apply the <br>work. One of the objectives of this book is to bridge this gap between theory and <br>practice by demonstrating in detail how the theory can be applied and by analyzing <br>the limitations of the theory.<br>Another interesting development in practical extrusion technology has been the <br>concept of feed-controlled extrusion. In this type of extrusion, the performance is <br>determined by the solids conveying zone of the extruder. By the use of grooves in <br>the first portion (close to the feed port) of the extruder barrel, the solids conveying <br>zone is capable of developing very high pressures and quite positive conveying char-<br>acteristics (i.e., throughput independent of pressure). In this case, the diehead pres-<br>sure does not need to be developed in the pumping or melt conveying zone; it is <br>developed in the solids conveying zone. The feed zone overrides both the plasticat-<br>ing and the melt conveying (pumping) zone.<br>Thus, the extruder becomes solids conveying or feed controlled. This concept has now <br>become an accepted standard in Western Europe, particularly in Germany. In the U.S., <br>this concept has been met with a substantial amount of reluctance and skepticism. <br>For the longest time, the few proponents of the feed-controlled extrusion were heavily <br>outnumbered by the many opponents. As a consequence, only a small fraction of the <br>extruders in the U.S. have been equipped with grooved barrel sections. However, <br>there does seem to be a trend towards increasing acceptance of this concept. This is <br>especially true in the blown film industry where the use of very high molecular weight <br>polyethylene has led many processors to use grooved feed extruders.<br>One approach to making the grooved feed extruder more attractive is to make it <br>adjustable. This can eliminate many of the disadvantages of current grooved feed <br>extruders. The grooved feed section can be made adjustable by changing the depth <br>of the grooves while the machine is operating. This makes the grooved feed extruder <br>more versatile and allows a greater degree of control of the extrusion process. <br>Adjustable grooved feed extruders are discussed in Chapter 7.<br>When a single screw extruder is used for demanding mixing and compounding <br>operations it is often preferred that the extruder is starve fed rather than flood fed. <br>Flood feeding often results in higher pressures inside the extruder and this can <br>lead to an agglomeration of powdered fillers. When this occurs it may be difficult or <br>
+<hr>
+<A name=24></a><b>8</b> <br>
+<b>1Introduction</b><br>
+impossible to disperse the agglomerates formed in the extruder. With starve feeding <br>the pressures inside the extruder are lower and can be controlled by adjusting the <br>feed rate and/or the screw speed. As a result, there is less risk of agglomeration <br>using starve feeding.<br>Around 2000, a new generation of mixing devices was developed to generate strong <br>elongational flow to improve mixing, particularly dispersive mixing. It has long been <br>known that elongational mixing is more effective than shear mixing. However, this <br>knowledge was not translated into actual mixing devices until recently. These new <br>mixing devices, such as the CRD mixer discussed in Chapter 7, use the same type <br>of mixing mechanism active in high-speed co-rotating twin screw extruders. As a <br>result, when these mixers are used in single screw extruders the mixing action can <br>be comparable to that of twin screw compounders. In fact, these mixers are now suc-<br>cessfully used in twin screw extruders as well.<br>The advent of effective new mixers for single screw extruders has improved the cap-<br>abilities of conventional single screw extruders. An interesting application may be <br>long single screw extruders (30 to 60D) with multiple downstream ports for com-<br>pounding and direct extrusion. When such machines are starve fed they can be used <br>in many applications now serviced by twin screw extruders. Because single screw <br>extruders are considerably less expensive to purchase and to operate, this approach <br>can offer substantial cost savings. This approach represents a significant departure <br>from conventional extrusion technology; as a result, it may take some time before it <br>is widely accepted in the extrusion industry. However, when a new technology makes <br>technical and economic sense and it works, then sooner or later it will be adopted.<br>Very high speed single screw extruders have been commercially available since <br>about 2005­2010. This is one of the most significant developments in single screw <br>extrusion over the past several decades. These relatively small extruders (50­75 mm) <br>that run at screw speeds as high as 1,000 to 1,500 rpm and achieve output rate about <br>an order of magnitude above the rate of conventional extruders.<br>
+References<br>1.  E.C. Bernhardt (Ed.), "Processing of Thermoplastic Materials," Reinhold, NY (1959)<br>2.  G. Schenkel, "Plastics Extrusion Technology and Theory," Illiffe Books Ltd., London <br>
+(1966), published in the USA by American Elsevier, NY (1966)<br>
+3.  G. Schenkel, "Kunststoff Extruder-Technik," Carl Hanser Verlag, Munich (1963)<br>4.  G. Schenkel, "Schneckenpressen fuer Kunststoffe," Carl Hanser Verlag, Munich (1959)<br>5.  Z. Tadmor and I. Klein, "Engineering Principles of Plasticating Extrusion," Van Nos-<br>
+trand Reinhold, NY (1970)<br>
+6.  H.R. Simonds, A.J. Weith, and W. Schack, "Extrusion of Rubber, Plastics and Metals," <br>
+Reinhold, NY (1952)<br>
+7.  E.G. Fisher, "Extrusion of Plastics," Illiffe Books Ltd., London (1954)<br>
+<hr>
+<A name=25></a> References <br>
+<b>9</b><br>
+8.  R. Jacobi, "Grundlagen der Extrudertechnik," Carl Hanser Verlag, Munich (1960)<br>9.  W. Mink, "Grundzuege der Extrudertechnik," Rudolf Zechner Verlag, Speyer am Rhein <br>
+(1963)<br>
+10.  A.L. Griff, "Plastics Extrusion Technology," Reinhold, NY (1968)<br>11.  R.T. Fenner, "Extruder Screw Design," Illiffe Books, Ltd., London (1970)<br>12.  N.M. Bikales (Ed.), "Extrusion and Other Plastics Operations," Wiley, NY (1971)<br>13.  P.N. Richardson, "Introduction to Extrusion," Society of Plastics Engineers, Inc. (1974)<br>14.  L.P.B.M. Janssen, "Twin Screw Extrusion," Elsevier, Amsterdam (1978)<br>15.  J.A. Brydson and D.G. Peacock, "Principles of Plastics Extrusion," Applied Science <br>
+Publishers Ltd., London (1973)<br>
+16.  F.G. Martelli, "Twin Screw Extrusion, A Basic Understanding," Van Nostrand Reinhold, <br>
+NY (1983)<br>
+17.  "Kunststoff-Verarbeitung im Gespraech, 2 Extrusion," BASF, Ludwigshafen (1971)<br>18.  "Der Extruder als Plastifiziereinheit," VDI-Verlag, Duesseldorf (1977)<br>19.  Levy, "Plastics Extrusion Technology Handbook," Industrial Press Inc., NY (1981)<br>20.  H. Potente, "Auslegen von Schneckenmaschinen-Baureihen, Modellgesetze und ihre <br>
+Anwendung," Carl Hanser Verlag, Munich (1981)<br>
+21.  H. Herrmann, "Schneckenmaschinen in der Verfahrenstechnik," Springer-Verlag, Ber-<br>
+lin (1972)<br>
+22.  W. Dalhoff, "Systematische Extruder-Konstruktion," Krausskopf-Verlag, Mainz (1974)<br>23.  E. Harms, "Kautschuk-Extruder, Aufbau und Einsatz aus verfahrenstechnischer Sicht," <br>
+Krausskopf-Verlag Mainz, Bd. 2, Buchreihe Kunststofftechnik (1974)<br>
+24.  J.M. McKelvey, "Polymer Processing," Wiley, NY (1962)<br>25.  R.M. Ogorkiewicz, "Thermoplastics: Effects of Processing," Illiffe Books Ltd., London <br>
+(1969)<br>
+26.  J.R.A. Pearson, "Mechanical Principles of Polymer Melt Processing," Pergamon, Oxford <br>
+(1966)<br>
+27.  S. Middleman, "The Flow of High Polymers," Interscience (1968)<br>28.  R.V. Torner, "Grundprozesse der Verarbeitung von Polymeren," VEB Deutscher Verlag <br>
+fuer Grundstoffindustrie, Leipzig (1973)<br>
+29.  S. Middleman, "Fundamentals of Polymer Processing," McGraw-Hill, NY (1977)<br>30.  H.L. Williams, "Polymer Engineering," Elsevier, Amsterdam (1975)<br>31.  J.L. Throne, "Plastics Process Engineering," Marcel Dekker, Inc., NY (1979)<br>32.  Z. Tadmor and C. Gogos, "Principles of Polymer Processing," Wiley, NY (1979)<br>33.  R.T. Fenner, "Principles of Polymer Processing," MacMillan Press Ltd., London (1979)<br>34.  N.S. Rao, "Designing Machines and Dies for Polymer Processing with Computer Pro-<br>
+grams," Carl Hanser Verlag, Munich (1981)<br>
+35.  J. Frados (Ed.), "Plastics Engineering Handbook," Van Nostrand Reinhold, NY (1976)<br>36.  S.S. Schwartz and S.H. Goodman, "Plastics Materials and Processes," Van Nostrand <br>
+Reinhold, NY (1982)<br>
+<hr>
+<A name=26></a><b>10</b>  <b>1Introduction</b><br>
+37.  M. Kaufman, Plastics &amp; Polymers, 37, 243 (1969)<br>38.  N.N., Engineering, 114, 606 (1922)<br>39.  H.S. Rowell and D. Finlayson, Engineering, 126, 249­250, 385­387, 678 (1928)<br>40.  W.H. Darnell and A.J. Mol, SPE Journal, 12, 20 (1956)<br>41.  J.F. Carley, R.A. Strub, R.S. Mallouk, J.M. McKelvey, and C.H. Jepson, 122nd Meeting <br>
+of the American Chemical Society, Atlantic City, NJ (1953). The seven papers were pub-<br>lished in Ind. Eng. Chem., 45, 970­992 (1953)<br>
+42.  Z. Tadmor, Polym. Eng. Sci., 6, 3, 1 (1966)<br>43.  B.H. Maddock, SPE Journal, 15, 383 (1959)<br>44.  F. Hensen, W. Knappe, and H. Potente (Eds.), "Plastics Extrusion Technology," Carl <br>
+Hanser Verlag, Munich (1988)<br>
+45.  F. Hensen, W. Knappe, and H. Potente (Eds.), "Handbuch der Kunststoff-Extrusions-<br>
+technik, Band I Grundlagen," Carl Hanser Verlag, Munich (1989)<br>
+46.  F. Hensen, W. Knappe, and H. Potente (Eds.), "Handbuch der Kunststoff-Extrusions-<br>
+technik, Band II Extrusionsanlagen," Carl Hanser Verlag, Munich (1989)<br>
+47.  J.L. White, "Twin Screw Extrusion," Carl Hanser Verlag, Munich (1991)<br>48.  M. Xanthos (Ed.) "Reactive Extrusion," Carl Hanser Verlag, Munich (1992)<br>49.  C. Rauwendaal, "Statistical Process Control in Extrusion," Carl Hanser Verlag, Munich <br>
+(1993)<br>
+50.  C. Rauwendaal (Ed.) "Mixing in Polymer Processing," Marcel Dekker, NY (1991)<br>51.  I. Manas-Zloczower and Z. Tadmor (Eds.), "Mixing and Compounding--Theory and Prac-<br>
+tice," Carl Hanser Verlag, Munich (1994)<br>
+52.  M.J. Stevens, "Extruder Principles and Operation," Elsevier Applied Science Publishers, <br>
+Essex, England (1985)<br>
+53.  T.I. Butler and E.W. Veasey, "Film Extrusion Manual, Process, Materials, Properties," <br>
+Tappi Press, Atlanta, GA (1992)<br>
+54.  C. Rauwendaal, "Polymer Mixing, A Self-Study Guide," Carl Hanser Verlag, Munich <br>
+(1998)<br>
+55.  C. Rauwendaal, "Understanding Extrusion," Carl Hanser Verlag, Munich (1998)<br>56.  C. Rauwendaal, "Statistical Process Control in Injection Molding and Extrusion," Carl <br>
+Hanser Verlag, Munich (2000)<br>
+57.  C.I. Chung, "Extrusion of Polymers, Theory and Practice," Carl Hanser Verlag, Munich <br>
+(2000)<br>
+58.  J.L. White and H. Potente (Eds.), "Screw Extrusion: Technology and Science," Carl <br>
+Hanser Verlag, Munich (2001)<br>
+59.  M. Noriega and C. Rauwendaal, "Troubleshooting the Extrusion Process," 1st edition, <br>
+Hanser Verlag, Munich (2001)<br>
+60.  G.A. Campbell and M.A. Spalding, "Analyzing and Troubleshooting Single-Screw <br>
+Extruders," Hanser Verlag, Munich (2013)<br>
+<hr>
+<A name=27></a><b>PART I</b><br>
+<b>Extrusion Machinery</b><br>
+<hr>
+<A name=28></a><hr>
+<A name=29></a><b>2 Different Types </b><br>
+<b>of Extruders</b><br>
+Extruders in the polymer industry come in many different designs. The main dis-<br>tinction between the various extruders is their mode of operation: continuous or <br>discontinuous. The latter type extruder delivers polymer in an intermittent fashion <br>and, therefore, is ideally suited for batch type processes, such as injection molding <br>and blow molding. As mentioned earlier, continuous extruders have a rotating mem-<br>ber, whereas batch extruders have a reciprocating member. A classification of the <br>various extruders is shown in Table 2.1.<br>
+<b></b><br>
+ <b>2.1The Single Screw Extruder</b><br>
+Screw extruders are divided into single screw and multi screw extruders. The single <br>screw extruder is the most important type of extruder used in the polymer industry. <br>Its key advantages are relatively low cost, straightforward design, ruggedness and <br>reliability, and a favorable performance/cost ratio. A detailed description of the hard-<br>ware components of a single screw extruder is given in Chapter 3.<br>The extruder screw of a conventional plasticating extruder has three geometrically <br>different sections; see Fig. 2.1.<br>This geometry is also referred to as a "single stage." The single stage refers to the fact <br>that the screw has only one compression section, even though the screw has three <br>distinct geometrical sections! The first section (closest to the feed opening) generally <br>has deep flights. The material in this section will be mostly in the solid state. This <br>section is referred to as the feed section of the screw. The last section (closest to the <br>die) usually has shallow flights. The material in this section will be mostly in the <br>molten state. This screw section is referred to as the metering section or pump sec-<br>tion. The third screw section connects the feed section and the metering section. <br>This section is called the transition section or compression section. In most cases, <br>the depth of the screw channel (or the height of the screw flight) reduces in a linear <br> <br>fashion, going from the feed section towards the metering section, thus causing a <br>compression of the material in the screw channel. Later, it will be shown that this <br>compression, in many cases, is essential to the proper functioning of the extruder.<br>
+<hr>
+<A name=30></a><b>14</b>  <b>2Different Types of Extruders</b><br>
+The extruder is usually designated by the diameter of the extruder barrel. In the <br>U.S., the standard extruder sizes are 3/4, 1, 1­1/2, 2, 2­1/2, 3­1/2, 4­1/2, 6, 8, 10, <br>12, 14, 16, 18, 20, and 24 inches.<br>Obviously, the very large machines are much less common than the smaller extrud-<br>ers. Some machines go up in size as large as 35 inches. These machines are used in <br>specialty operations, such as melt removal directly from a polymerization reactor. <br>In Europe, the standard extruder sizes are 20, 25, 30, 35, 40, 50, 60, 90, 120, 150, <br>200, 250, 300, 350, 400, 450, 500, and 600 millimeters. Most extruders range in <br>size from 1 to 6 inches or from 25 to 150 mm. An additional designation often used <br>is the length of the extruder, generally expressed as length to diameter (L/D) ratio. <br>Typical L/D ratios range from 20 to 30, with 24 being very common. Extruders used <br>for extraction of volatiles (vented extruders, see Section 2.1.2) can have an L/D ratio <br>as high as 35 or 40 and sometimes even higher.<br>
+<b>Table 2.1</b>Classification of Polymer Extruders<br>
+Melt fed<br>Plasticating<br>
+Single screw extruders<br>
+Single stage<br>Multi stage<br>
+Screw extruders <br>
+(continuous)<br>
+Compounding<br>Twin screw extruders<br>Gear pumps<br>
+Multi screw extruders<br>
+Planetary gear extruders<br>Multi (&gt;2) screw extruders<br>Spiral disk extruder<br>Drum extruder<br>
+Viscous drag extruders<br>
+Disk or drum extruders <br>
+Diskpack extruder<br>
+(continuous)<br>
+Stepped disk extruder<br>Screwless extruder<br>
+Elastic melt extruders<br>
+Screw or disk type melt extruder<br>Melt fed extruder<br>
+Ram extruders<br>
+Plasticating extruder<br>
+Reciprocating extruders <br>
+(discontinuous)<br>
+Capillary rheometer<br>
+Reciprocating single <br>
+Plasticating unit in injection molding machines<br>
+screw extruders<br>
+Compounding extruders such as the Kneader<br>
+<hr>
+<A name=31></a> <br>
+2.1The Single Screw Extruder <br>
+<b>15</b><br>
+<b>2.1.1Basic Operation</b><br>
+The basic operation of a single screw extruder is rather straightforward. Material <br>enters from the feed hopper. Generally, the feed material flows by gravity from the <br>feed hopper down into the extruder barrel. Some materials do not flow easily in dry <br>form and special measures have to be taken to prevent hang-up (bridging) of the <br>material in the feed hopper.<br>As material falls down into the extruder barrel, it is situated in the annular space <br>between the extruder screw and barrel, and is further bounded by the passive and <br>active flanks of the screw flight: the screw channel. The barrel is stationary and the <br>screw is rotating. As a result, frictional forces will act on the material, both on the <br>barrel as well as on the screw surface. These frictional forces are responsible for the <br>forward transport of the material, at least as long as the material is in the solid state <br>(below its melting point).<br>
+Feed section<br>
+Compression<br>
+Metering section<br>
+<b>Figure 2.1</b>Geometry of conventional extruder screw<br>
+As the material moves forward, it will heat up as a result of frictional heat genera-<br>tion and because of heat conducted from the barrel heaters. When the temperature <br>of the material exceeds the melting point, a melt film will form at the barrel surface. <br>This is where the solids conveying zone ends and the plasticating zone starts. It <br>should be noted that this point generally does not coincide with the start of the <br> <br>compression section. The boundaries of the functional zones will depend on poly-<br>mer properties, machine geometry, and operating conditions. Thus, they can change <br>as operating conditions change. However, the geometrical sections of the screw are <br>determined by the design and will not change with operating conditions. As the <br>material moves forward, the amount of solid material at each location will reduce as <br>a result of melting. When all solid polymer has disappeared, the end of the plasticat-<br>ing zone has been reached and the melt conveying zone starts. In the melt-convey-<br>ing zone, the polymer melt is simply pumped to the die.<br>As the polymer flows through the die, it adopts the shape of the flow channel of the <br>die. Thus, as the polymer leaves the die, its shape will more or less correspond to <br>the cross-sectional shape of the final portion of the die flow channel. Since the die <br>exerts a resistance to flow, a pressure is required to force the material through the <br>die. This is generally referred to as the diehead pressure. The diehead pressure is <br>determined by the shape of the die (particularly the flow channel), the temperature <br>
+<hr>
+<A name=32></a><IMG src="Polymer Extrusion by Chris Rauwendaal-32_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-32_11.jpg"><br>
+<b>16</b>  <b>2Different Types of Extruders</b><br>
+of the polymer melt, the flow rate through the die, and the rheological properties of <br>the polymer melt. It is important to understand that the diehead pressure is caused <br>by the die, and not by the extruder! The extruder simply has to generate sufficient <br>pressure to force the material through the die. If the polymer, the throughput, the <br>die, and the temperatures in the die are the same, then it does not make any differ-<br>ence whether the extruder is a gear pump, a single screw extruder, a twin screw <br>extruder, etc.; the diehead pressure will be the same. Thus, the diehead pressure is <br>caused by the die and by the flow process, taking place in the die flow channel. This <br>is an important point to remember.<br>
+<b>2.1.2Vented Extruders</b><br>
+Vented extruders are significantly different from non-vented extruders in design <br>and in functional capabilities. A vented extruder is equipped with one or more open-<br>ings (vent ports) in the extruder barrel, through which volatiles can escape. Thus, <br>the vented extruder can extract volatiles from the polymer in a continuous fashion. <br>This devolatilization adds a functional capability not present in non-vented extrud-<br>ers. Instead of the extraction of volatiles, one can use the vent port to add certain <br>components to the polymer, such as additives, fillers, reactive components, etc. This <br>clearly adds to the versatility of vented extruders, with the additional benefit that <br>the extruder can be operated as a conventional non-vented extruder by simply plug-<br>ging the vent port and, possibly, changing the screw geometry.<br>A schematic picture of a vented extruder is shown in Fig. 2.2.<br>
+Feed housing<br>
+Vent port<br>
+Breaker plate<br>
+Screw<br>
+Cooling channel<br>
+Heaters<br>
+Barrel<br>
+Die<br>
+<b>Figure 2.2</b>Schematic of vented extruder<br>
+The design of the extruder screw is very critical to the proper functioning of the <br>vented extruder. One of the main problems that vented extruders are plagued with <br>is vent flow. This is a situation where not only the volatiles are escaping through the <br>vent port, but also some amount of polymer. Thus, the extruder screw has to be <br>designed in such a way that there will be no positive pressure in the polymer under <br>the vent port (extraction section). This has led to the development of the two-stage <br>
+<hr>
+<A name=33></a> <br>
+2.1The Single Screw Extruder <br>
+<b>17</b><br>
+extruder screw, especially designed for devolatilizing extrusion. Two-stage extruder <br>screws have two compression sections separated by a decompression/extraction <br>section. It is somewhat like two single-stage extruder screws coupled in series along <br>one shaft. The details of the design of two-stage extruder screws will be covered in <br>Chapter 8. Vented extruders are used for the removal of monomers and oligomers, <br>reaction products, moisture, solvents, etc.<br>The devolatilization capability of single screw extruders of conventional design is <br>limited compared to twin screw extruders. Twin screw extruders can handle solvent <br>contents of 50% and higher, using a multiple-stage extraction system, and solvent <br>content of up to 15% using singlestage extraction. Single screw vented extruders <br>of conventional design usually cannot handle more than 5% volatiles; this would <br>require multiple vent ports. With a single vent port, a single screw vented extruder <br>of conventional design can generally reduce the level of volatiles only a fraction of <br>one percent, depending, of course, on the polymer/solvent system.<br>Because of the limited devolatilization capacity of single screw extruders of con-<br>ventional design, they are sometimes equipped with two or more vent ports. A draw-<br>back of such a design is that the length of the extruder can become a problem. <br>Some of these extruders have a L/D ration of 40 to 50! This creates a problem in <br>handling the screw, for instance when the screw is pulled, and increases the chance <br>of mechanical problems in the extruder (deflection, buckling, etc.). If substantial <br>amounts of volatiles need to be removed, a twin screw extruder may be more cost-<br>effective than a single screw extruder. However, some vented single screw extruders <br>of more modern design have substantially improved devolatilization capability and <br>deserve equal consideration; see Section 8.5.2.<br>
+<b>2.1.3Rubber Extruders</b><br>
+Extruders for processing elastomers have been around longer than any other type of <br>extruder. Industrial machines for rubber extrusion were built as early as the second <br>half of the nineteenth century. Some of the early extruder manufacturers were John <br>Royle in the U.S. and Francis Shaw in England. One of the major rubber extruder <br>manufacturers in Germany was Paul Troester; in fact, it still is a producer of extrud-<br>ers. Despite the fact that rubber extruders have been around for more than a cen-<br>tury, there is limited literature on the subject of rubber extrusion. Some of the hand-<br>books on rubber [1­5] discuss rubber extrusion, but in most cases the information <br>is very meager and of limited usefulness. Harms' book on rubber extruders [13] <br>appears to be the only book devoted exclusively to rubber extrusion. The few publi-<br>cations on rubber extrusion stand in sharp contrast to the abundance of books and <br>articles on plastic extrusion. Considering the commercial significance of rubber <br>extrusion, this is a surprising situation.<br>
+<hr>
+<A name=34></a><b>18</b>  <b>2Different Types of Extruders</b><br>
+The first rubber extruders were built for hot feed extrusion. These machines are fed <br>with warm material from a mill or other mixing device. Around 1950, machines <br>were developed for cold feed extrusion. The advantages of cold feed extruders are <br>thought to be:<br>
+ <br>
+ Less capital equipment cost<br>
+ <br>
+ Better control of stock temperature<br>
+ <br>
+ Reduced labor cost<br>
+ <br>
+ Capable of handling a wider variety of compounds<br>However, there is no general agreement on this issue. As a result, hot feed rubber <br>extruders are still in use today.<br>Cold feed rubber extruders, nowadays, do not differ too much from thermoplastic <br>extruders. Some of the differences are:<br>
+ <br>
+ Reduced length<br>
+ <br>
+ Heating and cooling<br>
+ <br>
+ Feed section<br>
+ <br>
+ Screw design<br>There are several reasons for the reduced length. The viscosity of rubbers is gener-<br>ally very high compared to most thermoplastics; about an order of magnitude higher <br>[5]. Consequently, there is a substantial amount of heat generated in the extru-<br>sion process. The reduced length keeps the temperature build-up within limits. The <br>specific energy requirement for rubbers is generally low, partly because they are <br>usually extruded at relatively low temperatures (from 20 to 120°C). This is another <br>reason for the short extruder length. The length of the rubber extruder will depend <br>on whether it is a cold or hot feed extruder. Hot feed rubber extruders are usually <br>very short, about 5D (D = diameter). Cold feed extruders range from 15 to 20D. <br>Vented cold feed extruders may be even longer than 20D.<br>Rubber extruders used to be heated quite frequently with steam because of the rela-<br>tively low extrusion temperatures. Today, many rubber extruders are heated like <br>thermoplastic extruders with electrical heater bands clamped around the barrel. Oil <br>heating is also used on rubber extruders and the circulating oil system can be used <br>to cool the rubber. Many rubber extruders use water cooling because it allows effec-<br>tive heat transfer.<br>The feed section of the rubber extruders has to be designed specifically to the feed <br>stock characteristics of the material. The extruder may be fed with either strips, <br>chunks, or pellets. If the extruder is fed from an internal mixer (e.g., Banbury, Shaw, <br>etc.), a power-operated ram can be used to force the rubber compound into the <br>extruder. The feed opening can be undercut to improve the intake capability of the <br>extruder. This can be useful, because the rubber feed stock at times comes in rela-<br>tively large particles of irregular shape. When the material is supplied in the form of <br>
+<hr>
+<A name=35></a><IMG src="Polymer Extrusion by Chris Rauwendaal-35_1.jpg"><br>
+ <br>
+2.1The Single Screw Extruder <br>
+<b>19</b><br>
+a strip, the feed opening is often equipped with a driven roll parallel to the screw to <br>give a "roller feed". Material can also be supplied in powder form. It has been shown <br>that satisfactory extrusion is possible if the powder is consolidated by "pill-making" <br>techniques. Powdered rubber technology is discussed in detail in [6].<br>The rubber extrusion technology appears to be considerably behind the plastics <br>extrusion technology. Kennaway, in one of the few articles on rubber extrusion [8], <br>attributes this situation to two factors. The first is the frequent tendency of rubber <br>process personnel to solve extrusion problems by changing the formulation of the <br>compound. The second is the widespread notion that the extrusion behavior of rub-<br>bers is substantially different from plastics, because rubbers crosslink and plastics <br>generally do not. This is a misconception, however, because the extrusion character-<br>istics of rubber and plastics are actually not substantially different [9].<br>When the rubber is slippery, as in dewatering rubber extruders, the feed section of <br>the barrel is grooved to prevent slipping along the barrel surface or the barrel I.D. <br>may be fitted with pins. This significantly improves the conveying action of the <br>extruder. The same technique has been applied to the thermoplastic extrusion, as <br>discussed in Section 1.4 and Section 7.2.2.2.<br>The extruder screw for rubber often has constant depth and variable decreasing <br>pitch (VDP); many rubber screws use a double-flighted design; see Fig. 2.3. Screws <br>for thermoplastics usually have a decreasing depth and constant pitch; see Fig. 2.1.<br>
+<b> Figure 2.3</b> <br>
+Typical screw geometry for rubber <br>
+extrusion<br>
+Another difference with the rubber extruder screw is that the channel depth is usu-<br>ally considerably larger than with a plastic extruder screw. The larger depth is used <br>to reduce the shearing of the rubber and the resulting viscous heat generation. <br>There is a large variety of rubber extruder screws, as is the case with plastic extruder <br>screws. Figure 2.4 shows the "Plastiscrew" manufactured by NRM.<br>
+<b> Figure 2.4</b> <br>
+The NRM Plastiscrew<br>
+Figure 2.5 shows the Pirelli rubber extruder screw. This design uses a feed section <br>of large diameter, reducing quickly to the much smaller diameter of the pumping <br>section. The conical feed section uses a large clearance between screw flight and <br>barrel wall. This causes a large amount of leakage over the flight and improves the <br>batch-mixing capability of the extruder.<br>
+<hr>
+<A name=36></a><IMG src="Polymer Extrusion by Chris Rauwendaal-36_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-36_8.jpg"><br>
+<b>20</b>  <b>2Different Types of Extruders</b><br>
+<b> Figure 2.5</b> <br>
+The Pirelli rubber extruder screw<br>
+Figure 2.6 shows the EVK screw by Werner &amp; Pfleiderer [14]. This design features <br>cross-channel barriers with preceding undercuts in the flights to provide a change <br>in flow direction and increased shearing as the material flows over the barrier or the <br>undercut in the flight. A rather unusual design is the Transfermix [10­13] extruder/<br>mixer, which has been used for compounding rubber formulations. This machine <br>features helical channels in both, the screw and barrel; see Fig. 2.7.<br>
+<b> Figure 2.6</b> <br>
+The EVK screw by Werner &amp; Pfleiderer<br>
+Feed<br>
+Vent<br>
+<b>Figure 2.7</b>The Transfermix extruder<br>
+By a varying root diameter of the screw the material is forced in the flow channel <br>of the barrel. A reduction of the depth of the barrel channel forces the material <br>back into the screw channel. This frequent reorientation provides good mixing. <br>However, the machine is difficult to manufacture and expensive to repair in case of <br>damage.<br>Another rubber extruder is the QSM extruder [7, 15­20]. QSM stands for the Ger-<br>man words "Quer Strom Misch," meaning cross-flow mixing. This extruder has <br>
+<hr>
+<A name=37></a><IMG src="Polymer Extrusion by Chris Rauwendaal-37_1.jpg"><br>
+ <br>
+2.1The Single Screw Extruder <br>
+<b>21</b><br>
+adjustable pins in the extruder barrel that protrude all the way into the screw chan-<br>nel; see Fig. 2.8.<br>
+<b>Figure 2.8</b>The QSM extruder (pin barrel extruder)<br>
+The screw flight has slots at the various pin locations. The advantages of this ex -<br>truder in rubber applications are good mixing capability with a low stock tempera-<br>ture increase and low specific energy consumption. This extruder was developed by <br>Harms in Germany and is manufactured and sold by Troester and other companies. <br>Even though the QSM extruder has become popular in the rubber industry, its appli-<br>cations clearly extend beyond just rubber extrusion. In thermoplastic extrusion, its <br>most obvious application would be in high viscosity, thermally less stable resins: <br>PVC could possibly be a candidate, although dead spots may create problems with <br>degradation. However, it can probably be applied wherever good mixing and good <br>temperature control are required.<br>Another typical rubber extrusion piece of hardware is the roller die. A schematic <br>representation is shown in Fig. 2.9.<br>
+<b>Figure 2.9</b>The roller die used in rubber extrusion<br>
+The roller die (B.F. Goodrich, 1933) is a combination of a standard sheet die and a <br>calender. It allows high throughput by reducing the diehead pressure; it reduces air <br>entrapment and provides good gauge control.<br>
+<hr>
+<A name=38></a><b>22</b>  <b>2Different Types of Extruders</b><br>
+<b>2.1.4High-Speed Extrusion</b><br>
+One way to achieve high throughputs is to use high-speed extrusion. High-speed <br>extrusion has been used in twin screw compounding for many years--since the <br>1960s. However, in single screw extrusion high speed extrusion has not been used <br>on a significant scale until about 1995. Twin screw extruders (TSE) used in com-<br>pounding typically run at screw speeds ranging from 200 to 500 rpm; in some cases <br>screw speeds beyond 1000 rpm are possible. Single screw extruders (SSE) usually <br>operate at screw speeds between 50 to 150 rpm.<br>Since approx. 1995, several extruder manufacturers have worked on developing <br>high-speed single screw extruders (HS-SSE). Most of these developments have taken <br>place at German extruder manufacturers such as Battenfeld, Reifenhäuser, Kuhne, <br>and Esde. Currently, HS-SSEs are commercially available and have been in use for <br>several years [108].<br>Battenfeld supplies a high speed 75-mm SSE that can run at screw speeds up to <br>1,500 rpm [109]. This machine can achieve throughputs up to 2,200 kg/hr. It uses a <br>four-motor CMG torque drive with 390 kW made by K&amp;A Knoedler.<br>
+<b>2.1.4.1Melt Temperature<br></b>One of the interesting characteristics of the HS-SSE is that the melt temperature <br>remains more or less constant over a broad range of screw speed [108, 109], see <br>Fig. 2.10.<br>
+<b>2500</b><br>
+<b>250</b><br>
+<b>75-mm extruder, PS 486</b><br>
+<b>P. Rieg, Battenfeld, H.J. Renner, BASF</b><br>
+<b>2000</b><br>
+<b>240 C]o</b><br>
+<b>1500</b><br>
+<b>230 ure [</b><br>
+<b>1000</b><br>
+<b>220</b><br>
+<b>Throughput [kg/hr] 500</b><br>
+<b>210 elt temperatM  Figure 2.10</b> <br>
+<b>0</b><br>
+<b>200</b><br>
+Throughput and melt <br>
+<b>0</b><br>
+<b>200</b><br>
+<b>400</b><br>
+<b>600</b><br>
+<b>800</b><br>
+<b>1000</b><br>
+<b>1200</b><br>
+temperature versus <br>
+<b>Screw speed [rev/min]</b><br>
+screw speed<br>
+This graph shows both the throughput in kg/hr and the melt temperature in °C. The <br>temperature curve indicates that there is less than a 2°C change in melt tempera-<br>ture from 200 rpm up to about 1000 rpm. At screw speeds from 110 rpm to 225 rpm <br>the melt temperature actually drops from about 223 to about 215°C.<br>
+<hr>
+<A name=39></a> <br>
+2.1The Single Screw Extruder <br>
+<b>23</b><br>
+In conventional extruders the melt temperature typically increases with screw <br>speed. This often creates problems when we deal with high-viscosity polymers, par-<br>ticularly when they have limited thermal stability. In the example shown in Fig. 2.10 <br>the melt temperature is essentially constant as screw speed increases from 200 to <br>1000 rpm. This is probably the result of two factors: the residence time of the poly-<br>mer melt is reduced with increasing screw speed and the volume of the molten poly-<br>mer likely reduces with increasing screw speed as the melting length becomes <br>longer with increased screw speed.<br>
+<b>2.1.4.2Extruders without Gear Reducer<br></b>An important advantage of high-speed SSE is that a conventional gear reducer is no <br>longer necessary. The gear reducer is one of the largest cost factors for a conven-<br>tional extruder. As a result, extruders without a gear reducer are significantly less <br>expensive compared to conventional extruders that achieve the same rate. A further <br>advantage of the elimination of the gear reducer is a significant reduction in noise <br>level. Contrary to what one would expect, HS-SSE actually run very quiet; the noise <br>level is substantially lower than it is for conventional extruders with gear reducers.<br>
+<b>2.1.4.3Energy Consumption<br></b>Another benefit of high-speed SSE is the reduction in energy consumption. Rieg <br>[108, 109] reports the following mechanical specific energy consumption.<br>The reduction in energy consumption listed in Table 2.2 is significant: between 35% <br>and 45%. If the extruder runs PP at 2000 kg/hr, a reduction in SEC of 0.10 kWh/kg <br>corresponds to a reduction in power consumption of 200 kW every hour. If the power <br>cost is $0.10 per kWh (for consistency), the savings per hour will be $20 per hour, <br>$480 per day, $3360 per week, and $168,000 per year at 50 weeks per year. Clearly, <br>this can have a significant effect on the profitability of an extrusion operation.<br>
+<b>Table 2.2</b>Specific Energy Consumption for Standard Extruder and High-Speed Extruder<br>
+Type of extruder<br>
+SEC with polystyrene [kWh/kg]<br>
+SEC with polypropylene [kWh/kg]<br>
+Standard extruder<br>
+0.19-0.21<br>
+0.27-0.29<br>
+High-speed extruder<br>
+0.10-0.12<br>
+0.17-0.19<br>
+<b>2.1.4.4Change-over Resin Consumption<br></b>Another important issue in efficient extrusion is the time and material used when a <br>change in resin is made. With high-speed extruders the change-over time is quite <br>short because the volume occupied by the plastic is small, while the throughput is <br>very high.<br>
+<hr>
+<A name=40></a><b>24</b>  <b>2Different Types of Extruders</b><br>
+Table 2.3 shows a comparison of the high-speed 75-mm extruder to a conventional <br>180-mm extruder running at the same throughput. The amount of material con-<br>sumed in the change-over is 150 kg for the 75-mm extruder and 1300 kg for the <br>180-mm extruder. If we assume a resin cost of $1.00 per kg, the savings are <br>$1,150.00 per change-over. If we have three change-overs per week the yearly sav-<br>ings in reduced resin usage comes to $172,500 per year. Clearly, this is not an insig-<br>nificant amount.<br>
+<b>Table 2.3</b>Comparison of Change-Over Time and Material Consumption<br>
+75-mm extruder<br>
+180-mm extruder<br>
+Volume in the extruder [liter]<br>
+4<br>
+40<br>
+Material consumption for change-over [kg]<br>
+150<br>
+1300<br>
+Change-over time [min]<br>
+5<br>
+40<br>
+<b>2.1.4.5  Change-over Time and Residence Time<br></b>Table 2.3 also shows that the change-over time for the 75-mm extruder is approx. <br>5 minutes, while it is approx. 40 minutes in the 180-mm extruder. Clearly, the <br>reduced change-over time results from the fact that the volume occupied by the plas-<br>tic is much smaller (4 liter) in the 75-mm extruder than in the 180-mm extruder <br>(40 liter). If the change-over time can be reduced from 40 to 5 minutes, this means <br>that 35 minutes are now available to run production, resulting in greater up-time of <br>the extrusion line.<br>The small volume of the HS-SSE also results in short residence times. The mean <br>residence time ranges from approx. 5 to 10 seconds. In conventional SSE the resi-<br>dence times are substantially longer; typically between 50 to 100 seconds. Since <br>HS-SSE can run at relatively low melt temperatures, the chance of degradation is <br>actually less in these extruders. There is ample evidence in actual high-speed ex -<br>trusion operations that the plastic is less susceptible to degradation in these opera-<br>tions.<br>
+<b></b><br>
+ <b>2.2The Multiscrew Extruder</b><br>
+<b>2.2.1The Twin Screw Extruder</b><br>
+A twin screw extruder is a machine with two Archimedean screws. Admittedly, this <br>is a very general definition. However, as soon as the definition is made more spe-<br>cific, it is limited to a specific class of twin screw extruders. There is a tremendous <br>variety of twin screw extruders, with vast differences in design, principle of opera-<br>
+<hr>
+<A name=41></a> <br>
+2.2The Multiscrew Extruder <br>
+<b>25</b><br>
+tion, and field of application. It is, therefore, difficult to make general comments <br>about twin screw extruders. The differences between the various twin screw extrud-<br>ers are much larger than the differences between single screw extruders. This is to <br>be expected, since the twin screw construction substantially increases the number <br>of design variables, such as direction of rotation, degree of intermeshing, etc. A clas-<br>sification of twin screw extruders is shown in Table 2.4. This classification is pri-<br>marily based on the geometrical configuration of the twin screw extruder. Some <br>twin screw extruders function in much the same fashion as single screw extruders. <br>Other twin screw extruders operate quite differently from single screw extruders <br>and are used in very different applications. The design of the various twin screw <br>extruders with their operational and functional aspects will be covered in more <br>detail in Chapter 10.<br>
+<b>Table 2.4</b>Classification of Twin Screw Extruders<br>
+Low speed extruders for profile extrusion<br>
+Co-rotating extruders<br>
+High speed extruders for compounding<br>
+Intermeshing extruders<br>
+Conical extruders for profile extrusion<br>
+Counter-rotating extruders<br>
+Parallel extruders for profile extrusion<br>High speed extruders for compounding<br>Equal screw length<br>
+Counter-rotating extruders<br>
+Unequal screw length<br>
+Co-rotating extruders<br>
+Not used in practice<br>
+Non-intermeshing <br>
+extruders<br>
+Inner melt transport forward<br>Inner melt transport rearward<br>
+Co-axial extruders<br>
+Inner solids transport rearward<br>Inner plasticating with rearward transport<br>
+<b>2.2.2The Multiscrew Extruder With More Than Two Screws</b><br>
+There are several types of extruders, which incorporate more than two screws. One <br>relatively well-known example is the planetary roller extruder, see Fig. 2.11.<br>
+Planetary screws<br>
+Discharge<br>
+Sun (main) screw<br>
+Melting and feed section<br>
+<b>Figure 2.11</b>The planetary roller extruder<br>
+<hr>
+<A name=42></a><b>26</b>  <b>2Different Types of Extruders</b><br>
+This extruder looks similar to a single screw extruder. The feed section is, in fact, <br>the same as on a standard single screw extruder. However, the mixing section of the <br>extruder looks considerably different. In the planetary roller section of the extruder, <br>six or more planetary screws, evenly spaced, revolve around the circumference of <br>the main screw. In the planetary screw section, the main screw is referred to as the <br>sun screw. The planetary screws intermesh with the sun screw and the barrel. The <br>planetary barrel section, therefore, must have helical grooves corresponding to the <br>helical flights on the planetary screws. This planetary barrel section is generally a <br>separate barrel section with a flange-type connection to the feed barrel section.<br>In the first part of the machine, before the planetary screws, the material moves <br>forward as in a regular single screw extruder. As the material reaches the planetary <br>section, being largely plasticated at this point, it is exposed to intensive mixing by <br>the rolling action between the planetary screws, the sun screw, and the barrel. The <br>helical design of the barrel, sun screw, and planetary screws result in a large sur-<br>face area relative to the barrel length. The small clearance between the planetary <br>screws and the mating surfaces, about ¼ mm, allows thin layers of compound to be <br>exposed to large surface areas, resulting in effective devolatilization, heat exchange, <br>and temperature control. Thus, heat-sensitive compounds can be processed with a <br>minimum of degradation. For this reason, the planetary gear extruder is frequently <br>used for extrusion/compounding of PVC formulations, both rigid and plasticized <br>[21, 22]. Planetary roller sections are also used as add-ons to regular extruders to <br>improve mixing performance [97, 98]. Another multiscrew extruder is the four-<br>screw extruder, shown in Fig. 2.12.<br>
+<b> Figure 2.12</b> <br>
+Four-screw extruder<br>
+This machine is used primarily for devolatilization of solvents from 40% to as low as <br>0.3% [23]. Flash devolatilization occurs in a flash dome attached to the barrel. The <br>polymer solution is delivered under pressure and at temperatures above the boiling <br>point of the solvent. The solution is then expanded through a nozzle into the flash <br>dome. The foamy material resulting from the flash devolatilization is then trans-<br>ported away by the four screws. In many cases, downstream vent sections will be <br>incorporated to further reduce the solvent level.<br>
+<hr>
+<A name=43></a> <br>
+2.2The Multiscrew Extruder <br>
+<b>27</b><br>
+<b>2.2.3The Gear Pump Extruder</b><br>
+Gear pumps are used in some extrusion operations at the end of a plasticating <br>extruder, either single screw or twin screw [99­106]. Strictly speaking, the gear <br>pump is a closely intermeshing counterrotating twin screw extruder. However, since <br>gear pumps are solely used to generate pressure, they are generally not referred to <br>as an extruder although the gear pump is an extruder. One of the main advantages <br>of the gear pump is its good pressure-generating capability and its ability to main-<br>tain a relatively constant outlet pressure even if the inlet pressure fluctuates con-<br>siderably. Some fluctuation in the outlet pressure will result from the intermeshing <br>of the gear teeth. This fluctuation can be reduced by a helical orientation of the gear <br>teeth instead of an axial orientation.<br>Gear pumps are sometimes referred to as positive displacement devices. This is not <br>completely correct because there must be mechanical clearances between the gears <br>and the housing, which causes leakage. Therefore, the gear pump output is depend-<br>ent on pressure, although the pressure sensitivity will generally be less than that of <br>a single screw extruder. The actual pressure sensitivity will be determined by the <br>design clearances, the polymer melt viscosity, and the rotational speed of the gears. <br>A good method to obtain constant throughput is to maintain a constant pressure dif-<br>ferential across the pump. This can be done by a relatively simple pressure feedback <br>control on the extruder feeding into the gear pump [102]. The non-zero clearances <br>in the gear pump will cause a transformation of mechanical energy into heat by vis-<br>cous heat generation, see Section 5.3.4. Thus, the energy efficiency of actual gear <br>pumps is considerably below 100%; the pumping efficiency generally ranges from <br>15 to 35%. The other 65 to 85% goes into mechanical losses and viscous heat gene-<br>ration. Mechanical losses usually range from about 20 to 40% and viscous heating <br>from about 40 to 50%. As a result, the polymer melt going through the gear pump <br>will experience a considerable temperature rise, typically 5 to 10°C. However, in <br>some cases the temperature rise can be as much as 20 to 30°C. Since the gear <br>pump has limited energy efficiency, the combination extruder-gear pump is not nec-<br>essarily more energy-efficient than the extruder without the gear pump. Only if the <br>extruder feeding into the gear pump is very inefficient in its pressure development <br>will the addition of a gear pump allow a reduction in energy consumption. This <br>could be the case with co-rotating twin screw extruders or single screw extruders <br>with inefficient screw design.<br>The mixing capacity of gear pumps is very limited. This was clearly demonstrated <br>by Kramer [106] by comparing melt temperature fluctuation before and after the <br>gear pump, which showed no distinguishable improvement in melt temperature uni-<br>formity. Gear pumps are often added to extruders with unacceptable output fluc-<br>tuations. In many cases, this constitutes treating the symptoms but not curing the <br>actual problem. Most single screw extruders, if properly designed, can maintain <br>
+<hr>
+<A name=44></a><b>28</b>  <b>2Different Types of Extruders</b><br>
+their output to within ± 1%. If the output fluctuation is considerably larger than 1%, <br>there is probably something wrong with the machine; very often incorrect screw <br>design. In these cases, solving the actual problem will generally be more efficient <br>than adding a gear pump. For an efficient extruder-gear pump system, the extruder <br>screw has to be modified to reduce the pressure-generating capacity of the screw.<br>Gear pumps can be used advantageously:<br>1.  On extruders with poor pressure-generating capability (e.g., co-rotating twin <br>
+screws, multi-stage vented extruders, etc.)<br>
+2.  When output stability is required, better than 1%, i.e., in close tolerance extru-<br>
+sion (e.g., fiber spinning, cable extrusion, medical tubing, coextrusion, etc.)<br>
+Gear pumps can cause problems when:<br>1.  The polymer contains abrasive components; because of the small clearances, the <br>
+gear pump is very susceptible to wear.<br>
+2.  When the polymer is susceptible to degradation; gear pumps are not self-clean-<br>
+ing and combined with the exposure to high temperatures this will result in <br>degraded product.<br>
+<b></b><br>
+ <b>2.3Disk Extruders</b><br>
+There are a number of extruders, which do not utilize an Archimedean screw for <br>transport of the material, but still fall in the class of continuous extruders. Some-<br>times these machines are referred to as screwless extruders. These machines <br>employ some kind of disk or drum to extrude the material. One can classify the disk <br>extruders according to their conveying mechanism (see Table 2.1). Most of the disk <br>extruders are based on viscous drag transport. One special disk extruder utilizes <br>the elasticity of polymer melts to convey the material and to develop the necessary <br>diehead pressure.<br>Disk extruders have been around for a long time, at least since 1950. However, at <br>this point in time the industrial significance of disk extruders is still relatively small <br>compared to screw extruders.<br>
+<b>2.3.1Viscous Drag Disk Extruders</b><br>
+<b>2.3.1.1Stepped Disk Extruder<br></b>One of the first disk extruders was developed by Westover at Bell Telephone Labo-<br>ratories: it is often referred to as a stepped disk extruder or slider pad extruder [24]. <br>A schematic picture of the extruder is shown in Fig. 2.13.<br>
+<hr>
+<A name=45></a> <br>
+2.3Disk Extruders <br>
+<b>29</b><br>
+In<br>
+<b> Figure 2.13</b> <br>
+Out<br>
+The stepped disk<br>
+The heart of the machine is the stepped disk positioned a small distance from a flat <br>disk. When one of the disks is rotated with a polymer melt in the axial gap, a pres-<br>sure build-up will occur at the transition of one gap size to another, smaller gap size; <br>see Fig. 2.14.<br>
+Pressure<br>
+Distance<br>
+v<br>
+<b> Figure 2.14</b> <br>
+Pressure generation in the step region<br>
+If exit channels are incorporated into the stepped disk, the polymer can be extruded <br>in a continuous fashion. The design of this extruder is based on Rayleigh's [25] <br>analysis of hydrodynamic lubrication in various geometries. He concluded that the <br>parallel stepped pad was capable of supporting the greatest load. The stepped disk <br>
+<hr>
+<A name=46></a><IMG src="Polymer Extrusion by Chris Rauwendaal-46_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-46_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-46_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-46_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-46_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-46_6.jpg"><br>
+<b>30</b>  <b>2Different Types of Extruders</b><br>
+extruder has also been designed in a different configuration using a gradual change <br>in gap size. This extruder has a wedge-shaped disk with a gradual increase in pres-<br>sure with radial distance.<br>A practical disadvantage of the stepped disk extruder is the fact that the machine is <br>difficult to clean because of the intricate design of the flow channels in the stepped <br>disk.<br>
+<b>2.3.1.2Drum Extruder<br></b>Another rather old concept is the drum extruder. A schematic picture of a machine <br>manufactured by Schmid &amp; Kocher in Switzerland is shown in Fig. 2.15.<br>
+In<br>
+In<br>
+Out<br>
+Out<br>
+<b> Figure 2.15</b> <br>
+The drum extruder by Schmid &amp; Kocher<br>
+Material is fed by a feed hopper into an annular space between rotor and barrel. By <br>the rotational motion of the rotor, the material is carried along the circumference of <br>the barrel. Just before the material reaches the feed hopper, it encounters a wiper <br>bar. This wiper bar scrapes the polymer from the rotor and deflects the polymer flow <br>into a channel that leads to the extruder die. Several patents [26, 27] were issued on <br>this design; however, these patents have long since expired.<br>A very similar extruder (see Fig. 2.16) was developed by Askco Engineering and <br>Cosden Oil &amp; Chemical in a joint venture; later this became Permian Research. Two <br>patents have been issued on this design [28, 29], even though the concept is very <br>similar to the Schmid &amp; Kocher design.<br>One special feature of this design is the capability to adjust the local gap by means <br>of a choker bar, similar to the gap adjustment in a flat sheet die, see Section 9.2. The <br>choker bar in this drum extruder is activated by adjustable hydraulic oil pressure. <br>Drum extruders have not been able to be a serious competition to the single screw <br>extruder over the last 50 years.<br>
+<hr>
+<A name=47></a><IMG src="Polymer Extrusion by Chris Rauwendaal-47_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-47_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-47_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-47_4.jpg"><br>
+ <br>
+2.3Disk Extruders <br>
+<b>31</b><br>
+Hopper<br>
+Hopper<br>
+Wiper bar<br>
+Wiper bar<br>
+Housing<br>
+Housing<br>
+Die<br>
+Die<br>
+Rotor<br>
+Rotor<br>
+<b> Figure 2.16</b> <br>
+The drum extruder by Asko/Cosden<br>
+<b>2.3.1.3Spiral Disk Extruder<br></b>The spiral disk extruder is another type of disk extruder that has been known for <br>many years. Several patented designs were described by Schenkel (Chapter 1, [3]). <br>Similar to the stepped disk extruder, the development of the spiral disk extruder <br>is closely connected to spiral groove bearings. It has long been known that spiral <br>groove bearings are capable of supporting substantial loads. Ingen Housz [30] has <br>analyzed the melt conveying in a spiral disk extruder with logarithmic grooves in <br>the disk, based on Newtonian flow behavior of the polymer melt. In terms of melt <br>conveying capability, the spiral disk extruder seems comparable to the screw ex -<br>truder; however, the solids conveying capability is questionable.<br>
+<b>2.3.1.4Diskpack Extruder<br></b>Another development in disk extruders is the diskpack extruder. Tadmor originated <br>the idea of the diskpack machine, which is covered under several patents [31­33]. <br>The development of the machine was undertaken by the Farrel Machinery Group of <br>Emgart Corporation in cooperation with Tadmor [34­39]. The basic concept of the <br>machine is shown in Fig. 2.17.<br>Material drops in the axial gap between relatively thin disks mounted on a rotating <br>shaft. The material will move with the disks almost a full turn, then it meets a chan-<br>nel block. The channel block closes off the space between the disks and deflects the <br>polymer flow to either an outlet channel, or to a transfer channel in the barrel. The <br>shape of the disks can be optimized for specific functions: solids conveying, melting, <br>devolatilization, melt conveying, and mixing. A detailed functional analysis can be <br>found in Tadmor's book on polymer processing (Chapter 1, [32]).<br>
+<hr>
+<A name=48></a><IMG src="Polymer Extrusion by Chris Rauwendaal-48_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-48_12.jpg"><br>
+<b>32</b>  <b>2Different Types of Extruders</b><br>
+Inlet<br>
+Inlet<br>
+Channel block<br>
+Channel block<br>
+Outlet<br>
+Outlet<br>
+Melt pool<br>
+Melt pool<br>
+Solid bed<br>
+Solid bed<br>
+Barrel<br>
+Barrel<br>
+Shaft<br>
+Shaft<br>
+v<br>v<br>
+<b> Figure 2.17</b> <br>
+v<br>
+v<br>
+The diskpack extruder<br>
+It is claimed that this machine can perform all basic polymer-processing operations <br>with efficiency equaling or surpassing existing machinery. Clearly, if the claim is <br>true and can be delivered for a competitive price, the diskpack extruder will become <br>an important machine in the extrusion industry. The first diskpack machines were <br>delivered to the industry in 1982, still under a joint development type of agreement. <br>Now, almost 20 years after the first diskpack machines were delivered, it is clear <br>that the acceptance of these machines in the industry has been very limited. In fact, <br>Farrel no longer actively markets the diskpack machine. In most of the publications, <br>the diskpack machine is referred to as a polymer processor. This term is probably <br>used to indicate that this machine can do more than just extruding, although the <br>diskpack is, of course, an extruder.<br>The diskpack machine incorporates some of the features of the drum extruder and <br>the single screw extruder. One can think of the diskpack as a single screw extruder <br>using a screw with zero helix angle and very deep flights. Forward axial transport <br>can only take place by transport channels in the barrel, with the material forced into <br>those channels by a restrictor bar, as with the drum extruder. The use of restrictor <br>bars (channel block) and transfer channels in the housing make it considerably <br>more complex than the barrel of a single screw extruder.<br>One of the advantages of the diskpack is that mixing blocks and spreading dams can <br>be incorporated into the machine as shown in Fig. 2.18(a).<br>
+<hr>
+<A name=49></a><IMG src="Polymer Extrusion by Chris Rauwendaal-49_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_48.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_49.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_50.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_51.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_52.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_53.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_54.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_55.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_56.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_57.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_58.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_59.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_60.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_61.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_62.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_63.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_64.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_65.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_66.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_67.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_68.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_69.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_70.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_71.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_72.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_73.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_74.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_75.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_76.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_77.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_78.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_79.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_80.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_81.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-49_82.jpg"><br>
+ <br>
+2.3Disk Extruders <br>
+<b>33</b><br>
+Channel block<br>
+Inlet<br>
+Inlet <br>
+Channel block<br>
+Outlet<br>
+Outlet<br>
+Mixing block<br>
+Mixing block<br>
+Disk<br>
+Disk<br>
+v<br>v<br>
+<b> Figure 2.18(a)</b> <br>
+v<br>
+v<br>
+Mixing blocks in the diskpack<br>
+Mixing blocks of various shapes can be positioned externally into the processing <br>chamber. This is similar to the QSM extruder, only that in the QSM extruder the <br>screw flight has to be interrupted to avoid contact. This is not necessary in the disk-<br>pack because the flights have zero helix angles; in other words, they run perpen-<br>dicular to the rotor axis. By the number of blocks, the geometry of the block, and the <br>clearance between block and disk, one can tailor the mixing capability to the par-<br>ticular application. The use of spreading dams allows the generation of a thin film <br>with a large surface area; this results in effective devolatilization capability. The <br>diskpack has inherently higher pressure-generating capability than the single <br>screw extruder does. This is because the diskpack has two dragging surfaces, while <br>the single screw extruder has only one, see Fig. 2.18(b).<br>
+v<br>
+Conveying mechanism<br>single screw extruder<br>
+v=0<br>
+v<br>
+Conveying mechanism<br>diskpack extruder<br>
+<b> Figure 2.18(b)</b> <br>
+Comparison of conveying mechanism in diskpack <br>
+v<br>
+and single screw extruder<br>
+<hr>
+<A name=50></a><b>34</b>  <b>2Different Types of Extruders</b><br>
+At the same net flow rate, equal viscosity, equal plate velocity, and optimum plate <br>separation, the theoretical maximum pressure gradient of the diskpack is eight <br>times higher than the single screw extruder [34]. Thus, high pressures can be gen-<br>erated over a short distance, which allows a more compact machine design. The <br>energy efficiency of the pressure generation is also better than in the single screw <br>extruder; the maximum pumping efficiency approaches 100%; see derivation in <br>Appendix 2.1. For the single screw extruder, the maximum pumping efficiency is <br>only 33% as discussed in Section 7.4.1.3. The energy efficiency of pressure genera-<br>tion is the ratio of the theoretical energy requirement (flow rate times pressure rise) <br>to the actual energy requirement (wall velocity times shear stress integrated over <br>wall surface). It should be noted that the two dragging plates' pumping mechanism <br>exists only in tangential direction. The pumping in axial (forward) direction occurs <br>only in the transfer channel in the housing. This forward pumping occurs by the one <br>dragging plate mechanism as in the single screw extruder. The maximum efficiency <br>for forward pumping, therefore, is only 33%. The overall pumping efficiency will be <br>somewhere between 33 and 100% if the power consumption in the disk and channel <br>block clearance is neglected.<br>Studies on melting in the diskpack were reported by Valsamis et al. [96]. It was <br>found that two types of melting mechanism could take place in the diskpack: the <br>drag melt removal (DMR) mechanism and the dissipative melt mixing (DMM) mech-<br>anism. The DMR melting mechanism is the predominant mechanism in single screw <br>extruders; this is discussed in detail in Section 7.3. In the DMR melting mechanism, <br>the solids and melt coexist as two largely continuous and separate phases. In the <br>DMM melting mechanism, the solids are dispersed in the melt; there is no conti-<br>nuous solid bed. It was found that the DMM mechanism could be induced by pro-<br>moting back leakage of the polymer melt past the channel block. This can be con-<br>trolled by varying the clearance between the channel block and the disks. The <br>advantage of the DMM melting mechanism is that the melting rate can be substan-<br>tially higher than with the DMR mechanism, reportedly by as much as a factor of 3 <br>[96].<br>Among all elementary polymer processing functions, the solids conveying in a disk-<br>pack extruder has not been discussed to any extent in the open technical literature. <br>Considering that the solids conveying mechanism is a frictional drag mechanism <br>(as in single screw extruders) and not a positive displacement type of transport (as <br>in intermeshing counterrotating twin screw extruders, see Sections 10.2 and 10.4), <br>it can be expected that the diskpack will have solids conveying limitations similar <br>to those of single screw extruders, see Section 7.2.2. This means that powders, <br>blends of powders and pellets, slippery materials, etc., are likely to encounter solids <br>conveying problems in a diskpack extruder unless special measures are taken to <br>enhance the solids conveying capability (e.g., crammer feeder, grooves in the disks, <br>etc.).<br>
+<hr>
+<A name=51></a><IMG src="Polymer Extrusion by Chris Rauwendaal-51_1.jpg"><br>
+ <br>
+2.3Disk Extruders <br>
+<b>35</b><br>
+Because of the more complex machine geometry, the cost per unit throughput of the <br>diskpack is higher than for the conventional single screw extruder. Therefore, the <br>diskpack does not compete directly with single screw extruders.<br>Applications for the diskpack are specialty polymer processing operations, such as <br>polymerization, post-reactor processing (devolatilization), continuous compound-<br>ing, etc. As such, the diskpack competes mostly with twin screw extruders. Pres-<br>ently, twin screw extruders are usually the first choice when it comes to specialty <br>polymer processing operations.<br>
+<b>2.3.2The Elastic Melt Extruder</b><br>
+The elastic melt extruder was developed in the late 1950s by Maxwell and Scalora <br>[40, 41]. The extruder makes use of the viscoelastic, in particular the elastic, pro-<br>perties of polymer melts. When a viscoelastic fluid is exposed to a shearing defor-<br>mation, normal stresses will develop in the fluid that are not equal in all directions, <br>as opposed to a purely viscous fluid. In the elastic melt extruder, the polymer is <br>sheared between two plates, one stationary and one rotating; see Fig. 2.19.<br>
+Solid polymer<br>
+Solid polymer<br>
+Hopper<br>
+Hopper<br>
+Heaters<br>
+Heaters<br>
+Die<br>
+Die<br>
+Rotor<br>
+Rotor<br>
+Extrudate<br>
+Extrudate<br>
+Polymer melt<br>
+Polymer melt<br>
+<b> Figure 2.19</b> <br>
+The elastic melt extruder<br>
+As the polymer is sheared, normal stresses will generate a centripetal pumping <br>action. Thus, the polymer can be extruded through the central opening in the sta-<br>tionary plate in a continuous fashion. Because normal stresses generate the pump-<br>ing action, this machine is sometimes referred to as a normal stress extruder.<br>This extruder is quite interesting from a rheological point of view, since it is prob-<br>ably the only extruder that utilizes the elasticity of the melt for its conveying. Thus, <br>
+<hr>
+<A name=52></a><b>36</b>  <b>2Different Types of Extruders</b><br>
+several detailed experimental and theoretical studies have been devoted to the elas-<br>tic melt extruder [42­47].<br>The detailed study by Fritz [44] concluded that transport by normal stresses only <br>could be as much as two orders of magnitude lower than a corresponding system <br>with forced feed. In addition, substantial temperature gradients developed in the <br>polymer, causing substantial degradation in high molecular weight polyolefins. The <br>scant market acceptance of the elastic melt extruder would tend to confirm Fritz's <br>conclusions.<br>Several modifications have been proposed to improve the performance of the elastic <br>melt extruder. Fritz [43] suggested incorporation of spiral grooves to improve the <br>pressure generating capability, essentially combining the elastic melt extruder and <br>the spiral disk extruder into one machine. In Russia [47], several modifications <br>were made to the design of the elastic melt extruder. One of those combined a screw <br>extruder with the elastic melt extruder to eliminate the feeding and plasticating <br>problem. Despite all of these activities, the elastic melt extruder has not been able to <br>acquire a position of importance in the extrusion industry.<br>
+<b>2.3.3Overview of Disk Extruders</b><br>
+Many attempts have been made in the past to come up with a continuous plasticat-<br>ing extruder of simple design that could perform better than the single screw ex -<br>truder. It seems fair to say that, at this point in time no disk extruder has been able <br>to meet this goal. The simple disk extruders do not perform nearly as well as the <br>single screw extruder. The more complex disk extruder, such as the diskpack, can <br>possibly outperform the single screw extruder. However, this is at the expense of <br>design simplicity, thus increasing the cost of the machine. Disk extruders, there-<br>fore, have not been able to seriously challenge the position of the single screw <br>extruder. This is not to say, however, that it is not possible for this to happen some <br>time in the future. But, considering the long dominance of the single screw extruder, <br>it is not probable that a new disk extruder will come along that can challenge the <br>single screw extruder.<br>
+<b></b><br>
+ <b>2.4Ram Extruders</b><br>
+Ram or plunger extruders are simple in design, rugged, and discontinuous in their <br>mode of operation. Ram extruders are essentially positive displacement devices and <br>are able to generate very high pressures. Because of the intermittent operation of <br>
+<hr>
+<A name=53></a> <br>
+2.4Ram Extruders <br>
+<b>37</b><br>
+ram extruders, they are ideally suited for cyclic processes, such as injection molding <br>and blow molding. In fact, the early molding machines were almost exclusively <br>equipped with ram extruders to supply the polymer melt to the mold. Certain limita-<br>tions of the ram extruder have caused a switch to reciprocating screw extruders or <br>combinations of the two. The two main limitations are:<br>1.  Limited melting capacity<br>2.  Poor temperature uniformity of the polymer melt<br>Presently, ram extruders are used in relatively small shot size molding machines <br>and certain specialty operations where use is made of the positive displacement <br>characteristics and the outstanding pressure generation capability. There are basic-<br>ally two types of ram extruders: single ram extruders and multi ram extruders.<br>
+<b>2.4.1Single Ram Extruders</b><br>
+The single ram extruder is used in small general purpose molding machines, but <br>also in some special polymer processing operations. One such operation is the ex -<br>trusion of intractable polymers, such as ultrahigh molecular weight polyethylene <br>(UHMWPE) or polytetrafluoroethylene (PTFE). These polymers are not considered to <br>be melt processable on conventional melt processing equipment. Teledynamik [48] <br>has built a ram injection-molding machine under a license from Th. Engel who <br>developed the prototype machine. This machine is used to mold UHMWPE under <br>very high pressures. The machine uses a reciprocating plunger that densifies the <br>cold incoming material with a pressure up to 300 MPa (about 44,000 psi). The fre-<br>quency of the ram can be adjusted continuously with a maximum of 250 strokes/<br>minute. The densified material is forced through heated channels into the heated <br>cylinder where final melting takes place. The material is then injected into a mold <br>by a telescoping injection ram. Pressures up to 100 MPa (about 14,500 psi) occur <br>during the injection of the polymer into the mold.<br>Another application is the extrusion of PTFE, with again the primary ingredient for <br>successful extrusion being very high pressures. Granular PTFE can be extruded at <br>slow rates in a ram extruder [49­51]. The powder is compacted by the ram, forced <br>into a die where the material is heated above the melting point and shaped into the <br>desired form. PTFE is often processed as a PTFE paste [52, 53]. This is small particle <br>size PTFE powder (about 0.2 mm) mixed with a processing aid such as naphta. PTFE <br>paste can be extruded at room temperature or slightly above room temperature. <br>After extrusion, the processing aid is removed by heating the extrudate above its <br>volatilization temperature. The extruded PTFE paste product may be sintered if the <br>application requires a more fully coalesced product.<br>
+<hr>
+<A name=54></a><IMG src="Polymer Extrusion by Chris Rauwendaal-54_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-54_8.jpg"><br>
+<b>38</b>  <b>2Different Types of Extruders</b><br>
+<b>2.4.1.1Solid State Extrusion<br></b>An extrusion technique that has slowly been gaining popularity is solid-state extru-<br>sion. The polymer is forced through a die while it is below its melting point. This <br>causes substantial deformation of the polymer in the die, but since the polymer is in <br>the solid state, a very effective molecular orientation takes place. This orientation is <br>much more effective than the one which occurs in conventional melt processing. As <br>a result, extraordinary mechanical properties can be obtained.<br>Solid-state extrusion is a technique borrowed from the metal industry, where solid-<br>state extrusion has been used commercially since the late 1940s. Bridgman [54] <br>was one of the first to do a systematic study on the effect of pressure on the mechan-<br>ical properties of metals. He also studied polymers and found that the glass tran-<br>sition temperature was raised by the application of pressure.<br>There are two methods of solid-state extrusion; one is direct solid-state extrusion, <br>the other is hydrostatic extrusion. In direct solid-state extrusion, a pre-formed solid <br>rod of material (a billet) is in direct contact with the plunger and the walls of the <br>extrusion die, see Fig. 2.20. The material is extruded as the ram is pushed towards <br>the die.<br>
+Plunger<br>
+Billet<br>
+Barrel<br>
+Die<br>
+<b> Figure 2.20</b> <br>
+Extrudate<br>
+Direct solid state extrusion<br>
+In hydrostatic extrusion, the pressure required for extrusion is transmitted from the <br>plunger to the billet through a lubricating liquid, usually castor oil. The billet must <br>be shaped to fit the die to prevent loss of fluid. The hydrostatic fluid reduces the fric-<br>tion, thereby reducing the extrusion pressure, see Fig. 2.21.<br>In hydrostatic extrusion, the pressure-generating device for the fluid does not neces-<br>sarily have to be in close proximity to the forming part of the machine. The fluid can <br>be supplied to the extrusion device by high-pressure tubes.<br>
+<hr>
+<A name=55></a><IMG src="Polymer Extrusion by Chris Rauwendaal-55_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-55_15.jpg"><br>
+ <br>
+2.4Ram Extruders <br>
+<b>39</b><br>
+Plunger<br>
+Billet<br>
+Oil<br>
+Barrel<br>
+<b> Figure 2.21</b> <br>
+Extrudate<br>
+Die<br>
+Hydrostatic solid state extrusion<br>
+Judging from publications in the open literature, most of the work on solid-state <br>extrusion of polymers is done at universities and research institutes. It is possible, <br>of course, that some companies are working on solid-state extrusion but are keeping <br>the information proprietary. A major research effort in solid-state extrusion has <br>been made at the University of Amherst, Massachusetts [50­64], University of <br>Leeds, England [65­72], Fyushu University, Fukuoka, Japan [73­76], Research <br>Institute for Polymers and Textiles, Yokohama, Japan [77­79], Battelle, Columbus, <br>Ohio [80­83], and Rutgers University, New Brunswick, New Jersey [84­86]. As <br> <br>mentioned earlier, publications from other sources are considerably less plentiful <br>[87­90]. Efforts have also been made to achieve the same high degree of orientation <br>in a more or less conventional extrusion process by special die design and tempera-<br>ture control in the die region [91].<br>Table 2.5 shows a comparison of mechanical properties between steel, aluminum, <br>solid state extruded HDPE, and HDPE extruded by conventional means.<br>Table 2.5 clearly indicates that the mechanical properties of solid-state extruded <br>HDPE are much superior to the melt extruded HDPE. In fact, the tensile strength of <br>solid-state extruded HDPE is about the same as carbon steel! There are some other <br>interesting benefits associated with solidstate extrusion of polymers. There is essen-<br>tially no die swell at high extrusion ratios (extrusion ratio is the ratio of the area in <br>the cylinder to the area in the die). Thus, the dimensions of the extrudate closely <br>conform to those of the die exit. The surface of the extrudate produced by hydro-<br>static extrusion has a lower coefficient of friction than that of the un-oriented poly-<br>mer. Above a certain extrusion ratio (about ten), polyethylene and polypropylene <br>become transparent. Further, solid-state extruded polymers maintain their ten-<br>sile properties at elevated temperatures. Polyethylene maintains its modulus up to <br>120°C when it is extruded in the solid state at a high extrusion ratio. The thermal <br>
+<hr>
+<A name=56></a><b>40</b>  <b>2Different Types of Extruders</b><br>
+conductivity in the extrusion direction is much higher than that of the un-oriented <br>polymer, as much as 25 times higher. The melting point of the solid-state extruded <br>polymer increases with the amount of orientation. The melting point of HDPE can be <br>shifted to as high as 140°C.<br>
+<b>Table 2.5</b>Comparison of Mechanical Properties<br>
+Material<br>
+Tensile modulus<br>
+Tensile strength<br>
+Elongation<br>
+Density<br>
+[MPa]<br>
+[MPa]<br>
+[%]<br>
+[g/cc]<br>
+Annealed SAE 1020<br>
+210,000<br>
+410<br>
+35<br>
+7.86<br>
+W-200 °F SAE 1020<br>
+210,000<br>
+720<br>
+6<br>
+7.86<br>
+Annealed 304 stainless <br>
+200,000<br>
+590<br>
+50<br>
+7.92<br>
+steel<br>Aluminum 1100­0<br>
+70,000<br>
+90<br>
+45<br>
+2.71<br>
+HDPE solid state <br>
+70,000<br>
+480<br>
+3<br>
+0.97<br>
+extruded<br>HDPE melt extruded<br>
+10,000<br>
+30<br>
+20­1000<br>
+0.96<br>
+A recent application of solid-state extrusion is the process developed by Synthetic <br>Hardwood Technologies, Inc. [107]. This company has developed an expanded, ori-<br>ented, wood-filled polypropylene (EOW-PP) that is about 300% stronger than regular <br>PP. The process is based on technology developed at Aluminum Company of Canada <br>Ltd. (Alcan) in the early 1990s to solidstate extrude PP. It involves ram extrusion/<br>drawing of billets of PP just below the melting point with very high draw ratio and <br>high haul-off tension (about 3 MPa) to freeze-in the high level of orientation. Alcan <br>did not pursue the technology and Symplastics Ltd. licensed the patented process; <br>this company started experimenting with adding small amounts of wood flour to the <br>PP. However, Symplastics found the research and development too costly and sold <br>the patents and lab extrusion equipment to Frank Maine who set up SHW to com-<br>mercialize applications for the EOW-PP. The key to the SHW process is that it com-<br>bines extrusion with drawing. As the polymer is oriented, the density drops about <br>50%, from about 1 g/cc to 0.5 g/cc. The die drawing also allowed substantial in -<br>creases in line speed, from about 0.050 m/min to about 9 m/min. The properties <br>achieved with EOW-PP are shown in Table 2.4 and compared to regular PP, wood, <br>and oriented PP.<br>Solid-state extrusion has been practiced with coextrusion of different polymers [93]. <br>Despite the large amount of research on solid-state extrusion and the outstanding <br>mechanical properties that can be obtained, there does not seem to be much interest <br>in the polymer industry. The main drawbacks, of course, are that solid-state extru-<br>sion is basically a discontinuous process, it cannot be done on conventional polymer <br>processing equipment, and very high pressures are required to achieve solid-state <br>extrusion. Also, one should keep in mind that very good mechanical properties can <br>be obtained by taking a profile (fiber, film, tube, etc.) produced by conventional, con-<br>
+<hr>
+<A name=57></a> <br>
+2.4Ram Extruders <br>
+<b>41</b><br>
+tinuous extrusion and exposing it to controlled deformation at a temperature below <br>the melting point. This is a well-established technique in many extrusion operations <br>(fiber spinning, film extrusion, etc.), and it can be done at a high rate. This method <br>of producing extrudates with very good mechanical properties is likely to be more <br>cost-effective than solid-state extrusion. It is possible that the die drawing process <br>developed by SHW can make solid-state extrusion more attractive commercially by <br>allowing large profiles to be processed at reasonable line speeds.<br>
+<b>Table 2.6</b>Mechanical Properties of PP, Wood, OPP, and EOW-PP<br>
+Flexural strength [MPa]<br>
+Flexural modulus [MPa]<br>
+Regular PP<br>
+50<br>
+1850<br>
+Wood<br>
+100<br>
+9000<br>
+Oriented PP<br>
+275<br>
+7600<br>
+EOW-PP<br>
+140<br>
+7600<br>
+<b>2.4.2Multi Ram Extruder</b><br>
+As mentioned before, the main disadvantage of ram extruders is their intermittent <br>operation. Several attempts have been made to overcome this problem by designing <br>multi-ram extruders that work together in such a way as to produce a continuous <br>flow of material.<br>Westover [94] designed a continuous ram extruder that combined four plunger- <br>cylinders. Two plunger-cylinders were used for plasticating and two for pumping. <br>An intricate shuttle valve connecting all the plunger cylinders provides continuous <br>extrusion.<br>Another attempt to develop a continuous ram extruder was made by Yi and Fenner <br>[95]. They designed a twin ram extruder with the cylinders in a V-configuration, see <br>Fig. 2.22.<br>The two rams discharge into a common barrel in which a plasticating shaft is rotat-<br>ing. Thus, solids conveying occurs in the two separate cylinders and plasticating <br>and melt conveying occurs in the annular region between the barrel and plasticat-<br>ing shaft. The machine is able to extrude; however, the throughput uniformity is <br>poor. The performance could probably have been improved if the plasticating shaft <br>had been provided with a helical channel. But, of course, then the machine would <br>have become a screw extruder with a ram-assisted feed. This only goes to demon-<br>strate that it is far from easy to improve upon the simple screw extruder.<br>
+<hr>
+<A name=58></a><IMG src="Polymer Extrusion by Chris Rauwendaal-58_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-58_22.jpg"><br>
+<b>42</b>  <b>2Different Types of Extruders</b><br>
+Main<br>
+Feed cylinder<br>
+Die<br>
+Breaker plate<br>
+block<br>
+Plasticating shaft<br>
+<b>Figure 2.22</b>Twin ram extruder<br>
+<b>2.4.3  Appendix 2.1</b><br>
+<b>2.4.3.1  Pumping Efficiency in Diskpack Extruder<br></b>The velocity profile between the moving walls is a parabolic function when the fluid <br>is a Newtonian fluid, see Section 6.2.1 and Fig. 2.23.<br>
+Large pressure gradient<br>
+Small pressure gradient<br>
+y<br>
+z<br>
+H<br>
+<b> Figure 2.23</b> <br>
+Velocity profiles between moving <br>
+Medium pressure gradient<br>
+walls<br>
+The velocity profile for a Newtonian fluid is:<br>
+<br>
+4y2  H 2 P<br>
+<br>
+v(y) = v - 1<br>
+<br> -<br>
+<br><br>
+ (1)<br>
+H 2<br>
+8<br>
+L<br>
+<br>
+ <br>
+µ<br>
+<hr>
+<A name=59></a> <br>
+2.4Ram Extruders <br>
+<b>43</b><br>
+where v is the velocity of the plates, H the distance between the plates, and  the <br>viscosity of the fluid. The flow rate can be found by integrating v(y) over the width <br>and depth of the channel; this yields:<br>
+.·<br>
+H3W P<br>
+<br>
+V = vWH -<br>
+ (2)<br>
+12<br>
+L<br>
+<br>
+µ<br>
+The first term on the right-hand side of the equalation is the drag flow term. The <br>second term is the pressure flow term. The ratio of pressure flow to drag flow is <br>termed the throttle ratio, rd (see Section 7.4.1.3):<br>
+H 2 P<br>
+<br>
+r<br>
+ (3)<br>
+d =  12v L<br>
+<br>
+µ<br>
+The flow rate can now be written as:<br>
+ (4)<br>
+The shear stress at the wall is obtained from:<br>
+dv<br>
+HP<br>
+ = µ  = <br>
+ (5)<br>
+dy<br>
+2L<br>
+0.5H<br>
+The power consumption in the channel is:<br>
+Z<br>
+ <br>
+=<br>
+  (6)<br>
+ch =  2 W Lv<br>
+vHW P<br>
+The energy efficiency for pressure generation is:<br>
+VP<br>
+ =<br>
+=1 - r  (7)<br>
+d<br>
+Zch<br>
+Equation 7 is not valid for rd = 0, because in this case both numerator and denomi-<br>
+nator become zero. From Eq. 7 it can be seen that when the machine is operated at <br>low rd values, the pumping efficiency can become close to 100%. This is considerably <br>
+better than the single screw extruder where the optimum pumping efficiency is 33% <br>at a throttle ratio value of 0.33 (rd = 1/3).<br>
+References<br>1.  J. LeBras, "Rubber, Fundamentals of its Science and Technology", Chemical Publ. Co, <br>
+NY (1957)<br>
+2.  W.S. Penn, "Synthetic Rubber Technology, Volume I", MacLaren &amp; Sons, Ltd., London <br>
+(1960)<br>
+3.  W.J.S. Naunton, "The Applied Science of Rubber", Edward Arnold Ltd., London (1961)<br>
+<hr>
+<A name=60></a><b>44</b>  <b>2Different Types of Extruders</b><br>
+4.  C.M. Blow, "Rubber Technology and Manufacture", Butterworth &amp; Co. Ltd., London <br>
+(1971)<br>
+5.  F.R. Eirich (Ed.), "Science and Technology of Rubber", Academic Press, NY (1978)<br>6.  C.W. Evans, "Powdered and Particulate Rubber Technology", Applied Science Publ. Ltd., <br>
+London (1978)<br>
+7.  G. Targiel et al., 10. IKV-Kolloquium, Aachen March 12­14, 45 (1980)<br>8.  A. Kennaway, Kautschuk und Gummi, Kunststoffe 17, 378­391 (1964)<br>9.  G. Menges and J.P. Lehnen, Plastverarbeiter 20, 1, 31­39 (1969)<br>10.  M. Parshall and A.J. Saulino, Rubber World, 2, 5, 78­83 (1967)<br>11.  S.E. Perlberg, Rubber World, 2, 6, 71­76 (1967)<br>12.  H.H. Gohlisch, Gummi, Asbest, Kunststoffe, 25, 9, 834­835 (1972)<br>13.  E. Harms, "Kautschuk-Extruder, Aufbau und Einsatz aus verfahrenstechnischer Sicht", <br>
+Krausskopf-Verlag Mainz, Bd 2, Buchreihe Kunststofftechnik (1974)<br>
+14.  G. Schwarz, Eur. Rubber Journal, Sept. 28­32 (1977)<br>15.  G. Menges and E.G. Harms, Kautschuk und Gummi, Kunststoffe 25, 10, 469­475 (1972)<br>16.  G. Menges and E.G. Harms, Kautschuk und Gummi, Kunststoffe 27, 5, 187­193 (1974)<br>17.  E.G. Harms, Elastomerics, 109, 6, 33­39 (1977)<br>18.  E.G. Harms, Eur. Rubber Journal, 6, 23 (1978)<br>19.  E.G. Harms, Kunststoffe, 69, 1, 32­33 (1979)<br>20.  E.G. Harms, Dissertation RWTH Aachen, Germany (1981).<br>21.  S.H. Collins, Plastics Compounding, Nov./Dec., 29 (1982)<br>22.  D. Anders, Kunststoffe, 69, 194­198 (1979)<br>23.  D. Gras and K. Eise, SPE Tech. Papers (ANTEC), 21, 386 (1975)<br>24.  R.F. Westover, SPE Journal, 18, 12, 1473 (1962).<br>25.  Lord Raleigh, Philosophical Magazine, 35, 1­12 (1918)<br>26.  German Patent: DRP 1,129,681<br>27.  British Patent: BP 759,354<br>28.  U.S. Patent: 3,880,564<br>29.  U.S. Patent: 4,012,477<br>30.  J.F. Ingen Housz, Plastverarbeiter, 10, 1 (1975)<br>31.  U. S. Patent: 4,142,805<br>32.  U.S. Patent: 4,194,841<br>33.  U.S. Patent: 4,213,709<br>34.  Z. Tadmor, P. Hold, and L. Valsamis, SPE Tech. Papers (ANTEC), 25, 193 (1979)<br>35.  P. Hold, Z. Tadmor, and L. Valsamis, SPE Tech. Papers (ANTEC), 25, 205 (1979)<br>36.  Z. Tadmor, P. Hold, and L. Valsamis, Plastics Engineering, Nov., 20­25 (1979)<br>37.  Z. Tadmor, P. Hold, and L. Valsamis, Plastics Engineering, Dec., 30­38 (1979)<br>
+<hr>
+<A name=61></a> References <br>
+<b>45</b><br>
+38.  Z. Tadmor et al., The Diskpack Plastics Processor, Farrel Publication, Jan. (1982)<br>39.  L. Valsamis, AIChE Meeting, Washington, D.C., Oct. (1983)<br>40.  B. Maxwell and A. J. Scalora, Modern Plastics, 37, 107, Oct. (1959)<br>41.  U.S. Patent: 3, 046,603<br>42.  L.L. Blyler, Ph.D. thesis, Princeton University, NJ (1966)<br>43.  H.G. Fritz, Kunststofftechnik, 6, 430 (1968)<br>44.  H.G. Fritz, Ph.D. thesis, Stuttgart University, Germany (1971)<br>45.  C.W. Macosko and J.M. Starita, SPE Journal 27, 30 (1971)<br>46.  P.A. Good, A.J. Schwartz, and C.W. Macosko, AIChE Journal, 20, 1, 67 (1974)<br>47.  V.L. Kocherov, Y.L. Lukach, E.A. Sporyagin, and G.V. Vinogradov, Polym. Eng. Sci., 13, <br>
+194 (1973)<br>
+48.  J. Berzen and G. Braun, Kunststoffe, 69, 2, 62­66 (1979)<br>49.  R.S. Porter et al., J. Polym. Sci., 17, 485­488 (1979)<br>50.  C.A. Sperati, Modern Plastics Encyclopedia, McGraw-Hill, NY (1983)<br>51.  S.S. Schwartz and S.H. Goodman, see Chapter 1, [36]<br>52.  G.R. Snelling and J.F. Lontz, J. Appl. Polym. Sci., 3, 9, 257­265 (1960)<br>53.  D.C.F. Couzens, Plastics and Rubber Processing, March, 45­48 (1976).<br>54.  P.W. Bridgman, "Studies in Large Plastic Flow and Fracture", McGraw-Hill, NY (1952)<br>55.  H.L.D. Push, "The Mechanical Behavior of Materials Under Pressure", Elsevier, Amster-<br>
+dam (1970)<br>
+56.  H.L.D. Push and A.H. Low, J. Inst. Metals, 93, 201 (1965/65)<br>57.  F. Slack, Mach. Design, Oct 7, 61­64 (1982)<br>58.  J.H. Southern and R.S. Porter, J. Appl. Polym. Sci., 14, 2305 (1970)<br>59.  J.H. Southern and R.S. Porter, J. Macromol. Sci. Phys., 3­4, 541 (1970)<br>60.  J.H. Southern, N.E. Weeks, and R.S. Porter, Macromol. Chem., 162, 19 (1972)<br>61.  N.J. Capiati and R.S. Porter, J. Polym. Sci., Polym. Phys. Ed., 13, 1177 (1975)<br>62.  R.S. Porter, J.H. Southern, and N.E. Weeks, Polym. Eng. Sci., 15, 213 (1975)<br>63.  A.E. Zachariades, E.S. Sherman, and R.S. Porter, J. Polym. Sci. Polym. Lett. Ed., 17, 255 <br>
+(1979)<br>
+64.  A.E. Zachariades and R.S. Porter, J. Polym. Sci. Polym. Lett. Ed., 17, 277 (1979)<br>65.  B. Parsons, D. Bretherton, and B.N. Cole, in: Advances in MTDR, 11th Int. Conf. Proc., <br>
+S.A. Tobias and F. Koeningsberger (Eds.), Pergamon Press, London, Vol. B, 1049 (1971)<br>
+66.  G. Capaccio and I.M. Ward, Polymer 15, 233 (1974)<br>67.  A.G. Gibson, I.M. Ward, B.N. Cole, and B. Parsons, J. Mater. Sci., 9, 1193­1196 (1974)<br>68.  A.G. Gibson and I.M. Ward, J. Appl. Polym. Sci. Polym. Phys. Ed., 16, 2015­2030 (1978)<br>69.  P.S. Hope and B. Parsons, Polym. Eng. Sci., 20, 589­600 (1980)<br>70.  P.S. Hope, I.M. Ward, and A.G. Gibson, J. Polym. Sci. Polym. Phys. Ed., 18, 1242­1256 <br>
+(1980)<br>
+<hr>
+<A name=62></a><b>46</b>  <b>2Different Types of Extruders</b><br>
+71.  P.S. Hope, A.G. Gibson, B. Parsons, and I.M. Ward, Polym. Eng. Sci., 20, 54­55 (1980)<br>72.  B. Parsons and I.M. Ward, Plast. Rubber Proc. Appl., 2, 3, 215­224 (1982)<br>73.  K. Imada, T. Yamamoto, K. Shigematsu, and M. Takayanagi, J. Mater. Sci., 6, 537­546 <br>
+(1971)<br>
+74.  K. Nakamura, K. Imada, and M. Takayanagi, Int J. Polym. Mater., 2, 71 (1972)<br>75.  K. Imada and M. Takayanagi, Int. J. Polym. Mater., 2, 89 (1973)<br>76.  K. Nakamura, K. Imada, and M. Takayanagi, Int. J. Polym. Mater., 3, 23 (1974)<br>77.  K. Nakayama and H. Kanetsuna, J. Mater. Sci., 10, 1105 (1975)<br>78.  K. Nakayama and H. Kanetsuna, J. Mater. Sci., 12, 1477 (1977)<br>79.  K. Nakayama and H. Kanetsuna, J. Appl. Polym. Sci., 23, 2543­2554 (1979)<br>80.  D.M. Bigg, Polym. Eng. Sci., 16, 725 (1976)<br>81. D.M. Bigg, M.M. Epstein, R.J. Fiorentino, and E.G. Smith, Polym. Eng. Sci., 18, 908 <br>
+(1978)<br>
+82.  D.M. Bigg and M.M. Epstein, "Science and Technology of Polymer Processing", N.S. Suh <br>
+and N. Sung (Eds.), 897, MIT Press (1979)<br>
+83.  D.M. Bigg, M.M. Epstein, R.J. Fiorentino, and E.G. Smith, J. Appl. Polym. Sci., 26, 395­<br>
+409 (1981)<br>
+84.  K.D. Pae and D.R. Mears, J. Polym. Sci., B-6, 269 (1968)<br>85.  K.D. Pae, D.R. Mears, and J.A. Sauer, J. Polym. Sci. Polym. Lett. Ed., 6, 773 (1968)<br>86.  D.R. Mears, K.P. Pae, and J.A. Sauer, J. Appl. Phys., 40, 11, 4229­4237 (1969)<br>87.  L.A. Davis and C.A. Pampillo, J. Appl., Phys., 42, 12, 4659­4666 (1971)<br>88.  A. Buckley and H.A. Long, Polym. Eng. Sci., 9, 2, 115­120 (1969)<br>89.  L.A. Davis, Polym. Eng. Sci., 14, 9, 641­645 (1974)<br>90.  R.K. Okine and N.P. Suh, Polym. Eng. Sci., 22, 5, 269­279 (1982)<br>91.  J.R. Collier, T.Y.T. Tam, J. Newcome, and N. Dinos, Polym. Eng. Sci, 16, 204­211 (1976)<br>92.  J.H. Faupel and F.E. Fisher, "Engineering Design", Wiley, NY (1981)<br>93.  A.E. Zachariades, R. Ball, and R.S. Porter, J. Mater. Sci., 13, 2671­2675 (1978)<br>94.  R.R. Westover, Modern Plastics, March (1963)<br>95.  B. Yi and R.T. Fenner, Plastics and Polymers, Dec., 224­228 (1975)<br>96.  A. Mekkaoui and L.N. Valsamis, Polym. Eng. Sci., 24, 1260­1269 (1984)<br>97.  H. Rust, Kunststoffe, 73, 342­346 (1983)<br>98.  J. Huszman, Kunststoffe, 73, 3437­348 (1983)<br>99.  J.M. McKelvey, U. Maire, and F. Haupt, Chem. Eng., Sept. 27, 94­102 (1976)<br>100.  K. Schneider, Kunststoffe, 68, 201­206 (1978)<br>101.  W.T. Rice, Plastic Technology, 87­91, Feb. (1980)<br>102.  Harrel Corp., "Melt Pump Systems for Extruders", Product Description TDS-264 (1982)<br>103.  J.M. McKelvey and W.T. Rice, Chem. Eng., 90, 2 89­94 (1983)<br>104.  K. Kapfer, K. Eise, and H. Herrmann, SPE ANTEC, Chicago, 161­163 (1983)<br>
+<hr>
+<A name=63></a> References <br>
+<b>47</b><br>
+105.  C.L. Woodworth, SPE ANTEC, New Orleans, 122­126 (1984)<br>106.  W.A. Kramer, SPE ANTEC, Washington, D.C., 23­29 (1985)<br>107.  J. Schut, "Die Drawing Makes Plastic Steel," Plastics Technology, Online Article, March <br>
+5 (2001)<br>
+108.  VDI Conference Extrusiontechnik 2006, "Der Einschnecken-Extruder von Morgen," VDI <br>
+Verlag GmbH, Düsseldorf (2006)<br>
+109. P. Rieg, "Latest Developments in High-Speed Extrusion," Plastic Extrusion Asia Con-<br>
+ference, Bangkok, Thailand, March 17­18, (2008); also presented at Advances in Ex -<br>trusion Conference, New Orleans, December 9­10 (2008)<br>
+<hr>
+<A name=64></a><b>3 Extruder Hardware</b><br>
+In this chapter, the hardware components of a typical single screw extruder will be <br>described. Each major component will be discussed with respect to its major func-<br>tion, the possible design alternatives, and how important the component is to the <br>proper functioning of the extruder.<br>
+<b></b><br>
+ <b>3.1Extruder Drive</b><br>
+The extruder drive has to turn the extruder screw at the desired speed. It should be <br>able to maintain a constant screw speed because fluctuations in screw speed will <br>result in throughput fluctuations that, in turn, will cause fluctuations in the dimen-<br>sions of the extrudate. Thus, constancy of speed is a very important requirement for <br>an extruder drive. The drive also has to be able to supply the required amount of <br>torque to the shank of the extruder screw. A third requirement for most extruder <br>drives is the ability to vary the speed over a relatively wide range. In most cases, one <br>would desire a screw speed that is continuously adjustable from almost zero to maxi-<br>mum screw speed. Over the years, various drive systems have been employed on <br>extruders. The main drive systems are:<br>
+ <br>
+ AC motor drive systems<br>
+ <br>
+ DC motor drive systems <br>
+ <br>
+ Hydraulic drives<br>
+<b>3.1.1AC Motor Drive System</b><br>
+The two AC drive systems used on extruders are the adjustable transmission ratio <br>drive and the adjustable frequency drive. The adjustable transmission ratio drive <br>can be either a mechanical adjustable speed drive or an electric friction clutch drive.<br>
+<hr>
+<A name=65></a><b>50</b>  <b>3Extruder Hardware</b><br>
+<b>3.1.1.1Mechanical Adjustable Speed Drive<br></b>There are four basic types of mechanical adjustable speed (MAS) drives: belt, chain, <br>wooden block, and traction type. The latter two are not used on extruders because <br>they are limited to low input speeds and easily damaged by shock loads [1, 2].<br>Belt drives use adjustable sheaves. The axial distance between the sheaves can be <br>varied: this changes the effective pitch at which the belt contacts the sheave. This, in <br>turn, changes the transmission ratio. The speed is usually varied by a vernier screw <br>mechanism, which is hand cranked or activated electrically. Belt drives are used up <br>to 100 hp. The largest speed ratio is about 10:1, and a maximum speed is typically <br>4000 rpm. Belt drives have a reasonable efficiency, tolerate shock leads, and provide <br>optimum smoothness in a mechanical drive. Disadvantages are heat generation, <br>possibility of slippage, and relatively poor speed control. In addition, belt drives are <br>subject to wear and, thus, are maintenance-intensive; belts generally have to be <br>replaced every 2000 hours.<br>Chain drives come in two different designs. One design uses a chain where each link <br>is composed of a number of laminated carriers through which a stack of hardened <br>steel slats slide. The slats fit in the grooves of conical, movable sheaves. The other <br>design uses a conventional sprocket chain, but uses extended pins to contact the <br>sheaves. Chain drives are more durable than belts and can transmit higher torques. <br>Additionally, their speed control is better than that of belt drives, about 1% of  setting. <br>Chain drives are more compact than belt drives and can operate at higher tempera-<br>tures. On the negative side, chain drives are about twice as expensive as belt drives, <br>they provide little shock load protection, they are suited only to relatively low speed <br>operation, and their speed ratio is about half that of belt drives. The efficiency of <br>both the chain and belt drive is about 90%.<br>Mechanical adjustable speed drives, nowadays, are rarely used in extruders because <br>they are maintenance intensive, have limited speed control and speed ratio, and <br>their power efficiency is not very good.<br>
+<b>3.1.1.2Electric Friction Clutch Drive<br></b>In the electric friction clutch drive, there is no direct mechanical connection between <br>input and output shaft, eliminating mechanical friction and wear. Electrical forces <br>are used to engage the input and output shaft. The three main types are hysteresis, <br>eddy-current, and magnetic particle clutches. In the extrusion industry, the eddy-<br>current drive has been widely applied in the past. The majority of the older extrud-<br>ers were equipped with eddy-current drives.<br>The popularity of this drive was, and still is to a large extent, due to the simplicity of <br>the drive. In simple terms, the eddy-current drive consists of a fixed speed AC motor <br>driving a steel drum; see Fig. 3.1.<br>
+<hr>
+<A name=66></a><IMG src="Polymer Extrusion by Chris Rauwendaal-66_1.jpg"><br>
+ <br>
+3.1Extruder Drive <br>
+<b>51</b><br>
+Wire wound rotor<br>
+Wire wound rotor<br>
+Steel drum<br>
+Steel drum<br>
+Fix Fixed speed motor<br>
+ed speed motor<br>
+Output shaft<br>
+Output shaft<br>
+<b> Figure 3.1</b><br>
+Solid state<br>
+Solid state<br>
+ <br>
+controller<br>
+controller<br>
+The eddy-current clutch<br>
+Inside this drum, a wire-wound rotor is positioned, with a small annular gap between <br>rotor and drum. When a low-level current is applied to the rotor, it is dragged by the <br>rotation of the drum at a somewhat lower rotational speed. When the voltage to <br>the rotor is reduced, the slippage between the rotor and drum will increase. Thus, <br>reduced voltage reduces the rotor speed, since the speed of the drum is constant. By <br>controlling the voltage to the rotor, the rotor speed can be varied or it can be main-<br>tained at a steady speed under varying loads.<br>Typical operating characteristics for eddy-current drives are [3]:<br>
+ <br>
+ 30:1 speed range at constant torque<br>
+ <br>
+ Intermittent torque to 200% of rated torque<br>
+ <br>
+ Speed regulation: 0.5% of maximum speed<br>
+ <br>
+ Drift: 0.05% of maximum speed per °C<br>
+ <br>
+ Ability to deliver rated torque at stall conditions<br>The efficiency of the eddy-current drive is proportional to the difference between <br>input and output speed. Thus, when an extruder is operated at low speeds for ex -<br>tended periods of time the eddy-current drive would not be a good candidate from an <br>energy consumption point of view. It is possible to reduce this problem by using a <br>two-speed AC motor to drive the eddy-current clutch [3, 4].<br>
+<b>3.1.1.3Adjustable Frequency Drive<br></b>Adjustable frequency drives use an AC squirrel cage induction motor connected to <br>a solid-state power supply capable of providing an adjustable frequency to the AC <br>motor. The AC squirrel cage induction motor has several advantages: low price, sim-<br>plicity, ruggedness, no commutators and brushes, low maintenance, and compact <br>construction. The cost of the adjustable frequency drive is mostly determined by <br>the solid-state power supply. The power supply converts AC power to DC power. It <br>then inverts the DC energy in order to provide the required frequency and voltage <br>for the AC motor. Thus, full power is handled through two required sets of solid-state <br>devices, unlike the single conversion used in the DC-thyristor system. Therefore, the <br>
+<hr>
+<A name=67></a><b>52</b>  <b>3Extruder Hardware</b><br>
+cost of an adjustable frequency drive used to be higher than a comparable DC drive, <br>even though the DC motor itself is more expensive than the AC motor.<br>A common power supply is the six-step variable voltage inverter; see Fig. 3.2.<br>
+Rectifier<br>
+Rectifier<br>
+Inv<br>
+In erter<br>
+verter<br>
+L<br>
+L 1<br>
+1<br>
+L<br>
+L 2<br>
+2<br>
+L<br>
+L 3<br>
+3<br>
+<b>Figure 3.2</b>Six-step variable voltage inverter<br>
+Incoming three-phase AC power is rectified and smoothed to generate a variable <br>voltage DC supply. This DC voltage is then switched among the three output phases <br>to generate a step waveform that approximates a sinusoidal waveform. The switch-<br>ing is done by six SCRs (silicon-controlled rectifiers), which are sequentially fired at <br>the proper frequency by a solid-state circuit.<br>The ratio of voltage to frequency must be held constant to maintain a constant torque <br>capability as the motor speed is varied. Almost any speed/torque characteristic can <br>be obtained by varying the voltage to frequency ratio. Because of limitations of the <br>SCR cells, the maximum rating of adjustable frequency drives is presently around <br>300 hp. As better SCR cells are developed, this maximum rating is likely to increase.<br>The flux vector (FV) controlled variable frequency AC drives represent a relatively <br>new technology made possible by developments in solid-state power switching <br>devices and microprocessors. An FV drive is a variable frequency AC drive, able to <br>control both the magnetizing current and torque producing current through vector <br>calculations. FV drives achieve torque and speed control better than DC drives. They <br>use a special AC motor, a "high efficiency" or "vector" duty motor, insulated suffi-<br>ciently to handle the voltage spikes to which they are subjected by the "chopping" <br>action of these drives. These motors are rugged, inexpensive, and low maintenance; <br>therefore, they have low operating cost. Figure 3.3 shows a schematic of an FV drive.<br>
+Variable<br>
+AC line<br>
+frequency AC<br>
+SCR<br>
+Constant<br>
+AC<br>
+Rectifier<br>
+DC voltage<br>
+Inverter<br>
+AC motor <b> Figure 3.3</b> <br>
+Speed<br>
+Encoder or<br>
+Schematic of flux <br>
+feedback<br>
+tachometer<br>
+ <br>vector  drive<br>
+<hr>
+<A name=68></a> <br>
+3.1Extruder Drive <br>
+<b>53</b><br>
+A speed regulation of 0.01% can be achieved with a 1000:1 speed range using an <br>encoder and with constant torque characteristics. FV drives have a better power fac-<br>tor over the low speed range than DC drives because there is not the long delay in <br>firing the SCRs to create a low DC voltage. FV drives provide constant torque up to <br>the base speed, usually 1750 rpm. The 1750 base speed motors can operate in an <br>extended speed range up to 3500 rpm. Low base speed vector duty motors with high <br>horsepower have not become widely available at a competitive price. The cost of <br>smaller FV drives is attractive; however, sizes over 100 hp tend to be more expen-<br>sive than DC drives. Flux vector drives are available up to 400 hp. A further advan-<br>tage of AC drives is that they do not require isolation transformers.<br>
+<b>3.1.2DC Motor Drive System</b><br>
+Some early DC extruder drives used fixed-speed AC motors to drive DC generators <br>that produced the variable voltage for the DC motor. Nowadays, the DC motor drives <br>usually operate from a solid-state power supply, since this power supply is generally <br>more cost-effective than the motor generator set. The DC motor drive can be simpler <br>and cheaper than the variable frequency drive, even when the higher cost of the DC <br>motor is included. The smaller number of solid-state devices tends to give the DC <br>drive a better reliability than the variable frequency drive. Brushes and commutator <br>maintenance is the principal drawback to the use of DC motors. If the drive has to be <br>explosion-proof, the additional expense associated with this option may be quite <br>large for a DC drive, more so than with a variable frequency AC drive or a hydraulic <br>drive. A schematic of the DC drive is shown in Fig. 3.4.<br>
+AC line<br>
+DC motor<br>
+SCR<br>
+Adjustable<br>
+Rectifier<br>
+DC voltage<br>
+Speed<br>
+Encoder or<br>
+<b> Figure 3.4</b> <br>
+feedback<br>
+tachometer<br>
+Schematic of DC drive<br>
+The DC drive can provide a speed range of up to 100:1. The DC motor can handle <br>either a constant torque or constant load, and in some cases both (with field weaken-<br>ing). Any overload capacity presented to the motor must be provided in the sizing of <br>the solid-state power supply. Generally, drives are provided with an overload capa-<br>city of 150% for one minute. The DC motor can be readily reversed by reversing the <br>armature of the motor. For rapid stopping, resistors can be connected across the <br>armature by a contactor, thereby providing dynamic braking at relatively low cost. <br>DC motors respond quickly to changes in control signal due to their high ratio of <br>torque to inertia.<br>
+<hr>
+<A name=69></a><b>54</b>  <b>3Extruder Hardware</b><br>
+The DC voltage from the solid-state power supply generally has a rather poor form <br>factor. The magnitude of the form factor is dependent on the configuration of the <br>rectifier circuitry. The poorer the form factor, the higher the ripple current in the DC <br>motor. This increases motor heating and reduces the power efficiency. Several three-<br>phase rectifier circuits are available for the AC line power into DC. Most drives over <br>5 hp use three-phase full wave circuitry. Figure 3.5 shows a three-phase half-con-<br>trolled full wave rectifier.<br>
+L1<br>L2<br>
+L3<br>
+<b>Figure 3.5</b>Three-phase half-controlled full wave rectifier<br>
+This circuit uses only three thyristors and four diodes. The drawback of this rectifier <br>circuit is its high ripple current; the typical form factor is 1.05, with the ripple cur-<br>rent frequency 180 Hz.<br>Another popular rectifier circuit is the full-controlled three-phase full wave rectifier. <br>This circuit is more expensive because six thyristors are used. However, the form <br>factor is much better, about 1.01, and the ripple current is 360 Hz. The higher fre-<br>quency makes it easier to filter the ripple current. The half-controlled three-phase <br>bridge rectifier circuit may require armature current smoothing reactors to reduce <br>the ripple current. Another problem associated with the non-uniform DC input to <br>the motor is the commutation. The motor must commutate under a relatively high <br>degree of leakage reactance.<br>A potential safety hazard is the fact that with armature current feedback, the arma-<br>ture current is connected to the operator control and potentiometers may be operat ed <br>at high potentials (500 V). This problem can be eliminated using isolation trans-<br>formers or DC to DC chopper circuits.<br>
+<hr>
+<A name=70></a> <br>
+3.1Extruder Drive <br>
+<b>55</b><br>
+<b>3.1.2.1Brushless DC Drives<br></b>Brushless DC drives have an advantage in that the motor does not contain brushes. <br>As a result, the drive is less maintenance intensive. The motor contains permanent <br>magnets; the size of the magnets determines the horsepower capability of the motor. <br>The maximum power available today is around 600 hp. A schematic of the brushless <br>DC drive is shown in Fig. 3.6.<br>
+Variable<br>
+Brushless<br>
+AC line<br>
+frequency AC<br>
+DC motor<br>
+SCR<br>
+Constant<br>
+AC<br>
+Rectifier<br>
+DC voltage<br>
+Inverter<br>
+Speed<br>
+Encoder or<br>
+feedback<br>
+tachometer<br>
+<b>Figure 3.6</b>Schematic of brushless DC drive<br>
+Brushless DC drives have been used frequently with extruders. With the advent of <br>flux vector AC drives, the brushless DC is used less; however, it still provides useful <br>characteristics as shown in Table 3.2 in Section 3.1.4.<br>
+<b>3.1.3Hydraulic Drive System</b><br>
+A hydraulic drive generally consists of a constant speed AC motor driving a hydrau-<br>lic pump, which, in turn, drives a hydraulic motor and, of course, the associated <br>controls. The entire package is often referred to as a hydrostatic drive. Some of the <br>advantages of a hydrostatic drive are stepless adjustment of speed, torque, and <br>power; smooth and controllable acceleration; ability to be stalled without damage; <br>and easy controllability.<br>Over the years, considerable improvements have been made to pumps and motors, <br>resulting in improved stability, controllability, efficiency, reduced noise, and reduced <br>cost. Hydrostatic drives are presently used in many demanding applications. At <br>least three types of output performance are commonly available. Variable power, <br>variable torque transmissions are based on a variable displacement pump supply-<br>ing a variable displacement motor. These transmissions provide a combination of <br>constant torque and constant power. These units are the most adjustable, most flex-<br>ible, and most expensive. Constant torque, variable power transmissions are based <br>on a variable displacement pump supplying fluid to a fixed displacement motor <br>under constant load. Speed is controlled by varying pump delivery. This is generally <br>considered the best general-purpose drive, with wide speed ranges, up to 40:1, and <br>simple controls. Constant power, variable torque transmissions are based on a vari-<br>
+<hr>
+<A name=71></a><b>56</b>  <b>3Extruder Hardware</b><br>
+able displacement pump with a power limiter, driving a fixed displacement motor. <br>The main strength of this transmission is its efficiency; however, the speed range is <br>usually limited to 4:1.<br>Hydrostatic transmissions are available where the pump and motor are combined in <br>a single rigid unit, usually referred to as close-coupled transmissions. This produces <br>a very compact drive that can be encased in a sealed housing to protect it from its <br>environment. By eliminating external plumbing, close coupling reduces noise, vib-<br>ration, flow losses, and leakage.<br>Hydraulic drives can be controlled quite well. Advances in control sophistication <br>have helped broaden the field in which hydrostatic drives are applied. Fast-acting <br>pressure compensators reduce heat generation, eliminate the need for crossport <br>relief valves, and simplify other control circuits. Load sensing controls increase <br>operating efficiency by reducing unnecessary loads on the pump. Brake and bypass <br>circuits eliminate the need for external actuation mechanical brakes. Power limiters <br>eliminate prime-mover stalls by limiting the transmission output to a maximum <br>value. And speed controls hold the output speed at a constant value, regardless of <br>the prime mover speed.<br>The overall efficiency of hydrostatic drives can be quite good. For a well-designed <br>system, the overall efficiency can be as high as 70% over a reasonably wide range. <br>The use of accumulators and logic-type valves (cartridge) is beneficial to the effi-<br>ciency of the drive, particularly when they are connected to programmable controls. <br>Typical efficiency curves for a high torque, low speed hydraulic motor [6] are shown <br>in Fig. 3.7.<br>
+98<br>
+96<br>
+ [%]<br>
+19.5<br>
+14.7<br>
+94<br>
+fi<br>ciency<br>
+9.8 MPa<br>
+92<br>
+Mechanical ef<br>
+90 0<br>
+50<br>
+100<br>
+150<br>
+Motor speed [rpm]<br>
+<b>Figure 3.7</b>Efficiency curves for high torque, low speed hydraulic motor<br>
+<hr>
+<A name=72></a> <br>
+3.1Extruder Drive <br>
+<b>57</b><br>
+It is seen that an excellent efficiency can be obtained in the range from zero to <br>150 rpm, which is the typical operating range for extruders. It can be further noticed <br>that higher operating pressures increase the mechanical efficiency. The drawback of <br>a higher working pressure is more strain on the seals, couplings, and other compo-<br>nents; this can increase the cost of the drive.<br>One significant advantage of hydrostatic drives for extruders is the elimination of <br>the need for a transmission between the hydraulic motor and extruder screw. Since <br>low speed hydraulic motors are readily available, one can do away with the bulky, <br>expensive gearbox found on most extruders. Therefore, when comparing the cost <br>of a hydrostatic drive to a DC drive, one should include the cost of the gearbox in <br>the cost of the DC drive. Another advantage is the fact that other components of <br>the machine or auxiliary equipment can be operated hydraulically from the same <br>hydraulic power supply. This is particularly advantageous for screen changers <br>where the screen is changed by a hydraulic cylinder or in molding machines where <br>the mold is opened and closed hydraulically. For these reasons, hydraulic drives <br>have become almost standard for reciprocating extruders used in injection molding <br>and blow molding.<br>It should be realized, however, that the operation of the reciprocating extruder as a <br>plasticating unit of a molding machine is considerably different from a conventional <br>rotating extruder. The screw rotation is stopped abruptly at the end of the plasticat-<br>ing cycle, then the screw is moved forward and remains in the forward position for <br>some time. Then the screw starts rotating again and as material accumulates at the <br>tip of the screw, the screw moves backward until enough material has accumulated. <br>Then the cycle repeats itself. Thus, in this operation there is a frequent stop and go <br>motion. The hydraulic drive is ideally suited for this type of operation. The operation <br>of a conventional extruder is much more continuous, and therefore its drive require-<br>ments are different from a reciprocating extruder.<br>Despite the attractive features of hydrostatic drives, they are rarely used on regular <br>(non-reciprocating) extruders. The reason for this situation is not obvious since <br>the hydrostatic drive is in many respects competitive with, for instance, the SCR DC <br>drive and in some respects better; e.g., there is no need for a gearbox. A possible <br>reason is that the hydrostatic drive is still regarded with suspicion by many people. <br>This is because early hydraulic drives were not very reliable and accurate. However, <br>this situation has changed dramatically over the years, but the hydrostatic drive still <br>seems to suffer from its early unfavorable reputation. Very few U.S. companies sup-<br>ply hydraulic drives for extruders: Feed Screws Division of New Castle Industries <br>and Wilmington Plastics Machinery. It is claimed [5] that hydraulic drives are less <br>expensive than DC drives on smaller extruders, up to about 90 mm.<br>
+<hr>
+<A name=73></a><b>58</b>  <b>3Extruder Hardware</b><br>
+<b>3.1.4Comparison of Various Drive Systems</b><br>
+Table 3.1 summarizes the functional requirements for drives used in extrusion <br>machinery. This comparison is based on a paper by Kramer [32].<br>
+<b>Table 3.1</b>Functional Requirement for Drives Used in Extrusion Machinery [32]<br>
+Machine<br>
+HP<br>
+Function<br>
+Torque<br>
+Speed<br>
+Speed<br>
+Regeneration<br>
+control<br>
+range<br>
+or braking<br>
+Extruders<br>
+5­800<br>
+Melting<br>
+Constant<br>
+0.1%<br>
+100:1<br>
+No<br>
+Melt pumps<br>
+1­20<br>
+Pumping<br>
+Constant<br>
+0.01%<br>
+100:1<br>
+No<br>
+Pullers<br>
+0.25­15<br>
+Pulling<br>
+Constant<br>
+0.01%<br>
+20:1<br>
+Both<br>
+Winders<br>
+0.25­10<br>
+Winding<br>
+Variable<br>
+1%<br>
+20:1<br>
+Braking<br>
+Cutters<br>
+0.25­5<br>
+Cycling<br>
+Starting<br>
+0.01%<br>
+100:1<br>
+Braking<br>
+Table 3.2 compares the most important drive systems.<br>
+<b>Table 3.2</b>Comparison of Different Drive Systems [32]<br>
+DC with<br>
+Brushless DC<br>
+AC servo<br>
+AC vector<br>
+encoder<br>
+with encoder<br>
+Power range [HP]<br>
+0.25­2000<br>
+0.25­600<br>
+0.25­20<br>
+0.25­400<br>
+Starting torque<br>
+150%<br>
+150%<br>
+175%<br>
+150%<br>
+Speed regulation<br>
+1.0­0.01%<br>
+0.01%<br>
+0.01%<br>
+0.5­0.01%<br>
+Constant torque<br>
+20:1<br>
+1000:1<br>
+40:1<br>
+1000:1<br>
+Regenerative braking<br>
+Yes<br>
+No<br>
+No<br>
+Limited<br>
+Dynamic braking<br>
+Yes<br>
+Yes<br>
+Yes<br>
+Yes<br>
+Relative cost<br>
+Moderate<br>
+Moderate<br>
+High<br>
+Moderate<br>
+Operating cost<br>
+Moderate<br>
+Low<br>
+Low<br>
+Low<br>
+EMI and line noise<br>
+Good<br>
+Poor<br>
+Poor<br>
+Poor<br>
+Power factor<br>
+0.20­0.85<br>
+0.98<br>
+0.98<br>
+0.98<br>
+The power factor is also an important factor to consider. The power factor of the DC <br>drive reduces slightly with load, but it drops drastically with speed. This becomes <br>more severe as the horsepower increases. The power factor of the variable frequency <br>AC drive is higher than the power factor of the DC drive and it is much less affected <br>by speed. In fact, for small to medium horsepower variable frequency AC drives, the <br>power factor is essentially independent of load and speed. The effect of the power <br>factor on overall energy cost cannot be simply calculated by multiplying the line-to-<br>shaft efficiency with the power factor.<br>The cost of electric power generally depends both on the actual power (KW) and <br>the apparent power (KVA). The power factor is the ratio of actual power to apparent <br>power. Most utilities incorporate a certain cost penalty if the power factor is consider-<br>
+<hr>
+<A name=74></a> <br>
+3.1Extruder Drive <br>
+<b>59</b><br>
+ably below one for extended periods of time. The actual cost penalty will differ from <br>one utility to another. Also, to determine the actual expenses for electric power for a <br>certain production facility, an energy survey must be made of the entire facility. It is <br>possible that the extruder drive has a small effect on the overall power factor. In that <br>case, the power factor of the extruder drive is of little concern. If the overall power <br>factor is strongly affected by the extruder drive(s), the power factor of the electric <br>motor of the extruder drive may be a serious concern. In this case, an energy man-<br>agement system should be used to keep the overall power factor as high as possible.<br>With DC electric motors, power factor correction is sometimes used to improve the <br>power factor of the drive [30]. This is done by incorporating capacitive components <br>into the circuit. Capacitors produce leading reactive power whereas the phase-con-<br>trolled rectifiers produce lagging reactive power. Thus, by adding appropriately <br>sized capacitors, the power factor of the drive can be improved.<br>The line-to-shaft efficiency of DC motors is around 0.85 and about 0.80 for variable <br>frequency AC motors. Considering that a typical two-stage gearbox has an efficiency <br>of about 0.95, the overall efficiency for a DC drive is about 0.80 and about 0.75 for <br>a variable frequency AC drive. The overall efficiency of a well-designed hydrostatic <br>drive is around 0.70. Thus, the hydrostatic drive compares reasonably well with the <br>AC and DC drives with regards to overall efficiency [31]. The drive efficiency of the <br>DC motor drive and the adjustable frequency drive increases with rated speed, <br>whereas the drive efficiency of the hydraulic drive is relatively independent of rated <br>speed. The efficiency of the DC motor is better than other motors in the range of <br>20 to 100% rated speed; below 20% rated speed the hydraulic drive is more efficient.<br>The mechanical drive has a reasonable overall efficiency at full load and full speed. <br>The advantage of this drive is that the efficiency does not change with speed. Thus, <br>at low speeds this drive can actually be more efficient that the DC or AC drive. A <br>drawback of the mechanical drive is the higher maintenance requirement.<br>Another consideration in choosing a drive is the speed drift. The eddy-current drive <br>has a speed drift of about 0.4% per °C. The adjustable frequency AC drive has a <br>speed drift of 0.05% or better. With a DC (SCR) drive using armature voltage control, <br>the speed can vary 10% for the initial warm-up period of about 15 to 30 minutes. <br>After the warm-up period, the drift will be about 1%. The speed drift can be reduced <br>by using tachometer feedback and regulated field; this reduced the drift of the DC <br>drive to about 0.25%.<br>
+<b>3.1.5Reducer</b><br>
+With AC or DC drives, a reducer is generally required to match the low speed of the <br>screw to the high speed of the drive. The typical reduction ratio ranges from 15:1 to <br>20:1. The type of reducer most frequently used is the spur gear reducer, often in a <br>
+<hr>
+<A name=75></a><b>60</b>  <b>3Extruder Hardware</b><br>
+two-step configuration, i.e., two sets of intermeshing gears. A popular type of spur <br>gear is the herringbone gear because the V-shaped tooth design practically elimi-<br>nates axial loads on the gears. The efficiency of these gears is high, about 98% at <br>full load and 96% at low load.<br>Some gearboxes are equipped with a quick-change gear provision, which allows one <br>to change gear ratio rather quickly and easily. This feature can add significantly to <br>the flexibility and versatility of the extruder. Of course, one has to make sure that <br>the quick-change gear unit is designed in such a way that it does not significantly <br>affect the transmission efficiency of the reducer.<br>Worm reduction gears have been used on rare occasions. Their advantage is low cost <br>and compactness, but the efficiency is rather poor, between 90 and 75%. Some <br>extruders do not have a direct connection between the drive and reducer but employ <br>either a chain or a belt transmission. This type of setup allows a relatively simple <br>change in overall reduction ratio by changing the sprocket or sheave diameter. An <br>advantage of the belt drive is the fact that it provides protection against excessive <br>torque. A distinct disadvantage is increased power consumption, as much as 5 to <br>10%. Another drawback is the fact that the chain or belt transmission is less reliable <br>than the spur gear reducer and requires more maintenance.<br>
+<b>3.1.6Constant Torque Characteristics</b><br>
+Most extruders have a so-called "constant torque" characteristic. This means that <br>the maximum torque obtainable from the drive, for all practical purposes, remains <br>constant over the range of screw speed. The torque-speed characteristic can be used <br>to determine the power-speed characteristic by using the well-known relationship <br>between torque and power:<br>
+ (3.1)<br>
+where T is torque, P power, N screw speed or motor speed, and C a constant (C = 2/ <br>60  0.1 when N is expressed in revolutions per minute).<br>Thus, if torque is constant with speed, then it follows directly from Eq. 3.1 that the <br>power is directly proportional to speed; see Fig. 3.8.<br>This means that the maximum power of the drive can only be utilized if the motor is <br>running at full speed. Whenever the extruder output is power-limited, it is good <br>practice to make sure the motor is running at full speed. If it is not, then a simple <br>gear change can often alleviate the problem.<br>
+<hr>
+<A name=76></a> <br>
+3.2Thrust Bearing Assembly <br>
+<b>61</b><br>
+Max. torque<br>
+Max. power<br>
+e<br>
+d<br>
+er<br>
+qu<br>
+d<br>
+Tor<br>
+Pow<br>
+Max. spee<br>
+Max. spee<br>
+Screw speed<br>
+Screw speed<br>
+<b>Figure 3.8</b>Torque and power versus screw speed<br>
+It is generally very expensive to solve a power limitation problem by installing a <br>more powerful motor. The reason is that most gearboxes are matched in terms of <br>power rating to the motor driving the gearbox. Thus, if the motor power is increased <br>significantly, quite often the gearbox has to be replaced concurrently. This makes <br>for a very expensive replacement. In fact, it could be more cost-effective to purchase <br>an entirely new extruder. Mechanical power consumption is, to a very large extent, <br>determined by the design of the extruder screw. There are many options to change <br>the screw design, which will reduce the power consumption of the drive (see Chap-<br>ter 8).<br>
+<b></b><br>
+ <b>3.2Thrust Bearing Assembly</b><br>
+The thrust bearing assembly is usually located at the point where the screw shank <br>connects with the output shaft of the drive, which is generally the output shaft of <br>the gearbox. Thrust bearing capability is required because the extruder generally <br>develops substantial diehead pressure in the polymer melt. This diehead pressure is <br>necessary to push the polymer melt through the die at the desired rate. However, <br>since action = reaction, this pressure will also act on the extruder screw and force it <br>towards the feed end of the extruder. Therefore, the thrust bearing capability has to <br>be available to take up the axial forces acting on the screw. Clearly, the load on the <br>thrust bearing is directly determined by the diehead pressure. The actual force on <br>the screw is obtained by multiplying the diehead pressure with the cross-sectional <br>area of the screw. Thus, when the size of the extruder increases, the load on the <br>thrust bearing will increase at least as fast. A 150 mm (6 inch) extruder running <br>with a diehead pressure of 35 MPa (about 5000 psi) will experience an axial thrust <br>of about 620 kN (about 140,000 lbf). This illustrates that significant forces are act-<br>
+<hr>
+<A name=77></a><IMG src="Polymer Extrusion by Chris Rauwendaal-77_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-77_48.jpg"><br>
+<b>62</b>  <b>3Extruder Hardware</b><br>
+ing on the screw and proper design and dimensioning of the thrust bearing is criti-<br>cal to the trouble-free operation of the extruder.<br>Figure 3.9 shows a typical thrust bearing arrangement for a single screw extruder.<br>
+<b>Figure 3.9</b>Thrust bearing assembly for single screw extruder<br>
+The screw shank is usually keyed or splined and fits into a driving sleeve in the <br>bearing housing. Thrust bearings are designed to last a certain number of revolu-<br>tions at a certain thrust load. Under normal operating conditions and reasonable <br>diehead pressure (0­35 MPa), the thrust bearing will generally last as long as the <br>life of the extruder. However, if the extruder operates with sharp fluctuations in <br>diehead pressure and/or if the diehead pressure is unusually high (40­70 MPa), the <br>life expectancy of the thrust bearing can reduce dramatically, particularly when the <br>extruder screw runs at high speed.<br>The statistical rated life of a bearing is generally predicted using the following for-<br>mula:<br>
+ (3.2)<br>
+where L10 = rated life in revolutions<br>C = basic load rating<br>P = equivalent radial load<br>K = constant, 3 for ball bearings, and 10/3 for roller bearings<br>It is important to notice that increased load, i.e., increased diehead pressure, <br>reduces the bearing life by a power of three or more! Also, the life L will be reached <br>more quickly when the extruder runs at high speed. The predicted life, Ly, expressed <br>
+in years, is obtained from the following expression:<br>
+<hr>
+<A name=78></a> <br>
+3.2Thrust Bearing Assembly <br>
+<b>63</b><br>
+ (3.3)<br>
+where Ly is predicted life in years and N the screw speed in revolutions per minute. <br>This is based on 24 hours per day, 365 days per year operation. Thus, the expected <br>lifetime of the thrust bearing is inversely proportional to the screw speed. Sharp <br>fluctuations to the thrust load can further reduce the thrust bearing life. The effect <br>of load fluctuation is usually assessed by means of load factors used in Eqs. 3.2 and <br>3.3. The handbook of the particular thrust bearing manufacturer or the extruder <br>manufacturer should be consulted for the proper values of these load factors.<br>Extruder manufacturers often give the rated life of the thrust bearing as a B-10 life. <br>This is expressed in hours at a particular diehead pressure, 35 MPa (5,000 psi), and <br>screw speed, 100 rev/min. The B-10 life represents the life in hours at a constant <br>speed that 90% of an apparently identical group of bearings will complete or exceed <br>before the first evidence of fatigue develops; i.e., 10 out of 100 bearings will fail be -<br>fore rated life. The B-10 life at normal operating load should be at least 100,000 <br>hours in order to get a useful life of more than 10 years out of the thrust bearing.<br>The predicted B-10 life at any diehead pressure and/or screw can be found by the <br>following relationship:<br>
+ (3.4)<br>
+where P is diehead pressure in MPa, N screw speed in rpm, constant K is 3 for ball <br>bearings and 10/3 for roller bearings, and B-10Std is the B-10 life at P = 35 MPa and <br>
+N = 100 rpm.<br>In single screw extruders, the design of the thrust bearing assembly is relatively <br>easy since the diameter of the bearings can be increased without much of a problem <br>in order to obtain the required load carrying capability. This situation is entirely dif-<br>ferent in twin screw extruders because of the close proximity of the two screws. This <br>severe space limitation makes the proper design of the thrust bearing assembly in <br>twin screw extruders considerably more difficult than in single screw extruders. <br>Older twin screw extruders were often limited in diehead pressure capability exactly <br>because of this thrust-bearing problem. A number of newer twin screw extruders <br>have solved this problem sufficiently and can generally withstand almost the same <br>diehead pressures as single screw extruders, although the rated life for the thrust <br>bearings of a twin screw extruder is generally lower.<br>Figure 3.10 shows an example of a thrust bearing assembly of a counter-rotating <br>twin screw extruder.<br>
+<hr>
+<A name=79></a><IMG src="Polymer Extrusion by Chris Rauwendaal-79_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-79_20.jpg"><br>
+<b>64</b>  <b>3Extruder Hardware</b><br>
+Axial self-aligning<br>
+Axial tandem<br>
+Anchor<br>
+Housing<br>
+roller bearing<br>
+roller bearing<br>
+bolt<br>
+<b>Figure 3.10</b>Thrust bearing assembly for counter-rotating twin screw extruder<br>
+The thrust bearing assembly consists of four or five roller bearings in tandem <br>arrangement with a special pressure balancing system. Advantages and disadvan-<br>tages of certain types of thrust bearings are listed in Table 3.3.<br>Fluid film thrust bearings have been applied to extruders on a few occasions. Their <br>load carrying capability at low speed is generally poor and a loss of fluid film would <br>have disastrous results. If a hydraulic drive is used to turn the screw, application of <br>hydraulic thrust bearings may deserve some consideration. Reference 28 describes <br>an extruder with hydraulic drive that incorporates a patented thrust bearing assem-<br>bly with hydraulic axial screw adjustment. By measuring the pressure of the hydro-<br>static chamber of the thrust bearing, the pressure in the polymer melt at the end of <br>the screw can be determined.<br>
+<b></b><br>
+ <b>3.3Barrel and Feed Throat</b><br>
+The extruder barrel is the cylinder that surrounds the extruder screw. The feed <br>throat is the section of the extruder where material is introduced into the screw <br>channel; it fits around the first few flights of the extruder screw. Some extruders do <br>not have a separate feed throat unit; in these machines, the feed throat is an integral <br>part of the extruder barrel. However, there are some drawbacks to this type of <br>design. The feed throat casting is generally water-cooled. This is done to prevent an <br>early temperature rise of the polymer. If the polymer temperature rises too high it <br>may stick to the surface of the feed opening, causing a restriction to the flow into the <br>extruder. Polymer sticking to the screw surface also causes a solids conveying prob-<br>
+<hr>
+<A name=80></a> <br>
+3.3Barrel and Feed Throat <br>
+<b>65</b><br>
+lem, because the polymer particles adhering to the screw will not move forward and <br>will restrict the forward movement of the other polymer particles.<br>
+<b>Table 3.3</b>Comparison of Various Types of Thrust Bearings<br>
+Type<br>
+Advantages<br>
+Disadvantages<br>
+Ball thrust bearing<br>
+High speed capability<br>
+Low thrust capability<br>
+Angular contact ball thrust <br>
+High speed capability, radial load <br>
+Lower thrust capability<br>
+bearing<br>
+capability<br>
+Cylindrical roller thrust <br>
+Higher capacity, minimum cost<br>
+Not true rolling contact, more heat <br>
+bearing<br>
+generation<br>
+Spherical roller thrust <br>
+Dynamic misalignment tolerability,  High cost/capacity ratio, difficult <br>
+ <br>bearing<br>
+almost true rolling contact, radial <br>
+to lubricate<br>
+load capability<br>
+Tapered roller thrust <br>
+True rolling contact, very high <br>
+Difficult to lubricate, minimum <br>
+ <br>bearing<br>
+capacity, low flange loading<br>
+sizes available, special alignment <br>
+considerations<br>
+Tapered roller thrust <br>
+True rolling contact<br>
+Higher flange loading<br>
+ <br>bearing,  V-flat<br>Tapered roller thrust <br>
+Greater size flexibility, least cost <br>
+ <br>bearing, V-flat, aligning<br>
+per L-10 life, easy to lubricate, <br>
+static misalignment tolerability<br>
+At the point where the feed throat casting connects with the barrel, a thermal bar-<br>rier should be incorporated to prevent heat from the barrel from escaping to the feed <br>throat unit. In a barrel with an integral feed opening this is not possible. Therefore, <br>heat losses will be greater and there is also the chance of overheating of the feed <br>throat.<br>The geometry of the feed port should be such that the material can flow into the <br>extruder with minimum restriction. Cross-sections of various feed port designs are <br>shown in Fig. 3.11.<br>Figure 3.11(a) shows the usual feed port design. Figure 3.11(b) shows an undercut <br>feed port as is often used on melt fed extruders. The danger with this design is the <br>wedging section between screw and feed opening. If the melt is relatively stiff and/<br>or highly elastic, significant lateral forces will act on the screw. These forces can be <br>high enough to deflect the screw and force it against the barrel surface. This, of <br>course, will lead to severe wear if the contact pressure is sufficiently high. This <br>problem is more severe when this geometry is used for feeding solid polymer pow-<br>der or pellets. This geometry, therefore, should only be used for feeding molten poly-<br>mers. A better geometry would be the one shown in Fig. 3.11(c). It has an undercut <br>to improve the intake capability, but the pronounced wedge is eliminated by a flat <br>section oriented more or less in the radial direction.<br>
+<hr>
+<A name=81></a><IMG src="Polymer Extrusion by Chris Rauwendaal-81_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-81_19.jpg"><br>
+<b>66</b>  <b>3Extruder Hardware</b><br>
+a<br>
+b<br>
+c<br>
+<b>Figure 3.11</b>Different feed port geometries<br>
+The shape of the inlet opening is usually circular or square. The smoothest transi-<br>tion from feed hopper to feed throat will occur if the cross-sectional shape of the <br>hopper is the same as the shape of the feed opening. Thus, a circular hopper should <br>feed into a circular feed port. A study done by Miller [7] on various feed port open-<br>ings did not reveal a noticeable advantage of increasing the length of the opening <br>beyond one diameter. Michaeli et al. [33] found that the throughput can increase <br>significantly when the axial length of the feed opening is increased beyond one <br>diameter. A length of 1.5D to 2.5D can offer advantages in many cases.<br>Extruders equipped with grooved barrel sections often have a specially designed <br>feed throat section to accommodate this grooved section. A schematic feed throat <br>section is shown in Fig. 3.12, where the effective length of the grooves may range <br>from three to five diameters.<br>The depth of the grooves varies with axial distance. The depth is maximum at the <br>start of the grooved bushing and reduces to zero where the grooved section meets <br>with the smooth extruder barrel.<br>
+Barrel<br>
+Hopper<br>
+Barrel<br>
+Hopper<br>
+Thermal barrier<br>
+Thermal barrier<br>
+A<br>
+A<br>
+A<br>
+A<br>
+Cooling channels<br>
+Cooling channels<br>
+Grooved sleeve<br>
+Grooved sleeve<br>
+Section A-A<br>
+Section A-A<br>
+<b>Figure 3.12</b>Grooved feed housing<br>
+<hr>
+<A name=82></a> <br>
+3.3Barrel and Feed Throat <br>
+<b>67</b><br>
+Several important requirements in this feed section design are<br>
+ <br>
+ Very good cooling capability<br>
+ <br>
+ Good thermal barrier between feed section and barrel<br>
+ <br>
+ Large pressure capability<br>The requirement of good cooling is due to the large amount of frictional heat gener-<br>ated in the grooved barrel section. If the heat is not carried away quickly enough, <br>the polymer will soften or even melt. This will severely diminish the effectiveness <br>of the grooved barrel section. The requirement of a good thermal barrier between <br>the grooved feed section and extruder barrel is to minimize the heat flow from the <br>barrel to the feed section, so as to maximize the cooling capacity of the feed throat <br>section. Very large pressures can be generated in the grooved feed section, from 100 <br>to 300 MPa (about 15,000 to 45,000 psi). The feed section should be designed with <br>the ability to withstand pressures of this magnitude otherwise spectacular modes of <br>failure may occur.<br>The stresses between the polymer and the grooves can be very high. As a result, <br>wear can be a significant problem, particularly when the polymer contains abrasive <br>components. The splines in the grooved bushing, therefore, are generally made out <br>of highly wear-resistant material (see Section 11.2.1).<br>The extruder barrel is simply a flanged cylinder. It has to withstand relatively high <br>pressures, as high as 70 MPa (10,000 psi), and should possess good structural rigid-<br>ity to minimize sagging or deflection. Many extruder barrels are made with a wear-<br>resistant inner surface to increase the service life. The two most common techniques <br>are nitriding and bimetallic alloying.<br>Nitriding can be done by ion-nitriding (by glow discharge or plasma) or by conven-<br>tional nitriding techniques (gas nitriding or liquid bath nitriding). It is generally <br>recognized that ion-nitriding yields superior results. In ion-nitriding, the barrel is <br>first hardened and tempered to achieve the desired core properties. The barrel is <br>then placed in a vacuum chamber and connected in a high voltage DC circuit with <br>the barrel as the cathode and the vacuum chamber as the anode. The chamber is <br>evacuated and nitrogen-bearing process gas is introduced. A potential difference of <br>about 400 to 1000 volts is applied between vacuum chamber and barrel. This causes <br>the nitrogen molecules to ionize, and the nitrogen ions collect on the barrel surface. <br>The dissipation of the kinetic energy of the ions heats the barrel surface to the <br>nitriding temperature. The nitrogen ions combine with the surface constituents to <br>form the nitrides that impart hardness to the surface. The surface layer consists of a <br>compound zone and a diffusion zone. The compound zone is usually about 5 to 8 <br>microns thick (0.2­0.3 milli-in); it can be tailored to be either wear-resistant or <br> <br>corrosion-resistant. The total nitriding depth is about 0.4 mm (16 milli-in).<br>Bimetallic barrels are made by centrifugally casting a bimetallic alloy onto the <br>inside of the barrel. The melting point of the bimetallic alloy is considerably lower <br>
+<hr>
+<A name=83></a><b>68</b>  <b>3Extruder Hardware</b><br>
+than the melting point of the barrel material. The barrel is charged with the alloy, <br>capped, and heated while rotating slowly. When the proper temperature is reached, <br>the barrel is rotated at a very high speed [27], forcing the molten alloy to form a <br>uniform layer with a strong bond with the barrel. The last finishing step is honing <br>to form a smooth surface. The depth of the bimetallic liner is usually about 1.5 to 2.0 <br>mm (60 to 80 milli-in) with a uniform consistency throughout the depth of the liner.<br>Comparative wear tests [27] indicate that the wear performance of bimetallic barrel <br>liners is better than nitrided barrel surfaces, with a predicted improvement in the <br>barrel life of about four to eight times the life of a nitrided barrel. An additional <br>drawback of the nitrided surface is that the hard compound zone is quite thin. Once <br>the compound zone is worn away, wear will increase more rapidly because the diffu-<br>sion zone is not as hard and wear-resistant. This becomes more severe as the diffu-<br>sion zone wears away more deeply.<br>
+<b></b><br>
+ <b>3.4Feed Hopper</b><br>
+The feed hopper feeds the granular material to the extruder. In most cases, the mate-<br>rial will flow by gravity, unaided, from the feed hopper into the extruder. Unfor-<br>tunately, this is not possible with all materials. Some bulk materials have very poor <br>flow characteristics and additional devices may be required to ensure steady flow <br>into the extruder. Sometimes this can be a vibrating pad attached to the hopper to <br>dislodge any bridges as soon as they form. In some cases, stirrers are used in the <br>feed hopper to mix the material (and prevent segregation) and/or to wipe material <br>from the hopper wall, if the bulk material tends to stick to the wall.<br>In order to achieve steady flow through the hopper there should be a gradual com-<br>pression in the converging region and the hopper should have a circular cross sec-<br>tion. Unfortunately, extruder manufacturers often make square feed hoppers with <br>rapid compression in the converging region because such hoppers are easier to <br>manufacture. Figure 3.13 shows both a good and poor hopper design.<br>Square feed hoppers with rapid compression usually work well with bulk materials <br>with uniform pellet size. However, when there is a large variation in particle size <br>and shape the square feed hopper is likely to cause conveying problems. This can <br>happen for instance when regrind is added to the virgin material. For this reason it <br>is better to use a circular hopper with gradual compression.<br>Crammer feeders are used for bulk materials that are very difficult to handle. Other <br>materials, particularly those with low bulk density, tend to entrap air. If the air cannot <br>escape through the feed hopper, it will be carried with the polymer and eventually <br>appear at the die exit. In most cases, this will cause surface imperfections of the ex -<br>trudate. In some cases, it causes small explosions when the air escapes from the die.<br>
+<hr>
+<A name=84></a><IMG src="Polymer Extrusion by Chris Rauwendaal-84_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-84_2.jpg"><br>
+ <br>
+3.4Feed Hopper <br>
+<b>69</b><br>
+Top view<br>
+Top view<br>
+Top view<br>
+Top view<br>
+Side view Isometric view<br>
+Isometric view<br>
+Side view Isometric <br>
+Isometr view<br>
+ic view<br>
+Poor design<br>
+Side view<br>
+Side view Good design<br>
+Poor design<br>
+Good design<br>
+<b>Figure 3.13</b>Poor and good feed hopper design<br>
+One method to overcome this air entrapment problem is to use a vacuum feed hop-<br>per. In principle, this is quite simple; however, in practice, a vacuum feed hopper is <br>not a trivial matter. The first problem to occur is how to load the hopper without <br> <br>losing vacuum. This has led to the development of double hopper vacuum systems, <br>in which material is loaded into a top hopper and the air is removed before the mate-<br>rial is dumped in the main hopper (see Fig. 3.14).<br>
+Stirrer drive<br>
+Stirrer drive<br>
+Inlet valv<br>
+Inlet v e<br>
+alve<br>
+Vacuum<br>
+Vacuum<br>
+Transfer<br>
+Transfer<br>
+valve<br>
+valve<br>
+Screw<br>
+Scre  drive<br>
+w drive<br>
+Vacuum<br>
+Vacuum<br>
+V Vacuum seal<br>
+acuum seal<br>
+Main extruder<br>
+Main extruder<br>
+<b> Figure  3.14</b> <br>
+Double feed hopper vacuum system<br>
+A second critical point is the rear vacuum seal around the screw shank. This seal is <br>exposed to sometimes-gritty materials. Leakage of air at this point can cause fluidi-<br>zation of material and adversely affect solids conveying in the extruder. Another <br>method to avoid air entrapment is to use a two-stage extruder screw with a vent port <br>in the barrel to extract air and any other volatiles that might be present in the poly-<br>mer.<br>
+<hr>
+<A name=85></a><b>70</b>  <b>3Extruder Hardware</b><br>
+An important bulk material property with respect to the design of the hopper is the <br>angle of internal friction (see Section 6.1). As a rule of thumb, the angle of the side-<br>wall of the hopper to horizontal should be larger than the angle of internal friction. <br>If the bulk material has a very large angle of internal friction, it will bridge in essen-<br>tially any hopper. In this case, force-feeding may be the only way to solve this.<br>
+<b></b><br>
+ <b>3.5Extruder Screw</b><br>
+The extruder screw is the heart of the machine. Everything revolves around the <br>extruder screw, literally and figuratively! The rotation of the screw causes forward <br>transport, contributes to a large extent to the heating of the polymer, and causes <br>homogenization of the material.<br>In simple terms, the screw is a cylindrical rod of varying diameter with a helical <br>flight(s) wrapped around it. The outside diameter of the screw, from flight tip to <br>flight tip, is constant on most extruders. The clearance between screw and barrel is <br>usually small. Generally, the ratio of radial clearance to screw diameter is around <br>0.001, with a range of about 0.0005 to 0.0020.<br>The details of screw design will be discussed Chapter 8. In the U.S., a very common <br>screw material is 4140 steel, which is a medium carbon, relatively low-cost material. <br>A table of common screw materials used in polymer extrusion with their chemical <br>composition is shown in Table 3.4. Table 3.6 shows some physical properties and <br>cost comparison data. The selection of the proper screw base material and hard- <br>facing material will be discussed in detail in Section 11.2.1.4.<br>Table 3.5 lists the European equivalents (or similar materials) to some of the mate-<br>rials listed in Table 3.4.<br>
+<b>Table 3.4</b>Composition of Various Materials Used in Extruded Screws<br>
+C<br>
+Si<br>
+Mn<br>
+F<br>
+S<br>
+Cr<br>
+Mo<br>
+Ni<br>
+V<br>
+Al<br>
+Cu<br>
+W<br>
+Co<br>
+Fe<br>
+<b>Low carbon steel<br></b>8620 0.21 0.30 0.80 0.035 0.035 0.50 0.20 0.55<br>
+97.37<br>
+<b>Medium carbon steels<br></b>4140 0.42 0.30 0.80 0.035 0.035 1.05 0.23<br>
+97.13<br>
+135<br>
+0.41<br>
+0.60 0.025 0.025 1.60 0.35<br>
+1.10<br>
+95.89<br>
+<b>Stainless steels<br></b>17-4<br>
+0.04 1.00 0.40<br>
+16.5<br>
+4.80<br>
+4.00<br>
+73.26<br>
+304<br>
+0.07 1.00 2.00 0.045 0.030 18.5<br>
+9.20<br>
+69.16<br>
+316<br>
+0.07 1.00 2.00 0.045 0.030 17.5 2.25 12.0<br>
+65.10<br>
+<hr>
+<A name=86></a> <br>
+3.5Extruder Screw <br>
+<b>71</b><br>
+C<br>
+Si<br>
+Mn<br>
+F<br>
+S<br>
+Cr<br>
+Mo<br>
+Ni<br>
+V<br>
+Al<br>
+Cu<br>
+W<br>
+Co<br>
+Fe<br>
+<b>Tool steels<br></b>H-13 0.40 1.00 0.35<br>
+5.35 1.35<br>
+1.00<br>
+90.55<br>
+D-2<br>
+1.50 0.25 0.30<br>
+12.0 0.80 0.60<br>
+84.55<br>
+D-7<br>
+2.35 0.40 0.40<br>
+12.5 0.95<br>
+4.00<br>
+79.40<br>
+<b>Nickel based materials (highly corrosion resistant)<br></b>276<br>
+0.02 0.05 1.00 0.030 0.030 15.5 16.0 55.5 0.35<br>
+4.00 2.50 5.00<br>
+Dni<br>
+3.00 1.00 0.50<br>
+0.010<br>
+94.9<br>
+0.60<br>
+<b>Hardfacing materials<br></b>St. 6 1.00 1.25<br>
+28.0<br>
+4.00 65.7<br>
+St. 12 1.25<br>
+29.0<br>
+8.00 61.7<br>
+<b>Table 3.5</b>European Equivalents or Similar Materials<br>
+US designation<br>
+European designation<br>
+8620<br>
+21NiCrMo2<br>
+4140 Heat treated<br>
+42CrMo4<br>
+Nitralloy 135M<br>
+41CrAlMo7<br>
+304 Stainless steel<br>
+X5CrNi189<br>
+316 Stainless steel<br>
+X5CrNi189<br>
+H-13 Tool steel<br>
+X40CrMoV51<br>
+D-2 Tool steel<br>
+X155CrVMo121<br>
+<b>Table 3.6</b>Properties of Various Screw Materials<br>
+Ultimate<br>
+Max. surface<br>
+Screw/<br>
+Used with<br>
+Used with<br>
+tensile<br>
+hardness<br>
+cost ratio<br>
+hard-facing<br>
+chrome<br>
+strength after<br>
+after [RC] HT<br>
+HT [MPa]<br>
+8620<br>
+900<br>
+60<br>
+1.5<br>
+No<br>
+Yes<br>
+4140 HT<br>
+2000<br>
+55­60<br>
+1.0<br>
+Yes<br>
+Yes<br>
+Nitralloy <br>
+1400<br>
+60­74<br>
+1.2<br>
+Yes<br>
+Not advisable<br>
+135M<br>17­4 PH<br>
+1400<br>
+65<br>
+2.0<br>
+Yes<br>
+No<br>
+304<br>
+1.5<br>
+Yes<br>
+No<br>
+316<br>
+1.5<br>
+Yes<br>
+No<br>
+H-13<br>
+1800<br>
+60­74<br>
+1.7<br>
+No<br>
+Yes<br>
+D-2<br>
+1650<br>
+1.7<br>
+No<br>
+Yes<br>
+D-7<br>
+1650<br>
+3.0<br>
+No<br>
+Yes<br>
+Hastelloy<br>
+3.0<br>
+Yes<br>
+No<br>
+Duranickel<br>
+1100<br>
+3.0<br>
+Yes<br>
+No<br>
+<hr>
+<A name=87></a><b>72</b>  <b>3Extruder Hardware</b><br>
+<b></b><br>
+ <b>3.6Die Assembly</b><br>
+In many extruders, a breaker plate is incorporated between the barrel and die <br>assembly. The breaker plate is a thick metal disk with many, closely spaced parallel <br>holes, parallel to the screw axis. There are two main reasons for using a breaker <br>plate. One reason is to arrest the spiraling motion of the polymer melt and to force <br>the polymer melt to flow in a straight-line fashion. Without a breaker plate, the spi-<br>raling motion could extend to the die exit and cause extrudate distortion. Another <br>reason is to put screens in front of the breaker plate; the breaker plate then acts as <br>a support for the screens. Screens are generally used for filtering contaminants out <br>of the polymer. Sometimes screens are used for the sole purpose of raising the die-<br>head pressure in order to improve the mixing efficiency of the extruder. This situa-<br>tion often indicates the use of an improper screw design. Another function of the <br>breaker plate is improved heat transfer between the metal and the polymer melt. <br>The reduced heat transfer distances in the breaker plate can improve the thermal <br>homogeneity of the polymer melt.<br>If the exit opening of the extruder barrel does not match up with the entry opening <br>of the die, an adaptor is used between barrel and die. Dies specifically designed for <br>a certain extruder will usually not require an adaptor. However, since there is little <br>standardization in extruder design and die design, the use of adaptors is quite com-<br>mon.<br>The die is one of the most critical parts of the extruder. It is here that the forming of <br>the polymer takes place. The rest of the extruder basically has only one task: to <br>deliver the polymer melt to the die at the required pressure and consistency. Thus, <br>the die forming function is a very important part of the entire extrusion process.<br>Analysis of the flow in extrusion dies is very difficult because of the nature of the <br>polymer melt. Die design, therefore, is largely still an empirical science. Flow be -<br>havior in the flow channels will be discussed in Section 7.5 and die design will be <br>discussed in detail in Chapter 9.<br>
+<b>3.6.1Screens and Screen Changers</b><br>
+The screens before the breaker plate are generally incorporated to filter out conta-<br>minants. The coarsest screen (lowest mesh number) is usually placed against the <br>breaker plate for support, with successively finer screens placed against it. A typical <br>screen pack is one 100-mesh screen followed by one 60-mesh and one 30-mesh <br>screen, with the 30-mesh placed against the breaker plate. Some extrusion opera-<br>tions use as many as twenty 325-mesh screens backed up by coarser screens, as <br>reported by Flathers [10].<br>
+<hr>
+<A name=88></a><IMG src="Polymer Extrusion by Chris Rauwendaal-88_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-88_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-88_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-88_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-88_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-88_6.jpg"><br>
+ <br>
+3.6Die Assembly <br>
+<b>73</b><br>
+There are three important types of metallic filter medium: wire mesh, sintered pow-<br>der, and random fiber. Wire mesh comes in a square weave or Dutch twill (woven <br>in parallel diagonal lines). The different filter media do not perform equally with <br>respect to their ability to hold contaminant, capture gels, etc. [11, 12]. A relative <br>performance comparison is shown in Table 3.7.<br>
+<b>Table 3.7</b>Performance Comparison of Different Filter Media<br>
+Wire mesh, <br>
+Wire mesh, <br>
+Sintered powder<br>
+Random metal <br>
+square weave<br>
+Dutch twill<br>
+fiber<br>
+Gel capture<br>
+Poor<br>
+Fair<br>
+Good<br>
+Very good<br>
+Contaminant  Fair<br>
+Good<br>
+Fair<br>
+Very good<br>
+capacity<br>Permeability Very good<br>
+Poor<br>
+Fair<br>
+Good<br>
+The commonly used square weave wire mesh has poor filtering performance; the <br>only redeeming quality is good permeability. It is clear, therefore, that if filtering <br>is very important, a different filter medium should be employed. Metal fibers stand <br>out in ability to capture gels and hold contaminants. Gel problems are particularly <br>severe in small gauge extrusion such as low denier fibers, thin films, etc. It is par-<br>ticularly for these applications that metal fiber filters have been applied. If the poly-<br>mer is heavily contaminated, the screen will clog rather quickly. If the screens have <br>to be replaced frequently, an automatic screen changer is often employed. In these <br>devices, the pressure drop across the screens is monitored continuously. If the pres-<br>sure drop exceeds a certain value, a hydraulic piston moves the breaker plate with <br>screen pack out of the way, and at the same time a breaker plate with fresh screens <br>is moved in position. These units are referred to as slide-plate screen changers; see <br>Fig. 3.15.<br>
+Soiled screen<br>
+Hydraulic cylinder rod<br>
+Screen located in melt channel<br>
+<b>Figure 3.15</b>Slide-plate screen changer<br>
+<hr>
+<A name=89></a><IMG src="Polymer Extrusion by Chris Rauwendaal-89_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-89_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-89_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-89_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-89_5.jpg"><br>
+<b>74</b>  <b>3Extruder Hardware</b><br>
+With some screen changers, the screen change operation can be performed without <br>having to shut down the extruder. The old screens can be removed and new screens <br>put in place and the screen changer is ready for a new cycle. In operations where <br>the polymer contains a high level of contaminants, screen changes may have to be <br>made as often as every 5 to 10 minutes. Usually, however, the cycle time is meas-<br>ured in hours and not in minutes.<br>The breaker plate allows only a limited surface area to be used for filtering. If a sub-<br>stantial amount of filtering is required, downstream filtering units can be used. <br>These devices use filter elements with large surface areas; a substantial amount of <br>contaminants can be filtered out before the filter clogs up. Various companies manu-<br>facture these filtration systems. The filter elements come in various forms: plain <br>cylinders, pleated cylinders, and leaf discs. Many of these devices can change over <br>from one filter to another without disrupting the flow, in other words, without hav-<br>ing to stop the extruder.<br>Another type of screen is the "autoscreen" system. This consists of a continuous <br>steel gauze, which moves very slowly across the melt stream in a continuous fashion <br>(see Fig. 3.16).<br>
+Breaker plate<br>
+Water<br>
+Heaters<br>
+cooling<br>
+Dirty screen<br>
+Clean screen<br>
+Screen supply<br>
+Heaters<br>
+<b>Figure 3.16</b>The autoscreen filter system<br>
+The movement occurs by the pressure drop over the screen; the higher the pressure <br>drop, the more lateral force will be exerted on the screen. In other designs, the <br>movement of the screen occurs by a motorized screen take-up. The seal is estab-<br>lished by solidified or partially solidified polymer. The autoscreen allows a small <br>amount of polymer to escape with the screen in a controlled fashion to carry the <br>contaminants out and to provide a seal. Keeping a good seal requires close tempera-<br>ture control. Reduction of temperature can cause hang-up of the screen and increased <br>temperatures can cause substantial leakage.<br>
+<hr>
+<A name=90></a> <br>
+3.7Heating and Cooling Systems <br>
+<b>75</b><br>
+Several attempts have been made to model the flow through porous media and to <br>predict the pressure drop as a function of flow rate and polymer flow properties. <br>A number of interesting articles [13­26] are listed in the references.<br>
+<b></b><br>
+ <b>3.7Heating and Cooling Systems</b><br>
+Heating of the extruder is required for bringing the machine up to the proper tem-<br>perature for start-up and for maintaining the desired temperature under normal <br>operations. There are three methods of heating extruders: electric heating, fluid <br>heating, and steam heating. Electric heating is the most common type of heating in <br>extruders.<br>
+<b>3.7.1Electric Heating</b><br>
+Electric heating has significant advantages over fluid and steam heating. It can cover <br>a much larger temperature range, and it is clean, easy to maintain, low cost, effi-<br>cient, etc. For these reasons, electric heating has displaced fluid and steam heating <br>in most applications. The electrical heaters are normally placed along the extruder <br>barrel grouped in zones. Small extruders usually have two to four zones, while <br>larger extruders have five to ten zones. In most cases, each zone is controlled in -<br>dependently so that a temperature profile can be maintained along the extruder. <br>This can be a flat profile, increasing profile, decreasing profile, and combinations <br>thereof, depending on the particular polymer and operation.<br>
+<b>3.7.1.1Resistance Heating<br></b>The most common barrel heaters are electric resistance heaters. This is based on <br>the principle that if a current is passed through a conductor, a certain amount of <br>heat is generated, depending on the resistance of the conductor and the current <br>passed through it. The amount of heat generated is:<br>
+ (3.5)<br>
+where I is the current, R the resistance, and V voltage. This equation is valid for <br>direct (DC) as well as single-phase alternating current (AC), provided the current <br>and voltage are expressed as root-mean-square (rms) values and the circuit is purely <br>resistive (phase difference zero). With three-phase circuits, the heat generation is:<br>
+ (3.6)<br>
+<hr>
+<A name=91></a><b>76</b>  <b>3Extruder Hardware</b><br>
+Early band heaters used a resistance wire insulated with mica strips and encased <br>in flexible sheet steel covers. These heaters are compact and low cost, but they are <br>also fragile, not very reliable, and have limited power density. The maximum load-<br>ing of these heaters is about 50 kW/m2 (30 W/in2) and maximum temperature about <br>500°C. Newer types of mica heaters reportedly can handle power densities up to <br>165 kW/m2 (100 W/in2). The efficiency of the heater and its life are largely deter-<br>mined by how good the contact is between the heater and the barrel over the entire <br>contact area. Improper contact will cause local overheating, and this will result in <br>reduced heater life or even premature burnout of the heater element. Special pastes <br>are commercially available to improve the heat transfer between heater and barrel.<br>Ceramic band heaters generally last much longer than the mica-insulated heaters <br>and they can withstand higher power densities, up to 160 kW/m2 (100 W/in2) or <br>higher, and block temperatures up to 750°C. The disadvantages of heaters with <br>ceramic insulation are that they are not flexible and tend to be bulky. However, some <br>ceramic band heaters have a thin-line design with minimal space requirements. <br>They usually come in halves that have to be bolted together around the extruder bar-<br>rel.<br>Another type of heater is the "cast-in" heater. In this heater, the heating elements <br>are cast in semicircular or flat aluminum blocks. The heat transfer in this heater is <br>very good. This heater is reliable and gives good service life. Cast aluminum heaters <br>have a maximum watt density of about 55 kW/m2 (35 W/in2) with a maximum oper-<br>ating temperature of about 400°C. Bronze castings can increase the power den-<br>sity to about 80 kW/m2 (50 W/in2) and a maximum operating temperature of about <br>550°C.<br>
+<b>3.7.1.2Induction Heating<br></b>In induction heating, an alternating electric current is passed through a primary <br>coil that surrounds the extruder barrel. The alternating current causes an alter-<br>nating magnetic field of the same frequency. This magnetic field induces an electro-<br>motive force in the barrel, causing eddy currents. The I2R losses of the circulating <br>current are responsible for the heating effect.<br>The depth of heating reduces with frequency. At normal frequencies of 50 or 60 Hz, <br>the depth is approximately 25 mm. This is similar to the thickness of a typical <br>extruder barrel. The advantage of this system, therefore, is the much reduced tem-<br>perature gradients in the extruder barrel because the heat is generated quite evenly <br>throughout the depth of the barrel as opposed to resistance-type barrel heaters.<br>Another advantage of inductive heating is reduced time lag in power input changes. <br>Local overheating because of poor contact does not occur. Power consumption is low <br>because of efficient heating and reduced heat losses, in spite of the fact that the <br>power factor is lower than that of resistance heaters. It is possible to have a cooling <br>
+<hr>
+<A name=92></a> <br>
+3.7Heating and Cooling Systems <br>
+<b>77</b><br>
+system directly on the barrel surface, allowing accurate temperature control with <br>fast response. A major drawback of induction heating is its high cost.<br>
+<b>3.7.2Fluid Heating</b><br>
+Fluid heating allows even temperatures over the entire heat transfer area, avoiding <br>local overheating. If the same heat transfer fluid is used for cooling, an even reduc-<br>tion in temperatures can be achieved. The maximum operating temperature of most <br>fluids is relatively low, generally below 250°C. A few fluids can operate at high <br>temperature; however, they often produce toxic vapors--this constitutes a consider-<br>able safety hazard. Fluid heating systems require considerable space, and installa-<br>tion and operating expenses are high. Another drawback with fluid heating is that if <br>several zones need to be maintained at different temperatures, several independent <br>fluid heating systems are required. This becomes rather expensive, bulky, and in -<br>effective.<br>Steam heating is rarely used on extruders anymore, although most of the very early <br>extruders were heated this way, particularly rubber extruders. Steam is a good heat <br>transfer fluid because of its high specific heat. However, it is difficult to increase the <br>temperature to sufficiently high temperatures (200°C and above) as required in <br> <br>polymer extrusion. This requires very high steam pressures; most extrusion plants <br>nowadays are not equipped with proper steam generating facilities to do this. Addi-<br>tional problems are bulkiness, chance of leakage, corrosion, heat losses, etc.<br>
+<b>3.7.3Extruder Cooling</b><br>
+Extruder cooling in most extrusion operations is a necessary evil. In all cases, cool-<br>ing should be minimized as much as possible; preferably, it should be eliminated <br>altogether. The reason is that any amount of extruder cooling reduces the energy <br>efficiency of the process, because cooling translates directly into lost energy. Heat-<br>ing of the extruder generally reduces the motor power consumption and thus <br> <br>contributes to the overall power requirement of the process. However, cooling does <br>not contribute to the overall power requirement, and the energy extracted by cooling <br>is wasted.<br>If an extrusion process requires a substantial amount of cooling, this is usually a <br>strong indication of improper process design. This could mean improper screw <br>design, excessive length-to-diameter ratio, or incorrect choice of extruder, e.g., <br> <br>single screw versus twin screw extruder.<br>The extrusion process is generally designed such that the majority of the total <br>energy requirement is supplied by the extruder drive. The rotation of the screw <br>
+<hr>
+<A name=93></a><IMG src="Polymer Extrusion by Chris Rauwendaal-93_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-93_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-93_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-93_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-93_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-93_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-93_7.jpg"><br>
+<b>78</b>  <b>3Extruder Hardware</b><br>
+causes frictional and viscous heating of the polymer, which constitutes a trans-<br>formation of mechanical energy from the drive into thermal energy to raise the <br>temperature of the polymer. The mechanical energy generally contributes 70 to 80% <br>of the total energy. This means that the barrel heaters contribute only 20 to 30%, <br>discounting any losses.<br>If the majority of the energy is supplied by the screw, there is a reasonable chance <br>that local internal heat generation in the polymer is higher than required to main-<br>tain the desired process temperature. Thus, some form of cooling is usually required. <br>Many extruders use forced-air cooling by blowers mounted underneath the extruder <br>barrel; see Fig. 3.17.<br>
+Heater<br>
+Heater<br>
+Ribbed spacer<br>
+Ribbed spacer<br>
+Adjustable restriction<br>
+Adjustable restriction<br>
+<b> Figure 3.17</b> <br>
+Blower<br>
+Blower<br>
+Extruder cooling by forced air<br>
+The external surface of the heaters or the spacers between the heaters is often made <br>with cooling ribs to increase the heat transfer area and thus the cooling efficiency. <br>Small extruders can often do without forced-air cooling because their barrel surface <br>area is quite large compared to the channel volume, providing a relatively large <br>amount of convective and radiative heat losses.<br>Some extruders operate without any forced cooling or heating. This is the so-called <br>"autogenous" extrusion operation, not to be confused with adiabatic operation. An <br>autogenous process is a process where the heat required is supplied entirely by the <br>conversation of mechanical energy into thermal energy. However, heat losses can <br>occur in an autogenous process. An adiabatic process is one where there is abso-<br>lutely no exchange of heat with the surroundings. Clearly, an autogenous extrusion <br>operation can never be truly adiabatic, rather only by approximation.<br>In practice, autogenous extrusion does not occur often because it requires a delicate <br>balance between polymer properties, machine design, and operating conditions. A <br>change in any of these factors will generally cause a departure from autogenous <br>conditions. The closer one operates to autogenous conditions, the more likely it is <br>
+<hr>
+<A name=94></a> <br>
+3.7Heating and Cooling Systems <br>
+<b>79</b><br>
+that cooling will be required. Given the large differences in thermal and rheological <br>properties of various polymers, it is difficult to design an extruder that can operate <br>in an autogenous fashion with several different polymers. Therefore, most extruders <br>are designed to have a reasonable amount of energy input from external barrel heat-<br>ers.<br>On the other hand, the energy input from the barrel heaters should not be too large. <br>The problem with external heating is that this is associated with relatively large <br>temperature gradients. In materials with low thermal conductivity, large tempera-<br>ture gradients are required to heat up the material by external heating at a reason-<br>able rate. Since polymers have a low thermal conductivity, raising the polymer <br> <br>temperature by external heating is a slow process and involves large temperature <br>gradients. Thus, locally high temperatures will occur at the metal/polymer inter-<br>face. The combination of high temperatures and long heating times makes for a high <br>chance of degradation. The heating by viscous heat generation is much more favor-<br>able in this respect, because the polymer is heated relatively uniformly throughout <br>its mass. Thus, one would generally want the mechanical energy input to be more <br>than 50% of the total energy requirement but less than about 90%.<br>Air cooling is a fairly gentle type of cooling, because the heat transfer rates are rela-<br>tively small. This is not good if intensive cooling is required. On the other hand, <br>there is an advantage in that when the air cooling is turned on, the change in tem-<br>perature occurs gradually. With water cooling, a rapid and steep change in tempera-<br>ture will occur as soon as the water cooling is activated. From a control point of view, <br>the latter situation can be more difficult to handle.<br>When substantial cooling is required, fluid cooling is used, with water being the <br>most common heat transfer medium. It was mentioned in Section 3.3 that grooved <br>barrel sections require intense cooling to be effective. In most cases, water is used to <br>cool the grooved barrel section, just as it is used to cool the feed throat casting. One <br>of the complications with water cooling is that evaporation can occur if the water <br>temperature exceeds the boiling temperature. This is an effective way to extract <br>heat, but it causes a sudden increase in cooling rate. From a control point of view, <br>this constitutes a non-linear effect, and it is more difficult to properly control the <br>extruder temperature if such sudden, non-linear effects occur. Thus, water cooling <br>may place much higher demands on the temperature control system as compared <br>to air cooling. The cooling efficiency of air can be increased by wetting the air; how-<br>ever, this requires cooling channels made out of corrosion-resistant material. This <br>technique is used in a patented vapor cooling system [8, 9]. The latent heat of a <br>vapor, which circulates around the extruder barrel, is extracted by a water cooling <br>system that surrounds a condensing chamber located away from the barrel. A sche-<br>matic of this vapor cooling system is shown in Fig. 3.18.<br>
+<hr>
+<A name=95></a><IMG src="Polymer Extrusion by Chris Rauwendaal-95_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-95_11.jpg"><br>
+<b>80</b>  <b>3Extruder Hardware</b><br>
+Cooling<br>
+Cooling<br>
+water in<br>
+water in<br>
+Cooling<br>
+Cooling<br>
+water out<br>
+water out<br>
+Heater<br>
+Jacket<br>
+Jacket<br>
+Barrel<br>
+Barrel<br>
+<b> Figure 3.18</b> <br>
+Condensation return<br>
+Condensation return<br>
+Extruder vapor cooling system<br>
+This type of vapor cooling is claimed to have a smooth operating characteristic and <br>good temperature control.<br>Oil- or air-cooled extruders can use stepless cooling control, using proportional <br>valves and positioning motors. These systems are relatively expensive, but they are <br>reliable and require little maintenance. With water cooling, the cooling power is <br>generally controlled by energizing a solenoid valve. For low temperatures (no flash-<br>ing), usually a constant cycle rate is used with variable pulse width. The pulse width <br>varies in proportion to the cooling power required. At high temperatures, where <br>water is flashed to steam, more intensive cooling is possible. In these cases, a differ-<br>ent cooling control can be used, known as a constant pulse width system. When <br>cooling is required, the solenoid is energized by a pulse signal of predetermined <br>length. The frequency of the pulse is varied in proportion to the cooling power <br>required.<br>Finally, it should be remembered that cooling is a waste of energy and should be <br>minimized as much as possible.<br>
+<b>3.7.4Screw Heating and Cooling</b><br>
+Thus far, the discussion has been focused on barrel heating and cooling. It is impor-<br>tant to realize, though, that the barrel/polymer interface constitutes only about 50% <br>of the total polymer/metal interface. Thus, with only barrel heating and/or cooling, <br>only about 50% of the total surface area available for heat transfer is being utilized. <br>The screw surface, therefore, constitutes a very important heat transfer surface. <br>Many extruders do not use screw cooling or heating; they run with a so-called "neu-<br>tral screw." If the external heating or cooling requirements are minor, then screw <br>
+<hr>
+<A name=96></a> <br>
+3.7Heating and Cooling Systems <br>
+<b>81</b><br>
+heating or cooling is generally not necessary. However, if the external heating or <br>cooling requirements are substantial, then screw heating or cooling can become <br>very important, sometimes a necessity.<br>It is obvious that heating or cooling of the screw is slightly more difficult than barrel <br>heating or cooling, because the screw is in motion. This means that one has to use <br>rotary unions, slip rings, or other devices to transfer energy in or out of the extruder <br>screw. These devices, however, have become rather standard in industry, and they <br>are commonly available. Water cooling can be done even without the use of a rotary <br>union. This involves running some copper tubing down into the bore of the extruder <br>screw and connecting a water supply to the copper tube. The water will flow towards <br>the end of the screw through the tube and will then flow back in the annular space <br>between the tube and the bore of the screw. As the water reaches the shank end of <br>the screw, it will simply drain away. This is a crude but effective type of screw cool-<br>ing (see Fig. 3.19).<br>
+Water in<br>
+Water out<br>
+<b>Figure 3.19</b>Simple screw cooling system<br>
+Screw cooling is generally arranged such that the fluid, water or oil, enters the screw <br>via a rotary union, flowing into a pipe in the bore of the screw. The screw is cooled <br>most at the discharge end of the pipe; further cooling occurs upstream as the fluid <br>flows back to the rotary union. The point of maximum cooling can be adjusted by <br>changing the location of the end of the pipe.<br>Sometimes the end of the pipe in the screw bore is made with an axially adjustable <br>seal. This allows cooling of a selected section of the screw. Other adjustments that <br>can be made are the flow rate and inlet temperature of the cooling medium. Thus, <br>a considerable degree of flexibility in cooling conditions is possible. This is why <br>screw cooling or heating adds significantly to the controllability of the process. In <br>fact, some extrusion operations are simply impossible without the use of screw cool-<br>ing. The two fluids most often used for cooling are water and oil.<br>In some cases, screw cooling is used to improve the pressure-generating capability <br>of the screw. The screw is cooled all the way into the metering section. The colder <br>screw surface freezes the polymer close to the screw or, at least, significantly in -<br>creases the viscosity of the polymer melt close to the screw surface. This reduces <br>the effective channel depth of the extruder screw and can result in improved pres-<br>sure-generating capacity if the screw was cut too deep to begin with. Therefore, if <br>
+<hr>
+<A name=97></a><b>82</b>  <b>3Extruder Hardware</b><br>
+screw cooling is required to obtain sufficient pressure at the die, this is a strong <br>indication that the design of the extruder screw is incorrect. Instead of cooling the <br>screw, a better solution would be to modify the screw design, specifically to reduce <br>the channel depth in the metering section. Determination of the optimum channel <br>depth for pressure generation will be discussed in Chapter 8.<br>An interesting concept of screw cooling is the application of a heat pipe in the <br>extruder screw. The heat pipe extends from the feed section to the metering section. <br>In most cases, the temperature of the metering section will be considerably higher <br>than the temperature of the feed section. This temperature difference will set up a <br>heat flow in the heat pipe trying to diminish this temperature difference. Thus, the <br>metering section will be cooled and the feed section will be heated. The advantage of <br>this system is that it is self-contained, sealed, and heat losses from cooling are very <br>small. A disadvantage is the fact that no external means of control is possible.<br>Screw heating is sometimes done with cartridge heaters located in the bore of the <br>extruder screw. Power is supplied to the heater by slip rings on the shank of the <br>screw. If the heater is axially adjustable, then the location of heating can be changed <br>as desired. With this type of heating, one has to be careful to establish good contact <br>between heater and screw. This almost requires installation of the heater in a pre-<br>heated screw. If the heater is installed in a cold screw, good contact between heater <br>and screw can be lost when the screw heats up. The requirement for tight contact <br>between screw and heater would make it difficult to have an axially adjustable <br>heater. In this case, one would have to use a heat transfer paste between screw and <br>heater that allows good heat transfer but still a reasonable axial movement of the <br>heater.<br>
+References<br>1.  K. Rape, Power Transmission Design, 8, 36­38 (1982)<br>2.  N.N., Modern Materials Handling, Oct. 6, 66­69 (1982)<br>3.  L.J. McCullough, Tech. Papers IEEE Meeting, 755­757 (1978)<br>4.  N.N., Generation Planbook, 144­147 (1982)<br>5.  S. Collings, Plastics Machinery &amp; Equipment, Sept. 26­29 (1982)<br>6.  Machine Design, Fluid Power Reference Issue (1982)<br>7.  R.L. Miller, SPE Journal, Nov., 1183­1188 (1964)<br>8.  U.S. Patent 2,796,632<br>9.  W.H. Willert, SPE Journal, 13, 6, 122­123 (1957)<br>10.  N.T. Flathers, et al., Int. Plast. Eng., 1, 256 (1961)<br>11.  H.M. Kennard, Plast. Eng., 30, 12, 59 (1974)<br>12.  J.S. Singleton, paper presented at Filtration Society Conference, London, Sept. (1973)<br>
+<hr>
+<A name=98></a> References <br>
+<b>83</b><br>
+13.  W.C. Smith, Ph.D. thesis, University of Colorado (1974)<br>14.  T.J. Sadowski and R.B. Bird, Trans. Soc. Rheol., 9, 2, 243 (1965)<br>15.  R.J. Marshall and A.B. Metzner, Ind. Eng. Chem. Fund., 6, 393 (1967)<br>16.  R.H. Christopher and S. Middleman, Ind. Eng. Chem. Fund., 4, 422 (1965).<br>17.  D.F. James and D.R. McLaren, J. Fluid Mech., 70, 733 (1975)<br>18.  R.E. Sheffield and A.B. Metzner, AIChE J., 22, 736 (1976)<br>19.  G. Laufer, C. Gutfinger, and N. Abuaf, Ind. Eng. Chem. Fund., 15, 77 (1976)<br>20.  E.H. Wissler, Ind. Eng. Chem. Fund., 10, 411 (1971)<br>21.  Z. Kamblowski and M.D. Ziubinski, Rheol. Acta, 17, 176 (1978)<br>22.  "Filtration of Polymer Melts," VDI Publication, Duesseldorf, Germany (1981)<br>23.  J.A. Deiber and W.R. Schowalter, AIChE J., 27, 6, 912 (1981)<br>24.  M.L. Booy, Polym. Eng. Sci., 22, 14, 895 (1982)<br>25.  S.H. Collins, Plast. Compounding, March/April, 57­70 (1982)<br>26.  D.S. Done and D.G. Baird, Techn. Papers 40th ANTEC, 454­457 (1982)<br>27.  K. O'Brien, Plast. Technology, Feb., 73­74 (1982)<br>28.  N.N., Int. Plast. Eng., 2, 92­95 (1962)<br>29.  A. Bres, VDI-Nachrichten, 19, 42, 14 (1965)<br>30.  R.G. Schieman, Reliance Electric Publication, D-7115, 1, 7 (1983)<br>31.  C.J. Ceroke, Chemical Engineering, Nov. 12, 133­134 (1984)<br>32.  W.A. Kramer, "Motors and Drives for Extrusion Applications," SPE ANTEC, 268­272 <br>
+(1999)<br>
+33.  C. Hiemenz, G. Ziegmann, B. Franzkoch, W. Hoffmanns, and W. Michaeli, "Verbesserung <br>
+am Einschneckenextruder," in "Handbuch des 8. Kunststofftechnischen Kolloquiums," <br>Aachen, p. 81­109 (1976)<br>
+<hr>
+<A name=99></a><b>4 Instrumentation </b><br>
+<b>and Control</b><br>
+<b></b><br>
+ <b>4.1Instrumentation Requirements</b><br>
+From a hardware point of view, extruder instrumentation is one of the most critical <br>components of the entire machine. An important reason for this is that the internal <br>workings of the extruder are totally obscured by the barrel and the die. In many <br>cases, the only visual observation that can be made is of the extrudate leaving the <br>die. When a problem is noticed in the extrudate, it is difficult to determine the <br>source and location of the problem. Instrumentation makes it possible to determine <br>what is happening inside the extruder. One can think of instrumentation as "the <br>window to the process."<br>Good instrumentation enables a continuous monitoring of the "vital signs" of the <br>extruder. These vital signs are pressure, temperature, power, and speed. These <br>important process parameters need to be measured for process control, but they are <br>also of vital importance in troubleshooting. Troubleshooting is only possible with <br>good instrumentation. At the very minimum, one needs to know pressures, screw <br>speed, and temperatures in order to properly diagnose an extrusion problem. A mini-<br>mum set of instrumentation should include:<br>1.  Diehead pressure before and after screen pack<br>2.  Rotational speed of the screw<br>3.  Temperature of polymer melt at the die<br>4.  Temperatures along barrel and die<br>5.  Cooling rate at each heat zone<br>6.  Power consumption of each heat zone<br>7.  Power consumption of the drive<br>8.  Temperature of cooling water in feed housing<br>9.  Flow rate of cooling water in feed housing<br>This is a minimum requirement. In many cases, additional measurements are re -<br>quired, e.g., in a vented extrusion operation the vacuum at the vent port should be <br>monitored continuously. In some cases, one may want to measure polymer melt tem-<br>perature at various locations in or outside the die to determine the melt temperature <br>distribution--remember there is not just one melt temperature!<br>
+<hr>
+<A name=100></a><b>86</b>  <b>4Instrumentation and Control</b><br>
+The parameters above relate just to the extruder. However, there are many more <br>process parameters for the entire extrusion line and they, of course, depend on the <br>line's specific components. Important parameters for any extrusion line are:<br>
+ <br>
+ Line speed<br>
+ <br>
+ Dimensions of the extruded product<br>
+ <br>
+ Cooling rate or cooling water temperature<br>
+ <br>
+ Line tension<br>Many other factors can influence the extrusion process, such as ambient tempera-<br>ture, relative humidity, air currents around the extruder, and plant voltage varia-<br>tions among others.<br>Incomplete instrumentation can severely hamper quick and accurate troubleshoot-<br>ing; in fact, it can turn troubleshooting from a logical step-by-step process into a <br>guessing game. Without good instrumentation it can be days, weeks, or even months <br>before a problem is located and solved. When an extrusion problem results in off-<br>quality product or downtime, it is very important to find the cause of the problem <br>quickly because such problems can be very costly. In some instances, a downtime of <br>just one day is more expensive than an entire new extruder! In most cases, trying to <br>save money on instrumentation is penny-wise and pound-foolish.<br>Good instrumentation allows problems to be detected early before becoming more <br>severe and causing substantial damage to the extruder hardware or to the extrudate <br>quality. It also allows process characterization for process development and opti-<br>mization. It is further important for production control and record keeping, and it <br>provides a means of interfacing the extruder to a computer.<br>
+<b>4.1.1Most Important Parameters</b><br>
+The most important process parameters are melt pressure and temperature. They <br>are the best indicators of how well or how poorly an extruder functions. Process <br>problems, in most cases, first become obvious from melt pressure and/or tempera-<br>ture readings. Just think what a doctor does when a patient comes into the office <br>with a problem. Usually, the first check of the patient's condition is made by taking <br>blood pressure and body temperature. These are two good indicators of the function-<br>ing of the human body. In the same fashion, melt pressure and temperature are good <br>indicators of how the extruder is functioning.<br>
+<hr>
+<A name=101></a> <br>
+4.2Pressure Measurement <br>
+<b>87</b><br>
+<b></b><br>
+ <b>4.2Pressure Measurement</b><br>
+<b>4.2.1The Importance of Melt Pressure</b><br>
+Measurement of melt pressure is important for two reasons:<br>1.  Process monitoring and control<br>2. Safety<br>The diehead pressure in the extruder determines the output from the extruder. It is <br>the pressure necessary to overcome the resistance of the die. When the diehead <br>pressure changes with time, the extruder output correspondingly changes and so do <br>the dimensions of the extruded product; see Fig. 4.1. As a result, when we monitor <br>how the pressure varies with time, we can see exactly how stable or unstable the <br>extrusion process is.<br>It is best, therefore, to plot pressure with a chart recorder or better, to monitor the <br>variation of pressure with a computer data acquisition system. A simple analog or <br>digital display of pressure is much less useful.<br>
+<b>Pressure</b><br>
+<b>Time</b><br>
+<b>Throughput</b><br>
+<b>Time</b><br>
+<b>Dimension</b><br>
+<b> Figure 4.1</b> <br>
+Pressure, throughput, and dimension as a <br>
+<b>Time</b><br>
+function of time<br>
+It is also critically important to measure pressure in the extruder to prevent serious <br>accidents that can happen when excessively high pressures are generated. <i>The very </i><br>
+<i>high pressures generated in the extruder can cause an explosion.</i> The barrel can crack <br>
+<hr>
+<A name=102></a><b>88</b>  <b>4Instrumentation and Control</b><br>
+open under excessive pressure or the die may be blown from the extruder. Both situ-<br>ations are extremely dangerous and should be avoided. All extruders should have an <br>over-pressure safety device, such as a rupture disk or a shear pin in the clamp hold-<br>ing the die against the extruder barrel. Even with such an over-pressure safety <br>device, the extruder should have at least one melt pressure measurement, in order <br>to overcome malfunctioning or disabled over-pressure devices. Pressure can build <br>up very quickly without warning and cause a catastrophic explosion.<br>When monitoring pressure, it is a good idea to use an automatic shutoff when the <br>pressure reaches a critical value. In pressure measurements, it is important to <br>determine the absolute level of pressure, but it is equally important, if not more <br>important, to determine changes in pressure with time. In most cases, pressure <br> <br>variation correlates closely with variation in the extruded product. Since high fre-<br>quency (cycle time less than one second) pressure fluctuations are quite common in <br>extrusion, a fast response measurement is important.<br>
+<b>4.2.2Different Types of Pressure Transducers</b><br>
+Pressure measurement on early extruders was done with grease-filled Bourdon <br>gauges. The reliability of these gauges was not very good. At high temperatures, the <br>grease tends to leak out; this causes inaccurate readings and product contamina-<br>tion. Polymer can hang up in the grease cavity; with time, it can form a hard plug, <br>again causing inaccurate readings. The temperature dependence of the Bourdon <br>gauge pressure measurement is also quite high. As a result, Bourdon gauges are <br>hardly used anymore.<br>Nowadays there are a number of different pressure transducers. The most common <br>ones in extrusion are the strain gauge transducer and the piezo-resistive transducer. <br>In a strain gauge pressure transducer a strain gauge is bonded to a diaphragm. The <br>melt pressure deforms the diaphragm and the strain gauge measures the deforma-<br>tion of the diaphragm; see Fig. 4.2. These units generally have good response and <br>resolution. Since the strain gauge cannot be exposed to high temperatures, it is <br>placed away from the heated polymer/barrel environment. Therefore, a mechanical <br>or hydraulic coupling is used to transmit the deflection of the diaphragm to the <br>strain gauge.<br>The strain gauge transducer can be either a capillary or a pushrod transducer. In <br>these transducers, there are two diaphragms, one in contact with the plastic melt <br>and the other some distance away from the hot plastic melt. The connection between <br>the first and second diaphragm is hydraulic in the capillary type and a pushrod in <br>the pushrod type; see Fig. 4.3. A strain gauge is attached to the second diaphragm to <br>measure the deflection that can be related to the pressure at the first diaphragm.<br>
+<hr>
+<A name=103></a><IMG src="Polymer Extrusion by Chris Rauwendaal-103_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-103_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-103_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-103_4.jpg"><br>
+ <br>
+4.2Pressure Measurement <br>
+<b>89</b><br>
+<b>Strain gage</b><br>
+<b>Thermally isolated</b><br>
+<b>diaphragm</b><br>
+<b>Diaphragm in</b><br>
+<b>contact with melt</b><br>
+<b>Connection between</b><br>
+<b>the two diaphragms</b><br>
+<b>Pressure</b><br>
+<b>Velocities</b><br>
+<b> Figure 4.2</b> <br>
+Principle of the strain gauge transducer<br>
+Capillary<br>
+Pushrod<br>
+<b> Figure 4.3</b> <br>
+Liquid fill<br>
+Capillary (left) and pushrod <br>
+Diaphragm<br>
+Diaphragm type (right) pressure transducer<br>
+Many capillary transducers are filled with mercury. Since the diaphragm of the <br>transducer is quite thin, there is a danger of the diaphragm rupturing and leaking <br>mercury into the plastic and into the workplace. Unfortunately, many transducers <br>do not carry a label indicating that the transducer is filled with mercury, so ade-<br>quate safety precautions are not always taken. It is important to make sure that <br>mercury-filled transducers are not used in the extrusion of medical products and <br>food packaging products. In these situations, other liquids can be used inside the <br>transducer; the most common alternative to mercury is NaK (sodium-potassium).<br>Another type of transducer is the pneumatic pressure transducer. It has good robust-<br>ness, but poor temperature sensitivity, poor dynamic response, and average measure-<br>ment error. The capillary transducer has fair robustness, fair temperature sensitivity, <br>and fair dynamic response. The total measurement error varies from 0.5 to 3% de -<br>pending on the quality of the transducer. The pushrod is similar to the capillary trans-<br>ducer, except that it tends to have poor temperature sensitivity and poor total error. <br>The piezo-resistive transducer has good robustness because of its relatively thick dia-<br>phragm, good temperature sensitivity, good dynamic response, and low measurement <br>error. A comparison of different pressure transducers is shown in Table 4.1.<br>
+<hr>
+<A name=104></a><b>90</b>  <b>4Instrumentation and Control</b><br>
+<b>Table 4.1</b>Comparison of Various Pressure Transducers<br>
+Transducer type<br>
+Robustness<br>
+Temperature<br>
+Dynamic<br>
+Total error<br>
+sensitivity<br>
+response<br>
+Pneumatic<br>
+Good<br>
+Poor<br>
+Poor<br>
+About 1.5%<br>
+Capillary strain gauge*<br>
+Fair<br>
+Fair<br>
+Fair<br>
+0.5 to 3%<br>
+Pushrod strain gauge<br>
+Fair<br>
+Poor<br>
+Fair<br>
+About 3%<br>
+Piezoelectric<br>
+Good<br>
+Poor<br>
+Good<br>
+0.5­1.5%<br>
+Piezo-resistive<br>
+Good<br>
+Good<br>
+Good<br>
+0.2 to 0.5%<br>
+Optical<br>
+Good<br>
+Good<br>
+Good<br>
+About 0.5%<br>
+* Concern with mercury<br>
+Another transducer uses compressed air. The regulated air pressure is controlled <br>such that there is a force balance between the measuring diaphragm and the balanc-<br>ing diaphragm. This is a purely pneumatic system. Pneumatic pressure transducers <br>have good robustness but poor temperature sensitivity and dynamic response. There <br>are also pressure transducers that use a measuring element with an unbonded wire <br>strain gauge. The wires are part of a full bridge circuit. The measuring element is <br>located directly behind the diaphragm, and the deflection of the diaphragm is trans-<br>ferred to the measuring element with a short rod. Temperature changes at the dia-<br>phragm affect all four arms of the bridge circuit and, therefore, have little effect on <br>the measurement. Some pressure transducers are available with an internal ther-<br>mocouple to provide temperature measurement capacity in addition to the pressure <br>measurement capability.<br>Other pressure transducers use a piezoelectric element. A piezoelectric material <br>has the ability to transform a very small mechanical deformation (input signal) into <br>an electric output signal (voltage or current) without any external electric power <br>supply. Quartz pressure transducers have been developed to measure large pres-<br>sures in high temperature polymer melts. The absence of a membrane allows a very <br>robust construction. This has led to widespread use in injection molding.<br>One of the main advantages of piezo pressure transducers is their outstanding <br>dynamic response. The natural frequency of the piezo transducers is about 40 kHz. <br>This allows accurate pressure measurements at a frequency up in the low kHz <br>range. This is about three orders of magnitude better than the capillary type strain <br>gauge pressure transducer. One of the main drawbacks of piezoelectric pressure <br>transducers is that they cannot measure steady pressure accurately because the <br>signal decays. Therefore, piezoelectric pressure transducers are limited to applica-<br>tions where the pressure changes over relatively short time frames (on the order of <br>a few seconds or less). Another drawback of the piezoelectric transducer is that it <br>cannot be exposed to high temperatures. The maximum use temperature is typically <br>120°C.<br>
+<hr>
+<A name=105></a><IMG src="Polymer Extrusion by Chris Rauwendaal-105_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-105_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-105_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-105_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-105_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-105_6.jpg"><br>
+ <br>
+4.2Pressure Measurement <br>
+<b>91</b><br>
+Some transducers use piezo-resistive semiconductors implanted into a small chip. <br>The resistance of piezo-resistive materials changes with stress or strain. When the <br>chip is bonded to a pressure-sensing diaphragm, the change in resistance can be <br>measured with a Wheatstone bridge. The piezo-resistive element can provide a <br>repeatable signal that is proportional to the pressure against the diaphragm. Figure <br>4.4 shows the sensing element of a piezo-resistive transducer.<br>
+Silicon resistance<br>
+Silicon dioxide<br>
+Metallizing<br>
+insulation layer<br>
+Silicon substrate <b> Figure 4.4</b> <br>
+Sensing element of a piezo-resistive <br>
+Diaphragm<br>
+transducer<br>
+Piezo-resistive pressure transducers offer a number of advantages [82]. The trans-<br>ducer is quite robust because of the relatively thick diaphragm. There is no liquid <br>fill inside the transducer; as a result, there is no concern about leakage of mercury. <br>The natural frequency of piezo-resistive transducers is about three orders of magni-<br>tude better than strain gauge type transducers. As a result, the dynamic response <br>of piezo-resistive transducers is very good compared to that of strain gauge trans-<br>ducers.<br>Optical pressure transducers can offer some important advantages. The German <br>company FOS Messtechnik GmbH has developed several pressure transducers that <br>use an optical method of measuring diaphragm deflection. This method allows very <br>precise measurement of diaphragm movement; the maximum deformation of the <br>diaphragm at normal pressure is only about 10 micron. This allows for a very rug-<br>ged construction of the transducer; the diaphragm is about ten times thicker than <br>that of conventional transducers. Also, the response time is quite short; the trans-<br>ducer can measure dynamic pressure up to 50 kHz. As a result, it can be applied in <br>very fast control circuits.<br>The optical pressure sensor measures the deformation of the diaphragm with a <br>quartz glass optical fiber. With an optical fiber a mirror on the backside of the dia-<br>phragm is illuminated and the reflected light intensity is measured with high reso-<br>lution. The sensor can be used to temperatures as high as 600°C, in some cases <br>even higher. Even though this type of transducer is not commonly used in the extru-<br>sion industry it offers some very important advantages over conventional pressure <br>transducers.<br>
+<hr>
+<A name=106></a><IMG src="Polymer Extrusion by Chris Rauwendaal-106_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-106_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-106_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-106_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-106_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-106_6.jpg"><br>
+<b>92</b>  <b>4Instrumentation and Control</b><br>
+<b>4.2.3Mechanical Considerations</b><br>
+Most diaphragms are made of stainless steel, 17-4 PH or Type 304. Type 17-4 stain-<br>less steel has a higher tensile strength and, under the same conditions, will be <br>stressed at a lower percentage of yield, thus improving fatigue life. Another impor-<br>tant design aspect is the connection between the diaphragm and the main body. <br>This is generally done by welding using two welding configurations; see Fig. 4.5.<br>
+<b> Figure 4.5</b> <br>
+Weld configuration in capillary <br>
+Stress plane<br>
+Liquid fill<br>
+Integral weld<br>
+Pushrod<br>
+weld<br>
+type (left) and pushrod type <br>
+Diaphragm<br>
+Diaphragm<br>
+transducer<br>
+In the "stress plane" weld configuration, the area of the diaphragm under maximum <br>stress coincides with the weld. This can create problems because the weld is differ-<br>ent from the other parts of the diaphragm. The weld and the area right around it will <br>be more susceptible to stress corrosion and brittleness problems. This is particu-<br>larly true of Type 304 stainless steel because of embrittlement from carbide precipi-<br>tation at the weld zone.<br>The integral weld places the weld zone in a non-stressed location. This configuration <br>is typical of the force-rod design and can readily be recognized by the lack of weld <br>bead at the circumference of the diaphragm. Some newer pressure transducers do <br>not employ the welded diaphragm construction but use a non-welded, one-piece dia-<br>phragm and tip assembly.<br>If the pressure transducer is exposed to highly corrosive substances, the diaphragm <br>can be made out of a highly corrosion-resistant material such as Hastelloy. This <br>improvement in corrosion resistance is generally at the expense of non-linearity and <br>hysteresis. These specifications can increase by as much as 0.5%. If the transducer <br>is exposed to highly abrasive components, a very thin wear-resistant coating can be <br>applied to the diaphragm. One manufacturer uses an electrolyzing process to deposit <br>a thin coating on the diaphragm similar to hard chrome, but more wear-resistant.<br>The transducers with hydraulic coupling between the diaphragm and the strain <br>gauge often use mercury; see Fig. 4.5 (left). The amount of mercury is very small. <br>The channel between the diaphragm and strain gauge is a small capillary. This <br>transducer, therefore, is sometimes referred to as a capillary-type transducer. Mer-<br>
+<hr>
+<A name=107></a> <br>
+4.2Pressure Measurement <br>
+<b>93</b><br>
+cury is used because of its low thermal expansion and high boiling point. The effect <br>of temperature on the pressure measurement is rather small. The capillary-type <br>transducer offers the advantage of a uniform liquid support behind the diaphragm. <br>A potential hazard is rupture of the diaphragm. This will cause the release of mer-<br>cury and will contaminate the extrudate and the production area.<br>Transducers with a mechanical coupling between diaphragm and strain gauge often <br>use a force rod; see Fig. 4.5 (right). The force rod design offers the dependability of <br>a direct mechanical link between the diaphragm and the strain gauge. The force rod <br>construction does not result in a uniform loading of the diaphragm.<br>Piezoelectric pressure transducers can be designed in a very compact package. The <br>required deflection to obtain a pressure reading can be very small because of the <br>high sensitivity of the piezoelectric element. The deflection is generally measured <br>in microns (m); full pressure can be reached with a deflection of less than 10 m <br>[1, 2]. The temperature range is primarily determined by the insulation. Transduc-<br>ers with ceramic insulation can be exposed to temperatures as high as 350°C. The <br>more common PTFE insulation allows temperatures up to 240°C. The linearity of <br>these transducers is better than 1%. The sensitivity is in the range of a few pC/bar at <br>pressures of up to 7500 bar (about 750 MPa or 110,000 psi).<br>
+<b>4.2.4Specifications</b><br>
+Specifications on pressure transducers from different manufacturers can vary sig-<br>nificantly. It is important to understand how and to what extent certain specifica-<br>tions affect the accuracy of the measurement. Some of the more important specifica-<br>tions will be reviewed.<br>An ideal transducer would have an exactly linear relationship between pressure and <br>output voltage. In reality, there will always be some deviation from the ideal linear <br>relationship; this is referred to as non-linearity. A "best straight line" (BSL) is fitted to <br>the non-linear curve. The deviation from BSL is quoted in the specifications and <br>expressed as a percent of full scale (FS). The non-linear calibration curve is deter-<br>mined in the ascending direction, i.e., with the pressure going from zero to full rating.<br>The pressure measured in the ascending mode will be slightly different from the <br>pressure measured in the descending mode; see Fig. 4.6.<br>This difference is termed hysteresis; it is expressed as a percent of full scale. The <br>non-linearity and hysteresis errors can be reduced by using a 75 to 80% shunt resis-<br>tor to calibrate the output indicator [3]. This procedure essentially reduces the non-<br>linearity error at the 75%-reading to zero by impressing an additional voltage on the <br>indicator that raises the non-linearity and hysteresis curves. The maximum devia-<br>tion now occurs at full scale and maximum precision occurs at mid-range, where the <br>transducer is most likely to be used in normal operating conditions.<br>
+<hr>
+<A name=108></a><b>94</b>  <b>4Instrumentation and Control</b><br>
+Hysteresis<br>
+Output [mV]<br>
+<b> Figure 4.6</b> <br>
+Pressure<br>
+Hysteresis in a pressure transducer<br>
+Repeatability is a measure of the ability of a transducer to reproduce output read-<br>ings when measuring pressure consecutively and in the same direction. It speci-<br>fies the maximum deviation obtained by comparing output readings for the three <br>ascending-descending full-scale loadings.<br>Temperature changes will cause variations in pressure reading. Thermal shift speci-<br>fications indicate the maximum deviations expected due to temperature changes <br>from room temperature to the specified limits of the operating range both for zero <br>and for up-scale conditions. Deviations are expressed as a percent of full-scale rating <br>for each degree of temperature above room temperature.<br>When a transducer is being selected, one should consider the accuracy of both the <br>transducer itself and the readout equipment. The meter readout should be readable <br>to an accuracy at least as good as the accuracy of the transducer.<br>In Germany, the VDMA (Verein Deutsche Maschinenbau Anstalten) has issued a <br>publication (VDMA 24456) describing in detail the various aspects of pressure <br>transducers. This publication also describes various test set-ups that can be used to <br>test pressure transducers; both static and dynamic testing are discussed.<br>The dynamic behavior of pressure transducers is of particular interest in the analy-<br>sis of extrusion instabilities. The procedure for dynamic testing described in VDMA <br>24456 was used to test various commercial pressure transducers. Puetz [4] pub-<br>lished some dynamic test data for different pressure transducers (see Table 4.2).<br>
+<b>Table 4.2</b>Dynamic Data of Various Pressure Transducers (Courtesy [4])<br>
+Company<br>
+Model<br>
+Range<br>
+Natural<br>
+Damping<br>
+Limiting<br>
+[bar]<br>
+frequency<br>
+[s­1]<br>
+frequency<br>
+[s­1]<br>
+[Hz]<br>
+Dr. Staiger<br>
+200<br>
+353<br>
+49<br>
+17<br>
+Dynisco<br>
+PT 422A<br>
+350<br>
+182<br>
+23<br>
+9<br>
+Dynisco<br>
+PT 420/12<br>
+350<br>
+628<br>
+69<br>
+30<br>
+Brosa<br>
+500<br>
+A.D.*<br>
+A.D.*<br>
+90<br>
+Rosemount<br>
+1401 A1<br>
+105<br>
+A.D.*<br>
+A.D.*<br>
+1<br>
+* A.D. is aperiodic damping<br>
+<hr>
+<A name=109></a> <br>
+4.2Pressure Measurement <br>
+<b>95</b><br>
+Dynamic testing of pressure transducers can be done by measuring the response to <br>a pulse input. If the pulse occurs over a small period of time, the system will respond <br>with a damped oscillation. A typical response is shown in Fig. 4.7.<br>
+<b> Figure 4.7</b> <br>
+Typical pressure transducer pulse <br>
+response<br>
+The response can be described by a function of the form:<br>
+ (4.1)<br>
+where:  A  = amplitude of oscillation<br> <br>
+  = damping constant<br>
+ <br>
+d = natural frequency of the system<br>
+The characteristic values of the system can be determined from the pulse response. <br>With these values, the amplitude-frequency (A-) and phase-frequency (-) char-<br>acteristics can be determined from the following relationships:<br>
+ (4.2)<br>
+ (4.3)<br>
+where D = degree of damping<br> <br>
+T = period of oscillation of undamped system<br>
+Typical amplitude- and phase-frequency characteristics are shown in Fig. 4.8.<br>It can be seen from the amplitude-frequency curve of the transducer shown in <br>Fig. 4.8 that an amplitude increase of 10% is reached at a frequency of about 80 rad/s <br>( 12.5 Hz). This means that pressure fluctuations with a frequency of 12.5 Hz will <br>be measured with an error in amplitude of 10%; with a frequency of 30 Hz the error <br>will be about 100%!<br>
+<hr>
+<A name=110></a><b>96</b>  <b>4Instrumentation and Control</b><br>
+<b> Figure 4.8</b> <br>
+Typical amplitude- and phase-frequency <br>
+curves<br>
+The limiting frequency shown in Table 4.2 represents the frequency at which the <br>error in amplitude reaches 10%. It should be noted that the limiting frequencies <br>listed in the table are considerably lower than the values sometimes claimed by <br>pressure transducer manufacturers. Some transducer suppliers claim a flat fre-<br>quency response up to 100 Hz; however, the data in Table 4.2 does not substantiate <br>that number. As mentioned before, the dynamic response of piezo pressure trans-<br>ducers is far better (several orders of magnitude) than membrane-type pressure <br>transducers.<br>
+<b>4.2.5Comparisons of Different Transducers</b><br>
+In general, electric transducers are more accurate than pneumatic transducers, <br>while pneumatic transducers, in turn, are more accurate than mechanical trans-<br>ducers. Typical accuracy of electric transducers is ± 0.5 to 1.0%, pneumatic trans-<br>ducers about ± 1.5%, and mechanical transducers about ± 3% [4].<br>The reproducibility for electric systems is about ± 0.1 to ± 0.2%, for pneumatic sys-<br>tems about ± 0.5%, and about ± 1 to ± 2% for mechanical pressure transducers. The <br>hysteresis with electric pneumatic transducers is about 0.1 to 0.2%, while it is as <br>high as 4 to 5% with mechanical systems.<br>
+<b></b><br>
+ <b>4.3Temperature Measurement</b><br>
+Temperature measurement occurs at various locations of the extruder: along the <br>extruder barrel, in the polymer melt, and at the extrudate once it has emerged from <br>the die. The choice of the type of temperature measurement will depend on what is <br>being measured and where. First, the methods of temperature measurement will be <br>reviewed.<br>
+<hr>
+<A name=111></a> <br>
+4.3Temperature Measurement <br>
+<b>97</b><br>
+<b>4.3.1Methods of Temperature Measurement</b><br>
+Temperature can be measured with resistive temperature sensors, thermocouple <br>temperature sensors, and radiation pyrometers. There are two types of resistive tem-<br>perature sensors: the conductive type and the semiconductor type. Both operate on <br>the principle that the resistance of sensor material changes with temperature.<br>The conductive-type temperature sensor (RTD) uses a metal element to measure <br>temperature. The resistance of most metals increases with temperature; thus, by <br>measuring resistance, one can determine the temperature. Platinum is used where <br>very precise measurements are required and where high temperatures are involved. <br>Platinum is available in highly purified condition; it is mechanically and electrically <br>stable and corrosion-resistant. Most of the RTD sensors have a wound wire configu-<br>ration; although for some applications, metal-film elements are used.<br>The semiconductor type sensor utilizes the fact that the resistance of a semiconduc-<br>tor decreases with temperature. The most common type of semiconductor tempera-<br>ture sensor is the thermistor shown in Fig. 4.9.<br>
+Rod thermistor<br>
+Bead<br>
+Disk<br>
+Rod<br>
+Shell<br>
+<b>Figure 4.9</b>Thermistor<br>
+Because of their small size, thermistors can be used where other temperature sen-<br>sors cannot be used. A typical resistance temperature (RT) curve is generally non-<br>linear; this is one of the drawbacks of thermistors. However, techniques to deal with <br>the thermistor non-linearity are now well established [14]; thus, the non-linearity is <br>not a major problem. Other disadvantages are the low operating currents (&lt; 100 A) <br>and the tendency to drift over time. An advantage is their quick response time.<br>Thermocouple (TC) temperature sensors are also known as thermoelectric trans-<br>ducers; a basic TC circuit is shown in Fig. 4.10.<br>A pair of wires of dissimilar metals are joined together at one end (hot junction or <br>sensing junction) and terminated at the other end by terminals (the reference junc-<br>tion) maintained at constant temperature (reference temperature). When there is a <br>temperature difference between sensing and reference junctions, a voltage is pro-<br>duced. This phenomenon is known as the thermoelectric effect. The amount of volt-<br>age produced depends on the temperature difference and the metals used.<br>
+<hr>
+<A name=112></a><IMG src="Polymer Extrusion by Chris Rauwendaal-112_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-112_17.jpg"><br>
+<b>98</b>  <b>4Instrumentation and Control</b><br>
+Metal A<br>
+Metal C<br>
+<b>(+)</b><br>
+<b>T</b><br>
+Metal B<br>
+mV<br>
+Metal C<br>
+<b>(-)</b><br>
+Temperature input<br>
+Millivolt<br>
+Ouput<br>
+<b>T0</b><br>
+(T-T0)<br>
+mV <b> Figure 4.10</b> <br>
+Reference Junction<br>
+Basic thermocouple circuit<br>
+One of the most common TCs is the iron-constantan TC. Thermocouples come in <br>various configurations, with exposed junction, grounded junction, ungrounded <br>junction, surface patch, etc.; see Fig. 4.11.<br>
+Extension wire<br>
+Metal sheath<br>
+Insulation<br>
+Exposed junction<br>
+Grounded junction<br>
+<b> Figure 4.11</b> <br>
+Various thermocouple <br>
+Insulated junction<br>
+ <br>configurations<br>
+Detailed information on thermocouples and their use in temperature measurements <br>can be found in the book by Pollock [69], the ASTM publication on thermocouples <br>[70], the NBS monograph on thermocouples [71], and the book by Baker and Ryder <br>[72]. A comparison of various temperature sensors is shown in Table 4.3.<br>For temperature measurements on the emerging extrudate, contacting-type meas-<br>urements are not suitable because of damage to the extrudate surface. For non-con-<br>tacting temperature measurements, infrared (IR) detectors can be used. The inten-<br>sity of the radiation depends on the wavelength and the temperature of a body. <br>Non-contact IR thermometers can be used to determine the temperature of the plas-<br>tic after it leaves the die. IR sensors can also be used to measure the melt tempera-<br>ture inside the extruder or die; see Section 4.3.3.2.<br>
+<hr>
+<A name=113></a> <br>
+4.3Temperature Measurement <br>
+<b>99</b><br>
+<b>Table 4.3</b>Comparison of Various Temperature Sensors<br>
+Thermocouple<br>
+RTD<br>
+Thermistor<br>
+Reproducibility<br>
+1­8°C<br>
+0.03­0.05°C<br>
+0.1­1°C<br>
+Stability<br>
+1­2 °C in 1 year<br>
+&lt;0.1% in 5 years<br>
+0.1­3 °C in 1 year<br>
+Sensitivity<br>
+0.01­0.05 mV/°C<br>
+0.2­10 Ohm/°C<br>
+100­1000 Ohm/°C<br>
+Interchangeability<br>
+Good<br>
+Excellent<br>
+Poor<br>
+Temperature range<br>
+­250 to 2300°C<br>
+­250 to 1000°C<br>
+­100 to 280°C<br>
+Signal output<br>
+0­60 mV<br>
+1­6 V<br>
+1­3 V<br>
+Minimum size<br>
+25 µm diameter<br>
+3 mm diameter<br>
+0.4 mm diameter<br>
+Linearity<br>
+Excellent<br>
+Excellent<br>
+Poor<br>
+Response time<br>
+Good<br>
+Fair<br>
+Good<br>
+Point sensing<br>
+Excellent<br>
+Fair<br>
+Excellent<br>
+Area sensing<br>
+Poor<br>
+Excellent<br>
+Poor<br>
+Cost<br>
+Low<br>
+High<br>
+Low<br>
+Unique features<br>
+Greatest economy, <br>
+Greatest accuracy, very <br>
+Greater sensitivity, <br>
+wide range, common in  stable, less common in  rarely used in extrusion<br>
+extrusion<br>
+extrusion<br>
+IR temperature measurement can give good temperature readings of the extrudate <br>provided the following conditions are met:<br>1.  The radiation from the extrudate must be wholly generated by the extrudate it -<br>
+self as a result of its temperature. Specifically, it must not contain significant <br>levels of transmitted radiation from hot objects behind it and reflected radiation <br>from hot objects in front of it.<br>
+2.  The stream of radiation from the extrudate to the IR thermometer must be trans-<br>
+mitted without absorption by the intervening atmosphere. This means that the <br>IR thermometer must not operate in the spectral regions of atmospheric absorp-<br>tion bands.<br>
+3.  The correct value for the emittance of the extrudate must be known and correctly <br>
+introduced into the IR thermometer calibration.<br>
+Non-contact IR thermometers are used in a large number of extrusion operations: <br>blown film, cast film, biaxially oriented film, sheet, extrusion coating, etc. In some <br>IR thermometers, an entire surface can be scanned and isotherms can be deter-<br>mined. With additional instrumentation, quantitative information on the tempera-<br>ture distribution can be obtained [5]. Portable infrared thermometers can be used to <br>spot check the process, maintain equipment, and do general plant maintenance; <br>they are also very useful tools in troubleshooting.<br>Non-contact thermometers offer a number of benefits. The product is never touched <br>or contaminated. Fast moving extruded products can be measured accurately and <br>quickly. Temperature measurements can be made of a large area or a small spot. <br>Many IR sensors have both analog and digital output; this allows temperature data <br>
+<hr>
+<A name=114></a><b>100</b>  <b>4Instrumentation and Control</b><br>
+to be integrated into a closed-loop control system for remote temperature monitoring <br>and analysis. An example of IR temperature sensing in extrusion coating is shown <br>in Fig. 4.12.<br>In extrusion coating, a molten web from a film die is applied to paper, film, or foil as <br>shown in Fig. 4.12. The distance between the die and the pressure and chill rolls is <br>usually quite short; between 75 and 125 mm. The polymer melt temperature in this <br>region has to be very hot for the melt to adhere to the substrate. A small IR sensor <br>can easily measure the polymer melt temperature in this small space. The operator <br>can monitor and adjust the die heater and the chill roll temperatures either manu-<br>ally or automatically.<br>
+Hopper<br>
+Extruder<br>
+Compact IR sensor<br>
+Film die<br>
+Wind-up roll<br>
+Pressure roll<br>
+Chill roll<br>
+Pay-off<br>
+Coated substrate<br>
+Uncoated substrate<br>
+<b>Figure 4.12</b>IR sensing in extrusion coating<br>
+<b>4.3.2Barrel Temperature Measurement</b><br>
+The barrel temperature needs to be measured to provide information on the axial <br>barrel temperature profile and to provide a signal for the controllers of the barrel <br>heaters and cooling devices. The temperature should be measured as close as pos-<br>sible to the inner barrel surface, since the polymer temperature is the primary con-<br>cern. The worst possible location of the temperature sensor would be in the barrel <br>heater itself. However, there are some commercial extruders where the temperature <br>sensor is placed in the barrel heater to reduce the thermal lag of the system. The <br>major drawback of this approach is that one controls the heater temperature and not <br>the temperature of the polymer in the extruder barrel. Some extruders are equipped <br>with a combination of deep-well and shallow-well temperature sensors to improve <br>
+<hr>
+<A name=115></a> <br>
+4.3Temperature Measurement  <b>101</b><br>
+the temperature control of the extruder [7, 8]. The advantages and disadvantages of <br>deep-well and shallow-well temperature sensors in terms of temperature control are <br>discussed in Section 4.5.2.4, as well as dual sensor temperature control.<br>In the measurement of barrel temperature, a temperature sensor is pressed into <br>a well in the extruder barrel; the sensor is generally spring-loaded; see Fig. 4.13. <br>Most temperature sensors are constructed with a metallic sheath to obtain sufficient <br>mechanical strength. As a result, significant thermal conduction errors can occur.<br>
+<b>Figure 4.13</b>Spring-loaded temperature sensor<br>
+Figure 4.14 shows how the accuracy of the temperature measurement depends on <br>the depth of the well and the type of temperature sensor [40].<br>
+<b> Figure 4.14</b> <br>
+Dependence of temperature meas-<br>
+urement on the depth of the well . <br>
+The true barrel temperature is 185°C; <br>
+the measurements were made in <br>
+still air<br>
+This figure illustrates quite clearly that the depth of the well should be at least <br>30 mm to minimize the measurement errors. The characteristics of the temperature <br>sensor itself have a strong effect on the accuracy of the measurement. If special pre-<br>cautions have been taken to minimize heat losses along the stem of the temperature <br>sensor, the measurement error can be greatly reduced as compared to standard tem-<br>perature sensors.<br>It is important to realize that barrel temperature measurement with a shallow well <br>can be, and most likely will be, inaccurate. With a well depth of 10 mm, the meas-<br>ured temperature will probably be about 10°C below actual temperature. When air <br>drafts occur around the extruder, the measured temperature can be as much as <br>25°C below the actual temperature. This is shown in Fig. 4.15.<br>
+<hr>
+<A name=116></a><b>102</b>  <b>4Instrumentation and Control</b><br>
+<b> Figure 4.15</b> <br>
+The effect of air current on the <br>
+ <br>measured  temperature<br>
+<b>4.3.3Stock Temperature Measurement</b><br>
+The measurement of the temperature of the polymer melt is of major importance. <br>Unfortunately, several factors complicate the stock temperature measurement sub-<br>stantially. It is very important to be aware of these complications in order to prop-<br>erly appreciate the measured value.<br>Measurement of stock temperatures along the extruder barrel is difficult because of <br>the rotation of the screw. To measure stock temperatures, the temperature sensor <br>has to protrude into the polymer. The temperature sensors cannot be placed in the <br>barrel because the protruding sensor would be damaged by the screw flight. One <br>alternative is to place the temperature sensors in the screw channel. However, this <br>is a rather complex operation [9­12]; it requires special thermocouple mountings <br>and sliding contacts. Another alternative is to place the temperature sensors in the <br>barrel and to machine slots in the screw flight at the location of the protruding sen-<br>sor [13]. The disadvantage of this method is the substantial modification of the flow <br>patterns in the extruder as a result of the slots in the screw flight. This, in turn, will <br>change the actual temperature distribution in the polymer melt.<br>When considering the stock temperatures along the length of the extruder, it should <br>be realized that the temperature, in most cases, varies much more strongly in the <br>radial direction than in the axial direction. Radial temperature gradients will be <br>particularly high in the plasticating region of the extruder. In this region, a thin <br>melt film, with a melt film thickness in the order of 1 mm, separates the solid bed <br>from the barrel. The temperature difference across this film is often in the range of <br>30 to 80°C. Thus, the radial temperature gradient will be of the order of 50,000°C <br>per meter. This is about three orders of magnitude higher than the typical axial tem-<br>
+<hr>
+<A name=117></a> <br>
+4.3Temperature Measurement  <b>103</b><br>
+perature gradient in an extruder. Therefore, if one would try to measure stock tem-<br>perature with a protruding temperature sensor, one would experience an extreme <br>sensitivity to radial location of the sensor. The movement of the solid bed is another <br>consideration. Any temperature sensor should avoid contact with the solid bed, <br>since this would likely result in damage of the sensor.<br>The solids conveying and plasticating zone generally extend about two-thirds of the <br>length of the extruder. This means that stock temperature measurements by immer-<br>sion probes are really only possible in the last one-third of the extruder. The polymer <br>temperature in the solids conveying and melting zone generally is maximum at or <br>close to the barrel wall, as will be shown in Chapter 7. This means that measurement <br>of barrel temperature is a good indication of the maximum stock temperature. Thus, <br>barrel temperature measurement may be more meaningful, and certainly much <br>easi er, than stock temperature measurement with a protruding temperature sensor.<br>For these reasons, stock temperatures along the extruder are generally not measured <br>on production extruders. Only a few, highly instrumented, development extruders <br>are equipped with stock temperature measurement capability along the barrel. The <br>situation is much simpler at the end of the barrel because there the screw flight is no <br>longer present. The temperature sensor can protrude freely into the melt stream <br>without danger of being damaged by the screw. However, even in this situation, there <br>are several complicating factors involved in the temperature measurement.<br>Numerous detailed studies have been devoted to the measurement of temperature <br>profiles in polymer melts flowing through channels. One of the most comprehensive <br>studies on the theoretical and experimental aspects of temperature measurement of <br>polymer melts was carried out by van Leeuwen [15­18]. Other studies on melt tem-<br>perature measurement are listed in the following references [19­23]. When a poly-<br>mer melt flows through a channel, a certain temperature profile will establish itself <br>in the polymer melt. The temperature profile in a steady-state process after some <br>time will become constant with respect to time; this is the so-called fully developed <br>temperature profile.<br>The temperature at any point can be predicted from the equations of mass, momen-<br>tum, and energy. When a temperature sensor, such as a thermocouple, is inserted <br>into the polymer melt stream to measure the temperature of the melt, the steady-<br>state flow is disturbed, and a new steady state will develop after a short time. There-<br>fore, the measured temperature will be different from the true, undisturbed melt <br>temperature. Thus, in order to determine the true melt temperature, certain correc-<br>tions have to be made to the measured (disturbed) melt temperature. The factors <br>that have to be taken into account are:<br>
+ <br>
+ Heat conduction along the probe<br>
+ <br>
+ Heat convection from the probe<br>
+ <br>
+ Energy dissipation at the probe due to shear heating<br>
+<hr>
+<A name=118></a><b>104</b>  <b>4Instrumentation and Control</b><br>
+The design of the temperature sensor should be such that the above-mentioned <br>errors are minimized. Various designs of temperature sensors for stock temperature <br>measurement are shown in Fig. 4.16.<br>
+Immersion probe<br>
+Flush mounted<br>
+Upstream tip,<br>
+Upstream tip,<br>
+straight protruding<br>
+fixed position<br>
+adjustable<br>
+Polymer melt flow<br>
+<b> Figure 4.16</b> <br>
+Various melt temperature <br>
+Bridge with<br>multiple probes<br>
+ <br>sensor  configurations<br>
+The flush-mounted temperature sensor does not disturb the flow in the channel. <br>How ever, the sensor does not protrude into the melt stream. The measured tempera-<br>ture, therefore, will be more representative of the metal wall temperature than the <br>polymer melt temperature. It should be remembered, though, that the temperature <br>of the polymer melt at the wall will equal the metal temperature at the wall. The <br>flush-mounted probe, therefore, will give a reasonably good indication of the tem-<br>perature of the polymer melt at the interface. The problem with this design is that <br>the maximum temperature of the polymer melt generally does not occur at the wall! <br>Unlike the situation in the solids conveying and melting zone, the maximum tem-<br>perature of the polymer melt in the melt conveying zone of the extruder (including <br>adaptor and die) generally occurs some distance away from the wall. For this reason, <br>a protruding sensor will yield information that is more meaningful.<br>The straight immersion sensor is simple and sturdy. It gives a reasonable indication <br>of the stock temperature in the flow channel. This design, however, results in sig-<br>nificant errors in the measured temperature because of shear heating and heat con-<br>duction along the probe. This is due to the perpendicular location of the probe with <br>respect to the direction of flow. As an improved design, van Leeuwen [15] proposed <br>the upstream probe. The sensor is oriented parallel to the flow, causing only a mini-<br>mal disturbance to the flow at the point where the temperature is measured. The <br>upstream probe is capable of measuring local temperatures. The parallel portion of <br>the probe should be long and thin to reduce heat conduction errors and to be able to <br>measure rapid changes in temperature. On the other hand, the mechanical strength <br>of the probe should be sufficient to withstand the forces that a melt probe is nor-<br>mally exposed to. Damage can easily occur during start-up or shut-down. To avoid <br>this problem, probes have been made with adjustable depth so that the probe is <br>inserted into the polymer when steady conditions have been reached. At high flow <br>
+<hr>
+<A name=119></a> <br>
+4.3Temperature Measurement  <b>105</b><br>
+rates and high polymer melt viscosity, the forces that the polymer melt exerts on the <br>immersion probe can be substantial. For these reasons, the parallel-to-flow portion <br>of the probe is often made into a conical shape or a short, small diameter tube.<br>The depth adjustment capability has another use besides avoiding damage. It allows <br>the measurement of the temperature profile across the depth of the flow channel <br>using only one probe. Adjustable upstream temperature probes are currently com-<br>mercially available, e.g., by Goettfert. However, their application in commercial ex -<br>truders is still rather limited.<br>The bridge with multiple probes has the advantage of being able to monitor various <br>temperatures at different locations at the same time. This allows a very careful mon-<br>itoring of the thermal conditions in the polymer melt throughout the material. It is <br>often possible to incorporate temperature probes in spider legs, torpedoes, etc., to <br>obtain information on the polymer temperature away from the outside wall.<br>In Germany, the VDMA (Verein Deutsche Maschinenbau Anstalten) has issued a <br>publication (VDMA 24485) to standardize evaluation and testing of temperature <br>sensors. In this publication, a test setup is described that allows determination of <br>the thermal conduction error as well as the transient response of the probe. From <br>the transient response, one can determine the delay time, recovery time, and 90% <br>time (T0.9). Puetz [4] describes evaluation and comparison of five different tempera-<br>
+ture probes according to the procedure described in VDMA 24485. The upstream <br>temperature probe proved to be much more accurate than the straight protruding <br>temperature probe with a difference in measured temperature of about 10 to 15°C! <br>Of the five probes, the only one able to determine temperature fluctuations occur-<br>ring in less than one minute was the upstream probe with a small thin parallel sec-<br>tion. The other temperature sensors could only measure temperature fluctuations <br>occurring in more than one or two minutes. This means that conventional thermo-<br>couples are not suited to determine high frequency (t &lt; 1 minute) extrusion insta-<br>bilities.<br>
+<b>4.3.3.1Ultrasound Transmission Time<br></b>Research at the IKV in Aachen, Germany, has shown that the measurement of ultra-<br>sound transmission time (UTT) yields a useful quantity to characterize the thermo-<br>dynamic condition of the polymer melt [68]. Advantages of the UTT measurement <br>are:<br>
+ <br>
+ The transducer does not disturb the polymer melt flow.<br>
+ <br>
+ The UTT measurement is a result of linearly integrating measurement across the <br>
+depth of the flow channel.<br>
+ <br>
+ The measurement is not affected by heat conduction errors or viscous heat dissi-<br>
+pation.<br>
+<hr>
+<A name=120></a><b>106</b>  <b>4Instrumentation and Control</b><br>
+Commercial applications of UTT measurement started in 1976; however, the method <br>has not found widespread use. An important application in extrusion is the non-<br>contacting temperature measurement of heat-sensitive materials. Protruding tem-<br>perature sensors can easily cause degradation with such materials. The UTT meas-<br>urement, pressure compensated, can be used for stock temperature control in a <br>similar fashion as with the conventional melt temperature sensor (see Section 4.6).<br>The UTT measurement can be used in total extrusion line control. The use of UTT <br>measurement reportedly has resulted in considerable technical improvements in <br>product performance in several extrusion operations [68]. It should be noted, how-<br>ever, that the pressure compensation is rather involved and that the accuracy of the <br>resulting temperature is only as good as the pressure measurement. In this respect, <br>the melt temperature measurement by IR radiation is less complicated (see Section <br>4.3.1).<br>
+<b>4.3.3.2Infrared Melt Temperature Measurement<br></b>IR probes that can be mounted in an extruder barrel or die are commercially avail-<br>able [6]. These probes are used to measure a more or less average stock temperature <br>over a certain depth of the polymer, about 1 to 5 mm for most unfilled polymers. The <br>actual depth of the measurement is determined by the optical properties of the poly-<br>mer melt, in particular the transmittance. The measurement is affected by varia-<br>tions in the consistency of the polymer melt. Thus, when fillers, additives, or other <br>polymeric components are added, the temperature readings will be affected.<br>An important advantage of the IR stock temperature measurement is the rapid <br>response time, which is about ten milliseconds. The response of conventional melt <br>thermocouples is several orders of magnitude slower. This means that rapid tem-<br>perature fluctuations can be made visible with IR allowing a more detailed study of <br>the dynamic behavior of extruders and injection molding machines [83, 84].<br>The elements of an infrared melt temperature sensor are a sapphire window, an opti-<br>cal fiber, and a radiation sensor with associated signal-conditioning electronics as <br>shown in Fig. 4.17. IR melt temperature probes are commercially available [85, 86] <br>and fit in standard pressure transducer mounting holes. Because the sapphire win-<br>dow is flush with the barrel or die, the sensor does not protrude into the polymer <br>melt. As a result, the sensor is less susceptible to damage, there is no chance of dead <br>spots behind the sensor, and the melt velocities are not altered around the sensor. <br>When melt velocities are changed, the melt temperatures will change as well. There-<br>fore, the melt temperatures measured with an IR sensor are less affected by the <br>actual measurement than with an immersion sensor.<br>The window material is usually sapphire, because it provides good abrasion resist-<br>ance and can withstand high pressures. Obviously, there is a potential problem of <br>build-up of material on the window, which could partially block the sensor window. <br>
+<hr>
+<A name=121></a><IMG src="Polymer Extrusion by Chris Rauwendaal-121_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-121_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-121_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-121_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-121_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-121_6.jpg"><br>
+ <br>
+4.4Other Measurements  <b>107</b><br>
+Calibration can be done using standard sources off-line or in-situ during static pre-<br>heat using a thermocouple placed in close proximity to the IR probe. The tempera-<br>ture reading for the IR system will be influenced by the emissivity of the polymer <br>melt and the stem heating effects.<br>
+optical fiber<br>
+IR diode<br>
+sapphire<br>window<br>
+die or barrel<br>
+<b>polymer melt</b><br>
+<b> Figure 4.17</b> <br>
+An infrared melt temperature sensor<br>
+<b></b><br>
+ <b>4.4Other Measurements</b><br>
+Pressure and temperature are two process parameters of major importance. There <br>are, however, various other parameters, which cannot be ignored.<br>
+<b>4.4.1Power Measurement</b><br>
+The basic method to measure electrical power consumed by a load is to connect an <br>ammeter in series with the load and a voltmeter across the load. In a DC circuit, the <br>power is obtained by multiplying current with voltage. The same is true for an AC <br>circuit where only a resistive load is concerned. In this case, the current and voltage <br>are in phase. In an AC circuit, where the load has an inductive or capacitive compo-<br>nent, the current and voltage are no longer in phase.<br>In an AC circuit with an inductive or capacitive component, there are two types of <br>power. One is true or useful power, which is capable of doing useful work, e.g., turn-<br>ing the rotor of a motor. The other power is reactive power that cannot do useful <br>work. Total power, which is known as apparent power, is the vectorial sum of true <br>power and reactive power. The ratio between true power and apparent power is <br>known as the power factor of the circuit.<br>
+<hr>
+<A name=122></a><b>108</b>  <b>4Instrumentation and Control</b><br>
+The power factor, which is always less than 100%, is a function of the phase differ-<br>ence between the current and voltage in a circuit. The apparent power in a circuit <br>can be measured by the basic volt and ammeter method. To distinguish apparent <br>power from true power, apparent power is stated in units of volts-amperes. The true <br>power, which is stated in units of watts, is measured by a wattmeter.<br>The power consumption of regular AC motors can be measured quite easily by using <br>a readily available wattmeter. The power consumption of SCR-controlled DC motors <br>and variable frequency AC motors is more difficult to measure because of the dis-<br>torted wave form going into the motor. If a conventional wattmeter is used to meas-<br>ure power consumption of an SCR-controlled DC motor, substantial measurement <br>error will be made, depending on the phase angle. On a DC motor, it is easier to <br>measure the AC power going into the drive before the rectifier circuit. This measure-<br>ment, of course, will include static losses in the drive and rectifier. Extruders with <br>DC drive often have an armature current readout on the instrument panel. The <br>approximate power consumption Z of the drive can be calculated from the armature <br>current Ia by using the following relationship:<br>
+ (4.4)<br>
+where Nact is the actual rpm, Nmax the maximum rpm, Ia the armature current, and <br>
+Va the armature voltage. This relationship is accurate to approximately 5% and does <br>
+not reflect the static losses in the drive.<br>A good method to measure the actual mechanical power consumed in the extrusion <br>process is to measure the torque transmitted through the shank of the extruder <br>screw. The actual power is obtained by multiplying the torque with the screw speed. <br>In fact, this is an excellent method to determine the overall energy efficiency of the <br>extruder drive system. This can be done by comparing the screw power to the total <br>power consumption of the drive. Unfortunately, this type of data is not readily avail-<br>able. The torque can be measured with a torque transducer. Torque transducers <br>generally measure the torsion of the shaft by means of a strain gauge on the shaft; <br>the torque is directly proportional to the angular torsion of the shaft. Unfortunately, <br>accurate torque transducers are quite expensive and, thus, can add significantly to <br>the cost of an extruder.<br>The power consumption of the barrel and die heaters can be determined by measur-<br>ing voltage and current to the heater. This works well in current proportioned tem-<br>perature control. It does not work well with on-off control or time-proportioning <br> <br>temperature control. In the latter case, a wattmeter should be used with a power <br>integrating function. In this case, the integrated power over a certain time period <br>can be measured so that the average power consumption of the barrel heater can be <br>established. Commercial extruders generally do not have sufficient instrumentation <br>
+<hr>
+<A name=123></a> <br>
+4.4Other Measurements  <b>109</b><br>
+to accurately determine the power consumption of the heaters; in many cases, it can-<br>not be determined at all!<br>
+<b>4.4.2Rotational Speed</b><br>
+The magnetic pickup is a very common method for measuring rotational speed. The <br>basic elements are a metallic-toothed wheel connected to the rotating shaft and a <br>magnetic pickup or coil; see Fig. 4.18.<br>
+Permanent magnet<br>
+Pickup<br>
+Toothed wheel<br>
+<b>Figure 4.18</b>Magnetic pickup to measure rotational speed<br>
+The teeth of the wheel pass near the pickup coil. The pickup typically consists of a <br>housing containing a small permanent magnet with a coil wound around it. A fixed <br>magnetic field surrounds the pickup. When the teeth of the wheel pass through the <br>field, a voltage pulse is induced in the coil. The frequency of the pulses depends on <br>the number of teeth and the speed of rotation. Since the number of teeth is known, <br>the pulse frequency can be related directly to the rotational speed. The pulses can <br>be measured by a frequency counter or they can be converted to a DC voltage.<br>Another method for measuring rotational speed is the rotating disk and light sensor. <br>Here, the basic elements are a perforated rotating disk connected to the rotating <br>shaft, a light source, and a light sensor; see Fig. 4.19. A fixed light source is placed <br>on one side of the disk in line with the holes. A light sensor is placed on the opposite <br>side of the disk in line with the light source. When the perforated disk rotates, a <br>pulsed output signal is produced. When the number of holes in the perforated disk <br>is known, the pulse frequency can be converted to rotational speed, as with the mag-<br>netic pickup.<br>
+<hr>
+<A name=124></a><b>110</b>  <b>4Instrumentation and Control</b><br>
+Perforated disk<br>
+Light source<br>
+Light sensor<br>
+<b> Figure 4.19</b> <br>
+Pulse output signal<br>
+Optical speed measurement<br>
+Another simple method is to use an electrical tachometer. The small DC generator <br>is coupled to the rotating shaft. The output voltage of the generator is fed to a volt-<br>meter. The generator output voltage is directly proportional to the rotational speed <br>of the shaft. Thus, the measured voltage can be directly converted into rpm.<br>
+<b>4.4.3Extrudate Thickness</b><br>
+A variety of methods are available to measure extrudate thickness. The methods can <br>be broadly classified into contacting and non-contacting techniques. The contacting <br>thickness measurement techniques are generally simple and inexpensive; however, <br>the contact of the transducer with the extrudate can adversely affect the extrudate <br>surface quality. In cases where the requirements for surface quality are very high, <br>non-contacting thickness measurement is generally preferred.<br>In the contacting measurement techniques, the micrometer caliper is a common <br>instrument. The micrometer, however, can only be used for pot measurements and <br>this is done manually. A spring-loaded dial gauge can be moved over the extrudate if <br>the thickness variations are small. Thus, the dial gauge can be used to monitor the <br>variation of thickness with time, i.e., in the extrusion direction. If an accurate travers-<br>ing mechanism is constructed, the dial gauge can also measure the thickness varia-<br>tion perpendicular to the extrusion direction. At the point of measurement, the oppo-<br>site side of the extrudate has to be firmly supported to avoid measurement errors.<br>The micrometer and dial gauge are simple mechanical devices. In many cases, one <br>would like to have a continuous record of the thickness measurement. The LVDT <br>(linear variable differential transformer) provides an electrical signal that can be <br>used to monitor the thickness on a recorder. The LVDT is a device in which the dis-<br>placement of an iron core changes the inductive coupling between primary and sec-<br>ondary coils; see Fig. 4.20.<br>
+<hr>
+<A name=125></a> <br>
+4.4Other Measurements  <b>111</b><br>
+Movement of the core produces an AC output signal that reflects the amount and <br>direction of movement. If an LVDT is used in thickness measurement, one has to <br>check the effect of temperature on the accuracy because the extrudate is generally <br>at an elevated temperature that may not be constant. The LVDT can be quite accu-<br>rate; it can measure to an accuracy of about 1 m.<br>
+Primary coil<br>
+Output voltage<br>
+Input voltage<br>
+Secondary coil<br>
+<b> Figure 4.20</b> <br>
+Linear variable differential transformer <br>
+(LVDT)<br>
+Another more or less contacting thickness measurement technique is pneumatic <br>gauging. The device consists of a nozzle fixed in position relative to a stop. Air at a <br>constant supply pressure passes through a restriction and discharges through the <br>nozzle; see Fig. 4.21.<br>
+Pressure regulator<br>
+Sample<br>
+Gage<br>
+Gauge<br>
+Gap<br>
+Stop<br>
+Nozzle<br>
+Restrictor<br>
+<b> Figure 4.21</b> <br>
+Pneumatic thickness gauge<br>
+<hr>
+<A name=126></a><b>112</b>  <b>4Instrumentation and Control</b><br>
+The nozzle back pressure P depends on the gap between the measured surface and <br>the nozzle opening. If the thickness increases, the gap decreases, restricting the <br>discharge of air, thus increasing pressure P. The pressure gauge indicates deviation <br>of the thickness from some normal value. With the proper design, this pressure <br>is directly proportional to the deviation, limited, however, to a range of about 100 <br>micron. The device is very sensitive, up to 0.0001 mm over a range of 0 to 2 mm. It <br>is rugged and, with periodic calibration, quite accurate. The gauge is adaptable to <br>automatic line control where the pressure signal is recorded or used to actuate an <br>alarm when the thickness exceeds a certain threshold value.<br>Another contacting thickness measurement is the capacitance measurement. Metal <br>plates are placed at either side of the polymer film. Thus, the material and plates <br>form a capacitor, with the polymer acting as a dielectric. Since the capacitance de -<br>pends on the thickness of the dielectric, the material thickness is determined by <br>measuring the capacitance. The problem in applying this technique to extrusion is <br>that it will be difficult to establish good contact between polymer and metal plates, <br>particularly in continuous monitoring of thickness.<br>The thickness of a polymer extrudate can also be measured ultrasonically. In this <br>type of measurement, the sensor uses mechanical vibrations of high frequencies, <br>beyond the audio range, i.e. more than about 15,000 vibrations per second. The <br>vibrations are produced by a transducer, which converts the electrical output of an <br>oscillator to ultrasonic vibrations of corresponding frequencies.<br>There are two kinds of ultrasonic transducers: the magneto-strictive type and the <br>piezoelectric type. The former consists of a metal rod placed in a coil driven by oscil-<br>lator signals. The alternating magnetic field alternately elongates and compresses <br>the rod. With one end of the rod fixed and the other end connected to a diaphragm, <br>ultrasonic sound waves are produced.<br>The piezoelectric ultrasonic transducer is more common. When a voltage is applied <br>to a piezoelectric material, it will compress or expand. If the voltage is alternating <br>at ultrasonic frequency, the piezoelectric will compress and expand at the same <br>frequency. The mechanical vibrations can be transferred to a diaphragm to produce <br>ultrasonic sound waves.<br>For thickness measurement, the transducer is placed against the material. Ultra-<br>sonic vibrations pass through the material to the surface and are reflected back to <br>the transducer. The time required for the vibrations to travel through the material <br>depends on the thickness of the material. When resonance occurs, there is a sudden <br>change in the load that the transducer offers the oscillator, producing a correspond-<br>ing change in oscillator current. By determining the resonant frequency, the thick-<br>ness of the material can be determined. Obviously, good contact is required between <br>transducer and extrudate. This good contact is difficult to achieve in continuous <br>thickness monitoring of a moving extrudate.<br>
+<hr>
+<A name=127></a> <br>
+4.4Other Measurements  <b>113</b><br>
+Ultrasonic measurements have also proven useful in the characterization of poly-<br>mer melts (see Section 4.3.3.1). The ultrasonic transit time ("Laufzeit") is dependent <br>on the elastic properties of the material, pressure, temperature, chemical composi-<br>tion, and structure. It has been found [37] that the ultrasonic transit time is a sensi-<br>tive measure of the condition of the polymer melt, in particular melt homogeneity. <br>In a process control system, the ultrasonic transit time can provide a more useful <br>feedback control signal than a single melt temperature measurement.<br>Thus far, the discussion has dealt with contacting thickness measurements. In addi-<br>tion to the fact that there is contact between the sensor and the extrudate, there is <br>another problem in that these methods cannot be applied to continuous thickness <br>monitoring of annular profiles, i.e., tubes and pipes. In automated extrusion lines, <br>non-contacting thickness measurement has become quite popular. Most of the non-<br>contacting thickness measurement techniques are based on a radiation sensor pick-<br>ing up a signal from a radiation source.<br>Various types of radiation are used: <i></i>-rays, -rays, -rays, X-rays, and infrared radia-<br>
+tion. A continuous stream of radiation is emitted from a constant radiation source <br>(X-ray tube or radioisotope), passes through the material whose thickness is being <br>measured, and strikes the radiation sensor. As radiation passes through the extru-<br>date, some of the radiation is absorbed and, as a result, the radiation reaching the <br>sensor is less intense. The amount of absorption depends on the material's density <br>and thickness. If the density is constant, the amount of radiation absorption is a <br>direct measure of thickness.<br>The absorption of radiation is governed by the following relationship:<br>
+ (4.5)<br>
+where I(x) is the intensity after transversing a distance x through a material with <br>absorption coefficient ; I(o) is the incident intensity. After proper calibration, very <br>high accuracies can be achieved, down to 0.2 m. Measurements can be made at <br>high speeds. These factors have made the radiation type thickness measurement an <br>almost standard tool on automated extrusion lines.<br>Nuclear radiation sensors cover a thickness range from 10 m to about 3 mm. Some <br>sensors come with air gap temperature sensors to compensate for changes in den-<br>sity of the air column as the temperature varies across the sheet. These sensors <br>can be designed to automatically correct for dirt build-up and drift. A disadvantage <br>of nuclear radiation is the potential health hazard. Very high standards have to be <br>applied to the design of the measuring device to ensure that all radiation is con-<br>tained within the instrument. With thick extrudate, relatively high radiation levels <br>are required because of the exponential decay of radiation intensity with distance. <br>Therefore, for relatively thick flat profiles, the LVDT type sensor may be more appro-<br>priate. Infrared sensors can be used for clear thin film in the thickness range of 2 to <br>
+<hr>
+<A name=128></a><b>114</b>  <b>4Instrumentation and Control</b><br>
+200 m. This sensor employs simultaneous valuation of the measurement and refer-<br>ence wavelength.<br>A problem with radiation sensors is that the measurement is affected by density or <br>compositional variations in the material. This is because most of these sensors <br>measure weight per unit area. Thus, when changes in density occur in the extru-<br>date, the accuracy of the thickness measurement is directly affected. Some sensors <br>have special compensation circuitry to eliminate the sensitivity to compositional <br>changes. These sensors are referred to as "nondiscriminatory," meaning that addi-<br>tives, base resin, and variations in composition are measured equally [73]. Of course, <br>the sensitivity to density variation can be used advantageously to detect flaws in the <br>material, such as voids, contaminants, etc.<br>-rays and high-frequency X-rays have a relatively high penetrating capability and <br>are almost exclusively used for thick parts, several millimeters up to as thick as <br>40 mm. However, they are also used for thicknesses down to about 100 m. -rays <br>have less penetrating capability and are used for relatively thin parts, less than <br>about 3 to 5 mm. Two isotopes are often used, Krypton-85 and Strontium-90. Kryp-<br>ton-85 is used in a measurement range of about 10 to 1000 g/m2, with a correspond-<br>ing thickness range of about 5 to 750 m. Strontium-90 is employed in a measure-<br>ment range of about 100 to 5000 g/m2, with a corresponding thickness range of <br>about 100 to 5000 m [79]. A disadvantage of -gauges is that they tend to drift and, <br>thus, require frequent recalibration.<br>-ray thickness detectors often employ the "-backscattering" technique [80]. This <br>type of measurement offers the advantage that the object is measured from one side, <br>thus allowing simple installation and measurement of relatively complicated shapes. <br>The isotope used is generally Americium-241, which has a half-life of about 450 <br>years as opposed to about 10 years for the isotopes used in X-ray thickness detec-<br>tors. Another advantage of this technique is that the gauge readings are relatively <br>insensitive to changes in the composition of the material, allowing relatively simple <br>electronics. Low-frequency X-rays (soft X-rays) are also used for thickness measure-<br>ment of relatively thin products.<br>
+<b>4.4.4Extrudate Surface Conditions</b><br>
+In the extrusion of sheet and film, it is often very important that the surface condi-<br>tions of the extrudate are maintained within a narrow range. Small irregularities, <br>such as specks, have to be avoided in the more demanding applications, e.g., record-<br>ing tapes, high-quality transparent sheet, etc. It is very difficult for the human <br>inspector to accurately detect a small dirt sport of 1 mm diameter moving at a speed <br>of 1 to 5 m/s, particularly if the web has a substantial width, more than 2 meters. <br>For these demanding applications, automatic inspection systems are available.<br>
+<hr>
+<A name=129></a> <br>
+4.4Other Measurements  <b>115</b><br>
+An automatic inspection system uses various transducers that produce electrical <br>signals representative of the surface condition of the web. These signals then have <br>to be analyzed and interpreted. If a signal, or a number of signals, exceeds a preset <br>detection threshold, an alarm is activated, and the area of concern can be further <br>analyzed to determine what corrective action should be taken. Where visible flaws <br>are to be detected, the transducers most commonly chosen are light-sensitive; they <br>produce signals that are parametric measures of such physical phenomena as reflec-<br>tion, transmission, and the like.<br>Several commercial automatic inspection systems have been developed [24, 25] <br>that utilize a laser scanning system. The web is scanned with a moving light beam <br>from a laser flying spot scanner. The reflected or transmitted beam, which has been <br>modified by the characteristics of the web, is picked up by a light-sensitive detector, <br>a photomultiplier tube. This design allows a very high scan rate (more than 5000 <br>scans per second) and permits 100% coverage of webs as wide as 4 m, moving at <br>speeds of 1 m/s and higher.<br>The measurement of haze and luminous transmittance of transparent plastics is <br>described in ASTM D1003 (American Society for Testing and Materials). The haze of <br>a specimen is defined as the percentage of transmitted light, which, in passing <br>through the specimen, deviates more than 2.5 degrees from the incident beam by <br>forward scattering. Luminous transmittance is defined as the ratio of transmitted to <br>incident light. The measurement of luminous reflectance, transmittance, and color <br>is described in ASTM E308. The measurement of gloss is described in ASTM D523 <br>and in DIN-Norm 67530 (Deutsche Industrie Norm is a German Industrial Stand-<br>ard).<br>The quantitative measurement of gloss of extruded sheet is described by Michaeli <br>[26]. In this study, a goniophotometer was used to measure both the gloss height <br>(maximum intensity of the reflected light beam) and the gloss sharpness expressed <br>as the reciprocal width of the gloss distribution curve; the width is measured in <br>degrees.<br>Quantitative characterization of the texture of extruded film was studied by Nadav <br>and Tadmor [27, 28]. The samples were characterized by measuring light transmis-<br>sion through a film sample. The results were analyzed using the concepts of scale of <br>segregation and intensity of segregation.<br>In the surface analysis of an extrudate, the irregularities in the size range of m to <br>mm (10­6 to 10­3 m) are of primary interest. These irregularities are responsible for <br>the gloss, roughness, and color properties of the surface. Most methods of surface <br>texture determination are based on reflection of light waves [29]. In transparent <br>materials, non-uniformities in the composition of the material can cause significant <br>changes in optical properties. These compositional variations cannot be assessed by <br>the measurement of light reflection; however, by measuring light transmission the <br>
+<hr>
+<A name=130></a><b>116</b>  <b>4Instrumentation and Control</b><br>
+effect of compositional variations can be determined accurately [30]. Compositional <br>variations, such as voids or cracks, can be measured ultrasonically or by using <br>microwaves or X-rays [31, 32]. The measurement with microwaves can be done in a <br>continuous fashion; however, the imperfections have to be rather large (about 1 mm) <br>to be detectable.<br>Color measurement requires the determination of the spectral distribution of the <br>light reflected from the surface. In continuous color measurement, usually a limited <br>number (e.g., three) of spectral regions are determined by measurement across a <br>filter. Various commercial color-measuring instruments, colorimeters and spectro-<br>photometers, are available today [33], including systems for monitoring continuous <br>webs. These systems automatically provide data on color, opacity, and yellowing at <br>one or several user-selected web locations.<br>Color measurement can also be tied in with the compound preparation step. The <br>spectrophotometer thus provides a feedback control signal to adjust the level of co -<br>lorants in order to obtain the desired color automatically [74, 75]. Some of the prob-<br>lems in assessing color and color difference were discussed by Osmer [34]. A good <br>basic text on the principles and theory of color is the book by Billmeyer and Saltz-<br>man [76].<br>The orientation of the polymer molecules in the extrudate has a large effect on the <br>physical properties. At the IKV in Aachen, Germany, a technique was developed to <br>measure the anisotropy of an extrudate in a continuous fashion [35, 36]. The meas-<br>urement is based on the compensation of the orientation birefringence; the phase <br>difference from the transmission of the light wave through the polymer is reduced <br>to zero by the use of a proper crystal. The birefringence value, thus obtained, has to <br>be divided by the sample thickness to determine the optical phase difference. There-<br>fore, the thickness has to be measured simultaneously. For practical purposes, the <br>degree of anisotropy is often used. This is the actual birefringence divided by the <br>maximum birefringence of the polymer. The degree of anisotropy, therefore, is no <br>longer dependent on the sample thickness.<br>
+<b></b><br>
+ <b>4.5Temperature Control</b><br>
+The dynamic behavior of an extruder is significantly determined by the temperature <br>control system on the extruder. It is, therefore, important to understand the basic <br>characteristics of the various temperature control systems. Most control systems are <br>closed-loop or feedback systems. The variable to be controlled is measured and this <br>information is sent to a control unit. From the control unit a signal is sent to an <br>actuator that adjusts the process such that the control variable is as close as possi-<br>
+<hr>
+<A name=131></a> <br>
+4.5Temperature Control  <b>117</b><br>
+ble to the desired value, the setpoint. Some systems are non-feedback or open-loop <br>systems. These are used when the effect of the input signal on the process can be <br>accurately predicted. This is generally not the case in extrusion and, as a result, <br>feedback control systems are commonly used in extruders.<br>There are basically two ways to keep the level of a variable within certain limits: the <br>on-off method and the modulating or continuous adjustment method. The on-off con-<br>trol is probably the simplest type of automatic control.<br>
+<b>4.5.1On-Off Control</b><br>
+In on-off control of temperature, the power to the extruder is full-on when the meas-<br>ured temperature is below the setpoint and completely off when the measured tem-<br>perature is above the setpoint. This situation is shown in Fig. 4.22.<br>
+Setpoint<br>
+e<br>
+Temperatur<br>
+Time<br>
+r<br>we<br>Po<br>
+<b> Figure  4.22</b> <br>
+Time<br>
+On-off temperature control<br>
+The temperature in most homes is controlled by thermostats using the on-off princi-<br>ple. On-off control is widely used in the industry to control temperature and other <br>variables. In fact, some missile guidance systems use on-off control, indicating that <br>accurate control in sophisticated systems can be achieved by on-off control. That is, <br>of course, if the system characteristics lend themselves to such type of control.<br>One of the problems of applying on-off control to the heating and cooling of extrud-<br>ers is the significant thermal lag in these machines. There is an inherent delay <br>between the time the controller calls for heat from the heater band, and the time <br>when the heat actually reaches the sensor. The same holds true when the controller <br>turns off. This thermal lag can be several minutes (about 5 minutes for a 90 mm <br>
+<hr>
+<A name=132></a><b>118</b>  <b>4Instrumentation and Control</b><br>
+extruder); the larger the extruder, the longer the thermal lag will be. Putting the <br>temperature sensor close to the heater (shallow well) reduces this thermal lag, but <br>makes the sensed temperature less representative of the temperature of most impor-<br>tance--the temperature of the polymer inside the barrel.<br>The temperature resulting from this type of on-off control will be fluctuating around <br>the setpoint. The frequency and amplitude of the temperature fluctuations will be <br>determined by the thermal lag of the particular machine. A problem of on-off control <br>is the possible effect of process disturbances and electrical noise interference, which <br>can cause the output to cycle rapidly as the temperature crosses the setpoint. This <br>condition can be detrimental to most final control elements such as contactors. To <br>prevent this, an on-off differential or hysteresis is added to the controller function. <br>This function requires that the temperature exceed the setpoint by a certain amount <br>(half the differential) before the output will turn on again. Hysteresis will prevent <br>the output from chattering if the peak-to-peak noise is less than the hysteresis. The <br>amount of hysteresis determines the minimum temperature variation possible. <br>However, process characteristics will add to this differential. Figure 4.23 shows a <br>time-temperature diagram for an on-off controller with hysteresis.<br>
+Setpoint<br>
+Hysteresis<br>
+e<br>
+Temperatur<br>
+Time<br>
+r<br>we<br>Po<br>
+<b> Figure  4.23</b> <br>
+Time/power-temperature diagram <br>
+Time<br>
+for on-off control with hysteresis<br>
+A different representation of the hysteresis curve is shown in the transfer function <br>of Fig. 4.24.<br>The transfer function describes the power-temperature relationship of a controller. <br>When the temperature is ascending, the power is turned off when the temperature <br>exceeds T2; when the temperature is descending, the power is turned on when the <br>
+temperature drops below T1.<br>
+<hr>
+<A name=133></a> <br>
+4.5Temperature Control  <b>119</b><br>
+100%<br>
+r<br>we<br>Po<br>
+T1<br>
+T2<br>
+<b> Figure  4.24</b> <br>
+Temperature<br>
+Transfer function of an on-off <br>
+Hysteresis<br>
+controller<br>
+<b>4.5.2Proportional Control</b><br>
+One of the drawbacks of on-off control is that there are only two power input levels <br>possible: fully on and fully off. In essentially all practical cases, the power level <br>required to maintain a certain temperature will be somewhere between 0 and 100% <br>power. Therefore, application of on-off control will invariably lead to fluctuations of <br>the actual temperature. To avoid this problem, a control system is needed that can <br>adjust the power input level to the exact level required to maintain the temperature <br>at the setpoint. Only then is it possible to maintain a steady temperature.<br>
+<b>4.5.2.1Proportional-Only Control<br></b>The proportional controller allows a continuous adjustment of the power input level <br>(from 0 to 100%) depending on the actual temperature. The range of temperature <br>over which the power is adjusted from 0 to 100% is called the proportional band. The <br>proportional band is usually expressed as a percentage of instrument span and is <br>often centered about the setpoint. Thus, in a controller with a 500°C span, a 5% <br> <br>proportional band would be 25 degrees about the setpoint. Sometimes, the setpoint <br>is located at the upper temperature limit of the proportional band.<br>Figure 4.25 shows the transfer function for a reverse-acting controller.<br>It is called reverse-acting because the output decreases with increasing tempera-<br>ture. If the temperature is below the lower boundary of the proportional band, T1, <br>
+the power to the heaters is on 100%. Above the upper boundary of the proportional <br>band, T2, the power to the heaters is completely off. Within the proportional band, <br>
+the power varies proportionally with temperature, from 100% power at T1 to 0% <br>
+power at T2.<br>
+<hr>
+<A name=134></a><b>120</b>  <b>4Instrumentation and Control</b><br>
+Power<br>
+Proportional band<br>
+100%<br>
+<b> Figure 4.25</b> <br>
+Temperature<br>
+0%<br>
+Transfer function for a reverse-acting <br>
+T1<br>
+T2<br>
+proportional controller<br>
+The proportional band in most controllers is adjustable to obtain stable control <br>under different process conditions. A narrower proportional band would result in a <br>steeper transfer function or power-temperature relationship. In the extreme case <br>when the width of the proportional band is reduced to zero, the proportional con-<br>troller would act simply as an on-off controller. In that case, all the advantages of the <br>proportional control would be lost.<br>The proportional band is often expressed as percent of span in the plastics industry, <br>but it is also expressed as controller gain in other industries. The proportional band <br>width in percent span and controller gain are related inversely:<br>
+ (4.6)<br>
+Thus, reducing the width of the proportional band will increase the gain. A propor-<br>tional band of 5% corresponds to a gain of 20, a proportional band of 4% corresponds <br>to a gain of 25, etc.<br>A block diagram of a proportional control system is shown in Fig. 4.26.<br>
+Setpoint, Ts +<br>
+e<br>
+K1<br>
+Load<br>
+-<br>
+Ta<br>
+Te mperature<br>
+<b> Figure 4.26</b> <br>
+sensor<br>
+Block diagram of a proportional control system<br>
+The control system contains a comparator, which compares the actual temperature, <br>Ta (measured by the temperature sensor), with the desired or setpoint temperature, <br>
+Ts, to provide an error or deviation signal, e. The signal is positive when the process <br>
+is below setpoint, zero when the process is at setpoint, and negative when the pro-<br>cess is above setpoint. The proportional term, K1, gives an output proportional to the <br>
+error:<br>
+ (4.7)<br>
+<hr>
+<A name=135></a> <br>
+4.5Temperature Control  <b>121</b><br>
+When the setpoint is centered in the proportional band, the power is 50% when the <br>error signal is zero, i.e., the process is at setpoint.<br>In a real process, it is rare that the power input required to maintain setpoint tem-<br>perature is exactly 50% of full power. Therefore, the temperature will increase or <br>decrease, adjusting the power level until an equilibrium condition exists. The tem-<br>perature difference between the stabilized temperature and the setpoint is called <br>offset or droop. The amount of offset can be reduced by narrowing the proportional <br>band. However, the proportional band can be narrowed only so far before instability <br>occurs. An illustration of a process coming up to temperature with an offset is shown <br>in Fig. 4.27.<br>
+<b> Figure 4.27</b> <br>
+Process coming up to <br>
+ <br>temperature with an offset<br>
+To understand the mechanism by which offset occurs with a proportional controller, <br>one should look at the controller transfer curve and the process transfer curve at the <br>same time, as in Fig. 4.28.<br>
+Power<br>
+100%<br>
+Power loss curve<br>
+(Process transfer curve)<br>
+Power input curve<br>
+(Controller transfer curve)<br>
+0%<br>
+T1<br>
+T2<br>
+Temperature <b> Figure 4.28</b> <br>
+Controller transfer curve and <br>
+Proportional band<br>
+ <br>process transfer curve<br>
+<hr>
+<A name=136></a><b>122</b>  <b>4Instrumentation and Control</b><br>
+The process transfer curve indicates the power-temperature characteristic of the <br>actual system, i.e., the extruder and its surroundings. This curve indicates how <br>much power is required to maintain a certain temperature level on the machine. The <br>higher the temperature level that needs to be maintained, the more power will <br>be required. For most machines, the relationship between temperature and power <br>requirement will be approximately linear. The power requirement is determined by <br>the heat losses in the system by conduction, convection, and radiation. By improv-<br>ing the thermal insulation of the extruder, the heat losses can be reduced. This will <br>directly affect the process transfer curve; adding insulation will reduce the power <br>requirements at a certain temperature, resulting in a reduced slope of the process <br>transfer curve. Obviously, this will also improve the energy efficiency of the entire <br>process.<br>Figure 4.28 shows the controller transfer curve (power input curve) superimposed <br>on the process transfer curve (power loss curve). The point where the two curves <br>intersect is the temperature where the input power to the heaters is in equilibrium <br>with the power losses. If the point of intersection occurs above the setpoint, there <br>will be a positive droop; if it occurs below the setpoint, the droop will be negative. <br>From Fig. 4.28 it is now clear how the offset can be eliminated without changing the <br>width of the proportional band. This is done by shifting the entire proportional band <br>to a higher or lower temperature. Figure 4.29 illustrates the effect of resetting the <br>proportional band.<br>This resetting can be done manually or automatically. With manual reset, a potentio-<br>meter is used to electrically shift the proportional band. The amount of shifting has <br>to be done in small increments until the controller power output matches the pro-<br>cess power demand as setpoint temperature.<br>
+Power<br>
+100%<br>
+Resetting proportional band<br>
+Power loss curve<br>
+Power input curve<br>
+0%<br>
+T1<br>
+T2<br>
+Temperature<b> Figure 4.29</b> <br>
+Effect of resetting the proportional <br>
+Proportional band<br>
+band<br>
+<hr>
+<A name=137></a> <br>
+4.5Temperature Control  <b>123</b><br>
+<b>4.5.2.2Proportional and Integral Control<br></b>Automatic reset is done by using an electronic integrator to perform the reset func-<br>tion. The deviation or error is integrated with respect to time, and the result is added <br>to the deviation signal to move the proportional band. The block diagram of propor-<br>tional control with automatic reset is shown in Fig. 4.30.<br>
+K2  edt<br>
+Setpoint +<br>
+e<br>
++<br>
++<br>
+K1<br>
+Load<br>
+-<br>
+Temperature<br>
+<b> Figure 4.30</b> <br>
+sensor<br>
+Block diagram for PI control system<br>
+The output now becomes:<br>
+ (4.8)<br>
+The K1 term operates exactly as in a simple proportional controller. The integration <br>
+term K2 has to be made long enough so that it will be negligible at the frequencies <br>
+when the control loop has 180 degrees phase shift. These are the critical frequen-<br>cies for loop stability. If this is done, the K2 term will not have an effect on the stabil-<br>
+ity of the loop, but it will accomplish its task, i.e., eliminate the offset.<br>The integrator keeps adjusting the level of the proportional band until the deviation <br>is zero. When this condition is achieved, the input to the integrator becomes zero <br>and its output stops changing. At this point, the correct amount of reset is held by <br>the integrator. If the process heat requirements should change, a deviation would <br>again occur, which the integrator would integrate, and corrective action would be <br>applied. This corrective action has to be applied rather slowly to avoid oscillations. <br>The automatic reset term can be thought of as a slow gain, as opposed to the propor-<br>tional term, which can be considered a fast gain.<br>One problem that can occur in a proportional controller with automatic reset is <br>"reset windup." This occurs when the integrator has acted on the error signal when <br>the temperature is outside the proportional band. The resulting large "woundup" <br>output of the integrator causes the proportional band to move so far that the set-<br>point is outside the band. The temperature must pass the setpoint before the con-<br>troller output will change. As the temperature crosses the setpoint, the deviation <br>signal changes sign and the integrator output starts to decrease or unwind. The <br>result can be a large temperature overshoot. This can be prevented by stopping the <br>integrator from acting if the temperature is outside the proportional band. This <br>function is referred to as reset windup inhibit or reset inhibit.<br>
+<hr>
+<A name=138></a><b>124</b>  <b>4Instrumentation and Control</b><br>
+One characteristic of all proportional plus integral (PI) controllers is that the tem-<br>perature often overshoots the setpoint on start-up. This occurs because the auto-<br>matic reset begins acting when the temperature reaches the lower boundary of the <br>proportional band. As the temperature reaches the setpoint, the reset action has <br>already moved the proportional band higher, causing excess heat output. As the <br>temperature exceeds the setpoint, the sign of the deviation signal reverses and the <br>integrator brings the proportional band back to the position required to eliminate <br>the offset. This situation is illustrated in Fig. 4.31.<br>
+Reset action stops<br>
+a<br>ture<br>
+Temper<br>
+Reset action starts<br>
+Proportional band<br>
+<b> Figure 4.31</b> <br>
+Temperature during start-up with <br>
+Time<br>
+PI   control<br>
+<b>4.5.2.3Proportional and Integral and Derivative Control<br></b>One drawback of the automatic reset is its relatively slow response. The response <br>time can be reduced by addition of a second corrective term, which acts on the rate <br>of change in temperature. A block diagram of a proportional plus integral plus deri-<br>vative (PID) control system is shown in Fig. 4.32.<br>
+K2  edt<br>
+Setpoint +<br>
+e<br>
++<br>
++<br>
+K<br>
+Load<br>
+1<br>
+-<br>
++<br>
+K3de/dt<br>
+Temperature<br>
+<b> Figure 4.32</b> <br>
+sensor<br>
+Block diagram of PID control system<br>
+The second corrective term yields a signal proportional to the rate of change in the <br>error signal. This rate of change is the derivative of the measured temperature with <br>respect to time, thus the term derivative control. The output of the PID control is:<br>
+ (4.9)<br>
+<hr>
+<A name=139></a> <br>
+4.5Temperature Control  <b>125</b><br>
+The derivative control comes into action when a transient occurs. This action is <br>immediate; it does not wait until the error builds up, but it responds directly to the <br>rate of change of the error. Corrective action will be taken in the shortest possible <br>time, and the magnitude of the temperature deviation, caused by an upset, will be <br>greatly reduced.<br>Derivative control helps to prevent overshooting or undershooting the setpoint. It <br>is an anticipatory function that adjusts the controller output, in advance, to antici-<br>pated needs. This reduces the time lag it takes for the controller to respond to a <br>change in the process. Derivative control is more important on machines with a long <br>thermal lag. Therefore, derivative control is important for large extruders. Small <br>extruders, with their inherently shorter thermal lag, may not benefit much from a <br>derivative type of temperature control.<br>A drawback of the rate control is the fact that it has a destabilizing influence on the <br>control loop. Therefore, it has to be sized carefully to maintain adequate stability in <br>the control loop. In spite of this, the derivative control can generally yield an <br>improvement in response time by a factor of two to four.<br>
+<b>4.5.2.4Dual Sensor Temperature Control<br></b>A few commercial temperature control systems are based on dual input from two <br>temperature sensors. One temperature sensor is located in a deep well and meas-<br>ures temperature close to the polymer. The other temperature sensor is located in a <br>shallow well and measures temperature close to the heater/cooler. The dual sensor <br>temperature control can combine the advantages of deep-well-only control and shal-<br>low-well-only control, but can largely eliminate the drawbacks of these types of con-<br>trol.<br>The deep-well-only control is well established and reliable; however, its main draw-<br>back is slow response to changes in external conditions, such as changes in ambient <br>temperature, heater line voltage variations, changes in cooling water temperature, <br>etc. On the other hand, the response to changes in internal conditions is quite rapid. <br>Examples are changes in screw speed, viscosity, changes in the polymer, tempera-<br>ture changes in the polymer, etc. The shallow-well-only control has the advantage of <br>being able to respond quickly to changes in external conditions but slowly to changes <br>in internal conditions.<br>One dual sensor control system uses a weighted average of the signals from the two <br>sensors. By doing this, the advantages of both deep-well and shallow-well can be <br>enjoyed to some extent; however, the same is true about the disadvantages. Another <br>system uses two different temperature control loops. The first one uses only the <br>deep-well sensor and controls the power to the heater. The second loop uses only the <br>shallow-well sensor. This is a cascade loop. It does not control the heaters directly, <br>but acts on the first loop in such a manner as to keep the temperature at the deep <br>
+<hr>
+<A name=140></a><b>126</b>  <b>4Instrumentation and Control</b><br>
+well at the setpoint. Both of these dual sensor temperature control systems have <br>been patented [38, 39].<br>
+<b>4.5.3Controllers</b><br>
+<b>4.5.3.1Temperature Controllers<br></b>There are two types of temperature controllers: analog and digital. An analog con-<br>troller contains a number of discrete components--resistors, capacitors, integrated <br>circuits, operational amplifiers--and performs its control algorithm through these <br>components. In a digital controller, a microprocessor replaces the discrete compo-<br>nents of the analog unit with integrated circuit chip logic. It takes analog input, <br>converts it into a digital signal, and then performs its control algorithm through a <br>stored computer program, which the microprocessor executes.<br>Microprocessor-based controllers offer great flexibility in that controller function <br>can be changed readily by simply changing a few steps in the program. Thus, the <br>controller function can be modified by changing the software without having to <br>make any modifications to the hardware.<br>The temperature controller is not capable of handling the high currents required to <br>power the heater bands. For this reason, a power controller is linked between the <br>temperature controller and the heater band. The temperature controller dictates to <br>the power controller how much power to supply to the heaters.<br>
+<b>4.5.3.2Power Controllers<br></b>A relatively inexpensive, time-proportioning power controller is the mercury con-<br>tactor. With this switch, there is no zero-crossing detection, so there will be some <br>noise in the circuit when the contactor is turned on and off.<br>This problem can be avoided by using a zero-crossover-fired power controller. This is <br>a time-proportioning system controlled by the temperature controller. Being zero-<br>voltage fired, this controller's circuit is free of RF noise. In other words, the switch-<br>ing occurs when no voltage is on the line.<br>A disadvantage of the contactors is their limited life span; they are usually replaced <br>after one million operations. In a typical operation, contactors are operated once <br>every 20 seconds. This gives a reasonable compromise between temperature ripple <br>and load life. Operation every 20 seconds means 4320 operations per day, or one <br>million operations in about 230 days. One will have to replace the contactor every <br>230 days. If there are eight controllers per machine, the machine will be down about <br>once a month on the average.<br>A newer type of power controller is the true proportional power controller--also <br>known as current proportioning controller or phase-angle-fired proportional con-<br>
+<hr>
+<A name=141></a><IMG src="Polymer Extrusion by Chris Rauwendaal-141_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_48.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_49.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_50.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_51.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_52.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_53.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_54.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_55.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_56.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_57.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_58.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_59.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_60.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-141_61.jpg"><br>
+ <br>
+4.5Temperature Control  <b>127</b><br>
+troller. The term "true proportional" means that the power to the heater is adjusted <br>on a continuous basis. This provides a smooth, stepless output of power. This is <br>particularly useful for extruder dies where rapid and smooth response is very <br>important. True power proportioning is less wearing on the heater bands because it <br>eliminates the on-off cycling.<br>True proportional control is obtained with solid-state switching of loads through <br>SCRs, triacs, and similar solid-state devices. This allows a reduction of cycle time to <br>the millisecond level. If the cycle time is reduced to one-half the power line period <br>(8.3 ms for 60 Hz), then the proportioning action is referred to as stepless control or <br>phase-fired control. A 50% phase-fired output is shown in Fig. 4.33.<br>
+<b> Figure 4.33</b> <br>
+A 50% phase-fired output<br>
+It is clear that SCR power controllers can also be used for time proportioning power <br>control by increasing the cycle time. However, this type of use does not utilize the <br>inherent advantages of the SCR power controller.<br>In a true proportional controller, the SCR switches extremely fast--in the order of <br>half a microsecond. This rapid switching causes radio frequency interference (RFI) <br>to be generated on the power lines. This can cause interference with nearby com-<br>puter and communications equipment. A further problem is that the power drawn <br>from the power lines has a severely distorted wave shape. This can result in extra <br>charges from the power company.<br>These problems can be avoided by the use of RFI-free or zero-crossing SCR circuitry. <br>A characteristic very much like time proportioning is used, but the power is always <br>turned on and off at the instant when the power line voltage is zero, as shown in <br>Fig. 4.34.<br>
+Time<br>
+<b>Figure 4.34</b>RFI-free power proportioning<br>
+<hr>
+<A name=142></a><b>128</b>  <b>4Instrumentation and Control</b><br>
+Another type of power controller is the solid-state relay power controller. This con-<br>troller is reliable and inexpensive. It is zero-crossover fired, thus it does not gener-<br>ate RFI. However, to use the controller at its rated current level, it must be provided <br>with a good heat sink.<br>
+<b>4.5.3.3Dual Output Controllers<br></b>Controlling the temperature of an extruder may require adding heat to the machine <br>by the barrel heaters or taking heat out of the machine by cooling. Since a substan-<br>tial amount of heat is generated internally in the extrusion process, cooling is often <br>required to maintain the desired process temperature. For this reason, it is impor-<br>tant to utilize dual output temperature controllers. These are controllers that control <br>both the heating as well as the cooling process.<br>Thus, if proportional temperature control is used in heating, then proportional tem-<br>perature control should also be used in cooling. This would seem to be an obvious <br>requirement. However, a large number of commercial extruders come standard <br>with PID heating control and on/off cooling control. The fact that such extruders are <br>built and the fact that customers buy such extruders illustrates that concern about <br>precise temperature control is not very widespread. Such extruders would only be <br>appropriate if the process is designed such that extruder cooling is not used in nor-<br>mal operations--a situation that is unlikely to occur in actual practice. Dual output <br>controllers, nowadays, are only slightly more expensive than single output control-<br>lers. Thus, cost is only a minor consideration.<br>
+<b>4.5.4Time-Temperature Characteristics</b><br>
+Thus far, the discussion has focused on the various types of temperature controllers. <br>However, the real issue is how the actual temperature of the extruder will change as <br>a result of the action of the temperature controller. In order to determine this, one <br>has to consider the thermal characteristics of the actual system, in this case the <br>extruder.<br>
+<b>4.5.4.1Thermal Characteristics of the System<br></b>The thermal characteristics of the system determine how the output signal (tempe-<br>rature) varies with changes in input (heat). One of the simplest and most common <br>methods to determine the system characteristics is to measure response of tempera-<br>ture to a step change in heating power; see Fig. 4.35.<br>This temperature/time curve is often referred to as the process reaction curve or <br>simply the response curve.<br>
+<hr>
+<A name=143></a> <br>
+4.5Temperature Control  <b>129</b><br>
+er<br>
+Pow<br>
+Zmax<br>
+Time<br>
+t d<br>
+t c<br>
+e<br>
+Tmax<br>
+Temperatur<br>
+Time<br>
+<b>Figure 4.35</b>Temperature response to step change in heater power<br>
+From the response curve, several parameters can be established that are important <br>in understanding the behavior of the system. The dead time (td) is the time to the <br>
+intersection of the maximum temperature gradient. The constant Ks indicates the <br>
+ratio of maximum temperature rise to the step change in power. (KS= Tmax/Zmax) <br>
+The time constant (tc) is the maximum change in temperature divided by the maxi-<br>
+mum temperature gradient.<br>The dead time in extruders can range from about 1 to 5 minutes. This constitutes <br>one of the main problems in temperature control, because this means that the effect <br>of a change in power input level is not felt until after at least 1 minute. This thermal <br>lag time is influenced by the depth of the temperature sensor, the thermal conduc-<br>tivity of the barrel, and the design of the barrel heaters. A typical time constant in <br>an extruder can range from 30 to 120 minutes. This value depends on the heating <br>capacity and the specific heat and mass of the extruder barrel.<br>Although these parameters do not fully describe the system, they can be used to <br>make approximate predictions of the degree of difficulty in temperature control [41].<br>td/tc  0.1 easy control<br>0.1 &lt; td/tc &lt; 0.3 reasonable control<br>td/tc  0.3 difficult control<br>
+<hr>
+<A name=144></a><b>130</b>  <b>4Instrumentation and Control</b><br>
+<b>4.5.4.2Modeling of Response in Linear Systems<br></b>In a linear control system, the relationship between input y and output x can be <br>described by a differential equation:<br>
+ (4.10)<br>
+The order of the differential equation determines the order of the control system. <br>Thus, a first order control system is described by:<br>
+ (4.11)<br>
+After a long time (t  ), the following relationship is valid:<br>
+ (4.12)<br>
+Thus, a change in input will produce a proportional change in the output. If tc = A1/<br>
+A0 and Ks = 1/A0, Eq. 4.11 can be written as:<br>
+ (4.13)<br>
+The solution to this differential equation, when the input variable is given a step <br>change of 1, is the well-known exponential relationship of a first order control sys-<br>tem.<br>
+ (4.14)<br>
+where tc is the system time constant and Ks the system gain. Various step response <br>
+functions are shown in Fig. 4.36. Figure 4.36(a) shows a first order system, <br>Fig. 4.36(b) a higher order system, and Fig. 4.36(c) a first order system with lag <br>time.<br>In reality, most systems do not behave in a truly linear fashion, but the assumption <br>of linearity can usually be justified over a small region near the operating point <br> <br>(linearization).<br>If a mathematical description of the system characteristics is not possible, the sys-<br>tem parameters have to be determined experimentally. If the step response function <br>is of a higher order than one (see Fig. 4.36[b]), the response can be approximated <br>by a first order response function with lag time.<br>
+<hr>
+<A name=145></a> <br>
+4.5Temperature Control  <b>131</b><br>
+X<br>
+t c<br>
+a.<br>
+t<br>
+X<br>
+t c<br>
+b.<br>
+t<br>
+t d<br>
+X<br>
+t c<br>
+c.<br>
+<b> Figure 4.36(a­c)</b> <br>
+t<br>
+t d<br>
+Various step response functions<br>
+If the following control is now considered (see Fig. 4.37):<br>
+Disturbance<br>
+Kp<br>
+z<br>
+Ks.tc<br>
+Setpoint, w +<br>
+Proportional<br>
+-y<br>
+x<br>
+System<br>
+control<br>
+-<br>
+Output<br>
+x<br>
+<b>Figure 4.37</b>First order control system<br>
+The response function of the system can be described by:<br>
+ (4.15)<br>
+For the controller:<br>
+ (4.16)<br>
+If Eqs. 4.15 and 4.16 are combined, the equation for the closed-loop control system <br>is obtained:<br>
+<hr>
+<A name=146></a><b>132</b>  <b>4Instrumentation and Control</b><br>
+ (4.17)<br>
+When the effect of a disturbance (step change) at a constant setpoint is considered <br>(z = 1 and w = 0):<br>
+ (4.18)<br>
+After a very long time (t  ), the output will attain the following value:<br>
+ (4.19)<br>
+The time constant of the closed-loop system is:<br>
+ (4.20)<br>
+Thus, one can conclude that the remaining deviation x (t  ) reduces with the <br>increased proportionality constant Kp of the controller. The time constant also <br>
+reduces with increased Kp.<br>A step change in setpoint (w = 1 and z = 0) is described by:<br>
+ (4.21)<br>
+After a very long time (t  ), the output will attain the following value:<br>
+ (4.22)<br>
+Thus, the change in output increases with the proportionality constant Kp of the <br>
+controller and the system gain Ks. In this example, it was assumed that there was no <br>
+dead time in the system. If a dead time is present, then output fluctuations will <br>occur upon changes in input. In this case, the control loop can become unstable, <br>depending on the actual dead time and the proportionality constant of the control-<br>ler. The effect of changes in setpoint or a disturbance is more difficult to analyze in <br>higher order systems. The controller parameters have to be selected such that the <br>control loop is stable in all cases.<br>
+<hr>
+<A name=147></a> <br>
+4.5Temperature Control  <b>133</b><br>
+<b>4.5.4.3Temperature Characteristics with On-Off Control<br></b>The approximate time/temperature characteristics of a simple on-off controller are <br>relatively easy to visualize. If a dual output on-off control is considered, then a typi-<br>cal temperature/time curve can be shown as in Fig. 4.38.<br>
+Temperature<br>
+t d<br>
+Tc1<br>
+T<br>
+T<br>
+h<br>
+c0<br>
+Setpoint<br>
+Ts<br>
+Tn<br>
+Th0<br>
+T<br>
+T<br>
+c<br>
+h1<br>
+t d<br>
+Time<br>
+r<br>we<br>
+Heating<br>
+Po<br>
+0<br>
+Cooling<br>
+<b> Figure 4.38</b> <br>
+Typical temperature/time curve for a <br>
+Time<br>
+dual output on-off control<br>
+If the system heats up from ambient temperature, the heating power will be turned <br>off when the temperature reaches:<br>
+ (4.23)<br>
+where Ts is the symmetrically located setpoint and Tn is the neutral temperature <br>
+zone in which no heating or cooling takes place, also referred to as a dead band. <br>If there is a dead time td in the system, the temperature will continue to rise for this <br>
+length of time. When the temperature reaches tc1, cooling is turned on. However, if <br>
+the dead time td is long enough, the actual temperature can continue to rise.<br>
+ (4.24)<br>
+where Th is the hysteresis loop width.<br>When the dead time has elapsed, the temperature will start to drop. The cooling will <br>be turned off when the temperature drops below Tco.<br>
+ (4.25)<br>
+<hr>
+<A name=148></a><b>134</b>  <b>4Instrumentation and Control</b><br>
+The temperature will continue to drop for a time period of td. When the temperature <br>
+drops below th1, the heating is turned on again. After another time period of td, the <br>
+temperature will start to increase again. This temperature cycle will repeat itself <br>if the thermal conditions of the process remain the same. The maximum tempera-<br>ture Tmax is:<br>
+ (4.26)<br>
+where vh is the rate of temperature rise upon heating.<br>The minimum temperature Tmin is:<br>
+ (4.27)<br>
+where vc is the rate of temperature drop upon cooling.<br>The maximum temperature swing Tmax is:<br>
+ (4.28)<br>
+If a typical value of the neutral zone is 10°C, a typical value of the dead time 2 min-<br>utes, and the rate of temperature change 6°C/min, then the maximum temperature <br>swing becomes 35°C! This assumes that the rate of temperature change in heating <br>is the same as in cooling (vh = vc). These values are typical of a dual output on-off <br>
+controller on an extruder. Because of the resulting large fluctuations in tempera-<br>ture, on-off control is not very desirable if accurate temperature control is required.<br>If the neutral zone is wide enough, the temperature may not reach the point at <br>which the cooling is turned on. Cooling can be avoided if:<br>
+ (4.29)<br>
+This situation is portrayed in Fig. 4.39.<br>
+Temperature<br>
+t d<br>
+Th0<br>Th1<br>
+t d<br>
+Time<br>
+er<br>
+Pow<br>
+0<br>
+Time<br>
+<b>Figure 4.39</b>Temperature/time curve without cooling<br>
+<hr>
+<A name=149></a> <br>
+4.5Temperature Control  <b>135</b><br>
+In this case, the maximum temperature swing is:<br>
+ (4.30)<br>
+The rate of temperature change upon cooling v 1c is much slower in this case because <br>
+it is passive cooling. In other words, the barrel cooling has not been activated; <br>the temperature drops simply as a result of heat losses in the system. If Th = 1°C, <br>
+t<br>
+1<br>
+d = 2 min, vh = 6°C/min, and vc  = 2°C/min, the maximum temperature swing is <br>
+Tmax = 17°C. Thus, the amplitude of the temperature fluctuation is cut in half as <br>
+compared to the first case. However, even a temperature fluctuation of 17°C would <br>still be considered unacceptable in most extrusion operations.<br>
+<b>4.5.5Tuning of the Controller Parameters</b><br>
+<b>4.5.5.1Performance Criteria<br></b>The proper selection, design, or tuning of controllers requires controller criteria <br>that are most appropriate for the particular application. Some response criteria <br>include overshoot, decay ratio, rise time, response time, frequency of oscillation, <br>phase and gain margin error integrals, etc. [42­48]. Three common error integrals <br>are integral of square error (ISE), integral of absolute error (IAE), and integral time <br>and absolute error (ITAE). Some of these criteria are illustrated in Fig. 4.40 showing <br>the response of a control system to a step change in input.<br>
+t c<br>
+A<br>
+R+5%<br>
+B<br>
+R<br>
+R-5%<br>
+Overshoot = A/R<br>Decay ratio = B/A<br>Rise time = tr<br>Response time = t95<br>
+Response<br>
+Frequency =  = 1/tc<br>
+t r<br>
+t95<br>
+Time<br>
+<b>Figure 4.40</b>Performance criteria from a response curve<br>
+<hr>
+<A name=150></a><b>136</b>  <b>4Instrumentation and Control</b><br>
+<b>4.5.5.2Effect of PID Parameters<br></b>Tuning of a PID controller should ideally lead to values of the P, I, and D terms of <br>the controller that result in the most favorable actual control performance. The <br>P-term is described by the proportional gain Kp or the proportional bandwidth Xp. <br>
+The effect of changing the proportional band is shown in Fig. 4.41.<br>
+Band too wide<br>
+e<br>
+Band OK<br>
+Temperatur<br>
+Band too narrow<br>
+Reduced heat demand<br>
+<b> Figure 4.41</b> <br>
+Time<br>
+Effect of bandwidth setting Xp<br>
+A narrow proportional band can result in strong oscillations, while a wide propor-<br>tional band will result in a large offset, in the absence of reset function. The I-term <br>provides the reset function and is described by the integration time constant ti or <br>
+reset time constant. The effect of different values of the reset time constant is shown <br>in Fig. 4.42.<br>
+No reset (ti=oo)<br>
+e<br>
+ti too long<br>
+ti correct<br>
+Setpoint<br>
+Temperatur<br>
+ti too short<br>
+Reduced heat demand<br>
+<b> Figure 4.42</b> <br>
+Time<br>
+Effect of reset time constant, ti<br>
+When the reset time constant is too long, the process will come back to the setpoint <br>very slowly. On the other hand, when the reset time constant is too short, oscilla-<br>tions will occur. The reset time constant is generally considered optimum when the <br>temperature returns to setpoint as rapidly as possible without overshoot.<br>The derivative term is characterized by the derivative time constant td or rate time <br>
+constant. The effect of different rate time constants is shown in Fig. 4.43.<br>
+<hr>
+<A name=151></a> <br>
+4.5Temperature Control  <b>137</b><br>
+td too short<br>
+e<br>ur<br>at<br>
+td correct<br>
+er<br>mp<br>
+Setpoint<br>
+Te<br>
+td too long<br>
+Reduced heat demand<br>
+<b> Figure 4.43</b> <br>
+Time<br>
+Effect of derivative time constant, td<br>
+When the rate time constant is too long, the temperature changes too rapidly, result-<br>ing in overshoot and oscillations.<br>When the rate time constant is too short, the temperature will return to setpoint too <br>slowly. The correct rate time constant will return the temperature to setpoint with a <br>minimum of oscillations.<br>
+<b>4.5.5.3Tuning Procedure When Process Model Is Unknown<br></b>The tuning technique that is applied will depend on whether or not the process <br>model is known. When the process model is not known, the most widely used tun-<br>ing techniques incorporate the ultimate-period method, the reaction-curve method, <br>and various search methods.<br>The ultimate-period method, also called Ziegler-Nichols method [42], starts by <br>obtaining dynamic response data. These data are obtained by tuning out the integral <br>and derivative actions of the controller and using only the proportional control <br>mode. The proportional gain is gradually increased until the closed-loop system is <br>forced to cycle continuously at the point of instability. The proportional gain at the <br>point of continuous cycling (ultimate gain) and the period of oscillation (ultimate <br>period) identify the frequency response of the open-loop system at one point. Re- <br>commended controller settings are shown in Table 4.4.<br>
+<b>Table 4.4</b>Recommended Controller Settings Based on Ultimate Gain Ku and Ultimate Period tu<br>
+Control<br>
+Criterion<br>
+Gain<br>
+Reset time<br>
+Rate time<br>
+P<br>
+1/4 decay<br>
+0.5Ku<br>
+­<br>
+­<br>
+PI<br>
+1/4 decay<br>
+0.45Ku<br>
+0.833tu<br>
+­<br>
+PID<br>
+1/4 decay<br>
+0.6Ku<br>
+0.5tu<br>
+0.125tu<br>
+PID<br>
+Some overshoot<br>
+0.33Ku<br>
+0.5tu<br>
+0.33tu<br>
+PID<br>
+No overshoot<br>
+0.2Ku<br>
+0.33tu<br>
+0.5tu<br>
+The reaction-curve method is based on the open-loop response of the process to a <br>step input. This response curve can be used to derive the dynamic characteristics of <br>the process. If the process can be described by a first-order lag and dead time, the <br>controller setting can be calculated.<br>
+<hr>
+<A name=152></a><b>138</b>  <b>4Instrumentation and Control</b><br>
+The controller is placed on manual control and a step change S is applied. With a <br>recorder, note the size of the step change and time of application. The reaction curve <br>should be approximately as shown in Fig. 4.44.<br>
+S<br>
+Maximum slope g<br>
+Process variation<br>
+max<br>
+<b> Figure 4.44</b> <br>
+Time<br>
+Typical reaction curve<br>
+Determine the rate of change per unit step change in the manipulated variable, <br>g1 = gmax/S. The recommended controller parameter settings, based on the quarter-<br>
+decay performance criterion, are shown in Table 4.5.<br>
+<b>Table 4.5</b>Recommended Controller Settings Based on Reaction Curve Process Test<br>
+Control<br>
+Gain<br>
+Reset time<br>
+Rate time<br>
+P<br>
+1/g1t1<br>
+­<br>
+­<br>
+PI<br>
+0.9/g1t1<br>
+3t1<br>
+­<br>
+PID<br>
+1.2/g1t1 to 2/g1t1<br>
+2.5t1 to 2t1<br>
+0.5t1 to 0.3t1<br>
+The two previous tuning techniques require a reasonably detailed control-loop ana-<br>lysis. In practice, many controllers are tuned by trial-and-error methods based on <br>process experience. Both the Ziegler-Nichols method and the reaction-curve method <br>are based on the assumption that the disturbances enter the process at one particu-<br>lar point. These methods, therefore, do not always give satisfactory results. In these <br>cases, the final adjustments must be made by trial-and-error search methods.<br>
+<b>4.5.5.4Tuning Procedure When Process Model Is Known<br></b>Various techniques are available for the experimental determination of process <br>model. Astron [77] and Eykhoff [78] have given a survey of different identification <br>techniques. The most common technique is to apply a step or impulse perturbation <br>and to evaluate the transfer function from the resulting transient response. A more <br>sophisticated technique is the stochastic identification technique. In this technique, <br>
+<hr>
+<A name=153></a> <br>
+4.5Temperature Control  <b>139</b><br>
+input variations of a known random form are applied. The similar statistical proper-<br>ties in input and output variables are then correlated, eventually yielding the pro-<br>cess transfer function. The transfer function is the ratio of the Laplace transform of <br>the responding variable (output) to the Laplace transform of the disturbing variable <br>(input).<br>The experimental determination of the process model requires good instrumenta-<br>tion with sufficient dynamic response. Once the transfer function is known, the <br>tuning can be based on Bode plots, Nyquist diagrams, or analytical methods. Bode <br>plots and Nyquist diagrams represent the transient performance characteristics <br>based on the open-loop frequency-response function. The logarithmic or Bode plot <br>shows how the phase angle and the magnitude of the direct-transfer function depend <br>on the frequency. In the polar or Nyquist diagram, the magnitude and phase angle <br>of the direct-transfer function are plotted as a vector with the frequency as a para-<br>meter. The normal design criterion for control systems using the frequency-response <br>approach is the specification of the gain margin and the phase margin of the open-<br>loop system. Common criteria are 30 degrees for phase margin and 1.7 to 3 for <br>the gain margin. The tuning or design of the controller is accomplished by adding <br>the controller frequency-response characteristics to the system characteristics to <br>achieve the desired phase and gain margins for the combined system.<br>Analytical techniques generally involve two areas. The first is the direct solution of <br>the system differential equations in the Aime domain, usually by state variables. <br>The second area is optimization of a specific performance criterion. The criteria for <br>optimization by analytical techniques usually involve minimum response time or an <br>integral time-cost function.<br>
+<b>4.5.5.5Pre-Tuned Temperature Controllers<br></b>Several controller manufacturers supply temperature controllers where the para-<br>meters are factory-tuned and non-adjustable. The parameter settings are based on <br>long-term experience with temperature controllers in extrusion applications. This is <br>possible because the thermal characteristics of most extruders are rather similar. <br>Advantages of this approach are easier installation and less chance of controller <br>tampering by unqualified personnel.<br>Controllers with non-adjustable parameters are not well suited to extrusion opera-<br>tions where changes in temperature are very rapid (water cooling, blown film, etc.) <br>or very slow combined with long dead times, as may occur in very large machines. <br>Typical controller settings for pre-tuned controllers are shown in Table 4.6.<br>In Table 4.6, Xp is the proportional band, ti the reset (integrating) time constant, and <br>
+td the rate (derivative) time constant.<br>
+<hr>
+<A name=154></a><b>140</b>  <b>4Instrumentation and Control</b><br>
+<b>Table 4.6</b>Typical Parameters for Pre-Tuned Controllers<br>
+Control<br>
+Non-adjustable<br>
+Adjustable<br>
+PD<br>
+Xp =  5%<br>
+Xp =  0­10%<br>
+td  =  0.5 min.<br>
+PID<br>
+Xp =  8%<br>
+Xp =  0­10%<br>
+ti  =  8 min.<br>td  =  0.5 min.<br>
+<b>4.5.5.6Self-Tuning Temperature Controllers<br></b>Self-tuning temperature controllers are also available in extrusion [49]. They have <br>become commercially available since around 1982. These controllers are micropro-<br>cessor-based; they are programmed to detect certain process conditions and adjust <br>the controller parameters when necessary. These adjustments are made internally <br>by the controller itself without the help of an operator.<br>The terminology used to describe these controllers is rather confusing. They are <br>referred to as adaptive-tuning controllers, self-tuning controllers, and automatic-<br>tuning controllers, but these terms mean different things to different suppliers. <br>Clearly, the ability of the controller to re-tune its parameter settings will be only as <br>good as the software. Some controllers tune their parameter settings during start-<br>up, but do not re-tune when operating conditions have been reached. Other cont-<br>rollers tune their parameter settings during start-up, but continue to re-tune--if <br>deemed necessary by the internal software--once operating conditions have been <br>reached. Again, other controllers require a predetermined process disturbance other <br>than start-up, to tune the controller parameters; they may or may not re-tune when <br>operating conditions have been reached.<br>To properly evaluate the goodness of the self-tuning controller, one would have to <br>know the details of the internal controller software. However, suppliers of self-tun-<br>ing controllers are not likely to give out such information because the software is <br>the main factor that sets one controller apart from another. Thus, the details of the <br>software will likely be treated as proprietary information. Therefore, the best method <br>to evaluate a self-tuning controller is to install the controller on an extrusion line <br>and closely monitor the actual performance.<br>
+<b></b><br>
+ <b>4.6Total Process Control</b><br>
+There is a definite trend in extrusion towards total process control. In a total extru-<br>sion control system, temperature measurement and control are tied in with pressure <br>control, thickness gauging, motor load and speed, and possibly other process func-<br>
+<hr>
+<A name=155></a> <br>
+4.6Total Process Control  <b>141</b><br>
+tions. Between the extremes of simple discrete temperature control and total pro-<br>cess control, there are many intermediate levels of controls. Melt temperature con-<br>trol systems have been used for quite some time already. In this type of control, the <br>settings of the first two or three zones closest to the point of melt temperature meas-<br>urement are continuously adjusted to maintain a constant melt temperature. The <br>temperature settings of the zones are changed automatically by a cascade-type con-<br>trol system. Only low frequency changes in melt temperature can be controlled in <br>this fashion because the response of the barrel temperature zones to changes in <br>setpoint is quite slow.<br>Melt pressure control systems are also quite common in extrusion. The screw speed <br>is varied continuously in order to maintain a constant pressure. Newer commercial <br>systems incorporate in one microprocessor-based unit combined control of melt <br>temperature, melt pressure, extrudate thickness and/or width, and possibly other <br>process variables. Other control systems are geared towards total plant control. In <br>addition to the regular extrusion control, such a control system can handle upstream <br>raw-material handling, metering of regrind and/or additives, auxiliary extruders in <br>a coextrusion system, drive and temperature control on a gear pump, biaxial orien-<br>tation in a tenter frame, in-line coating of the extruded web, tension control, corona <br>treating stations, slitting, rewinding, etc.<br>
+<b>4.6.1True Total Extrusion Process Control</b><br>
+Concentrating on the control of the extrusion process, it is clear from the literature <br>[54­68] that true total extrusion process control is quite complicated and has not <br>been fully achieved in practice. Under true total process control, the process is con-<br>sidered a multivariable system and the interaction between the variables is known <br>and fully taken into account in the control scheme. One can therefore assume that <br>many commercial microprocessor-based controllers that control melt temperature, <br>pressure, and extrudate dimensions are most likely built with more or less inde-<br>pendent control loops, each controlling only one variable. These controllers more or <br>less consolidate multiple discrete controllers into one convenient package without <br>major changes to the individual control characteristics. This may reduce cost but <br>does not necessarily improve overall control performance.<br>The first requirement in the development of a true extrusion process control is a <br>dynamic process model. The goodness of the process control will depend very <br>strongly on the accuracy of the process model. However, obtaining a good dynamic <br>process model is quite complicated in practice. The dynamic process model can, in <br>principle, be derived from extrusion theory, and various attempts in that direction <br>have indeed been made [50­54]. As will be discussed in Chapter 7, extrusion theo ry <br>to date has not been developed to a point that the entire process can be predicted <br>
+<hr>
+<A name=156></a><b>142</b>  <b>4Instrumentation and Control</b><br>
+with a sufficient degree of accuracy. Also, the theoretical prediction of extruder per-<br>formance requires a substantial amount of computation. Thus, following the phe-<br>nomenological approach, to derive a dynamic process model from extrusion theory <br>is likely to be quite complex, relatively inaccurate, and requires much computation. <br>However, it is possible that a simplified theoretical model in combination with <br>experimental data could yield an accurate dynamic process model.<br>Most work on the development of dynamic process models has been empirical; <br>this work is usually referred to as process identification. As mentioned earlier, two <br>classes of empirical identification techniques are available: one uses deterministic <br>(step, pulse, etc.) functions, the other stochastic (random) identification functions. <br>With either technique, the process is perturbed and the resulting variations of the <br>response are measured. The relationship between the perturbing variable and the <br>response is expressed as a transfer function. This function is the process model. <br>Empirical identification of process models by the deterministic method has been <br>reported by various workers [55­58]. A drawback of this method is the difficulty in <br>obtaining a measurable response while restricting the process to a linear response <br>(small perturbation). If the perturbation is large, the process response will be non-<br>linear and the representations of the process with a linear process model will be <br>inaccurate.<br>Stochastic identification techniques, in principle, provide a more reliable method of <br>determining the process transfer function. Most workers have used the Box and <br>Jenkins [59] time-series analysis techniques to develop dynamic models. An intro-<br>duction to these methods is given by Davies [60]. In stochastic identification, a low <br>amplitude sequence (usually a pseudorandom binary sequence, PRBS) is used to <br>perturb the setting of the manipulated variable. The sequence generally has an <br>implementation period smaller than the process response time. By evaluating the <br>auto- and cross-correlations of the input series and the corresponding output data, a <br>quantitative model can be constructed. The parameters of the model can be deter-<br>mined by using a least squares analysis on the input and output sequences. Because <br>this identification technique can handle many more parameters than simple first-<br>order plus dead-time models, the process and its related noise can be modeled more <br>accurately.<br>Identification of the noise and its probable causes usually leads to the most effective <br>method of removing these disturbances.<br>The time series method also contains the means for determining the goodness of <br>the model fit by examining the cross- and auto-correlations between the residual <br>and the input sequence. From Box and Jenkins's models, a minimum variance con-<br>trol strategy can be determined, resulting in a minimum deviation of the controlled <br> <br>variable. Tuning procedures can also be accurately determined from the model. The <br>structure and initial parameters for self-tuning regulators can be determined from <br>the minimum variance controller.<br>
+<hr>
+<A name=157></a> References <br>
+<b>143</b><br>
+A drawback of the stochastic identification technique is its complexity and the sub-<br>stantial computational requirements. Only a limited number of investigators have <br>applied this technique to the extrusion process. Parnaby et al. [61­63] did the first <br>work on stochastic identification of extrusion process models. A hierarchical auto-<br>matic optimal control scheme was developed and evaluated on a laboratory extru-<br>sion line [63]. The only operator input required was the desired output rate and die <br>inlet melt temperature. A variable die restriction was used to adjust diehead pres-<br>sure and throughput. Considerable improvements in control were obtained, parti-<br>cularly in the control of diehead pressure. Other applications of stochastic identifica-<br>tion to extrusion have been made by Patterson et al. [64, 65], Costin [66, 67], and at <br>the IKV in Aachen [81].<br>
+References<br>1.  H. Recker et al., Plastverarbeiter, 28, 1, 1­8 (1977)<br>2.  J.D. Lenk, "Handbook of Controls and Instrumentation," Prentice-Hall Inc. (1980)<br>3.  D.B. Hoffman and N. Sarasohn, Plastics Design and Process, July, 20­23 (1972)<br>4.  H. Puetz in "Messen und Regeln beim Extrudieren," VDI-Verlag GmbH, Duesseldorf <br>
+(1982)<br>
+5.  K.H. School in "Der Extruder als Plastifiziereinheit," VDI-Verlag GmbH, Duesseldorf <br>
+(1977)<br>
+6.  A. Galskoy and K.K. Wang, Plastics Engineering, Nov., 42­45 (1978)<br>7.  H.E. Harris, Plastic Technology, Feb., 22­29 (1982)<br>8.  S.H. Walton, The Wire Association International (1977)<br>9.  H. Janeschitz-Kriegl, Proc 7th Int'l Congress on Rheology, Brown Univ., Providence, RI <br>
+(1963)<br>
+10.  H. Janeschitz-Kriegl and J. Schijf, Plastics &amp; Polymers, 37, Dec., 523­527 (1969)<br>11.  D.I. Marshall, I. Klein, and R.H. Uhl, Soc. Plastics Engrs. J., 20, 329 (1964)<br>12.  L.P.B.M. Janssen, G.H. Noomen, and J.M. Smith, Plastics &amp; Polymers, 43, Aug., <br>
+135­140 (1975)<br>
+13.  N. Wheeler, Techn. Papers ANTEC, Pittsburgh, PA, Jan. (1962)<br>14.  H.L. Trietly, EDN, 28, 2, 93­96 (1983)<br>15.  J. van Leeuwen, M. Goren, A.A. Grimbergen, and J.C. Molijn, Plastica, 17, 6, 269­277 <br>
+(1969)<br>
+16.  J. van Leeuwen, M. Goren, A.A. Grimbergen, and J.C. Molijn, Plastica, 17, 7, 319­326 <br>
+(1969)<br>
+17.  J. van Leeuwen, M. Goren, A.A. Grimbergen, and J.C. Molijn, Plastica, 17, 8, 371­386 <br>
+(1969)<br>
+18.  J. van Leeuwen, Polym. Eng. Sci., 7, 98­109 (1967)<br>19.  W. Schlaeffer, J. Schijf, and H. Janeschitz-Kriegl, Plastics &amp; Polymers, 39, June, 193­199 <br>
+(1971)<br>
+<hr>
+<A name=158></a><b>144</b>  <b>4Instrumentation and Control</b><br>
+20.  M. Hulatt and W.L. Wilkinson, Plastics and Rubber Processing, March, 15­18 (1977)<br>21.  H.J. Kim and E.A. Collins, Polym. Eng. Sci., 11, 83 (1971)<br>22.  E.A. Collins and F.E. Filisko, AICHE J., 16, 399 (1970)<br>23.  H.T. Kim and J.P. Darby, SPE J., 26, 31­36 (1970)<br>24.  F. T. Farrace, "Standardization of an Automatic Inspection System," Nondestructive <br>
+Testing Standards--A Review, ASTM STP 624, Harold Berger (Ed.), American Society for <br>Testing and Materials, 231­245 (1977)<br>
+25.  H. Droscha, Kunststoffe, 72, 5, 257­259 (1982)<br>26.  W. Michaeli, "Zur Analyse der Flachfolien und Tafelextrusions-Prozesse," Ph. D. Thesis <br>
+IKV, TH Aachen (1975)<br>
+27.  N. Nadav, M.Sc. Thesis, Technion, Israel Institute of Technology<br>28.  N. Nadav and Z. Tadmor, Chem. Eng. Sci., 28, 2115­2126 (1973)<br>29.  H. Recker et al., Plastverarbeiter 28, 1, 1­8 (1977) and 28, 3 137­142 (1977)<br>30.  F.L. Binsberger, J. Polym. Sci. 11, 10, 1915­1929 (1967)<br>31.  H. Hermann, Kunststoffe, 61, 11, 839­842 (1971)<br>32.  G. Wittig, Material Pruefung, 16, 10, 327 (1974)<br>33.  N.N., Plastics Technology, 28, 11, 18 (1982)<br>34.  D. Osmer, Plastics Compounding, Jan/Feb., 14­24 (1983)<br>35.  H. Hensel, "Die Orientierungsdoppelbrechung," Ph.D. thesis IKV, TM Aachen (1975)<br>36.  G. Menges, et al., Plastverarbeiter, 24, 2, 73­77 (1973)<br>37.  P. Fischer, Plastverarbeiter, 29, 5, 231­243 (1978)<br>38.  U.S. Patent 3,866,669<br>39.  U.S. Patent 4,272,466<br>40.  H. Goermar and H. Puetz, Kunststoffe, 69, 9, 588­590 (1979)<br>41.  H. Recker, "Messen und Regeln beim Extrudieren," VDI-Verlag GmbH, Duesseldorf, <br>
+(1982), p. 46<br>
+42.  J.G. Ziegler and N.B. Nichols, Trans. ASME, 64, 11, 759 (1942)<br>43.  K.L. Chien, J.A. Hrones, and S.B. Resweck, Trans. ASME, 74, 175 (1952)<br>44.  E. Pester and E. Kollman, Elektrische Ausruestung, 3­8 and 45­52 (1960)<br>45.  J.A. Miller, A.M. Lopez, C.L. Smith and P.W. Murrill, Control Engineering, 14, 12, 72 <br>
+(1967)<br>
+46.  P.D. Roberts, Measurement and Control, 9, 227, June (1976)<br>47.  C.L. Smith, Instruments &amp; Control Systems, 43­46, Sept. (1976)<br>48.  R.F. Bayless, Plastics Engineering, 24, 3, 59­72, March (1978)<br>49.  R. Chostner, Plastics Technology, 68­70, Feb. (1983)<br>50.  R.B. Kirby, SPE Journal, 18, 1273 (1962)<br>51.  W.L. Krueger, SPE Journal, 18, 1282 (1962)<br>52.  D.H. Reber, R.E. Lynn Jr., and E.J. Freech, Polym. Eng. Sci., 13, 346 (1973)<br>
+<hr>
+<A name=159></a> References <br>
+<b>145</b><br>
+53.  Z. Tadmor, S.D. Lipshitz, and R. Lavie, Polym. Eng. Sci., 14, 112 (1974)<br>54.  N. Brauner, R. Lavie, and Z. Tadmor, Int. IFAC Conference on Instrumentation and Auto-<br>
+mation in the Paper, Rubber, and Plastics Ind., 3rd Proc., Brussels, 6, 353 (1978)<br>
+55.  N.R. Schott, Ph.D. Dissertation, Univ. of Arizona (1971)<br>56.  W. Fontaine, Ph.D. Dissertation, Ohio State Univ. (1975)<br>57.  S. Dormeier, SPE ANTEC Tech. Papers, 25, 216 (1979)<br>58.  D. Fingerle, J. Elastomers Plast., 10, 293 (1978)<br>59.  G.E.P. Box and G.M. Jenkins, "Time Series Analysis Forecasting and Control," Revised <br>
+Edition, Holden-Day, San Francisco, CA (1976)<br>
+60.  W.D.T. Davies, "System Identification for Self-Adaptive Control," Wiley-Interscience, <br>
+NY (1970)<br>
+61.  J. Parnaby, A.K. Kochhar, and B. Woods, Polym. Eng. Sci., 15, 594 (1975)<br>62.  A.K. Kochhar and J. Parnaby, Automatica, 13, 177 (1977)<br>63.  A.K. Kochhar and J. Parnaby, Int. Mech. Eng. (Lond.), Proc., 192, 299 (1978)<br>64.  G.A. Hassan and J. Parnaby, Polym. Eng. Sci., 21, 276 (1981)<br>65.  I. Patterson, P. Brandin, and J. Parin, SPE ANTEC Tech. Papers, 25, 166 (1979)<br>66.  M.H. Costin, M.Sc. Thesis, McMaster University, Hamilton, Canada (1981)<br>67.  M.H. Costin, P.A. Taylor, and J.D. Wright, Polymer Eng. Sci., 22, 393 (1982)<br>68.  H.G. Wiegand, "Prozessautomatisierung beim Extrudieren und Spritzgiessen von <br>
+Kunststoffen," Carl Hanser Verlag (1979)<br>
+69.  D.D. Pollock, "The Theory and Properties of Thermocouple Elements," ASTM publica-<br>
+tion STP 492<br>
+70.  "Manual on the Use of Thermocouples in Temperature Measurements," ASTM publica-<br>
+tions STP 470 B<br>
+71.  "Thermocouple Reference Tables," NBS Monograph 125.<br>72.  H.D. Baker, E.A. Ryder, and N.H. Baker, "Temperature Measurement in Engineering," <br>
+Vol. I and II, Omega Press<br>
+73.  E.L. Sarber, Plastics Technology, 50­55, June (1983)<br>74.  N.N., Plastics Compounding, 21­32, March/April (1980)<br>75.  E. Galli, Plastics Compounding, 18­26, March/April (1983)<br>76.  F.W. Billmeyer and M. Saltzman, "Principles of Color Technology," John Wiley &amp; Sons <br>
+(1966)<br>
+77.  K.J. Astrom, "An Introduction to Stochastic Control Theory," Academic Press, NY (1970)<br>78.  P. Eykhoff, "System Identification," Wiley-Interscience (1974)<br>79.  H. Marchand, Plast. Techn., Feb., 67­70 (1984)<br>80.  A. Kirkland, Plast. Techn., Feb., 97­99 (1981)<br>81.  12. Kunststofftechnisches Kolloquium, IKV Aachen, Germany 176­185 (1984)<br>82.  W. Michaeli, U. Langkamp, B. Schäfer, and J. Kraack, Kunststoffe, 85, 1912­1914 (1995)<br>
+<hr>
+<A name=160></a><b>146</b>  <b>4Instrumentation and Control</b><br>
+83.  W. Obendrauf, C. Kukla, and G.R. Langecker, Kunststoffe, 83, 971­974 (1993)<br>84.  T. Nietsch, P. Cassagneu, and A. Michel, International Polymer Processing, XII, 4, <br>
+307­315 (1997)<br>
+85.  J. Coughlin, Modern Plastics, April, 101­105 (1992)<br>86.  X. Shen, R. Malloy, and J. Pacini, SPE ANTEC Technical Papers, 918­926 (1992)<br>
+<hr>
+<A name=161></a><b>PART II</b><br>
+<b>Process Analysis</b><br>
+<hr>
+<A name=162></a><hr>
+<A name=163></a><b>5 Fundamental </b><br>
+<b>Principles</b><br>
+Before going into a detailed process analysis of extrusion, it may be useful to review <br>the basic principles that will be applied in analysis. One can think of the basic prin-<br>ciples as the tools used in the process analysis. This chapter is meant to be a review, <br>not an exhaustive dissertation on the subject. For more in-depth and detailed infor-<br>mation, the reader is referred to general texts, such as the one by Bird et al. [1] and <br>others [2­4].<br>
+<b></b><br>
+ <b>5.1Balance Equations</b><br>
+In extrusion, as well as in many other processes, one deals with the transport of <br>mass, momentum, and energy. Balance equations are used to describe the transport <br>of these quantities. They are universal physical laws that apply to all media (solids <br>and fluids). Matter is considered as a continuum. Thus, the volume over which the <br>balance equation is formulated must be large enough to avoid discontinuities.<br>
+<b>5.1.1The Mass Balance Equation</b><br>
+The mass balance equation, also referred to as the equation of continuity, is simply <br>a formulation of the principles of the conservation of mass. The principle states that <br>the rate of mass accumulation in a control volume equals the mass flow rate into the <br>control volume minus the mass flow rate out of the control volume. In Cartesian <br>coordinates (x, y, z), the mass balance equation for a pure fluid can be written as:<br>
+ (5.1)<br>
+<hr>
+<A name=164></a><b>150</b>  <b>5Fundamental Principles</b><br>
+If a steady-state process is analyzed, the first term of Eq. 5.1 (the time derivation of <br>density) will equal zero; the same is true for Eq. 5.2. The mass balance equation <br>expressed in cylindrical coordinates (r, , z) is:<br>
+ (5.2)<br>
+If the fluid is considered incompressible, the density terms () disappear in Eqs. 5.1 <br>and 5.2. In one-dimensional steady state flow problems (velocity components in only <br>one direction), the mass balance equation is automatically satisfied and does not <br>enter into the calculations. In two-dimensional flow problems (no velocity compo-<br>nent in the third coordinate direction), the mass balance is satisfied by the introduc-<br>tion of a stream function [1].<br>
+<b>5.1.2The Momentum Balance Equation</b><br>
+The momentum of a body is the product of its mass and velocity. Since velocity is a <br>vector, momentum is also a vector. The momentum balance equation describes the <br>conservation of momentum; it is also referred to as the equation of motion.<br>Momentum can be transported by convection and conduction. Convection of momen-<br>tum is due to the bulk flow of the fluid across the surface; associated with it is a <br>momentum flux. Conduction of momentum is due to intermolecular forces on each <br>side of the surface. The momentum flux associated with conductive momentum <br>transport is the stress tensor. The general momentum balance equation is also <br>referred to as Cauchy's equation. The Navier-Stokes equations are a special case of <br>the general equation of motion for which the density and viscosity are constant. The <br>well-known Euler equation is again a special case of the general equation of motion; <br>it applies to flow systems in which the viscous effects are negligible.<br>In polymer flow systems, the inertia and body forces are generally negligible. For <br>these systems, the momentum balance equation in Cartesian coordinates can be <br>written as:<br>
+ (5.3a)<br>
+ (5.3b)<br>
+ (5.3c)<br>
+<hr>
+<A name=165></a> <br>
+5.1Balance Equations  <b>151</b><br>
+or in cylindrical coordinates:<br>
+ (5.4a)<br>
+ (5.4b)<br>
+ (5.4c)<br>
+The analysis of many flow problems can be simplified by considering only one com-<br>ponent of the equation of motion, the one in the direction of flow. Further simplify-<br>ing assumptions are often necessary in order to solve the problem. In the analysis of <br>isothermal processes, only two balance equations are needed, the mass and momen-<br>tum balance. In order to solve the problem, additional information is required. This <br>is information on how the fluid deforms under application of various stresses. This <br>information is described by the constitutive equation of the fluid; see also Section <br>6.2 on melt flow properties.<br>An example of the use of momentum balance in pipe flow of a Newtonian fluid is <br>given in Appendix 5.1.<br>
+<b>5.1.3The Energy Balance Equation</b><br>
+The energy balance equation states that the rate of increase in specific internal <br>(thermal) energy in a control volume equals the rate of energy addition by conduc-<br>tion plus the rate of energy dissipation. The principle of energy conservation is also <br>described by the first law of thermodynamics; see Section 5.2. If a constant density <br>is assumed, the energy equation can be written as:<br>
+ (5.5)<br>
+where<br>
+ (5.5a)<br>
+ (5.5b)<br>
+<hr>
+<A name=166></a><b>152</b>  <b>5Fundamental Principles</b><br>
+ (5.5c)<br>
+ (5.5d)<br>
+acc is the accumulation term, conv the convection term, cond the conduction term, <br>
+and diss the dissipation term. Equation 5.5(c) is an expression of Fourier's law of <br>
+heat transfer; see also Section 5.3.1. In cylindrical coordinates, only the convection, <br>conduction, and dissipation terms change:<br>
+ (5.6a)<br>
+ (5.6b)<br>
+ (5.6c)<br>
+The energy equation has to be used to analyze non-isothermal processes. In these <br>situations, there are generally four unknown variables: velocity, stress, pressure, <br>and temperature. In order to solve such a non-isothermal problem, one more equa-<br>tion is needed in addition to the three balance equations. The missing relationship <br>is the constitutive equation of the fluid; this relationship basically describes the <br>relationship between stress and deformation. In polymer extrusion, the material <br>undergoes large changes in temperatures as it is transported along the extruder. <br>Consequently, the energy equation is used extensively in the analysis of the extru-<br>sion process.<br>
+<hr>
+<A name=167></a> <br>
+5.2Basic Thermodynamics  <b>153</b><br>
+<b></b><br>
+ <b>5.2Basic Thermodynamics</b><br>
+Thermodynamics is concerned with energy, the exchange and transportation of <br>energy in a system. The first law of thermodynamics is a statement of the principle <br>of conservation of energy. For a closed (constant mass) system, the first law of ther-<br>modynamics is expressed by:<br>
+ (5.7)<br>
+where E is the total energy change of the system, Q is heat added to the system, <br>and W is work done by the system. The total energy change E can be split up into <br>several terms, each representing the change in energy of a particular form:<br>
+ (5.8)<br>
+where Ek is the change in kinetic energy, Ep is the change in gravitational poten-<br>
+tial energy, and U is the change in internal energy. By definition, the kinetic energy <br>is:<br>
+ (5.9)<br>
+where m is mass and v is velocity.<br>The gravitational potential energy is:<br>
+ (5.10)<br>
+where z is the elevation above a reference level and ag the local acceleration of grav-<br>
+ity. These energy functions are common to both mechanics and thermodynamics.<br>The internal energy function U, however, is peculiar to thermodynamics. It repre-<br>sents the kinetic and potential energies of the molecules, atoms, and subatomic <br> <br>particles that make up the system on a microscopic scale. There is no known way to <br>determine absolute values of U. Fortunately, only changes of U are needed and <br>these can be derived experimentally. When the state of the system is fixed, the inter-<br>nal energy U is fixed. If Eq. 5.8 is used in Eq. 5.7, the first law of thermodynamics <br>can be written as:<br>
+ (5.11)<br>
+If the sum of the kinetic and potential energies of the system does not change, this <br>equation becomes<br>
+ (5.12)<br>
+<hr>
+<A name=168></a><b>154</b>  <b>5Fundamental Principles</b><br>
+or in differential form:<br>
+ (5.13)<br>
+If, in addition, the process is adiabatic (Q = 0), Eq. 5.12 becomes<br>
+ (5.14)<br>
+Note that in Eq. 5.13 the differential signs on Q and W are written , whereas the <br>differential sign of U is d. There is a fundamental difference between a property like <br>U and the quantities Q and W. A property like U always has a value dependent only <br>on the state. A process that changes the state of a system changes U. Thus, dU re- <br>presents an infinitesimal change in U, and integration gives a difference between <br>two values of the property:<br>
+ (5.14a)<br>
+On the other hand, Q and W are not properties of the system and depend on the path <br>of the process. Thus  is used to denote an infinitesimal quantity. Integration gives a <br>finite quantity and not a difference between the two values:<br>
+ (5.14b)<br>
+Thus, integration of Eq. 5.13 yields Eq. 5.12.<br>Special thermodynamic functions are defined as a matter of convenience. The sim-<br>plest such function is the enthalpy H, explicitly defined for any system by the math-<br>ematical expression:<br>
+ (5.15)<br>
+where P is pressure and V is volume.<br>The enthalpy has units of energy and is a system property like U, P, and V. When-<br>ever a differential change occurs in a system, its properties change by:<br>
+ (5.16)<br>
+The amount of heat in a closed PVT system that must be added to accomplish a <br>given change of state depends on how the process is carried out. Only for a rever-<br>sible process where the path is fully specified is it possible to relate the heat to a <br>property of the system. On this basis, the specific heat at a constant volume can be <br>defined:<br>
+ (5.17)<br>
+<hr>
+<A name=169></a> <br>
+5.2Basic Thermodynamics  <b>155</b><br>
+It represents the amount of heat required to increase the temperature by dT when <br>the system is held at a constant volume. Cv is a property of the system; this follows <br>
+from Eq. 5.12. For a constant volume, reversible process dU = Q, because no work <br>can be done without volume changes. Thus, the specific heat at a constant volume <br>can be related to the internal energy by:<br>
+ (5.18)<br>
+Thus, for a constant volume process<br>
+ (5.19)<br>
+Equation 5.19 is a useful relationship. If specific heat and temperature changes are <br>known, then the amount of heat required to accomplish this change in temperature <br>can be determined. If the amount of heat and the specific heat are known, then the <br>resulting change in temperature can be calculated. This relationship is indispensa-<br>ble in the analysis of the extrusion process. For instance, if the amount of viscous <br>heat generation in a certain amount of polymer is known, then the resulting adiaba-<br>tic temperature rise can be determined if the specific heat of the polymer is known.<br>The specific heat at constant pressure is defined as:<br>
+ (5.20)<br>
+It represents the amount of heat required to increase the temperature by dT when <br>the system is heated in a reversible process at a constant pressure. Using Eqs. 5.13 <br>and 5.16 it can be shown that:<br>
+ (5.21)<br>
+Thus, Cp is also a property of the system. Further<br>
+ (5.22)<br>
+In polymer melts, the material is generally considered to be incompressible; in this <br>case Cv = Cp. The actual relationship between Cv and Cp is given by Eq. 6.94.<br>Entropy is an intrinsic property of a system. For a reversible process, changes in <br>entropy are given by:<br>
+ (5.23)<br>
+<hr>
+<A name=170></a><b>156</b>  <b>5Fundamental Principles</b><br>
+The second law of thermodynamics states that the entropy change of any system <br>and its surroundings, considered together, is positive and approaches zero for any <br>process, which approaches reversibility. The mathematical expression of the second <br>law is simply:<br>
+ (5.24)<br>
+The left term is the entropy transfer, Q/T; it forms a direct link with the heat trans-<br>fer (Q) if the temperature at the system boundary is T. Entropy transfer as a con-<br>cept makes the distinction between heat transfer and work transfer as parallel forms <br>of energy transfer. Only the transfer of energy as heat is accompanied by entropy <br>transfer. The work transfer interaction is not accompanied by entropy transfer. The <br>term on the right of the inequality sign is the entropy change; it is a thermodynamic <br>property. The inequality sign in Eq. 5.24 expresses the essence of the second law of <br>thermodynamics. The change from state 1 to state 2 can occur over various paths. <br>The difference between possible paths is described by the strength of the inequality <br>sign. The entropy generation expresses this difference quantitatively:<br>
+ (5.25)<br>
+Thus, a thermodynamic process is accompanied by entropy generation. When <br>Sgen &gt; 0, the process is considered irreversible; when Sgen = 0, the process is consid-<br>
+ered reversible. It should be noted that the entropy generation Sgen is, of course, path <br>
+dependent and, therefore, not a thermodynamic property as opposed to the entropy <br>change S2­S1. In fluid flow the volumetric rate of entropy generation is:<br>
+ (5.26)<br>
+<hr>
+<A name=171></a> <br>
+5.2Basic Thermodynamics  <b>157</b><br>
+The first bracketed (square) term is the conductive energy flux, which is the same as <br>used in the energy Eq. 5.5(c). The other bracketed terms represent the dissipative <br>energy flux, as used also in the energy Eq. 5.5(d). Thus, the volumetric rate of <br>entropy generation can be written as:<br>
+ (5.27)<br>
+<b>5.2.1Rubber Elasticity</b><br>
+The theory of rubber elasticity is largely based on thermodynamic considerations. It <br>will be briefly discussed as an example of how thermodynamics can be applied in <br>polymer science. For more detailed information the reader is referred to the various <br>textbooks [10­13]. It is assumed that there is a three-dimensional network of chains, <br>that the chain units are flexible and that individual chain segments rotate freely, <br>that no volume change occurs upon deformation, and that the process is reversible <br>(i.e., true elastic behavior). Another usual assumption is that the internal energy U <br>of the system does not change with deformation. For this system the first law of <br>thermodynamics can be written as:<br>
+ (5.28)<br>
+If the equilibrium tensile force is F and the displacement dl then the work done by <br>the system is:<br>
+ (5.29)<br>
+The change in U with respect to l at a constant temperature and volume is:<br>
+ (5.30)<br>
+Thus, the equilibrium tensile force F is determined by the change in internal energy <br>with deformation and the change in entropy with deformation. An ideal rubber is <br>defined as a material for which the change in internal energy with deformation <br>equals zero. Thus, the only contribution to the force F is from the entropy term:<br>
+ (5.31)<br>
+When a rubber is deformed, its entropy S changes. The long chains tend to adopt a <br>most probable configuration; this is a highly coiled configuration. When the mate-<br>rial is stretched, the chains uncoil, resulting in a less probable chain configuration. <br>When the force is removed, the system wants to return to the more probable coiled-<br>
+<hr>
+<A name=172></a><b>158</b>  <b>5Fundamental Principles</b><br>
+up state. Thus, the entropy of the system increases. The basic problem in the theory <br>of rubber elasticity is a statistical mechanical problem of determining the change in <br>entropy in going from an undeformed state to a deformed state. The extension ratio <br> is defined as:<br>
+ (5.32)<br>
+where lo is the original length, l the increase in length, and  the elongational <br>
+strain.<br>If a cube of unit length dimension is considered, the force will equal the stress . <br>Equation 5.31 can now be written as:<br>
+ (5.33)<br>
+The most probable state of a system is the state that has the greatest number of <br>ways, Y, of being realized. The entropy S is related to this number of complexions Y <br>by the Boltzmann relation:<br>
+ (5.34)<br>
+where CB is the Boltzmann constant (1.38E­23 J/°K).<br>The change in entropy upon extension can be expressed as:<br>
+ (5.35)<br>
+where N is the number of freely orienting chain segments. Substituting this expres-<br>sion into Eq. 5.33 yields the stress-extension ratio relationship:<br>
+ (5.36)<br>
+The number of chains per unit volume can be related to the density  and the aver-<br>age molecular weight between crosslinks Mc:<br>
+ (5.37)<br>
+where NA is Avogadro's number (6.025E23 mol­1).<br>Considering that NACB is the gas constant R (8.314 J/mol °K), Eq. 5.36 can be writ-<br>
+ten as:<br>
+<hr>
+<A name=173></a> <br>
+5.2Basic Thermodynamics  <b>159</b><br>
+ (5.38)<br>
+The elastic modulus can be determined by:<br>
+ (5.39)<br>
+For small extensional strains the elastic modulus becomes:<br>
+ (5.40)<br>
+For a linear and isotropic material the shear modulus G is related to the extensional <br>modulus by:<br>
+ (5.41)<br>
+where <i>v</i> is Poisson's ratio; <i>v</i> = ½ for ideal rubbers.<br>Thus, the shear modulus can be obtained from:<br>
+ (5.42)<br>
+<b>5.2.2Strain-Induced Crystallization</b><br>
+It has been observed that the crystallization behavior of polymers is modified when <br>the material is strained. This behavior has been found in rubbers and in thermo-<br>plastics [14]. In thermoplastics, the effect of strain on crystallization behavior has <br>been studied quite extensively in solutions, in melt, and in solid state. For instance, <br>it has been found that flowing polymer melts can crystallize at temperatures that <br>are substantially above the crystallization temperature of the same material in a <br>quiescent state. This strain-induced crystallization is generally explained in terms <br>of thermodynamic processes. During a phase change, the Gibbs free energy is:<br>
+ (5.43)<br>
+where S is the conformational entropy.<br>For an equilibrium process G = 0. This would be the case for crystallization or <br>melting at the melting point. The melting point Tm, therefore, is:<br>
+ (5.44)<br>
+<hr>
+<A name=174></a><b>160</b>  <b>5Fundamental Principles</b><br>
+When the polymer is being deformed, the molecules are oriented to some extent and <br>this reduces the conformational entropy. If the material is considered to be entropy-<br>elastic, the energy expended in the deformation of the polymer will reduce the <br>entropy but not affect the internal energy. This point was discussed in some detail <br>by Astarita [15]. Thus, if the enthalpy is unaffected by orientation and the entropy <br>reduced, the melting point will increase with increasing orientation. This melting <br>point elevation will increase the degree of super-cooling, the driving force of crystal-<br>lization. The anisotropy of the oriented polymer favors crystallization in the direc-<br>tion of orientation and discourages it orthogonally. This explains the change in <br> <br>crystal growth mechanism from the three-dimensional (spherulitic) to the unidi-<br>mensional (fibrillar) growth.<br>
+<b></b><br>
+ <b>5.3Heat Transfer</b><br>
+Heat transfer takes place by three mechanisms: conduction, convection, and radia-<br>tion. In conductive heat transfer, the heat flows from regions of high temperature to <br>regions of low temperature. The transfer takes place due to motion at the molecular <br>level. Matter must be present in order for conduction to occur. The material itself <br>does not need to be in motion for conduction to take place; in fact, many times the <br>conducting medium will be stationary. In a solid material, the only mode of heat <br>transfer is conduction [16]. In convection, heat transfer is due to the bulk motion of <br>the fluid. Convective heat transfer only occurs in fluids. In radiation, heat or radiant <br>energy is transferred in the form of electromagnetic waves.<br>
+<b>5.3.1Conductive Heat Transfer</b><br>
+The most important relationship in conductive heat transfer is Fourier's law; for <br>conduction in the x direction:<br>
+ (5.45)<br>
+where  x is the heat flow (rate of conduction), kx the thermal conductivity, Ax the <br>
+area normal to heat flow, and T temperature.<br>For conduction in the y and z directions, Fourier's law is simply:<br>
+ (5.46)<br>
+<hr>
+<A name=175></a> <br>
+5.3Heat Transfer  <b>161</b><br>
+and<br>
+ (5.47)<br>
+Fourier's law states that the heat will flow from high to low temperatures. The heat <br>flow is proportional to the thermal conductivity, the temperature gradient, and the <br>cross-sectional area normal to heat flow. Thus, in order to calculate the heat flow, <br>one has to know the thermal conductivity of the material and the temperature distri-<br>bution within the material. The temperature distribution has to be determined by <br>solving the energy equation (Eq. 5.5) as discussed in Section 5.1.3.<br>
+<b>5.3.2Convective Heat Transfer</b><br>
+Convective heat transfer is considerably more difficult to analyze than conductive <br>heat transfer in a stationary material. This is simply due to the fact that more terms <br>have to be carried in the energy equation (Eq. 5.5) that has to be solved in order to <br>find the temperature distribution. Many practical problems encountered in polymer <br>processing are described by equations that do not allow simple analytical solutions. <br>In many cases, therefore, one has to use numerical techniques to obtain solutions to <br>the problem.<br>
+<b>5.3.3Dimensionless Numbers</b><br>
+It is quite common in process engineering to describe certain phenomena by re -<br>lating dimensionless combinations of physical variables. These combinations are <br>referred to as dimensionless groups or dimensionless numbers. This approach offers <br>a number of advantages. One is assured that the equations are dimensionally homo-<br>geneous. By using dimensionless numbers, the number of variables describing the <br>problem can be reduced. One can predict the effect of a change in a certain variable <br>even if the problem cannot be completely solved. If the dimensionless numbers <br>describing the problem remain the same, then the solution to the problem will <br>remain unchanged, even if individual variables are varied. The latter characteristic <br>is very useful in scale-up problems; see also Section 8.8.<br>
+<b>5.3.3.1Dimensional Analysis<br></b>Dimensionless numbers can be derived by making the appropriate balance equa-<br>tions dimensionless when the problem can be fully described. (See the example on <br>heat transfer in Newtonian fluid between two plates later in this section.) Dimen-<br>sionless numbers can also be derived from dimensional analysis; this approach is <br>
+<hr>
+<A name=176></a><b>162</b>  <b>5Fundamental Principles</b><br>
+used if the problem cannot be completely described mathematically. An example of <br>dimensional analysis is the determination of the force F that a sphere of diameter D <br>encounters in a fluid with viscosity  when the relative velocity between sphere and <br>fluid is v. From physical arguments, it can be deduced that F has to be a function of <br>diameter, velocity, fluid density, and viscosity:<br>
+ (5.48)<br>
+Even though the actual form of the equation is not known, the equation has to be <br>dimensionally homogeneous. This is also true if the force is written in the following <br>form:<br>
+ (5.49)<br>
+The dimensions of the these variables can be expressed in length L, time t, and <br>mass M; thus, F(L t­2 M), D(L), v(Lt­1), (L­3 M), and (L­1t­1 M). The requirement for <br>dimensional homogeneity yields:<br>
+ (5.50a)<br>
+ (5.50b)<br>
+ (5.50c)<br>
+From the three Eqs. 5.50(a) to 5.50(c), three of the unknowns (a, b, c, and d) can be <br>expressed as a function of the fourth. If a, b, and c are expressed as a function of d, <br>this yields:<br>
+ (5.51a)<br>
+ (5.51b)<br>
+ (5.51c)<br>
+With this relationship, Eq. 5.49 can be rewritten as:<br>
+ (5.52)<br>
+The original number of variables has now been reduced from 5 to 2 dimensionless <br>numbers. The dimensionless number on the right-hand side of Eq. 5.52 is the well-<br>known Reynolds number:<br>
+ (5.53)<br>
+<hr>
+<A name=177></a><IMG src="Polymer Extrusion by Chris Rauwendaal-177_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-177_18.jpg"><br>
+ <br>
+5.3Heat Transfer  <b>163</b><br>
+The Reynolds number represents the ratio of inertia forces (vD) to viscous forces <br>(). In the flow of a fluid through a flow channel, turbulent flow will occur when the <br>Reynolds number is above the critical Reynolds number, which is about 2100. Below <br>the critical Reynolds number, laminar flow takes place; this is also referred to as <br>streamline flow or viscous flow. In polymer processing, the polymer melt viscosity is <br>generally very high. As a result, the Reynolds number in polymer processing is very <br>small, typically 10­3. Therefore, in polymer processing, the process melt flow is <br>always laminar. The low Reynolds number generally allows one to neglect the effect <br>of inertia and body force, as was done in the formulation of the momentum balance <br>equation, Eq. 5.3.<br>
+<b>5.3.3.2Important Dimensionless Numbers<br></b>Several important dimensionless numbers in combined heat and momentum trans-<br>fer in fluids can be derived by considering the simple flow of a Newtonian fluid <br>between two flat plates, one stationary and one moving at velocity, v; see Fig. 5.1.<br>
+v<br>
+y<br>
+H<br>
+<b> Figure 5.1</b> <br>
+x<br>
+Drag flow between two flat <br>
+ <br>parallel  plates<br>
+If it is assumed that only conduction and viscous dissipation play a role of import-<br>ance, the energy balance can be written as:<br>
+ (5.54)<br>
+When the fluid is Newtonian, the shear stress can be written as (see Section 6.2.1):<br>
+ (5.55)<br>
+Thus, Eq. 5.54 becomes:<br>
+ (5.56)<br>
+<hr>
+<A name=178></a><b>164</b>  <b>5Fundamental Principles</b><br>
+If the pressure gradient in flow direction is assumed to be zero, i.e. only drag flow, <br>the velocity gradient becomes:<br>
+ (5.57)<br>
+Thus, Eq. 5.56 becomes<br>
+ (5.58)<br>
+By integrating twice, the temperature profile is obtained:<br>
+ (5.59)<br>
+If the temperature at the stationary wall is T0, T(y = 0) = T0 and T1 at the moving wall, <br>
+T(y = H) = T1, the integration constants can be calculated:<br>
+ (5.60a)<br>
+ (5.60b)<br>
+The temperature profile can now be written as:<br>
+ (5.61)<br>
+This is essentially the same equation as Eq. 7.92 describing the temperature profile <br>in the melt film in the melting region of an extruder. The equation can now be writ-<br>ten in the following dimensionless form:<br>
+ (5.62)<br>
+If the dimensionless temperature is T0 and dimensionless distance y0, then Eq. 5.62 <br>can be written as:<br>
+ (5.63)<br>
+The Brinkman number NBr is a measure of the importance of viscous heat genera-<br>
+tion relative to the heat conduction resulting from the imposed temperature differ-<br>ence T (= T1­T0):<br>
+<hr>
+<A name=179></a> <br>
+5.3Heat Transfer  <b>165</b><br>
+ (5.64)<br>
+If the Brinkman number is larger than 2, there is a maximum temperature at a posi-<br>tion intermediate between the two walls; see Fig. 5.2.<br>
+1.0<br>
+0.8<br>
+0.6<br>
+NBr=0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+0.4<br>
+0.2<br>
+Dimensionless distance<br>
+00<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+1.2<br>
+Dimensionless temperature<br>
+<b>Figure 5.2</b>Temperature profiles at various Brinkman numbers<br>
+In the previous problem, only conduction and dissipation were considered to play a <br>role. If the analysis is now extended to include the effect of convection, the energy <br>equation becomes:<br>
+ (5.65)<br>
+This equation can be made dimensionless by introducing the following dimension-<br>less variables:<br>
+ (5.66a)<br>
+ (5.66b)<br>
+ (5.66c)<br>
+ (5.66d)<br>
+<hr>
+<A name=180></a><b>166</b>  <b>5Fundamental Principles</b><br>
+This results in the following expression:<br>
+ (5.67)<br>
+The first dimensionless group is the Graetz number:<br>
+ (5.68)<br>
+where  is the thermal diffusivity; see Section 6.3.5. Equation 5.67 can now be writ-<br>ten as:<br>
+ (5.69)<br>
+Because this equation does not have a simple analytical solution it is generally <br>solved by some numerical technique. However, regardless of the actual solution, as <br>long as the Graetz and Brinkman numbers are constant, the solution to the problem <br>will remain unchanged.<br>The Graetz number can be considered to be a ratio of two time values, one being the <br>time required to reach thermal equilibrium through conduction in the direction <br> <br>normal to the flow direction (dimension H), the other time being the average resi-<br>dence time in the flow channel of length L. Thus, the Graetz number is a measure of <br>the importance of conduction normal to the flow, relative to the thermal convection <br>in the direction of flow. If the Graetz number is large, the conduction normal to the <br>flow is large relative to the convection in flow direction. This situation often occurs <br>in extruders in the flow through the screw channel and die flow channels.<br>A dimensionless number closely related to the Graetz number is the Peclet number:<br>
+ (5.70)<br>
+The Peclet number provides a measure of the importance of thermal convection rela-<br>tive to thermal conduction. The Peclet number in polymer processing is often quite <br>large, typically of the order of 103 to 105. This indicates that convective heat trans-<br>port is often quite important in polymer melt flow.<br>Another important dimensionless number is the Nusselt number:<br>
+ (5.71)<br>
+where h is the interfacial heat transfer coefficient.<br>
+<hr>
+<A name=181></a> <br>
+5.3Heat Transfer  <b>167</b><br>
+The Nusselt number is basically a dimensionless temperature gradient averaged <br>over the heat transfer surface. The Nusselt number represents the ratio of the heat <br>transfer resistance estimated from the characteristic dimension of the object (L/k) <br>to the real heat transfer resistance (1/h). In many convective heat transfer problems, <br>the Nusselt number is expressed as a function of other dimensionless numbers, <br>e.g., the Reynolds number and the Prandtl number.<br>The Prandtl number is:<br>
+ (5.72)<br>
+The Prandtl number is simply the ratio of kinematic viscosity (/) to thermal diffu-<br>sivity (). Physically, the Prandtl number represents the ratio of the hydrodynamic <br>boundary layer to the thermal boundary layer in the heat transfer between fluids and <br>a stationary wall. In simple fluid flow, it represents the ratio of the rate of impulse <br>transport to the rate of heat transport. It is determined by the material properties; for <br>high viscosity polymer melts, the number is of the order of 106 to 1010.<br>The Nahme number or Griffith number is:<br>
+ (5.73)<br>
+where T is the temperature coefficient of viscosity  as defined in Eq. 6.40.<br>The Nahme number can be considered to be the ratio of viscous dissipation to ther-<br>mal conduction in the direction perpendicular to flow. Large values of the Nahme <br>number (&gt;1) indicate that the temperature non-uniformities created by the viscous <br>dissipation have a substantial effect on the resulting velocity profile. Thus, if the <br>Nahme number is large, a non-isothermal flow analysis has to be made in order to <br>maintain sufficient accuracy. If the Nahme number is small, an isothermal analysis <br>can yield relatively accurate results. The Nahme number is also referred to as the <br>Griffith number.<br>The Biot number is:<br>
+ (5.74)<br>
+The Biot number is a dimensionless heat transfer number. When the Biot number <br>is zero, there is no exchange of heat; i.e., adiabatic conditions exist. When the Biot <br>number is infinitely large, the wall temperature Tw equals the melt temperature Tm; <br>
+this corresponds to isothermal conditions. Normal values for the Biot number in <br>extrusion dies range from 1 to 100, depending strongly on the presence and amount <br>of insulation.<br>
+<hr>
+<A name=182></a><b>168</b>  <b>5Fundamental Principles</b><br>
+The Fourier number is:<br>
+ (5.75)<br>
+The Fourier number is a convenient number in the analysis of transient heat trans-<br>fer problems, i.e., where the temperature changes with time. The Fourier number <br>can be considered to be a ratio of two time values: one is the actual time value, the <br>other is the time necessary to reach thermal equilibrium in the sample by conduc-<br>tion. If the Fourier number is large (&gt;1), the sample will reach thermal equilibrium <br>within the considered time frame. If the Fourier number is small (&lt;0.1), only the <br>skin of the sample will have changed in temperature, while the bulk of the material <br>will be largely unaffected. In many cases, the temperature distribution is about 90% <br>uniform when the Fourier number equals unity.<br>The thermal diffusivity of most polymers is about 10­7 m2/s. If a polymer slab 1 mm <br>thick is heated from two sides (H = 0.0005 m), after approximately 2.5 s the tem-<br>perature in the slab will be quite uniform. If the slab is 10 mm thick, the uniform <br>temperature conditions will not be approached until after 250 s or 4.17 min. This <br>explains, at least partly, why it is difficult to obtain uniform melt temperatures in <br>an extruder where the screw has very deep channels. In fact, if the channels are <br>too deep, it is unlikely that the melting process can be completed in the extruder. A <br>typical average residence time in a single screw extruder is about 1 to 3 min.<br>
+<b>5.3.4Viscous Heat Generation</b><br>
+Viscous heat generation is the dissipation of mechanical energy in a viscous fluid. <br>The last term in the energy Eq. 5.5(d) shows the viscous dissipation in the most <br>general case. In the simpler case of unidirectional shear, the viscous heat generation <br>per unit volume is:<br>
+ (5.76)<br>
+If the flow properties of the fluid can be described by a power law equation (see Sec-<br>tion 6.2.2), the viscous heat generation per unit volume is:<br>
+ (5.77)<br>
+where m is the consistency index and n the power law index. The power law index is <br>unity for a Newtonian fluid and between one and zero for a pseudo-plastic fluid such <br>as polymer melts. From Eq. 5.77 it can be seen that the viscous heat generation <br>increases more than proportionally with the shear rate. This has important implica-<br>tions in the extrusion process; this will be discussed in Chapter 7.<br>
+<hr>
+<A name=183></a> <br>
+5.3Heat Transfer  <b>169</b><br>
+Viscous heat generation occurs throughout a fluid. The local rate of heat generation <br>depends on the local shear rate. If the shear rate is constant throughout the entire <br>volume of a fluid, the viscous heat generation will be uniform throughout the fluid. <br>This is the case in pure drag flow (Couette flow), i.e., flow without the presence of <br>pressure differences in the flow direction; see Section 6.2.1. If the shear rate is not <br>uniform throughout the volume, the viscous heat generation will not be uniform <br>either. This is the case in pure pressure flow (Poiseuille flow) through a pipe. In this <br>flow situation, the shear rate in the center is zero and maximum at the wall. Conse-<br>quently, the viscous heat generation in the center is zero and maximum at the wall <br>just as with the shear rate. Since viscous heat generation occurs throughout a fluid, <br>it is an effective way of heating a polymer melt because it will result in a relatively <br>uniform temperature increase if the shear rate is approximately constant through-<br>out the fluid.<br>
+<b>5.3.5Radiative Heat Transport</b><br>
+Heat radiation consists of electromagnetic waves with a wavelength () range of 0.5 <br>to 10 microns. All bodies emit electromagnetic waves as a result of the thermal agi-<br>tation of their molecules. The rate at which a body emits radiant energy depends <br>mostly on its temperature. Between two surfaces, an exchange of radiation can take <br>place if the intermediate space is transparent to the radiation spectrum. If the tem-<br>peratures of the two surfaces are different, the sum of the two opposite heat flows <br>generally will not be equal to zero.<br>The rate of emission of radiant energy is given by the Stefan-Boltzmann law:<br>
+ (5.78)<br>
+where e is the emissivity, CSB the Stefan-Boltzmann constant, A the area of the <br>
+object, and T the absolute temperature (°K). For a perfectly "black" body, the emis-<br>sivity equals unity. The precise value of the Stefan-Boltzmann constant CSB can be <br>
+derived from other physical constants by the relationship:<br>
+ (5.79)<br>
+where CB is the Boltzmann constant (1.38054E­23 J/°K), v1 the speed of light <br>
+(2.997925E8 m/s), and CPl the Planck constant of action (6.6256E­34 Js).<br>For a "black" body, the spectral distribution of energy flux is given by Planck's law <br>of radiation. The wavelength at which this intensity is maximumal is inversely pro-<br>portional to the absolute temperature. This is Wien's law; it can be formulated as:<br>
+ (5.80)<br>
+<hr>
+<A name=184></a><b>170</b>  <b>5Fundamental Principles</b><br>
+At room temperature max = 10 m (infrared) and at 6000 °K max = 0.5 m (green). <br>
+The fact that the color of a body depends on its temperature is used in optical tem-<br>perature measurements. This is often referred to as infrared temperature measure-<br>ment even though some measurements may occur outside the infrared region of the <br>spectrum. The infrared region ranges from a wavelength of 0.7 m to about 400 m.<br>A perfectly black body emits the maximum amount of radiation based on its tem-<br>perature; its emissivity is unity. According to Kirchhoff's law, its absorptivity will <br>also be unity. In reality, surfaces have emissivities and absorptivities for infrared <br>radiation that are less than unity. The actual value will depend on the material, the <br>surface roughness, the temperature, and the wavelength of the radiation.<br>Absorptivity values are usually given as average values. For most non-metallic sur-<br>faces, this value is larger than 0.8. Clean polished metal surfaces have absorptivities <br>ranging between 0.05 and 0.20.<br>The radiative heat transfer between two surfaces is primarily determined by their <br>emissivities, absorptivities, and temperatures. For each surface, the amount of radi-<br>ation leaving the surface is the sum of the emitted radiation (eCSBT4 = eqb) plus the <br>
+reflected radiation (1­ a)qi:<br>
+ (5.81)<br>
+where a is the absorptivity and q the heat flux; qb is the black body radiation and qi <br>
+is the incident radiation.<br>Also, the net energy flux leaving the surface equals its own emission minus the frac-<br>tion of the incident radiation that is absorbed by the wall:<br>
+ (5.82)<br>
+If the wall is at the same temperature as its surroundings, the net energy flux will <br>be zero and qb = qi = CBT4. In this case, the emissivity has to equal the absorptivity <br>
+(e = a); this is known as Kirchhoff's law. From Eq. 5.81, it can be deduced that the <br>energy flux through any plane in a space equals qb = CSBT4.<br>In order to calculate the incident radiation qik reaching wall k, one has to know the <br>
+fractions fjk, indicating what part of the total radiation of wall j reaches wall k. The <br>
+geometrical factors, called view factors, are calculated by integration. In doing this, <br>one has to take into account the fact that the radiation intensity depends on the <br>angle with which the radiation hits the surface. In the simple case of two large par-<br>allel surfaces (1 and 2), both fractions are equal to unity: f12 = f21 = 1. Consider the <br>
+case where one body (body 1 with area A1 and temperature T1) is totally enclosed by <br>
+another body (body 2 with area A2 and temperature T2). The incident radiation on <br>
+wall 1 (A1qi1) equals f21 multiplied with the total radiation of wall 2 (f21A2q2). Thus, <br>
+the radiation heat flux reaching body 1 (qi1) equals the total radiation heat flux leav-<br>
+ing body 2 (q2). Therefore:<br>
+<hr>
+<A name=185></a> <br>
+5.3Heat Transfer  <b>171</b><br>
+ (5.83a)<br>
+and<br>
+ (5.83b)<br>
+The next radiation transport from body 1 (qn1A1) and from body 2 (qn2A2) are related <br>
+by:<br>
+ (5.84)<br>
+This is valid because the energy lost by body 1 is gained by body 2. The net radiation <br>flux leaving body 1 can be expressed as:<br>
+ (5.85)<br>
+The heat transfer coefficient for radiation hs can now be expressed as:<br>
+ (5.86)<br>
+At room temperature and relatively small temperature differences, the value of hs <br>
+will be above 5 W/m2°C. This value is large enough that it cannot be neglected rela-<br>tive to free convection. Obviously, at higher temperatures the contribution of the <br>radiative heat transport increases substantially.<br>Since radiative heating at elevated temperatures (above 300°C) generally occurs in <br>the infrared region, it is often referred to as infrared (IR) heating. The applications <br>of IR heating in polymer processing are numerous: thermoforming, film extrusion, <br>orientation, embossing, coating, laminating, ink drying, and fusing. It is also used in <br>curing filament-wound structures and in the manufacture of slit polypropylene <br>yarns out of polypropylene film. Paint drying and baking is the largest single use for <br>infrared heating. In polymer processing, thermoforming is probably the largest out-<br>let for infrared heating. A good series of papers on infrared heating of plastics was <br>written by Kraybill [19­21].<br>
+<b>5.3.5.1Dielectric Heating<br></b>Dielectric heating occurs when a dielectric material is placed in an electric field <br>that alternates at high frequency. A dielectric material is an electric insulator and it <br>has a low conductivity (high resistivity). Materials with a resistivity higher than 109 <br>ohm-cm are generally considered to be dielectric; most polymers fall into this cate-<br>
+<hr>
+<A name=186></a><b>172</b>  <b>5Fundamental Principles</b><br>
+gory. Dielectric energy, also referred to as radio-frequency energy, occupies the fre-<br>quency spectrum of about 10 to 100 megahertz.<br>In dielectric heating, heat is generated throughout the mass of material. In plastics, <br>this can be a very beneficial feature, considering the low thermal conductivity of <br>plastics. Since heat will be transferred away most quickly at the walls, dielectric <br>heating often results in a temperature profile where the highest temperature occurs <br>in the center and the lowest temperature at the wall. This is opposite to conductive <br>heating where the highest temperature will occur at the wall. Also, because heat is <br>generated throughout the material, temperature gradients are likely to be small in <br>dielectric heating as compared to conductive heating. Dielectric heating is uniform <br>throughout a mass of material because all of the polar molecules are oriented by an <br>electric field. The oscillations of the molecules resulting from the alternating field <br>produce heat through molecular friction. The rate at which electrical energy can be <br>dissipated in a dielectric material per unit volume   is proportional to the frequency <br>of the electric field f and to the square of the electric field strength E.<br>
+ (5.87)<br>
+where o is absolute permittivity of free space (8.854E­12 farad/m),  relative per-<br>
+mittivity or dielectric constant of the material, and tan the loss tangent or dissi-<br>pation factor. From Eq. 5.87, it is clear that fast heating can be accomplished most <br>easily by increasing the field strength; the heating rate increases with the field <br>strength square! The maximum field strength that can be applied is determined by <br>the dielectric strength of the material to be heated. If the field strength is too high, <br>dielectric breakdown will occur. This will result in sparking and can cause severe <br>damage to the material. If the heating rate at the maximum allowable field strength <br>is too slow, further increases can be obtained by increasing the frequency. Most <br>polymers have a dielectric strength that ranges between 100 and 200 kV/cm, a di -<br>electric constant that ranges between 2 and 4, and a dissipation factor that ranges <br>between 0.01 and 0.0001.<br>Dissipation factor, power factor, loss angle, etc. are important terms in dielectric <br>heating. They are defined as follows:<br>
+loss angle =  = 90 = <br>phase angle = <br>power factor = cos = sin<br>dissipation factor = cotan = tan (also loss tangent)<br>loss factor = tan= cotan<br>
+For most polymers, the loss angle is quite small, thus sin  tan; in other words the <br>power factor and dissipation factor are almost equal.<br>
+<hr>
+<A name=187></a> <br>
+5.3Heat Transfer  <b>173</b><br>
+When components of a material have different loss factors, selective heating will <br>occur. The loss factor of most materials increases with moisture content. Regions <br>with high moisture content will heat faster than others; thus, more water will be <br>removed from high moisture regions. This will result in a uniform moisture distri-<br>bution in the material. Non-polar polymers such as polyethylene will not heat well at <br>all in a high frequency field. The relative response to dielectric heating can be made <br>to respond better by adding additives to the polymer. An example of this is the radio <br>frequency heating of ultrahigh molecular weight polyethylene (UHMWPE), contain-<br>ing small amounts of Frequon [18].<br>
+<b>Table 5.1</b>Relative Response of Various Polymers to Dielectric Heating<br>
+Polymer<br>
+Loss factor<br>
+Response<br>
+ABS<br>
+0.025<br>
+Fair<br>
+Acetal<br>
+0.025<br>
+Fair<br>
+Cellulose acetate<br>
+0.15<br>
+Fair<br>
+Epoxy resins<br>
+0.12<br>
+Fair<br>
+Polyamide<br>
+0.16<br>
+Fair<br>
+Polycarbonate<br>
+0.03<br>
+Fair<br>
+Polyester<br>
+0.05<br>
+Fair<br>
+Polyethylene<br>
+0.0008<br>
+None<br>
+Polyimide<br>
+0.013<br>
+Poor<br>
+Polymethyl-methacrylate<br>
+0.09<br>
+Fair<br>
+Polypropylene<br>
+0.001<br>
+None<br>
+Polystyrene<br>
+0.001<br>
+None<br>
+Polytetrafluoroethylene<br>
+0.0004<br>
+None<br>
+Polyvinylchloride<br>
+0.4<br>
+Good<br>
+Rubber, compounded<br>
+0.13<br>
+Fair<br>
+Silicones<br>
+0.009<br>
+None<br>
+Urea-formaldehyde<br>
+0.2<br>
+Good<br>
+Water<br>
+0.4<br>
+Good<br>
+Dielectric or radio frequency heating is used in various parts of the polymer pro-<br>cessing industry. Some examples are preheating for molding, curing of thermo-<br>setting resins, heat-sealing of film, drying of coatings on web substrates, flow mold-<br>ing, etc.<br>
+<b>5.3.5.2Microwave Heating<br></b>Microwave heating is a close cousin of dielectric heating; the main difference being <br>the higher frequencies of microwaves, ranging from about 1000 to 100,000 MHz. <br>This is about two to three orders of magnitude higher than the frequency spectrum <br>
+<hr>
+<A name=188></a><b>174</b>  <b>5Fundamental Principles</b><br>
+of dielectric energy. By definition, the wavelength of microwave energy must lie in <br>the range of the spectrum between 1 m and 1 mm. This corresponds to a frequency <br>range of 300 MHz (3E8 Hz) to 300 GHz (3E11 Hz). The frequencies that can be used <br>in the U.S. are controlled by the Federal Communications Commission. For indus-<br>trial applications, the two most important microwave frequencies are 915 MHz and <br>2450 MHz. The lower frequency is generally used for high-powered systems (over <br>200 kW) where the power factor of the material is reasonably high. The higher fre-<br>quency is used for low-power systems (less than 100 kW) where the material has a <br>relatively low power factor. Consumer microwave ovens operate at 2450 MHz.<br>When a dielectric material is placed in a microwave field, the dipolar molecules will <br>tend to align their dipole moment along the field intensity vector. When the field <br>intensity vector varies sinusoidally with time, the direction of the vector will reverse <br>every half cycle. This will cause a realignment of the polar molecules. The internal <br>friction that has to be overcome involves a loss of energy from the electromagnetic <br>wave. This results in the conversion of a portion of the electromagnetic energy into <br>thermal energy. In this case, the heat generation is proportional to the number of <br>reversals of the electric field vector, i.e., the frequency. The amount of displacement <br>that occurs during each reversal is determined by the electric field strength. Thus, the <br>heat generation is also a function of the electric field strength, just as with RF heating.<br>The rate of heating by microwave energy is described by the same equation used for <br>radio frequency heating, Eq. 5.87. Thus, the amount of heating depends on the field <br>strength, frequency, and loss factor. The latter factor is a material property; the first <br>two factors are dependent on the details of the hardware.<br>The heat is generated throughout the material; however, the power level reduces <br>with depth of penetration. The depth at which the power is reduced to one-half is <br>given by:<br>
+ (5.88)<br>
+where o is the wavelength in free space,  the permittivity or dielectric constant of <br>
+the material, and  the loss angle. Considering that wavelength is the speed of light <br>divided by the frequency, Eq. 5.88 can also be written as:<br>
+ (5.89)<br>
+From these expressions, one can see that the depth of penetration reduces with <br>increasing frequency and with increasing loss factor.<br>If heat losses due to conduction, convection, radiation, or change of state are neg-<br>lected, the rate of increase in temperature   from the absorption of microwave <br>energy can be determined from the following equation:<br>
+<hr>
+<A name=189></a> <br>
+5.4Basics of Devolatilization  <b>175</b><br>
+ (5.90)<br>
+where   is the rate of energy dissipation per unit volume from Eq. 5.87, Cp is the <br>
+specific heat, and  the material density.<br>The penetrating action of microwave energy enables rapid and uniform heating of <br>large cross-sections. With conventional heating methods (hot air, steam, infrared, <br>fluidized bed, etc.), the rate of heating is limited by the poor thermal conductivity of <br>polymers. This is not the case in microwave heating; very short heating chambers <br>can be used.<br>Applications of microwave heating are drying, continuous curing of polymers (rub-<br>bers, filled polyethylenes, etc.), preheating for compression or transfer molding, <br>bonding, etc.<br>
+<b></b><br>
+ <b>5.4Basics of Devolatilization</b><br>
+In devolatilization, one or more volatile components are extracted from the polymer. <br>The polymer can be either in the solid state or in the molten state. Two processes <br>occur in the devolatilization process. First, the volatile components diffuse to the <br>polymer-vapor interface; then the volatile components evaporate at the interface <br>and are carried away. Thus, the first part of the process is a diffusional mass trans-<br>port and the second part a convective mass transport. If the diffusional mass flow <br>rate is less than the convective mass flow rate, the process is diffusion-controlled. In <br>polymer-volatile systems, the diffusion constants are generally very low, and, there-<br>fore, in many polymer devolatilization processes the process is diffusion-controlled.<br>The important relationship in diffusional mass transport is Fick's law. It states that <br>in a one-dimensional diffusion, the positive mass flux of component A is related to a <br>negative concentration gradient. It can be written as:<br>
+ (5.91)<br>
+where JA is the diffusional mass flow rate, CA the local concentration of component <br>
+A, and D'AB the binary diffusivity.<br>Fick's law is valid for constant densities and for relatively low concentrations of <br>component A in component B. The term binary mixture is used to describe a two-<br>component mixture. A binary diffusivity is the diffusion constant of one component <br>of a binary mixture. The diffusional mass transport is driven by a concentration <br>
+<hr>
+<A name=190></a><b>176</b>  <b>5Fundamental Principles</b><br>
+gradient, as described by Fick's law. This is very similar to Fourier's law, which <br>relates heat transport to a temperature gradient; see Eq. 5.45. It is also very similar <br>to Newton's law, which relates momentum transport to a velocity gradient; see <br>Eq. 6.16. Because of the similarities in diffusional mass transport, heat transport, <br>and momentum transport, many problems in diffusion are described with equations <br>of the same form as used in heat transfer problems or momentum transfer prob-<br>lems. Also, several of the dimensionless numbers that are used in heat transfer <br>problems (see Section 5.3.3) are also used in diffusional mass transfer problems.<br>For a binary system of constant density, where a low concentration component A is <br>diffusing through the other component, the equation of continuity for component A <br>can be written as:<br>
+ (5.92)<br>
+This equation of continuity, which incorporates Fick's law, is used to describe diffu-<br>sional transport problems. In most analyses of diffusion processes, it is assumed <br>that the concentration at the liquid-vapor or solid-vapor interface is the equilibrium <br>concentration between the vapor phase and the liquid or solid phase. When a liquid <br>phase of a mixture is in equilibrium with a vapor phase of that mixture, the partial <br>pressure of one component depends on the temperature, pressure, and entire com-<br>position of the mixture. Partial pressure  A of component A is defined as:<br>
+ (5.93)<br>
+where xA is the mole fraction of component A in the gas mixture and P is the total <br>
+pressure on the mixture.<br>The partial pressure for ideal gases is described by Dalton's law:<br>
+ (5.94)<br>
+where Ro is the gas constant, T is the absolute temperature, n is the number of <br>
+weight moles of gas, and V is the volume.<br>For a binary mixture, the composition is completely specified by x'A, which is the <br>
+mole fraction of component A in the liquid in equilibrium. In this case, the partial <br>pressure of component A will be a function of pressure, temperature, and x'A. If the <br>
+properties of the liquid are pressure-independent, and if the gases behave as ideal <br>gases, then the partial pressure of component A at constant temperature can be <br>written as a function of only x'A. If the liquid phase consists of only one component, <br>
+the partial pressure of A equals the vapor pressure of pure A. The partial pressure <br>of component A, P­A, is described as a function of x' by Henry's law. It states that PA <br>
+is directly proportional to x'A at low concentrations of component A:<br>
+<hr>
+<A name=191></a> <br>
+5.4Basics of Devolatilization  <b>177</b><br>
+ (5.95)<br>
+where HA is the Henry's law constant.<br>The Henry's law constant depends on the temperature, the volatility, and the pres-<br>sure. It is not valid for substances such as electrolytes, which dissociate in solution. <br>For ideal solutions, Henry's law is valid over the entire range of concentrations <br>(0­100%), and the Henry's law constant equals the vapor pressure of that component.<br>Polymer-solvent mixtures are highly non-ideal. Because of the very long polymer <br>molecules, the polymer exerts an influence far in excess of its molar concentration. <br>This behavior is often described by the Flory-Huggins relations [22, 23]:<br>
+ (5.96)<br>
+where P is the effective partial pressure of the volatile component, Po is the vapor <br>
+pressure of the pure volatile component, DP is the degree of polymerization, Vp is <br>
+the volume fraction of the polymer, and  is an interaction parameter.<br>For polymer-solvent systems where the solvent is chemically similar, the interaction <br>parameter generally falls within the range of 0.3 to 0.5. If the volatile component is <br>fully miscible, a first approximation of the interaction parameter is  = 0.4. If the <br>volatile component is not fully miscible, the interaction parameter  = 0.5.<br>For polymers with a high degree of polymerization and relatively small concentra-<br>tions of the volatile component A, the Flory-Huggins relationship can be simplified to:<br>
+ (5.97)<br>
+where VA is the volume fraction of the volatile component A. According to Eq. 5.97, <br>
+the ratio of partial pressure to vapor pressure is directly proportional to VA and <br>
+depends exponentially on the interaction parameter . The  /Po ratio as a function <br>
+of VA at various values of the interaction parameter is shown in Fig. 5.3.<br>Figure 5.3 is based on Eq. 5.96 with a degree of polymerization DP = 1000.<br>It can be seen that when the concentration of the volatile component is below 5%, <br>the relationship between   and VA is essentially linear. In this range, Henry's law <br>
+can be applied with reasonable accuracy and Eq. 5.97 can be used. At concentra-<br>tions above 5% considerable deviations from Henry's law (linear behavior) occur and <br>the Flory-Huggins (F-H) relationship, or a similar relationship, should be used. In <br>many cases, it has been observed that the interaction parameter  of the F-H rela-<br>tionship is concentration dependent to the extent that this concentration depend-<br>ence cannot be neglected. This indicates that the basic assumptions underlying the <br>F-H relationship are not fulfilled.<br>
+<hr>
+<A name=192></a><b>178</b>  <b>5Fundamental Principles</b><br>
+1.0<br>
+0.1<br>
+=0.5<br>
+o<br>
+=0.4<br>
+P/P<br>
+<br>
+0.01<br>
+=0.3<br>
+0.001<br>
+<b> Figure 5.3</b> <br>
+0.001<br>
+0.01<br>
+0.1<br>
+1.0<br>
+P<br>/P0 ratio versus VA for three values of the <br>
+VA<br>
+interaction parameter<br>
+An improved theory for vapor-liquid equilibrium of mixtures based on free-volume <br>considerations was proposed by Prigogine [24, 25]. This theory has been further <br>developed by various workers, e.g., Flory [26]. Bonner and Prausnitz [27] discuss <br>the new theory in detail and describe its application with a number of examples.<br>The viscosity of a polymer melt generally reduces with increased amounts of vola-<br>tile component. Figure 5.4 shows the viscosity of polystyrene as a function of the <br>solvent concentration. In this example the solvent is ethylbenzene [28].<br>
+<b> Figure 5.4</b> <br>
+Viscosity of polystyrene as a function <br>
+of solvent concentration<br>
+It can be seen that in this example the viscosity reduces exponentially with the sol-<br>vent concentration. A 10% change in solvent concentration causes approximately a <br>5× change in viscosity. Thus, if the initial solvent concentration is 20% and the final <br>
+<hr>
+<A name=193></a> <br>
+5.4Basics of Devolatilization  <b>179</b><br>
+concentration almost zero, the viscosity increase as a result of devolatilization will <br>be about 25×! This indicates that when substantial amounts of volatiles are removed <br>from a polymer melt, very large increases in viscosity can occur as a result of the <br>devolatilization.<br>The flow of concentrated polymer solutions and polymer melts is essentially always <br>laminar as a result of the high viscosity. Heat transfer in such flow systems is quite <br>poor because the heat transfer occurs primarily by conduction and the thermal con-<br>ductivity in most cases is very low; see also Section 6.3.1.<br>The diffusion in concentrated polymer solutions is much slower than in low viscos-<br>ity (low molecular weight) liquids. The diffusion coefficients for concentrated poly-<br>mer solutions range from about 10­8 m2/s to 10­12 m2/s. For low viscosity liquids, <br>the diffusion coefficients generally range from about 10­6 m2/s to 10­7 m2/s. The <br>difference is several orders of magnitude! The diffusion rate is highly temperature-<br>dependent. At higher temperatures, the vibration of segments of the polymer mole-<br>cules becomes more pronounced and the density of the polymer reduces. As a result, <br>diffusion of a volatile component will occur at a higher rate. The rate of diffusion will <br>generally also depend on the actual concentration of the volatile component. The <br>presence of a low molecular weight component increases the mobility of the poly-<br>mer molecules. Thus, the rate of diffusion will tend to be higher at large concentra-<br>tions of the volatile component. Figure 5.5 shows the diffusion coefficient as a func-<br>tion of the solvent concentration at various temperatures for a system of PMA and <br>methylacetate [29].<br>This figure clearly shows the temperature and concentration dependence of the dif-<br>fusion coefficient.<br>
+<b>Figure 5.5</b>Diffusion coefficient as a function of solvent concentration<br>
+<hr>
+<A name=194></a><b>180</b>  <b>5Fundamental Principles</b><br>
+<b>5.4.1Devolatilization of Particulate Polymer</b><br>
+Theoretical description of devolatilization of particulate polymer can generally be <br>achieved with a relatively high degree of accuracy. In most cases, the process will be <br>diffusion controlled. The diffusion coefficients in solid polymers are very low, rang-<br>ing from about 10­12 m2/s to 10­14 m2/s. The temperature in the polymeric particle <br>can usually be taken as constant since the thermal diffusivity (  10­7 m2/s) is <br>many orders of magnitude higher than the diffusion coefficient.<br>In the case of spherical particles with low concentrations of volatile components for <br>which the concentration dependence of the diffusion coefficient can be neglected, <br>the diffusion equation in spherical coordinates can be written as:<br>
+ (5.98)<br>
+where C is the concentration of the volatile component and D the diffusion coeffi-<br>
+cient.<br>If Ce is the equilibrium concentration at the interface and Co the initial concentra-<br>
+tion, then the solution to Eq. 5.98 can be written in terms of the average concentra-<br>tion   as a function of time [30]:<br>
+ (5.99)<br>
+where R is the radius of the spherical particle.<br>The equilibrium concentration Ce is usually very small relative to the initial concen-<br>
+tration Co, therefore, the Ce term is often neglected. In this case:<br>
+ (5.99a)<br>
+If the temperature cannot be assumed constant, then the equations have to be solved <br>numerically. The same is true if the diffusion coefficient is dependent on the concen-<br>tration.<br>In many cases, however, one can reasonably assume a linear dependence on the dif-<br>fusion coefficient on concentration:<br>
+ (5.100)<br>
+where c is the coefficient describing the concentration dependence of the diffusion <br>
+coefficient.<br>
+<hr>
+<A name=195></a> <br>
+5.4Basics of Devolatilization  <b>181</b><br>
+Figure 5.6 shows the dimensionless concentration  /Co as a function of dimension-<br>
+less time D'ot/R2 at various values of the parameter cCo/D as determined by Meier <br>
+[31].<br>The top curve, for which this parameter is zero, represents the case for which the <br>diffusion coefficient is independent of concentration and is described by Eq. 5.99.<br>The dimensionless Dot/R2 can be considered a Fourier number for diffusion.<br>
+<br>
+<b> Figure 5.6</b> <br>
+Dimensionless concentration versus <br>
+dimensionless time<br>
+<b>5.4.2Devolatilization of Polymer Melts</b><br>
+In the devolatilization of polymer solutions and polymer melts, the diffusion of the <br>volatile component is in many cases the rate-controlling part of the process. It is <br>generally assumed that the concentration at the interface is at the equilibrium con-<br>centration corresponding to the partial pressure of the volatile component in the <br>vapor. A concentration gradient will form in the melt film, and the diffusion rate will <br>be determined by the slope of the concentration gradient. If the volatile concentra-<br>tion is large, the viscosity of the liquid will be relatively low and the mass transport <br>of the volatile component will often occur by bubble transport. This is frequently <br>referred to as foam devolatilization. This causes a rather rapid reduction in volatile <br>concentration and results in a rapid increase in viscosity of the liquid. The increas-<br>ing viscosity inhibits the formation of bubbles, and as the volatile concentration <br>becomes low, the mass transport will be governed solely by molecular diffusion. The <br>
+<hr>
+<A name=196></a><b>182</b>  <b>5Fundamental Principles</b><br>
+devolatilization process of polymer melts is usually analyzed as a diffusion-con-<br>trolled process. Relatively little work has been done to study and analyze foam <br>devolatilization [32, 35­37]. Devolatilization in single screw extruders generally <br>occurs at relatively low levels of volatiles; therefore, foam devolatilization is usually <br>not considered to play a role of importance in devolatilizing extrusion. However, <br>research at Farrel Corp. indicates that foam devolatilization occurs quite readily and <br>may determine the devolatilization process to a large extent [38, 39].<br>A technique that is often employed in devolatilization of polymer solutions is flash <br>devolatilization [34]. In this technique, the polymer solution is delivered to a flash <br>point under high pressure and at temperatures above the boiling point of the vola-<br>tile component. The solution is then expanded through a nozzle; large amounts of <br>volatiles can thus be extracted rather quickly. The foamy liquid that results from <br>this operation is often exposed to another devolatilization step to remove residual <br>amounts of volatiles. This second step is generally a conventional melt film devola-<br>tilization, where the material in the film is continuously renewed to obtain an effec-<br>tive extraction of the volatiles.<br>Stripping agents such as water are often added to the polymer to enhance the devol-<br>atilization process. The improvement is obtained by bubble formation, which sub-<br>stantially improves the devolatilization process.<br>Consider a liquid polymer film with surface area A and depth H moving in direction <br>x in plug flow with a volumetric flow rate  f. The film is losing a volatile solute by <br>
+evaporation in the y direction at a rate of . If the mass transport in the y direction <br>occurs by molecular diffusion only, and if both dispersion in the x direction and <br>changes in  f due to loss of volatile are neglected, an expression for  can be devel-<br>
+oped. The exposure time of the film f is defined as:<br>
+ (5.101)<br>
+The characteristic time for diffusion is defined as:<br>
+ (5.102)<br>
+where D is the molecular diffusivity of the volatile solute in the liquid polymer.<br>It can be shown [30] that if f/D  0.1, the film can be considered of infinite depth. <br>
+In this case, the layer in which the concentration is varying is much thinner than <br>the total thickness of the melt film H. The stage efficiency X for this situation can be <br>expressed as:<br>
+ (5.103)<br>
+<hr>
+<A name=197></a> <br>
+5.4Basics of Devolatilization  <b>183</b><br>
+The stage efficiency is the actual rate of evaporation divided by the maximum pos-<br>sible rate of evaporation:<br>
+ (5.104)<br>
+where Co is the initial volatile concentration and Ce the concentration of the volatile <br>
+component in the liquid phase, which is in equilibrium with the vapor phase.<br>If f /D &gt; 0.1, the melt film cannot be considered infinite. The stage efficiency in this <br>
+case can be described by:<br>
+ (5.105)<br>
+If the film can be considered infinite, the concentration profile can be described by:<br>
+ (5.106)<br>
+With boundary conditions C(0) = Ce and C() = Co, the actual concentration profile <br>
+as a function of time becomes:<br>
+ (5.107)<br>
+The same problem in conductive heat transport will be discussed in Section 6.3.5. <br>The error function erf(x) is defined by Eq. 6.99. Figure 5.7 shows the concentration <br>profile at various values of the parameter Dt.<br>The penetration depth is about 0.1 mm when Dt = 2510­6 [m]. If the diffusion <br>
+coefficient is assumed to be D  10­8 m2/s, then the corresponding time is 0.0625 s. <br>
+If the diffusion coefficient is assumed to be D = 10­6 m2/s, then the corresponding <br>
+time is 6.2510­4 s. Since these are typical values of the diffusion coefficient, the <br>assumption of infinite melt film thickness may not be valid in the analysis of devo-<br>latilization in extrusion equipment. A typical exposure time of the melt film in <br>an extruder is of the order of one second; a typical melt film thickness is 0.1 mm <br>( 0.004 in).<br>The rate of diffusion J at the interface per unit area and per unit time is determined <br>by:<br>
+ (5.108)<br>
+<hr>
+<A name=198></a><b>184</b>  <b>5Fundamental Principles</b><br>
+Thus, the diffusional transport reduces with 1/t. The initial diffusional mass trans-<br>port will be the highest; thereafter, it will reduce with time according to Eq. 5.108. <br>Thus, in order to maintain high devolatilization efficiency, it is very important that <br>the surface through which the volatile is escaping is frequently renewed. This can <br>be achieved by feeding the film into a mixer.<br>
+1.0<br>
+0.1<br>
+0.8<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+4D't = 1.0 mm<br>
+) e 0.6<br>
+-C o<br>
+)/(C e 0.4<br>(<br>C-C<br>
+0.2<br>
+<b> Figure 5.7</b> <br>
+Concentration profile versus <br>
+0 0 0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+depth at various values of <br>
+Film thickness [mm]<br>
+(Dt)0 .5<br>
+The material leaving the mixer will have a homogeneous composition with the same <br>average volatile level as the film entering the mixer. The material leaving the mixer <br>can be spread into a new film with more volatiles diffusing out and evaporating. The <br>process can be repeated many times. It can be demonstrated [33], assuming ideal <br>mixing, that for each n-th stage:<br>
+ (5.109)<br>
+The improved devolatilization efficiency with surface renewal can be assessed by <br>comparing the residual concentration Cn after n ideal surface renewals with expo-<br>
+sure time between film renewal f to the residual concentration C(nf) with the <br>
+same total exposure time but without surface renewal by:<br>
+ (5.110)<br>
+Figure 5.8 shows the ratio Cn/C(nf) for various values of f/D, assuming that the <br>
+value of Ce is very small and negligible.<br>It is clear from Fig. 5.8 that surface renewal can yield a considerably lower residual <br>concentration compared to the case without surface renewal. The improvement in <br>devolatilization efficiency becomes more pronounced when the ratio f/D becomes <br>
+<hr>
+<A name=199></a> <br>
+5.4Basics of Devolatilization  <b>185</b><br>
+larger. When the ration f/D &gt; 0.1, a single curve represents the Cn/C(nf) versus n <br>
+relationship. In this case, Eq. 5.110 can be written as:<br>
+ (5.111)<br>
+1.0<br>
+0.9<br>
+f/D=0.001<br>
+)<br> f 0.8<br>
+<br>
+/C(n<br>
+f/D=0.01<br>
+nC<br>
+0.7<br>
+f/D&gt;0.1<br>
+0.6<br>
+0.5<br>
+2<br>
+4<br>
+6<br>
+8<br>
+10<br>
+Number of ideal surface renewals<br>
+<b>Figure 5.8</b>Ratio of Cn/C(nf) for various values of f/D<br>
+When the ratio f/D is very small (&lt; 0.001), the benefits of surface renewal are rela-<br>
+tively minor. In actual polymer processing equipment, the surface renewal process <br>will generally not be ideal, because only a fraction of the bulk material will be spread <br>out into a thin film. Therefore, the actual devolatilization efficiency would be <br>expected to be less than predicted by Eqs. 5.109 through 5.111.<br>
+<b>Appendix 5.1</b><br>
+<b>Example: Pipe Flow of Newtonian Fluid</b><br>
+In pipe flow, the fluid moves as a result of a pressure gradient along the pipe; see <br>Fig. 5.9.<br>The velocity at the wall is zero and maximum at the center. The velocity gradient <br>(shear rate) is zero in the center and maximum at the wall. The momentum balance <br>for this case can be determined by taking a force balance on a small fluid element as <br>shown in Fig. 5.9; this gives:<br>
+ (1)<br>
+<hr>
+<A name=200></a><b>186</b>  <b>5Fundamental Principles</b><br>
+Velocity profile<br>
+r<br>
++d<br>
+P+dP<br>
+P dr<br>
+<br>
+z<br>
+dz<br>
+R <b> Figure 5.9</b> <br>
+Flow through a circular pipe<br>
+This results in:<br>
+ (2)<br>
+This same result can be obtained directly from momentum balance Eq. 5.4(c) by <br>eliminating the stress derivatives in the tangential () and axial direction (z); this <br>yields:<br>
+ (3)<br>
+Verify that Eqs. 2 and 3 are the same when /r = d/dr, which is true in this case. <br>For a Newtonian fluid, the shear stress can be related to the velocity gradient by:<br>
+ (4)<br>
+where  is the Newtonian viscosity; see also Section 6.2. Inserting Eq. 4 into Eq. 2 or <br>3 gives:<br>
+ (5)<br>
+where gz is the axial pressure gradient                     .<br>
+By integrating once, the velocity gradient is obtained:<br>
+ (6)<br>
+Since the velocity gradient is zero at the center (r = 0), the integration constant C1 <br>
+has to be zero. The velocity profile is obtained by integrating Eq. 6:<br>
+ (7)<br>
+<hr>
+<A name=201></a> <br>
+5.4Basics of Devolatilization  <b>187</b><br>
+The integration constant C2 can be determined from the condition that the velocity <br>
+at the wall is zero, i.e., v(R) = 0. This yields:<br>
+ (8)<br>
+Thus, the velocity profile becomes:<br>
+ (9)<br>
+The velocity at the center (r = 0) is maximum and is given by:<br>
+ (10)<br>
+The velocity is positive when the pressure gradient is negative. The velocity profile <br>can now be expressed as:<br>
+ (11)<br>
+The volumetric flow rate can now be determined by integrating the velocity over the <br>cross-sectional area of the pipe:<br>
+ (12)<br>
+With Eq. 9, the flow rate can be determined to be:<br>
+ (13)<br>
+This is the well-known Poiseuille equation, first published in 1840 [40]. The flow <br>rate is directly proportional to the pressure gradient and inversely proportional to <br>the fluid viscosity. The flow rate depends strongly on the radius; it increases with <br>the radius to the fourth power.<br>
+<hr>
+<A name=202></a><b>188</b>  <b>5Fundamental Principles</b><br>
+References<br>1.  R.B. Bird, W.E. Stewart, and E.N. Lightfoot, "Transport Phenomena," Wiley, NY (1960)<br>2.  J.R. Welty, C.E. Wicks and R.E. Wilson, "Fundamentals of Momentum, Heat and Mass <br>
+Transport," Wiley, NY (1969)<br>
+3.  C. Truesdell and R.A. Toupin, "The Classical Field Theories," in Handbuch der Physik, <br>
+Vol. III, Springer, Berlin (1960)<br>
+4.  W.J. Beek and K.M. Muttzall, "Transport Phenomena," Wiley, NY (1975)<br>5.  L.E. Sisson and D.R. Pitts, "Elements of Transport Phenomena," McGraw-Hill, NY <br>
+(1972)<br>
+6.  W.C. Reynold and H.C. Perkins, "Engineering Thermodynamics," 2nd Edition, McGraw-<br>
+Hill, NY (1977)<br>
+7.  G.J. Van Wylen and R.E. Sonntag, "Fundamentals of Classical Thermodynamics," 2nd <br>
+Edition, Wiley, NY (1973)<br>
+8.  R.W. Haywood, "Equilibrium Thermodynamics," Wiley, NY (1980)<br>9.  A. Bejan, "Entropy Generation through Heat and Fluid Flow," Wiley, NY (1982)<br>10.  P.J. Flory, "Principles of Polymer Chemistry," Cornell University Press, Ithaca, NY <br>
+(1953)<br>
+11.  L.R.G. Treloar, "The Physics of Rubber Elasticity," 2nd Edition, Oxford Univ. Press, <br>
+Oxford (1958)<br>
+12.  F. Bueche, "Physical Properties of High Polymers," Wiley­Interscience, NY (1962)<br>13.  A.V. Tobolsky, "Properties and Structure of Polymers," Wiley, NY (1960)<br>14.  R.L. Miller (Ed.) "Flow-Induced Crystallization in Polymer Systems," Gordon and Breach <br>
+Science Publishers, NY (1979)<br>
+15.  G. Astarita, Polym. Eng. Sci., 14, 730­733 (1974)<br>16.  H.S. Carslaw and J.C. Jaeger, "Conduction of Heat in Solids," 2nd Edition, Oxford Univ. <br>
+Press, Oxford (1959)<br>
+17.  R. Siegel and J.R. Howell, "Thermal Radiation Heat Transfer," 2nd Edition, McGraw-Hill <br>
+(1981)<br>
+18.  B. Miller, Plastics World, March, 99­104 (1981)<br>19.  R.R. Kraybill, SPE ANTEC, Vol. 27, 590­592 (1981)<br>20.  R.R. Kraybill and W.J. Hennessee, SPE ANTEC, Vol. 28, 826­829 (1982)<br>21.  R.R. Kraybill, SPE ANTEC, Vol. 29, 466­468 (1983)<br>22.  P.J. Flory, J. Chem. Phys., 10, 51 (1942)<br>23.  M.L. Huggins, Ann. NY Acad. Sci., 43, 9 (1942)<br>24.  I. Prigogine, N. Trappeniers and V. Mathot, Disc. Farad. Soc., 15, 93 (1953); J. Chem. <br>
+Phys., 21, 559 (1953)<br>
+25.  I. Prigogine, "The Molecular Theory of Solutions," North Holland, Amsterdam (1957)<br>26.  P.J. Flory, J. Am. Chem. Soc., 87, 1833 (1965)<br>
+<hr>
+<A name=203></a> References <br>
+<b>189</b><br>
+27.  D.C. Bonner and J.M. Prausnitz, AIChE J., 19, 943 (1973)<br>28.  E. Schumacher, M.Sc. Thesis, Univ. of Stuttgart, Germany (1966)<br>29.  H. Fujita, A. Kishimoto and K. Matsumoto, Trans. Faraday Soc., 56, 424 (1960)<br>30.  I. Crank, "The Mathematics of Diffusion," 2nd Edition, Clarendon Press, Oxford (1975)<br>31.  E. Neier, Chemie Ing. Techn., 42, 20 (1970)<br>32.  R.E. Newman and R.H.M. Simon, 73rd Annual AIChE Meeting, Chicago (1980)<br>33.  J.A. Biesenberger, Polym. Eng. Sci., 20, 1015­1022 (1980)<br>34.  M.H. Pahl in "Entgasen von Kunststoffen," VDI-Verlag GmbH, Duesseldorf (1980)<br>35.  K.G. Powell and C.D. Denson, Paper No. 41a presented at the Annual Meeting of the <br>
+AIChE in Washington, DC (1983)<br>
+36.  H.J. Yoo and C.D. Han, Paper No. 41b presented at the Annual Meeting of the AIChE in <br>
+Washington, DC (1983)<br>
+37.  M. Amon and C.D. Denson, Polym. Eng. Sci., 24, 1026­1034 (1984)<br>38.  M.A. Rizzi, P. Hold, M.R. Kearney, and A.D. Siegel, Paper No. 41e presented at the <br>
+Annual Meeting of the AIChE in Washington, DC (1983)<br>
+39.  P.S. Mehta, L.N. Valsamis, and Z. Tadmor, Polym. Process Eng., 2, 103­128 (1984)<br>40.  J.L. Poiseuille, Compte Rendus, 11, 961 and 1041 (1840); 12, 112 (1841)<br>
+<hr>
+<A name=204></a><b>6 Important Polymer </b><br>
+<b>Properties</b><br>
+To understand the extrusion process, it is not enough just to know the hardware <br>aspects of the machine. To fully understand the entire process, one also has to know <br>and appreciate the properties of the material being extruded. The characteristics of <br>the polymer determine, to a large extent, the proper design of the machine and the <br>behavior of the process. There are two main classes of properties important in the <br>extrusion process: the rheological properties and the thermal properties. The rheo-<br>logical properties describe how the material deforms when a certain stress is <br>applied. The rheological properties of the bulk material are of importance in the <br>feed hopper region of the extruder. The rheological properties of the polymer melt <br>are important in the plasticating zone, the melt conveying zone, and the die forming <br>region. Thermal properties allow prediction of temperature changes in the polymer <br>and how the polymer reacts to these temperature changes.<br>
+<b></b><br>
+ <b>6.1Properties of Bulk Materials</b><br>
+Some of the most important properties of the bulk material are the bulk density, the <br>coefficient of friction, and particle size and shape. From these properties, the trans-<br>port behavior of the bulk material can be described with reasonable accuracy. These <br>properties will be discussed in more detail in the following section.<br>
+<b>6.1.1Bulk Density</b><br>
+The bulk density is the density of the polymeric particles, including the voids <br>between the particles. It is determined by filling a container of certain volume <br>(1 liter or more) with the bulk material without applying pressure or tapping. The <br>content is then weighed and the bulk density is obtained by dividing the material <br>weight by the volume. In order to get reproducible results, the dimensions of the <br>container should be several orders of magnitude larger than the particular dimen-<br>
+<hr>
+<A name=205></a><IMG src="Polymer Extrusion by Chris Rauwendaal-205_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-205_35.jpg"><br>
+<b>192</b>  <b>6Important Polymer Properties</b><br>
+sion.* Low bulk density materials (b &lt; 0.2 g/cc) tend to cause solids conveying <br>
+problems, either in the feed hopper or in the feed section of the extruder. Materials <br>with irregularly shaped particles tend to have a low bulk density; examples are fiber <br>scrap or film scrap (flakes). When the bulk density is low, the mass flow rate will be <br>low as well. Thus, the solids conveying rate may be insufficient to supply the down-<br>stream zones (plasticating and melt conveying) with enough material. Special <br>devices and special extruders have been designed to deal with these low bulk den-<br>sity materials. A crammer feeder, as shown in Fig. 6.1, is a device used to improve <br>the solids transport from the feed hopper into the extruder barrel.<br>Special extruders have been designed with the diameter of the feed section larger <br>than the transition and metering section. Two possible configurations, both found <br>commercially, are shown in Fig. 6.2.<br>
+Hopper<br>
+Crammer<br>
+auger<br>
+<b> Figure 6.1</b> <br>
+Example of crammer feeder<br>
+<b> Figure 6.2</b> <br>
+Two examples of <br>
+ <br>extruders designed to <br>
+handle low bulk density <br>
+feed<br>
+* ASTM D1895 describes standard test methods for apparent density (bulk density), bulk factor <br>
+(ratio bulk density to actual density), and pourability of plastic materials .<br>
+<hr>
+<A name=206></a> <br>
+6.1Properties of Bulk Materials  <b>193</b><br>
+Since scrap or regrind is more difficult to handle, it is often blended with the virgin <br>material to reduce the handling problems.<br>The bulk density at atmospheric pressure is useful but limited information. It is <br>very important to know how the bulk density changes with pressure, because the <br>compressibility of the bulk material determines, to a large extent, the solids convey-<br>ing behavior. Compaction occurs by a rearrangement of the particles and an actual <br>deformation of the particles. The difference between the loosely packed (untapped) <br>bulk density and the packed or tapped bulk density is sometimes referred to as com-<br>pressibility. This is actually the compression due to rearrangement of the particles. <br>Thus, it would be better to refer to this property as rearrangement compressibility. <br>When the rearrangement compressibility is large, this indicates that the material is <br>prone to packing in storage. This can result in discharge problems. The difference <br>between free-flowing and non-free-flowing is said to occur at a rearrangement com-<br>pressibility of approximately 20% [1]. Other workers [2] put the boundary between <br>free-flowing and non-free-flowing at an angle of repose of about 45°, with the non-<br>free-flowing material having an angle repose of more than 45° and the free-flowing <br>materials an angle of repose less than 45°. The angle of repose is the included angle <br>formed between the side of a cone-shaped pile of material and the horizontal plane; <br>see Fig. 6.3.<br>
+<b> Figure 6.3</b> <br>
+Angle of repose<br>
+The above-mentioned boundaries between free-flowing and non-free-flowing are <br>only approximate indicators. They are used because of their simplicity and ease of <br>measurement. However, neither rearrangement compressibility nor the angle of <br>repose are true measures of the flowability of particulate materials. Free-flowing <br>materials are also referred to as non-cohesive materials and non-free-flowing mate-<br>rials as cohesive materials. The shear stress at incipient internal shear deformation <br>in non-cohesive materials can be uniquely related to the normal stress. Conse-<br>quently, the coefficient of cohesion (see Eq. 6.6) is zero for non-cohesive particulate <br>materials.<br>When the rearrangement compressibility is about 40%, the material will have a very <br>strong tendency to pack in the feed hopper, and the chance of discharge problems <br>will be very high. The tendency towards packing can be assessed in a qualitative <br>fashion by the hand squeeze test. Material is squeezed in the hand, and the condi-<br>tion of the material is observed after squeezing. If the material has formed a hard <br>
+<hr>
+<A name=207></a><b>194</b>  <b>6Important Polymer Properties</b><br>
+clump that cannot be easily broken up, this indicates a moderately compressible <br>material. If the material does not clump at all but flows after squeezing, this indi-<br>cates a low compressibility and a relatively free-flowing material.<br>Many investigators have studied the compression characteristics of bulk materials <br>[3­14]; a number of these studies have dealt with polymeric bulk materials. The <br>compaction process is quite complicated for a number of reasons. The distribution of <br>stresses in the material during compaction is rather complex and depends very <br>much on the geometry and surface conditions of the compression apparatus and the <br>detailed characteristics of the bulk material. The effect of pressure on bulk density <br>is often described by an empirical relationship:<br>
+ (6.1)<br>
+where  is the porosity, 0 the porosity at zero pressure, P the pressure, and  the <br>
+compressibility coefficient.<br>It should be realized that Eq. 6.1 is an approximate relationship. The actual compac-<br>tion behavior will depend strongly on the details of the compaction apparatus and <br>the compacting procedures and conditions.<br>
+<b>6.1.2Coefficient of Friction</b><br>
+The coefficient of friction of the bulk material is another very important property. <br>One can distinguish both internal and external coefficient of friction. The internal <br>coefficient of friction is a measure of the resistance present when one layer of parti-<br>cles slides over another layer of particles of the same material. The external coeffi-<br>cient of friction is a measure of the resistance present at an interface between the <br>polymeric particles and a wall of a different material of construction. The coefficient <br>of friction is simply the ratio of the shear stress at the interface to the normal stress <br>at the interface. Friction itself is the tangential resistance offered to the sliding of <br>one solid over another.<br>In discussing the coefficient of friction, one has to specify whether it is a static or <br>dynamic coefficient of friction. The static coefficient of friction, f*, is determined by<br>
+ (6.2)<br>
+where *ij is the maximum shear stress just before sliding occurs and ii is the cor-<br>
+responding normal stress.<br>The dynamic coefficient of friction f is determined by<br>
+ (6.3)<br>
+<hr>
+<A name=208></a> <br>
+6.1Properties of Bulk Materials  <b>195</b><br>
+where ij is the actual shear stress during sliding motion and ii is the corresponding <br>
+normal stress.<br>A common way to determine the static coefficient (external) is the measurement of <br>the angle of slide. The object is put on a surface and the angle of the surface with <br>the horizontal plane is increased until the object just begins to slide. The angle that <br>corresponds to the onset of sliding is the slide angle and the coefficient of static fric-<br>tion equals the tangent of the angle of slide. Thus,<br>
+ (6.4)<br>
+where f* is the external coefficient of static friction and s the angle of slide.<br>Bowden and Tabor [23] attributed friction to two factors, one factor being the ad -<br>hesion that occurs at the regions of real contact. The actual area of contact is several <br>orders (about four) of magnitude smaller than the apparent contact area. If sliding <br>is to take place, the local regions of adhesion have to be sheared. The second factor <br>is the plowing, grooving, or cracking of one surface by the asperities of the other. <br>In static friction, the only factor of importance is the adhesion at the contact sites. In <br>dynamic friction, the plowing factor starts to play a role whereas the adhesion factor <br>reduces in significance.<br>Measurement of the external coefficient of friction of particulate polymers is very <br>difficult because of the very large number of variables that influence the coefficient <br>of friction. Many investigators have made elaborate measurements on the external <br>coefficient of friction [24­32]. The result of this work is that many variables have <br>been identified that affect the frictional behavior; however, most measurement tech-<br>niques do not yield accurate and reproducible results that can be used in the analy-<br>sis of the extrusion process. The most elaborate measurements and the most mean-<br>ingful results have probably been obtained at the DKI in Darmstadt, Germany [95]. <br>It is possible to obtain reproducible results by very careful experimental techniques <br>and special surface preparation of the metal wall. However, the frictional coefficients <br>determined in this fashion are hardly representative of the frictional process condi-<br>tions occurring in an extruder.<br>Some of the variables that affect the coefficient of friction are temperature, sliding <br>speed, contact pressure, metal surface conditions, particle size of polymer, degree of <br>compaction, time, relative humidity, polymer hardness, etc.<br>The coefficient of friction is very sensitive to the condition of the metal surface. The <br>coefficient of friction of a polymer against an entirely clean metal surface is very low <br>initially, as low as 0.05 or less. However, after the polymer has been sliding on the <br>surface for some time, the coefficient of friction will increase substantially and may <br>stabilize at a value about an order of magnitude higher than the initial value. This <br>behavior was described in detail by Schneider [24, 25] for a variety of polymers. <br>This effect is attributed to the transfer of polymer to the metal surface. Instead of <br>
+<hr>
+<A name=209></a><b>196</b>  <b>6Important Polymer Properties</b><br>
+pure polymer-metal friction, the actual situation is a polymer-metal/polymer fric-<br>tion. It has also been found that the measured coefficient of friction is changed if the <br>metal surface is accidentally touched by hand. The finger greases actually change <br>the metal surface conditions and the resulting coefficient of friction. The unavail-<br>ability of accurate and appropriate data on coefficient of friction is one of the main <br>stumbling blocks in being able to accurately predict extruder performance. Predic-<br>tions of the solids conveying rate and pressure development from theory are very <br>sensitive to the actual values of the coefficient of friction; e.g. see Fig. 7.16. Thus, <br>for accurate prediction, the coefficient of friction should be known to at least a one <br>percent accuracy; however, this is usually not feasible.<br>A comprehensive survey of the work on polymer friction was put together by Barte-<br>nev and Lavrentev [32]. An exhaustive study on frictional properties was under-<br>taken at the DKI (Deutsches Kunststoff-Institut) in Darmstadt, Germany, sponsored <br>by the VDMA (Verband Deutscher Maschinen- und Anlagenbau). The data was com-<br>piled in a book [95] that contains frictional properties of 27 different polymers, with <br>the coefficient of friction given as a function of temperature, sliding velocity, and <br>normal pressure. The measurements were made on the universal disk-Tribometer <br>(see Section 11.2.1.2, Fig. 11.9), using conditions that closely resemble the friction <br>process in a screw extruder. This publication is probably the most complete compila-<br>tion of frictional properties determined under controlled and meaningful conditions <br>with good reproducibility (better than 10%).<br>A typical plot of external coefficient of friction versus temperature at various pres-<br>sures is shown in Fig. 6.4(a).<br>
+<b> Figure 6.4(a)</b> <br>
+Coefficient of friction versus temperature<br>
+The material is polyethylene (Lupolen 5261 Z) and the sliding velocity is 0.60 m/s. <br>At low pressures, the coefficient of friction increases with temperature, reaches a <br>peak at the melting point, and then starts to drop rapidly. At high pressures, the <br>coefficient of friction drops monotonically with temperature.<br>If the transport of particulate polymer occurs by plug flow, then the only frictional <br>coefficient of importance is the external coefficient of friction. This condition is usu-<br>
+<hr>
+<A name=210></a> <br>
+6.1Properties of Bulk Materials  <b>197</b><br>
+ally assumed in the solids conveying zone of an extruder with a smooth barrel sur-<br>face. However, if there is any internal deformation occurring within the particulate <br>material, the internal coefficient of friction will also start to play a role of impor-<br>tance. This is the case when one analyzes the flow of material through a feed hopper <br>or in the solids conveying zone of an extruder when the barrel surface is grooved at <br>the feed section or when the channel is not fully filled so that no pressure increase <br>and no compacting can take place.<br>The flowability of a particulate material is determined by its shear properties. When <br>internal shear deformation is just about to occur, the local shear stress is called the <br>shear strength. The shear strength is a function of the normal stress; this functional <br>relationship is referred to as the yield locus (YL). For a free-flowing material, the <br>yield locus under fully mobilized friction conditions is<br>
+ (6.5)<br>
+where f*i is the internal static coefficient of friction, i the angle of internal friction <br>
+(i = arctan f*i),  the shear strength, and  the normal stress.<br>The shear strength of non-free-flowing (cohesive) materials is not a unique function <br>of the normal stress. The shear strength of these materials increases with pressure. <br>The YL is a function of consolidation pressure and consolidation time. Thus, the <br>shear strength has to be described by a series of yield loci, each curve representing <br>a certain consolidation pressure and time. These curves can often be described by:<br>
+ (6.6)<br>
+where a is an apparent tensile strength, which is obtained by extrapolating the <br>
+yield locus to zero shear stress; see Fig. 6.4(b).<br>
+<b>Figure 6.4(b)</b>Unconfined yield strength of a cohesive material<br>
+<hr>
+<A name=211></a><b>198</b>  <b>6Important Polymer Properties</b><br>
+The actual tensile strength is usually less than the apparent tensile strength. The <br>value of the shear stress at zero normal stress is often referred to as the coefficient <br>of cohesion c = a tani. This coefficient is a measure of the magnitude of the cohe-<br>
+sive forces in the particulate material that must be overcome for internal shearing to <br>occur.<br>In a state of incipient failure, the yield locus is tangent to the Mohr circle. The Mohr <br>circle graphically represents the equilibrium stress condition at a particular point <br>at any orientation for a system in a condition of static equilibrium in a two-dimen-<br>sional stress field. The equilibrium static conditions can also be applied to suffi-<br>ciently slow steady flows. The maximum principal stress c in Fig. 6.4(b) is called <br>
+the unconfined yield strength. This is the maximum normal stress, under incipient <br>failure conditions, at a point where the other principal stress becomes zero. Such a <br>situation occurs on the exposed surface of an arch or dome in a feed hopper at the <br>moment of failure; see Fig. 7.5(b). In the analysis of bridging in feed hoppers, the <br>unconfined yield strength becomes a very important parameter. The magnitude of <br>the unconfined yield strength is determined by the YL and depends, therefore, on <br>the consolidation pressure and time.<br>The principal stresses in cohesive materials can be related by:<br>
+ (6.7)<br>
+where max is the maximum normal stress and min the minimum normal stress.<br>For a cohesive particulate material, each YL curve ends at a point where the normal <br>stress equals the consolidation pressure. Mohr circles can now be drawn that are <br>tangent to the end point of the various yield loci. The envelope of these circles is <br>called the effective yield locus (EYL). This is generally a straight line passing through <br>the origin; see Fig. 6.5.<br>
+Shear stress <br>
+Ef ective yield locus<br>
+e<br>
+<b> Figure 6.5</b> <br>
+0<br>
+Normal stress  Effective yield locus<br>
+<hr>
+<A name=212></a> <br>
+6.1Properties of Bulk Materials  <b>199</b><br>
+The angle between the EYL and the normal stress axis is called the effective angle <br>of friction, e. The EYL describes the shear stress-normal stress characteristics of a <br>
+particulate material that is consolidated and being sheared under the same stress <br>conditions. This applies directly to a steady flow situation because, in this case, <br>shearing takes place throughout the particulate material. The Mohr circle describ-<br>ing the stress condition at any point must be tangent to the EYL. When the stress <br>field is such that the Mohr circle describing the stress field is below the EYL, no <br>shearing flow will occur. The effective yield locus for a non-cohesive material will <br>coincide with the yield locus. Thus, the effective angle of friction will equal the <br>internal angle of friction (e = i for a non-cohesive material).<br>A shearing cell was developed by Jenike [42] to measure shear properties of parti-<br>culate solids; see Fig. 6.6.<br>
+Fn<br>
+Fs<br>
+<b> Figure  6.6</b> <br>
+The shearing cell developed by Jenike [42]<br>
+In addition to the determination of the various YL curves and the EYL, the shearing <br>cell can also be used to measure the YL curve between the particulate solids and the <br>confining wall. This is referred to as the wall yield locus (WYL), and it generally lies <br>considerably below the YL. If the WYL is a straight line, it can be described by:<br>
+ (6.8)<br>
+where w is the wall angle of friction, f*w the static coefficient of friction at the wall <br>
+(f*w = tanw), w the shear stress at the wall, and wa the adhesive shear stress at the <br>
+wall.<br>Rautenbach and Goldacker [43, 44] described an apparatus developed to measure <br>the internal frictional properties of particulate solids under steady shear. They <br>found that after exceeding the shear strength, the deformation occurred in one or <br>more discrete shear planes. The thickness of these planes was in the order of only a <br>few particle diameters. Thus, they did not observe the development of a continuous <br>velocity gradient throughout the material. These observations were made with non-<br>cohesive polymeric powders. Based on these observations, a dynamic internal coeffi-<br>cient of friction fi was defined, representing the ratio of steady shear stress to steady <br>
+normal stress. It was found that the dynamic internal coefficient of friction was <br>independent of velocity and pressure, slightly dependent on temperature and par-<br>ticle size, and strongly dependent on particle shape. In all cases, the dynamic inter-<br>nal coefficient of friction was much higher (about five times) than the dynamic <br>
+<hr>
+<A name=213></a><b>200</b>  <b>6Important Polymer Properties</b><br>
+external coefficient of friction. The dynamic angle of internal friction (i = arctan fi) <br>
+was found to relate reasonably well with the angle of repose r according to the <br>
+expression:<br>
+ (6.9)<br>
+This expression is empirical and should only be used to estimate the approximate <br>value of i.<br>
+<b>6.1.3Particle Size and Shape</b><br>
+The range of polymeric particles used in extrusion is quite wide, from about 1 micron <br>to about 10 mm. Figure 6.7 shows the nomenclature generally used to describe par-<br>ticulate solids of a certain particle size range.<br>
+<b>Particulate solids</b><br>
+<b>    Powders</b><br>
+<b>Granular solids</b><br>
+<b>Broken solids</b><br>
+<b>    &lt;100</b>µ<b>m</b><br>
+<b>100-5000</b>µ<b>m</b><br>
+<b>&gt;5000</b>µ<b>m</b><br>
+<b>Fine powders</b><br>
+<b>Granules</b><br>
+<b>10-100</b>µ<b>m</b><br>
+<b>100-1000</b>µ<b>m</b><br>
+<b>Superfine powders</b><br>
+<b>Pellets</b><br>
+<b>1-10</b>µ<b>m</b><br>
+<b>1000-5000</b>µ<b>m</b><br>
+<b>Ultrafine powders<br>0.1-1</b>µ<b>m</b><br>
+<b>Figure 6.7</b>Nomenclature in particulate materials<br>
+Particle size can be determined by a variety of techniques, microscopic measure-<br>ment being one of the most common techniques. If the particles have a considerable <br>particle size distribution, one would like to measure this distribution. In this case, <br>microscopic measurement becomes very time consuming, unless it is tied in to an <br>automatic or semi-automatic image analyzer that can generate the distribution <br>curve. Sieving is a simple and popular technique; however, particles should be larger <br>than about 50 micron. Values obtained by sieving of non-spherical particles must be <br>modified to conform with those obtained by methods that yield data on equivalent <br>spherical diameter (esd). The esd is the diameter of a sphere having the same volume <br>as the non-spherical particle. Sedimentation methods are used for particles less <br>than 50 micron. Light transmission or scattering is another method used for particle <br>size measurement.<br>
+<hr>
+<A name=214></a> <br>
+6.1Properties of Bulk Materials  <b>201</b><br>
+Other important parameters in particle analysis are surface area, pore size, and <br>volume. The basic method for measuring surface area involves determining the <br>quantity of an inert gas, usually nitrogen, required to form a layer one molecule <br>thick on the surface of a sample at cryogenic temperature. Many techniques are <br>used for pore size measurement: impregnation with molten metal, particle beam <br>transmission, water absorption, freezing point depression, microscopy, mercury <br>intrusion, and gas condensation and evaporation. The last three techniques are most <br>often utilized.<br>The particle shape can generally be established by simple visual observation or by <br>using a microscope. The transport characteristics of particulate solids are quite sen-<br>sitive to the particle shape. Both the internal and external coefficient of friction can <br>change substantially with variations in particle shape even if the major particle <br>dimensions remain unchanged. Small differences in the pelletizing process can <br>cause major problems in a downstream extrusion process. Variations in the ratio of <br>regrind to virgin polymer can cause variations in the extrusion process.<br>The ease of solids transport is often determined by the particle size. Pellets are <br> <br>generally free-flowing and do not have a strong tendency to entrap air. From a solids <br>conveying point of view, pelletized materials are the easiest to work with. Granules <br>are often free-flowing, sometimes semi-free-flowing; they are more likely to entrap <br>air. Semi-free-flowing granules may require special feeding devices (such as a <br>vibrating pad on the hopper) to ensure steady flow. Powders tend to be cohesive and <br>also tend to entrap air. Therefore, in most cases, special precautions have to be taken <br>to successfully extrude powder material. The degree of difficulty in extrusion of <br>powders generally increases with reducing particle size. Broken solids usually con-<br>sist of fiber or film scrap. The particles are generally of irregular shape and the bulk <br>density is often very low. This type of particulate solid is also problematic from a <br>solids conveying point of view, because the particles tend to interlock and resist <br>vibration. Static build-up can also be a problem in these materials; this can be solved <br>by the use of a static eliminator.<br>
+<b>6.1.4Other Properties</b><br>
+A variety of other properties can affect the conveying characteristics of the bulk <br>material. Hygroscopic materials tend to absorb moisture; this may cause agglomera-<br>tion and reduce the flowability of the material. Additives that act as external lubri-<br>cants can change the frictional characteristics and adversely affect the solids trans-<br>port in the extruder.<br>
+<hr>
+<A name=215></a><b>202</b>  <b>6Important Polymer Properties</b><br>
+<b></b><br>
+ <b>6.2Melt Flow Properties</b><br>
+Knowledge of the flow properties of the polymer melt is very important in the ana-<br>lysis of the extrusion process. The first traces of melt generally appear only a few <br>diameters from the feed opening of the extruder. The metering end of the extruder <br>is, in many cases, completely filled with polymer melt. The polymer melt flow prop-<br>erties determine to a large extent the characteristics of the extrusion process. <br>Knowledge of the melt flow properties allows accurate optimization of the screw <br>design and the process operating conditions. If the melt flow properties are not <br>known, the selection of the extruder screw and the determination of the process <br>operating conditions becomes a trial and error process at best.<br>
+<b>6.2.1Basic Definitions</b><br>
+Before going into detail on the flow behavior of polymer melts, it may be useful to <br>describe and define some of the basic terminology used in fluid flow.<br><b>Drag Flow:</b> Flow caused by the relative motion of one or more boundaries with <br>respect to the other boundaries that contain the fluid.<br>This is also referred to as Couette flow, although Couette flow is only a specific type <br>of drag flow. Drag flow is important in extrusion. The two major boundaries that <br>contain the polymer in the extruder are the barrel surface and the screw surface. <br>Since the screw is rotating in a stationary barrel, one boundary is moving relative to <br>the other; this causes drag flow to occur.<br><b>Pressure Flow:</b> Flow caused by the presence of pressure gradients in the fluid; in <br>other words, local differences in the pressure.<br>One of the most common examples of pressure flow (pressure-driven flow) is the <br>flow of water that occurs when one opens a water faucet. This flow occurs because <br>the pressure upstream is higher than the pressure at the faucet. There is no relative <br>motion of the fluid boundaries (wall of the water pipe); thus, this is pure pressure <br>flow. In most extruder dies, the flow through is a pure pressure-driven flow. The <br>polymer melt flows through the die as a result of the fact that the pressure at the die <br>inlet is higher than the pressure at the outlet. The flow rate is determined by the <br>pressure at the die inlet, often referred to as diehead pressure. In some extruder <br>dies, the polymer coats a part that moves through the die, e.g., a wire coating die. In <br>such a die, the flow is not a pure pressure flow but a combination of drag flow (as a <br>result of the moving wire) and pressure flow (as a result of the diehead pressure).<br><b>Shear:</b> Occurrence of velocity differences in a direction normal to flow.<br>A fluid is sheared when velocity differences in normal direction occur in the fluid, <br>as shown in Fig. 6.8.<br>
+<hr>
+<A name=216></a><IMG src="Polymer Extrusion by Chris Rauwendaal-216_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-216_10.jpg"><br>
+ <br>
+6.2Melt Flow Properties  <b>203</b><br>
+Velocity profile<br>
+A(t)<br>
+A(t+t)<br>
+xA<br>
+A<br>
+vA<br>
+vA<br>
+<br>
+B<br>
+vB<br>
+v<br>
+x<br>
+B<br>
+B<br>
+<b> Figure 6.8</b> <br>
+B(t)<br>
+B(t+t)<br>
+Shearing of a fluid in flow through a pipe<br>
+<b>Elongation:</b> Occurrence of velocity differences in the direction of flow.<br>Elongational deformation of a fluid occurs when the velocity changes in the direc-<br>tion of flow, as shown in Fig. 6.9.<br>
+A(t+t)<br>
+v(t+t)<br>
+A(t)<br>
+v(t)<br>
+<b> Figure 6.9</b> <br>
+Elongational flow in a converging <br>
+after time t <br>
+flow channel<br>
+<b>Plug Flow:</b> A flow situation where all fluid elements move at the same velocity, i.e., <br>flow without shear.<br>Plug flow generally does not happen in polymer melts, except in the case of wall slip <br>(PVC). However, it does occur with granular polymeric solids. The solids conveying <br>theory of single screw extruders is based on the assumption of plug flow of the solid <br>polymer.<br><b>Shear Rate ( ):</b> The difference in velocity per unit normal distance (normal to the <br>direction of flow).<br>The rate of shearing or shear rate is one of the most important parameters in poly-<br>mer melt processing. If the process is to be described quantitatively, the shear rate <br>in the fluid at any location needs to be known. The shear rate is generally written <br>with the Greek letter gamma,  , with the dot above the gamma indicating a time <br>derivate                  .. In terms of Fig. 6.8, the shear rate between points A and B can be <br>approximated as:<br>
+ (6.10)<br>
+<hr>
+<A name=217></a><b>204</b>  <b>6Important Polymer Properties</b><br>
+Equation 6.10 is only valid for very small values of the normal distance AB. More <br>accurately, the shear rate is:<br>
+ (6.11)<br>
+From Eq. 6.11, it can be seen that the local shear rate equals the local gradient of the <br>velocity profile. Thus, if the velocity profile is known, the shear rate at any location <br>can be determined.<br><b>Shear Strain (</b><b>):</b> Displacement (in the direction of flow) per unit normal distance <br>over a certain time period.<br>The shear strain is generally written with the Greek letter gamma (), this time <br>without the dot! The relationship between shear rate ( ) and shear strain () is:<br>
+and<br>
+ (6.12)<br>
+In terms of Fig. 6.8, the shear strain can be written as:<br>
+ (6.13)<br>
+The units of shear rate are s­1 and the shear strain is a dimensionless number.<br><b>Shear Stress ():</b> The stress required to achieve a shearing type of deformation.<br>When a fluid is sheared, a certain force will be required to bring about that deforma-<br>tion. This force divided by the area over which it works is the shear stress. The shear <br>stress is generally written with the Greek letter tau (). In a simple example, shown <br>in Fig. 6.10, the shear stress is:<br>
+ (6.14)<br>
+and the shear rate is:<br>
+ (6.15)<br>
+<b>Shear Viscosity (s):</b> The resistance to shear flow. Quantitatively, the shear vis-<br>
+cosity is determined from the ratio of shear stress and shear rate.<br>
+ (6.16a)<br>
+<hr>
+<A name=218></a> <br>
+6.2Melt Flow Properties  <b>205</b><br>
+The shear viscosity is generally written with the Greek letter eta (); the units of <br>viscosity are stress x time. The viscosity is usually expressed in Poise (= dyne-s/<br>cm2) or Pas (= 10 Poise). In order to determine the shear viscosity of a fluid, one has <br>to determine the shear rate in a certain shear deformation and the corresponding <br>shear stress. Special instruments are available to determine the viscosity of polymer <br>melts; these are referred to as rheometers.<br>
+<b> Figure 6.10</b> <br>
+Simple shear deformation<br>
+<b>Elongational Viscosity (e):</b> The resistance to elongational flow. Quantitatively, the <br>
+elongational viscosity is determined from the ratio of elongational stress and elon-<br>gation rate.<br>
+ (6.16b)<br>
+The elongational viscosity is substantially higher than the shear viscosity. It is at <br>least three times higher than the shear viscosity but in many cases much higher.<br><b>Newtonian Fluid:</b> A fluid whose viscosity is <i>in</i>dependent of the shear rate.<br>Most low viscosity liquids and gases behave as a Newtonian fluid. In a plot of shear <br>stress versus shear rate, a Newtonian fluid will exhibit a linear relationship; see <br>Fig. 6.11, curve b. Therefore, Newtonian fluids are also referred to as linear fluids.<br>A plot of shear stress versus shear rate is generally referred to as a "flow curve."<br><b>Non-Newtonian Fluid:</b> A fluid whose viscosity is <i>de</i>pendent on the shear rate.<br>High viscosity polymer melts behave as non-Newtonian fluids, with the viscosity <br>reducing with increase shear rate.<br>Another type of non-Newtonian fluid is a dilatant fluid. The viscosity of a dilatant <br>fluid increases with increasing shear rate; see Fig. 6.11, curve a.<br>The reduction of viscosity with increasing shear rate is called pseudo-plastic <br>behavior; see Fig. 6.11, curve c. The shear stress-shear rate relationship of non- <br>Newtonian fluids is non-linear. Therefore, non-Newtonian fluids are also referred to <br>as non-linear fluids.<br>The concepts of shear rate, shear stress, and viscosity are extremely important in <br>developing a thorough understanding of the extrusion process (and other polymer <br>processing operations). Therefore, a few examples will be given to illustrate how <br>shear rate and shear stress can be determined in simple geometries.<br>
+<hr>
+<A name=219></a><b>206</b>  <b>6Important Polymer Properties</b><br>
+<b> Figure 6.11</b> <br>
+Flow curves of a dilatant fluid, a Newtonian <br>
+fluid, and a pseudo-plastic fluid<br>
+Example: Co-axial Cylinders; see Fig. 6.12.<br>
+T<br>
+N<br>
+Rotating inner cylinder<br>
+Fluid in annular space<br>
+Ri<br>
+H<br>
+Ro<br>
+L<br>
+Stationary outer cylinder<br>
+<b> Figure 6.12</b> <br>
+Co-axial cylinders: <br>
+outer cylinder stationary, <br>
+inner cylinder rotating<br>
+The fluid is contained in the annular space, with one boundary formed by the inner <br>cylinder and the other boundary formed by the outer cylinder. Since the inner <br>boundary is moving with respect to the outer boundary, a drag flow will be set up in <br>the fluid and the fluid will be sheared. The shear rate in the fluid will be the differ-<br>ence in velocity divided over the normal distance. Thus,<br>
+ (6.17)<br>
+<hr>
+<A name=220></a> <br>
+6.2Melt Flow Properties  <b>207</b><br>
+This expression is reasonably accurate as long as the radial clearance (H) is small <br>relative to the radius. The shear rate equals the circumferential velocity of the inner <br>cylinder (velocity of the outer cylinder is zero, vo = 0) divided by the radial clear-<br>
+ance. Thus, if the geometry and rotational speed are known, the shear rate can be <br>directly determined from that information. The shear rate will be high when the <br>diameter of the inner cylinder is large, when the rotational speed is high, or when <br>the radial clearance is small.<br>The shear stress acting on the fluid is obtained from the torque T that is necessary <br>to rotate the inner cylinder. The total shear force F acting on the inner cylinder is the <br>shear stress  multiplied with the area of the inner cylinder (2RiL):<br>
+ (6.18)<br>
+Thus, the shear stress is:<br>
+ (6.19)<br>
+Measurement of the torque, therefore, allows the determination of the shear stress. <br>The shear viscosity can now be determined as well:<br>
+ (6.20)<br>
+The co-axial cylinder geometry can thus be used to determine the viscosity of a <br>fluid. In practice, this geometry is mostly used to determine flow properties of low <br>viscosity liquids.<br>Example: Screw Extruder; see Fig. 6.13.<br>
+Barrel<br>
+Screw<br>
+Db<br>
+Ds<br>
+H<br>
+<br>
+<b> Figure 6.13</b> <br>
+Screw channel<br>
+Flight clearance<br>
+The screw extruder<br>
+<hr>
+<A name=221></a><b>208</b>  <b>6Important Polymer Properties</b><br>
+The geometry of a screw extruder is quite similar to the c-oaxial cylinder set-up. The <br>difference is the presence of the helical flight wrapped around the core of the screw. <br>It is often assumed that the screw root and barrel surfaces can be approximated by <br>flat plates ­ this is called the flat plate approximation. With this approximation the <br>shear rate in the screw channel is:<br>
+ (6.21a)<br>
+where D is the O.D. of the screw, H the channel depth, and N the rotational speed of <br>the screw in rev/s.<br>In most analyses of flow in screw extruders using the flat plate system (FPS) it is <br>assumed that the barrel rotates relative to a stationary screw. This actually yields <br>more accurate results than moving the screw relative to a stationary barrel; this <br>will be discussed in more detail in Section 7.4.3.4. When the barrel moves relative to <br>a stationary screw the shear rate becomes:<br>
+ (6.21b)<br>
+The approximately equal sign () is used because Eqs. 6.21(a) and 6.21(b) are based <br>on the FPS and neglect the effect of curvature. For a more accurate analysis one has <br>to use cylindrical or helical coordinates; this will be discussed in Section 7.4. Equa-<br>tion 6.21 is essentially the same expression as found for the co-axial cylinder prob-<br>lem, Eq. 6.17. The polymer melt between the screw flight and the barrel is exposed <br>to a different shear rate:<br>
+ (6.22)<br>
+The channel depth H is, in most cases, much larger than the radial flight clearance <br>. Therefore, the shear rate in the clearance will be much higher than the shear rate <br>in the screw channel. A typical value of D/H is 20 and a typical value of D/ is 1000. <br>Thus, the shear rate in the flight clearance will be approximately 50 times higher <br>than the shear rate in the screw channel. This has important implications for the <br>operation of the extruder, as will be discussed in more detail in the next two chap-<br>ters.<br>
+<b>6.2.2Power Law Fluid</b><br>
+In the previous section, it was discussed that polymer melts are pseudo-plastic fluids. <br>The fact that the polymer melt viscosity reduces with shear rate is of great impor-<br>tance in the extrusion process. It is, therefore, important to know the extent of <br>
+<hr>
+<A name=222></a> <br>
+6.2Melt Flow Properties  <b>209</b><br>
+change that will occur in a particular polymer. The general shape of the viscosity-<br>shear rate curve for a pseudo-plastic polymer melt will look as shown in Fig. 6.14.<br>
+Power law<br>
+approximation<br>
+Normal range of polymer<br>
+processing operations<br>
+o<br>
+a.s]<br>
+Log viscosity [P<br>
+oo<br>
+Log shear rate [s-1]<br>
+<b>Figure 6.14</b>General pseudo-plastic behavior<br>
+The viscosity at very low shear rates is essentially independent of shear rate. Thus, <br>the fluid behaves as a Newtonian fluid at low shear rates. The low shear rate plateau <br>value o is often referred to as the low shear limiting Newtonian viscosity. The high <br>
+shear rate plateau value  is often referred to as the high shear limiting Newtonian <br>
+viscosity. This value is difficult to determine experimentally because the effects of <br>pressure and temperature become very pronounced at these high shear rates (over <br>106 s­1).<br>The range of shear rates encountered in most polymer processing operations is <br>approximately 1 to 10,000 s­1. It can be seen in Fig. 6.14 that within this range the <br>viscosity-shear rate curve can be reasonably approximated with a straight-line rela-<br>tionship. This is true for most polymers. It should also be noted that Fig. 6.14 uses <br>a double logarithmic scale. The log-log scale is convenient because the viscosity <br>changes about 4 to 5 orders of magnitude over more than 10 orders of magnitude <br>change in shear rate. A straight-line relationship on a log-log plot indicates that the <br>variables can be related by a power law equation. This is generally written as:<br>
+or<br>
+ (6.23)<br>
+where m is the consistency index and n the power law index. This law is often <br>referred to as the power law of Ostwald and de Waele [15, 16]. The power law index <br>
+<hr>
+<A name=223></a><b>210</b>  <b>6Important Polymer Properties</b><br>
+indicates how rapidly the viscosity reduces with shear rate. For pseudo-plastic <br> <br>fluids, the power law index ranges from 1 to 0. When the power law index is unity, <br>the fluid is Newtonian and the consistency index becomes the Newtonian viscosity. <br>The power law index indicates the degree of non-Newtonian behavior. If the power <br>law index ranges from 0.8 to 1.0, the fluid is almost Newtonian. If the power law <br>index is less than 0.5, the fluid is strongly non-Newtonian. It turns out that most <br>large volume commodity polymers fall into this latter category, e.g., polyethylene, <br>polyvinylchloride, polystyrene, styrene acrylonitrile, acrylonitrile butadiene sty-<br>rene, etc. Examples of polymers with a relatively high power law index are poly-<br>carbonate, polyamide, polyethylene terephthalate, polysulfone, and polyphenylene <br>sulphide. The approximate power law index for a number of polymers is shown in <br>Table 6.1 at the end of this chapter.<br>Equation 6.23 can be used if the shear rate is positive throughout the flow channel <br>being considered. If the shear rate changes sign at some point in the flow channel, <br>a more general power law equation should be used:<br>
+<br>
+(6.24a)<br>
+Another form of the power law equation that is used quite often is:<br>
+ (6.24b)<br>
+where  is the specific fluidity and s the pseudo-plasticity index. The pseudo-plasti-<br>city index s is the reciprocal of the power law index n = 1/s. The specific fluidity  is <br>related to the consistency index by:<br>
+ (6.24c)<br>
+Power law equations (Eqs. 6.23­6.24) can be used to describe simple viscometric <br>flow, i.e., flow with velocity components in only one direction. For more complicated <br>flow situations, a more general power law expression should be used. In order to do <br>this, the rate of deformation tensor ij has to be introduced. The components of ij in <br>
+Cartesian coordinates are:<br>
+ <br>
+ (6.25a)<br>
+ <br>
+ (6.25b)<br>
+ <br>
+ (6.25c)<br>
+<hr>
+<A name=224></a> <br>
+6.2Melt Flow Properties  <b>211</b><br>
+The components of ij in cylindrical coordinates are:<br>
+ <br>
+ (6.25d)<br>
+ <br>
+ (6.25e)<br>
+ <br>
+ (6.25f)<br>
+Because the viscosity is a scalar, it can be a function only of the scalar invariants of <br>the rate of deformation tensor. There are three combinations of the components of <br>the rate of deformation tensor (ij), which are scalar invariants. They define any <br>
+state of deformation rate independently of the coordinate system. They are referred <br>to as the principal invariants of the rate of deformation tensor:<br>
+ (6.26a)<br>
+ (6.26b)<br>
+ (6.26c)<br>
+where "det" means the determinant of the enclosed matrix.<br>Equation 6.26 uses the summation convention on repeated subscripts.<br>In Cartesian coordinates:<br>
+ (6.27a)<br>
+ (6.27b)<br>
+ (6.27c)<br>
+If a fluid can be considered incompressible, the first principal invariant of the rate of <br>deformation tensor will be zero, I1 = 0. The third principal invariant vanishes in <br>
+many simple flow situations, like axial flow in pipe, tangential flow between concen-<br>tric cylinders, etc. In more general terms, the third invariant is zero in rectilinear <br>flow and in two-dimensional flow.<br>The power law expression can now be written in general terms:<br>
+ (6.28)<br>
+<hr>
+<A name=225></a><b>212</b>  <b>6Important Polymer Properties</b><br>
+In Cartesian coordinates, 0.5I2 is obtained by using Eqs. 6.27(b) and 6.25.<br>
+ (6.29)<br>
+Similarly, in cylindrical coordinates:<br>
+ (6.30)<br>
+With these more general expressions (Eqs. 6.28­6.30), more complicated flow situ-<br>ations can be described, i.e., flow with velocity components in two or three direc-<br>tions. It should be remembered that the power law description is an approximation; <br>it is not accurate over the entire range of shear rate. However, in most practical poly-<br>mer processing problems, the use of the power law equation yields sufficiently accu-<br>rate results. The major advantage of the power law equation is its simplicity, despite <br>the appearance of Eqs. 6.28­6.30. The relationship between stress and rate of defor-<br>mation can be described with only two fluid properties, the consistency index m and <br>power law index n. A drawback of the power law is that it does not allow construc-<br>tion of a time constant from the constants m and n. This is a problem in the analysis <br>of transient flow phenomena where a characteristic time constant is necessary to <br>describe the flow situation.<br>The truncated power law of Spriggs [17] allows a more accurate description. It is <br>written as:<br>
+<br>
+ (6.31a)<br>
+  <br>
+ (6.31b)<br>
+In this model, there are three constants: a zero shear rate viscosity 0, a characteris-<br>
+tic time 1/ 0, and a dimensionless power law index n. This model contains the hori-<br>
+zontal asymptote for small   and the power law for large  .<br>
+<hr>
+<A name=226></a> <br>
+6.2Melt Flow Properties  <b>213</b><br>
+<b>6.2.3Other Fluid Models</b><br>
+The sinh law was proposed by Ehring [18] and can be written as:<br>
+ (6.32)<br>
+where 0 is a characteristic stress and t0 a characteristic time. Other workers have <br>
+modified the Ehring model to improve its flexibility and accuracy in describing <br>stress-deformation rate relationships:<br>
+ (6.33)<br>
+A polynomial relationship was proposed by Rabinowitsch and Weissenberg; it can <br>be written as [20a]:<br>
+ (6.34)<br>
+where 1 and 3 are rheological constants depending on the nature of the fluid.<br>The Carreau model [19] has the useful properties of the truncated power law model <br>but avoids the discontinuity in the first derivative; it can be written as:<br>
+ (6.35)<br>
+where 0 is the zero shear rate viscosity,  is the infinite shear rate viscosity,  is a <br>
+time constant, and n is the dimensionless power law index. The Ellis model [20b] <br>describes the viscosity as a function of shear stress:<br>
+ (6.36)<br>
+where 0 is the zero shear rate viscosity, 1/2 the value of the shear stress at which <br>
+ = 0/2, and ­1 is the slope of (0/)­1 versus /1/2 on log-log paper. The Ellis <br>
+model is relatively easy to use and many analytical results have been obtained with <br>this model; a recent example is the analysis of flow in screw extruders by Steller <br>[99]. Actually, the Ellis model is a more general form of the Rabinowitsch equation. <br>The latter is a special case of the Ellis model when the constant  = 3.<br>Another model in which the viscosity is described as a function of shear stress is the <br>Bingham model [20]. This model is used for fluids with a yield stress 0. Below this <br>
+yield stress, the viscosity is infinite (no motion); above the yield stress, the viscosity <br>is finite (motion occurs). The Bingham Fluid model is written as:<br>
+<hr>
+<A name=227></a><b>214</b>  <b>6Important Polymer Properties</b><br>
+<br>
+ (6.37a)<br>
+  <br>
+ (6.37b)<br>
+This model is primarily used for slurries and pastes. The parameters 0 and 0 can <br>
+be related empirically to the volume fraction of solids , the particle diameter Dp, <br>
+and the viscosity of the suspending fluid s:<br>
+ (6.38)<br>
+ (6.39)<br>
+where Dp is measured in m and 0 in Pascal.<br>Many other fluid models have been proposed. For a more detailed discussion, the <br>reader is referred to the literature [17, 20­22, 54, 94].<br>
+<b>6.2.4Effect of Temperature and Pressure</b><br>
+The effect of shear rate on viscosity has been discussed in some detail in the pre-<br>vious sections. However, there are some other variables that also affect the viscosity. <br>Two important variables that influence the viscosity are temperature and pressure. <br>The effect of these variables is generally not as strong as the effect of shear rate; <br>however, in many cases, the effect of temperature and/or pressure on viscosity can-<br>not be neglected.<br>When the viscosity is plotted against shear rate at several temperatures, the curve <br>generally lowers with increasing temperature; see Fig. 6.15.<br>This is a result of the increased mobility of the polymer molecules. For the time <br>being, it is assumed that no irreversible changes occur as a result of degradation. <br>However, whenever experiments or processes are conducted at elevated tempera-<br>tures, the possible effects of degradation have to be taken into account. There will be <br>more on degradation in Section 11.3.<br>It is convenient to plot viscosity as a function of shear stress to evaluate the effect of <br>temperature. This is shown in Fig. 6.16 where the same data shown in Fig. 6.15 is <br>plotted in terms of viscosity and shear stress.<br>For many polymers, the shape of the viscosity-shear stress curve does not change <br>appreciably with temperature. With many polymers, the curves can be shifted along <br>lines of constant shear stress to produce a master curve. By comparing Fig. 6.15 to <br>
+<hr>
+<A name=228></a> <br>
+6.2Melt Flow Properties  <b>215</b><br>
+Fig. 6.16, it is clear that a shift along lines of constant shear rate would not produce <br>a good fit. Figure 6.15 shows a line of constant shear stress; it makes an angle of 45° <br>with both axes. The curves should be shifted in the direction of this constant shear <br>stress line to produce a good master curve. Figure 6.16 also shows lines of constant <br>shear rate. It can be seen in Fig. 6.16 that the effect of temperature is greater at <br>lower temperature; this is true for many polymers. It should be mentioned that most <br>polymers do not have as strong a temperature dependence as the polymer shown in <br>Figs. 6.15 and 6.16.<br>
+105<br>
+PMMA, Altuglas GR 7E LIL<br>
+210 C<br>
+104<br>
+230 C<br>
+ [Pa.s]<br>
+250 C<br>
+Viscosity<br>
+103<br>
+102 1<br>
+10<br>
+102<br>
+103<br>
+104<br>
+Shear rate [s-1]<br>
+<b>Figure 6.15</b>Viscosity versus shear rate at various temperatures<br>
+105<br>
+210 C<br>
+104<br>
+230 C<br>
+ [Pa.s]<br>
+103<br>
+250 C<br>
+Viscosity<br>
+102<br>
+PMMA, Altuglas GR 7E LIL<br>
+10<br>
+104<br>
+105<br>
+106<br>
+Shear stress [Pa]<br>
+<b>Figure 6.16</b>Viscosity versus shear stress at various temperatures<br>
+<hr>
+<A name=229></a><b>216</b>  <b>6Important Polymer Properties</b><br>
+The shift factor aT is a function of the temperature. For polyolefins, the relationship <br>
+can be written as:<br>
+ (6.40)<br>
+where E is the activation energy, R the universal gas constant, and Tr the reference <br>
+temperature in degrees Kelvin. Equation 6.40 is known as Andrade's Law [98]. It is <br>applicable to semi-crystalline and amorphous polymers above Tg + 100°C.<br>For amorphous polymers, the Williams-Landel-Ferry (WLF) equation is often used:<br>
+ (6.41)<br>
+where C1 and C2 are material constants.<br>If the reference temperature Tr is taken about 43 °K above the glass transition point <br>
+Tg, the constants C1 and C2 are essentially the same for a large number of amor-<br>
+phous polymers (C1 = 8.86 and C2 = 101.6). This results in the following equation:<br>
+ (6.41b)<br>
+The glass transition temperature of a number of polymers is shown in Table 6.2. <br>Equation 6.41(b) gives a reasonable description of the temperature dependence of <br>the viscosity in the range of Tg to Tg + 100°C.<br>A popular empirical form of the temperature dependence of viscosity is:<br>
+ (6.42)<br>
+where T is a temperature coefficient that can be considered constant as long as the <br>
+temperature range considered is relatively small.<br>The power law equation including the temperature effect can then be written as:<br>
+ (6.43a)<br>
+or<br>
+ (6.43b)<br>
+The temperature sensitivity of the viscosity varies widely for different polymers. As <br>a general rule, amorphous polymers have a high temperature sensitivity, while <br>semi-crystalline polymers have a relatively low temperature sensitivity. Polyvinyl-<br>chloride (PVC) and polymethyl methacrylate (PMMA) are two polymers with a very <br>
+<hr>
+<A name=230></a> <br>
+6.2Melt Flow Properties  <b>217</b><br>
+high temperature sensitivity of the viscosity. Polyethylene and polypropylene both <br>have quite low temperature sensitivity.<br>The relative change in viscosity per degree of temperature can be determined from:<br>
+ (6.44)<br>
+If the expression for aT is used for amorphous polymers (Eq. 6.41), one obtains:<br>
+ (6.45)<br>
+This relationship is shown in Fig. 6.17 where the relative viscosity change is plotted <br>against T­Tg for C1 = 8.86, C2 = 101.6, and Tr = Tg + 43.<br>It can be seen that the temperature sensitivity drops dramatically (several decades) <br>when T­Tg increases. The closer a polymer is to its glass transition temperature, <br>
+the larger the temperature sensitivity of the viscosity. This explains why polymers <br>whose normal process temperatures are close to their glass transition temperature <br>exhibit a high temperature sensitivity in processing. Examples are polystyrene, poly-<br>vinylchloride, and polymethyl methacrylate. In general, polymers that are processed <br>considerably above their glass transition temperature (more than 150°C above Tg) <br>
+show a relatively small temperature sensitivity. Examples are polyethylene, polypro-<br>pylene, and polyamide.<br>The effect of pressure on viscosity is relatively insignificant in most polymer pro-<br>cessing operations, where pressures generally do not exceed 35 MPa (5000 psi). It <br>has been found, however, that the effect of pressure on viscosity becomes quite sig-<br>nificant at pressures substantially above 35 MPa. In fact, in careful rheological <br>measurements, the effect of pressure on both viscosity and density has to be consid-<br>ered even at pressures around 35 MPa.<br>Special rheometers have been constructed to measure the effect of pressure on viscos-<br>ity. Various workers have presented data on the pressure dependence on viscosity <br>[33­40]. The viscosity as a function of pressure is generally written as:<br>
+ (6.46)<br>
+The values of the pressure sensitivity term p vary considerably from one polymer <br>
+to another. For polystyrene, increases in viscosity at fixed shear stress and about <br>150°C have been reported [34, 36, 40] of 200 to 1000 times over a pressure rise of <br>100 MPa (15,000 psi). For polyethylene at the same temperature and pressure con-<br>ditions, the viscosity increased only 4 to 5 fold. At a temperature of 200°C and a <br>pressure rise of 100 MPa (15,000 psi), the increase in viscosity of polystyrene was <br>reported to be about 30 to 50 fold, about 10 to 20 times lower than at 150°C!<br>
+<hr>
+<A name=231></a><b>218</b>  <b>6Important Polymer Properties</b><br>
+Data on the pressure sensitivity of the viscosity is quite scarce. It has been found <br>empirically [41] that the relative change in viscosity with pressure divided by the <br>relative change in viscosity with temperature is approximately constant for many <br>polymers:<br>
+ (6.47)<br>
+T - Tg [K]<br>
+0<br>
+50<br>
+100<br>
+150<br>
+103<br>
+102<br>
+10<br>
+e<br>
+1<br>
+ chang<br>
+10-1<br>
+iscosity<br>
+10-2<br>
+e v<br>
+10-3<br>
+Relativ<br>
+10-4<br>
+10-50<br>
+50<br>
+100<br>
+150<br>
+T - T<br>
+T -  g<br>
+T -43  [K]<br>
+- 43 [K]<br>
+g <br>
+<b>Figure 6.17</b>Relative change in viscosity versus T­Tg<br>
+The numerator can be determined from Eqs. 6.40 through 6.45. Thus, Eq. 6.47 pro-<br>vides a convenient, though approximate, method to determine the pressure sensitiv-<br>ity of the viscosity of a polymer. From Eq. 6.47, it is clear that a polymer with a high <br>temperature sensitivity of the viscosity will also have a high pressure sensitivity of <br>the viscosity. This explains the large differences in pressure sensitivity of the vis-<br>cosity between polystyrene and polyethylene, as mentioned earlier.<br>
+<b>6.2.5Viscoelastic Behavior</b><br>
+Thus far, the polymer melt has been considered as a purely viscous fluid. In a purely <br>viscous fluid, the energy expended in deformation of the fluid is immediately dissi-<br>pated and is non-recoverable. The other extreme is the purely elastic material where <br>
+<hr>
+<A name=232></a> <br>
+6.2Melt Flow Properties  <b>219</b><br>
+the energy expended in deformation of the fluid is not dissipated at all; the deforma-<br>tion is completely reversible and the energy completely recoverable.<br>Polymers are partly viscous and partly elastic. In the molten state, polymers are <br>primarily viscous but will be elastic to some extent. This behavior is generally re -<br>ferred to as viscoelastic behavior. This characteristic is responsible for the swelling <br>of the extrudate as it emerges from an extruder die. The swelling is caused by elastic <br>recovery of strain imparted to the polymer in and before the die. The swelling is not <br>instantaneous, but takes a finite time to fully develop. This indicates that the re -<br>arrangement of the polymer structure takes a certain amount of time; this can range <br>from a fraction of a second to several minutes or even hours, depending on the poly-<br>mer and the temperature. The polymer properties, therefore, are a function of time <br>and depend on the deformation history of the polymer. The deformation history is <br>often referred to as the shear history; however, it is not only shearing deformation <br>that affects the polymer properties but elongational deformation as well.<br>In fluids with time-dependent behavior, the effects of time can be either reversible <br>or irreversible. If the time effects are reversible, the fluids are either thixotropic <br>or rheopectic. Thixotropy is the continuous decrease of apparent viscosity with <br>time under shear and the subsequent recovery of viscosity when the flow is discon-<br>tinued. Rheopexy is the continuous increase of apparent viscosity with time under <br>shear; it is also described by the term anti-thixotropy. A good review on thixotropy <br>was given by Mewis [45]. Polymer melts do exhibit some thixotropic effects; how-<br>ever, thixotropy can also occur in inelastic fluids. The time scale of thixotropy is not <br>necessarily associated with the time scale for viscoelastic relaxation.<br>For a proper description of the flow of a polymer melt, the viscoelastic properties <br>have to be taken into account, including the dependence on deformation history. <br>Some experimental and theoretical work on time-dependent effects is covered in the <br>following references [46­53]; many other publications have been devoted to this <br>subject. Unfortunately, the viscoelastic models that include memory effects (i.e., the <br>dependence on deformation history) are quite complex and difficult to apply. Also, <br>there does not seem to be any model that is widely accepted as being able to describe <br>polymer melt flow accurately over a wide range of flow geometries and conditions. <br>Practicing process engineers probably will find these models difficult to apply to <br>actual extrusion problems. As a result, the workers in this field generally are spe-<br>cialists in rheology.<br>In the quantitative analysis of most extrusion problems, the polymer melt generally <br>is considered to be a viscous, time-independent fluid. This assumption is, of course, <br>a simplification, but it usually allows one to find a relatively straightforward solu-<br>tion to the problem. This assumption will be used throughout the rest of this book, <br>unless indicated otherwise. In the analysis of any flow problem, however, it should <br>be remembered that elastic effects may play a role. Also, some flow phenomena, <br>such as extrudate swell, clearly cannot be analyzed unless the elastic behavior of <br>
+<hr>
+<A name=233></a><IMG src="Polymer Extrusion by Chris Rauwendaal-233_1.jpg"><br>
+<b>220</b>  <b>6Important Polymer Properties</b><br>
+the polymer melt is taken into account. For more information on the rheology of <br>viscoelastic fluids, the reader is referred to the literature [17, 20­22, 54­64, 100, <br>101]. Six of these references [58­63] do not go into great mathematical complexities <br>and are relatively easy to understand for people not specialized in rheology.<br>
+<b>6.2.6Measurement of Flow Properties</b><br>
+Whenever a process engineer uses flow property data, he should know on what <br>instrument and how these data were determined in order to properly assess the <br>validity of the data. Instruments to determine flow properties are generally referred <br>to as rheometers. The rheometers that will be briefly described in the next few sec-<br>tions are the capillary rheometer, the melt indexer, the cone and plate rheometer, <br>the slit die rheometer, and dynamic mechanical rheometers. For a more detailed <br>description of these and other rheometers, the reader is referred to the literature <br>[64­66]. A brief but good survey of commercial rheometers was presented by Dealy <br>[92].<br>
+<b>6.2.6.1Capillary Rheometer<br></b>A capillary rheometer is basically a ram extruder with a capillary die at the end; <br>see Fig. 6.18.<br>As the piston moves down, it forces the molten polymer through the capillary. The <br>shear stress in the capillary at the wall (cw) can be related to the pressure drop <br>
+along the capillary (Pc) by the following equation:<br>
+Fp<br>
+vp<br>
+Piston<br>
+Dp<br>
+Reservoir<br>
+Dc<br>
+Lc<br>
+Capillary<br>
+Extruded strand <b> Figure 6.18</b> <br>
+Schematic of capillary rheometer<br>
+ (6.48)<br>
+<hr>
+<A name=234></a> <br>
+6.2Melt Flow Properties  <b>221</b><br>
+If the piston diameter is much larger than the capillary diameter (Dp &gt; Dc) and if <br>
+entrance effects are neglected, then:<br>
+ (6.49)<br>
+By inserting Eq. 6.49 into Eq. 6.48, the wall shear stress in the capillary can be <br>related to the force on the piston. Thus, by measuring the force on the piston, the <br>wall shear stress in the capillary can be determined. To avoid problems with <br>entrance effects, it is good practice to do measurements with a long capillary (high <br>L/D, 20 to 40). It is even better to do measurements with two capillaries of the same <br>diameter but different length, one having a length of almost zero. The actual pres-<br>sure drop along the capillary of length Lc is now:<br>
+ (6.50)<br>
+The apparent shear rate at the capillary wall can be determined from the flow rate   <br>through the capillary. This can be determined from Eqs. 6 and 13 in Appendix 5.1.<br>
+ (6.51)<br>
+The flow rate is determined by the area and the velocity of the piston:<br>
+ (6.52)<br>
+The apparent shear rate at the capillary wall can now be expressed as a function of <br>the piston velocity:<br>
+ (6.53)<br>
+Thus, by measuring the piston velocity one can determine the apparent shear rate <br>at the capillary wall. At this point, the apparent viscosity can be determined by <br>dividing the shear stress by the apparent shear rate:<br>
+ (6.54)<br>
+<hr>
+<A name=235></a><b>222</b>  <b>6Important Polymer Properties</b><br>
+The terms apparent shear rate and apparent viscosity are used because Eq. 6.51 is <br>valid only for Newtonian fluids. Therefore, if the fluid is non-Newtonian, the actual <br>value of the shear rate at the capillary wall will be different. If the fluid behaves as a <br>power law fluid with power law index n, the actual shear rate at the capillary wall is:<br>
+ (6.55)<br>
+Thus, for a power law fluid, the actual viscosity is related to the apparent viscosity <br>by:<br>
+ (6.56)<br>
+If the capillary rheometer is used to compare different polymers, it is not necessary <br>to go through the various correction procedures. However, if one wants to know <br>the absolute values of the viscosity, it is important to apply the various correction <br>factors. The most important corrections are the correction of the shear rate for non-<br>Newtonian fluid behavior (often referred to as Rabinowitsch correction) and the cor-<br>rection of the shear stress for entrance effects (often referred to as Bagley correc-<br>tion). These are the most common corrections applied to capillary rheometers. Other <br> <br>corrections that are sometimes considered are corrections for viscous heating, cor-<br>rections for the effect of pressure on viscosity, corrections for compressibility, <br> <br>correction for time effects, etc. If many corrections are applied to the data, the whole <br>measurement and data analysis procedure can become very complex and time con-<br>suming.<br>Figure 6.19 shows how the relative flow rate ( / o) through a capillary depends on <br>
+the relative pressure drop along the capillary die (P/Po) for three different values of <br>
+the power law index: n = 1, n = 1/2, and n = 1/3.<br>In all cases, the flow rate increases as the die pressure increases. However, there is <br>a very large difference in behavior between fluids of different power law index. For <br>a Newtonian fluid (n = 1), a 10-fold increase in pressure results in a 10-fold increase <br>in flow rate. For a power law fluid with n = 1/2, a 10-fold increase in pressure results <br>in a 100-fold increase in flow rate. For a power law fluid with n = 1/3, a 10-fold in -<br>crease in pressure results in a 1000-fold increase in flow rate! Essentially, the same <br>results are valid for any extrusion die. It is clear, therefore, that the power law index <br>of a polymer melt, to a large extent, will determine its extrusion behavior.<br>It is very important to know the power law index of a material. This is why the vis-<br>cosity has to be determined over a wide range of shear rates. The shear rate range <br>should be representative of the shear rates encountered in polymer processing <br>equipment, which is usually 0 to 10,000 s­1.<br>
+<hr>
+<A name=236></a> <br>
+6.2Melt Flow Properties  <b>223</b><br>
+1000<br>
+n = 1/3<br>
+100<br>
+ rate<br>
+e flow<br>
+ n = 1/2<br>
+Relativ<br>
+10<br>
+n = 1<br>
+1<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+6<br>
+7<br>
+8<br>
+9<br>
+10<br>
+Relative Pressure<br>
+<b>Figure 6.19</b>Relative flow rate versus relative pressure drop<br>
+Advantages of the capillary rheometer are:<br>1.  Ability to measure very high shear rates ( 106 s­1).<br>2.  Ability to measure extrudate swell characteristics.<br>3.  Ability to measure melt fracture characteristics.<br>4.  Relatively easy to use.<br>Disadvantages of the capillary rheometer are:<br>1.  The polymer is not exposed to a uniform shear rate.<br>2.  Various corrections have to be applied to the data.<br>3.  It does not yield an accurate description of viscoelastic behavior.<br>4.  It is unreliable at high shear rates (temperature effects).<br>
+<b>6.2.6.2Melt Index Tester<br></b>The melt index tester is essentially a simple capillary rheometer. The piston is <br>pushed down by a weight; see Fig. 6.20(a) [99].<br>The melt index is the number of grams of polymer extruded in a time period of <br>10 minutes. Details of the geometry and test procedures are described in ASTM <br>D1238.<br>The melt index tester is used in many companies to quickly test the polymer melt. <br>Unfortunately, many times MI data is the only information available on the polymer <br>melt flow properties.<br>
+<hr>
+<A name=237></a><b>224</b>  <b>6Important Polymer Properties</b><br>
+Dp = 9.5504 mm<br>
+D<br>
+Weight<br>
+c = 2.0955 mm<br>
+Lc = 8.000 mm<br>
+Heater<br>
+Temperature<br>
+Barrel<br>
+sensor<br>
+Piston<br>
+Reservoir<br>
+MI die<br>
+Extrudate<br>
+<b> Figure 6.20(a)</b> <br>
+The melt index tester<br>
+From the dimensions of the MI apparatus, the weight on the plunger, and the MI <br>value, one can determine the approximate shear stress, shear rate, and viscosity. By <br>using Eqs. 6.48 and 6.49, the shear stress at the capillary wall can be determined <br>from:<br>
+ (6.57)<br>
+where Fp is the weight on the plunger in grams. The flow rate through the capillary <br>
+can be expressed as:<br>
+ (6.58)<br>
+where MI is expressed in grams per 10 minutes and the density  in g/cm3. The <br>apparent shear rate can now be determined by using Eq. 6.51:<br>
+ (6.59a)<br>
+The apparent shear rate is thus directly proportional to the MI value and inversely <br>proportional to the polymer melt density. The apparent viscosity can now be deter-<br>mined from:<br>
+ (6.59b)<br>
+From Eq. 6.59(b) it can be seen that the polymer melt viscosity is inversely propor-<br>tional to the MI value; see Fig. 6.20(b).<br>The melt index can be measured at a number of different conditions as far as load <br>and temperature are concerned. A number of standardized conditions are listed in <br>Table 6.1.<br>
+<hr>
+<A name=238></a> <br>
+6.2Melt Flow Properties  <b>225</b><br>
+15000<br>
+10,000 gr<br>
+10000<br>
+ [Pa.s.cc/gr]<br>
+ / density<br>
+5,000 gr<br>
+5000<br>
+scosity<br>Vi<br>
+2160 gr<br>
+325 gr<br>
+00<br>
+2<br>
+4<br>
+6<br>
+8<br>
+10<br>
+Melt Index [gr/10 min]<br>
+<b>Figure 6.20(b)</b>Viscosity/density ratio versus MI at various plunger weights<br>
+<b>Table 6.1</b>Standardized Conditions for Melt Index Testing<br>
+T<br>
+Temperature<br>
+Load*<br>
+T<br>
+Temperature<br>
+Load<br>
+Condition<br>
+[°C]<br>
+[g]<br>
+Condition<br>
+[°C]<br>
+[g]<br>
+A<br>
+125<br>
+325<br>
+L<br>
+230<br>
+2160<br>
+B<br>
+125<br>
+2160<br>
+M<br>
+190<br>
+1050<br>
+C<br>
+150<br>
+2160<br>
+N<br>
+190<br>
+10000<br>
+D<br>
+190<br>
+325<br>
+O<br>
+300<br>
+1200<br>
+E<br>
+190<br>
+2160<br>
+P<br>
+190<br>
+5000<br>
+F<br>
+190<br>
+21600<br>
+Q<br>
+235<br>
+1000<br>
+G<br>
+200<br>
+5000<br>
+R<br>
+235<br>
+2160<br>
+H<br>
+230<br>
+1200<br>
+S<br>
+235<br>
+5000<br>
+I<br>
+230<br>
+3800<br>
+T<br>
+250<br>
+2160<br>
+J<br>
+265<br>
+12500<br>
+U<br>
+310<br>
+12500<br>
+K<br>
+275<br>
+325<br>
+* This includes the piston weight of 325 g.<br>
+A high MI value corresponds to a low polymer melt viscosity, and a low MI value <br>corresponds to a high polymer melt viscosity. The term "fractional melt index mate-<br>rial" refers to a polymer with a melt index less than one. These are materials with <br>high melt viscosities, and they generally have higher power consumption and die-<br>head pressure in extrusion as compared to polymers with higher melt index values.<br>
+<hr>
+<A name=239></a><b>226</b>  <b>6Important Polymer Properties</b><br>
+As an example, consider a polymer with MI = 0.2 g/10 min and density  = 1.0 g/cm3; <br>the MI is determined under condition E (Fp = 2160 g). The approximate viscosity of <br>
+this material is 52,488 Pas at a shear rate of about 0.37 s­1. If the MI = 20 g/10 min <br>with everything else the same, the viscosity would be about 525 Pas at a shear rate <br>of about 37 s­1. It should be noted that these figures are very approximate and should <br>be regarded as estimates rather than firm numbers. Considering that the standard <br>MI capillary is quite short (about 4 L/D), the entrance effect will be considerable. <br>Therefore, the error in the expression for the wall shear stress (Eq. 6.57) can be <br>considerable. Consequently, the error in the expression for the apparent viscosity <br>(Eq. 6.59(b)) can also be considerable.<br>A large drawback of the MI test is that it yields single-point data. Thus, it does not <br>give any idea about the degree of non-Newtonian behavior of the material. This <br>drawback can be negated by running the melt indexer with several different weights <br>on the plunger; however, this is not done very often. Another drawback is the rela-<br>tively poor reproducibility of the melt indexer; under the best circumstances it is <br>about 15% (plus or minus).<br>When the melt index is measured with two different weights on the plunger, the two <br>values are sometimes referred to as the high and low load melt index. When the <br>high load melt index is measured with a 10 kg weight and the low load melt index <br>with 2.16 kg, the ratio of melt index values is often reported as the I10/I2 ratio­the <br>two decimal points of the 2.16 are generally omitted. This value allows a determina-<br>tion of the degree of shear thinning of the polymer. When the I10/I2 ratio is about <br>4.6, the fluid is Newtonian; when I10/I2 is between 10 and 20, the fluid is weakly <br>shear thinning. When the I10/I2 ratio is greater than 50, the fluid is strongly shear <br>thinning.<br>An advantage of the melt indexer is its low costs and ease of operation. It should be <br>noted that the melt indexer generally operates at low shear rates (see Eq. 6.59(a)) <br>that are not representative of shear rates usually encountered in the extrusion pro-<br>cess. Thus, the MI value is not a good indicator of the extrusion processing behavior <br>of a polymer.<br>Advantages of the melt indexer are:<br>1.  The instrument is simple and inexpensive.<br>2.  It is easy to operate.<br>3.  It is widely available.<br>Disadvantages of the melt indexer are:<br>1.  It yields only single-point data.<br>2.  It has limited accuracy and reproducibility.<br>3.  It is not a good indicator of processability.<br>4.  It is not an accurate description of viscoelastic behavior.<br>
+<hr>
+<A name=240></a> <br>
+6.2Melt Flow Properties  <b>227</b><br>
+There are two melt index methods (ASTM D1238, ISO 1133). In method A only the <br>temperature is controlled and the operator times the MI and weighs the extruded <br>strand. Method A yields a melt flow rate (MFR) expressed in g/10 min. In method B <br>a sensor measures the position of the piston, and the volume of extruded plastic is <br>determined automatically. Method B yields a melt volumetric rate or MVR; this is <br>expressed in cm3/10 min. The MFR is related by the MVR by the following expres-<br>sion:<br>
+ (6.60)<br>
+where  is the melt density in g/cm3.<br>
+<b>6.2.6.3Cone and Plate Rheometer<br></b>In a cone and plate rheometer, the polymer melt is situated between a flat and a <br>conical plate. In most rheometers, the cone is rotating and the plate is stationary; <br>however, this is not absolutely necessary. The basic geometry is shown in Fig. 6.21.<br>
+T<br>
+<br>
+<br>
+<b> Figure 6.21</b> <br>
+R<br>
+Cone and plate rheometer<br>
+If the cone rotates with an angular velocity  and the cone angle  is very small, the <br>shear rate in the fluid is given by [57]:<br>
+ (6.61)<br>
+Because of the conical geometry, the shear rate is uniform throughout the fluid. The <br>torque necessary to rotate the cone is related to the shear stress by:<br>
+ (6.62)<br>
+The viscosity can now be determined from:<br>
+ (6.63)<br>
+<hr>
+<A name=241></a><b>228</b>  <b>6Important Polymer Properties</b><br>
+Thus, the viscosity can be directly determined from measurement of torque and <br>rotational speed. If the fluid between the cone and plate is viscoelastic, the plates <br>will be pushed apart when the fluid is sheared. The force with which the plates are <br>pushed apart F is related to the first normal stress difference N1 in the fluid:<br>
+ (6.64)<br>
+The first normal stress difference is an accurate indicator of the viscoelastic be havior <br>of a fluid. Thus, with the cone and plate rheometer, one can accurately determine <br>some viscoelastic characteristics of a fluid.<br>The cone and plate rheometer is susceptible to irregularities at the liquid-air interface <br>and to secondary flows. As a result, the shear rate in steady shear measurements has <br>to be quite low to avoid the above-mentioned problems. In general, the shear rate <br>should not exceed 1 s­1; data above this rate should be regarded with caution.<br>Advantages of the cone and plate rheometer:<br>1.  Its ease of operation.<br>2.  It has uniform shear rate distribution.<br>3.  It also measures first normal stress difference.<br>4.  Its uniform temperature distribution.<br>Disadvantages of the cone and plate rheometer:<br>1.  It is limited to low shear rates (&lt; 1 s­1).<br>2.  These instruments tend to be expensive.<br>The disadvantage of the low shear rate limitation can be negated by applying an <br>oscillatory rotary motion to the moving plate. Thus, one can measure complex vis-<br>cosity as a function of frequency; see also Section 6.2.6.5 on dynamic analysis. In <br>this fashion, frequency levels up to about 500 radians/s are possible; this corre-<br>sponds to shear rates of about 500 s­1.<br>
+<b>6.2.6.4Slit Die Rheometer<br></b>A slit die rheometer is an extruder die with a rectangular flow channel with provi-<br>sions to measure pressures at various axial locations. The slit die is either directly <br>connected to an extruder or to a gear pump, which, in turn, is connected to an <br>extruder. A typical slit die geometry is shown in Fig. 6.22.<br>By measuring the flow rate   through the flow channel, the apparent shear rate at <br>the wall can be determined:<br>
+ (6.65)<br>
+where H is the slit height and W the slit width (W&gt;&gt;H).<br>
+<hr>
+<A name=242></a> <br>
+6.2Melt Flow Properties  <b>229</b><br>
+For a power law fluid with power law index n, the actual shear rate at the wall is:<br>
+ (6.66)<br>
+The shear stress at the wall can be determined from the gradient of the measured <br>pressure profile (dP/dz):<br>
+ (6.67)<br>
+The apparent viscosity can be determined from:<br>
+ (6.68)<br>
+For a power law fluid, the actual viscosity is:<br>
+ (6.69)<br>
+<b>Figure 6.22</b>Slit die viscometer<br>
+These equations are valid provided the ratio of width to height is large; in most <br>cases W/H &gt; 10. If the pressure gradient is constant, the pressure can be extra-<br>polated to the exit of the die. Several workers have found positive exit pressure by <br>using this procedure [54]. This exit pressure can be related to the first normal stress <br>difference provided the velocity profile remains fully developed right up to the die <br>exit:<br>
+ (6.70)<br>
+<hr>
+<A name=243></a><b>230</b>  <b>6Important Polymer Properties</b><br>
+However, the theoretical justification of this relationship seems to be questionable, <br>as discussed by Boger and Denn [67]. From a practical view, it is very difficult to <br>extrapolate from pressure readings that range from several MPa up to about 30 or <br>40 MPa down to exit pressures that range from less than 0.1 MPa to about 0.2 MPa. <br>This is particularly true if only two pressure transducers are situated in the fully <br>developed region, as appears to have been the case in several experimental results <br>reported in the literature [68­70]. Another problem is the assumption of a constant <br>pressure gradient. It has been shown [71] that the effects of temperature and pres-<br>sure will generally cause a significant non-linearity. This raises serious doubts <br>about the validity of a linear extrapolation of the pressure profile.<br>An interesting aspect of the slit die rheometer is the fact that the polymer has a sig-<br>nificant temperature and shear history by the time it reaches the slit die. This can <br>affect the rheological properties, as reported by the author [71]. On the other hand, <br>if viscosity data is to be used for die design purposes, the slit die viscometer is most <br>likely to produce pertinent viscosity data.<br>Advantages of the slit die rheometer:<br>1.  It operates in a useful shear rate range.<br>2.  It yields accurate shear viscosity versus shear rate data.<br>3.  It yields data representative of behavior during actual extrusion.<br>Disadvantages of the slit die rheometer:<br>1.  Commercial rheometers tend to be expensive.<br>2.  Proper and frequent calibration of pressure transducers is important to ensure <br>
+accurate results.<br>
+3.  It is difficult to determine data that can be related to the viscoelastic behavior of <br>
+the polymer.<br>
+<b>6.2.6.5Dynamic Analysis<br></b>Several rheometers do not subject the polymer to a steady rate of deformation but to <br>an oscillatory deformation, usually sinusoidal simple shear. If the angular frequency <br>is , and the shear strain amplitude 0, the shear strain  can be written as a func-<br>
+tion of time:<br>
+ (6.71)<br>
+The shear rate is determined by differentiating the shear strain with respect to time. <br>The shear rate is:<br>
+ (6.72)<br>
+<hr>
+<A name=244></a> <br>
+6.2Melt Flow Properties  <b>231</b><br>
+Dynamic analysis is generally used to study the linear viscoelastic properties of <br>polymers. The region of linear viscoelastic behavior is where a material function, <br>such as shear modulus or shear viscosity, is independent of the amplitude of the <br>strain or strain rate. Polymers follow linear viscoelastic behavior when the strain or <br>strain rate is sufficiently small. Thus, if the strain amplitude is sufficiently small, <br>the shear stress can be written as:<br>
+ (6.73)<br>
+where 0 is the amplitude of the shear stress and  the phase angle between stress <br>
+and strain. The phase angle is often referred to as loss angle. For a purely elastic <br>material, there is no phase shift between stress and strain, thus  is zero. For a <br>purely viscous material, there will be a maximum phase shift between stress and <br>strain, or  is 90°.<br>Equation 6.73 can be rewritten by introducing an in-phase modulus G' (real) and a <br>90° out-of-phase modulus G'' (imaginary):<br>
+ (6.74)<br>
+The storage modulus G' represents the elastic contribution associated with energy <br>storage; it is a function of the stress and strain amplitude and the phase angle:<br>
+ (6.75)<br>
+Similarly, the loss modulus G'' represents the viscous contribution associated with <br>energy dissipation; it is:<br>
+ (6.76)<br>
+Both G' and G'' are components of the complex modulus G*. The magnitude of the <br>complex modulus, as shown in Fig. 6.23, is:<br>
+ (6.77)<br>
+G'<br>
+G*<br>
+<b> Figure 6.23</b> <br>
+G&quot;<br>
+Complex modulus G*<br>
+<hr>
+<A name=245></a><b>232</b>  <b>6Important Polymer Properties</b><br>
+A complex notation is frequently used to describe the relationship between stress <br>and strain. If stress and strain are written as:<br>
+ (6.78a)<br>
+ (6.78b)<br>
+then the complex modulus can be written as:<br>
+ (6.79)<br>
+Equation 6.73 can be rewritten in yet a different form by introducing an in-phase <br>viscosity ' and out-of-phase viscosity ''. The real part ', the dynamic viscosity, <br>represents the viscous contribution associated with energy dissipation. The imagi-<br>nary part '' represents the elastic contribution associated with energy storage. The <br>shear stress can be written as a function of ' and '' as follows:<br>
+ (6.80)<br>
+The dynamic viscosity is related to the loss modulus by:<br>
+ (6.81)<br>
+The imaginary part of the viscosity is related to the storage modulus by:<br>
+ (6.82)<br>
+Both ' and '' are components of the complex viscosity *. The magnitude of the <br>complex viscosity is:<br>
+ (6.83)<br>
+In complex notation, the complex viscosity can be expressed as a function of ' and <br>'':<br>
+ (6.84)<br>
+The tangent of the phase angle is often used in characterization of viscoelastic mate-<br>rial. The "tan " can be determined from:<br>
+ (6.85)<br>
+<hr>
+<A name=246></a> <br>
+6.3Thermal Properties  <b>233</b><br>
+The attractiveness of dynamic analysis is that an accurate determination of the vis-<br>coelastic behavior can be made. A common geometry for dynamic measurements <br>is the cone and plate rheometer. In dynamic analysis, the viscosity components can <br>be measured up to an angular frequency of about 500 radians/s. Cox and Merz [72] <br>found empirically that the steady shear viscosity corresponds to the complex viscos-<br>ity if the shear rate in s­1 is plotted on the same scale as the angular frequency in <br>radians/s. This can be stated as:<br>
+ (6.86)<br>
+This empirical rule seems to hold up quite well for most polymers. Using this rule, <br>it is possible to determine viscosity data up to 500 s­1 with a cone and plate rheometer <br>by applying an oscillatory motion to the cone. This would be impossible if a steady <br>rotational motion was applied to the cone. In steady shear measurements on a cone <br>and plate rheometer, the maximum shear rate that can be measured is around 1 s­1, <br>which is much too low for applications to extrusion problems. The same is true for <br>measurements in the parallel plate test geometry. Thus, the dynamic measurement <br>extends the shear rate measurement range considerably, while still being able to <br>take advantage of the cone and plate geometry.<br>Dynamic mechanical analysis is not limited to just shear deformation; it is also used <br>with elongational deformation. Further, dynamic mechanical analysis is employed <br>in the characterization of solids as well as liquids. In 1982, a new standard was <br>established, ASTM D4065, to standardize procedures for testing all types of mate-<br>rials.<br>
+<b></b><br>
+ <b>6.3Thermal Properties</b><br>
+By the nature of the plasticating extrusion process, thermal properties are very <br>important. In the early portion of the extruder, solid polymer particles are heated to <br>the melting point. In the midportion of the extruder, the molten polymer is raised in <br>temperature to a level considerably above the melting point while the remaining <br>solid particles continue to heat up and melt. In the last portion of the extruder, the <br>molten polymer has to reach a thermally homogeneous state. When the extrudate <br>leaves the extruder die, it has to be cooled down, usually to room temperature. <br>Through this whole process, the polymer experiences a complicated thermal his-<br>tory. The thermal properties of the polymer are crucial to being able to describe and <br>analyze the entire extrusion process.<br>
+<hr>
+<A name=247></a><b>234</b>  <b>6Important Polymer Properties</b><br>
+<b>6.3.1Thermal Conductivity</b><br>
+The thermal conductivity of a material is essentially a proportionality constant be -<br>tween the conductive heat flux and the temperature gradient driving the heat flux. <br>The thermal conductivity of polymers is quite low, about two to three orders of mag-<br>nitude lower than most metals. From a processing point of view, the low thermal <br>conductivity creates some real problems. It very much limits the rate at which poly-<br>mers can be heated and plasticated. In cooling, the low thermal conductivity can <br>cause non-uniform cooling and shrinking. This can result in frozen-in stresses, <br>deformation of the extrudate, delamination, shrink voids, etc.<br>The thermal conductivity of amorphous polymers is relatively insensitive to tempe-<br>rature. Below the Tg, the thermal conductivity increases slightly with temperature; <br>
+above the Tg, it reduces slowly with temperature. The thermal conductivity above <br>
+the Tg as a function of temperature can be approximated by [41]:<br>
+ (6.87)<br>
+where the temperature is expressed in °K.<br>In most extrusion problems, however, the thermal conductivity of an amorphous <br>polymer can be assumed to be independent of temperature.<br>The thermal conductivity of semi-crystalline polymers is generally higher than <br>amorphous polymers. Below the crystalline melting point, the thermal conductivity <br>reduces with temperature; above the melting point, it remains relatively constant. <br>The thermal conductivity increases with density and, thus, with the level of crystal-<br>linity. The thermal conductivity at constant temperature as a function of density can <br>generally be written as:<br>
+ (6.88)<br>
+The change in thermal conductivity with temperature is relatively linear at tempe-<br>ratures above 0°C. The thermal conductivity as a function of temperature can be <br>described with:<br>
+ (6.89)<br>
+where T is in °C and k0 = k (T = D).<br>The temperature coefficient of the thermal conductivity CT for a particular polymer <br>
+seems to be relatively independent of the actual density. Thus, the combined density <br>and temperature dependence can be described by:<br>
+ (6.90)<br>
+<hr>
+<A name=248></a> <br>
+6.3Thermal Properties  <b>235</b><br>
+For polyethylene, the thermal conductivity as a function of density and temperature <br>can be described by:<br>
+ (6.91)<br>
+where T is in °C and  in g/cm3.<br>Equations 6.89 through 6.91 become less accurate as the temperature approaches <br>the melting point because non-linearities become significant. Therefore, these ex -<br>pressions should be used as approximations. Above the melting point, the thermal <br>conductivity of polyethylene is about 0.25 J/ms °C.<br>The thermal conductivity is dependent on the orientation of the polymer. If the poly-<br>mer is highly oriented, substantial differences in thermal conductivity can occur in <br>the direction of orientation and perpendicular to it. The difference can be as high <br>as almost 100% in PMMA as shown by Eiermann and Hellwege [73]. Hansen and <br>Bernier [74] found differences in thermal conductivity as much as 20-fold in HDPE.<br>Compacted polymer particles have a lower thermal conductivity because of the pres-<br>ence of voids between the particles. Based on experimental data, Yagi and Kunii [75] <br>proposed a model for the thermal conductivity of the bed. For fine particles and low <br>temperatures, the thermal conductivity of the bed can be written as:<br>
+ (6.92)<br>
+where kg is the thermal conductivity of the gas occupying the voids, kp the thermal <br>
+conductivity of the polymer particles, b the bulk density, p the density of the poly-<br>
+mer particles, and F a function of the density ratio, b/p (a power law relationship). <br>
+Langecker [96, 97] performed extensive measurements of the thermal conductivity <br>of isotropically compressed polymeric powders. He found an essentially linear rela-<br>tionship between the thermal conductivity and the bulk density. In the range of b/<br>
+p from 0.5 to 1.0, the data on polyethylene can be approximated by:<br>
+ (6.93)<br>
+The dependence of thermal conductivity and diffusivity of polyethylene on tempe-<br>rature, density, and molecular parameters was investigated by Kamal, Tan, and <br>Kashani [93]. Their publication contains a good review of prior experimental work <br>on thermal conductivity.<br>
+<hr>
+<A name=249></a><b>236</b>  <b>6Important Polymer Properties</b><br>
+<b>6.3.2Specific Volume and Morphology</b><br>
+The polymer density  is a function of pressure, temperature, and cooling rate. Spe-<br>cific volume ^<br>
+<i>V</i> is the reciprocal of density, ^<br>
+<i>V</i> = 1/. The general P, ^<br>
+<i>V</i>, T diagram of an <br>
+amorphous polymer is shown in Fig. 6.24.<br>If the material is cooled very slowly, the specific volume will reach a lower value than <br>at a relatively high cooling rate. In simple terms, at a low cooling rate the polymer <br>molecules, because of their thermal motion, have more opportunity to position them-<br>selves closer together. This reduces the free volume of the polymer, i.e., the volume <br>fraction not occupied by polymer molecules. Below the glass transition temperature, <br>the thermal motion of the polymer molecules is drastically reduced and the free vol-<br>ume remains approximately constant. Therefore, the change in specific volume with <br>temperature is much larger above Tg than below Tg. The reduction in specific volume <br>
+below Tg is primarily due to the reduced thermal motion of the polymer molecules.<br>Increasing the molecular weight increases the glass transition temperature as <br>shown in the Fox and Flory equation:<br>
+ (6.94)<br>
+1.04<br>
+Lacqrene 1450<br>
+1.02<br>
+0.1 MPa<br>
+g]<br>
+20<br>
+/k<br>
+1.00<br>
+3 m<br>
+40<br>
+60<br>
+0.98<br>
+100<br>
+0.96<br>
+Specific Volume [E-3 <br>
+0.94<br>
+0.92 0<br>
+50<br>
+100<br>
+150<br>
+200<br>
+Temperature [C]<br>
+<b>Figure 6.24</b>P, ^V, T diagram of an amorphous polymer<br>
+where Tg is the glass transition temperature for infinite molecular weight and con-<br>
+stant K is a parameter of the polymer.<br>
+<hr>
+<A name=250></a> <br>
+6.3Thermal Properties  <b>237</b><br>
+When the ^<br>
+<i>V</i>, T curve is determined at a higher pressure, the specific volume will <br>
+reduce. The reduction above Tg will be more significant than below Tg because of the <br>
+larger free volume above Tg. Another interesting phenomenon is the shift of the Tg to <br>
+a higher temperature when the pressure is increased. In the normal range of pro-<br>cessing pressures (P &lt; 100 MPa), the pressure dependence of Tg can generally be <br>
+neglected.<br>When the cooling rate is fast, the material goes through the transition region at a <br>higher temperature and has a larger specific volume at temperatures below this <br>region. This is because the polymer molecules have not had sufficient time to posi-<br>tion themselves in a preferred configuration. As the polymer relaxes, it will reduce <br>in specific volume until it eventually reaches the specific volume corresponding to a <br>low cooling rate.<br>A general P, ^<br>
+<i>V</i>, T diagram of a semi-crystalline polymer is shown in Fig. 6.25.<br>
+The behavior in the liquid state is essentially the same as amorphous polymers. In <br>the transition region, an abrupt change in slope occurs as crystallization begins to <br>take place. This is the crystallization temperature Tc. If the material is cooled very <br>
+rapidly, the crystallization rate can be depressed, depending on the crystallization <br>kinetics. In fact, in some materials with sufficiently slow crystallization kinetics, the <br>crystallization can be almost completely suppressed by rapid cooling. A well-known <br>example is polyethylene terephthalate (PET). If PET is rapidly quenched, it is almost <br>completely amorphous with a density of about 1.33 g/­cm3. If it is cooled slowly, it <br>will crystallize with a resulting higher density of about 1.40 g/­cm3.<br>
+0.70<br>
+Kynar Flex 2750<br>
+0.1 MPa<br>
+g]<br>
+20<br>
+/k<br>
+0.65<br>
+3<br>
+40<br>
+m<br>
+80<br>
+120<br>
+0.60<br>
+Specific Volume [E-3 <br>
+0.55 0<br>
+50<br>
+100<br>
+150<br>
+200<br>
+250<br>
+Temperature [C]<br>
+<b>Figure 6.25</b>P, ^V, T diagram of a semi-crystalline polymer<br>
+<hr>
+<A name=251></a><b>238</b>  <b>6Important Polymer Properties</b><br>
+The percent crystallinity of polymers with rapid crystallization kinetics is not much <br>affected by the cooling rate; however, the crystallite morphology may be strongly <br>affected. These differences in morphology can cause significant changes in physical <br>properties. As the extrudate is cooled, the skin will cool most rapidly and the inte-<br>rior region of the extrudate will cool more slowly. This will cause corresponding <br>changes in polymer morphology. Annealing can also modify the polymer morphol-<br>ogy, particularly the crystalline regions. Annealing is the process of exposing the <br>polymer to an elevated temperature for a certain period of time. This is sometimes <br>done as a post-extrusion operation to control the polymer morphology and physical <br>properties. In polymers with a high level of crystallinity, annealing causes thicken-<br>ing of the lamellae and an increase of the melting point. The relationship between <br>crystallite size and melting point is given by Hoffman and Lauritzen's equation [81]:<br>
+ (6.95)<br>
+where <br>
+0<br>
+e is the surface free energy, Tm  the equilibrium melting temperature, Hf <br>
+the heat of fusion, and L the lamella thickness.<br>The application of stress can further cause significant changes in the polymer mor-<br>phology; see also Section 5.2.2. In flow-induced crystallization of dilute polymer <br>solutions, a shish-kebab crystal morphology develops. In melt-crystallized polymers, <br>a spherulitic crystal morphology develops, made up of folded chain lamellae. Defor-<br>mation of the polymer below the melting point can cause very effective orientation <br>of the polymer molecules. This can result in a high degree of anisotropy in the mate-<br>rial with very good mechanical properties in the orientation direction. Examples are <br>solid-state extrusion and fiber drawing; many other examples are available.<br>The change in specific volume below Tc is primarily due to the increasing degree of <br>
+crystallinity. Therefore, the volume change in semi-crystalline polymers is consider-<br>ably larger than those experienced with amorphous polymers. When the semi-crys-<br>talline polymer is cooled from the melt at elevated pressure, the Tc shifts to a higher <br>
+temperature. This phenomenon can be described by the Clausius-Clapeyron equa-<br>tion, which relates the equilibrium melting point at any pressure to the melting <br>point at atmospheric pressure:<br>
+ (6.96)<br>
+where  ^<br>
+<i>V</i>a and ^<i>V</i>c are the amorphous and crystalline specific volumes, P is the hydro-<br>
+static pressure in atmospheres, and Hf is the heat of fusion of the polymer at atmo-<br>
+spheric pressure. The melting point elevation can be significant in polymer melts, i.e., <br>for polyethylene about 7°C per 20 MPa ( 3000 psi). A polymer melt under high pres-<br>
+<hr>
+<A name=252></a> <br>
+6.3Thermal Properties  <b>239</b><br>
+sure is thus under a higher degree of supercooling. This will affect the crystallization <br>behavior of the material, an effect most noticeable in processes where high pressure <br>commonly occurs, such as injection molding. As mentioned earlier and also in Sec-<br>tion 5.2.2, this effect can be further magnified if the polymer melt is under stress. <br>The melting point elevation with pressure explains the large compressibility values <br>for polymer melts found by some investigators [76]. Actually, the sudden increase of <br>compressibility at elevated pressures indicates the onset of crystallization.<br>The P, ^<br>
+<i>V</i>, T diagram of HDPE is shown in Fig. 6.26.<br>
+Various workers have developed empirical relationships between P, ^<br>
+<i>V</i>, and T. Among <br>
+those are the equations of Spencer and Gilmore [77], Breuer and Rehage [78], Kamal <br>and Levan [79], and Simha and Olabisi [80].<br>
+1.4<br>
+GUR 4113<br>(Generic PE-HMW)<br>
+0.1 MPa<br>
+1.3<br>
+20<br>
+40<br>
+g]<br>/k3<br>
+80<br>
+m<br>
+1.2<br>
+120<br>
+160<br>
+200<br>
+1.1<br>
+Specific Volume [E-3 <br>
+1.0<br>
+0.9 0<br>
+50<br>
+100<br>
+150<br>
+200<br>
+250<br>
+Temperature [C]<br>
+<b>Figure 6.26</b>P, ^V, T diagram of high density polyethylene (HDPE)<br>
+<b>6.3.3Specific Heat and Heat of Fusion</b><br>
+The specific heat is the amount of energy required to raise a unit mass of a material <br>one degree in temperature. In S.I. units, specific heat is expressed in J/kg °K. It can <br>be measured either at constant pressure Cp or at constant volume Cv. The two values <br>
+are related by:<br>
+ (6.97a)<br>
+<hr>
+<A name=253></a><b>240</b>  <b>6Important Polymer Properties</b><br>
+where the expansion coefficient v is:<br>
+ (6.97b)<br>
+and the compressibility:<br>
+ (6.97c)<br>
+The specific heat at constant pressure is larger than the specific heat at constant <br>volume because additional energy is required to bring about the volume change <br>against external pressure P.<br>The specific heat of amorphous polymers increases with temperature in approxi-<br>mately a linear fashion below and about Tg. A step-like change occurs around the <br>
+glass transition temperature as shown in Fig. 6.27(a). With semi-crystalline poly-<br>mers, the step change at Tg is much less pronounced; however, a very distinct maxi-<br>
+mum occurs at the crystalline melting point. At the melting point, the specific heat <br>is theoretically infinite for a material with a perfectly uniform crystalline structure, <br>as shown in Fig. 6.27(b). Since this is not the case in semi-crystalline polymers, <br>these materials exhibit a melting peak of certain width as show in Fig. 6.27(c). The <br>narrower the peak, the more uniform the crystallite morphology.<br>The specific heat above the melting point increases slowly with temperature. The <br>area under the melting peak of the Cp, T curve equals the heat of fusion Hf multi-<br>
+plied with the crystalline weight fraction. Both the heat of fusion and the percent <br>crystallinity are dependent on the thermomechanical history of the polymer, as dis-<br>cussed in the previous section.<br>
+Amorphous polymer<br>
+100% Crystalline material<br>
+Semi-crystalline polymer<br>
+Specific heat<br>
+Specific heat<br>
+Specific heat<br>
+Temperature<br>
+Temperature<br>
+Temperature<br>
+(a)<br>
+(b)<br>
+(c)<br>
+<b>Figure 6.27</b>Specific heat as a function of temperature<br>
+<hr>
+<A name=254></a> <br>
+6.3Thermal Properties  <b>241</b><br>
+<b>6.3.4Specific Enthalpy</b><br>
+A very useful thermal property in polymer processing is the specific enthalpy. It is <br>defined by:<br>
+ (6.98)<br>
+If T1 is taken as ambient temperature and T2 as the process temperature, the specific <br>
+enthalpy indicates how much energy is required to accomplish this temperature <br>rise. This can be considered to be the theoretical minimum specific energy require-<br>ment in the extrusion process.<br>Figure 6.28 shows , T curves for several amorphous and semi-crystalline polymers.<br>
+0.10<br>
+<b>PA</b><br>
+<b>HDPE</b><br>
+0.15<br>
+Enthalpy<br>
+<b>LDPE</b><br>
+Enthalpy<br>
+[kW.hr/kg]<br>
+<b>PS</b><br>
+[Hp.hr/lb]<br>
+0.10<br>
+<b>PC</b><br>
+0.05<br>
+<b>PVC</b><br>
+0.05<br>
+0<br>
+<b> Figure 6.28</b> <br>
+0<br>
+50<br>
+100<br>
+150<br>
+200<br>
+250<br>
+Enthalpy-temperature curves for <br>
+Temperature [degrees C]<br>
+several polymers<br>
+The first observation is that amorphous polymers have a continuous rise in  while <br>semi-crystalline polymers exhibit an abrupt change in slope at the melting point. <br>The second observation is that amorphous polymers generally have much lower  <br>values than semi-crystalline polymers over the same T. If one compares PVC to <br>LDPE from T1 = 20°C to T2 = 150°C, the  (PVC)  0.05 kWhr/kg (0.03 hphr/lb), <br>
+while  (LDPE)  0.13 kWhr/kg (0.08 hphr/lb). Thus, if the throughput of the extru-<br>sion process is 100 kg/hr and the process temperature 150°C, the theoretical power <br>requirement is 5 kW for PVC and 13 kW for LDPE.<br>Thus, based on the thermal properties of the polymers, there is a very significant <br>difference in power requirement between PVC and LDPE, about a 3:1 ratio! A stand-<br>ard 24 L/D extruder that runs fine with LDPE is likely to have too high a power <br>consumption for PVC. This will result in high stock temperatures and increased <br>chance of degradation, particularly in the case of PVC, and especially with rigid <br>
+<hr>
+<A name=255></a><b>242</b>  <b>6Important Polymer Properties</b><br>
+PVC. The thermal properties of PVC dictate an extrusion process with low specific <br>energy consumption and short residence times to minimize high melt temperatures <br>and high temperature exposure time. This is the main reason that closely intermesh-<br>ing twin screw extruders have become so popular for RPVC. These machines have <br>low specific energy consumption, short residence times, narrow residence times <br>distribution, and good control over stock temperatures. Long single screw extruders <br>are generally not well suited for RPVC extrusion. As a result, a higher level of stabi-<br>lizers has to be added to the polymer to enable it to survive the extrusion process <br>without too much degradation. This increases the compound cost and may be more <br>expensive in the long run than processing the material on a more suitable machine.<br>
+<b>6.3.5Thermal Diffusivity</b><br>
+The thermal diffusivity is a property derived from thermal conductivity, specific <br>heat, and density. The relationship is:<br>
+ (6.99)<br>
+The thermal diffusivity is a very useful quantity in transient heat transfer problems, <br>as discussed in Section 5.3.1.<br>The thermal diffusivity can be calculated from the values k, , and Cp; however, in <br>
+most cases it is measured directly. In fact, the thermal diffusivity can be measured <br>more easily and accurately than the thermal conductivity. If a thick slab of material, <br>initially at To, is suddenly exposed to an elevated temperature T1 at one wall and <br>
+maintained at this temperature, the temperature distribution can be described by:<br>
+ (6.100)<br>
+if it is assumed that  is independent of temperature.<br>Equation 6.100 is a shortened version of the general energy balance equation <br>(Eq. 5.5) valid for simple unidirectional conduction. This is a standard handbook <br>problem; the solution is (see [1] of Chapter 5):<br>
+ (6.101)<br>
+In Eq. 6.101 erf (x) stands for error function; this is defined as:<br>
+ (6.102)<br>
+<hr>
+<A name=256></a> <br>
+6.3Thermal Properties  <b>243</b><br>
+This integral cannot be solved easily; however, tabular or graphical representations <br>of the error function are available in various handbooks, e.g. [82]. Figure 6.29 shows <br>the function 1­erf (x), the complementary error function.<br>Equation 6.101 describes conductive heating of a semi-infinite slab. It is valid as <br>long as thermal penetration thickness t is less than the slab thickness. Essentially <br>
+all temperature change (99%) takes place within the thermal penetration thickness, <br>which is given by:<br>
+ (6.103)<br>
+1.0<br>
+0.8<br>
+x)<br>
+0.6<br>
+unction, 1-erf(<br>rf<br>rro<br>
+0.4<br>
+ye<br>
+0.2<br>
+Complimentar<br>
+0<br>
+<b> Figure 6.29</b> <br>
+0<br>
+0.5<br>
+1.0<br>
+1.5<br>
+Variable x<br>
+Complementary error function<br>
+The time, temperature profiles in a slab with double-sided heating are given by <br>Eq. 7.99.<br>With Eqs. 6.101 and 6.102, the temperature as a function of time and distance is <br>completely described. Thus, by doing simple temperature measurements on a slab <br>of material with one wall suddenly exposed to a higher temperature, the thermal <br>diffusivity can be determined in a relatively straightforward fashion.<br>This example illustrates how the thermal diffusivity can be measured and how it is <br>used to describe heat conduction problems. For amorphous polymers and polymer <br>melts, diffusivity is approximately linear with the velocity of sound vs; the propor-<br>
+tionality constant is about 6E­13 [41]:<br>
+ (6.104)<br>
+The sound velocity is related to the molar sound velocity function FR or Rao function, <br>
+the molar volume per structural unit Vm, and the Poisson ratio <i>v</i>. It can be written <br>
+as:<br>
+ (6.105)<br>
+<hr>
+<A name=257></a><b>244</b>  <b>6Important Polymer Properties</b><br>
+The Poisson ratio for liquids is 1/2 and for isotropic solids 1/3. The Poisson ratio for <br>polymers below Tg is approximately 1/3 and above Tg about 1/2. The ratio of the Rao <br>
+function and the molar volume is approximately constant for many amorphous poly-<br>mers [41]; the ratio is about 55. Thus, the thermal diffusivity can be approximated <br>with:<br>
+ (6.106)<br>
+with  expressed in m2/s.<br>The ratio FR/Vm is related to the compressibility  and density  by:<br>
+ (6.107)<br>
+With Eq. 6.107, one can write the expression for the sound velocity as follows:<br>
+ (6.108)<br>
+The thermal diffusivity of most polymers is around 10­7 m2/s. In the analysis of <br>most extrusion problems, the thermal diffusivity is considered to be constant. In <br>reality, however, the thermal diffusivity depends on pressure, temperature, and orien-<br>tation. The anisotropy of the thermal diffusivity of uniaxially stretched polyethylene <br>was studied by Kilian and Pietralla [83]. They found large differences between the <br>thermal diffusivity in the orientation direction and perpendicular to the orientation <br>direction, as high as 20:1. The pressure dependence of the thermal conductivity, <br>thermal diffusivity, and specific heat of some polymers was studied by Andersson <br>and Sundqvist [84]. They found that the thermal conductivity and thermal diffusiv-<br>ity increase with pressure. At very high pressures ( 4000 MPa), the thermal con-<br>ductivity and thermal diffusivity about double. However, at pressures within the <br>range of normal polymer processing, pressures less than 100 MPa ( 15,000 psi), <br>the changes are less than 5%. At 30 MPa ( 5000 psi), the expected change in ther-<br>mal conductivity and thermal diffusivity is about 1 to 2%; this will be negligible in <br>most cases. The specific heat reduces with increasing pressure; however, in the <br>poly mer processing range, the changes are quite small ­ less than 0.5% at 100 MPa <br>( 15,000 psi).<br>
+<hr>
+<A name=258></a> <br>
+6.3Thermal Properties  <b>245</b><br>
+<b>6.3.6Melting Point</b><br>
+The melting point is the temperature at which the crystallites melt. Since the crys-<br>tallites are not perfectly uniform, there is really not one single melting point but a <br>melt temperature range. The melting point is often taken as the temperature at the <br>peak of the DSC curve; see Section 6.3.8.<br>The melting point is dependent on the pressure and crystallite morphology as dis-<br>cussed in Section 6.3.3. It can be measured quite easily and accurately. The melting <br>point dictates, to some extent, the process temperatures necessary in extrusion. As <br>a general rule, the process temperatures are about 50°C above the melting point. If <br>the process temperature is too close to the melting point, the polymer melt viscosity <br>will be too high, resulting in excessive power consumption. If the process tempera-<br>ture is too far above the melting point, the polymer may degrade.<br>For homopolymers, the melting temperature depends on the molecular weight of the <br>polymer:<br>
+ (6.109)<br>
+where Tm is the melting temperature for an infinite length polymer molecule and <br>
+M0 is the molecular weight of the monomer.<br>
+<b>6.3.7Induction Time</b><br>
+The induction time of a polymer is a very useful quantity in process design, process <br>optimization, and troubleshooting. It represents the amount of time elapsed at a <br>certain temperature and in a certain atmosphere before the effects of degradation <br>become measurable. Essentially, the induction time indicates how long a polymer <br>can be exposed to a certain temperature before it starts to degrade.<br>In extrusion, one would like to make sure that the longest residence time in the <br>machine at a certain process temperature is less than the induction time at the <br>same temperature. Thus, if one knows the residence times to be expected in the <br>extrusion process and if one knows how the induction time varies with temperature, <br>the process temperature at which degradation will be avoided can be accurately <br>determined. This is a very useful tool in process engineering, particularly if one <br>deals with a polymer of limited thermal stability.<br>If degradation occurs during the extrusion process, there are two approaches that <br>one can take to the problem. One is to modify the process so as to reduce the chance <br>of degradation. The other approach is to modify the polymer to improve its thermal <br>stability. The changes to the process should result in lower stock temperatures, and/<br>
+<hr>
+<A name=259></a><b>246</b>  <b>6Important Polymer Properties</b><br>
+or reduced exposure time to elevated temperatures, and possibly exclusion of degra-<br>dation-promoting substances (e.g., oxygen, certain additives, certain metal compo-<br>nents of the tooling or substrate, etc.). If the changes to the process cannot alleviate <br>the degradation problem, one has to consider the polymer itself. The thermal stabil-<br>ity (induction time) of most polymers can be improved by adding stabilizers to the <br>polymer, such as antioxidants. If process changes cannot solve the problem of degra-<br>dation, the thermal stability of the polymer should be improved by adding stabi-<br>lizers to it. Of course, one may opt to select a different polymer altogether.<br>In some cases, the thermal stability of a polymer or a compound is so poor that it <br>cannot be extruded without degradation under any conditions. This can be conclu-<br>sively determined if induction time data are available. The process engineer can <br>then go back to the polymer chemist and explain exactly what changes should be <br>made to the polymer. This procedure eliminates the question of whether the prob-<br>lem is caused by the polymer or by the process. Thus, induction time data can act as <br>a bridge between the process engineer and polymer chemist and allow them to com-<br>municate and cooperate in a useful fashion.<br>It is clear that the induction time is a strong function of temperature. For many poly-<br>mers, a plot of induction time against reciprocal absolute temperature will form <br>approximately a straight line on semi-log paper, as shown in Fig. 6.30.<br>This indicates that the induction time reduces exponentially with temperature. <br>Curve A in Fig. 6.29 is an HDPE as received from the manufacturer and curve B is <br>an ethylene acrylic acid (EAA) as received from the manufacturer. The HDPE can be <br>processed at 200°C without noticeable degradation; however, EAA shows clear signs <br>of degradation at that temperature. It should be processed at about 160°C to avoid <br>degradation.<br>
+Temperature [degrees C]<br>
+260 240 220 200<br>
+180<br>
+160<br>
+10.0<br>
+1.0<br>
+HDPE<br>
+EAA<br>
+Induction time [minutes]<br>
+0.1<br>
+0.0018<br>
+0.0020<br>
+0.0022<br>
+0.0024 <b> Figure 6.30</b> <br>
+Reciprocal temperature [K-1]<br>
+Induction time versus temperature<br>
+<hr>
+<A name=260></a> <br>
+6.3Thermal Properties  <b>247</b><br>
+The relationship between induction time tind and temperature T can generally be <br>
+expressed as an Arrhenius equation:<br>
+ (6.110)<br>
+where A is a time constant, E the activation energy, and R the universal gas con-<br>stant.<br>The induction time can be conveniently determined on a TGA, but other instruments <br>can be used as well, for instance a DSC. Thermal characterization will be discussed <br>in the next section.<br>
+<b>6.3.8Thermal Characterization</b><br>
+In thermal characterization, a controlled amount of heat is applied to a sample and <br>its effect measured and recorded. In isothermal operations, the effect is recorded as <br>a function of time at constant temperature. In a programmed temperature operation, <br>the temperature is changed in a predetermined fashion, e.g., at a certain rate, and <br>the effect is recorded as a function of temperature. General texts on thermal charac-<br>terization include Wendlandt [85], Daniels [86], and Turi [87].<br>
+<b>6.3.8.1DTA and DSC<br></b>Differential thermal analysis (DTA) and differential scanning calorimetry (DSC) are <br>similar techniques. They measure change in the heat capacity of a sample. These <br>techniques can be used to determine various transition temperatures (Tm, Tg, T, T, <br>
+etc.), specific heat, heat of fusion, percent crystallinity, onset of degradation tempera-<br>ture, induction time, reaction rate, crystallization rate, etc. A DSC instrument oper-<br>ates by compensating electrically for a change in sample heat. The power for heating <br>is controlled in such a way that the temperature of the sample and the reference is <br>the same. The vertical axis of a DSC temperature scan shows the heat flow in cal/s.<br>A DTA instrument operates by measuring the change in sample temperature with <br>respect to an inert reference. Newer DTA instruments with externally mounted ther-<br>mocouple and reproducible heat path have a precision comparable to the DSC. Older <br>DTA instruments with the thermocouple placed in the sample were less accurate <br>and reproducible.<br>
+<b>6.3.8.2TGA<br></b>A thermogravimetric analyzer measures the change in weight of a sample due to <br>volatilization, reaction, or absorption from the gas phase. With polymers, the TGA is <br>used to measure the amount and loss of moisture or diluent, and rates and tempera-<br>
+<hr>
+<A name=261></a><b>248</b>  <b>6Important Polymer Properties</b><br>
+tures of reactions. It is a convenient instrument to determine the polymer induction <br>time, as discussed in Section 6.3.7. Sample size is usually less than one gram, thus <br>the amount of polymer required for characterization is minimal.<br>
+<b>6.3.8.3TMA<br></b>In thermomechanical analysis (TMA), the change in mechanical properties is meas-<br>ured as a function of temperature and/or time. A probe in contact with the sample <br>moves as the sample undergoes dimensional changes. The movement of the probe is <br>measured with an LVDT. The sample deformations that can be measured are com-<br>pression, penetration, extension, and flexure or bending.<br>
+<b>6.3.8.4Other Thermal Characterization Techniques<br></b>While TMA refers to a measurement of a static mechanical property, there are also <br>techniques that employ dynamic measurement. In the torsional braid analysis <br>(TBA), a sample is subjected to free torsional oscillation. The natural frequency and <br>the decay of oscillations are measured. This provides information about the visco-<br>elastic behavior of materials. However, these measurements are elaborate and time <br>consuming. In dynamic mechanical analysis (DMA), a sample is exposed to forced <br>oscillations. A large number of useful properties can be measured by this technique; <br>see also Section 6.2.6.5.<br>In thermal optical analysis (TOA), the conversion of plane-polarized light to ellipti-<br>cally polarized light is measured in semi-crystalline polymers. The intensity of the <br>depolarized light transmitted through a sample is a function of the level of crystal-<br>linity. Melting and recrystallization phenomena can be analyzed; the technique does <br>not appear to be sensitive to glass transitions [88]. The TOA technique is also <br>referred to as thermal depolarization analysis (TDA) and depolarized light intensity <br>method (DLI).<br>
+<b></b><br>
+ <b>6.4Polymer Property Summary</b><br>
+In Table 6.2 a number of rheological and thermal properties have been tabulated for <br>several important generic polymers. These data have been gathered from numerous <br>sources, including the author's own measurements. The data should be used as esti-<br>mates only, because measurement techniques may differ and because considerable <br>differences in properties can occur in one particular polymer as a result of varia-<br>tions in molecular weight distribution, additives, thermomechanical history, etc. <br>Actual measurement of polymer properties should always be preferred above pub-<br>lished data. However, actual measurement is not always possible, in which case the <br>table may provide useful information.<br>
+<hr>
+<A name=262></a> <br>
+6.4Polymer Property Summary  <b>249</b><br>
+Finally, some useful references should be mentioned containing data on polymer <br>properties. Nielson's book [89] on polymer properties is an exhaustive survey of a <br>large number of physical properties. The book by van Krevelen [41] is an excellent <br>book on polymer properties and their relationship to chemical structure. The VDMA <br>series on properties for polymer processing [90, 91, 95] contains a large amount of <br>data on thermal properties [90], melt flow properties [91], and frictional properties <br>[95]. Other useful data can be found in the yearly issues of the International Plastics <br>Selector Books, the yearly Modern Plastics Encyclopedia, the Plastics Technology <br>Manufacturing Handbook and Buyers' Guide, etc.<br>Nowadays, a substantial amount of information is available on the internet, and poly-<br>mer properties are no exception. Many resin suppliers have data available on their <br>website, and there are a number of electronic polymer databases available. One of <br>the most useful databases is CAMPUS, acronym for Computer Aided Material Pre-<br>selection by Uniform Standards. CAMPUS has become the most successful and <br>widely used materials database for plastics. More than 25 international resin suppli-<br>ers provide technical data on their products; more than 100,000 copies have been <br>distributed in Europe alone.<br>
+<b>Table 6.2</b>Useful Properties of a Number of Generic Polymers<br>
+Polymer<br>
+k<br>
+Cp<br>
+<b></b><br>
+Tg<br>
+Tmp<br>
+n<br>
+d<b></b>/<b></b>dT<br>
+[J/ms°C]<br>
+[J/g°C]<br>
+[g/cm3]<br>
+[°C]<br>
+[°C]<br>
+[­]<br>
+[°C­1]<br>
+PS<br>
+0.12<br>
+1.20<br>
+1.06<br>
+101<br>
+­<br>
+0.30<br>
+0.08<br>
+PVC<br>
+0.21<br>
+1.10<br>
+1.40<br>
+80<br>
+­<br>
+0.30<br>
+0.20<br>
+PMMA<br>
+0.20<br>
+1.45<br>
+1.18<br>
+105<br>
+­<br>
+0.25<br>
+0.20<br>
+SAN<br>
+0.12<br>
+1.40<br>
+1.08<br>
+115<br>
+­<br>
+0.30<br>
+0.20<br>
+ABS<br>
+0.25<br>
+1.40<br>
+1.02<br>
+115<br>
+­<br>
+0.25<br>
+0.20<br>
+PC<br>
+0.19<br>
+1.40<br>
+1.20<br>
+150<br>
+­<br>
+0.70<br>
+0.05<br>
+LDPE<br>
+0.24<br>
+2.30<br>
+0.92<br>
+­120/­90<br>
+120<br>
+0.35<br>
+0.03<br>
+LLDPE<br>
+0.24<br>
+2.30<br>
+0.92<br>
+­120/­90<br>
+125<br>
+0.60<br>
+0.02<br>
+HDPE<br>
+0.25<br>
+2.25<br>
+0.95<br>
+­120/­90<br>
+130<br>
+0.50<br>
+0.02<br>
+PP<br>
+0.15<br>
+2.10<br>
+0.91<br>
+­10<br>
+175<br>
+0.35<br>
+0.02<br>
+PA-6<br>
+0.25<br>
+2.15<br>
+1.13<br>
+50<br>
+225<br>
+0.70<br>
+0.02<br>
+PA-6.6<br>
+0.24<br>
+2.15<br>
+1.14<br>
+55<br>
+265<br>
+0.75<br>
+0.03<br>
+PET<br>
+0.29<br>
+1.55<br>
+1.35<br>
+70<br>
+275<br>
+0.60<br>
+0.03<br>
+PBT<br>
+0.21<br>
+1.25<br>
+1.35<br>
+45<br>
+250<br>
+0.60<br>
+0.03<br>
+PVDF<br>
+0.16<br>
+1.38<br>
+1.76<br>
+­40<br>
+170<br>
+0.38<br>
+0.03<br>
+FEP<br>
+0.20<br>
+1.18<br>
+2.15<br>
+70<br>
+275<br>
+0.60<br>
+0.04<br>
+k is the thermal conductivity<br>
+CP is the specific heat at constant pressure<br>
+ is the density<br>
+Tg is the glass transition temperature<br>
+Tmp is the crystalline melting point<br>
+n is the power law index<br>
+d/dT is the relative change in viscosity with temperature<br>
+<hr>
+<A name=263></a><b>250</b>  <b>6Important Polymer Properties</b><br>
+The data in the CAMPUS database has been obtained with uniform, standardized <br>test methods as descriribed in ISO 10350, ISO 11403-1, and ISO 11403-2. CAMPUS <br>is distributed free of charge to customers directly from the resin manufacturers. In <br>fact, CAMPUS data from a number of resin suppliers can be downloaded from their <br>websites at no cost. CAMPUS is available in five languages: English, German, French, <br>Spanish, and Italian [102]. Some of the data in this chapter are actually from this <br>database.<br>M-Base Engineering + Software GmbH in Aachen, Germany (www.m-base.de) makes <br>available a program, MCBase, that allows the user to search, compare, and perform <br>queries of all CAMPUS data that the user has loaded into the databank. MCBase has <br>several features not available in CAMPUS such as WLF and power-law curve fitting <br>of viscosity data, substitute grade search, exclude function, and enhanced text <br>search options.<br>
+References<br>1.  R.C. Wahl, Plastics World, 64­67, Nov. (1978)<br>2.  J.R. Mitchell, Chem. Eng. (U.K.), 177­183, Feb. (1977)<br>3.  D. Train and C.J. Lewis, 3rd Congress European Federation of Chemical Engineers, <br>
+ <br>London, June 20­29 (1962)<br>
+4.  A.W. Jenike, P.J. Elsey, and R.H. Woolley, Proc. Am. Soc. Test Mater., 60, 1168 (1960)<br>5.  R.S. Spencer, G.D. Gilmore, and R.M. Wiley, J. Appl. Phys., 21, 527­531 (1950)<br>6.  D. Train, Trans. Inst. Chem. Eng. 35, 262­265 (1957)<br>7.  W.M. Long, Powder Metall., No. 6, 73­86 (1980)<br>8.  K. Schneider, Chem. Eng. Techn., 41, 142 (1969)<br>9.  E. Goldacker, Ph.D. thesis, IKV, Aachen, Germany (1971)<br>10.  K. Umeya and R. Hara, Polym. Eng. Sci., 18, 366­371 (1978)<br>11.  K. Umeya and R. Hara, Polym. Eng. Sci., 20, 778­782 (1980)<br>12.  N.M. Smith and J. Parnaby, Polym. Eng. Sci., 20, 830­833 (1980)<br>13.  K. Kawakita and K.H. Luedde, Powder Techn., 4, 61 (1970)<br>14.  R.J. Crawford, Polym. Eng. Sci., 22, 33­306 (1982)<br>15.  W. Ostwald, Kolloid-2, 36, 99­117 (1925)<br>16.  A. de Waele, Oil and Color Chem. Assoc. J., 6, 33­88 (1923)<br>17.  R.B. Bird, R.C. Armstrong, and O. Hassager, "Dynamics of Polymeric Liquids," Vol. I, p. <br>
+209, Wiley, NY (1977)<br>
+18.  H. Eyring, Ind. Eng. Chem., 50, 1036­1040 (1958)<br>19.  P.J. Carreau, Ph.D. thesis, Univ. of Wisconsin, Madison (1968)<br>20.  M. Reiner, "Deformation Strain and Flow," a) p. 258, b) p. 246, Interscience Publishers, <br>
+NY (1960)<br>
+<hr>
+<A name=264></a> References <br>
+<b>251</b><br>
+21.  J.M. McKelvey, "Polymer Processing," Wiley, NY (1962)<br>22.  A.H.P. Skelland, "Non-Newtonian Flow and Heat Transfer," Wiley, NY (1967)<br>23.  F.P. Bowden and D. Tabor, "Friction and Lubrication of Solids," Oxford Univ. Press, <br>
+ <br>London  (1950)<br>
+24.  K. Schneider, Ph.D. thesis, IKV, Aachen, Germany (1968)<br>25.  K. Schneider, Kunststoffe, 59, 97­102 (1969)<br>26.  C.I. Chung, W.J. Hennessee, and M.H. Tusim, Polym. Eng. Sci., 17, 9­20 (1977)<br>27.  J. Huxtable, F.N. Cogswell, and J.D. Wriggles, Plast. Rubber Proc. Appl., 1, 87­93 (1981)<br>28.  H. Chang and R.A. Daane, SPE 32nd ANTEC, San Francisco, p. 335, May (1974)<br>29.  R.B. Gregory, SPE J., 25, 55­59 (1969)<br>30.  G.M. Gale, SPE 39th ANTEC, Boston, p. 669, May (1981)<br>31.  J.A.D. Emmanuel and L.R. Schmidt, SPE 39th ANTEC, Boston, p. 672, May (1981)<br>32.  G.M. Bartenev and V.V. Lavrentev, "Friction and Wear of Polymers," Elsevier, NY (1981)<br>33.  B. Maxwell and A. Jung, Modern Plastics, 35, 3, 174­180 (1957)<br>34.  R.F. Westover, SPE Trans. 1, 14­20 (1962)<br>35.  V. Semjonov, Rheologica Acta, 2, 138­142 (1962); 4, 133­137 (1965); 6, 154­170 <br>
+(1967)<br>
+36.  R.C. Penwell, R.S. Porter, and S. Middleman, J. Polym. Sci. A2, 9, 4, 731­745 (1971).<br>37.  P.H. Goldblatt and R.S. Porter, J. Appl. Polym. Sci., 20, 1199­1208 (1976)<br>38.  I.J. Duvdevani and I. Klein, SPE J., Dec., 41­45 (1967)<br>39.  S.T. Choi, J. Polym. Sci. A2, 6, 2043­2049 (1968)<br>40.  F.N. Cogswell, Plastics and Polymers, 41, 30­43 (1973)<br>41.  D.W. van Krevelen, "Properties of Polymers, Correlations with Chemical Structure," <br>
+Elsevier, NY (1972)<br>
+42.  A.W. Jenike, "Gravity Flow of Bulk Solids," Bulletin No. 108 of the Utah Engineering <br>
+Experimental Station, Univ. of Utah, Salt Lake City (1961)<br>
+43.  R. Rautenbach and E. Goldacker, Kunststoffe, 61, 104­107 (1971)<br>44.  E. Goldacker and R. Rautenbach, Chemie Ing. Techn. 44, 405­410 (1972)<br>45.  J. Mewis, J. Non-Newtonian Fluid Mech., 6, 1­20 (1979)<br>46.  J.M. Dealy and W.K.W. Tsang, J. Appl. Polym. Sci., 26, 1149­1158 (1981)<br>47.  T.Y. Liu, D.S. Soong, and M.C. Williams, Polym. Eng. Sci., 21, 675­687 (1981)<br>48.  J.L. White and W. Minoshima, Polym. Eng. Sci., 21, 1113­1121 (1981)<br>49.  C.M. Vrentas and W.W. Graessley, J. Non-Newtonian Fluid Mech., 9, 339­355 (1981).<br>50.  G. De Cleyn and J. Mewis, J. Non-Newtonian Fluid Mech., 9, 91­105 (1981)<br>51.  J. Mewis and G. De Cleyn, AIChE J., 28, 900­907 (1982)<br>52.  J. Mewis and M.M. Denn, J. Non-Newtonian Fluid Mech., 12, 69­83 (1983)<br>53.  D. Acierno, F.P. LaMantia, G. Marrucci, and G. Titomanlio, J. Non-Newtonian Fluid Mech., <br>
+1, 125­146 (1976)<br>
+<hr>
+<A name=265></a><b>252</b>  <b>6Important Polymer Properties</b><br>
+54.  C.D. Han, "Rheology in Polymer Processing," Academic Press, NY (1976)<br>55.  J.D. Ferry, "Viscoelastic Properties of Polymers," 3rd edition, Wiley, NY (1980)<br>56.  J.J. Aklonis, W.J. MacKnight, and M. Shen, "Introduction to Polymer Viscoelasticity," <br>
+Wiley-Interscience, NY (1972)<br>
+57.  K. Walters, "Rheometry," Wiley, NY (1975)<br>58.  R.S. Lenk, "Polymer Rheology," Applied Science Publ. LTD, London (1978)<br>59.  J.A. Brydson, "Flow Properties of Polymer Melts," 2nd edition, George Godwin Limited, <br>
+London (1981)<br>
+60.  F.N. Cogswell, "Polymer Melt Rheology," George Godwin Limited, London (1981)<br>61.  "Praktische Rheologie der Kunststoffe," VDI-Verlag GmbH, Duesseldorf (1978)<br>62.  O. Plajer, "Praktische Rheologie fuer Kunststoffschmelzen," Zechner &amp; Huethig Verlag <br>
+GmbH, Speyer, Germany (1970)<br>
+63.  L.E. Nielsen, "Polymer Rheology," Marcel Dekker, NY (1977)<br>64.  "Rheometry: Industrial Applications," K. Walters (Ed.), Research Studies Press, Chiches-<br>
+ter, England (1980)<br>
+65.  J.R. Van Wazer, J.W. Lyons, K.Y. Kim, and R.E. Colwell, "Viscosity and Flow Measure-<br>
+ment, A Laboratory Handbook of Rheology," Interscience Publishers, NY (1963)<br>
+66.  J.M. Dealy, "Rheometers for Molten Plastics," Van Nostrand Reinhold Co., NY (1982)<br>67.  D.V. Boger and M.M. Denn, J. Non-Newtonian Fluid Mech., 6, 163­185 (1980)<br>68.  R.D. Pike, D.E. Baird, and M.D. Read, SPE 41st ANTEC, Chicago, 312­315 (1983)<br>69.  C.D. Han, J. Appl. Polym. Sci., 15, 2567­2577 (1971)<br>70.  J.L.S. Wales, J.L. den Otter, and H. Janeschitz-Kriegl, Rheol. Acta, 4, 146­152 (1965)<br>71.  C. Rauwendaal and F. Fernandez, SPE 42nd ANTEC, 282­287, New Orleans (1984)<br>72.  W.P. Cox and E.H. Merz, J. Polym. Sci., 28, 619­622 (1958)<br>73.  K. Eiermann and K.H. Hellwege, J. Polym. Sci., 57, 99 (1962)<br>74.  D. Hansen and G.A. Bernier, Polym. Eng. Sci., 12, 204 (1972)<br>75.  S. Yagi and D. Kunii, AIChE J., 3, 373 (1957)<br>76.  B. Maxwell et al., SPE Trans., 4, 165 (1964)<br>77.  R.S. Spencer and G.D. Gilmore, J. Appl. Phys., 20, 502 (1949), 21, 523 (1950)<br>78.  H. Breuer and G. Rehage, Kolloid Z.Z. Polym., 216, 166 (1967)<br>79.  M.R. Kamal and N.T. Levan, Polym. Eng. Sci., 13, 131 (1973)<br>80.  O. Olabisi and R. Simha, Macromolecules, 8, 206 (1975); 211 (1975)<br>81.  W. Thompson, Phil. Mag., 42, 448 (1971)<br>82.  H.B. Dwight, "Tables of Integrals and Other Mathematical Data," 3rd edition, p. 275 <br>
+MacMillan, NY (1957)<br>
+83.  H.G. Kilian and M. Pietralla, Polymer, 19, 664­672 (1978)<br>84.  P. Andersson and B. Sundqvist, J. Polym. Sci., 13, 243­251 (1975)<br>85.  W.W. Wendlandt, "Thermal Methods of Analysis," Wiley, NY (1979)<br>
+<hr>
+<A name=266></a> References <br>
+<b>253</b><br>
+86.  T. Daniels, "Thermal Analysis," Anchor Press, London (1973)<br>87. E.A. Turi (Ed.), "Thermal Characterization of Polymeric Materials," Academic Press, <br>
+NY (1981)<br>
+88. G.W. Miller and R.S. Porter, "Analytical Calorimetry," Vol. 2, p. 407 Plenum Press, <br>
+NY (1970),<br>
+89.  L.E. Nielson, "Mechanical Properties of Polymers and Composites," Vol. 1 and 2, Marcel <br>
+Dekker, NY (1974)<br>
+90.  "Kenndaten fuer die Verarbeitung Thermoplastischer Kunststoffe, Teil I, Thermodyna-<br>
+mik," Carl Hanser, Munich (1979)<br>
+91. "Kenndaten fuer die Verarbeitung Thermoplastischer Kunststoffe, Teil II, Rheologie," <br>
+Carl Hanser, Munich (1982)<br>
+92.  J.M. Dealy, Plast. Eng., March, 57­61 (1983)<br>93.  M.R. Kamal, V. Tan, and F. Kashani, Adv. Polym. Tech., 3, 89­98 (1983)<br>94.  C.J.S. Petrie, "Elongational Flows," Pitman (1979)<br>95.  "Kenndaten fuer die Verarbeitung Thermoplastischer Kunststoffe, Teil III, Tribologie," <br>
+Carl Hanser, Munich (1983)<br>
+96.  G.R. Langecker and R. Rautenbach, Powder Techn., 15, 39­42 (1976)<br>97.  G.R. Langecker, Dissertation, IKV, Aachen, Germany (1977)<br>98.  H. Schott, J. Appl. Polym. Sci., 6, 529 (1962)<br>99. R. Steller and J. Iwko, "Generalized Flow of Ellis Fluid in the Screw Channel," Inter-<br>
+national Polymer Processing, Mar. (2001) pp. 241­248<br>
+100.  R.I. Tanner, "Engineering Rheology," Oxford University Press, New York (1985)<br>101.  G.V. Gordon and M.T. Shaw, "Computer Programs for Rheologists," Carl Hanser Verlag, <br>
+Munich (1994)<br>
+102.  H. Breuer, et al., "CAMPUS Set for its Global Breakthrough," Kunststoffe, 84, 8, 1003­<br>
+1012 (1994)<br>
+<hr>
+<A name=267></a><b>7 Functional Process </b><br>
+<b>Analysis</b><br>
+Chapters 5 and 6 on fundamental principles and important polymer properties are <br>meant to be a preparation for the material covered in this chapter. In this chapter, <br>the central part of this book, the six main functions of an extruder will be described <br>and analyzed. The main functions are solids conveying, plasticating or melting, melt <br>conveying or pumping, devolatilization, mixing, and die forming. The material in <br>this chapter is key to developing a thorough understanding of the extrusion process <br>and indispensable in taking an engineering approach to screw design, die design, <br>and troubleshooting.<br>Each function will be discussed separately. The mechanism behind each function <br>will be described in detail. The emphases will be on developing a thorough under-<br>standing and quantitative description of the entire extrusion process from a func-<br>tional point of view. In later chapters, this knowledge will be applied to practical <br>aspects, such as screw design, die design, troubleshooting, etc. It should be realized <br>that in dividing the extruder into functional zones, these zones are not discrete but, <br>to some extent, overlapping. Their boundaries can shift when polymer properties or <br>operating conditions change. For instance, in the melt conveying zone, there will be <br>a certain amount of mixing taking place as well; thus, the mixing zone will overlap <br>the melt conveying zone. On the other hand, the geometrical sections of the screw <br>are fixed; they are discrete and non-overlapping. The geometrical sections of a stand-<br>ard screw are the feed, compression, and metering sections.<br>
+<b></b><br>
+ <b>7.1Basic Screw Geometry</b><br>
+The analysis of the functional zones of the extruder requires knowledge of the basic <br>relationships describing the geo metry of an extruder screw. The geo metry of the <br>flight along the screw surface can be constructed by unrolling the flight onto a flat <br>plane. On a flat surface, the flight along the screw surface will form a right triangle <br>as shown on the right-hand triangle in Fig. 7.1. The base of the triangle is one-half <br>of the flight pitch if the height of the triangle is half the circumference at the screw <br>surface. The flight lead or pitch S is the axial distance between two points on the <br>
+<hr>
+<A name=268></a><b>256</b>  <b>7Functional Process Analysis</b><br>
+flight separated by a full turn of the flight. The flight pitch is the same as the flight <br>lead for a single-flighted screw. However, for a multi-flighted screw the pitch is the <br>lead divided by the number of flights. The top angle of the triangle is the helix angle <br>of the screw surface s. If points along the screw surface are connected to corre-<br>
+sponding points along the altitude of the triangle, one constructs the helical geo-<br>metry of the flight along the surface of the root of the screw.<br>The same procedure can be used to construct the helical geo metry of the flight along <br>the O.D. of the screw as shown in the left-hand side of Fig. 7.1.<br>
+<b>Figure 7.1</b>Construction of a basic screw geo metry<br>
+It should be noted that the helix angle at the O.D. of the screw b is different from <br>
+the helix angle at the root of the screw s. The screw geo metry is usually repre-<br>
+sented by straight flights, as shown in Fig. 7.2.<br>
+<b> Figure 7.2</b> <br>
+Screw with flights drawn <br>
+straight<br>
+<hr>
+<A name=269></a> <br>
+7.1Basic Screw Geometry  <b>257</b><br>
+However, in reality the flights are S-shaped, as seen in Fig. 7.1. If a cross-section is <br>made perpendicular to the flights, it can be seen that the screw channel is not a true <br>rectangle. The bottom and top surface of the screw channel are curved and the flight <br>flanks diverge. Thus, the channel width is larger at the screw O.D. than at the root <br>of the screw.<br>Important geometrical relationships will be given next. The pitch of the screw equals <br>the sum of the axial channel width and the axial flight width.<br>
+ (7.1)<br>
+The lead L is the pitch times the number (p) of parallel flights.<br>
+ (7.2)<br>
+The helix angle at the barrel surface b is determined by the ratio of the pitch and <br>
+the circumference at the barrel surface.<br>
+ (7.3)<br>
+Similarly, the helix angle m at the middle of the channel is:<br>
+ (7.4)<br>
+The helix angle s at the root of the screw is:<br>
+ (7.5)<br>
+The perpendicular channel width at the barrel surface is:<br>
+ (7.6)<br>
+The perpendicular channel width at the middle of the channel is:<br>
+ (7.7)<br>
+The perpendicular channel width at the root of the screw is:<br>
+ (7.8)<br>
+The channel width at the barrel surface is related to the pitch and the helix angle at <br>the barrel surface as follows:<br>
+ (7.9)<br>
+<hr>
+<A name=270></a><b>258</b>  <b>7Functional Process Analysis</b><br>
+The channel width in the middle of the channel can be expressed as:<br>
+ (7.10a)<br>
+Subscript m refers to the dimensions at the midpoint of the channel, and the dia-<br>meter at the midpoint of the channel equals the barrel diameter minus the flight <br>height (Dm = Db­H).<br>Equation 7.10(a) is valid only for single-flighted screws (p = 1).<br>If the screw has p parallel flights, then:<br>
+ (7.10b)<br>
+The channel dimensions used without subscripts are related to the O.D. of the screw.<br>
+<b></b><br>
+ <b>7.2Solids Conveying</b><br>
+The solids conveying zone extends from the feed hopper down several diameters <br>into the extruder barrel. The first traces of molten polymer generally do not appear <br>until about three or four diameters into the barrel, measured from the feed port. <br>Since the conveying process in the feed hopper is considerably different from the <br>conveying process in the screw channel, the solids conveying zone will be divided <br>into the gravity induced solids conveying zone (feed hopper) and the drag induced <br>solids conveying zone (extruder screw).<br>Some feed hoppers are equipped with crammer feeders; see Fig. 7.3.<br>
+<b> Figure 7.3</b> <br>
+Crammer feeder<br>
+<hr>
+<A name=271></a> <br>
+7.2Solids Conveying  <b>259</b><br>
+This is a feed hopper with a rotating screw in the discharge region. The screw is <br>incorporated to augment the solids conveying rate by the action of the rotating <br>screw. In such a feed hopper, gravity induced solids conveying and drag induced <br>solids conveying occur simultaneously at the same location. In this case, the two <br>types of solids conveying cannot be separated but must be analyzed together.<br>
+<b>7.2.1Gravity Induced Solids Conveying</b><br>
+Most feed hoppers have a cylindrical top section and a truncated conical section at <br>the bottom; see Fig. 7.4.<br>
+<b> Figure 7.4</b> <br>
+Typical feed hopper<br>
+The driving force for solids conveying is gravity. Based on the bulk properties and <br>the hopper geo metry, one would like to be able to calculate the stress distribution in <br>the hopper, the velocity profiles in the hopper, and finally the discharge rate. Unfor-<br>tunately, the analysis of flow of granular materials in hoppers is rather complicated <br>as a result of the complex flow behavior of granular material, as discussed in Section <br>6.1. In most cases, the results of the analysis are approximate and apply only to a <br>limited number of cases. Even the relatively simple problem of a non-cohesive bulk <br>material flowing through a hopper has not been completely solved. Not surprisingly, <br>the situation is worse for the more realistic problem of a cohesive particulate mate-<br>rial flowing through a hopper.<br>Theoretical and experimental work on the transport of bulk material started in ear-<br>nest around 1960. A collection of experimental and theoretical work was presented <br>at a joint ASME-CSME Conference on Mechanics Applied to the Transport of Bulk <br>Materials [1] and the U.S.-Japan seminars on Continuum Mechanical and Statistical <br>Approaches in the Mechanics of Granular Materials [2]. Books on particulate solids <br>have been written by Orr [3] and Brown and Richards [4]. Review articles have been <br>published by Richards [5], Wieghardt [6], and Savage [7]. Some of the pioneering <br>work on flow of bulk solids was done by Jenike [8, 9].<br>
+<hr>
+<A name=272></a><b>260</b>  <b>7Functional Process Analysis</b><br>
+In the flow of bulk materials through a hopper, one generally distinguishes between <br>two types of flow. In mass flow or hopper flow, the entire volume of particulate solids <br>moves down toward the exit; there are no stagnating regions. The other type of flow <br>is funnel flow. In funnel flow, the bulk material flows out through a flow channel, the <br>wall of the flow channel being formed by stationary particles of the bulk material. In <br>funnel flow, therefore, there is at least one stagnating region; see Fig. 7.5.<br>
+No flow<br>
+v=0<br>
+v=0<br>
+<b> Figure 7.5</b> <br>
+Various types of flow that can occur <br>
+Mass flow<br>
+Arching<br>
+Funnel flow<br>
+in feed hoppers<br>
+A common flow problem in hopper flow is arching or bridging. The particles form a <br>natural bridge able to support the material above it. As mentioned in Section 6.1, <br>highly compressible materials have a strong tendency towards bridging. With such <br>materials, the hopper is often equipped with a vibrating pad to dislodge any bridges <br>that might form by a continuous mechanical vibration of the hopper. It is possible to <br>derive criteria to avoid arching; these can be used in the design of feed hoppers and <br>will be discussed later. Piping is a disturbance that occurs in funnel flow. An annu-<br>lar ring of stationary bulk material is formed. The material in this stagnating region <br>is able to support the material above it and the exposed surface of the internal, <br>empty channel.<br>Both in arching and in piping, the material is consolidated to the extent that it can <br>support the material above it and form an exposed surface. Thus, both these flow <br>problems are typical of cohesive (non-free-flowing) particulate solids. This is partic-<br>ularly true for materials with a high, unconfined yield strength c; see Section 6.1.2 <br>
+and Fig. 6.4(b).<br>In the analysis of flow of granular material, two types of flow can be distinguished. <br>The first is slow frictional flow where the particles remain in continuous contact <br>with each other; the internal forces result from Coulomb friction between contacting <br>particles. The second type of flow is much more rapid; the particles are not in con-<br>stant contact with their neighbors. The energy associated with the velocity fluctua-<br>tions is comparable to that of the mean motion. In this type of flow, the internal <br>forces arise because of the transfer of momentum during collisions between par-<br>ticles. The constitutive relations for this rapid flow are rate-dependent. This type of <br>flow, therefore, is referred to as viscous flow (sometimes just rapid flow). Steady, <br>
+<hr>
+<A name=273></a> <br>
+7.2Solids Conveying  <b>261</b><br>
+viscous flows are generally described by elliptic partial differential equations <br>[10­13].<br>In gravity flow through feed hoppers, it can be assumed that the flow is sufficiently <br>slow that the particles are in constant contact and that momentum transfer by colli-<br>sions between particles is negligible. The flow through a feed hopper can thus be <br>considered to be frictional flow. Various workers [14­21] have made experimental <br>studies of the flow patterns in feed hoppers. In some studies, dyed particles were <br>used, e.g., [17]; in others, X-ray techniques were employed to determine the flow <br>patterns in the hopper [15, 16]. In other studies [18, 19, 21], a stereoscopic tech-<br>nique was used. This involves taking photographs of the flow field at short time <br>intervals. Sequential photographs are analyzed using a stereocomparator; this <br>results in a three-dimensional model of the displacement field from which the velo-<br>city field can be determined. This technique is limited to two-dimensional flows, but <br>does not require tracers and enables determination of the entire flow field.<br>The flow of bulk material in a feed hopper is generally quite different in the various <br>sections of the hopper; see Fig. 7.6.<br>
+Plug flow<br>
+Rupture zone<b> Figure  7.6</b> <br>
+Freefall zone<br>
+Various flow regimes in a feed hopper<br>
+The flow in the cylindrical hopper section tends to be plug flow. The plug flow zone <br>is bound by a rupture zone at the bottom of the plug flow zone. The rupture zone is <br>situated approximately at the cylinder-cone transition. This zone is characterized by <br>intense relative deformation of the granular material. Below the rupture zone, fur-<br>ther down in the conical section, there may be local regions of plug flow. These local <br>plug flow regions are generally bound by the hopper wall and the rupture zone. <br>Finally, in the bottom part of the conical section the particles flow out freely; this is <br>referred to as the free-flow zone.<br>Unfortunately, at this point in time, the understanding of the mechanics of flow of <br>bulk solids is not well enough developed that this flow behavior can be predicted <br>theoretically. The main area of uncertainty is the proper constitutive equation for <br>bulk materials relating stress and strain rates. Because the theory is not well devel-<br>oped, only a few aspects of gravity flow in hoppers will be discussed further. These <br>
+<hr>
+<A name=274></a><b>262</b>  <b>7Functional Process Analysis</b><br>
+are pressure distribution in feed hoppers, criteria to avoid flow disturbances, and <br>flow rate predictions. It should be remembered that the following relationships have <br>limited applicability and accuracy in terms of their predictive ability.<br>
+<b>7.2.1.1Pressure Distribution<br></b>In the cylindrical portion of a hopper the pressure distribution can be derived if the <br>following assumptions are made:<br>1.  The vertical compressive stress is constant over any horizontal plane.<br>2.  The ratio of horizontal and vertical stresses is constant and independent of <br>
+depth.<br>
+3.  The bulk density is constant.<br>4.  The wall friction is fully mobilized, meaning that the particulate material is in <br>
+incipient slip conditions at the wall.<br>
+A force balance over a differential element (see Fig. 7.7) gives:<br>
+ (7.11)<br>
+<b> Figure 7.7</b> <br>
+Illustration of force balance<br>
+where b is bulk density, g the gravitational acceleration, f*w the coefficient of fric-<br>
+tion at the wall, and k the ratio of compressive stress in the horizontal direction to <br>compressive stress in the vertical direction. The shear stress at the wall is deter-<br>mined by Eq. 6.8. An expression for the ratio k can be found if it is assumed that the <br>maximum principal stress is in the vertical direction. The ratio k can be determined <br>from Eq. 6.7.<br>
+ (7.12)<br>
+where e is the effective angle of internal friction.<br>
+<hr>
+<A name=275></a> <br>
+7.2Solids Conveying  <b>263</b><br>
+Equation 7.12 applies if the particulate solids are in a condition of steady flow; it <br>also applies to cohesionless materials in a condition of incipient flow. Integration of <br>Eq. 7.11 yields the pressure distribution:<br>
+ (7.13)<br>
+If the pressure at h = H is taken as zero and if the adhesive wall shear stress is zero, <br>Eq. 7.13 reduces to the well-known Janssen Equation derived in 1895 [22]:<br>
+ (7.14)<br>
+If the value of H is sufficiently large, the pressure becomes independent of vertical <br>distance; this limiting pressure value is:<br>
+ (7.15)<br>
+The maximum pressure is directly proportional to the bulk density and cylinder <br>radius and inversely proportional to the coefficient of friction at the wall and the <br>ratio k.<br>Walker [23] made a more rigorous analysis of the pressure distribution in vertical <br>bins. He assumed a plastic equilibrium in the particulate solids with the Mohr cir-<br>cles representing the stress condition at a certain level touching the effective yield <br>locus. Walker derived the following expression for the pressure profile in a vertical <br>cylinder:<br>
+ (7.16)<br>
+where D* is defined as a distribution factor relating the average vertical stress with <br>the vertical stress near the wall. In principle, this distribution function D* can be <br>evaluated by solving the entire stress field, as discussed by Walters and Nedderman <br>[24]. However, as a first approximation, the distribution factor can be assumed to be <br>unity. The ratio of shear stress to normal stress at the wall, B, is given by:<br>
+ (7.17)<br>
+where * is the angle between the major principal plane and the cylinder wall.<br>The angle * is related to the effective angle of friction e and the wall angle of fric-<br>
+tion w = arctan f*w by:<br>
+<hr>
+<A name=276></a><b>264</b>  <b>7Functional Process Analysis</b><br>
+ (7.18)<br>
+where the arcsin value is to be larger than 90°.<br>Walker also derived equations for the stress distribution in a conical hopper section. <br>These equations have clear practical importance because most feed hoppers are <br>designed with conical sections. The pressure distribution for mass flow conditions is <br>given by:<br>
+ (7.19)<br>
+and<br>
+ (7.20)<br>
+where ho is the height where the vertical pressure is Po. This height can be taken as <br>
+the height of the conical hopper section, as shown in Fig. 7.8.<br>The coefficient c for conical hoppers is:<br>
+ (7.21)<br>
+where  is the hopper half-apex angle; see Fig. 7.8.<br>
+<b> Figure 7.8</b> <br>
+Conical hopper configuration<br>
+The coefficient c for wedge-shaped hoppers is:<br>
+ (7.22)<br>
+<hr>
+<A name=277></a> <br>
+7.2Solids Conveying  <b>265</b><br>
+The stress ratio B is given by:<br>
+ (7.23)<br>
+In the convergent hopper section, angle * is given by:<br>
+ (7.24)<br>
+where the arcsin value is to be less than 90°!<br>If the initial pressure in the conical section is zero, the maximum pressures will <br>occur somewhere along the conical section. In most cases, however, a cylindrical <br>hopper section is placed on top of the conical hopper section. In these situations, the <br>initial pressure distribution in the conical section will be determined by the final <br>pressure distribution in the cylindrical section. If the stress distributions do not <br>match, rupture zones may form in the transition region as observed by Lee et al. <br>[15]. Instabilities of stress conditions at the transition region have been discussed <br>by Bransby and Blair-Fish [25].<br>
+<b>7.2.1.2Flow Rate<br></b>The flow rate of granular materials is independent of the head if the head is suffi-<br>ciently large. This experimental result was known as early as 1852, when Hagen <br>presented an equation for the flow rate through a circular opening [26]:<br>
+ (7.25)<br>
+where Ja is a parameter called the non-dimensional axisymmetric flow rate, Da is the <br>
+diameter of the aperture, dp is the particle diameter, and fc is a correction factor of <br>
+the order one. This result can be obtained from a dimensional analysis; see Section <br>5.3.3. For a two-dimensional slot of length L and width W, the flow rate is:<br>
+ (7.26)<br>
+where J is the non-dimensional flow rate for slot outlets.<br>The flow rate seems to be determined, to a large extent, by what happens in the <br>vicinity of the discharge opening. This concept forms the basis of most of the early <br>analyses of flow rate. Brown [27] derived the following expression for the non-<br>dimensional flow rate for a slot outlet in case of mass flow:<br>
+ (7.27)<br>
+<hr>
+<A name=278></a><b>266</b>  <b>7Functional Process Analysis</b><br>
+Johanson [28] developed the following expression for steady flow of non-cohesive <br>bulk materials:<br>
+ (7.28)<br>
+This result is obtained by assuming the flow in the hopper to be one-dimensional <br>and further assuming that at the orifice the convective acceleration in the upper <br>converging flow is equal to the gravitational acceleration g, appropriate for the <br>freely falling particles below the orifice level.<br>Savage [29] derived flow rate equations for a frictional Coulomb material by assum-<br>ing radial body forces and neglecting wall friction:<br>
+ (7.29)<br>
+where k is given by Eq. 7.12.<br>Equation 7.29 generally overestimates the flow rate by about 40 to 100%, as deter-<br>mined by comparison to experimental results. Savage [30] extended his analysis to <br>include the effect of wall friction by solving the equations of motion by the method <br>of integral relations. The flow rates were lower than predicted with Eq. 7.29, but still <br>higher than experimental values. Savage and Sayed [31] improved the earlier analy-<br>sis [30] and derived an expression for the flow rate in a two-dimensional wedge-<br>shaped hopper:<br>
+ (7.30)<br>
+where the constants are given in Appendix 7.1.<br>The closed-form solution of the flow rate was derived by using a mean normal stress <br>averaged for the width of the hopper. A more accurate analysis considered the <br>detailed variations of the stresses throughout the hopper. Predictions based on Eq. <br>7.30 agreed well with the numerical results from the more accurate analysis; the <br>differences were generally less than one percent. An interesting theoretical pre-<br>diction is the increase in flow rate when the wall friction is increased at large hop-<br>per half-angles. This result is not intuitively obvious but has been experimentally <br>observed [32­34].<br>
+<b>7.2.1.3Design Criteria<br></b>Jenike [8, 9] and his coworkers did extensive work on gravity flow of bulk solids, <br>both experimental and theoretical. He developed design methods and criteria for <br>hoppers and bins with steady mass flow without disturbances. In determining vari-<br>ous flow criteria, Jenike used a function termed "flow factor". This flow factor is the <br>ratio of the consolidating pressure 1 to the stress acting on an exposed surface  1:<br>
+<hr>
+<A name=279></a> <br>
+7.2Solids Conveying  <b>267</b><br>
+ (7.31)<br>
+The stress acting on an exposed surface is also the only non-zero principal stress, <br>because the exposed surface is assumed self-supporting and traction-free (i.e., no <br>shear stresses acting on the surface). The flow factor is determined by the geo metry <br>of the hopper and the properties of the bulk material. Another function used by <br>Jenike is the "flow function". This flow function is the ratio of the consolidating pres-<br>sure 1 to the unconfined yield strength c, as defined in Section 6.1.2:<br>
+ (7.32)<br>
+The flow function is a material property; it gives an indication of the flowability of a <br>bulk material:<br> <br>
+FF &gt;  10  free flowing material<br>
+10 &gt;  FF &gt;  4 <br>
+easy flowing material<br>
+4 &gt;  FF &gt;  1.6  cohesive material<br> <br>
+FF &lt;  1.6  very cohesive, non-flowing material<br>
+As a general rule, solids that do not contain particles smaller than approximately <br>0.2 mm are free flowing; thus, most granular solids are free flowing and most pow-<br>ders are to some extent cohesive.<br>The exposed surface, whether it is an arch or a pipe, is stable when the unconfined <br>yield strength c is higher than the stress acting on the exposed surface  1 and <br>
+unstable when c is less than 1. The condition for no arching or piping, therefore, is:<br>
+ (7.33)<br>
+In order to obtain quantitative results, the flow factor has to be determined; this <br>requires knowledge of the stress field in the hopper. Closed-form expressions for ff <br>are not available, except for the simplest case of flow through a straight cylinder. <br>Results from numerical analysis are given by Jenike [8, 9] in graphical form for <br>plane symmetry and axial symmetry.<br>If steady flow is desired, the hopper geo metry should be designed such that mass <br>flow will take place; no stagnating regions should occur. In this case, the solids flow <br>along the walls of the hopper. The wall, therefore, must be sufficiently steep and the <br>flow channel must not have any sharp corners, abrupt transitions, or discontinuities <br>in frictional properties at the wall. As a rule, the hopper half-angle a should not <br>exceed max, with max determined from:<br>
+ (7.34)<br>
+where e is the effective angle of internal friction.<br>
+<hr>
+<A name=280></a><b>268</b>  <b>7Functional Process Analysis</b><br>
+The minimum outlet dimension to avoid doming was formulated by Jenike [8] as:<br>
+ (7.35)<br>
+where c = 2 for circular outlets and c = 1.8 for square outlets. The critical yield stress <br>c is determined from the point of intersection of the appropriate flow function and <br>
+flow factor. Walker [35] and Eckhoff [36] published experiments showing that the <br>Jenike method considerably overdesigns the critical outlet dimensions, as much as a <br>hundred percent or more. Engstad [37] has formulated a relationship for the critical <br>outlet dimension that is claimed to be more accurate and results in less overdesign. <br>The critical outlet dimension, according to Engstad's analysis, is given by:<br>
+ (7.36)<br>
+where the various terms of Eq. 7.36 are given in Appendix 7.2.<br>From an analysis of the flow field in a hopper, criteria can be formulated for prevent-<br>ing arching and funneling. These criteria place restrictions either on the dimen-<br>sions of the outlet or the slope of the walls. Another approach is to modify the hop-<br>per geo metry into a nonlinear, curved shape in order to obtain optimum flow <br>conditions. Lee [38] designed a hyperbolic hopper by making certain assumptions <br>about the rate of change of the horizontal cross-section with respect to axial dis-<br>tance. Richmond [39] suggested that in an optimum hopper, the material would be <br>on the verge of arching at any point. By using a one-dimensional analysis, this con-<br>dition could be achieved, resulting in an exponential profile. Gardner [40] proposed <br>a solution based on having a single surface at any level in the hopper on the verge of <br>arching. Richmond and Morrison [41] used a modified procedure based on arching <br>being imminent only along the axis of the hopper. For this case, positive pressures <br>exist at all other points in the hopper. This approach leads to smoothly curved fun-<br>nel-shaped hoppers. However, the optimum shape will have to be modified if the <br>converging hopper section is connected to a cylindrical hopper section. A practical <br>drawback of this approach is that such a carefully tailored curved hopper geo metry <br>is difficult to manufacture and could become quite expensive.<br>
+<b>7.2.2Drag Induced Solids Conveying</b><br>
+Once the particulate solids have reached the feed port of the extruder, the material <br>will flow down until it is situated in the screw channel. At this point, the gravity <br>induced flow mechanism will essentially cease. In most extruders, the screw and <br>barrel are placed in a horizontal direction and the role of gravity becomes a very <br>minor one. In fact, in most analyses of solids conveying in single screw extruders, <br>
+<hr>
+<A name=281></a><IMG src="Polymer Extrusion by Chris Rauwendaal-281_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-281_21.jpg"><br>
+ <br>
+7.2Solids Conveying  <b>269</b><br>
+the effect of gravity is assumed to be negligible. The material in the screw channel <br>will move forward as a result of the relative motion between boundaries of the sol-<br>ids, i.e., the screw surface and the barrel surface. The flow rate of the solids is deter-<br>mined by the forces acting on the solids. These forces are largely determined by the <br>frictional forces acting on the solids at the boundaries.<br>It has been found experimentally that, in most cases, polymeric particulate solids <br>compact readily in the early portion of the screw channel. As a result, the solids <br>form into a solid bed and the solids move down the screw channel in plug flow; thus, <br>at any cross-section of the solid bed all elements move at the same velocity. In other <br>words, there is no internal deformation taking place inside the solid bed. This com-<br>paction of the particulate solids into a solid bed can occur only if there is a sufficient <br>amount of pressure generation in the screw channel.<br>If there is not enough pressure generation in the screw channel, the particulate <br> <br>solids will not form a solid bed. In this case, plug flow will not occur; there will be <br>internal deformation in the solid material. As a result, the solids conveying process <br>will be less steady compared to plug flow. The type of non-plug flow that occurs <br>when the channel is only partially filled with polymer particles has been referred to <br>as "Archimedean transport" [42]. Archimedean transport will occur if the intake <br>rate of the extruder is less than the plug flow solids conveying rate. In metered <br>starve fed extrusion, the Archimedean transport is created intentionally. In some <br>cases, Archimedean transport occurs accidentally if the flow rate through the feed <br>hopper is too low or if the feed port geo metry of the extruder is too small. Archi-<br>medean solids transport is also likely to occur if the pressure at the end of the solids <br>conveying zone is low. An example would be a screw type solids conveying device <br>(a screw feed) where the discharge pressure at the end of the solids conveying zone <br>is zero. Archimedean transport is essentially always associated with a partially <br>filled screw channel; see Fig. 7.9.<br>
+<b> Figure 7.9</b> <br>
+Archimedean transport in the first <br>
+few turns of the screw<br>
+Pressure generation cannot take place if the screw channel is not fully filled and <br>consequently compaction and plug flow cannot occur.<br>It has been observed [44] that even under normal extrusion conditions, Archime-<br>dean transport can occur for a short length. Observations of the filling action in the <br>
+<hr>
+<A name=282></a><b>270</b>  <b>7Functional Process Analysis</b><br>
+feed port have shown that the flow of material from the hopper to the screw channel <br>is such that little pressure is likely to be transmitted. During some parts of the fill-<br>ing cycle, empty spaces in the screw channel have been observed. This indicates <br>that virtually no pressure exists for some time periods. Thus, Archimedean trans-<br>port can occur over a short distance in the feed port region.<br>The solids transport resulting from the relative motion of the boundaries of the solid <br>material is referred to as drag induced solids conveying. For optimum solids convey-<br>ing, the particulate solids should compact easily and move in plug flow. It is impor-<br>tant to note that properties of the bulk material that are advantageous in the gravity <br>flow in the feed hopper can be detrimental in the drag induced solids conveying in <br>the screw channel. An example is the flow function, defined by Eq. 7.22. For optimal <br>gravity flow in the feed hopper, one would like to have a bulk material with a large <br>value of the flow function. However, such a material will not form a strong solid bed <br>because the yield strength will be low relative to the compacting pressure. Thus, <br>such a material may well be problematic in the drag induced solids conveying zone <br>because of internal deformation of the solid bed.<br>The first comprehensive analysis of solids transport in single screw extruders was <br>made by Darnell and Mol in 1956 [43]. Later, numerous workers extended the work <br>of Darnell and Mol; however, the basic analysis has remained relatively unchanged. <br>In order to come to a quantitative description of the drag induced solids conveying <br>process, the following assumptions were made by Darnell and Mol:<br>1. The particulate solids behave like a continuum.<br>2. The solid bed is in contact with the entire channel wall, i.e., the barrel surface, <br>
+the root of the screw, the flank of the active flight, and the flank of the passive <br>flight.<br>
+3. The channel depth is constant.<br>4. The flight clearance can be neglected.<br>5. The solid bed moves in plug flow.<br>6. The pressure is a function of down-channel distance only.<br>7. The coefficient of friction is independent of pressure.<br>8. Gravitational forces are neglected.<br>9. Centrifugal forces are neglected.<br>10. Density changes in the plug are neglected.<br>The first five assumptions are made in most analyses of solids conveying. The last <br>five assumptions have been relaxed by various workers. The general approach to <br>solids conveying analysis is to consider an element of the solid bed in the screw <br>channel and determine all forces that are acting on this solid bed element. The most <br>important forces are the frictional forces at the boundaries and the forces resulting <br>from pressure gradients in the solid bed.<br>
+<hr>
+<A name=283></a> <br>
+7.2Solids Conveying  <b>271</b><br>
+Figure 7.10 shows the various forces acting on the solid bed element; the screw is <br>considered stationary and the barrel rotating.<br>Fr is the friction force between the solid bed and the root of the screw:<br>
+ (7.37)<br>
+where fs is the dynamic coefficient of friction on the screw surface.<br>
+<b> Figure 7.10</b> <br>
+Forces acting on a solid plug element <br>
+and corresponding velocity diagram<br>
+Fna is the normal force on the solid bed on the active flight flank:<br>
+ (7.38)<br>
+where F* is an extra normal force, which is unknown.<br>Fnp is the normal force on the solid bed on the passive flight flank:<br>
+ (7.39)<br>
+Ffa is the frictional force between the solid bed and the active flight flank:<br>
+ (7.40)<br>
+Ffp is the frictional force between the solid bed and the passive flight flank:<br>
+ (7.41)<br>
+Fp1 is the force against the face of the solid plug element resulting from the local <br>
+pressure P:<br>
+ (7.42)<br>
+FP2 is the force against the face of the solid plug element resulting from the local <br>
+pressure Pp2+ dP:<br>
+ (7.43)<br>
+<hr>
+<A name=284></a><b>272</b>  <b>7Functional Process Analysis</b><br>
+Obviously, if the pressure gradient in the down-channel direction is zero, then Fp1 = <br>
+Fp2. The final force Fb is the frictional force between the solid bed and the barrel <br>
+surface:<br>
+ (7.44)<br>
+where fb is the dynamic coefficient of friction on the barrel surface.<br>The force Fb makes an angle  with the plane perpendicular to the screw axis; see <br>
+Fig. 7.10. The direction of force Fb is determined by the direction of the vectorial <br>
+velocity difference between the barrel and the solid bed:<br>
+ <br>
+(7.45)<br>
+where  b is the barrel velocity and  sz the solid bed velocity vector. From the velo city <br>
+diagram in Fig. 7.10, the direction of   and Fb becomes clear. The angle  is the <br>
+solids conveying angle. If the solids conveying angle can be determined, then the <br>solid bed velocity can be calculated directly from it:<br>
+ (7.46a)<br>
+Equation 7.46(a) can be rewritten as follows:<br>
+ (7.46b)<br>
+The vectorial velocity difference v between the barrel velocity vb and the solid bed <br>
+velocity vsz is:<br>
+ (7.47)<br>
+v as a function of vb, , and  is given in Eq. 7.61.<br>Once the solid bed velocity is known, the solids conveying rate is simply determined <br>from:<br>
+ (7.48)<br>
+where  is the solid bed density and p the number of parallel flights.<br>At this point, there are two unknowns: the extra force F* and the solids conveying <br>angle . The solution procedure followed by Darnell and Mol was to break up all <br>forces into their axial and tangential components. The sum of all forces in the axial <br>direction is taken to be zero, assuming that acceleration is negligible. The tangential <br>force components are used in a torque balance with the net torque also assumed to <br>be zero. The extra force F* is then eliminated from the two balance equations and an <br>expression for the solids conveying angle results. Darnell and Mol also included the <br>
+<hr>
+<A name=285></a> <br>
+7.2Solids Conveying  <b>273</b><br>
+dependence of the helix angle and flight width on radial distance. The same proce-<br>dure has been followed by many other workers, e.g., Tadmor and Broyer [45, 46].<br>Considering that in most extruder screws the screw diameter is much larger than <br>the channel depth (D/H &gt;&gt; 1, usually about 5), the change in channel width and <br>helix angle over the depth of the channel will be rather small. If it is assumed that <br>the channel curvature can be neglected, the screw channel can be unrolled onto a <br>flat plane. The error that is made in this process may be acceptable considering the <br>limited accuracy and reproducibility of most data on the coefficient of friction, as <br>discussed in Section 6.1.2. Two simplifications result from this assumption. The <br>first one is that now the channel width and helix angle are constant over the depth <br>of the channel. The second simplification is that the extra force F* can be deter-<br>mined directly from a force balance in the cross-channel direction:<br>
+ (7.49)<br>
+An expression for the solids conveying angle  is obtained from a force balance in <br>the down-channel direction:<br>
+ (7.50)<br>
+Equation 7.50 can be integrated to give the pressure as a function of down-channel <br>distance. If the pressure at z = 0 is taken as P(z = 0) = Po, the solution is:<br>
+ (7.51)<br>
+Equation 7.51 indicates that at a certain solids conveying angle the pressure will <br>increase exponentially with down-channel distance. This means that very high <br> <br>pressures can be generated in the solids conveying zone, at least theoretically. Equa-<br>tion 7.51 can be worked out further to yield a closed-form expression for the solids <br>conveying angle:<br>
+ (7.52)<br>
+where:<br>
+ (7.52a)<br>
+If curvature is taken into account the expression for the solids conveying angle can <br>be written as:<br>
+ (7.53)<br>
+<hr>
+<A name=286></a><b>274</b>  <b>7Functional Process Analysis</b><br>
+where:<br>
+ (7.53a)<br>
+and:<br>
+ (7.53b)<br>
+ (7.53c)<br>
+ (7.53d)<br>
+ (7.53e)<br>
+Equation 7.52 is considerably more compact than Eq. 7.53. Figure 7.11 compares <br>the two solutions for a 75-mm (3-inch) extruder with a square pitch at various  values <br>of the channel depth.<br>
+0.4<br>
+Darnell and Mol model<br>
+Darnell and Mol model<br>Eq. 7.53<br>
+Eq. 7.53<br>
+0.2<br>
+0<br>
+angle [radians]<br>
+ing<br>ey<br>
+Flat plate model<br>
+Flat plate model<br>
+-0.2<br>
+Eq. 7.52<br>
+Eq. 7.52<br>
+Solids conv<br>
+-0.4<br>
+0<br>
+2.5<br>
+5.0<br>
+7.5<br>
+10.0<br>
+12.5<br>
+15.0<br>
+Channel depth [mm]<br>
+<b>Figure 7.11</b>Solids conveying angle vs . channel depth calculated with Eqs . 7 .52 and 7 .53<br>
+The results from the two equations are very close for small values (H &lt; 0.01 D) of the <br>channel depth. However, for larger values of the channel depth (H &gt; 0.05 D) the results <br>are quite different. Considering that in most extruder screws H &gt; 0.05 D, Eq. 7.52 will <br>not yield accurate results for typical values of the channel depth in the feed section.<br>
+<hr>
+<A name=287></a> <br>
+7.2Solids Conveying  <b>275</b><br>
+Figure 7.11 shows that the solids conveying angle reduces with the channel depth of <br>the screw. An increase in channel depth increases the surface area of the screw <br>while the barrel surface area stays the same. This means that the retarding force <br>increases while the driving force for conveying does not change. As a result, the <br>solids conveying angle reduces with increasing channel depth.<br>From Eqs. 7.52 and 7.53 the solids conveying angle, and thus the solids conveying <br>rate, can be calculated if the pressure gradient is known. This process can also be <br>reversed. If the actual solids conveying rate is known, the pressure gradient can be <br>calculated using Eq. 7.51. However, the solids conveying angle in Eq. 7.51 must be <br>expressed as a function of the solid bed velocity. The pressure profile derived from <br>Eq. 7.52 (flat plate model) can be written as:<br>
+ (7.54a)<br>
+where:<br>
+The pressure profile derived from Eq. 7.53 (Darnell and Mol) can be written as:<br>
+ (7.54b)<br>
+From Eq. 7.54 it can be seen that the exponential term will increase with fb and <br>
+decrease with fs. Thus, the pressure rise will be most rapid when fb is large and fs is <br>
+small. The exponential term is inversely proportional to the channel depth H; thus, <br>the pressure will rise more slowly when the channel depth is increased.<br>The transport of the solids down the screw channel can be compared to a nut located <br>on a long threaded rotating shaft. If the nut can freely rotate with the shaft, it will <br>not move in the axial direction. However, if the nut is kept from rotating with the <br>shaft, it will move in the axial direction. In the extruder, the frictional force on the <br>barrel wall will keep the solid bed from freely rotating with the screw. The frictional <br>force on the barrel, therefore, constitutes the driving force of the solid bed. The fric-<br>tional force on the screw surface constitutes a retarding force on the solid bed. If the <br>frictional force on the barrel is zero, no forward transport will occur. If the frictional <br>force on the screw is zero, maximum forward transport will occur. From the velocity <br>diagram of this extreme case, Fig. 7.10, it can be seen that the maximum solids con-<br>veying angle in this case is:<br>
+ (7.55)<br>
+<hr>
+<A name=288></a><b>276</b>  <b>7Functional Process Analysis</b><br>
+The maximum solid bed velocity in this case is:<br>
+ (7.55a)<br>
+And thus the maximum solids conveying rate is:<br>
+ (7.55b)<br>
+The approximately equal sign is used because the flight width is neglected in the <br>right-hand expression. For optimum solids conveying, the frictional force on the bar-<br>rel should be maximum and the frictional force on the screw should be minimum. <br>This is clear from simple qualitative arguments without any elaborate analysis or <br>equations. One would like, therefore, to have a low coefficient of friction on the <br>screw and a high coefficient of friction on the barrel. In many instances, the screw is <br>chrome-plated or nickel-plated and highly polished to minimize the friction on the <br>surface. Special platings are available where the surface is impregnated with a fluo-<br>ropolymer to give a low coefficient of friction; see also Section 11.2.1.4. On the other <br>hand, the surface of the barrel should be rough to increase the frictional force on the <br>barrel. Many extruders have grooves machined into the internal barrel surface in <br>the solids conveying zone to improve the solids conveying performance. This will be <br>discussed in more detail in Section 7.2.2.2.<br>With the equations developed so far, the solids conveying performance can be ana-<br>lyzed as a function of screw geo metry and polymer properties. The effect of channel <br>depth on solids conveying rate is shown in Fig. 7.12; the results are from the flat <br>plate model.<br>
+Channel depth [mm]<br>
+2.5<br>
+5.0<br>
+7.5<br>
+10.0<br>
+12.5<br>
+1500<br>
+675<br>
+]<br>
+1000<br>
+450<br>
+rate [kg/hr<br>
+ing rate [lbs/hr]<br>
+ing<br>
+ey<br>
+ey<br>
+500<br>
+225<br>
+Solids conv<br>
+Solids conv<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Channel depth [inch]<br>
+<b>Figure 7.12</b>Solids conveying rate versus channel depth for 75-mm extruder<br>
+<hr>
+<A name=289></a> <br>
+7.2Solids Conveying  <b>277</b><br>
+The curve shown is for a 75-mm extruder running at 100 rpm, assuming fs = 0.2, <br>
+fb = 0.3, and the pressure ratio is 25:1. At low values of the channel depth, the solids <br>
+conveying rate increases with channel depth. However, if the channel depth is fur-<br>ther increased, the solids conveying rate reaches a maximum and then reduces with <br>further increase in the channel depth. The result can be explained by considering <br>the forces acting on the solid bed. If the channel depth is increased, the frictional <br>force on the screw will increase, while the frictional force on the barrel will remain <br>unchanged. Thus, the retarding force increases while the driving force stays the <br>same. This has to result in a reduction of the solids conveying angle. The reduction <br>of the solids conveying angle causes a reduction in the solids conveying rate. How-<br>ever, the increase in channel depth increases the cross-sectional area of the screw <br>channel and this causes an increase in the solids conveying rate. This explains why <br>the solids conveying rate increases at first and then decreases. The solids conveying <br>angle as a function of channel depth is shown in Fig. 7.11. It is clear that the solids <br>conveying angle drops monotonically with channel depth.<br>It should be noted, that the effect of the channel depth on the solids conveying <br>rate as determined from a flat plate analysis is different when the curvature of the <br>channel is taken into account. In reality, when the channel depth increases, the area <br>of the flight flanks will increase but the area of the root of the screw will decrease; <br>this decrease is not taken into account in the flat plate model. The contact area Ab <br>
+between the differential element of the solid bed and the barrel is:<br>
+ (7.56)<br>
+In the flat plate model, the contact area between the solid bed and the screw is:<br>
+ (7.57a)<br>
+while in the curved plate model the contact area is:<br>
+ (7.57b)<br>
+Both the helix angle s and the down-channel incremental distance zs are different <br>
+in the curved plate model. The flat plate model considerably overestimates the con-<br>tact area between the solid bed and the screw. Therefore, when the curvature is <br>taken into account, the solids conveying rate will increase in a monotonic fashion <br>with the channel depth as long as the pressure rise is sufficiently small. This dem-<br>onstrates that the assumptions underlying a model have to be critically evaluated <br>each time the model is used to analyze the influence of a certain parameter.<br>Equations 7.57(a) and (b) assume a zero radius of curvature between the flight <br>flank and the screw root. If the radius Rc is taken into account, the screw contact <br>
+area is reduced by Rc(4­). From this point of view one would like to use a large <br>
+radius; i.e., Rc = H. The resulting flight geo metry is shown in Fig. 7.13 and also in <br>
+<hr>
+<A name=290></a><b>278</b>  <b>7Functional Process Analysis</b><br>
+Fig. 8.4(a) of Section 8.2.3. In this case, the screw contact area in the flat plate <br>model becomes:<br>
+ (7.57c)<br>
+Considering that the ratio of channel depth to channel width is usually about 1/10, <br>using a radius of curvature equal to the channel depth can reduce the contact area <br>between solid bed and screw by about 6 to 7%, while the contact area between solid <br>bed and barrel remains unchanged. Another advantage of the large flight radius is <br>that the flight width at the screw O.D. can be reduced so that the channel cross-<br>sectional area can be at least as large as with a small flight radius. If the flight width <br>at the screw O.D. is maintained at the same value, a flight curvature of Rc = H will <br>
+reduce the channel cross-sectional area by 2H2(/4­1). With a usual ratio of chan-<br>nel width to channel depth, this will result in a reduction in cross-sectional area of <br>about 4%. Thus, the beneficial effect of a large flight radius (reduced screw contact <br>area) is larger than the adverse effect (reduced cross-sectional area of the screw <br>channel). An additional benefit of the large flight radius is the reduced chance of <br>solid bed deformation; see Section 8.2.3.<br>The effect of flight radius on extruder performance was studied by Spalding et al. <br>[238]. They found that the solids conveying rate with large radius was improved at <br>high levels of pressure development and reduced at low levels of pressure. The <br>authors postulate that the improvement in solids conveying rate at high pressures <br>is due to a change in forwarding forces when a large radius is employed. When the <br>pushing radius is small the extra force F* is parallel to the barrel surface as shown <br>in the left-hand side of Fig. 7.13. However, with a large pushing radius the extra <br>force F* points to the barrel at an angle  and this causes the normal force at the <br>barrel to be increased by F* sin as shown in the right-hand side of Fig. 7.13. This <br>will increase the driving force in the conveying process.<br>
+<b>Figure 7.13</b>Schematic of forces with small (left) and large (right) flight flank radius<br>
+Spalding et al. recommend using a flight flank radius in the feed section of about <br>1/4 of the channel depth. However, the basis for this recommendation is not entirely <br>clear because the radii tested were all larger (0.54 H and 1.71 H) than 1/4 of the <br>channel depth (0.25 H). This recommendation may be appropriate for smooth bore <br>extruders where the pressure development in solids conveying is modest. This re -<br>commendation most likely is not appropriate for grooved feed extruders because the <br>pressure development in these extruders is usually substantial.<br>
+<hr>
+<A name=291></a> <br>
+7.2Solids Conveying  <b>279</b><br>
+Another method to reduce the screw contact area is to use flat slanted flight flanks, <br>i.e., a trapezoidal flight geo metry; see also Fig. 8.4(b). If the flight flank angle is 45°, <br>the screw contact area in the flat plate model becomes:<br>
+ (7.57d)<br>
+When the channel width is about ten times the channel depth, the screw contact <br>area will be reduced by about 10%. However, the cross-sectional area of the screw <br>channel will be reduced by the same percentage. Thus, the net effect will be less <br>beneficial than the curved flight geo metry.<br>A multi-flighted screw geo metry can adversely effect the solids conveying per-<br>formance for two reasons. An additional flight will reduce the open cross-sectional <br>area of the screw channel, causing a reduction in solids conveying rate at a constant <br>solid bed velocity. An additional flight will also change the wetted surface area of <br>the screw. When the channel depth is small the wetted screw surface will reduce <br>with additional flights. However, with large values of the channel depth the wetted <br>screw surface will increase with multiple flights. The combined effect can cause a <br>substantial reduction in solids conveying rate. This is shown in Fig. 7.14 for a 75-mm <br>extruder screw under the same conditions as described in Fig. 7.11.<br>
+675<br>
+Single flighted screw<br>
+Single flighted screw<br>
+]<br>
+450<br>
+rate [kg/hr<br>
+ing<br>ey 225<br>
+Double f<br>
+Doub<br>
+lighted screw<br>
+le flighted screw<br>
+<b> Figure 7.14</b> <br>
+Solids conveying rate versus channel <br>
+Solids conv<br>
+depth for 75-mm screw showing both a <br>
+0<br>
+0<br>
+5<br>
+10<br>
+15<br>
+20<br>
+25<br>
+ <br>single- and a double-flighted geo metry; <br>
+Channel depth [mm]<br>
+results based on model with curvature<br>
+When the channel depth is small the double-flighted screw actually has a higher <br>solids conveying rate than the single-flighted screw. At larger values of the channel <br>depth the single-flighted screw has a higher rate than the double-flighted screw. <br>This effect tends to be more severe on smaller extruders as shown in Fig. 7.15. <br>At very small values of the channel depth (H &lt; 0.5 mm) the solids conveying rate of <br>the double-flighted screw is again greater than for the single-flighted screw. When <br>the channel depth increases beyond 1 mm, the rate for the double-flighted screw <br>drops rapidly and becomes zero when the depth H = 1.6 mm. Considering that the <br>feed depth for any screw will have to be at least the size of a typical polymer pellet <br>(3­4 mm), it is obvious that for a small diameter screw a double-flighted geo metry in <br>the feed section can result in very low output or even no output at all.<br>
+<hr>
+<A name=292></a><b>280</b>  <b>7Functional Process Analysis</b><br>
+9.00<br>
+]<br>
+6.75<br>
+Single flighted screw<br>
+Single flighted screw<br>
+rate [kg/hr 4.50<br>
+ing<br>ey<br>
+2.25<br>
+Double f<br>
+Doub lighted screw<br>
+le flighted screw <b> Figure 7.15</b> <br>
+Solids conveying rate versus channel <br>
+Solids conv<br>
+depth for single- and double-flighted <br>
+0 0<br>
+0.5<br>
+1.0<br>
+1.5<br>
+2.0<br>
+2.5<br>
+3.0<br>
+screw with 25-mm diameter; results <br>
+Channel depth [mm]<br>
+based on model with curvature<br>
+At larger values of the channel depth the difference between a single- or double-<br>flighted screw geo metry can mean the difference between output or no output at all. <br>This has been experimentally observed by the author in unpublished studies on <br>solids conveying in single screw extruders. A short 4 L/D 19-mm extruder, used just <br>for solids conveying, was outfitted with a single-flighted screw and yielded reason-<br>able output with open discharge and with moderate discharge pressures. The same <br>extruder with a double-flighted screw and all other dimensions unchanged did not <br>yield any output at all, even with open discharge.<br>Clearly, this adverse effect of multiple flights on solids conveying performance will <br>be less severe as the screw diameter increases. Nevertheless, the adverse effect is <br>there and it should be taken into account. It can also be seen in Fig. 7.15 that the <br>optimum channel depth becomes smaller as the geo metry is changed from a single-<br>flighted to a double-flighted screw geo metry. The effect of the number of flights will <br>reduce as the coefficient of friction on the barrel becomes larger relative to the fric-<br>tion on the screw. In extruders with grooved barrel sections, therefore, one would <br>expect a much smaller adverse effect due to multiple flights as compared to an ex -<br>truder with a standard smooth barrel.<br>The predicted solids conveying rate is very sensitive to the values of the coefficient <br>of friction; changes in coefficient of friction of 20% can cause changes in rate of 100 <br>to 1000% as shown in Fig. 7.16. At high values of the barrel friction the rate does <br>not change much with the coefficient of friction. However, at low values of the barrel <br>friction the rate changes substantially with coefficient of friction.<br>This has several important implications. If accurate predictions are required for <br>application of the theory to an actual extrusion problem, very accurate data on the <br>coefficient of friction will be required. However, measurement of accurate and mean-<br>ingful data on coefficient of friction is very difficult, as discussed in Section 6.1.2. <br>The reproducibility of measured data on the coefficient of friction generally ranges <br>from 10 to 50%. This means that the solids conveying rate predicted from these data <br>
+<hr>
+<A name=293></a> <br>
+7.2Solids Conveying  <b>281</b><br>
+cannot be very accurate in a quantitative sense. The predicted results should be <br>analyzed in a qualitative sense. In that respect, the solids conveying theory is very <br>useful in analyzing extrusion problems and in screw design. Thus, the theory should <br>be used to uncover the important trends.<br>
+1350<br>
+1125<br>
+0.40<br>
+]<br>
+0.35<br>
+900<br>
+0.30<br>
+rate [kg/hr<br>
+675<br>
+fb=0.25<br>
+ing<br>ey<br>
+450<br>
+<b> Figure 7.16</b> <br>
+Solids conveying rate versus channel <br>
+225<br>
+fb=0.20<br>
+depth for 75-mm screw for several values <br>
+Solids conv<br>
+of the coefficient of friction against the <br>
+0<br>
+0<br>
+5<br>
+10<br>
+15<br>
+20<br>
+25<br>
+barrel; the coefficient of friction against <br>
+Channel depth [mm]<br>
+the screw is 0 .2<br>
+If the actual solids conveying performance is as sensitive to the coefficient of fric-<br>tion as the theory indicates, small changes in the actual coefficient of friction can <br>have a substantial effect on the entire extrusion process. This is particularly true for <br>low values of the coefficient of friction against the barrel. The high sensitivity to the <br>coefficient of friction will tend to result in unstable extruder performance. At high <br>values of the barrel coefficient of friction the extruder performance will be less <br>affected by changes in friction and the extruder will be more stable under these <br>conditions. This explains why grooved feed extruders tend to be more stable than <br>smooth bore extruders.<br>An external means of altering the coefficient of friction is the temperature setting. <br>By changing the barrel temperature, the coefficient of friction on the barrel will <br>change, and by changing the screw temperature the coefficient of friction on the <br>screw will change. Thus, small changes in the temperature in the feed section can <br>have a large effect on the overall extruder performance. This behavior has been <br>observed by several workers. Kessler, Bonner, Squires, and Wolf [47] reported a <br>rather convincing case concerning the extrusion of nylon on a 82.55-mm (3.25-inch) <br>diameter extruder. A 28°C (50°F) increase in the rear barrel temperature reduced <br>the diehead pressure fluctuation from about 2.8 MPa (400 psi) down to about 0.4 <br>MPa (60 psi).<br>Ideally, the barrel temperature should be set to the temperature at which fb is maxi-<br>
+mum, and the screw temperature to the temperature at which fs is minimum. If <br>
+the coefficient of friction is known as a function of temperature, then the optimum <br>barrel and screw temperature can be determined directly from the friction data. <br>Unfortunately, in most practical situations, data on the coefficient of friction as a <br>
+<hr>
+<A name=294></a><b>282</b>  <b>7Functional Process Analysis</b><br>
+function of temperature are not known. Thus, in most cases, the optimum barrel and <br>screw temperatures have to be determined by trial and error. The effect of the rear <br>barrel temperature on extruder performance is generally larger than any other tem-<br>perature zone. In practice, therefore, considerable attention should be paid to the <br>proper fine-tuning of the rear barrel temperature zones.<br>
+<b>7.2.2.1Frictional Heat Generation<br></b>If one would want to determine optimum temperature settings from data on the co -<br>efficient of friction at various temperatures, there is an additional complication that <br>should be taken into account. This complication is the frictional heat generation that <br>occurs in the solids conveying zone. As a result of this frictional heat generation, the <br>temperature at the interface between solid bed and barrel may be substantially <br>higher than the barrel temperature setting indicates.<br>In any sliding motion where a frictional force is operational, there will be frictional <br>heat generation between the two bodies. The skeptic can experimentally verify this <br>fact by climbing up a rope for a reasonable distance and then sliding down while <br>keeping the hands tightly clamped around the rope. In a very short distance, the <br>effects of frictional heat generation between the rope and the hands will become <br>noticeable. Most likely, the experimenter will develop a healthy respect for what <br>frictional heat generation is capable of doing. The rate of frictional heat generation <br>equals the product of the frictional force Ff and the relative velocity v:<br>
+ (7.58)<br>
+where Fn is the normal force and f the coefficient of friction.<br>On the screw surface, the relative velocity between the solid bed and screw is sim-<br>ply the solid bed velocity. Thus, the frictional heat generation on the screw is:<br>
+ (7.59)<br>
+where F* = PfbWsin( + )dz.<br>Written in different form:<br>
+ (7.60)<br>
+On the barrel surface, the relative velocity between solid bed and barrel can be <br> <br>written  as:<br>
+ (7.61)<br>
+The frictional heat generation on the barrel surface is:<br>
+ (7.62)<br>
+<hr>
+<A name=295></a> <br>
+7.2Solids Conveying  <b>283</b><br>
+In most cases, the solid bed velocity vsz will be small compared to the relative velo-<br>
+city between solid bed and barrel v. Therefore, the frictional heat generation will <br>generally be higher on the barrel surface than on the screw surface. The frictional <br>heat generation of the barrel surface is dissipated into two fluxes, one conducting <br>the heat into the solid bed and the other conducting the heat into the extruder bar-<br>rel. The actual temperature profile of the solid bed will depend strongly on the heat <br>flux in the barrel. If the barrel is intensely cooled, the majority of the frictional heat <br>generation will conduct away through the barrel.<br>This tends to slow down the temperature development at the interface and extends <br>the length of the solids conveying zone. When the temperature at the interface <br>reaches the melting point, the solids conveying zone will terminate because poly-<br>mer melt will form at the interface and the solid-to-solid frictional mechanism will <br>cease to operate.<br>It should be noted in Eqs. 7.59 through 7.62 that the frictional heat generation is <br>directly proportional to the local pressure. Earlier it was determined that the local <br>pressure increases exponentially with down-channel distance; see Eqs. 7.51 and <br>7.54. Therefore, the frictional heat generation will increase exponentially with <br>down-channel distance. As a result, the temperature at the interface will closely <br> <br>follow the local pressure. This is shown qualitatively in Fig. 7.17.<br>
+e<br>
+Pressure<br>
+Temperatur <b> Figure 7.17</b> <br>
+Pressure and temperature <br>
+ <br>profile along the length of the <br>
+Length from feed opening<br>
+extruder<br>
+When the local pressure becomes sufficiently high, the temperature at the interface <br>will reach the melting point. This can happen even if the barrel is not heated. In this <br>case heat required for melting is supplied only by the frictional heating. This occurs <br>in auto-thermal extrusion operations; these are operations without active barrel <br>heating or cooling.<br>When a melt film forms at the barrel surface the exponential rise in pressure will <br>terminate because the solid-to-solid frictional mechanism breaks down. In fact, this <br>pressure/temperature relationship constitutes a built-in safety mechanism against <br>the development of very high pressures. The temperature rise resulting from the <br>
+<hr>
+<A name=296></a><b>284</b>  <b>7Functional Process Analysis</b><br>
+pressure rise limits the maximum pressure that can develop in the solids conveying <br>zone. Thus, the extruder self-regulates the maximum pressure that develops in the <br>solids conveying zone. By intensely cooling the barrel, the maximum pressure can <br>be increased considerably because the onset of melting will be delayed. Obviously, <br>the maximum pressure that can develop will also depend on the actual coefficients <br>of friction and the screw geo metry.<br>Because of the inherent non-isothermal nature of the solids conveying process, <br>accurate prediction of the actual solids conveying process becomes quite difficult. <br>Not because the mathematics are so complicated--they are relatively straight-<br>forward--but because the coefficient of friction should be known as a function of <br>temperature and pressure. This information is generally not available. Unless this <br>information is available, a complete non-isothermal solids conveying analysis would <br>not be very useful. Detailed calculations of non-isothermal solids conveying were <br>performed by Tadmor and Broyer [46]. Their numerical calculations of pressure and <br>temperature profiles showed that the temperature rise very closely follows the pres-<br>sure rise.<br>
+<b>7.2.2.2Grooved Barrel Sections<br></b>In Section 7.2.2, it was discussed how solids conveying can be improved by increas-<br>ing the roughness of the internal barrel surface. This conclusion can be reached <br>without detailed theoretical analysis; it is obvious from simple qualitative argu-<br>ments. The desirability of a large coefficient of friction at the barrel was recognized <br>as early as 1941 by Decker [48] as a result of a very simplified analysis. One of the <br>simplest methods of ensuring a high coefficient of friction on the barrel is to machine <br>grooves into the barrel surface.<br>In the late 1960s, several theoretical and experimental studies were made on the <br>effect of grooved barrel sections on solids conveying performance and overall <br>extruder performance; see for example references 49 through 51. This work was <br>mostly done in Germany. It was soon realized that substantial improvements could <br>be made to the extruder performance by using grooved barrel sections. The main <br>benefits of the grooved barrel section were found to be:<br>1.  Substantially improved output<br>2.  Substantially improved extrusion stability<br>3.  Lower pressure sensitivity of the output<br>Since these benefits appeal to most extrusion processors, grooved barrel sections <br>have become quite popular. Also, the use of grooved barrel sections allows proces-<br>sors to extrude materials that cannot be processed on conventional extruders, e.g., <br>very high molecular weight polyethylenes and powders. A grooved feed housing is <br>shown in Fig. 7.18.<br>
+<hr>
+<A name=297></a> <br>
+7.2Solids Conveying  <b>285</b><br>
+Barrel<br>
+Hopper<br>
+Barrel<br>
+Hopper<br>
+Thermal<br>
+Ther barrier<br>
+mal barrier<br>
+A<br>
+A<br>
+A<br>
+A<br>
+Cooling channels<br>
+Cooling channels<br>
+Groov<br>
+Groo ed sleev<br>
+v<br>
+e<br>
+ed sleeve<br>
+Section A-A<br>
+Section A-A<br>
+<b>Figure 7.18</b>A grooved barrel section<br>
+In a typical grooved feed extruder the groove length from the feed port is about 3 to <br>5 D. The groove depth generally reduces in a linear fashion, reaching zero depth at <br>the end of the grooved section. Cooling channels are located relatively close to the <br>internal barrel surface. This is important because the cooling capacity of the grooved <br>barrel section has to be high. The high cooling capacity is necessary to avoid too <br>high a temperature rise at the internal barrel surface. Melting should be avoided if <br>the effectiveness of the grooved barrel section is to be maintained. From the discus-<br>sions in the previous section it is clear that with a grooved barrel section there will <br>be a very large frictional heat generation at the barrel surface. Thus, good cooling <br>is crucial to the proper operation of a grooved barrel section. For the same reason, <br>a thermal barrier is generally designed between the grooved barrel section and the <br>smooth barrel section.<br>In applying the grooved barrel concept to existing extruders, a word of caution is in <br>order. The high effective coefficient of friction at the barrel surface results in a rapid <br>rise in pressure; this is obvious from Eq. 7.54. However, when the solids conveying <br>section is intensely cooled, the built-in safety mechanism against high pressures <br>can break down. In fact, these grooved barrel sections are designed to ensure that <br>the safety mechanism breaks down because melting has to be avoided in the grooved <br>barrel section. As a result, extremely high pressures can develop in the grooved bar-<br>rel section. Pressures of 100 to 300 MPa (15,000 to 45,000 psi) are not uncommon. <br>Therefore, these extruders have to be specially designed to withstand these high <br>pressures. Otherwise, mechanical failure of the barrel will occur. Thus, if a grooved <br>barrel section is used in an existing extruder, it is prudent to keep the length of the <br>
+<hr>
+<A name=298></a><b>286</b>  <b>7Functional Process Analysis</b><br>
+grooved section reasonably short (1 to 2 D) in order to avoid excessive pressures. <br>Obviously, this will limit the benefits that one can derive from a grooved barrel sec-<br>tion.<br>Another practical consideration is the wear of the grooves. Since the active edge of <br>the groove is exposed to very high stresses, considerable wear can occur, particu-<br>larly if the polymer contains abrasive components. Therefore, the grooved barrel <br>section is generally made out of a strong wear-resistant material in order to main-<br>tain optimum performance over a long period of time.<br>The benefits of a grooved barrel section can be analyzed from the theory of drag <br>induced solids conveying developed in Section 7.2.2. Figure 7.19 shows how the <br>solids conveying rate varies with the coefficient of friction on the barrel fb, for a situ-<br>
+ation where the pressure gradient is zero.<br>
+0.4<br>
+fs=0.1 0.20.3<br>
+ate<br>
+0.4<br>
+gr<br>
+0.2<br>
+Normalized conveyin<br>
+0<br>
+<b> Figure 7.19</b> <br>
+0<br>
+1<br>
+2<br>
+Solids conveying rate versus <br>
+Barrel coefficient of friction<br>
+barrel coefficient of friction<br>
+The figure shows curves of constant coefficient of friction on the screw fs. Four typi-<br>
+cal values are shown: fs = 0.1, fs = 0.2, fs = 0.3, and fs = 0.4. It can be seen that the <br>
+curve rises steeply when fb is small, but later reaches a plateau at high fb values. Two <br>
+major problems are evident when barrel coefficient of friction is close to the screw <br>coefficient of friction. One is that the solids conveying rate is considerably below <br>the theoretical maximum value. The second, more important, problem is that small <br>changes in fb will result in very large changes in the solids conveying rate when fb <br>
+ fs. This will lead directly to surging of the extruder. Since small variations in the <br>
+friction on the barrel are bound to occur by the nature of the process, there is a high <br>possibility of extrusion instabilities when fb is approximately equal to fs. Thus, this <br>
+constitutes an unstable operating point.<br>When the coefficient of friction at the barrel is increased to a value about two or <br>three times the coefficient of friction at the screw, the solids conveying rate increases <br>substantially. At the same time, the slope of the curve is reduced. When fb &gt;&gt; fs, <br>
+<hr>
+<A name=299></a> <br>
+7.2Solids Conveying  <b>287</b><br>
+small variations in the barrel friction will result only in small changes in the  solids <br>conveying rate. Thus, the process will be inherently much more stable when fb is <br>
+much larger than fs. This explains how grooved barrel sections can substantially <br>
+improve extrusion stability.<br>Figure 7.20 shows how the solids conveying rate varies with the pressure gradient <br>along the solids conveying zone.<br>
+<b>_</b><br>
+2500<br>
+2000<b>_</b><br>
+]<br>
+fb=0.6<br>
+1500<b>_</b><br>
+rate [kg/hr<br>ing<br>ey 1000<b>_</b><br>
+fb=0.4<br>
+500<b>_</b><br>
+Solids conv<br>
+0.30<br>
+0.25<br>
+0.20<br>
+0<br>
+<b> Figure 7.20</b> <br>
+<b>I</b><br>
+<b>I</b><br>
+<b>I</b><br>
+<b>I</b><br>
+<b>I</b><br>
+<b>I</b><br>
+1<br>
+1E2<br>
+1E4<br>
+1E6<br>
+Solids conveying rate versus <br>
+Pressure ratio [P(10)/P(0)]<br>
+pressure ratio<br>
+When fb = fs, the solids conveying rate at a zero pressure gradient is quite small and <br>
+the rate drops off quickly as the pressure gradient increases. At a relatively small <br>pressure gradient, the rate becomes zero. As fb is increased at constant fs, the rate at <br>
+zero pressure gradient increases and the fall-off with pressure gradient becomes <br>less severe. At relatively high values of fb, the fall-off with pressure gradient becomes <br>
+very small; in fact, the output becomes essentially independent of back-pressure. <br>This indicates a high degree of positive displacement behavior, which is quite un -<br>usual in conventional extruders. However, many workers, e.g., [52, 53], have experi-<br>mentally verified the fact that the output is essentially independent of back-pressure <br>when an extruder is equipped with a grooved barrel section. Thus, the three main <br>benefits of grooved barrel sections, high output, good stability, and pressure-inde-<br>pendent output, can be predicted directly from theory.<br>Besides the high pressures and the wear problems, there are a few other disadvan-<br>tages of grooved barrel sections. The main disadvantage is probably the fact that a <br>substantial amount of energy is lost through the intensive cooling of the grooved <br>barrel section. Nowadays, with the increasing cost of and concern about energy, this <br>energy loss is more of a factor than it was in the past. Detailed measurements of <br>energy consumption in various sections of the grooved barrel extruder were made <br>
+<hr>
+<A name=300></a><b>288</b>  <b>7Functional Process Analysis</b><br>
+by Menges and Hegele [54]. They found that as much as 30 to 40% of the mechanical <br>energy is lost through the cooling water; about 60% of the mechanical energy is dis-<br>sipated in the solids conveying zone. In the worst case, the specific energy loss <br>through the cooling water is about 150 KJ/kg = 0.042 kWhr/kg (0.07 Hphr/lb). By <br>improving the thermal barrier between the grooved barrel section and smooth <br> <br>barrel section, the specific energy loss can be reduced to about 100 KJ/kg = 0.028 <br>kWhr/kg (0.046 Hphr/lb). However, considering that the specific enthalpy (see <br> <br>Section 6.3.4) of the polymer is generally around 0.06 kWhr/kg (0.10 Hphr/lb), <br>the losses in the grooved barrel section are quite substantial. In a later publication, <br>Menges [55] reports a total mechanical energy loss through the cooling water of <br>about 14%. The mechanical energy is the energy supplied by the screw, which is <br>transformed into heat by frictional and viscous heat generation. Helmy [229] <br>reported lower specific energy consumption with grooved barrel extrusion with <br>HMWPE and MMWPE than with smooth barrel extrusion. He also found lower melt <br>temperatures with grooved barrel extrusion. However, with normal polyethylene, <br>Helmy found the specific energy consumption of the grooved barrel extruder to be <br>about 10 to 25% higher than the smooth barrel extruder.<br>Energy losses in the grooved barrel section can be reduced by reducing the amount <br>of cooling. This can be achieved by a closed-loop temperature control of the grooved <br>barrel section, as discussed by Menges, Feistkorn, and Fischback [237]. They found <br>that the energy efficiency in the grooved feed section could be increased from 45 to <br>80% by increasing the cooling water temperature from 5 to 70°C.<br>Another related drawback of the grooved barrel extruder is its higher torque require-<br>ment. On a modification of an existing extruder, this generally requires a gear <br>change; in some cases, a new high torque drive may be necessary. However, this <br>should not be a problem on a new extruder designed for operation with a grooved <br>barrel section.<br>Another disadvantage of the grooved barrel section is that material can accumulate <br>in the grooves; this can cause problems with a product changeover. Also, the screw <br>geo metry has to be adapted to the presence of a grooved barrel section. Screw design <br>rules that work for conventional extruders do not work for extruders with grooved <br>barrel sections. In extruders with grooved barrel sections, the extruder screws gen-<br>erally have a much lower compression ratio (if at all), a shallower feed section, and <br>a deeper metering section. The melting and mixing capability must also be greater <br>because complete melting and thermal homogeneity are more difficult to achieve; <br>higher output at the same screw speed means shorter residence time, thus, less <br>time for completion of melting and for mixing (lower shear strain).<br>Despite the disadvantages, extruders with grooved barrel sections have found wide-<br>spread acceptance in Europe. In many cases, the grooved barrel section has become <br>a standard instead of an option. Somewhat surprisingly, the acceptance of grooved <br>barrel sections in the U.S. has been quite slow. One reason may be that U.S. extruder <br>
+<hr>
+<A name=301></a> <br>
+7.2Solids Conveying  <b>289</b><br>
+manufacturers have not promoted the grooved barrel concept vigorously. As of 2001, <br>the number of grooved barrel extruders in the U.S. is still relatively small.<br>Most grooved barrel sections used in the past had axial grooves running parallel to <br>the axis of the screw. A relatively recent development is the barrel section with heli-<br>cal grooves. One of the first publications on the subject was an article by Langecker <br>et al. [56]. They claimed a higher conveying efficiency as compared to axial grooves <br>and a 20% reduction in energy consumption; this corresponded to a 45% reduction in <br>energy loss through cooling of the grooved barrel section. Langecker et al. also found <br>that screws with a very small compression ratio were most suitable for use with heli-<br>cally grooved barrel sections; in some cases, compression ratios of unity or slightly <br>less than unity (actually a decompression screw) gave optimal performance. Langecker <br>filed for a patent [57] on the helical groove idea as far back as 1972. Another patent on <br>barrel sections with helical grooves was issued to Maillefer [58] in 1979.<br>Grünschloß [59] presented an analysis of a barrel section with helical grooves, <br>attempting to explain the advantages of the helical grooves over axial grooves. Grün-<br>schloß examines a situation where the grooves in the barrel are quite wide and <br>deep, similar to the channel in the screw. Thus, transport occurs in the barrel chan-<br>nel as well as in the screw channel. He also assumes no shearing between the bulk <br>in the screw channel and in the barrel channel.<br>The effect of the helical grooves can be explained by considering the velocity dia-<br>gram, shown earlier in Fig. 7.10. Figure 7.21(left) shows a typical velocity diagram <br>with a smooth barrel. The frictional heat generation is direction-determined by <br>the relative velocity between the barrel and the solid bed v and the coefficient of <br>friction between the barrel and the solid bed fb. Figure 7.21(right) shows a velocity <br>
+diagram with axial grooves in the barrel.<br>
+Groove direction<br>
+Vsz<br>
+Vsz<br>
+Fl<br>
+F<br>
+ig<br>
+li<br>
+h<br>
+gh<br>
+<br>
+<br>
+t d<br>
+<br>
+<br>
+t <br>
+i<br>
+d<br>
+re<br>
+i<br>
+v<br>
+r<br>
+c<br>
+e<br>
+v<br>
+t<br>
+c<br>
+io<br>
+ti<br>
+n<br>
+on<br>
+<b> Figure 7.21</b> <br>
+Velocity diagram with smooth barrel <br>
+(left) and axially grooved barrel <br>
+Vb<br>
+Vb<br>
+(right)<br>
+With the grooves in the axial direction, there will be no movement of the material in <br>the barrel grooves. The frictional heat generation in this situation will be deter-<br>mined again by v and the effective coefficient of friction on the barrel surface feb. <br>
+With axial grooves, the relative velocity difference between the barrel and solid bed <br>will be somewhat smaller than with a smooth barrel. However, the effective coeffi-<br>
+<hr>
+<A name=302></a><b>290</b>  <b>7Functional Process Analysis</b><br>
+cient of friction on the barrel surface feb will be much higher as a result of the <br>
+grooves. The net effect is that the frictional heat generation will be much higher <br>than with a smooth barrel.<br>The effective coefficient of friction will be partially determined by the contact of the <br>solid bed in the screw channel and the barrel flight tip surface, and partially by the <br>contact of the solid bed in the screw channel and the solid bed in the barrel grooves. <br>Friction at the barrel flight tip surface is similar to friction at a smooth barrel wall. <br>However, friction at the solid bed in the barrel grooves will be of an entirely different <br>nature. To a certain extent the internal friction of the polymer will determine the <br>friction, but the friction will be augmented by the action of the active edge of the <br>barrel grooves. If the groove is relatively wide compared to the barrel flight width, <br>one can assume that an effective coefficient of friction is roughly determined by <br>the internal coefficient of friction of the polymer. Since the internal coefficient of <br>friction is generally two to three times higher than the external coefficient of fric-<br>tion, it can be assumed that the effective coefficient of friction with axial grooves is <br>about two to three times as high as compared to a smooth barrel. This explains the <br>high frictional heat generation in axially grooved barrels.<br>The frictional heat generation can be reduced by reducing the effective coefficient of <br>friction or by reducing the velocity difference between the solid bed in the screw <br>channel and the barrel. Reducing the effective coefficient of friction will adversely <br>affect the solids conveying rate and pressure generating capability. However, the <br>velocity difference can be reduced by using helical grooves. This is explained by the <br>velocity diagrams in Fig. 7.22.<br>
+b<br>
+roove direction<br>
+G<br>
+VszFlight direction<br>
+<br>
+s<br>
+v1<br>
+<b> Figure 7.22</b> <br>
+Velocity diagram with helical <br>
+ <br>barrel grooves without (left) and <br>
+with (right) transport in the barrel <br>
+Vb<br>
+groove<br>
+The velocity diagram in the case of no material movement in the barrel grooves is <br>shown in the left-hand side of Fig. 7.22. If the material in the barrel grooves is sta-<br>tionary with respect to the barrel, the effective barrel velocity equals the actual bar-<br>
+<hr>
+<A name=303></a> <br>
+7.2Solids Conveying  <b>291</b><br>
+rel velocity. The effective velocity difference v1 in the case of no material move-<br>
+ment in the barrel grooves is:<br>
+ (7.63)<br>
+However, the situation will change significantly if there is movement of the material <br>in the barrel grooves. The right-hand side of Fig. 7.22 shows the situation where the <br>velocity of the material in the barrel grooves is vsg. In this case, the effective barrel <br>
+velocity becomes vbe; this is determined from vectorial addition of the barrel velocity <br>
+vb and the solid bed velocity in the barrel grooves vsg:<br>
+ (7.64)<br>
+The magnitude of vbe is determined by the following relationship:<br>
+ (7.65)<br>
+where b is the barrel groove helix angle.<br>The relative velocity between the solid bed in the screw channel and the material <br>in the barrel grooves v2 is determined from the vectorial differences between vbe <br>
+and vsz:<br>
+ (7.66)<br>
+From Fig. 7.22 it is clear that the effective relative velocity can be reduced substan-<br>tially if there is movement of the material in the barrel grooves. The magnitude of <br>the effective velocity difference v2 is:<br>
+ (7.67)<br>
+where s is the helix angle of the screw and b the helix angle of the barrel.<br>It can be easily seen from Fig. 7.22 that the effective velocity difference v2 is mini-<br>
+mized when:<br>
+ (7.68)<br>
+This is the case when the helix angle on the barrel b equals the solids conveying <br>
+angle of the solid bed in the screw channel s. Thus, the effective velocity difference <br>
+and the frictional heat generation is minimized when:<br>
+ (7.69)<br>
+If this condition can be achieved, the conveying efficiency of the solid bed in the <br>screw channel will be high because of a large frictional force acting on it at the bar-<br>rel surface. At the same time, however, the velocity difference between the solid bed <br>
+<hr>
+<A name=304></a><b>292</b>  <b>7Functional Process Analysis</b><br>
+in the screw channel and the material in the barrel groove is minimized, resulting <br>in a substantially reduced frictional heat development in the grooved section. This <br>approach to the conveying mechanism in grooved barrel sections demonstrates that <br>the claimed advantages of helical grooves can be confirmed by an engineering analy-<br>sis. Therefore, it does seem to make sense to use helical grooves instead of axial <br>grooves. Helical grooves have the potential to eliminate one of the main drawbacks <br>of axial grooves, that is, the substantial loss of energy as a result of the very high <br>frictional heat generation and the need for intensive cooling, causing loss of energy <br>through cooling of the grooved barrel section.<br>In the actual extrusion process, the solids conveying angle s will not be absolutely <br>
+constant along the length of the screw. In the initial portion of the channel, a sub-<br>stantial amount of compacting will often take place, resulting in corresponding <br>reductions in the solid bed velocity, and, thus, in the solids conveying angle s; see <br>
+Eq. 7.48. The theoretically optimum barrel helix angle b, therefore, will vary along <br>
+the axial length of the grooved barrel section. This would be quite difficult to <br>machine, however, and could make the grooved section relatively expensive. From a <br>practical point of view, it would seem reasonable to make the barrel helix angle <br>equal to the solids conveying angle based on a fully compacted solid bed. With a <br>fully compacted solid bed, the pressure and frictional heat generation are very high <br>and, thus, more of a concern.<br>In order to obtain an expression for the optimum helix angle of the barrel groove(s), <br>the forces acting on the solid bed in the screw channel and on the solid bed in <br>the barrel groove can be analyzed in similar fashion as done for the smooth barrel. <br>Figure 7.23 shows the forces acting on the solid bed in the screw channel and the <br>corresponding velocity diagram.<br>
+<b> Figure 7.23</b> <br>
+Force and velocity diagram of a <br>
+solid bed in the screw channel<br>
+<hr>
+<A name=305></a> <br>
+7.2Solids Conveying  <b>293</b><br>
+The frictional force acting on the barrel surface Fbs makes an angle  with the tan-<br>
+gential direction. This angle is determined by the direction of the velocity vector <br>v2. If it is assumed that the frictional force Fbs is determined by the internal friction <br>
+of the bulk material, then the force can be expressed as:<br>
+ (7.70)<br>
+The extra force F*s can be determined from a force balance in the cross-channel <br>
+direction:<br>
+ (7.71)<br>
+The frictional force on the screw Ffs can be determined from:<br>
+ (7.72)<br>
+The net pressure force acting on the solid bed element is:<br>
+ (7.73)<br>
+The relationship between angle  and the pressure gradient in the screw down-<br>channel direction is obtained by a force balance in the screw down-channel direc-<br>tion:<br>
+ (7.74)<br>
+This yields the following equation:<br>
+ (7.75)<br>
+Figure 7.24 shows the forces acting on the solid bed in the barrel groove and the <br>corresponding velocity diagram.<br>
+<b> Figure 7.24</b> <br>
+Force and velocity diagram of <br>
+solid bed element in a barrel <br>
+groove<br>
+<hr>
+<A name=306></a><b>294</b>  <b>7Functional Process Analysis</b><br>
+In using the velocity diagram, it should be kept in mind that the barrel is now taken <br>as being stationary and the screw moving at a tangential velocity vs = ­vb. The solid <br>
+bed is moving at velocity vsg. The velocity of the solid bed in the screw channel with <br>
+respect to the stationary barrel is vse; this is determined by the vectorial addition of <br>
+vs and vsz:<br>
+ (7.75a)<br>
+The frictional force acting on the screw surface Fbb makes the same angle  with the <br>
+tangential direction. The angle, in this case, is determined by the vectorial differ-<br>ence of vse and vsg:<br>
+ (7.76)<br>
+The frictional force Fbb is determined from:<br>
+ (7.77)<br>
+The extra force F*b is determined from a force balance in the cross-groove direction:<br>
+ (7.78)<br>
+The frictional force on the barrel surface is:<br>
+ (7.79)<br>
+The force resulting from the pressure gradient is:<br>
+ (7.80)<br>
+As before, the relationship between angle  and the pressure gradient in the barrel <br>down-groove direction is obtained by a force balance on the solid bed element in the <br>down-groove direction:<br>
+ (7.81)<br>
+The optimum barrel helix angle *b is the one for which the velocity difference v2 is <br>
+minimized; this occurs when b = s and  = 0. At this point, the pressure gradient <br>
+in the down-channel direction can be expressed in known quantities:<br>
+ (7.82)<br>
+<hr>
+<A name=307></a> <br>
+7.2Solids Conveying  <b>295</b><br>
+This pressure gradient can now be inserted into Eq. 7.81 to yield an expression for <br>the optimum barrel helix angle. It should be remembered that the relationship <br>between the down-channel coordinate zs and the down-groove coordinate zb is:<br>
+ (7.83)<br>
+The expression for the optimum barrel helix angle now takes the following form:<br>
+ (7.84)<br>
+where:<br>
+ (7.84a)<br>
+and<br>
+ (7.84b)<br>
+The solution to Eq. 7.84 can be expressed in the now familiar form:<br>
+ (7.85)<br>
+Equation 7.85 allows the calculation of the optimum barrel helix angle if the screw <br>geo metry is known and if the various coefficients of friction, internal and external, <br>are known as well. The solution for the optimum helix angle is not completely ana-<br>lytical because A2 contains a term Wb that is dependent on the barrel helix angle. <br>
+This can be solved by initially guessing a value of Wb, then calculating *b according <br>
+to Eq. 7.85. The Wb can be calculated with:<br>
+ (7.85a)<br>
+where wbg is the perpendicular barrel flight width and pb the number of parallel <br>
+grooves in the barrel. The calculated value of Wb can then be used to calculated *b <br>
+again. This process can be repeated until the initial value of Wb and the calculated <br>
+value of Wb are within a certain tolerance. Convergence is very rapid and accurate <br>
+values of *b are generally obtained in two or three iterations.<br>Figure 7.25 shows *b as a function of the screw helix angle when fb = fs = 0.2, <br>
+fi = 0.6, and Hs = 15.24 mm (0.6 in). The optimum barrel helix angle increases with <br>
+the screw helix angle and with reducing barrel groove depth. The optimum barrel <br>angle is relatively insensitive to the internal coefficient of friction as shown in Fig. <br>7.26.<br>
+<hr>
+<A name=308></a><b>296</b>  <b>7Functional Process Analysis</b><br>
+]<br>
+70<br>
+H<br>
+Hb=2.5 mm<br>
+=2.5 mm<br>
+b<br>
+ angle [degr.<br>
+60<br>
+5.0 mm<br>
+5.0 mm<br>
+e helix<br>
+r<br>oov<br>
+7.5 mm<br>
+7.5 mm<br>
+lg<br>
+50<br>
+Opt. barre<br>
+4010<br>
+15<br>
+20<br>
+25<br>
+Screw flight helix angle [degrees]<br>
+<b>Figure 7.25</b>Optimum barrel groove helix angle versus the screw flight helix angle at various <br>
+values of the barrel groove depth<br>
+65<br>
+]<br>
+64<br>
+63<br>
+ angle [degr.<br>
+62<br>
+helix<br>
+61<br>
+<b> Figure 7.26</b> <br>
+60<br>
+Optimum barrel groove helix angle <br>
+Opt. barrel<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+versus the internal coefficient of <br>
+Internal coefficient of friction<br>
+ <br>friction<br>
+<b>7.2.2.3Adjustable Grooved Barrel Extruders<br></b>Grooved feed extruders offer considerable advantages over conventional extruders, <br>such as higher throughput, better stability, and the ability to process very high <br>molecular weight polymers. There are some important disadvantages as well, for <br>instance, higher motor load, wear is more likely, high pressures in the grooved <br>region, and the screw design has to be adapted.<br>The disadvantages of the grooved feed extruder disappear when the grooved feed <br>extruder is made with a mechanism that allows adjustment of the groove depth. <br>Recent developments in grooved feed extruders incorporate an adjustment mecha-<br>nism that allows the depth of the grooves to be changed during actual operation <br>from zero to full depth. These developments will be described and some operational <br>data from actual extrusion experiments will be presented.<br>
+<hr>
+<A name=309></a> <br>
+7.2Solids Conveying  <b>297</b><br>
+<b>7.2.2.3.1Problems with Grooved Feed Extruders</b><br>
+The use of a conventional compression screw in a grooved feed extruder often results <br>in poor extruder performance and rapid wear of the equipment. This happens when <br>the solids conveying efficiency is too high for the melting and melt conveying zones <br>of the extruder to keep up. It is advantageous to have a means of controlling the <br>solids conveying efficiency of the extruder, so that it can be adjusted to achieve <br>the most efficient and consistent operation of the extruder. One common method of <br>adjusting the solids conveying efficiency is to change the temperature of the barrel <br>in the feed section of the extruder. The drawback of a temperature adjustment is <br>that it generally only has a weak effect on the solids conveying efficiency. It is also <br>possible to change the temperature of the screw in the feed section of the extruder. <br>However, this suffers from the same problem as barrel temperature adjustment and, <br>further, screw temperature control is more complicated than barrel temperature <br>control.<br>
+<b>7.2.2.3.2Eliminating Drawbacks with Grooved Feed Extruders</b><br>
+A more effective method of controlling the solids conveying efficiency is by adjust-<br>ing the groove geo metry. The solids conveying efficiency is determined by the num-<br>ber of grooves, the length of the grooves, the orientation of the grooves, and the <br>depth of the grooves. A continuous adjustment of the number of grooves is not pos-<br>sible. Adjustment of the length or the orientation of the grooves is possible, but <br>likely to be mechanically complex. The typical axial length of the grooves is three to <br>five barrel diameters. Thus, the adjustment length would have to be in the same <br>range; this is a rather long length. The most convenient method of controlling the <br>solids conveying efficiency would appear to be to adjust the depth of the grooves. <br>The groove depth usually varies from about 2 to 3 mm to zero. Therefore, the adjust-<br>ment has to be only about 2 to 3 mm.<br>
+<b>7.2.2.3.3The Adjustable Grooved Feed Extruder</b><br>
+The adjustable grooved feed extruder developed at Rauwendaal Extrusion Engineer-<br>ing [246] uses a grooved feed section in which the depth of the grooves can be con-<br>tinuously adjusted while the extruder is in operation. Parallel developments have <br>taken place in Poland at the Technical University of Lublin [244, 245]. Earlier con-<br>cepts have been developed [247], however, due to the complexity of the adjustment <br>mechanism, these have not been applied on a wide scale.<br>There are two basic methods by which the groove depth can be adjusted:<br>1.  By moving an insert (key) in the barrel groove in radial direction.<br>2.  By moving a tapered key along a barrel groove with reducing depth in axial <br>
+direction.<br>
+A prototype of the first mechanism has been constructed; experimental results will <br>be discussed later in this section.<br>
+<hr>
+<A name=310></a><IMG src="Polymer Extrusion by Chris Rauwendaal-310_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-310_35.jpg"><br>
+<b>298</b>  <b>7Functional Process Analysis</b><br>
+<b>7.2.2.3.4Groove Depth Adjustment by Radial Movement of Keys</b><br>
+Figure 7.27 depicts an example of a mechanism where the keys are moved radially, <br>showing the axial and perpendicular cross-section of a feed housing (4) with three <br>axial grooves. The groove depth is adjustable by means of electromechanical actua-<br>tors directly connected to keys. The keys can move radially, allowing a simple and <br>direct adjustment of the groove depth. Only one actuator is necessary for each <br>groove. Other adjustment mechanisms are possible as well.<br>The depth of each groove is individually adjustable with electromechanical actua-<br>tors (1) located at the outside of the feed housing. The actuator stem (2) is connected <br>to the back end of the key, allowing a radial movement of the key in the housing. The <br>key is located in a slot (7) in the feed housing. Both the key and the key slot have <br>a shoulder to ensure that the minimum groove depth can be no less than zero. In <br>other words, the key cannot move beyond the internal diameter of the feed housing. <br>Figure 7.27 shows the position of the keys with zero groove depth.<br>
+Section B-B<br>
+Section A-A<br>
+A<br>
+B<br>
+2<br>
+1<br>
+3<br>
+4<br>
+7<br>
+5<br>
+3<br>
+6<br>
+7<br>
+A<br>
+B<br>
+<b>Figure 7.27</b>Feed housing with adjustable groove depth, zero groove depth position<br>
+The keys are made to pivot at the end of the grooves, so that the downstream end of <br>the grooves always tapers to a zero depth. This avoids hang-up of material at the end <br>of the grooves. Another advantage of this arrangement is that only one actuator is <br>required for each groove, thus minimizing the cost of the adjustment mechanism. <br>The position of the keys in the position with maximum groove depth is shown in Fig. <br>7.28. In this position the keys have been pulled into the key slots as far as possible.<br>
+Section B-B<br>
+A<br>
+B<br>
+Section A-A<br>
+2<br>
+1<br>
+3<br>
+4<br>
+7<br>
+5<br>
+6<br>
+7<br>
+A<br>
+B<br>
+<b>Figure 7.28</b>Feed housing with adjustable groove depth, maximum groove depth position<br>
+<hr>
+<A name=311></a> <br>
+7.2Solids Conveying  <b>299</b><br>
+The feed housing can have a standard feed opening (6). The feed opening in Fig. 7.27 <br>is offset to improve the feeding capability. The feed housing also has cooling chan-<br>nels to keep the temperatures low enough to avoid melting of the plastic particles in <br>the grooved feed housing. Cooling channels are easy to incorporate since the groove <br>depth adjustment mechanism takes up little space.<br>
+<b>7.2.2.3.5Groove Depth Adjustment by Axial Movement of Tapered Keys</b><br>
+One of the drawbacks of the pivoting key mechanisms is that there is a chance of <br>material getting between the key and the keyway. When this happens, the key can-<br>not move radially outward and mechanical problems may occur. One way of circum-<br>venting this problem is to make the keys and keyways tapered, and adjust the groove <br>depth by sliding the keys along the base of the keyways. This is illustrated schema-<br>tically in Fig. 7.29.<br>
+Feed opening<br>
+Feed housing<br>
+Conveying direction<br>
+Key in rearward position, Key in forward position, Lubricating surface Tapered key<br>
+maximum groove depth<br>
+zero groove depth<br>
+<b>Figure 7.29</b>Groove depth adjustment by sliding key axially<br>
+The keys can be moved individually or together. One possible method of moving the <br>keys is to use a rack-and-pinion mechanism. The advantage of the sliding key mech-<br>anism is that there is very little chance that material will get between the keys and <br>the keyways. Another advantage is that the sliding key mechanism has few parts <br>and is easy to manufacture.<br>The bottom of the key can be a self-lubricating material to ensure good sliding <br>action. The key cross-section (at least part of it) should be made such that the keys <br>will stay in place, even if they are in a vertical position. Some possible groove geo-<br>metries that can accomplish this are shown in Fig. 7.30.<br>There are several advantages of the adjustable groove depth:<br>
+ <br>
+ The efficiency of the feed throat can be adjusted to the conveying characteristics of <br>
+the material as well as the conveying behavior of the screw.<br>
+ <br>
+ Since the groove depth can be adjusted to zero depth, the feed throat can be easily <br>
+cleaned out upon material change-over; in other words, no material can get trapped <br>in the grooves.<br>
+<hr>
+<A name=312></a><b>300</b>  <b>7Functional Process Analysis</b><br>
+<b> Figure 7.30</b> <br>
+Various groove geometries to secure key<br>
+ <br>
+ The depth of the grooves can be adjusted while the machine is running, allowing <br>
+optimization of the groove depth under actual operating conditions.<br>
+ <br>
+ With electronic actuators the groove depth can be adjusted quickly, automatically, <br>
+and precisely; the groove depth can be automatically optimized to produce the <br>smallest pressure variation at the discharge end of the extruder or to achieve <br>other process objectives.<br>
+ <br>
+ The adjustable depth of the grooves allows a grooved feed to be easily used on <br>
+vented extruders. The groove depth can be adjusted to make sure that vent flow <br>does not occur.<br>
+ <br>
+ Being able to adjust the depth of the grooves in a feed housing will greatly expand <br>
+and improve process adjustment capability of screw extruders.<br>
+Current screw extruders are quite limited in the ability to control the process. The <br>primary control parameters are screw speed and machine temperatures (barrel, <br>die, and screw). Screw speed is directly linked to output; as a result, rapid and sub-<br>stantial changes in screw speed will result in output changes. This limits the use of <br>screw speed as a process control variable to optimize the process under semi-steady-<br>state conditions. Machine temperatures can only be changed slowly; thus, they can-<br>not be used to make fast adjustments to the process. Also, temperatures usually do <br>not have a strong effect on the conveying characteristics of the extruder. As a result, <br>machine temperature changes have only limited effect on the extruder conveying <br>performance. With the ability to influence the conveying characteristics rapidly and <br>strongly, groove depth adjustment is likely the most powerful method to optimize <br>extruder performance. It will allow a single screw extruder to run a wider range of <br>plastics by adjusting the conveying characteristics of the machine to the character-<br>istics of the plastic. Similarly, a single extruder can use a wider range of extruder <br>screws and still perform well.<br>
+<b>7.2.2.3.6Experimental Results</b><br>
+An experimental adjustable grooved feed extruder was developed and manufactured <br>with a diameter of 25 mm and a length-to-diameter ratio of 18:1. The feed housing <br>was equipped with two grooves containing pivoting keys, allowing a continuous <br>adjustment of the groove depth during operation of the extruder. The angle over <br>which the keys can be moved ranges from 0 to 0°54 (0.0157 radians or 0.90 <br>
+<hr>
+<A name=313></a> <br>
+7.2Solids Conveying  <b>301</b><br>
+degrees). The screw speed ranged from 177 to 279 rpm. No heat was applied to the <br>barrel from the barrel heaters; in fact, the barrel heaters were switched off after <br>start-up of the extruder. The material processed was an MDPE.<br>Figure 7.31 shows how the throughput varies with the angle of inclination at four <br>different screw speeds (177, 211, 248, and 279 rpm). In all cases, the throughput <br>increases with the angle of inclination; the increase in throughput is greater at <br>higher screw speeds (about 12%).<br>
+14<br>
+279 rpm<br>
+12<br>
+]<br>
+248 rpm<br>
+10<br>
+211 rpm<br>
+8<br>
+177 rpm<br>
+Throughput [kg/hr<br>
+6 0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<b> Figure 7.31</b> <br>
+Taper angle [degrees]<br>
+Throughput versus taper angle<br>
+Figure 7.32 shows the specific energy consumption, SEC, at again four screw speeds. <br>The SEC increases with the angle of inclination. This is to be expected since the <br>effective coefficient of friction at the barrel increases with the groove depth. Figure <br>7.33 shows barrel, die, and melt temperatures plotted against the angle on inclina-<br>tion. All temperatures increase with the groove depth. The results show that the <br>extrusion process can be strongly influenced by adjustment of the groove depth.<br>
+1350<br>
+177 rpm<br>
+1300<br>
+211 rpm<br>
+248 rpm<br>279 rpm<br>
+1250<br>
+ [Joule/gram]<br>
+1200<br>
+Specific energy 1150<br>
+<b> Figure 7.32</b> <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Specific energy consump-<br>
+Taper angle [degrees]<br>
+tion versus taper angle<br>
+<hr>
+<A name=314></a><b>302</b>  <b>7Functional Process Analysis</b><br>
+225<br>
+melt temperature<br>
+] 195<br>
+barrel temperature<br>
+165<br>
+head temperature<br>
+Temperature [C<br>
+<b> Figure 7.33</b><br>
+135<br>
+ <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+Temperatures versus the groove <br>
+Taper angle [degrees]<br>
+taper angle<br>
+<b>7.2.2.3.7Outlook for Adjustable Grooved Feed Extruders</b><br>
+The adjustable grooved feed extruder offers the advantages of a conventional <br>grooved feed extruder while largely eliminating its disadvantages. The advantages <br>of a grooved feed extruder are higher output, better process stability, and the ability <br>to process very high molecular weight polymers, such as VHMWPE. Additional <br>advantages are that the conveying efficiency of the grooved feed section can be <br>matched to the characteristics of the polymer and the screw, the feed housing can <br>be easily cleaned out upon material change-over, and adjusting the groove depth <br>during operation allows process optimization under actual operating conditions. <br>Further, optimization of the groove depth can be done automatically by feedback of <br>the head pressure fluctuations (and/or other process parameters) to the groove <br>depth adjustment. Additionally, the adjustable grooved feed extruder can be used on <br>vented extruders or extruders with downstream feed port without fear of vent flow.<br>The main advantage of the adjustable grooved feed extruder is that it offers an <br>increased level of control and versatility that has never before been possible. As a <br>result, the extruder can process a wider range of materials and can operate with a <br>greater number of different screw geometries and still maintain good process stabil-<br>ity and product quality. The adjustable grooved feed extruder may increase accept-<br>ance of grooved feed extruders in the United States.<br>
+<b>7.2.2.4Starve Feeding Versus Flood Feeding<br></b>Most single screw extruders are flood fed; this means that the bottom section of the <br>feed hopper is completely filled with material and the screw will take in as much <br>material as it can handle. When an extruder is flood fed the output is determined <br>primarily by the screw speed. Flood feeding is illustrated in Fig. 7.34.<br>With flood feeding, high pressures are generated in the solids conveying and plasti-<br>cating zones of the extruder. These high pressures tend to agglomerate ingredients <br>that later need to be dispersed and distributed [247­249]. As a result, flood feeding <br>can be detrimental to the mixing capability of the extruder.<br>
+<hr>
+<A name=315></a><IMG src="Polymer Extrusion by Chris Rauwendaal-315_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-315_40.jpg"><br>
+ <br>
+7.2Solids Conveying  <b>303</b><br>
+Hopper<br>
+Hopper<br>
+<b> Figure 7.34</b> <br>
+Flood feeding<br>
+In starve feeding the material is metered into the extruder with a feeder. As a result, <br>there is no accumulation of material at the feed opening and the throughput is <br>determined by the feeder and not by the screw speed. The first several turns of the <br>screw are partially filled with material without any pressure development in this <br>part of the extruder. The screw channel does not become completely filled until <br>some distance from the feed opening; at this point the pressure will start building <br>up in the extruder. In effect, starve feeding reduces the effective length of the <br>extruder. Starve feeding is illustrated in Fig. 7.35.<br>
+Feede<br>
+F<br>
+r<br>
+eeder<br>
+Hopper<br>
+Hopper<br>
+<b> Figure 7.35</b> <br>
+Starve feeding<br>
+One of the benefits of starve feeding is that the pressures along the extruder are <br>lower than in flood feeding. Therefore, there is less chance of agglomeration, result-<br>ing in improved mixing action in the extruder. Recently, a number of workers have <br>analyzed the effect of starve feeding on the mixing capability of extruders and in -<br>jection molding machines [250­253]. Without exception these investigators found <br>major improvements in mixing quality in starve feeding compared to flood feeding. <br>Starve feeding has been the standard mode of operation for twin screw extruders <br>used in compounding. However, the benefits of starve feeding are not limited to twin <br>screw extruders. Mixing in single screw extruders can be improved significantly <br>by using starve feeding. With starve feeding single screw extruders can be used for <br>more demanding mixing jobs, in some cases competing with twin screw extruders.<br>
+<hr>
+<A name=316></a><b>304</b>  <b>7Functional Process Analysis</b><br>
+In starve feeding the screw speed can be varied at constant throughput. Also, the <br>throughput can be varied at constant screw speed. As a result, there is a greater <br>degree of control of the extrusion process. This is beneficial as long as the extruder <br>is long enough to achieve complete melting and mixing of the polymer. Without suf-<br>ficient machine length, starve feeding will not result in acceptable performance. In <br>most cases the length of the extruder should be 30 D or longer to use starve feeding <br>successfully. With grooved feed extruders the feed intensity can be controlled by the <br>degree of starvation. The effect of a grooved feed section will depend on the length <br>over which the grooved section is completely filled. Since this can be controlled by <br>the degree of starvation (actual feed rate divided by flood feed rate) starve feeding in <br>a sense allows control of the effective length of a grooved feed section. As such, this <br>is an alternative to the adjustable grooved feed extruder discussed in Section 7.2.2.3.<br>Another benefit of starve feeding has to do with melting. In single screw extruders <br>the solid bed is usually compacted into a dense, continuous solid bed that extends <br>over a considerable length of the extruder. This solid bed forms a helical ribbon that <br>reduces in size as melting progresses along the extruder. At the point where the <br>solid ribbon disappears the melting process is completed and the melt conveying <br>process starts. This type of melting is referred to as contiguous solids melting <br>(CSM)--it is typical of single screw extruders. A typical melting length in CSM is <br>from 10 to 15 D.<br>In an intermeshing twin screw extruder (TSE) it is not possible for the solids to form <br>a continuous solid bed because there are no continuous channels along the screws. <br>As a result, the melting in an intermeshing TSE is different from the CSM type melt-<br>ing. In the melting zone of TSEs the solid particles generally maintain their in -<br>dividuality as melting progresses. The individual solid particles are suspended in a <br>melt matrix and the particle size reduces as melting progresses. This type of melt-<br>ing is referred to as dispersed solids melting (DSM). The melting length in DSM is <br>often quite short, about 2 to 3 D, about five times shorter than the melting length in <br>CSM.<br>The high melting efficiency of DSM makes twin screw extruders very versatile <br>machines because it makes more machine length available for other tasks, such as <br>mixing, degassing, chemical reactions, etc. Clearly, if DSM can be achieved on single <br>screw extruders it will greatly increase the utility of single screw extruders. It <br>seems safe to say that one of the important ingredients for achieving DSM on single <br>screw extruders is starve feeding. To achieve DSM high pressures in solids convey-<br>ing have to be avoided; this can be readily accomplished with starve feeding. Ob -<br>viously, other ingredients will be required as well, such as good screw design with <br>effective mixing elements--these issues will be discussed in Chapter 8.<br>
+<hr>
+<A name=317></a> 7.3Plasticating  <b>305</b><br>
+<b></b><br>
+ <b>7.3Plasticating</b><br>
+The second functional zone in the extruder is the plasticating zone or melting zone. <br>The melting zone starts as soon as melt appears, usually after 3 to 5 diameters from <br>the feed opening. Since most of the frictional heat generation in the solids convey-<br>ing zone generally occurs at the barrel-solid bed interface, the first traces of melt <br>usually appear at the barrel surface. It should be noted that the start of melting does <br>not necessarily occur where the measured barrel temperature profile exceeds the <br>melting point. The temperature at the barrel-solid bed interface can be quite differ-<br>ent from the measured barrel temperature. In fact, melting can often be initiated <br>without any external heating simply by the action of the frictional heat generation.<br>As melting proceeds, the initial melt film at the barrel surface will grow in thick-<br>ness. This is particularly true in the very early stages of melting because the thin <br>melt film is subjected to a high rate of shearing, causing a rapid temperature rise in <br>the material and a high melting rate. Once the thickness of the melt film exceeds the <br>radial flight clearance, the melt will flow into the screw channel, displacing the con-<br>tiguous solid bed. In most cases, the solid bed will be pushed against the passive <br>flight flank and the melt will start to accumulate in a melt pool between the solid <br>bed and the active flight flank.<br>Maddock [60] was the first to accurately describe the melting behavior in single <br>screw extruders. His observations were based on screw extraction experiments, <br>where the screw is stopped with material still in the extruder and the machine is <br>rapidly cooled. The screw is then pushed out of the extruder barrel, generally with a <br>hydraulic piston. The material in the screw channel can then be analyzed at various <br>axial locations along the screw. Maddock's description was accurate but only quali-<br>tative; he did not attempt to model the melting process to allow a quantitative <br>description of the melting process. This type of melting is described as the contigu-<br>ous solids melting (CSM) model.<br>Several years later, Tadmor and coworkers did extensive experimental work on melt-<br>ing in single screw extruders [61]. In addition to the experimental work, Tadmor <br>performed a theoretical analysis of the plasticating process and developed the now <br>classic Tadmor melting model [62]. This was a major contribution to the extrusion <br>theory, particularly since melting was the one major extruder function for which a <br>theoretical model had not been developed. Theoretical models for solids conveying <br>and melt conveying or pumping had already been developed in the 1950s or earlier. <br>However, the melting theory was not developed until the mid-1960s.<br>As discussed, the melting in twin screw extruders occurs by dispersed solids melt-<br>ing. A theoretical model of dispersed solids melting will be presented in Section 7.3.5.<br>
+<hr>
+<A name=318></a><b>306</b>  <b>7Functional Process Analysis</b><br>
+<b>7.3.1Theoretical Model of Contiguous Solids Melting</b><br>
+An idealized cross-section of the screw channel in the melting zone is shown in <br>Fig. 7.36. It is assumed that the screw is stationary; thus, in a cross-section perpen-<br>dicular to the screw flights, the barrel moves towards the active flight at velocity vbx, <br>
+which is the cross-channel component of the barrel velocity vb. As a result of this <br>
+motion, the thin melt film between the solid bed and the barrel is sheared at a high <br>rate. This will cause substantial viscous heat generation in the melt film; see also <br>Section 5.3.4. Since the melt film is quite thin, the effect of pressure gradients on <br>the velocity profile in the melt film will be quite small. Therefore, the flow in the <br>melt film will be essentially drag flow, and the shear rate and viscous heat genera-<br>tion will be relatively uniform along the depth of the melt film.<br>
+vsz<br>
+<br>
+v<br>
+<br>
+<br>
+y<br>.<br>
+x<br>
+z '<br>
++<br>
++0<br>
+x ' z<br>
+v<br>
+Melt film<br>
+H<br>
+b<br>
+m(x)<br>
+Melt pool<br>
+v<br>
+<br>
+bx<br>
+Trailing flight flank<br>
+Solid bed<br>
+x<br>
+Ws<br>
+Pushing flight flank<br>
+W<br>
+<b> Figure 7.36</b> <br>
+The Tadmor melting model<br>
+The melt flows from the melt film towards the active flight flank. Only a small frac-<br>tion of the material can flow through the clearance. As a result, the majority of the <br>melt will flow into the melt pool. A circulating flow will be set up in the melt pool as <br>a result of the barrel velocity. Since most of the viscous heat generation occurs in <br>the upper melt film, it is generally assumed that all melting takes place at the upper <br>solid bed-melt film interface. As melting proceeds, the cross-sectional area of the <br>solid bed will reduce and the cross-sectional area of the melt pool will tend to <br>increase. The melt pool, therefore, will exert considerable pressure on the solid bed. <br>This reduces the width of the solid bed, while the melt film between the solid bed <br>
+<hr>
+<A name=319></a> 7.3Plasticating  <b>307</b><br>
+and the barrel remains relatively constant. In order for this mechanism to work, <br>the solid bed has to be continuously deformable. The width of the solid bed reduces <br>as more material melts away from the barrel side of the solid bed. Thus, the material <br>in the solid bed close to the solid-melt interface moves towards the interface as melt-<br>ing proceeds. This occurs at a velocity vsy, the solid bed melting velocity. This velo-<br>
+city determines the melting rate of the solid bed.<br>There are basically two sources of energy utilized for melting in the extruder. The <br>first and generally the most important one is the mechanical energy supplied by the <br>screw, which is transformed into heat by a process of viscous heat generation. The <br>second source of energy is the heat supplied by the external barrel heaters and <br> <br>possibly by screw heaters. In most extruders, the majority of the energy will be sup-<br>plied by the screw, about 80 to 90% or more. There is a very good reason for this. <br>Energy supplied by the screw will be dissipated primarily in the melt film. The <br>resulting viscous heat generation will be relatively uniform throughout the mate-<br>rial. Thus, the temperature rise in the polymer melt will be relatively uniform and <br>the heat transfer distances will be small.<br>The heat supplied by the barrel heaters has to be conducted through the entire <br>thickness of the barrel and through the entire thickness of the melt film before it <br>can reach the solid bed. Problems with this energy transport are considerable heat <br>losses by conduction, convection, and radiation. Another, probably more severe, <br>problem is the low thermal conductivity of the polymer. The heat has to be trans-<br>ferred across the entire melt film thickness. Therefore, the conductive heat flux will <br>be small, particularly when the melt film thickness is large. Increasing the barrel <br>temperature can accelerate the heating process; however, this temperature is lim-<br>ited by the possibility of degradation of the polymer.<br>If melting only occurs by heating from the barrel heaters without viscous heat gen-<br>eration, the rate of melting is unacceptably slow. This is precisely why ram extrud-<br>ers have such a poor plasticating capability; there is little or no viscous heat genera-<br>tion in these types of extruders; see Section 2.4. This also explains why reciprocating <br>single screw extruders have become so popular on injection molding machines, <br>even though they are more complex than ram extruders. Thus, it should be clear <br>that the key to the plasticating ability of screw extruders is the viscous heat genera-<br>tion in the polymer melt.<br>In order to be able to predict the melting rate, the amount of heat flowing towards <br>the solid-melt interface has to be known. This can be determined if the temperature <br>profiles in the melt film and in the solid bed are known. The temperature profile in <br>the melt film can be determined from the velocity profile in the melt film. In order to <br>derive an expression for the temperature profile in the melt film, the following <br>assumptions are made:<br>1.  The process is a steady-state process.<br>2.  The polymer melt density and thermal conductivity are constant.<br>
+<hr>
+<A name=320></a><b>308</b>  <b>7Functional Process Analysis</b><br>
+3.  Convective heat transfer is neglected.<br>4.  Conductive heat transfer occurs only in a direction normal to interface.<br>With these assumptions, the general energy equation (Eq. 5.5) can be simplified to:<br>
+ (7.86)<br>
+The shear stress can be determined from the equation of motion. In order to derive <br>an expression for the shear stress, additional assumptions are made:<br>1.  The polymer melt flow is laminar.<br>2.  Inertia and body forces are negligible.<br>3.  There is no slip at the walls.<br>4.  There are no pressure gradients in the melt film.<br>5.  The temperature dependence of the viscosity can be neglected.<br>With these assumptions, the general equation of motion (Eq. 5.3) can be simplified <br>to:<br>
+ (7.87)<br>
+The direction of coordinate x is determined by the vectorial velocity difference <br>
+between the barrel and the solid bed; see Fig. 7.36. The magnitude of this velocity <br>difference v, the relative velocity, is given by Eq. 7.47, and the angle  with the <br>tangential direction is:<br>
+ (7.88)<br>
+This angle  is essentially the same angle as the solids conveying angle discussed in <br>Section 7.2.2. If the polymer melt behaves as a Newtonian fluid, Eq. 7.87 becomes:<br>
+ (7.89)<br>
+By integrating twice and taking the boundary conditions vx (0) = 0 and vx(Hm) = v, <br>
+the following linear velocity profile is obtained:<br>
+ (7.90)<br>
+Equation 7.86 now becomes:<br>
+ (7.91)<br>
+<hr>
+<A name=321></a> 7.3Plasticating  <b>309</b><br>
+The boundary conditions are T(0) = Tm and T(Hm) = Tb. By integrating twice, a quad-<br>
+ratic temperature profile is obtained:<br>
+ (7.92)<br>
+where Tb = Tb--Tm.<br>From Eq. 7.92, the heat flow into the interface   can now be determined. By using <br>Fourier's law, Eq. 5.45, the heat flow per unit area (heat flux) becomes:<br>
+ (7.93)<br>
+The temperature profile in the solid bed can be determined from the energy equa-<br>tion (Eq. 5.5) applied to a moving solid slab:<br>
+ (7.94)<br>
+If the temperature at the interface is taken at the melting point, T(0) = Tm, and the <br>
+temperature far away from the interface is taken as a reference temperature T(­) = <br>Tr, then the temperature profile in the solid bed becomes:<br>
+ (7.95)<br>
+where Tr = Tm--Tr.<br>The reference temperature Tr is typically the temperature at which the solid poly-<br>
+mer particles are introduced into the extruder. In Eq. 7.95, it is assumed that the <br>bulk of the solid bed is at Tr and that only a relative thin skin is heating up. The <br>
+validity of this assumption depends primarily on the total thickness of the solid bed <br>and the residence time of the solid bed in the extruder. The validity of the assump-<br>tion can be tested by analyzing the Fourier number as discussed in Section 5.3.3. <br>If the thickness of the solid bed is about 10 mm and the residence time less than <br>one or two minutes, the assumption is probably valid. However, if the solid bed <br>thickness is in the range of 3 to 4 mm at the same residence times, the assumption <br>is clearly not valid. Thus, the assumption should not be used when the thickness of <br>the solid bed is less than about 5 mm (0.2 in) because, in reality, the solid bed is <br>heated from all sides. This means that Eq. 7.95 should not be used for small dia-<br>meter extruders with shallow channels and should not be used towards the end of <br>melting if the solid bed thickness has reduced to less than 5 mm.<br>The actual temperature profile in the solid bed will change with axial location. The <br>developing temperature profile can be described with Eq. 6.98 for one-sided heating, <br>
+<hr>
+<A name=322></a><b>310</b>  <b>7Functional Process Analysis</b><br>
+as discussed in Section 6.3.5. The temperature profile in the center for two-sided <br>heating is shown in Fig. 7.37. The vertical axis shows the dimensionless tempera-<br>ture and the horizontal axis shows the Fourier number, as discussed in Section <br>5.3.3. Figure 7.37 is a graphical representation of the equation describing the devel-<br>oping temperature profile in a finite slab heated from both sides when y = 0 ([16] of <br>Chapter 5).<br>
+1.0<br>
+0.8<br>
+)<br>
+0.6<br>
+0-T<br>
+Temperature at center<br>
+1<br>
+Temperature at center<br>
+)/(T<br>
+0.4<br>
+0-T(T<br>
+0.2<br>
+<b> Figure 7.37</b> <br>
+0<br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0 Temperature profile in a solid bed with <br>
+Fourier number<br>
+two-sided heating<br>
+ (7.96)<br>
+where b is half the slab thickness.<br>Within an accuracy of about 10 to 20%, the following approximation can be made for <br>the temperature profile at the center of the slab:<br>
+ (7.97)<br>
+This approximation can be used to obtain a corrected reference temperature T*r, <br>
+where T*r increases with axial distance along the screw:<br>
+ (7.98)<br>
+where the term in parentheses represents the Fourier number of the solid bed.<br>This corrected reference temperature T*r can be used in Eq. 7.95 instead of Tr.<br>
+<hr>
+<A name=323></a> 7.3Plasticating  <b>311</b><br>
+The effect of this correction will be an accelerated melting towards the end of the <br>melting zone. The heat flux from the interface into the solid bed qout is again deter-<br>
+mined from Fourier's law:<br>
+ (7.99)<br>
+A more correct determination of the heat flux into the solid bed could be made by <br>using Eq. 7.96 and differentiating the temperature with respect to the normal dis-<br>tance y.<br>An additional complication is that the solid bed is assumed to be freely deformable. <br>This means that the heat transfer situation is no longer determinate. Equations 7.97 <br>through 7.99 can only be used if the rate of deformation of the solid bed is small <br>relative to the rate of heat conduction into the solid bed. A typical solid bed melt <br>velocity (vsy) is 0.2 mm/s. The thermal penetration thickness (4t, see Eq. 6.99[a]) <br>
+is about 1.3 mm in one second when the thermal diffusivity () is 10­7 m2/s. Thus, <br>the rate of heat conduction into the solid bed is about one order of magnitude higher <br>than the rate of deformation of the solid bed. Thus, the rate of deformation of the <br>solid bed, in most cases, is relatively small compared to the rate of heat conduction <br>into the solid bed.<br>The temperature profiles in the melt film and solid bed are shown in Fig. 7.38.<br>
+<b>Tb</b><br>
+Barrel<br>
+1<br>
+<b>T</b><br>
+Melt film<br>
+<b>m</b><br>
+<b>q1</b><br>
+<b>q2</b><br>
+2<br>
+Solid bed<br>
+<b> Figure 7.38</b> <br>
+<b>Tr</b><br>
+Temperature profiles and heat fluxes in CSM type <br>
+Temperature profile<br>
+melting<br>
+From a heat balance of the interface, the melting rate can be determined. The heat <br>used to melt the polymer at the interface is determined by the heat flux into the <br>interface minus the heat flux out of the interface:<br>
+ (7.100)<br>
+The melting velocity vsy now becomes:<br>
+ (7.101)<br>
+where H = Hf + Cs Tr.<br>
+<hr>
+<A name=324></a><b>312</b>  <b>7Functional Process Analysis</b><br>
+The factor H can be considered the heat sink; it is the enthalpy difference between <br>Tm and Tr. As melting occurs along the direction of relative motion of the solid bed, <br>
+more molten polymer will accumulate in the melt film. Along a length of dx, the <br>
+increase in mass flow as a result of melting is:<br>
+ (7.102)<br>
+The prime with the mass flow indicates mass flow per unit length. The prime with <br>the x-coordinate indicates the direction determined by the vectorial difference <br>between the barrel and solid bed; see Fig. 7.36. This increase in mass flow will re -<br>quire an increase in melt film thickness. The corresponding increase in drag flow <br>rate is:<br>
+ (7.103)<br>
+It can be assumed that the changes in melt film thickness occur only in direction x <br>
+of solid bed relative motion. In that case, by using Eqs. 7.101 through 7.103, a dif-<br>ferential equation describing the melt film thickness can be formulated:<br>
+ (7.104)<br>
+If the melt film thickness at x = 0 is taken as the local radial clearance  between <br>
+flight and barrel, then the melt film thickness can be expressed as:<br>
+ (7.105)<br>
+The amount of polymer melting over the entire solid bed width can be found by:<br>
+ (7.106)<br>
+where W´s is the solid bed width in direction x.<br>This represents the melting rate per unit length in direction z (see Fig. 7.36); it can <br>
+be written as:<br>
+ (7.107)<br>
+If the clearance is considered to be negligible, Hm(0) = 0, the melting rate is:<br>
+ (7.108)<br>
+<hr>
+<A name=325></a> 7.3Plasticating  <b>313</b><br>
+Equation 7.108 expresses the melting per unit length in direction z. The direction <br>
+x is determined by the relative velocity v. The relationship between x and the <br>
+cross-channel coordinate x is: x = xsin( + ). The relationship between z and the <br>
+down-channel coordinate z is: z= zsin(+ ). Thus, the melting rate per unit down-<br>
+channel distance is:<br>
+ (7.109a)<br>
+The relationship between z and axial coordinate l is l = zcos. Thus, the melting <br>
+rate per unit axial length is:<br>
+ (7.109b)<br>
+The relationship between the solid bed width Ws in x-direction and Ws in cross-<br>
+channel direction is Ws = Ws sin( + ). With this relationship, the melting rate per <br>
+unit down-channel length can be written as:<br>
+ (7.109c)<br>
+where:<br>
+ (7.109d)<br>
+This result will become obvious when it is realized that vsin( + ) = vbsin; <br>
+see Eq. 7.61.<br>From Eq. 7.107, it can be seen that the effect of a non-zero radial flight clearance is <br>to reduce the local melting rate. This means that screw or barrel wear in the plasti-<br>cating zone will reduce the melting performance of the extruder; see also Section <br>8.2.2.3. From Eqs. 7.108 and 7.109, the contribution from heat conduction 2kmTb <br>
+and the contribution from the viscous heat generation v2 can be clearly distin-<br>guished. As mentioned earlier, the viscous heat generation term is generally larger <br>than the heat conduction term.<br>These equations can also explain why sometimes an increase in barrel temperature <br>does not result in improved melting performance. When the barrel temperature Tb is <br>
+increased, the heat conduction term increases; however, the viscous heat generation <br>term will decrease because the viscosity in the melt film will decrease with increas-<br>ing temperature of the melt film. If the reduction in the viscous heat generation is <br>larger than the increase in heat conduction, the net result will be a reduced melting <br>rate as shown in Fig. 7.39.<br>
+<hr>
+<A name=326></a><b>314</b>  <b>7Functional Process Analysis</b><br>
+100<br>
+total melting rate<br>
+total melting rate<br>
+50 viscous heating<br>
+viscous heating<br>
+<b> Figure 7.39</b> <br>
+Melting Rate [%]<br>
+conductive heating<br>
+conductiv<br>
+0<br>
+Effect of barrel temperature on melting rate <br>
+150<br>
+200<br>
+250<br>
+300<br>
+when the viscosity is highly temperature <br>
+Barrel Temperature<br>
+sensitive<br>
+This can occur in polymers whose melt viscosity is very sensitive to temperature, <br>such as PMMA, PVA, PVC, etc. For temperature dependence of melt viscosity, see <br>Section 6.2.4 and Table 6.1.<br>When the melt viscosity is not very sensitive to temperature the reduction in vis-<br>cous heating with barrel temperature will be small. As a result, the melting rate will <br>likely increase with barrel temperature as shown in Fig. 7.40.<br>
+100<br>
+total melting rate<br>
+total melting rate<br>
+50<br>
+viscous heating<br>
+viscous heating<br>
+<b> Figure 7.40</b> <br>
+conductive heating<br>
+Melting Rate [%]<br>
+conductive heating<br>
+Effect of barrel temperature on melting rate <br>
+0150<br>
+200<br>
+250<br>
+300<br>
+when the viscosity is not highly tempera-<br>
+Barrel Temperature<br>
+ture sensitive<br>
+Another interesting observation is that the local melting rate is directly determined <br>by the width of the solid bed. Obviously, the maximum solid bed width is the chan-<br>nel width. The solid bed width in the very early part of melting can be assumed to be <br>equal to the channel width. As melting proceeds, the solid bed width will generally <br>reduce and, consequently, the local melting rate will reduce with it. In most cases, <br>the highest melting rate is achieved at the start of melting; the melting rate then <br>reduces monotically with axial distance as the solid bed width reduces. This is an <br>important consideration in screw design. For good melting performance, one would <br>like to maintain a relatively wide solid bed over a substantial length in order to <br>maintain the highest possible melting rate. This point will be discussed in more <br>detail in Chapter 8 on screw design.<br>It should be noted that Eqs. 7.102 through 7.109 are slightly different from the equa-<br>tions developed by Tadmor [62]; see also reference 5 of Chapter 1. The reason is that <br>Tadmor assumed a constant melt film thickness. However, it is clear that the melt film <br>
+<hr>
+<A name=327></a> 7.3Plasticating  <b>315</b><br>
+thickness has to increase with cross-channel distance to accommodate the increased <br>amount of melt. Both Shapiro [63] and Vermeulen [64, 65] have analyzed this point in <br>detail and demonstrated that in a consistent model the melt film thickness must vary <br>with cross-channel distance. If the melt film thickness is assumed to be constant, very <br>high cross-channel pressure gradients must occur to allow additional material in the <br>melt film. These pressure gradients are of such high magnitude as to be unrealistic. <br>If the thickness of the melt film is assumed constant across the width of the solid bed, <br>the predicted melting rate will be lower by a factor of 2 compared to the case where <br>the melt film thickness varies across the width of the solid bed.<br>The one unknown left at this point is the width of the solid bed. A relationship for <br>the change in solid bed width with down-channel distance can be obtained from a <br>mass balance of the solid bed in the down-channel direction. This can be written as:<br>
+ (7.110)<br>
+In the melting zone, the channel is generally tapered; thus, the channel depth varies <br>linearly with down-channel distance:<br>
+ (7.111)<br>
+where Hf is the channel depth of the feed section and Az the degree of taper in the <br>
+down-channel direction.<br>From Eqs. 7.109 and 7.110, a differential equation is obtained describing the change <br>in solid bed:<br>
+ (7.112)<br>
+The thickness of the solid bed Hs is primarily determined by the channel depth. <br>
+Thus, it can be assumed that the change in Hs with distance equals the change in <br>
+channel depth with distance:<br>
+ (7.113)<br>
+This is an important point because this means that without melting, the compres-<br>sion in the channel would cause an increase in solid bed width. In this case:<br>
+ (7.113a)<br>
+In the absence of melting the solid bed width will increase directly proportional to <br>distance. Thus, there are two mechanisms affecting the width of the solid bed. Melt-<br>ing will cause a reduction of the solid bed width, but at the same time, channel taper <br>will cause an increase in solid bed width. The reduction in Ws from melting should <br>
+<hr>
+<A name=328></a><b>316</b>  <b>7Functional Process Analysis</b><br>
+always exceed the increase in Ws from compression. If the compression is too rapid, <br>
+with large Az, the melting cannot reduce the solid bed width fast enough. As a result, <br>
+the solid bed width will increase and plug the channel as it reaches the width of the <br>screw channel. This puts an upper bound on the maximum compression ratio that <br>can be applied in the plasticating section of an extruder screw; see more on this in <br>Section 8.2.2.<br>From Eqs. 7.112 and 7.113, the following differential equation for the solid bed <br>width is obtained if it is assumed that H  Hs:<br>
+ (7.114)<br>
+The solution for this equation is:<br>
+ (7.115)<br>
+where W1 is the solid bed width at z = 0.<br>The total length for melting can be obtained by setting Ws = 0:<br>
+ (7.116)<br>
+From Eq. 7.115, it can be seen that an extreme condition is reached when the term <br>1/(AzvszsW1) becomes unity. In this case, the solid bed width becomes independ-<br>
+ent of down-channel distance, and the shortest possible melting length is obtained <br>ZT = Hf/Az. However, this condition cannot be achieved in practice because there is <br>
+no room for the polymer melt. Thus, in practical extrusion operations, the term 1/<br>
+(AzvszsW1) has to be larger than unity to ensure a continuous reduction in solid <br>
+bed width with distance and to avoid plugging.<br>
+<b>7.3.1.1Non-Newtonian, Non-Isothermal Case<br></b>A more realistic prediction of the melting performance can be obtained if the poly-<br>mer melt is considered non-Newtonian and non-isothermal. However, this extension <br>of the analysis results in coupled energy and momentum equations. Such problems <br>generally do not allow analytical solutions. One approach to this problem, as sug-<br>gested by Tadmor [61], is to assume a certain temperature profile and solve the <br>equations. If a quadratic temperature profile is assumed, the solution becomes quite <br>elaborate containing many error functions. Evaluation of the solutions requires sub-<br>stantial numerical analysis and number crunching; for details, the reader is referred <br>to reference 5 of Chapter 1. If a linear temperature profile is assumed in the melt <br>film, the solution becomes more manageable. The constitutive equation is:<br>
+<hr>
+<A name=329></a> 7.3Plasticating  <b>317</b><br>
+ (7.117)<br>
+where aT = exp[T(Tm--T)]; see also Eq. 6.40.<br>The equation of motion (Eq. 5.3) becomes:<br>
+ (7.118)<br>
+This equation can be integrated to give:<br>
+ (7.119)<br>
+where K1 is an integration constant.<br>The assumed temperature profile can be written as:<br>
+ (7.120)<br>
+When this expression is substituted in Eq. 7.117, the velocity gradient will become <br>dependent on normal distance y. Equation 7.117 becomes:<br>
+ (7.121)<br>
+where K2 = sT Tb and s is the reciprocal power law index s = 1/n.<br>With boundary conditions vx(0) = 0 and vx(Hm) = v, the solution becomes:<br>
+ (7.122)<br>
+The shear rate distribution is determined by taking the first derivative of vx(y) with <br>
+respect to y:<br>
+ (7.123)<br>
+The velocity gradient as a function of y can now be inserted into the energy equa-<br>tion, Eq. 7.86. By taking boundary conditions T(0) = Tm and T(Hm) = Tb, the solution <br>
+becomes:<br>
+ (7.124a)<br>
+where:<br>
+ (7.124b)<br>
+<hr>
+<A name=330></a><b>318</b>  <b>7Functional Process Analysis</b><br>
+For a temperature-independent fluid (T = 0), the factor K2 becomes zero. When K2 <br>
+approaches zero:<br>
+ (7.124c)<br>
+and<br>
+ (7.124d)<br>
+Thus, for a temperature-independent power law fluid, the temperature profile is:<br>
+ (7.124e)<br>
+For a temperature-independent Newtonian fluid (n = 1), the temperature profile <br>becomes:<br>
+ (7.124f)<br>
+Equation 7.124(f) corresponds, of course, to Eq. 7.92 derived earlier and also to <br>Eq. 5.62. From Eq. 7.124(a), the heat flux from the melt film into the solid melt inter-<br>face can be calculated. The first derivative of T(y) with respect to y is:<br>
+ (7.125)<br>
+The heat flux from the melt film into the interface is:<br>
+ (7.126)<br>
+The heat balance for the interface now becomes:<br>
+ (7.127)<br>
+where:<br>
+ (7.128)<br>
+Equation 7.127 corresponds to Eq. 7.100, describing the Newtonian, temperature-<br>independent fluid. From Eq. 7.127, the solid bed melting velocity can be obtained:<br>
+ (7.129)<br>
+<hr>
+<A name=331></a> 7.3Plasticating  <b>319</b><br>
+Following the same procedure that was used to derive Eq. 7.105, the melt film thick-<br>ness can be expressed as:<br>
+ (7.130)<br>
+For a temperature-independent power law fluid, factor B3 becomes:<br>
+ (7.128a)<br>
+and the melt film thickness becomes:<br>
+ (7.130a)<br>
+Equation 7.130(a) becomes equal to Eq. 7.105 when the power law index n is set to <br>unity.<br>The melting rate per unit-down channel length is still described by Eq. 7.107. If the <br>clearance between flight and barrel is assumed zero, the melting rate is:<br>
+ (7.131)<br>
+By the same transformations as used before, the melting rate per unit length in z <br>
+direction can be rewritten as the melting rate per unit down-channel length z:<br>
+ (7.132)<br>
+where:<br>
+ (7.132a)<br>
+This value 1* can be used in the equations describing the solid bed width profile <br>
+along the melting zone.<br>
+<b>7.3.1.1.1Melting of Temperature-Dependent Power Law Fluid</b><br>
+Rauwendaal [270] published a melting theory for temperature-dependent fluids and <br>presented an exact analytical solution of power law fluids. This represents an impor-<br>tant extension of the previous theories in that it allows accurate assessment of the <br>effect of the temperature dependence of the melt viscosity. The following assump-<br>tions are made:<br>1.  The process is steady state<br>2.  Polymer melt density and thermal conductivity are constant<br>
+<hr>
+<A name=332></a><b>320</b>  <b>7Functional Process Analysis</b><br>
+3.  Convective heat transfer is negligible<br>4.  Conductive heat transfer only in normal direction<br>5.  Polymer melt flow is laminar<br>6.  Inertia and body forces are negligible<br>7.  No slip at the walls<br>8.  No pressure gradient in the melt film<br>9.  The temperature profile in the melt film is fully developed<br>The equation of motion takes the same form as Eq. 7.118. The constitutive equation <br>of the polymer melt is expressed as:<br>
+ (7.133)<br>
+where:<br>
+m = m0exp[(T0--T)]<br>
+When the dimensionless temperature is defined as  = (T­T0)/n and dimensionless <br>
+normal coordinate  = y/Hm the energy equation can be written as:<br>
+ (7.134)<br>
+where:<br>
+ (7.135)<br>
+and:<br>
+ (7.136)<br>
+When the maximum in the temperature profile in the melt film occurs at * &gt; 0 the <br>temperature profile can be determined to be [271]:<br>
+ (7.137)<br>
+where<br>
+ (7.138)<br>
+ (7.139)<br>
+ (7.140)<br>
+<hr>
+<A name=333></a> 7.3Plasticating  <b>321</b><br>
+Temperature m is the dimensionless melt temperature. The value of A determines <br>
+the maximum temperature * = ln A; it can be determined from:<br>
+ (7.141)<br>
+where:<br>
+ (7.142)<br>
+and the Nahme number:<br>
+ (7.143)<br>
+The analytical solution, Eq. 7.137, has been compared to finite element solutions <br>with very good agreement between analytical and numerical results [271]. From <br>Eq. 7.137 the heat flux from the melt film into the interface can be determined. <br>Using Fourier's law the heat flux becomes:<br>
+ (7.144)<br>
+The heat flux from the interface into the solid bed qout is determined from the tem-<br>
+perature profile in the solid bed. This is given by Eq. 7.99. The melting rate can be <br>determined from a heat balance at the interface. This can be written as:<br>
+ (7.145)<br>
+where Hf is the latent heat of fusion. The melt velocity becomes:<br>
+ (7.146)<br>
+where:<br>
+ (7.147)<br>
+and:<br>
+ (7.148)<br>
+Factor H is the enthalpy difference between Tm and Tr. The melt film thickness Hm <br>
+can be determined from a mass balance in direction x. This leads to the following <br>
+expression:<br>
+ (7.149)<br>
+<hr>
+<A name=334></a><b>322</b>  <b>7Functional Process Analysis</b><br>
+In the derivation of Eq. 7.149 it is assumed that the dependence of A on x can be <br>
+neglected. The melting rate per unit length in direction z can be determined by <br>
+integrating the melting velocity over the width of the solid bed as shown in <br>Eq. 7.106. The integral can be evaluated using Eqs. 7.103 and 7.146, and results in:<br>
+ (7.150)<br>
+The melting rate per unit length in down-channel direction z can be determined by <br>considering that z = z sin ( + ) and v sin ( + ) = vbsin. Thus, the melting rate <br>
+per unit length z becomes:<br>
+ (7.151)<br>
+The solid bed width can be determined from a mass balance of the solid bed in the <br>down-channel direction as expressed by Eq. 7.110. This leads to the following dif-<br>ferential equation:<br>
+ (7.152)<br>
+where:<br>a = mvbx1<br>b  = 0.5 mvbx<br>c1 = svsz<br>c2 = svszAz<br>The solution to Eq. 7.152 can be written as:<br>
+( (7.153)<br>
+where:<br>
+It is assumed that at z = 0 the width of the solid bed is W1 and the depth of the solid <br>
+bed H1. The problem is simpler when the flight clearance is taken as zero. In this <br>
+case, the width of the solid bed can be described by:<br>
+ (7.154)<br>
+<hr>
+<A name=335></a> 7.3Plasticating  <b>323</b><br>
+With Eq. 7.154 the solid bed profile can be written as:<br>
+ (7.155)<br>
+The length required to complete melting z0 can be determined by setting the solid <br>
+bed width equal to zero. This results in the following expression for the melting <br>length:<br>
+ (7.156)<br>
+When the flight clearance is taken as zero the melting length becomes:<br>
+ (7.157)<br>
+The shortest melting length occurs when a = c2W1. In this case, the melting <br>
+length becomes z0 = H1/Az. From Eq. 7.155 it can be seen that in this situation the <br>
+solid bed width becomes independent of down-channel distance z. In other words, <br>the solid bed width remains constant. Clearly, this minimum length cannot be <br>achieved in practice because there will be no room for the polymer melt to accumu-<br>late.<br>With these equations the melting process of a temperature-dependent power law <br>fluid can be completely described. From Eq. 7.151 it can be seen that the melting <br>rate decreases with increasing flight clearance. Figure 7.41 shows how the melting <br>rate is affected by the flight clearance.<br>
+<b> Figure 7.41</b> <br>
+Melting rate versus flight <br>
+clearance<br>
+<hr>
+<A name=336></a><b>324</b>  <b>7Functional Process Analysis</b><br>
+These results are for a 63-mm extruder running a 0.2 melt index HDPE at a screw <br>speed of 60 rpm. It is clear that increased flight clearance significantly reduces <br>melting performance. It is important, therefore, to keep the flight clearance small in <br>the melting zone of the extruder.<br>The effect of barrel temperature on melting performance can be predicted in a quan-<br>titative way. Figure 7.42 shows how the melting rate changes with barrel tempera-<br>ture for two values of the Nahme number as expressed by Eq. 7.143. Low Nahme <br>numbers indicate little viscous dissipation, while high Nahme numbers correspond <br>to high levels of viscous dissipation.<br>
+<b> Figure 7.42</b> <br>
+Melting rate versus <br>
+ <br>barrel  temperature <br>
+(m = 0 and n = 0 .5)<br>
+At low Nahme numbers the melting rate increases with barrel temperature. How-<br>ever, at high values of the Nahme number the melting rate reduces with increasing <br>barrel temperature. There is a critical Nahme number above which increasing barrel <br>temperature results in reduced melting rate. At a power law index of n = 0.5 the <br>critical Nahme number is about 4.5. The critical Nahme number increases with the <br>power law index as shown in Fig. 7.43.<br>
+<b> Figure 7.43</b> <br>
+Critical Nahme number versus power law <br>
+index<br>
+<hr>
+<A name=337></a> 7.3Plasticating  <b>325</b><br>
+The temperature coefficient of the melt viscosity has a strong effect on the tempera-<br>ture profile in the melt film and, thus, on the melting rate. The melting rate reduces <br>when the temperature coefficient increases. This is shown in Fig. 7.44.<br>It is clear from Fig. 7.44 that predictions of melting performance based on an analy-<br>sis of a temperature-independent fluid will significantly overestimate the melting <br>performance. As a result, such predictions should be treated with caution.<br>
+]<br>.m<br>
+.<br>in]<br>
+Melting rate [kg/s<br>
+Melting rate [lbs/s<br>
+Temperature coefficient [C-1]<br>
+<b>Figure 7.44 </b>l i<br>
+Melting rate ver h<br>
+sus the temper<br>
+ffi<br>
+atur i<br>
+e coefficient<br>
+In screw design it is important to make sure that the compression in the transition <br>section of the screw is gentle enough to avoid plugging. This is a situation where the <br>melting cannot keep up with the channel compression, resulting in an increase of <br>the width of the solid bed. When this happens the solid bed will get stuck in the <br>channel and severe instabilities can occur. From Eq. 7.155 it can be determined that <br>plugging can be avoided when the following inequality is satisfied:<br>
+ (7.158)<br>
+If the length of the transition section is Lc, plugging can be avoided when:<br>
+ (7.159)<br>
+where Xc is the compression ratio of the screw.<br>From Eq. 7.159 it can be seen that the length of the transition section has to increase <br>with the channel depth of the feed section H1, with the compression ratio Xc, with <br>
+the throughput svszH1W1, and with the helix angle . The maximum possible com-<br>
+pression ratio is directly determined by Lc. A short transition section requires a low <br>
+compression ratio to avoid plugging.<br>
+<hr>
+<A name=338></a><IMG src="Polymer Extrusion by Chris Rauwendaal-338_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-338_13.jpg"><br>
+<b>326</b>  <b>7Functional Process Analysis</b><br>
+<b>7.3.2Other Melting Models</b><br>
+After Tadmor's first publication on melting, many others started to study melting in <br>single screw extruders. As a result, many publications appeared in the technical <br>literature in the 1970s and beyond. This section will review the various melting <br>models proposed and analyze their advantages and disadvantages.<br>Many workers have repeated the screw extraction experiments as done by Maddock <br>and later by Tadmor and coworkers; the majority of the workers confirmed the basic <br>Tadmor melting model. Two exceptions are the finds of Klenk [66­68] and Dekker <br>[69]. Klenk observed the melting of PVC and noticed the melt pool located against <br>the passive flight flank. This melting behavior was also observed by Gale [81] and <br>Mennig [82] in PVC extrusion. Klenk attributed the unusual melting behavior to the <br>slippage of the PVC melt along the wall. It should be mentioned that other workers <br>have found the normal melting model with PVC [61, 70]. Yet other workers [71] have <br>found the melt pool location shifting from the passive flight to the active flight <br>with increasing screw speed in PVC extrusion. Cox, et al. [80] observed the unusual <br>melting behavior with the melt pool at the passive flight in extrusion of LDPE <br> <br>powder as well as the normal melting behavior. The unusual behavior occurred par-<br>ticularly at low screw speeds, with shallow channels, and with high barrel tempera-<br>tures. Thus, the Tadmor melting model may not be valid in all cases.<br>Dekker [69] made observations of the melting behavior of polypropylene. He did not <br>detect a clear melt pool at any side of the channel, but the solid bed was more or less <br>suspended in a melt film. Lindt [72, 73] developed a mathematical model describing <br>this melting behavior, which is shown in Fig. 7.45.<br>
+Upper melt film<br>
+Barrel surface<br>
+y<br>
+z<br>
+x<br>
+Lower melt film<br>
+Solid bed<br>
+Transport direction<br>
+<b>Figure 7.45</b>The Dekker/Lindt melting model<br>
+Some of the important assumptions in this model are:<br>1.  The solid bed does not deform.<br>2.  The solid bed is completely surrounded by a melt film and melting occurs at the <br>
+entire circumference of the solid bed.<br>
+3.  The melt film is constant in the cross-channel direction.<br>
+<hr>
+<A name=339></a> 7.3Plasticating  <b>327</b><br>
+Assumption 3 causes some difficulties because a constant melt film thickness requires <br>the presence of excessively large pressure gradients to accommodate the increased <br>amount of polymer melt. This conflicts directly with Assumption 1. A detailed analy-<br>sis of the Lindt melting model was made by Gieskes [74] and Meijer [75]. It appears <br>that the Lindt melting model is unlikely to occur in the early stages of melting, but <br>it may possibly occur in the later stages of melting.<br>A melting model in which the solid bed is considered to be subjected to a limited <br>amount of deformation was proposed by Edmondson and Fenner [76­79]. However, <br>their analysis contains certain inconsistencies as pointed out by Meijer [75]. Dono-<br>van [83, 84] extended the Tadmor analysis by removing the assumption of constant <br>solid bed velocity and by incorporating the gradual heating of the solid bed along <br>the melting zone. As mentioned earlier, Shapiro [63] and Vermeulen [64, 65] <br>included the increasing melt film thickness with cross-channel distance. Shapiro, <br>Pearson, and coworkers have developed one of the most elaborate extensions of <br>the Tadmor analysis [63, 85­87]. Their model is a five-zone model as shown in <br>Fig. 7.46.<br>The assumptions are more realistic and broader than most other analyses of melt-<br>ing. This can improve the accuracy of the model, but, unfortunately, it also increases <br>the complexity of the computations quite substantially.<br>
+Barrel<br>
+Zone 3<br>
+4<br>
+Zone 2<br>
+Zone 1<br>
+Zone <br>
+Zone 5<br>
+Screw<br>
+<b>Figure 7.46</b>Illustration of the five-zone melting model<br>
+Hinrichs and Lilleleht [88] included the effect of flight clearance and used a helical <br>coordinate system to account for the channel curvature. Sundstrom and Young [89] <br>examined the effect of convective heat transfer and found that it can play an impor-<br>tant role in the melting process, resulting in a melting rate about 20% higher than <br>the predictions based on conductive heat transfer only. Sundstrom and Lo [89a] <br>studied melting of amorphous polymers. They assumed the polymer-melt interface <br>to be at the glass transition temperature and used a modified WLF equation to deter-<br>mine the shift factor for the temperature dependence of the viscosity. Chung [90, <br>91] analyzed the effect of finite solid bed thickness, varying solid bed density, and <br>varying solid bed velocity. Later Chung developed a screw simulator to study melt-<br>ing behavior of polymers. A substantial amount of experimental and theoretical <br>work [94­97] on melting was done using the screw simulator.<br>
+<hr>
+<A name=340></a><b>328</b>  <b>7Functional Process Analysis</b><br>
+An interesting approach to the analysis of melting in single screw extruders was <br>taken by Viriyayuthakorn and Kassahun [234]. They developed a new plasticating <br>extrusion analysis program based on a finite element method capable of simulating <br>three-dimensional flow with phase change. This program contains a number of new <br>features that had not been incorporated in earlier work (prior to 1984) on melting in <br>single screw extruders. The solid-melt phase change in this program was handled <br>differently from prior work on melting. The conventional approach is to determine <br>the location of the interface and use an energy balance of the interface to calculate <br>the melting rate, as exemplified by Eq. 7.100. This method can introduce errors if <br>the location of the interface is not accurately known and if the material does not <br>have a sharp melting point, as is the case for most polymers. The technique used by <br>Viriyayuthakorn and Kassahun is to absorb the latent heat of fusion term in the <br>specific heat capacity and formulate the problem as though there is no phase <br>change. Thus, a functional dependence of the specific heat capacity on temperature <br>is used. This eliminates the need to first assume a certain melting model. The melt-<br>ing model is actually predicted from the calculations, something earlier workers <br>have not been able to do.<br>Results from computer simulations for a HDPE polymer show that the Maddock/<br>Tadmor melting mechanism occurs in the early stages of melting. At the beginning <br>of the compression section, the solid bed tends to become totally encapsulated, while <br>towards the end of the melting, the solid bed breaks apart into several pieces. At <br>the time of publication of the paper by Viriyayuthakorn and Kassahun, no direct <br>comparison was available between theoretical predictions and experimental results. <br>Therefore, no statement can be made about the accuracy of the predictions. However, <br>regardless of the accuracy of the predictions, this model provides new capabilities <br>that no doubt will prove very useful in future work on the analysis of plasticating <br>extrusion.<br>A drawback of the program is the need for very large computational capability. The <br>simulations were performed on a Cray-1 computer, a computer not readily available <br>to most process engineers. Even on this extremely powerful computer, one simula-<br>tion took as long as several hours. However, considering the dramatic improvement <br>in the number-crunching capability of new computers, the limitations in computa-<br>tional capabilities of computers in 2013 are less of a concern than they were in 1984 <br>when Viriyayuthakorn and Kassahun presented their paper.<br>Chung and coworkers have developed simple analytical expressions to predict the <br>melting behavior of polymers [95]. They developed analytical expressions valid for <br>non-Newtonian fluids with temperature-dependent viscosity, following an approach <br>very similar to Pearson's [87].<br>The melting rate per unit area is:<br>
+ (7.160)<br>
+<hr>
+<A name=341></a> 7.3Plasticating  <b>329</b><br>
+where M0p is the dimensionless melting efficiency; it represents the melting capa-<br>
+city per unit melting area per unit sliding distance. Various functional forms of  0p <br>
+are given in [95]. A simple expression that yields accurate results is:<br>
+ (7.161)<br>
+where K2 is defined in Eq. 7.121 and F1(K2) is:<br>
+ (7.162)<br>
+By curve fitting, the function F1(K2) can be approximated by:<br>
+ (7.163)<br>
+If the melt viscosity is temperature independent, K2 = 0 and F1(K2) = 1. By inserting <br>
+Eq. 7.161 in Eq. 7.160, the melting rate per unit melt area becomes:<br>
+ (7.164)<br>
+The melting rate per unit length normal to the sliding direction is obtained by multi-<br>plying Eq. 7.164 with Ws:<br>
+ (7.165)<br>
+Equation 7.165 is comparable to Eq. 7.131; both equations are closed-form analy-<br>tical solutions. However, Eq. 7.165 is more compact and easier to use. A word of <br>caution is in order. Equation 7.165 has been experimentally verified with the screw <br>simulator. The results of the screw simulator may not fully apply to actual melting in <br>a single screw extruder. For instance, the screw simulator uses a molded solid poly-<br>mer sample of one cubic inch. The solid bed in the extruder consists of compressed, <br>partially sintered, polymeric particles. It is clear that an actual solid bed, as occurs <br>in the melting zone of an extruder, may have different characteristics in terms of <br>heat transfer properties and deformation behavior, as compared to a molded solid <br>block of the same material.<br>Equation 7.165 can be written in terms of melting rate per unit down-channel by <br>making the same coordinate transformation as discussed earlier in Eq. 7.109:<br>
+ (7.166)<br>
+<hr>
+<A name=342></a><b>330</b>  <b>7Functional Process Analysis</b><br>
+<b>7.3.3Power Consumption in the Melting Zone</b><br>
+The mechanical power consumption in the melting zone can be determined by <br>breaking down the power consumption in three parts: the power consumed in the <br>melt film dZmf, the power consumed in the melt pool dZmp, and the power consumed <br>
+in the clearance between flight and barrel dZcl. The power consumed in shearing the <br>
+melt film is described by:<br>
+ (7.167)<br>
+where yx is the shear stress in the direction x of the relative velocity v between <br>
+the solid bed and the barrel; see also Fig. 7.23(a). If the material can be described as <br>a power law fluid, the shear stresses can be written as:<br>
+ (7.168)<br>
+and:<br>
+ (7.169)<br>
+It is assumed that the flow in the melt film is a drag flow, i.e., pressure gradients are <br>neglected in this case:<br>
+ (7.170)<br>
+and:<br>
+ (7.171)<br>
+The melt film thickness can be written according to Eq. 7.130:<br>
+ (7.172)<br>
+where:<br>
+ (7.173)<br>
+Factor B3 is given by Eq. 7.128. The power consumption in the melt film can now be <br>
+expressed as:<br>
+ (7.174)<br>
+<hr>
+<A name=343></a> 7.3Plasticating  <b>331</b><br>
+By using Eq. 7.172, the integral can be rewritten as:<br>
+ (7.175)<br>
+where Hms = Hm(Ws), which is the maximum melt film thickness. The solution to <br>
+Eq. 7.175 can be written as:<br>
+ (7.176)<br>
+With z = z sin( + ), the power consumption in the melt film can be written as:<br>
+ (7.177)<br>
+From Eqs. 7.176 and 7.177, it can be seen clearly that the power consumption in the <br>melt film reduces with increasing clearance.<br>The power consumption in the melt pool will be relatively small compared to the <br>other two terms. Because of the relatively complicated flow pattern in the melt pool, <br>the derivation of the power consumption for a power law fluid will be rather involved. <br>To simplify matters considerably, the power consumption in the melt pool can be <br>approximated with the power consumption in a screw channel of width Wm = W--Ws <br>
+with a Newtonian fluid:<br>
+ (7.178)<br>
+The derivation of this expression will be discussed in Section 7.4.1.3 on melt con-<br>veying.<br>The power consumption in the clearance can be easily determined if it is assumed <br>that the velocity profile in the clearance is dominated by drag flow. In that case, the <br>power consumption in the clearance can be written as:<br>
+ (7.179)<br>
+The total mechanical power consumption in the melting zone now becomes:<br>
+ (7.180)<br>
+<hr>
+<A name=344></a><b>332</b>  <b>7Functional Process Analysis</b><br>
+<b>7.3.4Computer Simulation</b><br>
+One of the general problems of the melting theories is that the most realistic models <br>are also the most complex. However, if the complexity goes beyond the level of the <br>equations developed earlier, analytical solutions become very difficult to obtain, if <br>not impossible. In this case, one has to use numerical techniques and computer <br>simulation to find solutions to the equations. The main question is whether the <br>more complex analysis will result in improved predictive ability. Relaxing certain <br>assumptions may result in an improvement in accuracy that is relatively insignifi-<br>cant compared to the inherent uncertainties in most melting models. These are, <br>among others, the actual location of the solid bed and melt pool, the shear strength <br>and tensile strength of the solid bed as a function of time, temperature, and pres-<br>sure, the actual melt film temperature, the actual temperature at the screw surface, <br>the contribution of melting at the sides of the solid bed and at the screw surface, etc. <br>The actual temperature in the melt film is most likely not fully developed, and con-<br>vective heat transport should be considered in determining the actual temperature <br>profile.<br>Therefore, one has to strike a balance between the degree of sophistication of the <br>analysis and the practical usefulness of the analysis. This balance will depend on <br>the particular interests of the individual. For industrial applications the degree of <br>sophistication of the analysis of melting, as described in Section 7.3.1, is probably <br>sufficient to analyze most practical extrusion problems. However, in some instances <br>one may want to go into much more detail on certain aspects of the melting process.<br>Analyses that require numerical techniques to arrive at solutions tend to be quite <br>time consuming and require skilled personnel to develop the computer programs <br>and to interpret the results of the computer simulations. Many people in the extru-<br>sion industry do not have the time or inclination to work through elaborate and com-<br>plex analyses of melting. In this case, the preferred action is to use a less complicated <br>analysis that yields analytical results. In most cases, actual predictions of melting <br>performance can be made with a relatively simple programmable calculator.<br>Another approach is to use a computer program developed elsewhere to analyze the <br>problem. Nowadays many commercial packages are available to simulate the plasti-<br>cating extrusion process. In most cases this offers the most expedient approach to <br>advanced analysis of melting. The danger is that if a person using such program is <br>not intimately familiar with the theory behind the program, the assumptions, and <br>the validity of the assumptions, it is possible that improper conclusions will be <br>drawn from the predicted results.<br>When using commercial simulation packages it is important that the theory behind <br>it is clearly described with assumptions and simplifications. If this is not the case <br>and the user does not really know what type of analysis they are actually using, the <br>value of the results will be questionable. Early extruder simulation packages had <br>
+<hr>
+<A name=345></a><IMG src="Polymer Extrusion by Chris Rauwendaal-345_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-345_21.jpg"><br>
+ 7.3Plasticating  <b>333</b><br>
+some serious deficiencies. In some cases suppliers of simulation software programs <br>made claims that simply were not correct. As a result, computer simulation in extru-<br>sion had a less than perfect reputation. It has taken many years for computer simu-<br>lation in extrusion to regain respectability. Modeling and computer simulation will <br>be covered in detail in Chapter 12.<br>
+<b>7.3.5Dispersed Solids Melting</b><br>
+As mentioned earlier, the CSM type melting is the one that has been most often <br>observed in single screw extruders. However, in some experimental studies it was <br>found that the solid particles are not contiguous but dispersed in a melt matrix <br>[256­258] as illustrated in Fig. 7.47.<br>
+<b> Figure 7.47</b> <br>
+Dispersed solids melting (DSM)<br>
+In fact, in several extruders, dispersed solids melting (DS) is more likely to occur <br>than contiguous solids melting. This is particularly true of extruders that do not <br>feature a continuous, uninterrupted channel for material flow. Examples are twin <br>screw extruders and pin barrel extruders.<br>CSM is more likely to occur in single screw extruders where the material is con-<br>veyed along a continuous screw channel. However, even in a single screw, DSM can <br>occur. In fact, certain screw geometries have been developed with the specific pur-<br>pose to initiate the dispersed solids melting process. Examples are the double wave <br>screw [254] and the HP screw [255]. It has been found experimentally that the DSM <br>process can produce more efficient melting than the CSM process. Despite the <br> <br>recognized importance of DSM in extrusion, very few theoretical analyses have been <br>devoted to this type of melting. Yucheng and Hanxiong [255] performed a theore-<br>tical analysis of DSM using a six-block model. The non-isothermal, non-Newtonian <br>model requires numerical techniques to obtain solutions. Unfortunately, no details <br>were presented on the theoretical description of the model. Some of the important <br>conclusions by Yucheng and Hanxiong are:<br>1.  DSM can reduce power consumed by the screw by as much as 30%<br>2.  The temperature is more uniform and lower<br>3.  DSM can shorten the total melting length<br>
+<hr>
+<A name=346></a><b>334</b>  <b>7Functional Process Analysis</b><br>
+Rauwendaal [259] presented the first theoretical model of DSM that allows analyti-<br>cal description of the melting process. In the following, this theory will be described <br>in detail. Predictions from the DSM theory will be compared to the CSM theory.<br>
+7.3.5.1.1.1Dispersed Solids Melting Theory<br>In the DSM model it will be assumed that the solid particles are uniform, spherical, <br>and dispersed in a melt matrix. This means a minimum volume of polymer melt has <br>to be available to fill the space between the solid particles. For the closest packing of <br>regular spheres the minimum polymer melt volume fraction is about 40%. For more <br>random packing configurations it will be closer to 50%. This means that the DSM <br>theory can only be applied after about half the solid material has already melted.<br>It is further assumed that the melting of solid particles is uniform, i.e., not dependent <br>on the location of the particle in the channel. The heat available for melting will be <br>determined by the net heat conducted into the channel (through barrel and screw) <br>and the viscous heat generation in the polymer melt. The viscous heat generation <br>is determined by using the dissipation model for filled polymers developed by <br> <br>Geisbüsch [260]. The system is considered a two-phase polymer system with the <br>shearing taking place in the polymer melt matrix. The viscosity of such a two-phase <br>system as determined by theological measurements yields "integral" values of the <br>viscosity, i, as a function of an "integral" shear rate,  i. The viscous heat generation <br>
+is determined by the product of shear stress and shear rate. In a two-phase system <br>the viscous dissipation has to be corrected by a factor Fd. This factor, according to <br>
+Limper [261], is given by:<br>
+ (7.181)<br>
+where  is the solids volume fraction.<br>The correction factor K according to Neumann [262] can be written as:<br>
+ (7.182)<br>
+The expression for Fd can thus be written as:<br>
+ (7.183)<br>
+where a = 6/ and b = 0.5(4/)1.5.<br>Function Fd(M) is graphically represented in Fig. 7.48.<br>The viscous heat generation per unit volume for a power law fluid can be written as:<br>
+ (7.184)<br>
+<hr>
+<A name=347></a> 7.3Plasticating  <b>335</b><br>
+<b>Figure 7.48</b>Correction factor Fd versus the solids volume fraction<br>
+At this point the integral viscosity has to be expressed as a function of the volume <br>fraction solid. A number of relationships have been proposed to describe the increase <br>in viscosity with volume fraction solid. Good reviews are incorporated in the two-<br>volume book on polymer blends by Paul and Newman [263]; some other references <br>are 264 through 267. A useful expression is the Maron-Pierce relationship, which <br>for a power law fluid results in the following expression for the consistency index:<br>
+ (7.185)<br>
+In this expression m0 is the consistency index of the unfilled polymer melt and max <br>
+the solids volume fraction at close packing. A typical value of max is about 0.6 for <br>
+spherical particles. Non-spherical particles usually have lower maximum solid frac-<br>tion. Figure 7.49 shows the consistency index ratio mi()/m0 as a function of the <br>
+solids volume fraction as expressed in Eq. 7.185.<br>The heat required to raise the temperature of the solids to the melting point and to <br>melt the solids comes from the viscous dissipation and the net heat conducted into <br>the polymer from the barrel and the screw. It is assumed that there is a gradual and <br>uniform reduction in the solid particles, thus reducing the solids volume fraction . <br>The change in the solids fraction over an increment of time can be related to the <br>heat added to the polymer in the same time increment. Thus, the energy balance <br>can be written as:<br>
+ (7.186)<br>
+where Êp is the enthalpy difference between the initial solids temperature and the <br>
+melting point.<br>
+<hr>
+<A name=348></a><b>336</b>  <b>7Functional Process Analysis</b><br>
+<b>Figure 7.49</b>Consistency index ratio versus solids volume fraction<br>
+The net heat flux per unit depth of the channel conducted into the screw channel is <br>obtained from:<br>
+ (7.187)<br>
+where the subscript r refers to the screw and b to the barrel.<br>At this point we will assume that the conductive heating term is negligible com-<br>pared to the viscous heating term (qc &lt;&lt; qv). This is a reasonable assumption for <br>
+high viscosity polymers, particularly when the extruder is operated at high screw <br>speed. With both Fd and m functions of the solids fraction, the melting time tp is <br>
+determined by integrating the -terms in Eq. 7.186 from 1 to 0 and the time part <br>
+of the equation from 0 to tp. Thus, we obtain the following equation:<br>
+ (7.188)<br>
+where 1 is the solids volume fraction at the beginning of the dispersed solids melt-<br>
+ing process.<br>The left-hand integral of Eq. 7.188 is a function of the initial solids volume fraction; <br>we will write it as I(1). Solution of the integral yields a closed-form, but lengthy <br>
+expression:<br>
+<hr>
+<A name=349></a> 7.3Plasticating  <b>337</b><br>
+ (7.189)<br>
+where c = max.<br>Figure 7.50 shows how I(1) depends on 1.<br>
+0.2<br>
+0.18<br>
+max = 0.6<br>
+<br>
+=0.6<br>
+max<br>
+0.16<br>
+max = 0.5<br>
+=0.5<br>
+max<br>
+0.14<br>
+<br>
+<br>
+=0.4<br>
+)<br>
+max = 0.4<br>
+max<br>
+1<br>) 0.12<br>
+F  1<br>
+0.1<br>
+0.08<br>
+Integral I( Integral I(<br>
+0.06<br>
+0.04<br>
+0.02<br>
+0<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+Initial solids volume fraction <br>
+Initial solids v<br>
+F<br>
+olume fraction <br>
+1<br>
+1<br>
+<b>Figure 7.50</b>Integral I(1) versus solids volume fraction<br>
+If the integral shear rate can be taken as the ratio of the barrel velocity and the <br>channel depth, the melting time can be expressed as:<br>
+ (7.190)<br>
+where it is assumed that the channel depth is constant.<br>The melting time values for DSM from Eq. 7.190 can be compared to the melting <br>time for CSM. The melting time in the CSM model to go from a solids fraction of 1 <br>
+to zero can be expressed as [259]:<br>
+ (7.191)<br>
+where:<br>
+<hr>
+<A name=350></a><b>338</b>  <b>7Functional Process Analysis</b><br>
+The axial melting length can be determined by relating the melting time to the aver-<br>age velocity of the polymer. This leads to the following expression for the axial melt-<br>ing length:<br>
+ (7.192)<br>
+With Eq. 7.192 the melting length in DSM can be compared to the melting length in <br>CSM, which can be written as [259]:<br>
+ (7.193)<br>
+7.3.5.1.1.2Comparison of CSM to DSM Model<br>At this point we can compare predictions from the CSM model to the DSM model. <br>First, the effect of power law index on melting time is shown in Fig. 7.51. The data <br>are for a 50 mm extruder running at 60 rpm with an output of 1.0E-5 m3/s. The <br>material data is typical for LDPE, melt density 780 kg/m3, solid density 920 kg/m3, <br>Êp is 3.0E5 Nm/kg, and the consistency index is m = 20,000 Pasn. The melting <br>
+times are compared based on an initial solids fraction of 0.5. For all the values of the <br>power law index the DSM time is significantly shorter than the CSM time; the differ-<br>ence becomes larger with smaller values of the power law index.<br>
+<b> Figure 7.51</b> <br>
+Melting time versus power law index for DSM and CSM<br>
+The effect of the consistency index on the melting time is shown in Fig. 7.52; the <br>power law index is 0.5. In this case, again, the DSM time is considerably shorter <br>than the CSM time. The effect of screw speed on melting time is shown in Fig. 7.53. <br>The melting time reduces rapidly with screw speed. Doubling the screw speed <br>reduces the melting time by a factor of almost three. Again, the DSM time is much <br>shorter than the CSM time. Thus, over a wide range of values for the screw speed, <br>power law index, and consistency index, the predicted melting efficiency with the <br>DSM model is consistently better than with the CSM model.<br>
+<hr>
+<A name=351></a> 7.3Plasticating  <b>339</b><br>
+<b> Figure 7.52</b> <br>
+Melting time versus consistency index for DSM <br>
+and CSM<br>
+<b> Figure 7.53</b> <br>
+Melting time versus screw speed for DSM and <br>
+CSM<br>
+7.3.5.1.1.3DSM versus CSM Type Melting<br>This theoretical model for dispersed solids melting allows analytical solutions for <br>a temperature-independent power law fluid. The model presented is based on the <br>assumption that the conductive heating term is negligible, the channel depth con-<br>stant, and the pellet size is uniform at a particular cross-section of the channel. <br>Obviously, for analysis of more realistic cases the assumptions will have to be <br>relaxed. When comparing the melting efficiency of the DSM model to the CSM <br>model, dispersed solids melting is considerably more efficient than contiguous solids <br>melting. This is true over a wide range of values of the consistency index, power law <br>index, and screw speed.<br>These results suggest that solid bed breakup is not necessarily bad for the melting <br>efficiency. However, the solid bed breakup would have to be complete, i.e., to the <br>level of the individual pellets making up the solid bed. For single screw extruders it <br>can be advantageous to modify the screw channel geo metry in the plasticating zone <br>to intentionally induce solid bed breakup. This approach has already been taken in <br>some instances [254, 255]. A new dispersive mixing device developed by Rauwendaal <br>[268, 269] has been successfully used to disperse unmelted polymer fragments (see <br>Section 8.7.1.2).<br>In single screw extrusion there is a commonly held belief that solid bed breakup <br>should be avoided if at all possible. Most barrier extruder screws are based on this <br>premise. They are designed to minimize the chance of solid bed breakup. The type <br>
+<hr>
+<A name=352></a><b>340</b>  <b>7Functional Process Analysis</b><br>
+of solid bed breakup that can be detrimental to extruder performance is partial solid <br>bed breakup. In this case the solid bed breaks up in large chunks, much larger than <br>the size of the individual pellet. It is this partial solid bed breakup that barrier <br>screws attempt to avoid. However, DSM type melting can substantially improve melt-<br>ing in a single screw extruder. If a consistent DSM can be achieved in single screw <br>extruders it will enable a quantum jump improvement in the performance of these <br>machines.<br>More work is necessary on dispersed solids melting, both theoretical and experi-<br>mental. Certainly the possibility to enhance melting efficiency by inducing solid bed <br>breakup in single screw extruders is potentially very attractive. The challenge will <br>be to initiate complete solid bed breakup without creating partial solid bed breakup <br>in single screw extruders. The melting process is often the rate-determining part <br>of the extrusion process. As a result, improved melting will allow corresponding <br>improvements in throughput.<br>There are probably a number of factors that will have to be combined to generate <br>DSM in single screw extruders. The pressure build-up in the solids will have to be <br>controlled and kept to a relatively low level to avoid making the solid bed difficult to <br>break up. As discussed in Section 7.2.2.4, starve feeding is an effective method of <br>controlling the pressure build-up in the solids. As a result, starve feeding is probably <br>a necessary ingredient to generate DSM in single screw extruders. It is probably <br>very difficult to generate DSM in a flood fed grooved feed extruder because of the <br>high pressure generated in the solids conveying zone.<br>The second ingredient is a screw geo metry that can break down agglomerated solids. <br>This will require special screw geometries. At this point in time, only little infor-<br>mation is available on screw geometries that can achieve efficient dispersion of <br>unmelted polymer (see Section 8.7.1.2). A third ingredient is a sufficient length of <br>the machine to allow for partially filled regions along the extruder. Most likely DSM <br>type single screw extruders will have to be at least 30 D long, and perhaps longer <br>than that.<br>
+<b></b><br>
+ <b>7.4Melt Conveying</b><br>
+The melt conveying zone of the extruder starts at the point where the melting <br> <br>process is just completed. The melt conveying zone is also referred to as a pumping <br>zone because, in most cases, the polymer melt has to be transported towards the die <br>against a considerable head pressure. The melt conveying in an extruder was the <br>subject of engineering studies as early as 1922 [98]. The early work on melt con-<br>veying in extruders dealt with Newtonian fluids with temperature-independent vis-<br>
+<hr>
+<A name=353></a> <br>
+7.4Melt Conveying  <b>341</b><br>
+cosity. This is a very convenient case to analyze because it is simple and yields <br>clean, straightforward analytical solutions. The reason is that the cross-channel flow <br>can be analyzed independent of the down-channel flow for a Newtonian fluid with <br>temperature-independent viscosity. Also, the pressure flow can be analyzed sepa-<br>rate from the drag flow and the results can be superimposed.<br>These simplifications cannot be made when the fluid is non-Newtonian or when the <br>viscosity is temperature dependent. Most analyses on melt conveying can be catego-<br>rized by the six most important assumptions made about the process:<br>1.  Shear stress-shear rate relationship<br>
+a)  Linear (Newtonian)<br>b)  Non-linear (non-Newtonian)<br>
+2.  Flow situation<br>
+a)  One-dimensional (down-channel only)<br>b)  Two-dimensional (down- and cross-channel)<br>c)  Three-dimensional (down-, cross-channel, and radial)<br>
+3.  Effect of the flight flanks<br>
+a)  Negligible (infinite channel width)<br>b)  Not negligible (finite channel width)<br>
+4.  Temperature effects<br>
+a)  Melt viscosity temperature independent<br>b)  Melt viscosity temperature dependent<br>
+5.  Flight clearance<br>
+a)  Negligible (zero flight clearance)<br>b)  Not negligible (finite flight clearance)<br>
+6.  Channel curvature<br>
+a)  Negligible (flat place approximation)<br>b)  Not negligible (cylindrical or helical analysis)<br>
+The infinite channel width assumption applies to shallow channels, channels with a <br>width-to-depth ratio higher than 10 (W/H &gt; 10). If the depth of the channel is large <br>relative to the width of the channel, the effect of the flight flanks on the down-chan-<br>nel velocity profile has to be taken into account. Several reviews of the work on melt <br>conveying in extruders have been written [101­106].<br>In addition to the six main assumptions of the analysis of melt conveying, there are <br>a few more that are sometimes considered, such as elastic effects, the influence of <br>an oblique channel end, etc. After the Newtonian analysis had been mostly worked <br>out, non-Newtonian fluids were analyzed in the early 1960s. This adds significantly <br>to the complexity of the analysis, and generally the equations cannot be solved ana-<br>
+<hr>
+<A name=354></a><b>342</b>  <b>7Functional Process Analysis</b><br>
+lytically, but have to be solved by numerical techniques. The simplest non-New-<br>tonian case that can be considered is the one-dimensional, isothermal flow of the <br>power law fluid in a channel of infinite width. This problem has been studied by <br>various investigators [107­113], but a closed-form solution was not obtained until <br>1983 [114]. An exact analytical solution to even this simple problem has not been <br>found to this date.<br>Relaxing other assumptions, i.e., temperature-dependent viscosity, two- or three-<br>dimensional flow, finite channel width, etc., substantially increases the degree of <br>difficulty. Such analyses have been the subject of several Ph.D. studies [102, <br>115­121]. The level of complexity can be increased almost at will by further relax-<br>ing the assumptions. However, this work can reach such a high level of complexity <br>and sophistication that the usefulness to practicing polymer processing engineers <br>becomes questionable. One should always try to balance the increased generality of <br>the predictions against the time and effort spent on solving increasingly more com-<br>plex problems. Somewhere, one can reach a point of diminishing return.<br>To the practicing process engineer, the most important question should be whether <br>the more general analysis will be applicable; e.g., are all the boundary conditions <br>well known, and will it result in more accurate predictions. For instance, it probably <br>does not make much sense to use a sophisticated non-isothermal melt conveying <br>analysis to predict the melt temperature at the end of the screw if the actual <br>screw temperature in the process is unknown. These practical considerations do not <br>necessarily apply to the academic. On the other hand, it would make sense to per-<br>form a non-isothermal analysis to predict the effect of barrel temperature fluctua-<br>tions on melt temperature or conveying rate. Besides the complications already <br> <br>discussed, there is the additional complication that the polymer melt is not a pure, <br>inelastic power law fluid and significant time-dependent effects can occur (e.g., <br>[122­128]). <br>The next section will start with an analysis of melt conveying of isothermal fluids. <br>This will be followed by a non-isothermal analysis of melt conveying of cases that <br>allow exact analytical solutions. More general analyses of the effect of tempe rature on <br>flow will be discussed in more detail in Chapter 12 on modeling and  computer simu-<br>lation. In the next section, melt conveying of Newtonian fluids and non-Newtonian <br>fluids will be analyzed. The non-Newtonian fluids will be described with the power <br>law equation (Eq. 6.23). The effect of the flight flank will be  discussed and the differ-<br>ence between one- and two-dimensional analysis will be demonstrated with parti-<br>cular emphasis on the implications for actual extruder performance.<br>
+<hr>
+<A name=355></a> <br>
+7.4Melt Conveying  <b>343</b><br>
+<b>7.4.1Newtonian Fluids</b><br>
+A very simple analysis of melt conveying can be made if the following assumptions <br>are made:<br>1.  The fluid is Newtonian<br>2.  The flow is steady<br>3.  The viscosity is temperature-independent<br>4.  There is no slip at the wall<br>5.  Body and inertia forces are negligible<br>6.  The channel width can be considered infinite<br>7.  The channel curvature is negligible (flat plate model)<br>The geo metry is now simplified to Fig. 7.54.<br>
+W<br>
+vbx<br>
+<br>
+. y<br>
+vbz<br>
+x<br>
+vb<br>
+z<br>
+<br>
+H<br>
+.vbz<br>
+vbx y .z<br>
+x<br>
+W<br>
+<b> Figure 7.54</b> <br>
+w<br>
+Flat plate model for melt conveying<br>
+A flat plate is moving at velocity vb over a flat rectangular channel with angle  <br>
+between vb and the flights of the channel.<br>The equation of motion in the down-channel direction for this problem can be writ-<br>ten as:<br>
+ (7.194)<br>
+The pressure is a function of down-channel coordinate z only. Therefore, Eq. 7.194 <br>can be integrated to give the shear stress profile in the radial direction (y):<br>
+ (7.195)<br>
+where o is the shear stress at the screw surface, as yet unknown.<br>
+<hr>
+<A name=356></a><b>344</b>  <b>7Functional Process Analysis</b><br>
+For a Newtonian fluid, Eq. 7.195 can be written as:<br>
+ (7.196)<br>
+By integration of Eq. 7.196, the down-channel velocity as a function of normal dis-<br>tance y is obtained. By using boundary conditions vz(0) = 0 and vz(H) = vbz, the fol-<br>
+lowing expression is obtained:<br>
+ (7.197)<br>
+The flow rate in the down-channel direction is obtained by integrating the down-<br>channel velocity over the cross-sectional area of the screw channel:<br>
+ (7.198)<br>
+where p is the number of parallel flights and W the perpendicular channel width.<br>This represents the volumetric throughput of the melt conveying zone. The first <br>term after the equal sign in Eq. 7.198 is the drag flow term. It represents the flow <br>rate in pure drag flow, i.e., without a pressure gradient in the down-channel direc-<br>tion:<br>
+ (7.199)<br>
+The second term of Eq. 7.198 is the pressure flow term. It represents the flow rate in <br>pure pressure flow, i.e., without relative motion between the screw and barrel (zero <br>screw speed):<br>
+ (7.200)<br>
+Note that the pressure flow reduces the output when the pressure gradient is positive, <br>which is often the case in actual extrusion operations.<br>Equation 7.198 is useful because the output of the extruder can be determined from <br>this equation. The effect of changes in screw geo metry becomes quite obvious from <br>Eq. 7.198. Particularly, the effect of channel depth is interesting. The drag flow rate <br>is directly proportional to the channel depth, but the pressure flow rate increases <br>with the channel depth cubed. Thus, the pressure flow increases much faster with <br>channel depth than drag flow. For this reason, the channel depth in the metering <br>section is usually made quite shallow. The optimum channel depth that will give the <br>highest output at a given screw speed and pressure gradient can be determined <br>directly from Eq. 7.198; see also Section 8.2.1.<br>
+<hr>
+<A name=357></a> <br>
+7.4Melt Conveying  <b>345</b><br>
+The optimum channel depth can be determined by setting:<br>
+ (7.201)<br>
+This results in the following optimum channel depth H*:<br>
+ (7.202)<br>
+The optimum helix angle can be determined from:<br>
+ (7.203)<br>
+This results in the following optimum helix angle *:<br>
+ (7.204)<br>
+where ga is the axial pressure gradient (ga = gz/sin). At the optimum channel depth <br>
+the output becomes:<br>
+ (7.205)<br>
+At the optimum channel depth H*, the optimum helix angle becomes:<br>
+ (7.206)<br>
+When the channel depth and helix angle are both at their optimum value, the output <br>becomes:<br>
+ (7.207)<br>
+where B is the axial channel width and N is the screw speed (B  Dtan and <br>vb = DN).<br>Another interesting observation is that the volumetric drag flow rate is independent <br>of any fluid property for Newtonian fluids. Thus, the drag flow rate for water will be <br>the same as oil, molten nylon, etc. The drag flow rate is directly proportional to <br>screw speed because vbz = DNcos, where N is the screw speed.<br>Many extruder screws are designed such that the drag flow is considerably larger <br>than the pressure flow. In these cases, the approximate output of the extruder is <br>determined from Eq. 7.199. If the mass flow rate is required, the volumetric flow <br>rate is simply multiplied with the polymer melt density. The reason that many <br>
+<hr>
+<A name=358></a><b>346</b>  <b>7Functional Process Analysis</b><br>
+screws are designed to have a relatively small pressure flow is that the extrusion <br>operation tends to become less sensitive to diehead pressure fluctuations. If the <br>pressure flow is only 10% of the total throughput, a 50% change in diehead pressure <br>will cause only a 5% change in output. On the other hand, if the pressure flow is 50% <br>of the total throughput, a 50% change in diehead pressure will cause a 25% change <br>in output.<br>There are two ways to achieve a relatively low-pressure sensitivity. One way is to <br>reduce the depth of the channel in the metering section of the screw. The other way <br>is to reduce the pressure gradient in the melt conveying zone by building up pres-<br>sure in earlier zones, the plasticating and/or the solids conveying zone. The most <br>common way to achieve the latter situation is by employing a grooved barrel section, <br>as discussed in Section 7.2.2.2. Down-channel velocity profiles for various values of <br>the pressure flow (pressure gradient) are shown in Fig. 7.55.<br>
+P&gt;&gt;0<br>
+P&gt;0<br>
+P=0<br>
+P&lt;0<br>
+Zero<br>output<br>
+pure<br>drag<br>flow<br>
+<b> Figure 7.55</b> <br>
+Down-channel velocity profiles<br>
+When the pressure change is zero (P = 0), the velocity profile is linear; this repre-<br>sents pure drag flow. When the pressure change is negative (P &lt; 0), the velocities <br>between the screw and barrel will be increased relative to the drag flow velocities. <br>When the pressure change is positive (P &gt; 0) the melt conveying zone is generat-<br>ing pressure and the velocities will be reduced relative to the drag flow velocities. <br>When the pressure drop is very large (P &gt;&gt; 0) the output becomes zero. In this <br>case, the forward drag flow rate equals the rearward pressure flow rate.<br>By the same procedure employed to derive the down-channel velocity, the cross-<br>channel velocity can be determined:<br>
+ (7.208)<br>
+In this case, gx can be determined from the condition that there is no net flow in the <br>
+cross-channel direction:<br>
+ (7.209)<br>
+The cross-channel pressure gradient is:<br>
+ (7.210)<br>
+<hr>
+<A name=359></a> <br>
+7.4Melt Conveying  <b>347</b><br>
+The resulting cross-channel velocity profile is:<br>
+ (7.211)<br>
+The cross-channel velocity profile is shown in Fig. 7.56.<br>
+Vbx<br>
+H<br>
+2H/3<br>
+<b> Figure 7.56</b> <br>
+Cross-channel velocity profile<br>
+It can be seen that the cross channel velocity at y = 2H/3 is zero. Thus, the material <br>in the top one-third of the channel moves towards the active flight flank and the <br>material in the bottom two-thirds of the channel moves towards the passive flight <br>flank. It is clear that in reality the situation becomes more complex at the flight <br>flanks because normal velocity components must exist to achieve the circulatory <br>flow patterns in the cross-channel direction. However, these normal velocity compo-<br>nents will be neglected in this analysis. Normal velocity components were analyzed <br>by Perwadtshuk and Jankow [129] and several other workers. The actual motion of <br>the fluid is the combined effect of the cross- and down-channel velocity profiles. <br>This is shown in Fig. 7.57.<br>
+vb<br>
+helix angle<br>
+<br>
+axial direction<br>
+<b>Figure 7.57</b>Fluid motion in melt conveying zone<br>
+<hr>
+<A name=360></a><b>348</b>  <b>7Functional Process Analysis</b><br>
+A fluid element follows a spiraling path along the length of the screw channel. Close <br>to the barrel surface it flows in the direction of the barrel. When it gets close to the <br>pushing flight flank the element will move down toward the lower portion of the <br>screw channel where it will cross the channel. When the element gets close to the <br>trailing flight flank it moves up along the flight flank until it gets close to the barrel <br>and then the cycle starts again. As more pressure is developed, the spirals are closer <br>together as shown in Fig. 7.58 where the path of a fluid element is shown viewed <br>from the top of the unrolled channel.<br>
+Low pressure<br>
+Medium pressure<br>
+High pressure<br>
+development<br>
+development<br>
+development<br>
+<b>Figure 7.58</b>Fluid motion at various levels of pressure development<br>
+As the melt conveying zone develops more pressure the spirals are pushed more <br>closely together. In the extreme case of closed discharge (zero output) there is no <br>axial component to the flow. In this case fluid elements move tangentially close to <br>the barrel surface and in the opposite direction close to the screw surface; the screw <br>flights act as a plough, so there is mixing but no forward flow.<br>
+<b>7.4.1.1Effect of Flight Flanks<br></b>If the channel width cannot be considered infinite, the equation of motion in the <br>down-channel direction becomes:<br>
+ (7.212)<br>
+Equation 7.212 is an expanded form of Eq. 7.194.<br>Considering the fluid Newtonian allows Eq. 7.212 to be written as:<br>
+ (7.213)<br>
+<hr>
+<A name=361></a> <br>
+7.4Melt Conveying  <b>349</b><br>
+The solution to this equation is more difficult than the solution to Eq. 7.194. The <br>case of pure pressure flow was first solved by Boussinesq [130] in 1868. The solu-<br>tion to the combined drag and pressure flow was first published in 1922 [98]; the <br>authorship of this publication remains a question. Since the 1922 publication, <br>numerous workers have presented solutions to this problem. Meskat [131] reviewed <br>various solutions and demonstrated that they were equivalent. The velocity profile <br>resulting from the drag flow can be written as:<br>
+ (7.214)<br>
+The velocity profile resulting from the pressure flow can be written as:<br>
+ (7.215)<br>
+The combined velocity is simply:<br>
+ (7.216)<br>
+The volumetric output can be conveniently expressed in the following form:<br>
+ (7.217)<br>
+where the shape factor for drag flow Fd is:<br>
+ (7.218)<br>
+and the shape factor for pressure flow Fp is:<br>
+ (7.219)<br>
+In the range of most extruder screws (H/W &lt; 0.6), the shape factors Fd and Fp can be <br>
+quite accurately approximated with the following expressions:<br>
+ (7.220)<br>
+and:<br>
+ (7.221)<br>
+<hr>
+<A name=362></a><b>350</b>  <b>7Functional Process Analysis</b><br>
+Equations 7.220 and 7.221 are considerably easier to evaluate than the exact expres-<br>sions, Eqs. 7.218 and 7.219. With the approximate expressions for the shape factor, <br>the volumetric output can be expressed as:<br>
+ (7.222)<br>
+In most extruder screws, the ratio of channel depth to channel width H/W will range <br>from 0.10 to 0.03, with the latter value being more common than the former. This <br>means that the correction as a result of the shape factors is usually less than 2% and <br>essentially always less than 5%.<br>
+<b>7.4.1.2Effect of Clearance<br></b>Another source of error that can become quite important is the leakage flow through <br>the clearance between the flight and the barrel. A normal design clearance (radial) <br>is 0.001 D, where D is the diameter of the screw. When the clearance is normal, the <br>flow through the clearance will be quite small. However, if the screw and/or barrel <br>is subject to wear, the actual clearance can increase substantially beyond the nor-<br>mal design clearance. This can cause a considerable reduction in output and it is <br>important to know how to evaluate the effect of clearance flow.<br>The clearance reduces the drag flow rate. The drag flow rate is reduced by a factor <br>/H. The corrected drag flow becomes:<br>
+ (7.223)<br>
+where  is the radial clearance.<br>The proper derivation of the pressure induced leakage flow is rather involved. For <br>details of the derivation, the reader is referred to the publications of Mohr and Mal-<br>louk [228], Tadmor [103], or Rauwendaal [271]. The total volumetric output includ-<br>ing the effect of leakage can be written as:<br>
+ (7.224)<br>
+The correction factor for pressure induced leakage through the flight clearance can <br>be written as:<br>
+ (7.225)<br>
+where:<br>
+ (7.225a)<br>
+<hr>
+<A name=363></a> <br>
+7.4Melt Conveying  <b>351</b><br>
+and:<br>
+v =  D N cos (7.225b)<br>
+bz<br>
+The viscosity in the clearance c1 is differentiated from the viscosity in the channel <br>
+ because, in reality, the viscosity in the clearance will be substantially different <br>from the viscosity in the channel as a result of differences in local temperature and <br>shear rate. When the flight clearance is close to the normal design clearance, the <br>value of fL will be very close to zero and can be neglected unless extreme accuracy is <br>
+required. However, when the radial clearance is considerably larger than the nor-<br>mal design clearance, for instance, as a result of wear, the actual value of fL should <br>
+be used in the expression for the total volumetric output, Eq. 7.224.<br>It is interesting to note that even when the down-channel pressure gradient gz is <br>
+zero, there is a pressure induced leakage flow. This results from the drag induced <br>cross channel pressure gradient.<br>A significantly simpler expression for the pressure induced leakage flow is obtained <br>by taking the following approach. The pressure induced leakage flow through the <br>flight clearance can be approximated by considering the flight clearance as a rectan-<br>gular slit of width Dcos, height , and depth w, with a pressure differential across <br>the flight of Pf. By using the equations in Table 7.1, Section 7.5.1, the leakage flow <br>
+can be written as:<br>
+ (7.226)<br>
+The pressure differential across the flight results from both the down-channel pres-<br>sure gradient gz and the cross-channel gradient gx. The latter is a drag induced pres-<br>
+sure gradient; thus, it is present even in pure drag flow. When the actual leakage <br>flow is very small, the cross-channel pressure gradient is given by Eq. 7.210. The <br>pressure differential across the flight can now be written as:<br>
+ (7.227)<br>
+where W is the cross-channel width.<br>The effect of down-channel and cross-channel pressure gradient of the pressure <br> <br>differential across the flight is illustrated in Fig. 7.59.<br>The leakage flow can now be written as:<br>
+ (7.228)<br>
+<hr>
+<A name=364></a><b>352</b>  <b>7Functional Process Analysis</b><br>
+S<br>
+b<br>
+B<br>
+zero axial pressure gradient (ga=0)<br>
+Pressure<br>
+Axial distance<br>
+Pressure<br>
+positive axial pressure gradient (ga&gt;0)<br>
+Axial distance<br>
+Pressure<br>
+<b> Figure 7.59</b> <br>
+negative axial pressure gradient (ga&lt;0)<br>
+Axial pressure profiles at various <br>
+Axial distance<br>
+pressure gradients<br>
+The difference between Eqs. 7.228 and 7.224 is less than 5% at normal values of the <br>clearance and about 10% at clearance values of about four times the normal value.<br>Equation 7.228 can be made more accurate by using an improved equation for the <br>cross-channel pressure gradient. Leakage through the flight clearance will reduce <br>the cross-channel pressure gradient to a value lower than the one given by Eq. 7.210. <br>In Section 10.5, a more accurate expression for the drag induced pressure gradient <br>is derived, taking into account the leakage flow over the flight clearance; see <br>Eqs. 10.108 through 10.112. If the viscosity in the clearance c1 is differentiated <br>
+from the viscosity in the channel , the same approach leads to the following drag <br>induced cross-channel pressure gradient:<br>
+ (7.229)<br>
+Thus, the more accurate expression of the leakage flow can be written as:<br>
+ (7.230)<br>
+For normal values of the flight clearance, however, Eq. 7.228 will generally give suf-<br>ficiently accurate results.<br>The drag induced pressure differential across the flight plays an important role <br>and cannot be neglected. The value of the leakage flow at normal clearance values <br>
+<hr>
+<A name=365></a> <br>
+7.4Melt Conveying  <b>353</b><br>
+( = 0.001 D) is about 0.01% of the drag flow rate; at four times the normal clearance, <br>the leakage flow is about 1% of the drag flow rate. Thus, the leakage flow becomes <br>significant only when the clearance is larger than about four times its normal value.<br>The total volumetric output, including the effect of leakage, Eq. 7.228, can be written:<br>
+ (7.231)<br>
+<b>7.4.1.3Power Consumption in Melt Conveying<br></b>The power consumption in the melt conveying zone is an important parameter to <br>consider in screw design and in the analysis of actual extrusion operations. The <br>power consumed for pumping in the channel is:<br>
+ (7.232)<br>
+where the shear stresses are related to the barrel surface.<br>If the material is Newtonian:<br>
+ (7.233)<br>
+and:<br>
+ (7.234)<br>
+The shear stresses can be evaluated from Eqs. 7.223 and 7.224 and the equations for <br>down-channel velocity profile, Eq. 7.197 or 7.216, and the-cross channel velocity <br>profile, Eq. 7.211. The power consumption in the screw channel can be written as:<br>
+ (7.235)<br>
+where rd is ratio of pressure flow to drag flow:<br>
+ (7.236)<br>
+Equation 7.236 is valid if the clearance is negligible. The ratio of pressure flow to <br>drag flow is often referred to as the throttle ratio (Drossel quotient in German). It <br>enables a compact expression for the combined drag and pressure flow:<br>
+ (7.237)<br>
+<hr>
+<A name=366></a><b>354</b>  <b>7Functional Process Analysis</b><br>
+The specific energy consumption in the channel can be determined by dividing the <br>power consumption by the throughput:<br>
+ (7.238)<br>
+The pumping efficiency is the ratio of the theoretical energy requirement to develop <br>pressure P (=  P) divided by the actual energy requirement (= dZch). Thus, the <br>
+pumping efficiency in the channel can be written as:<br>
+ (7.239)<br>
+The optimum pumping efficiency can be determined by setting the first derivative of <br>the pumping efficiency with respect to the throttle ratio equal to zero:<br>
+ (7.240)<br>
+This yields the following expression for the optimum throttle ratio r*d:<br>
+ (7.241)<br>
+By inserting the optimum throttle ratio from Eq. 7.241 into Eq. 7.239 for the pumping <br>efficiency, one can determine the optimum pumping efficiency in the channel. When <br>the helix angle is zero, the optimum throttle ratio is 1/3 or ­1 and the optimum <br>pumping efficiency is also 1/3. A throttle ratio of ­1 represents a large negative <br>pressure gradient; this is not a situation that is likely to occur in practice. When the <br>helix angle increases, the optimum throttle ratio increases, but the optimum pump-<br>ing efficiency decreases. This is shown in Fig. 7.60.<br>
+throttle ratio<br>
+pumping efficiency<br>
+<b> Figure 7.60</b> <br>
+Optimum throttle ratio and <br>
+pumping efficiency versus <br>
+helix angle<br>
+<hr>
+<A name=367></a> <br>
+7.4Melt Conveying  <b>355</b><br>
+Thus, the highest pumping efficiency of the channel that can possibly be obtained is <br>only 33.33%. The other 66.67% of the energy is dissipated in the fluid as heat. In <br>practice, the actual pumping efficiency will be around 10% or less. The screw pump, <br>therefore, is rather inefficient in developing pressure. Other types of pumps, such as <br>a gear pump, can be more efficient in generating pressure. However, in many extru-<br>sion operations, the energy dissipated in the fluid is not wasted but effectively used <br>to bring the polymer melt to the required melt temperature. Heating the polymer <br>melt by viscous heat generation is more effective than heating by external barrel <br>heaters.<br>The power consumption in the clearance can be determined quite easily if it is <br>assumed that the velocity profile in the clearance is dominated by drag flow. The <br>power consumption in the clearance can then be written as:<br>
+ (7.242)<br>
+where w is the perpendicular flight width (w = bcos).<br>The power consumption in the clearance is directly proportional to the number of <br>parallel flights, the local viscosity, and the flight width, and is inversely proportional <br>to the radial clearance. It will be shown later that a substantial portion (in some <br>cases 50% or more) of the total power consumption is consumed in the clearance. <br>Therefore, the geo metry of the flight clearance becomes an important geometrical <br>variable when it comes to minimizing power consumption.<br>The power consumption necessary to build up pressure in the polymer is deter-<br>mined by multiplying the volumetric flow rate with the pressure rise:<br>
+ (7.243)<br>
+The total pumping efficiency can now be determined from:<br>
+ (7.244)<br>
+The total pumping efficiency will usually be about 10% or less. The amount of power <br>that is dissipated in the fluid as heat is:<br>
+ (7.245)<br>
+This power should be used to calculate the temperature increase in the polymer <br>melt. Since the pumping efficiency is usually less than 10%, the amount of power <br>dissipated in the screw channel will generally be more than 90% of the total power <br>input.<br>
+<hr>
+<A name=368></a><b>356</b>  <b>7Functional Process Analysis</b><br>
+<b>7.4.2Power Law Fluids</b><br>
+In this section, the effect of the pseudo-plastic behavior of the polymer melt on the <br>conveying characteristics will be analyzed by describing the polymer melt as a <br>power law fluid. As before, the flow is considered to be steady, fully developed, and <br>isothermal, and leakage flow is neglected. The effect of the flight flanks on the down-<br>channel velocity profile will be neglected. At first, the analysis will be one-dimen-<br>sional, considering only the down-channel velocity. Thus, the effect of cross-channel <br>flow will initially be neglected. Later, the analysis will be extended to a two-dimen-<br>sional case, considering both the down-channel and cross-channel velocity.<br>
+<b>7.4.2.1One-Dimensional Flow<br></b>For the one-dimensional analysis, the equation of motion is described by Eqs. 7.194 <br>and 7.195. The constitutive relationship, the power law equation, is written as:<br>
+ (7.246)<br>
+Equation 7.246 combined with Eq. 7.195 describes the basic problem. It is convenient <br>to write the resulting equation in dimensionless form. For this purpose, the follow-<br>ing dimensionless quantities are defined: the dimensionless depth  = y/H, the <br>dimensionless down-channel velocity v0z = vz/vbz, and a reduced pressure gradient <br>
+R. The reduced pressure gradient is defined as:<br>
+ (7.247)<br>
+where s is the reciprocal power law index (s = 1/n).<br>Equation 7.246 can be integrated to give:<br>
+ (7.248)<br>
+Variable  in Eq. 7.248 represents the location where the shear rate is zero, which is <br>also the location of the extremum in the velocity profile. This value needs to be <br>known to eliminate the absolute value in Eq. 7.248. For the time being, only positive <br>pressure gradients gz will be considered. If the extremum occurs in the screw chan-<br>
+nel, then 0    1. When   , Eq. 7.248 can be written as:<br>
+ (7.249)<br>
+<hr>
+<A name=369></a> <br>
+7.4Melt Conveying  <b>357</b><br>
+By integration and by using the appropriate boundary condition v0z(1) = 1, the fol-<br>
+lowing expression is obtained:<br>
+ (7.250)<br>
+Similarly, for   , the expression becomes:<br>
+ (7.251)<br>
+At  = , the two velocities from Eqs. 7.250 and 7.251 should be the same. From this <br>equality, the following equation for  results:<br>
+ (7.252)<br>
+From this equation, the value of  can be determined. The condition for the exist-<br>ence of an extremum within the actual flow region is R  s + 1. The maximum <br>
+velocity can be written as:<br>
+ (7.253)<br>
+The extremum falls outside of the actual flow regime when R &lt; s + 1. By using the <br>
+same procedure, the equation for  when R &lt; s + 1 becomes:<br>
+ (7.254)<br>
+Both the dimensionless velocity v0z and the dimensionless flow rate  0 can be written <br>
+into single expressions:<br>
+ (7.255)<br>
+and:<br>
+ (7.256)<br>
+The dimensionless flow rate is the actual flow rate divided by the drag flow rate, <br>thus:<br>
+ (7.257)<br>
+<hr>
+<A name=370></a><b>358</b>  <b>7Functional Process Analysis</b><br>
+Later in Section 8.2.1, it will be shown that if the channel depth is optimized to give <br>the highest output, the corresponding value of  = 0. Thus, the corresponding opti-<br>mum dimensionless flow rate is:<br>
+ (7.258)<br>
+And thus the optimum actual flow rate is:<br>
+ (7.259)<br>
+For a Newtonian fluid (n = 1), Eq. 7.259 becomes equal to Eq. 7.205.<br>The velocity profiles at  = 0.1 and various values of the power law index n are <br>shown in Fig. 7.61.<br>
+elocity<br>
+Dimensionless v<br>
+<b> Figure 7.61</b> <br>
+Dimensionless velocity versus <br>
+Dimensionless normal distance<br>
+ <br>dimensionless normal distance<br>
+It is clear from Fig. 7.61 that the velocities reduce and the velocity profiles start to <br>approach plug flow as the power law index becomes smaller. As a result, the through-<br>put reduces with reducing power law index.<br>In order to determine the velocity profile and the flow rate, the value of  has to be <br>known. This involves solving Eq. 7.252 or Eq. 7.254. This is normally done by using <br>a numerical technique, e.g., Newton-Raphson. Exact analytical solutions are only <br>possible for the special case when s is a positive integer. However, by rewriting the <br>equations and by performing a series expansion, a closed-form solution can be <br>obtained. If a new variable x is introduced in Eq. 7.252 with x = ­0.5, then the <br>equation can be rewritten as:<br>
+ (7.260)<br>
+<hr>
+<A name=371></a> <br>
+7.4Melt Conveying  <b>359</b><br>
+By performing a series expansion of the first two terms of Eq. 7.260 and neglecting <br>the x terms of order four and higher, the following equation is obtained:<br>
+ (7.261)<br>
+This is a standard third order (cubic) equation that can be readily solved. There is <br>only one root that is real; the solution for  when R  s + 1 is:<br>
+ (7.262)<br>
+where:<br>
+ (7.263)<br>
+and:<br>
+ (7.264)<br>
+This solution will be accurate when x is close to zero, which will generally be the <br>case. In the region for which Eq. 7.260 applies, the values of  will range between 0 <br>and 0.5, thus x will range between ­0.5 and 0.<br>Equation 7.254 can be rewritten by introducing a new variable x =  + 1. After series <br>expansion and neglecting x terms of order three and higher, a quadratic equation is <br>obtained. The solution for  when R &lt; s + 1 is:<br>
+ (7.265)<br>
+where:<br>
+ (7.266)<br>
+ (7.267)<br>
+ (7.268)<br>
+Again, this solution will be accurate when x is close to zero. In the region where <br>Eq. 7.254 applies, the value of  will range from 0 (where R = s + 1) to ­ when the <br>
+pressure gradient is zero. Therefore, Eq. 7.265 cannot give accurate results at very <br>small pressure gradients. However, at larger pressure gradients, Eq. 7.265 will give <br>accurate results. The limited accuracy of the solution at small pressure gradients <br>does not have much practical significance. In the analysis of real extruders, one is <br>
+<hr>
+<A name=372></a><b>360</b>  <b>7Functional Process Analysis</b><br>
+primarily concerned about the effect of large pressure gradients, not about the effect <br>of small pressure gradients. The flow rate at small pressure gradients will be nearly <br>equal to the drag flow rate. The predictions of the analytical solutions can now be <br>plotted in the often-used dimensionless form.<br>This is shown in Fig. 7.62, where the dimensionless output is plotted as a function <br>of the dimensionless pressure gradient g0z, where:<br>
+ (7.269)<br>
+<b> Figure 7.62</b> <br>
+Dimensionless throughput versus <br>
+dimensionless pressure gradient from <br>
+closed-form solution<br>
+For a Newtonian fluid (n = 1), the familiar linear output-pressure relationship is <br>found. However, when the power law index is less than unity, substantial deviations <br>from Newtonian characteristics occur. The deviations increase as the material <br>becomes more pseudo-plastic (more strongly non-Newtonian). The result is that for <br>a pseudo-plastic fluid, the pressure generating capability is drastically reduced com-<br>pared to a Newtonian fluid. Or, at the same pressure gradient, the output is drasti-<br>cally reduced. For a fluid with a power law index less than 0.8, the use of the equa-<br>tions for Newtonian fluids will result in large errors!<br>Comparisons of Fig. 7.62 to similar figures determined by other workers from <br>numerical techniques, e.g., [132] or [106], reveal essentially indistinguishable <br>results. This is a first indication that the analytical solutions for  are quite accurate. <br>Up to this point, only positive pressure gradients have been considered. From <br>Eq. 7.256, it can be demonstrated rather easily that the output values for negative <br>pressure gradients can be obtained from the following relationship:<br>
+ (7.270)<br>
+<hr>
+<A name=373></a> <br>
+7.4Melt Conveying  <b>361</b><br>
+The dashed line in Fig. 7.62 shows at what point along each curve  becomes zero. <br>When  is zero, the extremum in the velocity profile occurs right at the screw sur-<br>face. The dimensionless throughput in this case is:<br>
+ (7.271)<br>
+This is the same value as the optimum dimensionless flow rate described by Eq. 7.258. <br>It is interesting to note that this throughput is determined only by the power law <br>index. The data above the dashed line have been determined with Eq. 7.265. In this <br>case, R &lt; s + 1 and no extremum occurs in the velocity profile. The data below the <br>
+dashed line have been determined with Eq. 7.262. In this case, R  s + 1 and an <br>
+extremum does occur in the velocity profile.<br>
+<b>7.4.2.2Two-Dimensional Flow<br></b>In this section, the previous analysis will be extended to include the effect of cross-<br>channel flow. The cross-channel flow does not directly affect the conveying rate, but <br>it does affect the total shear rate to which the polymer is exposed. Therefore, the <br>viscosity will be affected and thus the actual flow rate is affected as well. If the helix <br>angle reduces to zero, there will be no cross-channel flow, and the results of the two-<br>dimensional analysis will be identical to the one-dimensional analysis. This limiting <br>case has been studied by Tadmor [133] and Dyer [134]. More general two-dimen-<br>sional analyses can be found in references 132 through 145. Steller [297] developed <br>an analytical solution for 2-D flow of a power law fluid and later [298] for an Ellis <br>fluid. The solution, however, requires numerical analysis to evaluate the solution. <br>Rauwendaal [271] studied the 2-D flow problem in screw extruders including the <br>effect of leakage flow; the effect of leakage flow increases significantly when the <br>polymer melt becomes more shear thinning.<br>The difference between the one-dimensional and the two-dimensional analysis will <br>increase with increasing helix angle and reducing power law index. From a practical <br>point of view, the use of a two-dimensional analysis becomes important when large <br>helix angles and strongly non-Newtonian fluids are analyzed. The equation of motion <br>in the down-channel direction is the same as used before; see Eq. 7.194. A similar <br>expression has to be used for the cross-channel direction. The shear stress profiles <br>can be written as:<br>
+ (7.272)<br>
+ (7.273)<br>
+The magnitude of the total shear stress is obtained from:<br>
+ (7.274)<br>
+<hr>
+<A name=374></a><b>362</b>  <b>7Functional Process Analysis</b><br>
+The power law equation for the two-dimensional flow can be written as:<br>
+ (7.274a)<br>
+The direction of shear stress  and velocity v is determined by yz and yx. At this <br>
+point, there are three unknowns: the cross-channel pressure gradient gx, the cross-<br>
+channel shear stress at the screw surface xo, and the down-channel shear stress at <br>
+the screw surface zo. At the time of writing, no analytical solutions to this problem <br>
+are known. In fact, an analytical solution does not seem possible. Therefore, some <br>numerical scheme has to be used in order to determine the unknowns. Because this <br>problem is of considerable importance to the proper analysis of melt conveying, it <br>will be discussed in some detail.<br>Some initial values of gx, xo, and zo can be selected, for instance, by calculating the <br>
+values for the Newtonian case. From those initial values, the corresponding velocity <br>profiles and the flow rates in the cross-channel direction can be determined. The <br>velocity profile in the x and z directions can be determined from:<br>
+ (7.275)<br>
+and:<br>
+ (7.276)<br>
+The net flow rate in the cross-channel direction should be zero if it is assumed the <br>leakage over the flight is negligible. The cross-channel flow rate is:<br>
+ (7.277)<br>
+The accuracy of the initial guesses of gx, xo, and zo can be evaluated by calculating <br>
+vx(H), vz(H) and  x. This can be done by using a standard numerical technique to <br>
+integrate Eqs. 7.275, and 7.276, e.g., Simpson's rule. The calculated values are then <br>compared to the actual values: vbx, vbz, and 0 respectively. Unless the initial values <br>
+are perfect, there will be residuals:<br>
+ (7.278)<br>
+ (7.279)<br>
+<hr>
+<A name=375></a> <br>
+7.4Melt Conveying  <b>363</b><br>
+ (7.280)<br>
+New values of gx, xo, and zo can be obtained by using a Newton-Raphson scheme. <br>
+The residuals can be expressed as:<br>
+ (7.281)<br>
+ (7.282)<br>
+ (7.283)<br>
+The partial derivatives can be determined by selecting a second set of data for gx, xo, <br>
+and zo with values very close to the first values. When this is done, gx, xo, and <br>
+zo can be calculated by solving the three linear equations, which is a straight-<br>
+forward operation. The new values of gx, xo, and zo can now be determined from:<br>
+ (7.284)<br>
+ (7.285)<br>
+ (7.286)<br>
+The iteration is repeated until the relative difference between the new and old value <br>is less than a certain value, depending on the accuracy required. The throughput is <br>determined from:<br>
+ (7.287)<br>
+A short Fortran program to perform this numerical procedure is given in Appendix <br>7.3. The program converges rapidly; the solution is usually obtained in five itera-<br>tions. Figure 7.63 shows the dimensionless throughput-pressure gradient relation-<br>ship from the two-dimensional analysis for a helix angle of zero degrees.<br>As mentioned before, these results should be the same as the results from the one-<br>dimensional analysis as shown in Fig. 7.62. By comparing the two figures, it is clear <br>that the results are virtually identical. This confirms the accuracy of the analytical <br>solution for the one-dimensional flow of a power law fluid. The difference is gener-<br>ally less than 1%, except at small values of the dimensionless pressure gradient <br>g0z &lt; 0.1. The loss of accuracy at low pressure gradients was predicted earlier and <br>
+should not pose a serious problem in the analysis of real extrusion problems.<br>
+<hr>
+<A name=376></a><b>364</b>  <b>7Functional Process Analysis</b><br>
+1.0<br>
+0.9<br>
+0.8<br>
+2D power law fluid<br>
+helix angle 0 degrees<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+0.3<br>
+0.2<br>
+Dimensionless throughput<br>
+power law index n = 0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+0.1<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Dimensionless pressure gradient<br>
+<b>Figure 7.63</b>Dimensionless throughput versus dimensionless pressure gradient from 2-D <br>
+analysis for helix angle equals zero<br>
+Figure 7.64 shows the dimensionless throughput versus pressure gradient for a <br>helix angle of 17.66° (square pitch screw).<br>
+1.0<br>
+0.9<br>
+0.8<br>
+2D Power law fluid<br>
+Helix angle 17.66 degrees<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+0.3<br>
+0.2<br>
+Dimensionless throughput<br>
+Power law index n = 0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+0.1<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Dimensionless pressure gradient<br>
+<b>Figure 7.64</b>Dimensionless throughput versus dimensionless pressure gradient from 2-D <br>
+analysis for helix angle equals 17 .66°<br>
+The dimensionless throughput at 17.66° helix angle is considerably lower than a <br>zero helix angle. The difference is about 10% at small pressure gradients and about <br>40% at large pressure gradients when the power law index is less than one-half. At <br>larger values of the power law index, the difference is generally less than 10%. This <br>means that application of the equations for the one-dimensional case will lead to <br>substantial errors for standard helix angles when the power law index is less than <br>one-half! When the power law index is larger than one-half, these equations will be <br>reasonably accurate.<br>
+<hr>
+<A name=377></a> <br>
+7.4Melt Conveying  <b>365</b><br>
+It should be noted that a reduction in the dimensionless throughput does not neces-<br>sarily mean that the actual throughput reduces as well. Obviously, when the helix <br>angle is zero, the actual throughput will be zero. The dimensionless throughput is <br>determined from:<br>
+ (7.288)<br>
+The actual volumetric throughput is related to the dimensionless throughput by:<br>
+ (7.289)<br>
+The second equality of Eq. 7.289 is correct if the flight width is negligible. Figure 7.65 <br>shows the dimensionless throughput versus pressure gradient for five helix angles <br>when the power law index is one-half.<br>The dimensionless throughput reduces with increasing helix angle over the entire <br>pressure gradient range. This demonstrates again that the equation for the one-<br>dimensional case should not be used for large helix angles and/or small values of <br>the power law exponent.<br>
+1.0<br>
+0.9<br>
+0.8<br>
+2D Power law fluid<br>
+Power law index n = 0.5<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0 degree helix angle<br>
+0.4<br>
+10<br>
+20<br>
+0.3<br>
+30<br>
+0.2<br>
+40<br>
+Dimensionless throughput<br>
+0.1<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Dimensionless pressure gradient<br>
+<b>Figure 7.65</b>Dimensionless throughput versus dimensionless pressure gradient for several helix <br>
+angles<br>
+From Figs. 7.62 through 7.65, it becomes clear that the Newtonian output-pressure <br>gradient relationship is unacceptably inaccurate when the power law index of the <br>polymer melt is less than 0.8. The one-dimensional power law (1-DPL) output-pres-<br>sure gradient relationship is accurate only for small helix angles. Thus, for accurate <br>results, a two-dimensional power law (2-DPL) analysis should be used. However, the <br>2-DPL analysis does not yield analytical solutions; numerical techniques have to be <br>
+<hr>
+<A name=378></a><b>366</b>  <b>7Functional Process Analysis</b><br>
+used to obtain results. One way to avoid complex calculations is to use the New-<br>tonian output-pressure gradient relationship with correction factors for non-New-<br>tonian behavior of the polymer melt. Figure 7.66 shows the dimensionless output <br>versus pressure gradient for an expression of  0 that incorporates such correction <br>factors.<br>
+1.0<br>
+0.9<br>
+0.8<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+0.3<br>
+0.2<br>
+n = 0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Dimensionless throughput<br>
+0.1<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Dimensionless pressure gradient<br>
+<b>Figure 7.66</b>Dimensionless throughput versus dimensionless pressure gradient using linear <br>
+approximation<br>
+The correction factors apply to helix angles in the range of 15 to 25° and are deter-<br>mined to minimize the difference with 2-DPL analysis. The dimensionless output is <br>written as:<br>
+ (7.290)<br>
+By using Eq. 7.269 for g0z and Eq. 7.257 for  0, the Newtonian output-pressure gra-<br>
+dient relationship with the correction factors for non-Newtonian behavior can be <br>written as:<br>
+ (7.291)<br>
+where  is calculated at the Couette shear rate in the channel (  = vbz/H); thus<br>
+ (7.292)<br>
+With this equation, the difference with results from a 2-DPL analysis will generally <br>be less than 10% when the power law index lies in the range of 0.3 to 1.0. Essentially <br>all polymers fall in this range of power law indici. Thus, Eq. 7.291 should provide a <br>
+<hr>
+<A name=379></a> <br>
+7.4Melt Conveying  <b>367</b><br>
+useful relationship between output and pressure gradient for use in the analysis of <br>practical extrusion problems. Of course, Eq. 7.291 is 100% accurate when the power <br>law index equals unity, i.e., for a Newtonian fluid. The correction factors in Eqs. 7.290 <br>and 7.291 have been determined for helix angles normally used in the extrusion <br>industry, with the helix angle usually ranging from about 15 to 25°. If helix angles <br>considerably below or above this range are used, then other correction factors will <br>be more appropriate.<br>Another method to avoid complex calculations is to use an effective viscosity to be <br>used in the Newtonian equation in order to predict output for a non-Newtonian fluid. <br>This method was proposed by Booy [132]. A drawback of this method is that it only <br>corrects the pressure flow term. As a result, the method does not work at small pres-<br>sure gradients. The effective viscosity ratio has to be obtained from a graph. The <br>ratio can be approximated by taking the effective viscosity ratio equal to the power <br>law index. However, the latter approximation will be considerably less accurate than <br>Eq. 7.291.<br>Potente [235] also proposed approximate equations to predict the output as a function <br>of pressure gradient for power law fluids. However, Potente's equations are only valid <br>in a limited range of the dimensionless output V0, for 0.6  V0  1. Thus, for a power <br>law fluid with index n = 0.3, the equations are valid only when the dimensionless <br>pressure gradient g0z is less than about 0.1. In practical extrusion problems, one is <br>
+generally more concerned about the effect of large pressure gradients than about the <br>effect of small pressure gradients. Therefore, the equations developed by Potente have <br>limited usefulness.<br>
+<b>7.4.3Non-Isothermal Analysis</b><br>
+We will now address the effect of temperature on melt conveying. First, we will ana-<br>lyze fully developed temperature profiles. These conditions exist when the tempera-<br>tures no longer change in the flow direction; the region before is called the region of <br>developing temperatures. We will address Newtonian fluids first and then analyze <br>power law fluids.<br>
+<b>7.4.3.1Newtonian Fluids with Negligible Viscous Dissipation<br></b>For a Newtonian fluid (n = 1) the momentum balance equation can be written as:<br>
+ (7.293)<br>
+The dimensionless velocity  is defined as:<br>
+ (7.294)<br>
+<hr>
+<A name=380></a><b>368</b>  <b>7Functional Process Analysis</b><br>
+The dimensionless temperature is defined as:<br>
+ (7.295)<br>
+The dimensionless normal distance is defined as:<br>
+ (7.296)<br>
+The throttle ratio r is given by Eq. 7.236 and the factor b is given by:<br>
+ (7.297)<br>
+where  is the temperature coefficient of the viscosity.<br>The moving surface, the barrel, is at temperature Tb and the stationary surface, the <br>
+screw, is at temperature Ts. The momentum balance equation can be integrated to <br>
+yield the velocity gradient:<br>
+ (7.298)<br>
+The integration constant 0 corresponds to the normal coordinate where the shear <br>
+stress is zero and the velocity profile has an extremum. If it is assumed that the <br>viscosity is determined by a linear temperature profile the velocity gradient can be <br>written as:<br>
+ (7.299)<br>
+The velocity profile can be obtained by integrating the velocity gradient; this results <br>in the following expression:<br>
+ (7.300)<br>
+The integration constants can be evaluated from the boundary conditions (0) = 0 <br>and (1) = 1. This leads to the expression for 0:<br>
+ (7.301)<br>
+Integration constant 0 becomes:<br>
+ (7.302)<br>
+<hr>
+<A name=381></a> <br>
+7.4Melt Conveying  <b>369</b><br>
+Figure 7.67 shows the velocity profiles for b = 1.0 and several values of the throttle <br>ratio r.<br>
+<b>Figure 7.67</b>Dimensionless velocity versus normal distance at b = 1<br>
+The velocity profile for pure drag flow (r = 0) is curved such that the velocities are <br>increased from the linear profile that occurs in isothermal flow. The lines at various <br>r-values are closely grouped together. Therefore, changes in pressure gradient will <br>have a relatively small effect on the velocity profile and, thus, on the flow rate.<br>The velocity profiles at b = ­1 are shown in Fig. 7.68.<br>
+<b>Figure 7.68</b>Dimensionless velocity versus normal distance at b = ­1<br>
+<hr>
+<A name=382></a><b>370</b>  <b>7Functional Process Analysis</b><br>
+In this case the velocity profile at r = 0 is curved such that the velocities are <br>decreased from the linear profile. Also, the lines at various r-values are spaced fur-<br>ther apart. As a result, changes in pressure gradient will have a relatively large <br>effect on the velocity profile and flow rate. This indicates less stable extrusion opera-<br>tion at negative values of b, in other words, when the barrel temperature is higher <br>than the screw temperature.<br>The effect of the value of b on the velocity profile is shown in Fig. 7.69.<br>Figure 7.69 compares the velocities at b-values of ­1, 0, and 1 at a throttle ratio of <br>r = 0. It is clear that the temperature effect is significant even in pure drag flow (r = 0). <br>The effect is greater for higher values of the throttle ratio. This is shown in Fig. 7.70.<br>
+<b>Figure 7.69</b>Dimensionless velocity versus normal distance for three b values at r = 0<br>
+<b>Figure 7.70</b>Dimensionless velocity versus normal distance for three b values at r = 0 .4<br>
+<hr>
+<A name=383></a> <br>
+7.4Melt Conveying  <b>371</b><br>
+The volumetric flow rate can be determined by integrating () from  = 0 to  = 1. <br>This leads to the following expression for the dimensionless flow rate:<br>
+ (7.303)<br>
+Figure 7.71 shows how the dimensionless flow rate changes with factor b.<br>
+<b>Figure 7.71</b>Dimensionless flow rate versus b for several values of the throttle ratio (r)<br>
+The flow rate increases with b for positive values of the throttle ratio. For r = ­0.2 <br>and r = ­0.4 the curves are fairly flat and the effect of temperature is slight. At posi-<br>tive values of the throttle ratio the effect of temperature is significant and increases <br>with the throttle ratio. For positive r-values the flow rate increases with b. This <br>means increasing the screw temperature at constant barrel temperature will <br>increase the flow rate. This points to a positive effect of internal screw heating. At <br>constant screw temperature, reducing the barrel temperature will increase the flow <br>rate.<br>At very high values of factor b the dimensionless flow rate will approach the value <br>of unity. This value corresponds to plug flow. Obviously, this condition will not be <br>reached in real operations.<br>When the viscosity is temperature independent (b = 0) the flow rate is:<br>
+ ­   (7.304)<br>
+The relative flow rate can be defined at the ratio of (b)/(b = 0). Figure 7.72 shows <br>how the relative flow rate varies with b for several values of the throttle ratio.<br>In Fig. 7.72 all the curves intersect at b = 0 where the relative flow rate equals unity. <br>The curve for r = ­0.2 is rather flat. For larger values of r the slope increases and, <br>thus, the effect of temperature. As a result, temperature related extrusion instabili-<br>
+<hr>
+<A name=384></a><b>372</b>  <b>7Functional Process Analysis</b><br>
+ties are more likely to occur at large positive pressure gradients. For positive pres-<br>sure gradients the temperature sensitivity of the flow rate reduces with b. As a <br>result, increasing the screw temperature relative to the barrel temperature (or <br>reducing the barrel temperature relative to the screw temperature) will not only <br>increase the flow rate but it will also improve extrusion stability.<br>
+<b>Figure 7.72</b>Relative flow rate versus b for several values of the throttle ratio r<br>
+An approximate solution for the temperature profile can be obtained by inserting <br>Eq. 7.298 for the velocity gradient into the energy equation. This leads to the follow-<br>ing expression for the second derivative of the temperature:<br>
+ (7.305)<br>
+where A1 is given by:<br>
+ (7.306)<br>
+The Brinkman number NBr is defined as:<br>
+ (7.307)<br>
+The temperature profile can be obtained by integrating twice.<br>
+ (7.308)<br>
+<hr>
+<A name=385></a> <br>
+7.4Melt Conveying  <b>373</b><br>
+Integration constants A2 and A3 can be evaluated from the thermal boundary condi-<br>
+tions. The most general thermal boundary conditions are obtained by prescribing the <br>local heat flux with the Biot modulus. The thermal boundary condition at the screw <br>surface can thus be written as:<br>
+ (7.309)<br>
+Similarly, the thermal boundary condition at the barrel surface becomes:<br>
+ (7.310)<br>
+Integration constant A3 becomes:<br>
+ (7.311)<br>
+and:<br>
+ (7.312)<br>
+where:<br>
+ (7.313)<br>
+ (7.314)<br>
+ (7.315)<br>
+ (7.316)<br>
+When the screw surface is isothermal the integration constants become A2 = ­B1 <br>
+and A3 = ­B2. When the screw is adiabatic and the barrel isothermal the integration <br>
+constants become A2 = ­B1 and A3 = B1­B4+1.<br>Figure 7.73 shows the dimensionless temperature versus normal distance for several <br>values of the throttle ratio.<br>The condition Ns = 0 corresponds to adiabatic conditions at the screw surface. As a <br>
+result, the temperature gradient at the screw surface ( = 0) is zero. The condition <br>Nb = 1000 corresponds to almost isothermal conditions at the barrel surface ( = 1). <br>
+As a result, the temperature at  = 1 equals unity for all curves. The screw tempera-<br>ture increases when the throttle ratio reduces as a result of higher shear rates at the <br>screw surface.<br>
+<hr>
+<A name=386></a><b>374</b>  <b>7Functional Process Analysis</b><br>
+<b> Figure 7.73</b> <br>
+Dimensionless temperature versus <br>
+ <br>normal distance with b = 1 and <br>
+for  several values of the throttle ratio <br>
+(Ns = 0 and Nb = 1000)<br>
+The solution for the velocity profile and flow rate is correct as long as the viscous <br>dissipation can be neglected. The solution for the temperature profile is approximate <br>because of the assumption of a linear temperature profile for viscosity determina-<br>tion. If the calculated temperature profile deviates sharply from the linear profile <br>the error in the temperature profile can be significant. This issue will be addressed <br>in the following section.<br>
+<b>7.4.3.2Non-Isothermal Analysis of Power Law Fluids<br></b>We will first analyze temperature profiles in pure drag flow of power law fluids with <br>negligible dissipation. This will be followed by an analysis of power law fluids with <br>the effects of dissipation included.<br>
+<b>7.4.3.2.1Power Law Fluid with Negligible Dissipation</b><br>
+The energy equation for pure drag flow can be written as:<br>
+ (7.317)<br>
+The corresponding velocity profile can be written as:<br>
+ (7.318)<br>
+The velocity profile at several values of b is shown in Fig. 7.74. When b &gt; 0 (screw <br>temperature higher than barrel temperature), the velocities increase; when b &lt; 0, <br>the velocities decrease.<br>The dimensionless flow rate is obtained by integrating () from  = 0 to  = 1. This <br>leads to:<br>
+ (7.319)<br>
+<hr>
+<A name=387></a> <br>
+7.4Melt Conveying  <b>375</b><br>
+<b> Figure 7.74</b>Dimensionless velocity <br>
+versus dimensionless normal distance<br>
+Figure 7.75 shows how the relative flow rate changes with b at several values of the <br>power law index.<br>
+<b>Figure 7.75</b>Dimensionless flow rate versus term b = (Ts­Tb)<br>
+The flow rate increases when b increases. The temperature sensitivity increases <br>when the power law index becomes smaller. This indicates that strongly shear thin-<br>ning fluids will be more sensitive to temperature fluctuations than weakly shear <br>thinning fluids. Based on the results with Newtonian fluids it can be expected that <br>the sensitivity to temperature fluctuations will get worse with large positive values <br>of the pressure gradient.<br>
+<hr>
+<A name=388></a><b>376</b>  <b>7Functional Process Analysis</b><br>
+<b>7.4.3.2.2Pure Drag Flow of Power Law Fluid with Dissipation Included</b><br>
+The power law fluid problem in pure drag flow was solved by Gavis and Laurence <br>[302] without the assumption of negligible viscous dissipation. Their analysis was <br>based on parallel plates of equal temperature. Rauwendaal [271] extended this ana-<br>lysis to parallel plates maintained at different temperatures; this analysis will be <br>presented next. The momentum equation for pure drag flow can be integrated to <br>express the shear stress as follows:<br>
+ (7.320)<br>
+This expression can be integrated to express the velocity as a function of m. With <br>boundary conditions vz(0) = 0 and vz(H) = v we obtain:<br>
+ (7.321)<br>
+where:<br>
+ (7.322)<br>
+The energy equation for a power law fluid with a fully developed temperature profile <br>can be written as:<br>
+ (7.323)<br>
+where:<br>
+ (7.324)<br>
+The energy equation can now be written as:<br>
+ (7.325)<br>
+With the dimensionless temperature 1 = s(T­T0) and the dimensionless normal <br>
+coordinate  = y/H the energy equation becomes:<br>
+ (7.326)<br>
+where:<br>
+ (7.327)<br>
+<hr>
+<A name=389></a> <br>
+7.4Melt Conveying  <b>377</b><br>
+This type of second-order ordinary differential equation can be solved by multi-<br>plying the equation by 1. By integrating each term we obtain:<br>
+ (7.328)<br>
+The first derivative of temperature 1 can thus be written as:<br>
+ (7.329)<br>
+This can be rewritten by evaluating the integral:<br>
+ (7.330)<br>
+where:<br>
+ (7.331)<br>
+The 1() curve has a maximum at *(1 = lnA). When Tb &gt; Ts, the first derivative <br>
+1 will be positive when  &lt; * and the plus sign will be used in Eq. 7.330. When <br>
+ &gt; *, the first derivative will be negative and the minus sign must be used in this <br>interval. When * &gt; 1, the solution with the plus sign will be valid over the entire <br>depth of the channel. When Ts &gt; Tb, the signs in Eq. 7.330 will be opposite from <br>
+those used when Tb &gt; Ts. When isothermal boundary conditions are used, 1(0) = <br>
+s(Ts­T0) and (1) = s(Tb­T0). When Tb &gt; Ts, Eq. 7.330 can be integrated to give:<br>
+ (7.332)<br>
+Equation 7.332 can also be used when Tb &lt; Ts, as long as * &gt; 0. The integrals in <br>
+Eq. 7.332 can be evaluated by considering that:<br>
+ (7.333)<br>
+By working out the integrals the temperature 1 can be expressed as a function of . <br>
+After some rearrangement the following expression is obtained:<br>
+ (7.334)<br>
+<hr>
+<A name=390></a><b>378</b>  <b>7Functional Process Analysis</b><br>
+where:<br>
+ (7.335)<br>
+By further rearrangement Eq. 7.334 can be cast in the same form as used by Gavis <br>and Laurence. The temperature thus becomes:<br>
+ (7.336)<br>
+where:<br>
+ (7.337)<br>
+and:<br>
+ (7.338)<br>
+The value of integration constant A is evaluated by using thermal boundary condi-<br>tion 1(1) = s(Tb­T0) = 1b. Thus, constant A is obtained by solving the following <br>
+equation:<br>
+ (7.339)<br>
+where:<br>
+ (7.340)<br>
+When the wall temperatures are equal such that 1s = 1b = 0, Eq. 7.339 reduces to:<br>
+ (7.341)<br>
+In this case y2 becomes y2 = ­0.5y1 and Eq. 7.336 can be written as:<br>
+ (7.342)<br>
+Equations 7.341 and 7.342 are identical to those derived by Gavis and Laurence; <br>they can be considered as describing a special case of the more general problem <br>whose solution is described by Eqs. 7.336 and 7.339.<br>In order to determine the actual temperature profile from Eq. 7.336, Eq. 7.339 has to <br>be solved first to obtain the value of constant A. Figure 7.76 graphically shows A as <br>a function of C at three values of 1, where 1 is the difference in temperature <br>
+between the moving wall 1b and the stationary wall 1s.<br>
+<hr>
+<A name=391></a> <br>
+7.4Melt Conveying  <b>379</b><br>
+<b> Figure  7.76</b> <br>
+Constant C versus constant A for various <br>
+values of 1<br>
+Below Cmax there are two values of A for each value of C. The value of Cmax reduces as <br>
+1 increases. There are no solutions to the problem for C &gt; Cmax. Since C has to be <br>
+equal to or smaller than Cmax there is a maximum shear stress that can be deter-<br>
+mined from the expression for C1 (Eq. 7.327). The maximum shear stress can be <br>
+written as:<br>
+ (7.343)<br>
+The maximum shear stress reduces as 1 increases because the value of Cmax <br>
+reduces as shown in Fig. 7.76. It appears that for each value of C below Cmax there <br>
+are two solutions for the temperature and velocity profile. However, there is an addi-<br>tional relationship that fixes the actual value of integration constant A. In order to <br>determine this relationship the velocity profile has to be determined first.<br>
+7.4.3.2.2.1Velocity Profile and Flow Rate<br>The velocity profile can be determined from Eqs. 7.321 and 7.322; it can be written <br>in dimensionless form as follows:<br>
+ (7.344)<br>
+where the dimensionless velocity () = vz()/v.<br>The flow rate   per unit width can be determined from:<br>
+ (7.345)<br>
+<hr>
+<A name=392></a><b>380</b>  <b>7Functional Process Analysis</b><br>
+Using Eq. 7.334 or 7.336 for the temperature profile the velocity profile can be written <br>as:<br>
+ (7.346)<br>
+where:<br>
+ (7.347)<br>
+and:<br>
+ (7.348)<br>
+Using the expression for the velocity profile the flow rate can be determined by inte-<br>gration. This leads to the following expression:<br>
+ (7.349)<br>
+Up to this point the actual plate velocity has not entered into the picture. By using <br>the plate velocity v the actual value of constant A can be determined. From Eq. 7.321 <br>the integral can be written as:<br>
+ (7.350)<br>
+Thus the numerator in Eq. 7.344 has to obey the same equality. This leads to the fol-<br>lowing expression:<br>
+ (7.351)<br>
+By using Eqs. 7.327 and 7.331 the shear stress 0 can be expressed as:<br>
+ (7.352)<br>
+By rearranging Eq. 7.339 constant C can be expressed as a function of A:<br>
+ (7.353)<br>
+<hr>
+<A name=393></a> <br>
+7.4Melt Conveying  <b>381</b><br>
+where:<br>
+ (7.354)<br>
+and:<br>
+ (7.355)<br>
+By expressing both C and 0 as a function of A and inserting these expressions in <br>
+Eq. 7.351, an equation for A is obtained that yields a unique solution of A in terms of <br>known variables, i.e., n, k, m0, , H, v, 1s, and 1b. Thus, the equation for A becomes:<br>
+ (7.356)<br>
+where NNa is the Nahme number. For a temperature dependent power law fluid this <br>
+can be expressed as:<br>
+ (7.357)<br>
+Equation 7.356 shows that the value of constant A depends on the Nahme number <br>NNa, the power law index n, and wall temperatures 1s and 1b.<br>There does not appear to be a general, explicit solution for constant A. Thus, in most <br>cases a numerical scheme has to be used to determine the value of A. A simple solu-<br>tion can be found for the special case of a Newtonian fluid with equal wall tempera-<br>tures. In this case, A depends only on the Nahme number with the following linear <br>relationship:<br>
+ (7.358)<br>
+The value of constant A is determined from Eq. 7.356 and, thus, a unique solution <br>for the temperature profile, velocity profile, and flow rate is obtained. Figure 7.77 <br>shows the right-hand term of Eq. 7.356 as a function of A at a power law index of <br>n = 0.5 and 1s = 0.<br>Three curves are shown for different values of the dimensionless barrel tempera-<br>ture: 1b = 0, 1b = ­1, and 1b = +1. Curves as shown in Fig. 7.77 can be used to find <br>
+a graphical solution to Eq. 7.356. We will take the following example:<br>v  =  0.2 m/s <br>
+ <br>
+H  =  0.005 m <br>
+ <br>
+  =  0.02 C­1 <br>
+ <br>
+n  =  0.5 (s = 2) <br>
+ <br>
+m0 = 104 Nsn/m2 <br>
+ <br>
+k  =  0.24 N/sC<br>
+<hr>
+<A name=394></a><b>382</b>  <b>7Functional Process Analysis</b><br>
+The left-hand term of Eq. 7.356 is 6.94; the corresponding value of constant A = 2.9 <br>when 1s = 1b = 0. The corresponding maximum temperature *1 = lnA = 1.07 and <br>
+the actual maximum temperature T*­ T0 = n*1/ = 26.64°C. Since 1s = 1b = 0 <br>
+the wall temperatures are Ts = Tb = T0. Thus, the maximum temperature in this case <br>
+is about 26.6°C above the wall temperatures. Clearly, in this case there is a signifi-<br>cant amount of viscous heat generation.<br>
+<b> Figure 7.77</b> <br>
+Right-hand term of equation Eq . 7 .356<br>
+When ls = 0 and lb = 1 the value of A = 4.24 as shown in Fig. 7.77. The maximum <br>
+temperature is *1 = 1.44 and T*­ T0 = 29.83°C. Thus, a significant increase in the <br>
+temperature of the moving wall (25°C higher) causes only a small increase in the <br>maximum temperature (about 3°C).<br>At this point the temperature and corresponding velocity profiles can be deter-<br>mined. Figure 7.78 shows the temperature profile at three values of A for the case of <br>equal plate temperatures.<br>
+<b> Figure 7.78</b> <br>
+Temperature profiles when 1s = 1b = 0 <br>
+at several values of A<br>
+<hr>
+<A name=395></a> <br>
+7.4Melt Conveying  <b>383</b><br>
+The temperature profile is symmetric around the  = 0.5 axis. The maximum tem-<br>perature increases with larger values of A. The corresponding velocity profiles are <br>shown in Fig. 7.79.<br>
+<b> Figure 7.79</b> <br>
+Velocity profiles when 1s = 1b = 0 at <br>
+ <br>several values of A<br>
+In the lower half of the channel ( &lt; 0.5) the velocities decrease with increasing A, <br>while in the upper half of the channel the velocities increase with A. Since the velo-<br>city curves are anti-symmetric the total flow rate does not change with varying A.<br>When the wall temperatures are different the effect of temperature changes signifi-<br>cantly. Figure 7.80 shows the temperature profile at three values of A when ls = 0 <br>
+and lb = 1.<br>
+<b> Figure 7.80</b> <br>
+Temperature profiles when 1s = 0 and <br>
+1b = 1 at several values of A<br>
+The temperature profiles are now no longer symmetric. The maximum temperature <br>still increases with larger values of A. The corresponding velocity profiles are shown <br>in Fig. 7.81.<br>
+<hr>
+<A name=396></a><b>384</b>  <b>7Functional Process Analysis</b><br>
+<b> Figure 7.81</b> <br>
+Velocity profiles when 1s = 0 and 1b = 1 <br>
+at several values of A<br>
+The velocities in the lower half of the channel are still reduced with increasing A. <br>However, there is no corresponding increase in the velocities in the upper half of the <br>channel. At low values of A the velocities are decreased throughout the channel, <br>while at higher values of A the velocities in the upper portion are increased relative <br>to the linear velocity profile. As a result, the flow rate is reduced when the barrel <br>temperature is higher than the screw temperature.<br>Temperature profiles for the case where the screw temperature is higher than the <br>barrel temperature are shown in Fig. 7.82.<br>
+<b> Figure 7.82</b> <br>
+Temperature profiles for several values of <br>
+A when 1s = 0 and 1b = ­1<br>
+The corresponding velocity profiles are shown in Fig. 7.83.<br>When the screw temperature is higher than the barrel temperature the velocities <br>are increased relative to the case with equal wall temperatures. As a result the flow <br>rate increases as the screw temperature is increased relative to the barrel tempera-<br>ture. The effect of applied temperature difference on flow rate is shown in Fig. 7.84.<br>
+<hr>
+<A name=397></a> <br>
+7.4Melt Conveying  <b>385</b><br>
+<b> Figure 7.83</b> <br>
+Velocity profiles for adiabatic screw at <br>
+various A values when 1s = 0 and <br>
+1b = ­1<br>
+<b> Figure 7.84</b> <br>
+Flow rate versus constant A at various <br>
+values of 1<br>
+The flow rate reduces as 1 increases, but it increases with the value of A. It is <br>
+clear from Fig. 7.84 that the temperature difference 1 can have a significant effect <br>
+on the flow rate.<br>The equations presented up to this point are valid when Tb &gt; Ts. However, the equa-<br>
+tions are also valid when Tb &lt; Ts as long as the location of the extremum of the tem-<br>
+perature *1 occurs at * &gt; 0. From Eq. 7.336 it can be determined that the extremum <br>
+occurs at:<br>
+ (7.359)<br>
+The extremum occurs at the screw surface when y2 = 0. In this case:<br>
+ (7.360)<br>
+<hr>
+<A name=398></a><b>386</b>  <b>7Functional Process Analysis</b><br>
+Thus, the equations presented are valid for cases where *  0 even when Tb &lt; Ts. <br>
+When * &lt; 0, Eq. 7.332 changes to:<br>
+ (7.361)<br>
+All subsequent equations remain the same except the equation for y2. When * &lt; 0 <br>
+the value of y2 can be determined from:<br>
+ (7.362)<br>
+7.4.3.2.2.2Adiabatic Screw and Isothermal Barrel<br>We will now consider thermal boundary conditions with an adiabatic screw and <br>isothermal barrel. In this case, the thermal boundary condition at the screw surface <br>can be written as:<br>
+ (7.363)<br>
+When this condition is inserted into Eq. 7.330 we obtain Eq. 7.360. The actual tem-<br>perature at the screw surface 1s = lnA. Equation 7.332 now becomes:<br>
+ (7.364)<br>
+The temperature profile can now be written as:<br>
+ (7.365)<br>
+In this case the previously developed expressions for the velocity profile and the <br>flow rate can be used by setting y2 = 0. Temperature profiles for an adiabatic screw <br>
+and isothermal barrel at various values of A are shown in Fig. 7.85.<br>The maximum temperature occurs at the screw surface. The maximum temperature <br>increases with A. The corresponding velocity profiles are shown in Fig. 7.86.<br>The shape of the velocity profiles with one adiabatic wall is considerably different <br>from the shape with two isothermal walls. In the latter case the velocity profile has <br>a typical s-shape while in the former case there is monotonic reduction in the slope <br>of the curves when A &gt; 1. The velocities with an adiabatic screw are higher than <br>with an isothermal screw. As a result, the flow rate is increased considerably com-<br>pared to the isothermal drag flow rate. When A = 1 the flow rate equals 0.5; the flow <br>rate increases with the value of A. When A becomes very large, the flow rate <br>approaches unity and the velocity profile approaches plug flow.<br>
+<hr>
+<A name=399></a> <br>
+7.4Melt Conveying  <b>387</b><br>
+<b> Figure 7.85</b> <br>
+Temperature profiles with adiabatic screw <br>
+for various values of A<br>
+<b> Figure 7.86</b> <br>
+Velocity profiles for several values of A<br>
+<b>7.4.3.3Developing Temperatures<br></b>In the previous section we analyzed the fully developed temperatures in drag flow <br>with corresponding velocity profiles and flow rates. Non-uniform temperatures can <br>clearly have a significant effect on velocities and flow rates. However, before using <br>expressions for the fully developed temperatures we should investigate whether the <br>temperatures are indeed fully developed. This can be determined from an analysis of <br>developing temperatures in drag flow. This problem was investigated in detail by <br>Rauwendaal [303]; some of the important elements of this analysis will be pre-<br>sented; for details, the reader is referred to the original reference.<br>The velocity profile for drag flow of a power law fluid between two flat plates can be <br>expressed as:<br>
+ (7.366)<br>
+<hr>
+<A name=400></a><b>388</b>  <b>7Functional Process Analysis</b><br>
+The volumetric flow rate per unit width can be written as:<br>
+ (7.367)<br>
+where 0 is the location where the shear stress equals zero and the velocity reaches <br>
+a extremum; s is the reciprocal power law index, s = 1/n.<br>The value of 0 is determined by solving:<br>
+ (7.368)<br>
+where gz is the pressure gradient, gz = dP/dz, and:<br>
+ (7.369)<br>
+The solution to Eq. 7.368 can be found graphically by plotting 0 versus g*/gz at <br>
+various values of the power law index. This is shown in Fig. 7.87.<br>
+<b>Figure 7.87</b>Graph of 0 versus g*/gz at various values of the power law index<br>
+The energy equation can be written in dimensionless form by using dimensionless <br>normal coordinate  = y/H and down-channel coordinate  = z/H; this yields:<br>
+ (7.370)<br>
+<hr>
+<A name=401></a> <br>
+7.4Melt Conveying  <b>389</b><br>
+The Peclet number is given by:<br>
+ (7.371)<br>
+where d is the thermal diffusivity of the fluid.<br>The Brinkman number is given by:<br>
+ (7.372)<br>
+The fully developed temperature profile can be determined by solving the energy <br>Eq. 7.370 without the first term. Using the general thermal boundary conditions <br>expressed by Eq. 7.309 and 7.310 the fully developed temperature can be written as:<br>
+ (7.373)<br>
+where:<br>
+ (7.374)<br>
+ (7.375)<br>
+ (7.376)<br>
+The developing temperature profile can be expressed analytically [303]; however, <br>the expressions are rather involved and not easily evaluated. Therefore, the solution <br>will not be presented here. The issue that is of most practical importance is the ther-<br>mal development length (TD). This is defined as the distance in flow direction over <br>
+which the difference between the original temperature and the fully developed tem-<br>perature reduces to one percent of its initial value. The thermal development length <br>depends largely on the Peclet number and the thermal boundary conditions. When <br>the heat flux at the boundaries is greater than zero the thermal development length <br>TD = 0.2NPe; these conditions are the most likely to occur in practice. When isother-<br>
+mal conditions exist at the walls the thermal development length can increase to <br>TD = 0.24NPe.<br>The thermal development length will depend on the plate velocity, plate separation, <br>and the thermal diffusivity since these variables determine the Peclet number; <br>see Eq. 7.371. Typical values of the Peclet number range from 103 to 106 due to the <br>low value of the thermal diffusivity. This means that the flow length will often be <br>
+<hr>
+<A name=402></a><b>390</b>  <b>7Functional Process Analysis</b><br>
+insufficient to reach fully developed thermal conditions, particularly when the plate <br>separation is large. This issue will be addressed in more detail in Chapter 12. In <br>practice this means that with large extruders (diameter greater than 100 mm) fully <br>developed thermal conditions are not likely to be reached in the extruder.<br>When adiabatic conditions occur at the walls (zero heat flux), the thermal develop-<br>ment length becomes infinite because the temperatures will continue to rise indefi-<br>nitely. However, the ultimate shape of the temperature profile (u) for this case can <br>
+be determined. For pure drag flow (zero pressure gradient) u becomes:<br>
+ (7.377)<br>
+This temperature profile is shown in Fig. 7.88 at several values of the Brinkman <br>number.<br>The temperature gradient at the wall is zero. The difference between the wall tem-<br>peratures increases with the Brinkman number. When the Brinkman number is <br>zero all temperatures are the same because there will be no dissipation. In this case, <br>the thermal development length is zero as well. For non-zero values of the Brink-<br>man number the thermal development length is infinite.<br>
+<b> Figure 7.88</b> <br>
+Ultimate shape of the temperature <br>
+ <br>profile with adiabatic walls<br>
+<b>7.4.3.3.1Temperature Development in Screw Extruders</b><br>
+Viscous dissipation and the resulting increase in melt temperature are important in <br>most polymer extrusion operations. Existing theories do not allow a simple predic-<br>tion of temperature development without resorting to numerical techniques. This <br>section describes the development of an analytical theoretical model that allows the <br>calculation of developing melt temperatures in a single screw extruder. The model is <br>based on simplified flow in screw extruders. Initially, we consider only flow in the <br>screw channel and assume the leakage flow through the flight clearance is negligible. <br>The dissipation in the flight clearance is added later.<br>
+<hr>
+<A name=403></a> <br>
+7.4Melt Conveying  <b>391</b><br>
+The polymer melt is considered a power law fluid with a temperature dependent <br>consistency index. We take into account viscous dissipation and heat transfer <br>through the barrel; these are the main factors affecting the melt temperatures. As a <br>result, the analytical results are useful in developing an understanding of the role of <br>the different variables that affect the melt temperature development in the polymer <br>extrusion process. The role of the power dissipated in the flight clearance is briefly <br>discussed and a simple method to include this effect in the temperature calculations <br>is presented. The effect of the flight clearance is significant for polymers that are not <br>strongly shear-thinning and for multi-flighted screws.<br>
+<b>7.4.3.3.2Power Consumption</b><br>
+The viscous dissipation of a power law fluid <br>
+n 1<br>
++ is determined from the consistency <br>
++<br>
+ <br>
+index (m), the shear r<br>
+q at=e ( n 1<br>
+DN<br>
+<br>
+m&amp; = m<br>
+<br>
+), and the power law index (n). If we neglect the power <br>
+s<br>
+<br>
+<br>
+consumed in the flight clearance t  H <br>
+he viscous dissipation can be written as:<br>
+n 1<br>
++<br>
+n 1<br>
++<br>
+ DN<br>
+q = <br>
+m&amp; = m<br>
+<br>
+s<br>
+<br>
+  (7.378)<br>
+ H <br>
+We will use as an example an extruder screw with an outside diameter (D) of 60 <br>mm, a channel depth (H) of 4 mm, a length (L) of 10 D, and running at a rotational <br>speed (N) of 100 rpm or 1.67 rev/sec. The polymer is a medium viscosity, low  density <br>polyethylene with the following properties: the consistency index of the polymer is <br>m = 25,120 Pasn and the power law index n = 0.3 [-]. With these data, the power dis-<br>sipation becomes qs = 7,324,273 W/m3 (7324.3 kW/m3). This is the specific power <br>
+consumption per unit volume. The volume between the barrel and the screw can be <br>approximated with:<br>
+ (7.379)<br>
+This means that the actual power consumption can be determined from:<br>
+ (7.380)<br>
+With a volume of V = 0.0004524 m3, the power consumption becomes Z = 3313.5 <br>and W = 3.313 kW. This power consumption will result in an increase in melt tem-<br>perature, which will be discussed next.<br>
+<b>7.4.3.3.3Temperature Increase</b><br>
+In order to determine the corresponding increase in melt temperature, we need to <br>know the specific heat and the melt density of the polymer. We will take the specific <br>heat Cp = 2300 J/kg°C and the melt density  = 750 kg/m3. The adiabatic tempera-<br>
+ture rise (no heat transfer at the walls) can be determined as follows:<br>
+<hr>
+<A name=404></a><IMG src="Polymer Extrusion by Chris Rauwendaal-404_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-404_2.jpg"><br>
+<b>392</b>  <b>7Functional Process Analysis</b><br>
+ (7.381)<br>
+where   is the volumetric flow rate in axial direction and   the mass flow rate. If we <br>have a mass flow rate of 92.7 kg/hr (0.02574 kg/s), then the adiabatic temperature <br>rise is Ta = 55.97°C. This mass flow rate is typical for a 60-mm extruder running <br>
+at a screw speed of 100 rpm. As a result, we can expect a considerable increase in <br>melt temperature by viscous dissipation over a length of 10 D. Equation 7.381 <br>assumes that the shear rate is determined only by the tangential velocity differ-<br>ences in the extruder screw and that the axial velocity gradients are negligible for <br>the determination of the melt temperature rise. These are generally reasonable <br>assumptions for screw extruders.<br>The flow rate for screw extruders can be predicted from the following equation:<br>
+ (7.382)<br>
+where rt is the throttle ratio (pressure flow divided by drag flow) and  is the flight <br>
+helix angle.<br>With Eq. 7.382 we can express the adiabatic temperature rise as follows:<br>
+ (7.383)<br>
+From this equation, the various parameters that influence the temperature rise by <br>viscous dissipation can be clearly distinguished. The temperature rise increases <br>with consistency index (m), the length (L), the screw diameter (D), the screw speed <br>(N), and the power law index (n). The temperature rise reduces with melt density <br>(), the specific heat (Cp), the channel depth (H), the throttle ratio (rt), and the helix <br>
+angle ().<br>
+<b>7.4.3.3.4Effect of Temperature Dependent Viscosity</b><br>
+The actual rise in melt temperature will generally be less than the values predicted <br>by Eqs. 7.381 and 7.383. One of the reasons for this is that the melt viscosity reduces <br>with increasing temperature. The temperature dependence of the viscosity for a <br>power law fluid can be written as follows:<br>
+ (7.384)<br>
+This means that the consistency index is made to be temperature dependent using <br>an exponential dependence of temperature with a temperature coefficient of a. The <br>consistency index mr is the value at reference temperature Tr.<br>
+<hr>
+<A name=405></a> <br>
+7.4Melt Conveying  <b>393</b><br>
+<b>7.4.3.3.5Adiabatic Case [Dissipation without Conductive Heat Transfer]</b><br>
+We can determine the temperature rise over an infinitesimally small axial distance <br>dx. This leads to the following differential equation:<br>
+ (7.385)<br>
+Constant B1 can be written as:<br>
+ (7.386)<br>
+This differential equation can be solved with the boundary condition T(x = 0) = T0. <br>
+The temperature as a function of axial distance x can now be written as:<br>
+ (7.387)<br>
+If we make the initial temperature T0 = 190°C equal to the reference temperature <br>
+and take the temperature coefficient to be a = 0.02°C--1, then the temperature <br>after 10 D becomes T(x = 0.6) = 227.56°C. This means that the adiabatic tempera-<br>ture rise with a temperature dependent viscosity is 37.56°C compared to 55.97°C <br>for a temperature independent viscosity. Therefore, this indicates that temperature <br>dependence of the viscosity results in a significantly lower temperature rise.<br>Figure 7.89 shows how the melt temperature changes over distance for several values <br>of the temperature coefficient (a). For very low values of "a," the melt temperature <br>increases linearly with distance. This corresponds to the temperature profile for a <br>temperature independent fluid.<br>
+250<br>
+240<br>
+a = 0.0001<br>
+230<br>
+]<br>
+a = 0.02<br>
+220<br>
+a = 0.05<br>
+mperature [C 210<br>
+ <br>Te<br>
+a = 0.10<br>
+Melt<br>
+200<br>
+1900<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Distance [m]<br>
+<b>Figure 7.89</b>Melt temperature vs . distance for several values of temperature coefficient "a"<br>
+<hr>
+<A name=406></a><b>394</b>  <b>7Functional Process Analysis</b><br>
+When the temperature coefficient increases, the melt temperatures reduce. Most <br>semi-crystalline polymers have a temperature coefficient of about 0.02; amorphous <br>polymers generally have a temperature coefficient of about 0.05. Figure 7.89 shows <br>that an increase in the temperature coefficient from 0.02 to 0.05 reduces the tem-<br>peratures significantly, about 10°C after a length of 10 D.<br>
+<b>7.4.3.3.6Effect of Screw Speed and Throttle Ratio</b><br>
+Figure 7.90 shows how the temperature development is affected by screw speed <br>assuming adiabatic conditions.<br>
+250<br>
+600 rpm<br>
+240<br>
+300 rpm<br>200 rpm<br>
+230<br>
+]<br>
+100 rpm<br>
+50 rpm<br>
+220<br>
+25 rpm<br>
+mperature [C 210<br>
+ <br>Te<br>
+Melt<br>
+200<br>
+1900<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Distance [m]<br>
+<b>Figure 7.90</b>Temperature vs . distance for several values of the screw speed<br>
+The effect of increasing screw speed diminishes at higher screw speeds. Increasing <br>the screw speed from 25 to 50 rpm increases the final temperature by about 5°C, <br>while an increase from 100 to 200 rpm increases the final temperature about 4°C.<br>Figure 7.91 shows how the temperature development is affected by the value of the <br>throttle ratio, again assuming adiabatic conditions. The values of the throttle ratio <br>range from ­0.5 to +0.5. The last number in the parentheses behind T represents <br>the throttle ratio. For instance, T(x, 0.02, 1.67, ­0.5) means temperature (T) as a <br>function of axial distance (x) at a temperature coefficient a = 0.02, a screw speed of <br>N = 1.67 rev/sec, and a throttle ratio of rt = ­0.5. The lowest value of throttle ratio <br>
+(­0.5) results in the lowest increase in melt temperature. Negative values of the <br>throttle ratio correspond to a negative axial pressure gradient, and a mass flow rate <br>that is higher than the drag flow rate. Higher mass flow rates result in shorter <br> <br>residence times and, thus, also in lower melt temperatures. When the polymer melt <br>is exposed to a certain shear rate for a shorter time the resulting increase in melt <br>temperature is reduced.<br>
+<hr>
+<A name=407></a> <br>
+7.4Melt Conveying  <b>395</b><br>
+250<br>
+r = -0.5<br>
+r = -0.3<br>
+240<br>
+r = 0.0<br>r = 0.1<br>
+230<br>
+]<br>
+r = 0.2<br>r = 0.3<br>
+220<br>
+mperature [C 210<br>
+ <br>Te<br>
+Melt<br>
+200<br>
+1900<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Distance [m]<br>
+<b>Figure 7.91</b>Temperature vs . distance for several values of the throttle ratio<br>
+When the throttle ratio is zero, the mass flow rate equals the drag flow rate; in this <br>case the axial pressure gradient is zero. When the throttle ratio is positive, the axial <br>pressure gradient is positive and the mass flow rate is less than the drag flow rate. <br>This results in longer residence times and higher melt temperatures.<br>Figure 7.92 shows how the adiabatic temperature development is affected by the <br>value of the throttle ratio and the screw speed. The top curve is actually two curves <br>almost completely overlapping. One curve corresponds to a screw speed of 1.67 rev/<br>sec (100 rpm) and a throttle ratio of 0.4, while the other curve corresponds to a <br>screw speed of 9.3 rev/sec (558 rpm) and a throttle ratio of 0. The bottom curve is <br>also two curves that are almost completely overlapping. One curve corresponds to a <br>screw speed of 1.67 rev/sec (100 rpm) and a throttle ratio of ­0.5, and the other <br>curve corresponds to a screw speed of 0.42 rev/sec (25 rpm) and a throttle ratio of 0.<br>
+250<br>
+240<br>
+560 rpm, r = 0.0<br>
+230<br>
+]<br>
+100 rpm, r = 0.4<br>
+220<br>
+100 rpm, r = -0.5<br>
+mperature [C 210<br>
+ <br>Te<br>
+25 rpm, r = 0.0<br>
+Melt<br>
+200<br>
+1900<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Distance [m]<br>
+<b>Figure 7.92</b>Temperature vs . distance for different values of screw speed and throttle ratio<br>
+<hr>
+<A name=408></a><b>396</b>  <b>7Functional Process Analysis</b><br>
+These two sets of curves indicate that an increase in screw speed can be offset by a <br>reduction in the throttle ratio. The throttle ratio is determined by the inlet and outlet <br>pressure of the melt conveying zone. It is clear, therefore, that reducing barrel pres-<br>sure will reduce polymer melt temperature. This can be achieved by using a less <br>restrictive screen pack, a less restrictive extrusion die, higher die temperatures, or <br>by using a melt pump to generate most of the diehead pressure.<br>
+<b>7.4.3.3.7Conductive Heat Transfer without Dissipation</b><br>
+We can also determine the temperature development for the case of zero viscous dis-<br>sipation and non-zero conductive heat transfer. If the heat flux is determined by the <br>temperature difference between the melt temperature, T, and the barrel coolant tem-<br>perature, Tc, at distance h away from the barrel I.D., the heat flux can be written as<br>
+ (7.388)<br>
+where kb is the thermal conductivity of the barrel.<br>The temperature gradient resulting from heat conduction can be expressed as fol-<br>lows:<br>
+ (7.389)<br>
+With boundary condition T(x = 0) = T0 the change in melt temperature resulting <br>
+from conductive heat transfer can be written:<br>
+ (7.390)<br>
+Equation 7.390 is useful for cases where the conductive heat transfer is important <br>and the viscous heat generation negligible. In most realistic polymer extrusion oper-<br>ations, both the viscous dissipation and the heat conduction are important. The <br>combined effect of these factors will be discussed next.<br>
+<b>7.4.3.3.8Temperature Development with Dissipation and Conduction</b><br>
+Equation 7.387 applies only to temperature development with viscous dissipation <br>under adiabatic conditions, that is, without heat transfer through the barrel or the <br>screw. In reality, however, there will be conductive heat transfer through the barrel. <br>If the heat transfer through the barrel is constant, the temperature gradient is deter-<br>mined by both viscous dissipation and conduction. In this case, the temperature <br>gradient can be expressed as:<br>
+ (7.391)<br>
+<hr>
+<A name=409></a> <br>
+7.4Melt Conveying  <b>397</b><br>
+Constant B1 is given by Eq. 7.386; constant B2 is given by:<br>
+ (7.392)<br>
+The units of B1 and B2 are [°C/m]; these are units of temperature gradient. Constant <br>
+B1 represents the contribution of viscous heating, and constant B2 represents the <br>
+contribution of conductive heat transfer. Variable qc is the heat flux through the bar-<br>
+rel wall. Subject to boundary condition T(x = 0) = T0 the differential equation can be <br>
+solved. The solution can be written as:<br>
+ (7.393)<br>
+For very small values of B2, the results of Eq. 7.393 become the same as the results <br>
+of Eq. 7.387. As such, Eq. 7.393 is a more general description of the developing tem-<br>perature than Eq. 7.387.<br>Equation 7.393 allows determination of the thermal development length, xfd. This is <br>
+the axial distance necessary for the temperature to reach fully developed conditions, <br>i.e., the temperature no longer changes with distance. Since the temperature gradi-<br>ent generally does not reach a zero value until x reaches infinity, it is more practical <br>to define a thermal development length based on the distance at which the tempera-<br>ture gradient falls below a certain value B0. We can set the value of this limiting <br>
+temperature gradient at B0 = 10[°C/m]. The fully developed temperature can be <br>
+determined from Eq. 7.390 as follows:<br>
+ (7.394)<br>
+This leads to the following equation for the fully developed temperature:<br>
+ (7.395)<br>
+When we insert this value of the temperature into Eq. 7.393, we can determine the <br>thermal development length; this can be written as:<br>
+ (7.396)<br>
+Equations for the fully developed temperature were developed by Rauwendaal [326]. <br>However, Eq. 7.395 yields more accurate results because it takes into account the <br>temperature dependence of the viscosity as well as the shear thinning behavior of <br>the polymer melt.<br>
+<hr>
+<A name=410></a><b>398</b>  <b>7Functional Process Analysis</b><br>
+<b>7.4.3.3.9Temperature Profiles with Dissipation and Conduction</b><br>
+With Eqs. 7.391 through 7.396, we can evaluate the combined effect of conduction <br>and viscous dissipation on the temperature development. Figure 7.93 shows the <br>temperature profiles for different screw speeds (0.42, 0.83, 1.67, 3.34, 5.00, and <br>10.00 rev/sec) with a heat flux of ­10,000 W/m2. The results of Fig. 7.93 can be <br>compared to those of Fig. 7.91, which does not include the effect of conduction.<br>
+250<br>
+240<br>
+600 rpm<br>
+300 rpm<br>
+230<br>
+]<br>
+200 rpm<br>
+220<br>
+100 rpm<br>
+mperature [C 210<br>
+ <br>Te<br>
+Melt<br>
+200<br>
+1900<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Distance [m]<br>
+<b>Figure 7.93</b>Temperature profiles at different screw speeds with heat flux of ­10,000 W/m2<br>
+Comparison of Figs. 7.91 and 7.93 shows that the melt temperatures reduce when <br>conduction is taken into account and the heat flux is negative (i.e., the melt is cooled). <br>The reduction in melt temperature is small for high screw speed. However, the reduc-<br>tion in temperature becomes significant at lower screw speed. At 100 rpm with a <br>heat flux of ­10,000 W/m2, the melt temperature drops about 14°C to about 214°C.<br>Figure 7.94 shows the temperature profiles at three different screw speeds with and <br>without conduction; the heat flux is ­10,000 W/m2. At a screw speed of 10 rev/sec <br>(600 rpm), the temperature drops about 2°C after a distance of 10 D. At a screw <br>speed of 3.34 rev/sec (200 rpm), the melt temperature drops about 6.5°C; at 0.83 <br>rev/sec (50 rpm) the melt temperature drops about 27°C. Clearly, at low screw <br>speeds, the melt temperature can be affected significantly by conduction through <br>the barrel. The reason for the large effect of barrel cooling at low screw speed is <br>because the residence time of the polymer increases with reducing screw speed. As <br>a result, more time is available to remove heat from the polymer melt at low screw <br>speed.<br>The thermal development length is plotted against screw speed in Fig. 7.95 at several <br>values of the temperature coefficient. The heat flux at the barrel is set at ­10,000 W/<br>m2 and the value of B0 is set at 10°C/m. This figure clearly shows that the thermal <br>
+development length reduces with increasing values of the temperature coefficient. <br>This is to be expected because increasing values of the temperature coefficient <br>reduce the amount of viscous dissipation when the temperature goes up.<br>
+<hr>
+<A name=411></a> <br>
+7.4Melt Conveying  <b>399</b><br>
+250<br>
+T(600 rpm)<br>
+240<br>
+Heat flux -10,000 W/m2 <br>
+Tc(600 rpm)<br>
+T is without heat transfer <br>
+Tc is with heat transfer<br>
+T(200 rpm)<br>
+230<br>
+]<br>
+Tc(200 rpm)<br>T(50 rpm)<br>
+220<br>
+mperature [C 210<br>
+ <br>Te<br>
+Melt<br>
+200<br>
+Tc(50 rpm)<br>
+1900<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+Distance [m]<br>
+<b>Figure 7.94</b>Temperature profiles at different screw speeds, with and without heat transfer . <br>
+T is the temperature without heat transfer and Tc is the temperature with heat transfer<br>
+For the case shown in Fig. 7.95 the thermal development length becomes longer <br>than the typical metering section of an extruder when the screw speed is greater <br>than about 1 rev/sec (60 rpm). This means that at high screw speeds it cannot be <br>expected that the melt temperatures become fully developed within the length of <br>the extruder.<br>
+8<br>
+temperature coefficient 0.04<br>
+_<br>
+]<br>
+6<br>
+_<br>
+4<br>
+temperature coefficient 0.02<br>
+_<br>
+2<br>
+temperature coefficient 0.01<br>
+Thermal Development Length [m<br>
+_<br>
+_<br>
+_<br>
+_<br>
+_<br>
+_<br>
+_<br>
+_<br>
+_<br>
+0 0<br>
+60<br>
+120<br>
+180<br>
+240<br>
+300<br>
+360<br>
+420<br>
+480<br>
+540<br>
+600<br>
+Screw Speed [rpm]<br>
+<b>Figure 7.95</b>Thermal development length vs . screw speed<br>
+Figure 7.95 shows that the thermal development length becomes zero at a screw <br>speed of about 0.7 rev/sec. It is evident from Eq. 7.396 that the thermal develop-<br>ment length is zero when B1 = (B2 + B0)exp[a(T0--Tr)]. From this relationship, we can <br>
+determine the screw speed at which the thermal development length becomes zero.<br>
+ (7.397)<br>
+<hr>
+<A name=412></a><b>400</b>  <b>7Functional Process Analysis</b><br>
+When B0 is taken as zero, Eq. 7.397 represents the screw speed at which the viscous <br>
+dissipation equals the heat transfer. At this critical screw speed, the melt tempera-<br>ture does not change at all along the length of the extruder screw. This critical value <br>of the screw speed is not dependent on the mass flow rate and specific heat; it can be <br>written as follows:<br>
+ (7.398)<br>
+Thus, the critical screw speed depends on the heat flux, channel depth, temperature <br>coefficient, initial melt temperature, reference temperature, consistency index, <br>screw diameter, and power law index. The critical screw speed increases with the <br>heat flux and channel depth, and reduces with the power law index, consistency <br>index, and screw diameter. The effect of the power law index is shown in Fig. 7.96.<br>Even relatively small increases in the power law index, e.g., from 0.3 to 0.5, can <br>reduce the critical screw speed significantly. This indicates that small increases in <br>the power law index can cause significant increases in viscous heating and melt <br>temperature. This is known in practice when we consider the extrusion characteris-<br>tics of LLDPE relative to LDPE [325]. The power law index of LLDPE is considerably <br>higher than that of LDPE. As a result, LLDPE tends to have more power consump-<br>tion, higher melt temperatures, higher diehead pressures, and is more susceptible <br>to melt fracture.<br>
+90<br>
+] 60 _<br>
+peed [rpm<br>
+30 _<br>
+Critical Screw S<br>
+0<br>
+I<br>
+I<br>
+I<br>
+I<br>
+I<br>
+I<br>
+I<br>
+I<br>
+I<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Power Law Index [-]<br>
+<b>Figure 7.96</b>Critical screw speed vs . power law index for heat flux of 10,000 W/m2<br>
+Equation 7.398 can be rewritten to determine the critical heat flux that is necessary <br>to carry away the heat generated by viscous dissipation. This critical heat flux can <br>be written as follows:<br>
+<hr>
+<A name=413></a> <br>
+7.4Melt Conveying  <b>401</b><br>
+ (7.399)<br>
+This equation indicates that the amount of cooling necessary to maintain constant <br>melt temperature increases with the consistency index, screw diameter, screw <br>speed, and power law index. The critical heat flux reduces with increasing channel <br>depth and temperature coefficient. Equation 7.399 provides a practical tool to deter-<br>mine the cooling capacity required to remove the amount of heat generated by vis-<br>cous dissipation.<br>
+<b>7.4.3.3.10Comparison to Numerical Calculations</b><br>
+To compare the results of the analytical solutions to numerical calculations, several <br>simulations were performed with a 60-mm square pitch extruder screw using the <br>Compuplast Flow 2000 simulation software. Figure 7.97 shows the increase in melt <br>temperature over the length of a 10 D long screw with an O.D. of 60 mm and a chan-<br>nel depth of 4 mm. The polymer melt flow properties are the same as used in the <br>earlier example. The temperature field is shown as a color contour plot.<br>
+225<br>
+10.0<br>
+210 -<br>
+- 8.00<br>
+]<br>
+195 -<br>
+-<br>
+-<br>
+6.00<br>
+]<br>
+180 -<br>
+- 4.00<br>
+-<br>
+165 -<br>
+- 2.00<br>
+-<br>
+Pressure [MPa<br>
+Melt temperature [C<br>
+-<br>
+-<br>
+-<br>
+-<br>
+- 0.00<br>
+1500.0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Axial Length [-]<br>
+<b>Figure 7.97</b>Melt temperature field for 60-mm extruder at 100 rpm<br>
+Figure 7.97 shows that the increase in melt temperature is about 24°C. The adiaba-<br>tic temperature rise (Eq. 7.387) is about 37°C. The difference between these two <br>values is largely due to the fact that the numerical predictions include heat transfer <br>to the barrel. The heat flux (qc) calculated in the numerical simulation is about <br>
+qc = ­10,000 W/m2. Considering that the total heat transfer area (DL) is 0.113 m2 <br>
+the total heat loss is about 1130 W. We can relate this heat loss with a reduction in <br>melt temperature by conductive heat transfer.<br>
+ (7.400)<br>
+<hr>
+<A name=414></a><b>402</b>  <b>7Functional Process Analysis</b><br>
+The reduction in melt temperature from this heat loss is about 19°C. This explains <br>to a large extent the difference in predicted melt temperatures seen when compar-<br>ing the analytical solution to the numerical solution.<br>The temperature rise with dissipation and heat conduction (Eq. 7.393) is about 23°C <br>with a heat flux of ­10,000 W/m2; see Fig. 7.93 with screw speed of 1.67 rev/sec <br>(100 rpm). These results compare favorably to the numerical results; thus, the ana-<br>lytical solutions correspond closely to the numerical calculations.<br>Figure 7.98 shows the numerical predictions of melt temperature and pressure <br>along the length of the extruder as a simple line graph. This graph can be compared <br>to Fig. 7.93; it shows that the predicted temperature profiles exhibit a pattern simi-<br>lar to that shown in Fig. 7.98. This indicates that the analytical equations appear to <br>yield reasonable results.<br>
+<b>Figure 7.98</b>Melt temperature and pressure profiles along the length of the extruder<br>
+The heat flux may not be constant along the length of the extruder. In this case, the <br>temperature profiles in different sections of the extruder may have to be determined <br>separately to take into account different heat fluxes in different parts of the extruder. <br>The same is true if the screw geo metry changes along the length of the melt convey-<br>ing part of the screw.<br>
+<b>7.4.3.3.11Discussion</b><br>
+The equations derived in this section allow a realistic calculation of developing melt <br>temperatures along an extruder screw. Many authors have studied developing melt <br>temperatures [327­329]; however, closed-form analytical solutions as presented in <br>this section have not been published before. Most realistic calculations of develop-<br>ing melt temperatures in screw extruders require numerical analysis. For instance, <br>
+<hr>
+<A name=415></a> <br>
+7.4Melt Conveying  <b>403</b><br>
+Derezinski solved the problem numerically with the barrel temperature as the <br>boundary condition and a power law viscosity [337]. Later the analysis was extended <br>to include a Carreau-Yasuda viscosity model [338].<br>Heat transfer through extruder barrels has been discussed by a number of authors <br>as well [330­334]. The equations developed here allow determination of axial tem-<br>perature profiles if the heat transfer rate through the barrel is known. With good <br>instrumentation on an extruder, it is possible to determine the amount of heat <br>removed through the barrel.<br>It should be noted that the melt temperatures calculated in this section represent <br>the bulk average temperature at a certain axial location along the extruder screw. At <br>a certain cross-section, the melt temperatures will vary in both radial and circum-<br>ferential directions. In order to predict both the axial and the cross-sectional tem-<br>perature profiles, a 3-D analysis has to be performed [335]. Such analyses invariably <br>require numerical techniques to solve the pertinent equations.<br>The analysis as developed to this point assumes that the effect of flight clearance <br>is negligible. This assumption is not unreasonable for the prediction of melt <br> <br>temperature as long as the flight clearance is small so that the flight provides an <br>efficient wiping action [336]. However, this assumption is less reasonable for the <br>prediction of power consumption (Eq. 7.380) because a significant amount of <br>power can be dissipated in the flight clearance. This is particularly true for poly-<br>mers with relatively high values of the power law index, i.e., weakly shear thin-<br>ning materials.<br>Equation 7.380 can be expanded to take into account the power dissipated in the <br>flight clearance. This leads to the following equation:<br>
+ (7.401)<br>
+where the power consumed in the channel is as follows:<br>
+ (7.402)<br>
+and the power consumed in the flight clearance is given by:<br>
+ (7.403)<br>
+where  is the radial flight clearance, Lc the axial channel length, and Lf the axial <br>
+flight length.<br>Equations 7.401 through 7.403 allow a more realistic calculation of the power con-<br>sumption. The melt temperature rise in the flight clearance is generally small <br>because the heat is effectively conducted away due to the fact that the clearance <br>
+<hr>
+<A name=416></a><b>404</b>  <b>7Functional Process Analysis</b><br>
+melt film thickness is very small compared to the channel depth [336]. With <br>Eqs. 7.401 through 7.403, the total power consumption can be written as follows:<br>
+ (7.404)<br>
+Usually the axial flight length is about one-tenth of the channel length (Lf = 0.1 Lc) <br>
+and the flight clearance is about 2% of the channel depth ( = 0.02 H). With these <br>numbers, the power consumed in the flight clearance is about 30% of the power <br> <br>consumed in the channel when n = 0.3, about 70% when n = 0.5, and about 150% <br>when n = 0.7. Clearly, for values of the power law index over 0.3, the power con-<br>sumption in the clearance cannot be neglected in most cases. For multi-flighted <br>screws, the power consumed in the flight clearance is even more significant and <br>certainly not negligible.<br>The bracketed term on the right-hand side of Eq. 7.404 can be used to as a correction <br>term in Eqs. 7.381 through 7.400 to incorporate the effect of the power dissipated in <br>the flight clearance. Obviously, for polymers with a relatively large value of the <br>power law index, the correction can be significant.<br>
+7.4.3.3.11.1Conclusions<br>This section describes the derivation of analytical equations of developing melt tem-<br>peratures in screw extruders. The analytical equations for temperature as a function <br>of axial distance are useful in predicting axial melt temperature profiles. The advan-<br>tage of analytical equations is that the factors influencing temperature development <br>can be easily identified and their effect determined in a quantitative fashion. Both <br>the temperature and shear rate dependence of the viscosity strongly affect the devel-<br>oping temperatures in the extruder.<br>The analytical predictions compare well to numerical predictions. This indicates <br>that the analytical equations can be useful in the analysis of melt temperature devel-<br>opment in single screw extruders. The results indicate that the melt temperatures <br>can become fully developed if the heat flux through the barrel is substantial and if <br>the screw speed is not too high. When the screw speed is high and the consistency <br>index large it is not likely that the melt temperatures will be fully developed at the <br>discharge end; this is particularly true for large diameter extruders.<br>
+<b>7.4.3.4Estimating Fully Developed Melt Temperatures<br></b>It is very useful to be able to predict melt temperature in extrusion, particularly in the <br>extrusion of temperature sensitive polymers. Examples are extrusion of crosslinkable <br>polymers, foamed polymers, and polymers that are susceptible to degradation. Unfor-<br>tunately, the proper calculation of melt temperature is rather involved and requires <br>the use of numerical techniques, the most popular being finite element analysis.<br>
+<hr>
+<A name=417></a> <br>
+7.4Melt Conveying  <b>405</b><br>
+In this section we will describe a method to predict the fully developed melt tem-<br>perature in screw extruders based on simple analytical expressions. The method is <br>easy to use and leads to quantitative results with a minimum of time expenditure.<br>
+<b>7.4.3.4.1Melt Temperature Calculation</b><br>
+The fully developed melt temperature is reached when the viscous heat generation <br>is balanced by the heat flux away from the polymer melt. The viscous dissipation in <br>the extruder will cause an increase in melt temperature resulting in reduced viscos-<br>ity, which will result in a reduction in viscous dissipation. The melt temperature will <br>reach a steady state value when the viscous dissipation has reduced to the point that <br>it equals the heat flux from the polymer melt.<br>The viscous dissipation is determined by the product of shear stress and shear rate. <br>The average shear rate in the screw channel can be approximated by the Couette <br>shear rate:<br>
+ (7.405)<br>
+where D is the barrel diameter, N the screw rotational speed, and H the depth of the <br>screw channel.<br>If the melt viscosity is described with a power law equation, the viscous dissipation <br>can be written as:<br>
+ (7.406)<br>
+where m is the consistency index and n the power law index of the polymer melt.<br>The consistency index is a function of melt temperature. The temperature depend-<br>ence of the consistency index can be written as:<br>
+ (7.407)<br>
+where T is the temperature, T0 the reference temperature, m0 the consistency index <br>
+at temperature T0, and T the temperature coefficient of the viscosity. The viscous <br>
+dissipation can thus be written as:<br>
+ (7.408)<br>
+<hr>
+<A name=418></a><b>406</b>  <b>7Functional Process Analysis</b><br>
+The viscous dissipation is a specific power consumption, i.e., power per unit volume. <br>In S.I. units it is expressed in Watts per cubic meter [W/m3]. The heat flux away <br>from the polymer melt is determined by the heat flux from the melt to the barrel and <br>screw. If the screw is neutral the heat flux to the screw is usually small and can be <br>assumed to be negligible. If screw cooling is used, this assumption will not be cor-<br>rect. The heat flux (heat flow per unit cross-sectional area) for cooling the polymer <br>melt is determined by Fourier's law of conductive heat transport:<br>
+ (7.409)<br>
+Where kb is the thermal conductivity of the barrel, T/rb is the temperature gradi-<br>
+ent in the barrel at the polymer/metal interface. If qc occurs over unit length, then <br>
+qc represents the power per unit volume. Thus, if qc is divided by the thickness, t, <br>
+over which the heat conduction occurs, it attains the units of specific power, just <br>as qv.<br>In many cases, the temperature at the inside and outside barrel surfaces will not be <br>known. In those situations we have to find another method to determine the heat <br>flux through the barrel. This issue was studied Radovich [277] who compared the <br>cooling capability of air- and water-cooled extruder barrels.<br>The melt temperature will rise initially and then level off as the viscous dissipation <br>reduces with increasing melt temperature. When a steady state is achieved, the melt <br>temperature no longer changes along the length of the extruder. This is called the <br>fully developed melt temperature or equilibrium melt temperature. This tempera-<br>ture Te can be determined from a simple energy balance equating the viscous dissi-<br>
+pation to the conductive heat loss:<br>
+ (7.410)<br>
+where Tbi is the inside barrel temperature, Tbo the outside barrel temperature, and tb <br>
+the thickness of the barrel.<br>This equation leads to the following expression for the fully developed melt tem-<br>perature:<br>
+ (7.411)<br>
+where qc is given by Eq. 409 and qv0 is the viscous dissipation in the polymer melt at <br>
+the reference temperature T0; it can be written as:<br>
+ (7.412)<br>
+<hr>
+<A name=419></a> <br>
+7.4Melt Conveying  <b>407</b><br>
+We now have a quantitative, analytical expression from which the fully developed <br>temperature can be calculated. If we define the equilibrium melt temperature rise <br>Te as the difference between the reference temperature and the fully developed <br>
+temperature, Te = Te--T0, we can write:<br>
+ (7.413)<br>
+When qc &gt; qv0 the equilibrium melt temperature rise will be negative; when qc &lt; qv0 <br>
+the equilibrium melt temperature rise will be positive. Equation 7.413 provides a <br>simple and convenient expression from which the effect of the various factors influ-<br>encing the melt temperature becomes clear. These effects can be easily quantified as <br>discussed next.<br>
+<b>7.4.3.4.2Influence of Important Parameters</b><br>
+There are several factors that affect the melt temperature. They can be categorized <br>in three main categories:<br>
+ <br>
+ <i>Material properties:</i> barrel thermal conductivity, melt consistency index, power <br>
+law index, and temperature coefficient of the melt viscosity<br>
+ <br>
+ <i>Operational parameters:</i> screw speed and barrel temperature difference<br>
+ <br>
+ <i>Machine design parameters:</i> barrel diameter, barrel thickness, and channel depth<br>The effect of the various parameters is shown qualitatively in the following table.<br>
+<b>Table 7.1</b>The Effect of Various Parameters on Equilibrium Melt Temperature Rise<br>
+kb<br>
+m<br>
+n<br>
+<b></b>T<br>
+N<br>
+Tb<br>
+D<br>
+tb<br>
+H<br>
+Te<br>
+<br>
+<br>
+<br>
+<br>
+<br>
+<br>
+<br>
+<br>
+<br>
+As shown in Table 7.1, an increase in the thermal conductivity of the barrel material <br>reduces the melt temperature. Considering that the thermal conductivity of various <br>metals varies substantially, it makes sense to consider using a highly conductive <br>barrel material if low melt temperatures are required. The thermal conductivity of <br>various metals is shown in Table 7.2. Corrosion-resistant metals tend to have a low <br>conductivity, while copper-containing alloys have a conductivity about three to five <br>times higher than carbon steel.<br>An increase in the consistency index will increase melt temperatures because the <br>viscous dissipation will increase; see Eqs. 7.406 and 7.408. The numeric value of <br>the consistency index is the same as the viscosity at shear rate one (  = 1) for a <br>power law fluid. The consistency index, therefore, is closely related to the melt index <br>(MI). A low melt index indicates a high value of the consistency index. Clearly, it will <br>be much more difficult to control melt temperature for a low MI material than for a <br>
+<hr>
+<A name=420></a><b>408</b>  <b>7Functional Process Analysis</b><br>
+high MI material. Figure 7.99 shows a graph of the equilibrium melt temperature <br>rise versus consistency index for two values of the cooling flux (qc = 10,000 W/m3 <br>
+and qc = 100,000 W/m3) and the coefficient of viscosity T = 0.02 [K­1].<br>
+<b>Table 7.2</b>Thermal Conductivity of Various Metals<br>
+Thermal conductivity<br>
+Material<br>
+[W/m°K]<br>
+[BTU ft/ft2hr°F]<br>
+Hastelloy C276<br>
+11.25<br>
+6.5<br>
+Inconel 718<br>
+11.42<br>
+6.6<br>
+Inconel 625<br>
+9.86<br>
+5.7<br>
+Monel 400<br>
+21.80<br>
+12.6<br>
+Monel 500<br>
+17.47<br>
+10.1<br>
+4140 steel<br>
+42.56<br>
+24.6<br>
+4340 steel<br>
+42.21<br>
+24.4<br>
+17­4 stainless<br>
+17.82<br>
+10.3<br>
+316 stainless<br>
+16.09<br>
+9.3<br>
+304 stainless<br>
+16.29<br>
+9.4<br>
+AlZnMgCu<br>
+160<br>
+92.5<br>
+CuBe-2<br>
+115<br>
+66.5<br>
+CuCoBe<br>
+210<br>
+122.4<br>
+Z 434<br>
+105<br>
+60.7<br>
+<b> Figure 7.99</b>Equilibrium <br>
+melt temperature rise <br>
+ <br>versus consistency index<br>
+The straight-line relationship using a semi-log scale indicates that the melt tempera-<br>ture increases exponentially with the consistency index. An increase in the cooling <br>flux by a factor of ten reduces the melt temperature rise by about 115°K for all <br> <br>values of the consistency index.<br>
+<hr>
+<A name=421></a> <br>
+7.4Melt Conveying  <b>409</b><br>
+The power law index is a measure of the degree of shear thinning. Lower values of <br>the power law index result in lower viscosity values at high shear rates. In other <br>words, as the power law index becomes smaller, the polymer becomes more shear <br>thinning. This will have a strong effect on the melt temperature rise, particularly at <br>high shear rates, as shown in Fig. 7.100.<br>Figure 7.100 shows that an increase in power law index from 0.3 to 0.6 will increase <br>the melt temperature about 35°K at a shear rate of 10 s­1 and about 70°K at a shear <br>rate of 100 s­1. This indicates that the power law index has a powerful effect on melt <br>temperature (excuse the pun!). Figure 7.100 explains why LDPE tends to run at <br>lower melt temperatures than LLDPE or metallocenes. LDPE has a power law index <br>of about 0.3, while LLDPE and metallocenes have a power law index of about 0.6.<br>
+300<br>
+250<br>
+shear rate 10 s^-1<br>
+]<br>
+shear rate 25 s^-1<br>
+200<br>
+shear rate 50 s^-1<br>shear rate 100 s^-1<br>
+150<br>
+100<br>
+50<br>
+0<br>
+-50<br>
+Melt Temperature Rise [K -100<br>
+-150<br>
+<b> Figure 7.100</b> <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1<br>
+Equilibrium melt temperature <br>
+Power Law Index<br>
+rise versus power law index<br>
+The effect of the temperature coefficient of the melt viscosity is quite clear from <br>Eq. 7.413. When the temperature coefficient of the viscosity (T) increases, the melt <br>
+temperature rise will reduce. Amorphous polymers generally have a much greater <br>temperature coefficient than semi-crystalline polymers; the difference is about a fac-<br>tor of ten. This means that the melt temperature rise in semi-crystalline polymers <br>will be about ten times greater than in amorphous polymers.<br>An increase in screw speed will raise the viscous dissipation and thus the melt tem-<br>perature. Since shear rate is directly proportional to screw speed (see Eq. 7.405), the <br>effect of screw speed can be seen from Fig. 7.100. However, it is more obvious by <br>re-plotting the data with the shear rate along the horizontal axis; see Fig. 7.101.<br>Increasing screw speed will increase shear rate correspondingly. Figure 7.101 <br>shows that the melt temperature increases rapidly at low screw speed and more <br>slowly at higher screw speed. As discussed earlier, higher values of the power law <br>index result in higher melt temperatures.<br>
+<hr>
+<A name=422></a><b>410</b>  <b>7Functional Process Analysis</b><br>
+Increasing the barrel diameter will increase the shear rate--other factors being <br> <br>constant. This will increase viscous dissipation and melt temperatures as discussed <br>earlier. Another problem with larger barrel diameters is that the heat transfer sur-<br>face area increases with the diameter squared, while the channel volume increases <br>with the diameter cubed. As a result, the heat transfer becomes less effective with <br>larger diameter extruders. It is well known in the extrusion industry that the ability <br>to influence melt temperature by changes in barrel temperature is very limited for <br>large extruders.<br>
+200<br>
+150<br>
+100<br>
+50<br>
+0<br>
+Melt Temperature Rise [K]<br>
+power law index n=0.3<br>
+-50<br>
+power law index n=0.5<br>power law index n=0.7<br>
+-100<br>
+0<br>
+20<br>
+40<br>
+60<br>
+80<br>
+100<br>
+Shear rate [s^-1]<br>
+<b>Figure 7.101</b>Equilibrium melt temperature rise versus shear rate<br>
+Increasing the barrel thickness will reduce the heat flux through the barrel assum-<br>ing that the inner and outer barrel temperatures are the same. As a result, a thicker <br>barrel will result in higher melt temperatures.<br>Finally, increasing the channel depth will reduce the shear rate and viscous heating <br>as discussed earlier; this will result in lower melt temperatures. In fact, the channel <br>depth is one of the most critical screw design parameters to control melt tempera-<br>ture. Deep-flighted screws are used when the viscous dissipation and melt tempera-<br>tures have to be minimized. That is why screws used to extrude rubbers generally <br>have deep channels. The same is true for cooling extruders in tandem extrusion <br>lines for foamed polymers.<br>
+<b>7.4.3.4.3Discussion</b><br>
+This analysis provides a simple and fast method to estimate the fully developed melt <br>temperature in screw extruders. The effect of material properties, processing condi-<br>tions, and machine design parameters can be determined quantitatively. As a result, <br>the analysis can be used to predict how melt temperature will change when another <br>
+<hr>
+<A name=423></a> <br>
+7.4Melt Conveying  <b>411</b><br>
+plastic is extruded and how the screw design or processing conditions should be <br>changed to change melt temperature.<br>It should be noted that there are several simplifying assumptions in the analysis. We <br>have assumed that the melt temperatures are uniform across the depth of the chan-<br>nel. In reality this is not the case; in fact, large melt temperature changes can occur <br>across the depth of the channel. We can assume that the melt temperature calcu-<br>lated with this analysis corresponds to a bulk average melt temperature. We have <br>also assumed that the fully developed melt temperature is reached before the end of <br>the extruder. This is a reasonable assumption for small diameter extruders; how-<br>ever, this may not be a good assumption for large diameter extruders as discussed in <br>the previous section.<br>It was also assumed that the viscous dissipation in the flight clearance does not <br>affect the melt temperature in the screw channel. This would appear to be a ques-<br>tionable assumption. However, finite element analysis has shown that the actual <br>melt temperature rise in the clearance region is relatively small [278]. The reason <br>for this is that the heat transfer in the flight clearance is very effective because the <br>clearance is generally quite thin.<br>Lastly, it was assumed that the details of the screw geo metry do not affect the heat <br>transfer to the barrel; this is not completely true. The number of flights, the flight <br>clearance, the flight helix angle, and the flight width all affect the heat transfer from <br>the polymer melt to the barrel. If we want to study the effect of these parameters in <br>detail we have to use a more complicated, numerical analysis.<br>
+<b>7.4.3.5Assumption of Stationary Screw and Rotating Barrel<br></b>The theory developed up to this point is based on a model where the screw is <br> <br>stationary and the barrel rotates around the screw. It is assumed that the flow that <br>results is the same as when the barrel is stationary and the screw rotates in the <br>opposite direction. This assumption was considered valid for over fifty years until <br>several workers challenged this assumption, first in the early 1990s [272­276], and <br>then more recently [323]. Because of the importance of this issue we will critically <br>analyze this assumption to determine to what extent the assumption is correct. Flow <br>will be analyzed using the parallel plate assumption with either the barrel or the <br>screw considered moving. Flow will also be analyzed without the parallel plate <br>assumption, using a cylindrical coordinate system, again considering both cases. <br>This analysis is based on a study by Osswald et al. [281].<br>
+<b>7.4.3.5.1Parallel Plate Analysis with Moving Screw</b><br>
+The parallel plate analysis with moving barrel and Newtonian fluid was already dis-<br>cussed in Section 7.4.1. If we take the barrel stationary and the screw moving at <br>velocity vst = DsN, the velocity profile can be determined from Eq. 7.196 using the <br>
+<hr>
+<A name=424></a><b>412</b>  <b>7Functional Process Analysis</b><br>
+following boundary conditions: vt(y = 0) = vst and vt(y = H) = 0. This results in the <br>
+following velocity profile in the tangential direction:<br>
+ (7.414)<br>
+The maximum tangential pressure gradient occurs when there is no net output from <br>the extruder. The maximum pressure gradient can be determined from:<br>
+ (7.415)<br>
+If it is assumed, again, that the leakage flow is negligible ( tl = 0), the pressure gra-<br>
+dient can be determined to be:<br>
+ (7.416)<br>
+With this, the tangential velocity profile at closed discharge can be written as:<br>
+ (7.417)<br>
+The corresponding velocity profile with moving barrel is:<br>
+ (7.418)<br>
+The tangential shear rate can be obtained by taking the first derivative of vt with <br>
+respect to normal distance; this results in the following expression:<br>
+ (7.419)<br>
+The corresponding expression with moving barrel is:<br>
+ (7.420)<br>
+It is clear that Eq. 7.419 is quite close to Eq. 7.420; the difference is in the tangential <br>velocity term. Equation 7.393 uses vbt, while Eq. 7.419 uses vst. The difference <br>
+between these two velocities is determined by the screw diameter D and the channel <br>depth H. In most plastic extruders, the channel depth is about 0.05 D; this will <br>result in a difference between vbt and vst of about 10%. Thus, there is a difference in <br>
+the tangential velocities and shear rates when we take the screw moving rather than <br>the barrel moving in the flat plate model. However, when the channel depth is small <br>relative to the screw diameter, the difference is not substantial.<br>
+<hr>
+<A name=425></a> <br>
+7.4Melt Conveying  <b>413</b><br>
+<b>7.4.3.5.2Flow Analysis Using Cylindrical Coordinates</b><br>
+To determine the velocities in the cylindrical coordinate system (CCS), the momen-<br>tum equation has to be expressed in cylindrical coordinates. Again, the flow is <br>assumed to be in steady state; inertia, centrifugal, and gravitational forces are <br>assumed negligible and the fluid viscosity constant. For the angular component this <br>can be written as:<br>
+ (7.421)<br>
+If we assume that the angular pressure gradient g is constant and that the angular <br>
+velocity depends only on normal distance r, the velocity can be obtained by double <br>integration:<br>
+ (7.422)<br>
+Constants c1 and c2 have to be determined from the boundary conditions. The cylin-<br>
+drical system is shown in Fig. 7.102.<br>
+<b>7.4.3.5.3Cylindrical System with Rotating Barrel</b><br>
+When the barrel rotates and the screw is stationary, the boundary conditions are <br>v(Rs) = 0 and v(Rb) = ­Rb, where  is the angular velocity ( = 2N); see Fig. 7.102. <br>
+From these conditions we can determine c1 and c2:<br>
+ (7.423)<br>
+ (7.424)<br>
+Rotating<br>
+Stationary<br>
+barrel<br>
+barrel<br>
+<br>
+<br>
+<br>
+<br>
+Stationary<br>
+Rotating<b> Figure 7.102</b> <br>
+screw<br>
+screw<br>
+The cylindrical system in <br>
+two kinematic conditions<br>
+When g = 0, Eq. 7.421 describes the drag flow between concentric cylinders with <br>
+the outer cylinder rotating around the stationary inner cylinder. The angular pres-<br>sure gradient at closed discharge is determined from a mass balance. In this case <br>
+<hr>
+<A name=426></a><b>414</b>  <b>7Functional Process Analysis</b><br>
+the tangential flow rate equals the tangential leakage flow, both per unit axial <br>length:<br>
+ (7.425)<br>
+If the leakage flow is considered negligible, the angular pressure gradient at closed <br>discharge can be determined from Eq. 7.425 using Eqs. 7.422 through 7.424:<br>
+ (7.426)<br>
+where  is the ratio of the radii:  = Rs/Rb.<br>When  approaches unity, g/Rb approaches the tangential pressure gradient in the <br>
+flat plate case. The angular shear rate is determined from the first derivative of the <br>angular velocity; this leads to:<br>
+ (7.427)<br>
+<b>7.4.3.5.4Cylindrical System with Rotating Screw</b><br>
+When the screw rotates and the barrel is stationary, the boundary conditions are <br>v(Rs) = RsN and v(Rb) = 0, where  is the angular velocity ( = 2N). From these <br>
+conditions we can determine c1 and c2:<br>
+ (7.428)<br>
+ (7.429)<br>
+When g = 0 these equations describe the drag flow between concentric cylinders <br>
+with the inner cylinder rotating inside the stationary outer cylinder.<br>The maximum angular pressure gradient is determined from a mass balance. The <br>tangential flow rate has to equal the tangential flow with the screw minus the tan-<br>gential leakage flow, both per unit axial length:<br>
+ (7.430)<br>
+If the leakage flow is considered negligible, the maximum angular pressure gradi-<br>ent can be determined to be exactly the same expression as for the moving barrel <br>case, i.e. Eq. 7.416. At this point the angular velocities for the barrel rotating, vb(r), <br>
+<hr>
+<A name=427></a> <br>
+7.4Melt Conveying  <b>415</b><br>
+can be compared to the angular velocities for the screw rotating, vs(r). The velocity <br>
+profiles are the same if the following condition is fulfilled:<br>
+ (7.431)<br>
+This condition states that the velocities with the barrel turning should be the same <br>as the velocities with the screw turning relative to a cylindrical coordinate system <br>rotating with the screw at 2N radians/second. In fact, the left-hand side of <br>Eq. 7.431 describes the velocities seen by an observer rotating with the screw. It <br>turns out that this condition is fulfilled exactly for all values of the angular pressure <br>gradient. Some investigators have neglected to make the correction expressed in <br>Eq. 7.431 and incorrectly concluded that the velocities in the two kinematic condi-<br>tions were different.<br>Figure 7.103 shows the velocities as a function of radial distance.<br>
+<b>Figure 7.103</b>Velocities vs and vb versus radial distance<br>
+The top curve shows the velocities with rotating screw and the bottom curve shows <br>the velocities with rotating barrel; the angular pressure gradient for both curves is <br>­1E5 [Pa/rad]. When the top curve is shifted according to the transformation in <br>Eq. 7.431, the corrected velocities fall exactly on the bottom curve. Even though only <br>one curve is visible, there are actually two curves in the graph: the two curves <br>overlay exactly.<br>
+<b>7.4.3.5.5Flow Rate</b><br>
+The flow rate can be determined by integrating the velocity profile (Eq. 7.422) from <br>Rs to Rb. This results in the following expression per unit axial width:<br>
+ (7.432)<br>
+<hr>
+<A name=428></a><b>416</b>  <b>7Functional Process Analysis</b><br>
+Equation 7.432, which is for the moving barrel case, is exactly the same as for the <br>moving screw case. This expression can be compared to the corresponding flow rate <br>expression for the parallel plate system. For the flat plate system with moving barrel <br>this can be written as:<br>
+ (7.433)<br>
+For the flat plate system with moving screw this can be written as:<br>
+ (7.434)<br>
+The three expressions are compared graphically in Fig. 7.104 where the flow rate <br>per unit width is plotted against the pressure gradient. The screw root radius is <br>0.090 m, the barrel radius 0.100 m, the viscosity 500 Pas, and the screw speed <br>60 rpm.<br>
+Rs = 0.090 m<br>
+Cylindrical sy<br>
+Cylindr<br>
+stem<br>
+R<br>
+ical system<br>
+b = 0.100 m<br>
+m = 500 Pa.s<br>
+N = 60 rev/min<br>
+0.003<br>
+Flat plate system, moving barrel<br>
+Flat plate system, moving barrel<br>
+]<br>
+2 /s<br>
+Flat plate system, moving screw<br>
+Flat plate system, moving screw<br>
+0.002<br>
+ rate [m<br>
+Flow<br>
+0.0010<br>
+2E5<br>
+4E5<br>
+6E5<br>
+8E5<br>
+10E5<br>
+Tangential pressure gradient [Pa/rad]<br>
+<b>Figure 7.104</b>Flow rate versus tangential pressure gradient for cylindrical and flat plate system; <br>
+flow rate is per unit axial width<br>
+The flow rate for the cylindrical system is slightly higher than for the flat plate sys-<br>tem with the moving barrel. The difference is quite small, less than 2% when the <br>channel depth is 10% of the barrel radius (5% of barrel diameter). This value is a <br>typical metering depth for extruder screws. However, the difference between the <br>cylindrical system and the flat plate with moving screw is much larger, between 10 <br>and 15%. This indicates that the flat plate analysis with moving screw results in <br>large errors.<br>
+<hr>
+<A name=429></a> <br>
+7.4Melt Conveying  <b>417</b><br>
+The difference between the cylindrical and flat plate system will increase when the <br>H/D ratio increases. This is shown in Fig. 7.105 where the flow rate is plotted <br>against the screw root radius.<br>Figure 7.105 illustrates that the difference between the cylindrical and flat plate <br>system increases when the screw root radius reduces. When the root radius is 80% <br>of the barrel radius, the flow rate in the flat plate system with moving barrel is about <br>4% lower than the cylindrical system. However, the flow rate in the flat plate system <br>with moving screw is about 30% lower. The flat plate system with moving screw <br>results in significant errors and should not be used in serious analysis of screw <br>extruders.<br>
+0.006<br>
+g = 105 Pa/rad<br>
+Cylindrical system<br>
+Cylindrical system<br>
+0.005<br>
+Flat plate system, moving barrel<br>
+Flat plate system, moving barrel<br>
+] 0.004<br>
+2 /s<br>
+0.003<br>
+0.002<br>
+Flow rate [m<br>
+Flat plate system, moving screw<br>
+Flat plate system, moving screw<br>
+0.001<br>
+<b> Figure 7.105</b> <br>
+00.08<br>
+0.085<br>
+0.09<br>
+0.095<br>
+0.10<br>
+Flow rate versus screw root radius for <br>
+Screw root radius [m]<br>
+cylindrical and flat plate system<br>
+The flow rate expressions can be expressed in terms of cross-channel (x) and down-<br>channel coordinates by considering that the tangential pressure gradient gt can be <br>
+expressed as:<br>
+ (7.435)<br>
+<b>7.4.3.5.6Discussion on Kinematic Conditions</b><br>
+In the flat plate system the velocities and shear rates for the case where the barrel <br>moves relative to a stationary screw are different from the case where the screw <br>moves relative to the stationary barrel. The difference is determined by the screw <br>root diameter Ds and the barrel diameter Db. In most single screw extruders where <br>
+the root diameter is close to the barrel diameter, the difference between the two <br>analyses is relatively small, about 10%. However, when the root diameter is consid-<br>erably smaller than the barrel diameter the flat plat analysis with moving screw <br>results in serious errors. The reason for the differences in the flat plate system is the <br>fact that it cannot account for the curvature of the channel.<br>In the cylindrical system there is no difference in velocities between the rotating <br>barrel and rotating screw cases; the velocities are exactly the same. When the velo-<br>
+<hr>
+<A name=430></a><b>418</b>  <b>7Functional Process Analysis</b><br>
+cities are the same the shear rates and flow rates are the same as well. The results <br>of 3-D boundary element analysis using the geo metry of an actual extruder screw <br>confirm that there is no difference between the two kinematic conditions [281]; the <br>same results were obtained with 3-D finite element analysis. Further, careful extru-<br>sion experiments with a 150-mm extruder confirmed that there is no difference <br>between rotating the barrel and rotating the screw [281]. The experimental 150-mm <br>extruder was capable of rotating either the screw or the barrel. The barrel was con-<br>structed of a cast acrylic tube and the screw machined from an aluminum cylinder. <br>The screw was filled with a 10,000 centistokes silicone fluid and back-pressure was <br>controlled with a valve. Tracer dyes were injected in the screw channel. The screw <br>was rotated one revolution, causing a distortion of the tracer material. The barrel <br>was then rotated one revolution in the same direction; this brought the tracer mate-<br>rial back to the original position and shape. This process was repeated several times <br>with negligible change in the shape of the tracer.<br>Several other workers have confirmed the validity of the moving barrel assumption. <br>A thorough study was conducted by Pape et al. [279, 280] using finite element ana-<br>lysis comparing the moving screw to the moving barrel case. The results of this <br>analysis indicate that velocities with the moving barrel are the same as with the <br>moving screw. Pape et al. also investigated non-isothermal flow conditions.<br>With these results there is strong evidence that rotating the barrel results in the <br>same flow as opposite rotation of the screw, as long as one is dealing with highly <br>viscous fluids (creeping flow). With the preponderance of evidence supporting the <br>moving barrel assumption one would expect that this issue would have been put to <br>rest once and for all. However, a recent study by Sikora and Sasimowski [276] using <br>a helical coordinate system (HCS) claims a difference between the two kinematic <br>conditions. Considering that the CCS is a special case of the HCS it would appear <br>that it is not possible for the HCS analysis to show a difference between the two <br> <br>kinematic conditions while the CCS shows no difference. At the time of this writing, <br>the source of this discrepancy has not yet been determined.<br>The deformation of the fluid depends on the magnitude of the strain rate tensor, <br>which, in turn, depends on its second invariant. The invariants of the strain rate ten-<br>sor are independent of their frame of reference. As a result, the deformation is the <br>same if the screw rotates (fixed coordinate system) or if the barrel rotates (rotating <br>coordinate system). The only difference between the two kinematic conditions arises <br>from centrifugal and Coriolis forces. Spalding et al. [282] demonstrated with a 3-D <br>finite element analysis of a single screw extruder that for polymer melts the centri-<br>fugal and Coriolis force effects are negligible. This means that with a helical coordi-<br>nate system the flow with a rotating barrel should be the same as with a rotating <br>screw. The only reason that the flat plate system indicates a difference between the <br>two kinematic conditions is that it does not account for the curvature of the channel.<br>
+<hr>
+<A name=431></a> <br>
+7.5Die Forming  <b>419</b><br>
+<b></b><br>
+ <b>7.5Die Forming</b><br>
+In this functional zone, the shaping of the polymer takes place. In many respects, <br>the die forming zone is the most important functional zone because the actual shape <br>of the final product develops in this zone. The die forming zone is essentially always <br>a pressure consuming zone. The pressure built up in the preceding functional zones <br>is used in the die forming zone. The diehead pressure is the pressure required to <br>force the polymer melt through the die. The diehead pressure is not determined by <br>the extruder but by the extruder die. The variables that affect the diehead pressure <br>are:<br>1.  The geo metry of the flow channel in the die<br>2.  The flow properties of the polymer melt<br>3.  The temperature distribution in the polymer melt<br>4.  The flow rate through the die<br>When these variables remain the same, the diehead pressure will be the same <br>whether a single screw extruder or a twin screw extruder is used. The main function <br>of the extruder is to supply a homogeneous polymer melt to the die at the required <br>rate and diehead pressure. The rate and diehead pressure should be steady and the <br>polymer melt should be homogeneous in terms of temperature and consistency. Die <br>design and analysis of flow in dies are two of the most complicated elements of poly-<br>mer process engineering. One of the reasons is that in a proper analysis of the die <br>forming process, the polymer melt can no longer be assumed to be purely viscous <br>because some important die flow phenomena such as extrudate swell (often errone-<br>ously referred to as die swell) cannot be explained with this simplifying assumption. <br>Therefore, the polymer melt has to be analyzed as a viscoelastic fluid, and this com-<br>plicates the analysis of the die forming process considerably.<br>Even if the polymer melt is assumed to be purely viscous, the analysis will generally <br>be quite complicated because many dies have flow channels of complex shape. As a <br>result, accurate analysis of flow in extrusion dies generally requires three-dimen-<br>sional flow analysis. This presents quite a challenge for simple Newtonian fluids. <br>Three-dimensional flow analysis of viscoelastic non-Newtonian fluids is beyond the <br>capabilities of most (if not all) die designers. Not surprisingly, die designers often <br>take an empirical approach to die design.<br>There are only two basic flow channel geometries that are rather easy to analyze: <br>the circular flow channel and the slit flow channel (rectangular cross section with <br>W &gt;&gt; H). Most other geometries are quite difficult to analyze.<br>
+<hr>
+<A name=432></a><b>420</b>  <b>7Functional Process Analysis</b><br>
+<b>7.5.1Velocity and Temperature Profiles</b><br>
+Velocity profiles and temperature profiles in extruder dies are intimately related <br>because of the high polymer melt viscosity and because the melt viscosity is tem-<br>perature dependent. It is important to understand and appreciate this interrelation-<br>ship in order to understand the die forming process and the variables that influence <br>this process. The relationship between velocity and temperature profiles can be <br>illustrated by considering the down-channel velocity profile in a circular die. Typical <br>velocity profiles are shown in Fig. 7.106 for several values of the power law index.<br>
+elocity<br>
+n = 1.0 .7 .5 .3<br>
+ed v<br>aliz<br>rm<br>No<br>
+<b> Figure 7.106</b> <br>
+Velocity versus radius for power law <br>
+fluid at various values of the power <br>
+Normalized radius<br>
+law index n<br>
+The more or less parabolic velocity profile is observed that is typical of pressure <br>driving flow (pipe flow). The velocity curve for the Newtonian fluid (n = 1) is an <br>exact parabola. The curves for the non-Newtonian fluid are not purely parabolic <br>(quadratic); they have a flattened center region and a larger gradient at the wall. <br>From the velocity profile, one can obtain the shear rate profile by determining the <br>local gradients of the velocity profile. This is described in Fig. 7.107 for both the <br>Newtonian and non-Newtonian fluid. For all curves the shear rate in the center of <br>the flow channel is zero and the highest shear rate occurs at the wall. The wall shear <br>rate for the non-Newtonian fluid, however, is considerably higher than for the New-<br>tonian fluid.<br>As a result of the velocity gradients, there will be heat generation in the fluid from <br>the viscous dissipation of energy. In rectilinear flow, the rate of energy dissipation <br>per unit volume is given by:<br>
+ (7.436)<br>
+<hr>
+<A name=433></a> <br>
+7.5Die Forming  <b>421</b><br>
+<b> Figure 7.107</b> <br>
+Shear rate versus radial distance for <br>
+several values of the power law index<br>
+This is a simplified version of the general expression for energy dissipation, Eq. 5.5d. <br>If the fluid can be described by the power law equation, then Eq. 7.436 becomes:<br>
+ (7.437)<br>
+Thus, the local viscous dissipation is determined by the local shear rate raised to the <br>power n + 1. Since the highest shear rate occurs at the wall, it is clear that the high-<br>est viscous dissipation will also occur at the wall. As a result of the non-uniform <br>shear rate distribution in the flow channel, there will be a non-uniform viscous heat <br>generation in the flow channel. The largest amount of viscous heat generation occurs <br>at the wall. As a result of the viscous heat generation, the temperature of the poly-<br>mer melt will increase. But since the viscous heat generation is non-uniform across <br>the flow channel, the temperature rise of the polymer melt will also be non-uniform <br>across the flow channel.<br>Because the die wall material usually has a thermal conductivity much higher than <br>polymer melts, adiabatic conditions are not likely to be achieved. On the other hand, <br>it is also not likely that the wall temperature will remain constant. In this case, the <br>heat flux through the wall would be such as to maintain a perfectly constant tem-<br>perature along the wall. This is referred to as an isothermal wall boundary condi-<br>tion. Because of the high thermal conductivity of the wall, the isothermal boundary <br>condition is more likely to occur than the adiabatic boundary condition. Adiabatic <br>conditions can be approached if the die is very well insulated. In most actual cases, <br>the true thermal boundary condition will be somewhere between isothermal and <br>adiabatic, depending on the design of the die and external conditions around the <br>die. A typical temperature profile resulting from the velocity profiles shown in <br>Fig. 7.106 is shown in Fig. 7.108.<br>Initially, the maximum temperature will occur close to the wall; later, this maximum <br>will shift towards the center. A quantitative method of evaluating temperature pro-<br>files will be discussed next.<br>
+<hr>
+<A name=434></a><b>422</b>  <b>7Functional Process Analysis</b><br>
+<b> Figure 7.108</b> <br>
+Temperature versus radial distance<br>
+Important relationships for shear stress, shear rate, velocity, and flow rate for the <br>pressure flow of power law fluids in a slit flow channel are given in Fig. 7.109. Figure <br>7.110 shows the same relationships for a circular flow channel. These relationships <br>are valid for fluids with temperature independent viscosity.<br>
+<b>Figure 7.109</b>Pressure flow of a power  <br>
+<b>Figure 7.110</b>Pressure flow of a power  <br>
+law fluid through a slit<br>
+law fluid through a circular  channel<br>
+<hr>
+<A name=435></a> <br>
+7.5Die Forming  <b>423</b><br>
+Dinh and Armstrong [146] have developed general analytical solutions for the local <br>temperature change due to viscous dissipation for non-Newtonian fluids. Their <br>results apply to fluids with flow properties that are insensitive to temperature. The <br>dimensionless normal distance is defined as y0 = 2y/H for a slit or y0 = 2y/D for a <br>circular flow channel. The dimensionless velocity is defined as v0z = vz/vmax; it can <br>
+be expressed in terms of dimensionless normal distance as follows:<br>
+ (7.438)<br>
+Since the analysis deals with rectilinear flow, the subscript of the velocity will be <br>deleted.<br>The dimensionless viscosity is defined as:<br>
+ (7.439)<br>
+where:<br>
+ (7.439a)<br>
+The energy equation expressed in terms of dimensionless quantities can be written <br>as:<br>
+ (7.440)<br>
+where:<br>
+ (7.440a)<br>
+ (7.440b)<br>
+The following boundary conditions will be considered:<br>
+ (7.440c)<br>
+ (7.440d)<br>
+ (7.440e)<br>
+<hr>
+<A name=436></a><b>424</b>  <b>7Functional Process Analysis</b><br>
+where N is the Biot number as defined in Eq. 5.74. When N is zero, there is no <br>exchange of heat; adiabatic conditions prevail. When N is infinitely large, the wall <br>temperature equals the temperature of the polymer melt; this corresponds to iso-<br>thermal conditions. Normal values for the Biot number in extruder dies range from <br>1 to 100. As long as the Biot number is non-zero, there will be a fully developed <br>temperature profile. However, when the Biot number is zero, the temperature in the <br>fluid will continue to rise without limit.<br>The fully developed temperature profile 1 is the solution to the following differen-<br>
+tial equation:<br>
+ (7.441)<br>
+with the following boundary condition:<br>
+ (7.441a)<br>
+The solution to this equation is:<br>
+ (7.442)<br>
+where s is the reciprocal power law index (s = 1/n).<br>Equation 7.442 is a very useful relationship for determining fully developed tem-<br>perature profiles in pipe flow law fluids. The maximum fully developed temperature <br>always occurs at the center of the flow channel as can be seen from Eq. 7.442 as well <br>as from Fig. 7.111, which illustrates the fully developed temperature profile under <br>isothermal wall conditions and at various values of the power law index.<br>
+<b> Figure 7.111</b> <br>
+Fully developed temperature <br>
+ <br>profiles, isothermal wall conditions<br>
+<hr>
+<A name=437></a> <br>
+7.5Die Forming  <b>425</b><br>
+As the fluid becomes more pseudo-plastic, the fully developed temperature profile <br>becomes more flattened in the center. The temperature remains almost constant in a <br>center region, which extends for about half the channel height. The larger tempera-<br>ture gradients occur in a relatively thin region at the walls.<br>The temperature profile as a function of normal distance and down-channel distance <br>is postulated to be of the form:<br>
+ (7.443)<br>
+For the details of the determination of eigenfunctions xi and eigenvalues ai, the <br>
+reader is referred to the paper by Dinh and Armstrong [146]. The eigenvalues are <br>given by:<br>
+ (7.444)<br>
+where j = 1, 2, 3,... and  is given by:<br>
+ (7.445)<br>
+where /3    2/3,  w is the dimensionless shear rate at the wall, and (p) is the <br>
+gamma function. The gamma function is defined as:<br>
+ (7.446)<br>
+where:<br> <br>
+(p+1) = p(p)  if p &gt; 0<br>
+ <br>
+(p+1) = p! <br>
+if p is a positive integer<br>
+ <br>
+(1) = 1<br>
+ <br>
+<hr>
+<A name=438></a><b>426</b>  <b>7Functional Process Analysis</b><br>
+In evaluating the eigenvalues using Eq. 7.444, a convenient relationship involving <br>the gamma function is:<br>
+ (7.447)<br>
+The eigenfunctions xj are given by:<br>
+ (7.448)<br>
+where:<br>
+ (7.449)<br>
+where Jv is the Bessel function of the first kind of order v.<br>The Bessel function Jv is given by:<br>
+ (7.450)<br>
+The expansion coefficients ci in Eq. 7.443 for the slit problem are given by:<br>
+ (7.451)<br>
+If the fluid is a power law fluid and the wall condition is isothermal, the eigenvalues <br>become:<br>
+ (7.452)<br>
+The dimensionless temperature as a function of dimensionless normal distance for <br>Newtonian fluid is shown in Fig. 7.112.<br>With isothermal conditions the fully developed temperature profile is reached when <br>z0 &gt; 5. With adiabatic conditions the temperature profile continues to increase along <br>the length of the channel.<br>
+<hr>
+<A name=439></a> <br>
+7.5Die Forming  <b>427</b><br>
+<br>
+<b>Figure 7.112</b>Dimensionless temperature versus normal distance for Newtonian fluid with <br>
+isothermal conditions (left) and adiabatic conditions (right)<br>
+Figure 7.113 shows the temperature profiles for a power law fluid with power law <br>index n = 0.5.<br>
+Power law fluid n=0.5<br>adiabatic wall<br>
+T<br>emperature<br>
+T<br>emperature<br>
+<b>Figure 7.113</b>Dimensionless temperature versus normal distance for power law fluid with power <br>
+law index n = 0 .5 with isothermal conditions (left) and adiabatic conditions (right)<br>
+The dimensionless temperature and down-channel distance in Figs. 7.112 and 7.113 <br>are related to the average fluid velocity. In comparing Fig. 7.112 to 7.113, it is seen <br>that increased pseudo-plasticity reduces the temperature build-up in the polymer <br>melt at equal volumetric flow rates. It is also quite apparent that the temperature <br>build-up under adiabatic conditions is substantially higher than under isothermal <br>conditions.<br>The fully developed temperature profile, Eq. 7.442, is rather easy to calculate. How-<br>ever, the developing temperature profile, Eqs. 7.443­7.452, involves some rather <br>lengthy and complex calculations. Although the solutions to the developing tem-<br>perature profiles are analytical, obtaining actual results still requires computations <br>that go beyond the capabilities of most pocket calculators. In many practical cases, <br>one would like to know to what extent the actual temperature profile approaches the <br>fully developed temperature profile. This can be determined by using a dimension-<br>less axial distance ZGz, defined as:<br>
+ (7.453)<br>
+<hr>
+<A name=440></a><b>428</b>  <b>7Functional Process Analysis</b><br>
+where L is the length of the channel and NGz the Graetz number as defined in Eq. 5.68. <br>
+If ZGz  1, the temperature profile will be essentially fully developed in most practi-<br>
+cal heat transfer situations as analyzed, for instance, by Winter [147]. For a slit flow <br>channel, the axial length Z1 at which the temperature profile will be fully developed <br>
+can now be expressed as:<br>
+ (7.454)<br>
+If we use some typical numbers H = 0.003 [m],   = 0.1 [m/s], and  = 1E­7 [m2/s] <br>then the length Z1 = 9 [m]. Considering that most dies are no longer than about <br>
+0.5 m it is clear that the temperatures in dies generally will not get close to fully <br>developed conditions. In most practical extrusion operations, the length of the die <br>land is much too short to reach a fully developed temperature profile. Thus, in order <br>to determine the actual stock temperatures, one must evaluate the developing tem-<br>perature profile.<br>The total amount of power dissipated in the flow channel of a die is simply deter-<br>mined by the product of flow rate and pressure drop along the flow channel. Thus:<br>
+ (7.455)<br>
+If it is assumed that all this power is used to raise the temperature of the polymer <br>melt, i.e., adiabatic conditions, then the volume average rise in melt temperature <br>can be determined from:<br>
+ (7.456)<br>
+Thus, the volume average adiabatic temperature rise is directly proportional to the <br>pressure drop. If the pressure drop is 30 MPa (= 4350 psi), a typical volume average <br>temperature rise is about 10°C. In most cases, however, the actual volume average <br>temperature rise will be less because heat transfer will take place at the die wall. In <br>other words, adiabatic conditions will not occur in actual extrusion operation. It is <br>important to realize, though, that local temperatures can be considerably higher <br>than the volume average temperature. The highest shear rates occur at the wall, and <br>consequently the highest viscous heat generation occurs at the wall. Therefore, the <br>polymer close to the wall will heat up much faster than the polymer in the center <br>region of the channel. Thus, it is quite possible that a local temperature rise close to <br>the wall can be considerably higher than the volume average temperature rise.<br>In extrusion, one is always concerned about temperature uniformity. One of the <br>important requirements of the extruder is to deliver to the die a polymer melt of <br>uniform consistency and temperature. However, it should be realized that, even if <br>the polymer melt entering the die is uniform in temperature, non-uniformities in <br>
+<hr>
+<A name=441></a> <br>
+7.5Die Forming  <b>429</b><br>
+temperature will develop in the die as a result of the non-uniform velocity gradients. <br>This is inherent to die flow. The temperature build-up and non-uniformities can be <br>reduced by reducing the shear rate. This can be achieved by lowering the flow rate <br>through the die or by opening up the die flow channel. Another possibility is to use <br>coextrusion with the outer layer being thin and of low viscosity. One can also use an <br>external lubricant in the polymer to reduce a die flow problem; however, this may <br>introduce other problems as well (e.g., solids conveying problems, loss of mechani-<br>cal properties, etc.).<br>
+<b>7.5.2Extrudate Swell</b><br>
+A well-known and typical phenomenon in polymer melt extrusion is the swelling of <br>the extrudate as it leaves the die. This is sometimes referred to as die swell; how-<br>ever, it is not the die but the polymer that swells. The elasticity of the polymer melt <br>is largely responsible for the swelling of the extrudate upon leaving the die. This is <br>primarily due to the elastic recovery of the deformation that the polymer was <br>exposed to in the die. The elastic recovery is time-dependent. A die with a short land <br>length will have a large amount of swelling, while a long land length will reduce the <br>amount of swelling. The polymer has what is often called a "fading memory." A <br>deformation can be recovered to a large extent shortly after the occurrence of the <br>deformation; however, after longer times the recoverable deformation reduces. Thus, <br>a certain amount of relaxation occurs in the die depending on the geo metry of the <br>flow channel.<br>It should be noted that extrudate swelling is not unique to viscoelastic fluids. It can <br>also occur in an inelastic or purely viscous fluid; this has been demonstrated experi-<br>mentally and theoretically. Obviously, in an inelastic fluid, the mechanism of extru-<br>date swell is not an elastic recovery of prior deformation. The swelling is caused by <br>a significant rearrangement of the velocity profile as the polymer leaves the die; this <br>is shown in Fig. 7.114.<br>
+<b> Figure 7.114</b> <br>
+Change in velocity profiles in the <br>
+die exit region<br>
+<hr>
+<A name=442></a><b>430</b>  <b>7Functional Process Analysis</b><br>
+The velocity profile changes from an approximately parabolic velocity profile in the <br>die to a straight velocity profile (plug flow) a short distance away from the die. In a <br>Newtonian fluid, this causes a small amount of extrudate swelling (about 10%) at <br>low Reynolds numbers (Nre &lt; 16). Viscoelastic fluids exhibit about the same amount <br>
+of swelling at low shear rates, but much larger amounts of swelling can occur at <br>high shear rates (over 200%!).<br>One of the main problems with extrudate swell is that it is generally not uniformly <br>distributed over the extrudate. This means that some areas of the extrudate swell <br>more than others. If the geo metry of the exit flow channel of the die is made to <br>match the geo metry of the required product, the uneven swelling will cause a distor-<br>tion of the extrudate and the required product geo metry cannot be obtained. Draw-<br>down cannot cure this problem! Therefore, the geo metry of the exit flow channel <br>must generally be different from the required product geo metry. This can be under-<br>stood by analyzing the velocity profiles in the flow channel. Figure 7.115 shows the <br>velocity profile in a flow channel with a square cross-section; the figure shows the <br>upper right quadrant with the solid lines indicating points of equal velocity.<br>
+<b> Figure  7.115</b> <br>
+Velocity profile in square channel<br>
+It can be seen that the shear rates at the wall vary significantly. The wall shear rate <br>in the corner is relatively low, while the highest shear rate occurs at the middle of <br>the wall. Therefore, the elastic recovery in the middle will be larger than the elastic <br>recovery at the corners. This results in "bulged" extrudate. It is not possible to <br>obtain a perfectly square extrudate with a perfectly square flow channel. To elimi-<br>nate this problem one has to modify the shape of the flow channel to compensate for <br>the uneven swelling of the extrudate. A good die designer must anticipate the <br>
+<hr>
+<A name=443></a> <br>
+7.5Die Forming  <b>431</b><br>
+amount of uneven swelling and design the flow channel accordingly. This is a diffi-<br>cult task, and the determination of the flow channel geo metry is often done by a <br>"trial and error" process.<br>Accurate mathematical prediction of the die swell profile is quite difficult and, <br>therefore, determination of the proper flow channel geo metry to minimize uneven <br>swelling by engineering calculations is generally not practical. The non-uniform <br>extrudate swell and the correction of the flow channel geo metry are illustrated in <br>Fig. 7.116.<br>
+<b> Figure  7.116</b> <br>
+Uneven swelling of extrudate and possible <br>
+Uncorrected die<br>
+Corrected die<br>
+correction<br>
+The amount of swelling is very much dependent on the nature of the material. Some <br>polymers exhibit considerable swell (100 to 300%), e.g., polyethylenes; other poly-<br>mers exhibit lower swell, e.g., polyvinylchloride. When PVC is extruded at relatively <br>low temperatures (165 to 175°C), the swell ranges from 10 to 20% only. This is one <br>of the reasons that PVC is such a popular material in profile extrusion; it conforms <br>quite well to the geo metry of the die flow channel and has good melt strength.<br>
+<b>7.5.3Die Flow Instabilities</b><br>
+In extrusion, certain die flow instabilities can occur that may seriously affect the <br>entire extrusion process and render the extruded product unacceptable. Two very <br>important die flow instabilities are shark skin and melt fracture.<br>
+<b>7.5.3.1Shark Skin<br></b>Shark skin manifests itself as a regular ridged surface distortion, with the ridges <br>running perpendicular to the extrusion direction. A less severe form of shark skin is <br>the occurrence of matness of the surface, where the glossy surface cannot be main-<br>tained. Shark skin is generally thought to be formed in the die land or at the exit. It <br>is dependent primarily on the temperature and the linear extrusion speed. Factors <br>such as shear rates, die dimensions, approach angle, surface roughness, L/D ratio, <br>and material of construction seem to have little or no effect on shark skin.<br>The mechanism of shark skin is postulated to be caused by the rapid acceleration of <br>the surface layers of the extrudate when the polymer leaves the die; this is illus-<br>
+<hr>
+<A name=444></a><b>432</b>  <b>7Functional Process Analysis</b><br>
+trated in Fig. 7.117. If the stretching rate is too high, the surface layer of the polymer <br>can actually fail and form the characteristic ridges of the shark skin surface [148]. <br>High-viscosity polymers with narrow molecular weight distribution (MWD) seem to <br>be most susceptible to shark skin instability [149, 150].<br>
+<b> Figure  7.117</b>Change in velocity <br>
+profile in the die exit region<br>
+The shark skin problem can generally be reduced by reducing the extrusion velocity <br>and increasing the die temperature, particularly at the land section. There is some <br>evidence that running at very low temperatures can also reduce the problem [151]. <br>Selection of a polymer with a broad MWD will also be beneficial in reducing shark <br>skin. Using an external lubricant can also reduce the problem. This can be done by <br>using an additive to the polymer or by coextruding a thin, low-viscosity outer layer.<br>
+<b>7.5.3.2Melt Fracture<br></b>Melt fracture is a severe distortion of the extrudate, which can take many different <br>forms: spiraling, bambooing, regular ripple, random fracture, etc.; see Fig. 7.118.<br>
+<b> Figure  7.118</b>Various forms of melt fracture<br>
+It is not a surface defect like shark skin, but is associated with the entire body of the <br>molten extrudate. However, many workers do not distinguish between shark skin <br>and melt fracture, but lump all these flow instabilities together under the term melt <br>fracture. There is a large amount of literature on the subject of melt fracture (e.g., <br>[152­164]). Despite the large number of studies on melt fracture, there is no clear <br>
+<hr>
+<A name=445></a> <br>
+7.5Die Forming  <b>433</b><br>
+agreement as to the exact cause and mechanism of melt fracture. It is quite possible <br>that the mechanism is not the same for different polymers and/or different flow <br>channel geometries [169]. Linear polymers tend to develop an instability of the <br>shear flow in the die land, while branched polymers tend to develop instabilities in <br>the converging region of the die flow channel.<br>However, there is relatively uniform agreement that melt fracture is triggered when <br>a critical wall shear stress is exceeded in the die. This critical stress is in the order <br>of 0.1 to 0.4 MPa (15 to 60 psi). A number of mechanisms have been proposed to <br>explain melt fracture. Some of the more popular ones are:<br>1.  Critical elastic deformation in the entry zone<br>2.  Critical elastic strain<br>3.  Slip-stick flow in the die<br>The effect of the entry zone has been demonstrated by many workers. In general, the <br>smaller the entry angle, the higher the deformation rate at which instability occurs. <br>Gleissle [230] has proposed a critical elastic strain as measured by recoverable <br>strain. Based on measurements with 11 fluids, he proposed the existence of a criti-<br>cal value of the ratio of first normal stress difference to the shear stress, the average <br>value being 4.63 for 11 widely different fluids with a standard deviation of about 5%. <br>Much larger differences were found in the critical shear stress, the average being <br>3.7E5 Pa with a standard deviation of about 55%. In 1961, Benbow, Charley, and <br>Lamb [232, 233] introduced the slip-stick mechanism to explain flow instability and <br>extrudate distortion. Above a certain critical stress, the polymer melt is believed to <br>experience intermittent slipping due to lack of adhesion between the melt and the <br>die wall in order to relieve excessive deformation energy absorbed as a result of flow <br>through a die. A large number of workers have observed slippage by various tech-<br>niques.<br>More recent work by Utracki and Gendron on pressure oscillations in extrusion of <br>polyethylenes [231] led them to conclude that the pressure oscillation does not seem <br>to be related to elasticity or slip. They conclude that the parameter responsible for <br>pressure oscillations is the critical strain (Hencky) value c of the melt. For LLDPE, <br>
+c &lt; 3, for HDPE, c &lt; 2, while for LDPE, c &gt; 3.5. The instability seems to be based on <br>
+the inability of the polymer melt to sustain levels of strain larger than the critical <br>strain.<br>Streamlining the flow channel geo metry has been found to reduce the tendency for <br>melt fracture in branched polymers. Increased temperatures, particularly at the wall <br>of the die land, enable higher extrusion rates before melt fracture appears. The <br> <br>critical wall shear stress appears to be relatively independent of the die length, <br>radius, and temperature. The critical stress seems to vary inversely with molecular <br>weight, but seems to be independent of MWD. Certain polymers exhibit a super-<br>extrusion region, above the melt fracture range, where the extrudate is not distorted <br>
+<hr>
+<A name=446></a><b>434</b>  <b>7Functional Process Analysis</b><br>
+[165]. This process is particularly advantageous with polymers that melt fracture at <br>relatively low rates, such as FEP. In superextrusion, the polymer melt is believed to <br>slip relatively uniformly along the die wall. The occurrence of slip in extruder dies <br>has been studied by a number of investigators, e.g., [166, 168]. However, it is still <br>not clear whether the slip is actual loss of contact of polymer melt and metal wall or <br>whether it is failure of a thin polymeric layer very close to the metal surface.<br>The melt fracture problem can be reduced by streamlining the die, increasing the <br>temperature at the die land, running at lower rates, reducing the MW or the polymer <br>melt viscosity, increasing the cross-sectional area of the exit flow channel, or by <br>using an external lubricant. In some instances, the melt fracture problem can be <br>solved by going to superextrusion; this process is used particularly often in the wire <br>coating industry where high line speeds are quite important for economic produc-<br>tion.<br>
+<b>7.5.3.3Draw Resonance<br></b>Draw resonance occurs in processes where the extrudate is exposed to a free sur-<br>face stretching flow, such as blown film extrusion, fiber spinning, and blow molding. <br>It manifests itself in a regular cyclic variation of the dimensions of the extrudate. An <br>extensive review [169] and an analysis [170] of draw resonance were done by Petrie <br>and Denn. Draw resonance occurs above a certain critical draw ratio while the poly-<br>mer is still in the molten state when it is taken up and rapidly quenched after take-<br>up.<br>Draw resonance will occur when the resistance to extensional deformation decreases <br>as the stress level increases. The total amount of mass between die and take-up may <br>vary with time because the take-up velocity is constant but not necessarily the <br>extrudate dimensions. If the extrudate dimensions reduce just before the take-up, <br>the extrudate dimensions above it have to increase. As the larger extrudate section <br>is taken up, a thin extrudate section can form above it; this can go on and on. Thus, <br>a cyclic variation of the extrudate dimensions can occur. Draw resonance does not <br>occur when the extrudate is solidified at the point of take-up because the extrudate <br>dimensions at the take-up are then fixed [171, 172]. Isothermal draw resonance is <br>found to be independent of the flow rate. The critical draw ratio for almost-Newtonian <br>fluids such as nylon, polyester, polysiloxane, etc., is approximately 20. The critical <br>draw ratio for strongly non-Newtonian fluids such as polyethylene, polypropylene, <br>polystyrene, etc., can be as low as 3 [173]. The amplitude of the dimensional varia-<br>tion increases with draw ratio and drawdown length.<br>Various workers have performed theoretical studies of the draw resonance problem <br>by linear stability analysis. Pearson and Shah [174, 175] studied inelastic fluids and <br>predicted a critical draw ratio of 20.2 for Newtonian fluids. Fisher and Denn [176] <br>confirmed the critical draw ratio for Newtonian fluids. Using a linearized stability <br>
+<hr>
+<A name=447></a> 7.6Devolatilization  <b>435</b><br>
+analysis for fluids that follow a White-Metzner equation, they found that the critical <br>draw ratio depends on the power law index n and a viscoelastic dimensionless num-<br>ber. The dimensionless number is a function of the die take-up distance, the tensile <br>modulus, and the velocity at the die. Through their analysis, Fisher and Denn were <br>able to determine stable and unstable operating regions. In some instances, draw <br>resonance instability can be eliminated by increasing the draw ratio, although under <br>most operating conditions draw resonance is eliminated by reducing the draw ratio.<br>White and Ide [177­180] demonstrated experimentally and theoretically that poly-<br>mers whose elongational viscosity increases with time or strain do not exhibit draw <br>resonance, but undergo cohesive failure at high draw ratios. A polymer that behaves <br>in such a fashion is LDPE. Polymers whose elongational viscosity decreases with <br>time or strain do exhibit draw resonance at low draw ratios and fail in a ductile <br> <br>fashion at high draw ratios. Examples of polymers that behave in such a fashion are <br>HDPE and PP. Lenk [181] proposed a unified concept of melt flow instability. His <br>main conclusions are that all flow instabilities originate at the die entrance and that <br>melt fracture and draw resonance are not distinct and separate flow phenomena; <br>both are caused by elastic effects that have their origin at the die entrance. Lenk's <br>analysis, however, is purely qualitative and does not offer much help in the engi-<br>neering design of extrusion equipment or in determining how to optimize process <br>conditions to minimize instabilities.<br>
+<b></b><br>
+ <b>7.6Devolatilization</b><br>
+Devolatilization is a function that is not performed on all extruders, as opposed to <br>the other functions, such as solids conveying, melting, melt conveying, and mixing. <br>Therefore, devolatilizing extruders are relatively specialized. However, as polymer <br>processing operations become more sophisticated, it is becoming less unusual for <br>the extruder to be used for continuous devolatilization. There have been relatively <br>few engineering analyses of the devolatilization process in extruders. The first major <br>effort seems to have been the work by Latinen [182]. Other analytical studies of <br>devolatilization in extruders have been made by Coughlin and Canevari [183], <br> <br>Roberts [184], Biesenberger [185, 186], and Denson [236]. The physical model of <br>devolatilization in a single screw extruder is shown in Fig. 7.119.<br>
+<hr>
+<A name=448></a><b>436</b>  <b>7Functional Process Analysis</b><br>
+<b> Figure 7.119</b> <br>
+Model for devolatilization in screw <br>
+ <br>extruders<br>
+The exposure time f of the wiped film as it travels with the barrel surface from the <br>
+flight clearance to the melt pool is:<br>
+ (7.457)<br>
+where N is the screw speed and Y the fraction of the channel width occupied by the <br>melt pool. If the channel is partially filled, then Y &lt; 1 and Y represents the degree of <br>fill in the extraction section.<br>The total volumetric flow rate of the wiped film is:<br>
+ (7.458)<br>
+where hm is the melt film thickness and Lb is the total axial length of the devola-<br>
+tilizing zone.<br>The feedback ratio nf is:<br>
+ (7.459)<br>
+where   is the total volumetric flow rate through the extruder.<br>The feedback ratio is the part of the material that splits off of the main material flow <br>to form a film through which volatiles are escaping. The feedback ratio nf can be <br>
+interpreted as the extent of the surface renewal of the melt film. As the melt film <br>
+<hr>
+<A name=449></a> 7.6Devolatilization  <b>437</b><br>
+enters the melt pool, again a certain amount of back mixing will take place, depend-<br>ing on the width of the melt pool and the helix angle. The melt pool will also lose <br>volatiles because of the exposed surface. The exposure time of the exposed surface <br>of the melt pool is limited because of the circulatory motion in the melt pool. Roberts <br>[184] approximated the exposure time of the melt pool by:<br>
+ (7.460)<br>
+where H is the channel depth and vbx the cross-channel component of the barrel <br>
+velocity.<br>The local bulk evaporation rate p(z) can be expressed as:<br>
+ (7.461)<br>
+where D is the diffusion coefficient, C(z) the local bulk average concentration of the <br>
+melt pool, and Ce the equilibrium concentration at the vapor-liquid interface. For a <br>
+derivation of Eq. 7.461, see Section 5.4.2.<br>The local film evaporation f(z) can be expressed as:<br>
+ (7.462)<br>
+where C(z) is the concentration of the film reentering the melt pool at point z, having <br>
+left the melt pool a distance z1 downstream; see Fig. 7.119. The formulation of the <br>
+model can be completed by taking a steady-state material balance on the volatile <br>component over a differential element of volume of a thickness z. The balance sim-<br>ply states that the reduction in convective transport of the volatile component equals <br>the film evaporation plus the bulk evaporation:<br>
+ (7.463)<br>
+Now the value C(z + z1) has to be determined to obtain C(z). Concentration C(z + <br>
+z1) is the initial concentration of the melt film. This melt film reenters the melt <br>
+pool at location z with concentration C(z). As suggested by Roberts [184], a Taylor's <br>
+series expansion about z can be used:<br>
+ (7.464)<br>
+Terms of order three and higher will be assumed to be negligible. The concentration <br>C(z) at which the polymer reenters the melt pool is the initial concentration C(z + <br>
+z1) of the film as it leaves the melt pool minus the amount of volatiles lost through <br>
+<hr>
+<A name=450></a><b>438</b>  <b>7Functional Process Analysis</b><br>
+evaporation in the film during film exposure time f. If Xf is the stage efficiency of <br>
+the diffusing film, the concentrations C(z) and C(z + z1) can be related by:<br>
+ (7.465)<br>
+By using Eqs. 7.464 and 7.465, the mass balance equation can be rewritten as:<br>
+ (7.466)<br>
+where:<br>
+ (7.466a)<br>
+Equation 7.439 can be written in dimensionless form:<br>
+ (7.467)<br>
+where:<br>
+ (7.467a)<br>
+ (7.467b)<br>
+ (7.467c)<br>
+ (7.467d)<br>
+ (7.467e)<br>
+ (7.467f)<br>
+<hr>
+<A name=451></a> 7.6Devolatilization  <b>439</b><br>
+The Peclet number NPe represents the effect of longitudinal backmixing. Backmixing <br>
+can be neglected when the Peclet number is very large (NPe &gt;&gt; 1). In the extreme <br>
+case of pure plug flow, the Peclet number NPe = ; when NPe = 0, the entire flow chan-<br>
+nel acts as an ideal mixer.<br>The extraction number Ex is a measure of the overall devolatilization efficiency. The <br>
+backmixing term disappears when the melt film thickness is zero. This is to be <br>expected because the backmixing is caused by the transport in the melt film. If the <br>film thickness is zero, there can be no transport in the melt film and consequently <br>no backmixing. If the film stage efficiency Xf = 1, the backmixing term will disap-<br>
+pear as well. This is also easy to understand because in this case the concentration <br>C(z) at which the melt film reenters the melt pool is known and will equal Ce. Thus, <br>
+the Taylor's series expansion is no longer necessary and the first-order differential <br>equation can be determined directly from the mass balance, Eq. 7.463.<br>The magnitude of the backmixing effect will be directly determined by the distance <br>z1. This distance is directly related to the degree of fill Y, and the channel width W, <br>
+and the helix angle :<br>
+ (7.468)<br>
+The channel width W, however, is also a function of the helix angle. If the screw is <br>single flighted and if the flight width is negligible, distance z1 can be written as:<br>
+ (7.469)<br>
+From Eq. 7.469, it can be seen that for a certain size extruder the backmixing effect <br>will increase with the degree of fill in the extraction section. Thus, the channel <br>depth in the extraction section should be significantly larger than the channel depth <br>of the preceding screw section, the metering section. The backmixing effect will also <br>increase when the helix angle becomes smaller. Thus, one would like to have a rela-<br>tively small helix angle in the extraction section to increase the backmixing effect.<br>If the backmixing effect can be neglected (NPe &gt;&gt; 1), Eq. 7.467 becomes simply:<br>
+ (7.470)<br>
+The concentration thus becomes an exponential function of distance. If (0) = 1, the <br>concentration profile becomes:<br>
+ (7.471)<br>
+where:<br>
+ (7.471a)<br>
+<hr>
+<A name=452></a><b>440</b>  <b>7Functional Process Analysis</b><br>
+The devolatilization efficiency of the machine XT is a function of the individual stage <br>
+efficiency X and the extent of surface renewal. In continuous equipment, such as <br>screw extruders, the extent of surface renewal is described by the factor nf (Eq. 7.459 <br>
+or 7.467e); in batch devolatilizers, the extent of surface renewal is described by n, <br>the discrete number of surface renewals. The film stage efficiency Xf is a function of <br>
+the surface-to-volume ratio and the exposure time f. It can generally be written as <br>
+a single function of the ratio f/D. Thus, the overall efficiency XT can be described <br>
+as a function of the surface renewal factor nf and the ratio f D:<br>
+ (7.472)<br>
+where <br>
+2<br>
+D is the characteristic time for diffusion in the film (D = hm /D).<br>
+The effectiveness of the devolatilization operation is strongly dependent on the <br>actual length of the devolatilization zone LB. In actual extrusion the length LB can be <br>
+considerably longer than the length of the extraction section of the screw Le. Thus, <br>
+the length LB can extend into the pump section of the extruder. This is determined <br>
+by the filled length of the pump section Lpf.<br>If the length of the pump section is Lp, the actual devolatilization length LB can be <br>
+determined from:<br>
+ (7.473)<br>
+where Le is the length of the extraction section, Lp the length of the pump section, <br>
+and Lpf the filled length of the pump section. The latter can be calculated from melt <br>
+conveying theory if the following parameters are known:<br>1.  The geo metry of the pump section<br>2.  The throughput<br>3.  The flow properties of the polymer<br>4.  The diehead pressure<br>From the melt conveying theory of Newtonian fluids, the length Lpf can be deter-<br>
+mined from:<br>
+ (7.474)<br>
+where P is the diehead pressure.<br>This equation does not take into account the leakage flow or the effect of the flight <br>flanks. The diehead pressure P is related to the total volumetric flow rate   by the <br>die constant K:<br>
+ (7.475)<br>
+<hr>
+<A name=453></a> 7.7Mixing  <b>441</b><br>
+In most two-stage devolatilizing extruder screws, the throughput   is determined <br>by the metering section as a result of its shallow channel depth Hm. If the through-<br>
+put is determined by the drag flow rate of the metering section and if the helix angle <br>and the number of parallel flights is constant, the expression for Lpf can be simpli-<br>
+fied to:<br>
+ (7.476)<br>
+Thus, in order to keep Lpf short and LB long, the depth of the metering section Hm <br>
+should be small compared to the depth of the pump section H. Also the restriction of <br>the die K should be made as small as possible.<br>
+<b></b><br>
+ <b>7.7Mixing</b><br>
+Mixing can be broadly defined as a process to reduce the non-uniformity of a compo-<br>sition. The basic mechanism of mixing is to induce physical relative motion of the <br>ingredients. The types of motion that can occur are molecular diffusion, turbulent <br>motion, and convective motion. The first two types of motion are essentially limited <br>to gases and low-viscosity liquids. Convective motion is the predominant motion in <br>high-viscosity liquids, such as polymer melts. As discussed in Section 5.3.3, poly-<br>mer melts are not capable of turbulent motion as a result of their high viscosity; <br>motion in polymer melts is always by laminar flow.<br>Convective mixing by laminar flow is referred to as laminar mixing. This is the type <br>of mixing that occurs in polymer melt extrusion. The mixing action generally occurs <br>by shear flow and elongational flow. If the components to be mixed are compatible <br>fluids and do not exhibit a yield point, the mixing is distributive. This is sometimes <br>referred to as extensive mixing. The process of distributive mixing can be described <br>by the extent of deformation or strain to which the fluid elements are exposed. The <br>actual stresses involved in this process are irrelevant in the description of the dis-<br>tributive mixing.<br>If the mixture contains a component that exhibits a yield stress, then the actual <br>stresses involved in the process become very important. If the component exhibiting <br>a yield point is a solid, this type of mixing is referred to as dispersive mixing, some-<br>times as intensive mixing. In dispersive mixing, a solid component needs to be <br> <br>broken down, but the breakdown only occurs after a certain minimum stress (yield <br>stress) has been exceeded. If the component exhibiting a yield point is a liquid, the <br>mixing process is referred to as homogenization. An example of dispersive mixing is <br>the manufacture of a color concentrate where the breakdown of the pigment agglo-<br>
+<hr>
+<A name=454></a><b>442</b>  <b>7Functional Process Analysis</b><br>
+merates below a certain critical size is of critical importance. An example of distribu-<br>tive mixing is the manufacture of a polymer blend, where none of the components <br>exhibit a yield point.<br>Distributive mixing and dispersive mixing are not physically separated. In disper-<br>sive mixing, there will always be distributive mixing. However, the reverse is not <br>always true. In distributive mixing, there can be dispersive mixing only if there is a <br>solid component with a yield stress and if the stresses acting on this component <br>exceed the yield stress.<br>A very important aspect of the study of mixing is the characterization of the mix-<br>ture. A complete characterization requires the specification of the size, shape, orien-<br>tation, and spatial location of every discrete element of the minor component. This, <br>of course, is generally impossible. Various theories and techniques have been <br>devised to describe and measure the goodness of mixing [187­200]. Some of the <br>characterization techniques are quite sophisticated and can be time consuming. <br>Quantitative characterization is very important to workers doing research on mixing. <br>However, such techniques are not always practical in actual polymer processing <br>operations. Visual observation, although qualitative, is often sufficiently accurate to <br>determine whether a product is acceptable or not. Therefore, the various characteri-<br>zation theories and techniques will not be discussed here. For more information on <br>this subject, the reader is referred to the literature [187­200, 301]. In this section, <br>the primary emphasis will be on the description of the mixing process in a screw <br>extruder.<br>
+<b>7.7.1Mixing in Screw Extruders</b><br>
+Mixing is an essential function of the screw extruder. It occurs in all screw extrud-<br>ers as opposed to devolatilization, which is done only on specialized machines. The <br>mixing zone in the extruder extends from the start of the plasticating zone to the <br>end of the die, assuming that significant mixing only takes place when the polymer <br>is in the molten state. The fact that mixing starts at the beginning of the melting <br>zone presents a practical and an analytical problem. It means that at the end of the <br>melting zone there will be a considerable non-uniformity in the mixing history of <br>the polymer.<br>A polymer element that melts early will be exposed to a significant mixing history <br>by the time it reaches the end of the melting zone. On the other hand, a polymer ele-<br>ment that melts at the very end of the melting zone will have hardly any mixing <br>history at the end of the melting zone.<br>Similar problems occur in the melt conveying zone. A polymer element at about 2/3 <br>of the height of the channel will have no cross-channel velocity component and as a <br>result will have a short residence time in the melt conveying section and little mixing <br>
+<hr>
+<A name=455></a> 7.7Mixing  <b>443</b><br>
+history. A polymer element at about 1/3 of the channel will have considerable cross-<br>channel velocity and relatively low down-channel velocity. As a result, this element <br>will have a long residence time in the melt conveying zone and a large mixing his-<br>tory. This can be verified by analyzing the velocity profiles in the metering section <br>as discussed in Section 7.4. It is clear, therefore, that the mixing action is not uni-<br>formly applied to all elements of the polymer melt. As a result of the inherent trans-<br>port process in a screw extruder there will be considerable non-uniformities in the <br>intensity of the mixing action and the duration of the mixing action. This is also true <br>for the extruder die. Fluid elements in the center of the flow channel are exposed to <br>a very low shear rate, and their residence is short because the velocities are highest <br>in the center. Fluid elements at the wall are exposed to high shear rates, and their <br>residence time is long because of the low velocities at the wall. Thus, even if a per-<br>fectly mixed fluid enters a die, non-uniformities can be expected as the fluid leaves <br>the die.<br>The mixing process in extruders is generally analyzed by determining the velocity <br>profiles occurring in the screw channel. From the velocity profiles, the deformation <br>at various locations in the fluid can be determined. In most analyses, the fluid is <br>considered Newtonian, the components have the same flow properties (i.e., a rheo-<br>logically homogeneous fluid mixture), and the flow through the flight clearance is <br>neglected. Another common assumption is a two-dimensional flow pattern in the <br>screw channel; only flow in down-channel and cross-channel directions are con-<br>sidered.<br>When two viscous liquids are mixed, the interfacial area increases and the striation <br>thickness decreases. Spencer and Wiley [201] have proposed to use the interfacial <br>area as a quantitative measure of the goodness of mixing. Mohr et al. [189] used <br>the striation thickness to describe the mixing process. If a surface element with <br> <br>arbitrary orientation is located in a simple shear flow field, the surface area A after <br>a total shear strain of  can be demonstrated to be [201]:<br>
+ (7.477)<br>
+where Ao is the original surface area, x the angle of the vector normal to Ao with <br>
+the x axis, and y the angle of the vector normal to Ao with the y axis.<br>Angles x, y, and z determine the initial orientation of the surface element under <br>
+consideration. The three angles are related by:<br>
+ (7.478)<br>
+If the total shear strain is very large ( &gt;&gt;1), then Eq. 7.477 becomes:<br>
+ (7.479)<br>
+<hr>
+<A name=456></a><b>444</b>  <b>7Functional Process Analysis</b><br>
+Equation 7.479 indicates that the increase in interfacial area is directly proportional <br>to the total shear strain and cosx. Thus, the total shear strain is an important vari-<br>
+able in the description of the mixing process in a shear flow field. The initial orienta-<br>tion x is also very important. If the initial surface is oriented parallel to the flow <br>
+field (x = 90°), then the increase in interfacial area is zero. However, if the initial <br>
+surface orientation is perpendicular to the flow field (x = 0), then the increase in <br>
+interfacial area is maximum. At low strains, it can be seen from Eq. 7.477 that the <br>interfacial area can increase or decrease with strain, depending on the initial orien-<br>tation.<br>If the interfaces are initially randomly oriented, the mean change in interfacial area <br>becomes [202]:<br>
+ (7.480)<br>
+Equation 7.480 is valid when the total strain is very large ( &gt;&gt; 1). The striation <br>thickness is defined as the total volume divided by half the total interfacial surface:<br>
+ (7.481)<br>
+If the minor component is initially introduced as randomly oriented cubes of height <br>H and with a volume fraction , the striation thickness can be expressed as:<br>
+ (7.482)<br>
+Thus, the striation thickness is directly proportional to the initial domain size of the <br>minor component and inversely proportional to the volume fraction and total shear <br>strain. This indicates that a small striation thickness is achieved more easily when <br>the initial domain size of the minor component is small and the volume fraction <br>large.<br>The striation thickness is a commonly used measure of mixing. In simple shear the <br>striation thickness reduces with the shear strain as follows:<br>
+ (7.483)<br>
+The relationship between the striation thickness and the total shear strain is shown <br>in Fig. 7.120. The initial orientation of the striation is perpendicular to the flow field <br>(x = 0); this is the optimum orientation.<br>
+<hr>
+<A name=457></a> 7.7Mixing  <b>445</b><br>
+1<br>
+0.9<br>
+0.8<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+0.3<br>
+Striation Thickness Ratio [-]<br>
+0.2<br>
+0.1<br>
+0<br>
+0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+100<br>
+Shear Strain [-]<br>
+<b>Figure 7.120</b>Striation thickness ratio versus total shear strain<br>
+It is interesting to note that the striation thickness reduces rapidly in the first 5 to <br>10 units of shear strain. After 10 units of shear, the striation thickness has reduced <br>to about 10% of its original value. However, after about 10 to 20 units of shear strain, <br>the striation thickness reduces only very slowly. There is very efficient mixing in <br>the first 10 to 20 units of shear strain and inefficient mixing beyond 20 units of <br>shear strain.<br>The reason that the mixing efficiency reduces with shear strain is that the orienta-<br>tion of the striation changes with shear strain. The striation becomes more and more <br>oriented in the direction of flow as the shear strain increases. As a result, mixing for <br>a long time does not make much sense because most of the mixing is achieved <br>within the first 20 units of shear strain. However, the distributive mixing efficiency <br>can be improved dramatically by reorienting the striations during the mixing pro-<br>cess.<br>If the simple shear field is disrupted by a short mixing section that produces a ran-<br>domly oriented minor component, the interfacial area at the outlet of the mixing <br>section is:<br>
+ (7.484)<br>
+<hr>
+<A name=458></a><b>446</b>  <b>7Functional Process Analysis</b><br>
+where 1 is the total shear strain the fluid is exposed to before the mixing section. It <br>
+is assumed that the shear strain in the mixing section itself is insignificant. If the <br>simple shear field is restored after the mixing section, then the total interfacial area <br>after another exposure to shear strain 1 becomes:<br>
+ (7.485)<br>
+Similarly, after n mixing sections and n shear strain exposures of the same magni-<br>tude 1, the total interfacial area will be:<br>
+ (7.486)<br>
+From Eq. 7.486, it can be seen that the generation of interfacial area can be increased <br>substantially by inclusion of mixing sections that randomize the minor component. <br>The improvement can be evaluated by comparing n mixing sections and n1 strain <br>
+exposures to simple shear mixing without mixing sections but the same total strain <br>exposure. The ratio of the interfacial area is:<br>
+ (7.487)<br>
+The striation thickness versus shear strain with various reorientation events is <br>shown in Fig. 7.121.<br>
+1.00E+00<br>
+1.00E-01<br>
+1.00E-02<br>
+]<br>
+no reorientation<br>
+1.00E-03<br>
+1.00E-04<br>
+1 reorientation<br>
+1.00E-05<br>
+1.00E-06<br>
+Striation thickness [-<br>
+2 reorientations<br>
+1.00E-07<br>
+1.00E-08<br>
+3 reorientations<br>
+1.00E-09<br>
+0<br>
+100<br>
+200<br>
+300<br>
+400<br>
+500<br>
+600<br>
+700<br>
+800<br>
+900<br>
+1000<br>
+Shear strain [-]<br>
+<b>Figure 7.121</b>Striation thickness versus shear strain with various reorientation events<br>
+<hr>
+<A name=459></a> 7.7Mixing  <b>447</b><br>
+In Fig. 7.121 reorientation occurs after 100 units of shear strain. Without reorienta-<br>tion the striation thickness reduces to about 10­3 after 1000 units of shear strain. <br>With one reorientation the striation thickness reduces to about 10­5, with two re- <br>orientations to about 10­7, etc. Clearly, reorientation can achieve improvements in <br>distributive mixing by orders of magnitude. It is a powerful tool in mixing operations.<br>It is obvious that randomizing mixing sections greatly improves the generation of <br>interfacial area and thus the mixing performance. Erwin and Ng [205] constructed <br>an experimental mixing apparatus by which the results of Eqs. 7.486 and 7.487 and <br>Fig. 7.121 have been experimentally verified. However, the mixing apparatus does <br>not lend itself to practical mixing operations.<br>If the mixing section is capable of orienting the minor component in the most favor-<br>able direction, i.e., perpendicular to the velocity, the total interfacial area after n <br>mixing sections and n shear strain exposures of magnitude 1 will be:<br>
+ (7.488)<br>
+where C = 1/2 if the initial orientation is random and C = 1 if the initial orientation <br>is most favorable.<br>Equations 7.486 and 7.487 demonstrate, at least qualitatively, that incorporation <br>of mixing devices can substantially improve laminar mixing performance. In a <br>dynamic mixing device, such as an extruder, it may be difficult to design a mixing <br>section that will orient the minor component in the most favorable orientation. How-<br>ever, random orientation may be more feasible. In a static mixing device, it is easier <br>to control the orientation of the minor component, and effective laminar mixing can <br>occur in such mixing devices; see Section 7.7.2.<br>
+<b>7.7.1.1Distributive Mixing in Screw Extruders<br></b>The shear rate in the polymer melt is found by taking the first derivative of the <br>velocity. The velocity profiles for Newtonian fluids were derived in Section 7.4.1. <br>From Eq. 7.197 we can determine the down-channel shear rate:<br>
+ (7.489)<br>
+The down-channel shear rates are shown in Fig. 7.122.<br>When the pressure gradient is positive the shear rates increase toward the barrel <br>surface, when the pressure gradient is zero the shear rate is constant, and when the <br>pressure gradient is negative the shear rates reduce toward the barrel surface.<br>The cross-channel shear rate is determined the same way:<br>
+ (7.490)<br>
+<hr>
+<A name=460></a><b>448</b>  <b>7Functional Process Analysis</b><br>
+1<br>
+r = -0.2<br>
+r = 0. 0<br>
+r = 0. 2<br>
+r = 0. 4<br>
+r = 0. 6<br>
+0.9<br>
+r =  -0.2<br>
+r = 0.0<br>
+r = 0.2<br>
+r = 0.4<br>
+r = 0.6<br>
+0.8<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+Normal coordinate 0.3<br>
+0.2<br>
+0.1<br>
+0<br>
+-1<br>
+-0.5<br>
+0<br>
+0.5<br>
+1<br>
+1.5<br>
+2<br>
+2.5<br>
+3<br>
+Down-channel shear rate [s-1]<br>
+<b>Figure 7.122</b>Down-channel shear rates for several values of the throttle ratio<br>
+The cross-channel shear rate profile is shown in Fig. 7.123.<br>
+<b>Figure 7.123</b>Cross-channel shear rate versus normal distance<br>
+In the bottom of the channel the fluid is exposed to negative shear rates, and in the <br>top of the channel the fluid is exposed to positive shear rates. This has important <br>consequences for the mixing that occurs in screw extruders. At the top of the chan-<br>nel the fluid elements travel in the direction of the barrel, while at the bottom of the <br>channel the fluid elements travel across the channel. The position of a fluid element <br>in the upper portion of the channel (y) corresponds to a complementary position in <br>the lower portion of the channel (yc) as shown in Fig. 7.124.<br>
+<hr>
+<A name=461></a> 7.7Mixing  <b>449</b><br>
+<b>Figure 7.124</b>Position y and complementary position yc in screw channel<br>
+The time that a fluid element spends in the upper portion of the channel (tu) is deter-<br>
+mined by the width of the channel and the cross-channel velocity:<br>
+ (7.491)<br>
+A similar expression is used to determine the time spent in the lower portion of the <br>channel. The total residence time over axial length L of the extruder screw is deter-<br>mined from Eq. 7.492.<br>
+ (7.492)<br>
+The residence time as a function of the normal distance is shown in Fig. 7.125.<br>
+100<br>
+90<br>
+80<br>
+70<br>
+60<br>
+50<br>
+40<br>
+Residence time [s] 30<br>
+20<br>
+10<br>
+0<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Normal coordinate<br>
+<b>Figure 7.125</b>Residence time versus normal distance (dimensionless)<br>
+<hr>
+<A name=462></a><b>450</b>  <b>7Functional Process Analysis</b><br>
+It is clear from Fig. 7.125 that the residence time in the center region of the channel <br>has the lowest residence time; the minimum occurs at y = 2H/3. There is a rather <br>broad region (from about 10 to 90% of the depth of the channel) where the residence <br>time is quite low. The residence time increases towards the screw and barrel surface <br>and reaches infinity at y/H = 0 (screw root) and y/H = 1 (barrel). The long residence <br>time layer on the screw surface is substantially thicker than at the barrel surface. <br>This indicates that problems due to long residence time, such as degradation, are <br>more likely to occur at the screw surface than at the barrel surface.<br>The time fraction spent in the upper portion of the channel is determined from:<br>
+ (7.493)<br>
+This fraction is shown in Fig. 7.126.<br>
+1<br>
+0.9<br>
+0.8<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+<b>Time fraction</b><br>
+0.3<br>
+0.2<br>
+0.1<br>
+0<br>
+0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1<br>
+<b>Normal coordinate</b><br>
+<b>Figure 7.126</b>The fraction of time in the lower and upper portion of the channel versus normal <br>
+distance<br>
+The lowest value fu = 0 occurs at y/H = 1; the highest value fu = 0.5 occurs at y = 2/3. <br>
+This indicates that the fluid elements spend much more time in the lower portion of <br>the channel than in the upper portion. It is worthwhile to analyze the cross-channel <br>flow in more detail. We can distinguish two types of re-circulation, one in the outer <br>region and one in the inner region of the channel. The outer region A is bounded by <br>0.91H  y  H (region Au) and 0  yc  H/3 (region Al). The inner region is bounded <br>
+by y  0.91 and yc  H/3; see Fig. 7.127.<br>Interestingly, the shearing in the lower region A is in the negative direction, while <br>the shearing in the upper region A is in the positive direction. This means that the <br>mixing that occurs in the lower portion of the channel is counteracted by the mixing <br>
+<hr>
+<A name=463></a> 7.7Mixing  <b>451</b><br>
+in the opposite direction in the upper portion of the channel. In fact, there is a demix-<br>ing action going on as a result of the exposure to positive and negative shear rates! <br>This is illustrated by the shear deformation of a rectangular element in Fig. 7.127.<br>
+Positive shear deformation<br>
+Negative shear deformation<br>
+<b>Figure 7.127</b>Cross-channel flow and resulting shear deformation<br>
+The situation is quite different in region B. Here the shearing in the lower part of <br>region B is in the same direction as in the upper part. As a result, the mixing in the <br>upper portion of region B enhances the mixing in the lower portion of region B. <br>Therefore, there are no demixing effects occurring in region B, the inner re-circulat-<br>ing region, only in region A, the outer re-circulating region.<br>The total cross-channel shear strain can be determined by adding the shear strain in <br>the upper portion of the channel to that in the lower portion of the channel. The total <br>cross-channel shear strain can thus be written as:<br>
+ (7.494)<br>
+Figure 7.128 shows the cross-channel shear strain as a function of normal distance <br>for one value of the throttle ratio rd = 0.1, assuming that DN/H = 1.<br>
+50<br>
+40<br>
+30<br>
+20<br>
+10<br>
+0<br>
+-10<br>
+-20<br>
+Crsss channel shear strain<br>
+-30<br>
+-40<br>
+-50<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Normal coordinate<br>
+<b>Figure 7.128</b>The cross-channel shear strain versus normal distance (dimensionless)<br>
+<hr>
+<A name=464></a><b>452</b>  <b>7Functional Process Analysis</b><br>
+The shear strain reaches ­ at the screw and barrel surface. The shear strain <br>
+becomes zero at y = 0.98H and yc = 0.16H; this corresponds to the streamline where <br>
+the shear strain in region Au is canceled exactly by the opposite shear in region Al. <br>
+The cross-channel shear strain reaches a maximum at y = (2/3)H; this is where the <br>cross-channel velocity becomes zero. The value of the maximum shear increases <br>with increasing throttle ratio rd, because the residence time increases when the <br>
+throttle ratio increases.<br>Similar to the total cross-channel shear strain, the total down-channel shear strain <br>can be written as:<br>
+ (7.495)<br>
+Figure 7.129 shows the down-channel shear strain as a function of the normal dis-<br>tance at several values of the throttle ratio rd.<br>
+100<br>
+90<br>
+80<br>
+70<br>
+60<br>
+50<br>
+40<br>
+30<br>
+Down channel shear strain<br>
+20<br>
+10<br>
+0<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Normal coordinate<br>
+<b>Figure 7.129</b>Down-channel shear strain versus normal distance for rd = 0 .1<br>
+When rd = 0, the down-channel shear strain curve has exactly the same form as the <br>
+residence time curve; this is due to the fact that the down-channel shear rate is con-<br>stant when rd = 0. When rd = 1/3, the down-channel shear strain becomes independ-<br>
+ent of the normal coordinate y. When rd &gt; 1/3 the down-channel shear strain becomes <br>
+negative close to the screw and barrel surface. This is due to the fact that the shear <br>rates at the screw surface become negative when rd &gt; 1/3.<br>The magnitude of the total shear strain is obtained by vectorial addition of the cross- <br>and down-channel shear strains. This leads to the following expression of the total <br>shear strain:<br>
+ (7.496)<br>
+<hr>
+<A name=465></a> 7.7Mixing  <b>453</b><br>
+With this expression the distributive mixing process in the melt conveying zone of <br>a single screw extruder can be evaluated quantitatively. The total shear versus nor-<br>mal distance for several values of the throttle ratio is shown in Fig. 7.130.<br>
+100<br>
+90<br>
+80<br>
+70<br>
+60<br>
+50<br>
+40<br>
+Total Shear Strain 30<br>
+20<br>
+r = 0. 3<br>
+10<br>
+r =  0<br>
+0<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Normal Coordinate<br>
+<b>Figure 7.130</b>Total shear strain versus normal distance (dimensionless) at two rd values<br>
+The total strain depends strongly on both the normal distance and on the throttle <br>ratio. Fluid elements close to the wall experience a high level of shear strain and, <br>thus, will be well mixed. Elements further away from the wall experience a lower <br>level of strain and will not be mixed as well as elements close to the wall.<br>The shear strain in the center region increases with the throttle ratio. When the <br>throttle ratio is one-third or greater, the shear strain reaches a minimum close to the <br>wall; this corresponds to the location where the down-channel shear strain <br>approaches zero. Mixing is improved by increasing the throttle ratio; this can be <br>done by increasing the restriction at the end of the screw, for instance, by increasing <br>the number of screens of the screen pack. Such an increase in mixing, however, is at <br>the expense of output. Since reduced output will increase residence time and poly-<br>mer melt temperatures, there will be an increased chance of degradation. Therefore, <br>increasing the restriction of the screen pack is often not the most efficient way of <br>improving mixing.<br>The shear strains in both x and z directions can be used to calculate the striation <br>thickness:<br>
+ (7.497)<br>
+where <i>v</i> is the volume fraction of the minor component and s0 the initial striation <br>
+thickness.<br>
+<hr>
+<A name=466></a><b>454</b>  <b>7Functional Process Analysis</b><br>
+Since x and z depend on the normal distance, the striation thickness will depend <br>
+on the normal distance as well. Thus, the non-uniform cross- and down-channel <br>shear strain will result in non-uniform mixing in the screw channel. Unfortunately, <br>poor mixing ability is typical of single screw extruders with straight conveying <br>screws, i.e., without mixing sections. The best method to really improve mixing in <br>single screw extruders is to incorporate mixing sections. This topic will be discussed <br>in Chapter 8. Another reason that mixing sections can be important is the substan-<br>tial non-uniformities in melt temperature that can occur in screw extruders. If a <br>melt with non-uniform melt temperatures is discharged into an extrusion die, a <br>number of problems can occur that affect the quality of the extruded product. Melt <br>temperature distributions will be discussed in Chapter 12.<br>The energy requirement to achieve a certain amount of increase in interfacial area <br>was studied by Erwin [204]. In uniaxial extension flow, the energy per unit volume <br>is related to the surface area increase as shown in Eq. 7.498.<br>
+ (7.498)<br>
+where   is the viscosity of the fluid and t the duration of the extensional defor-<br>mation.<br>In biaxial extensional flow, the energy per unit volume is:<br>
+ (7.499)<br>
+In plane strain elongation (two-dimensional elongation), the energy per unit volume <br>is:<br>
+ (7.500)<br>
+In simple shear, the energy per unit volume is:<br>
+ (7.501)<br>
+Figure 7.131 shows the normalized energy per unit volume versus the ratio of inter-<br>facial areas for uniaxial extension and simple shear flow.<br>For the same increase in interfacial area, extensional flow is substantially more <br>energy efficient than shear flow. When the area ratio becomes larger than 100, the <br>energy requirement in simple shear is several orders of magnitude higher than <br>extensional flow. This is an important advantage of extensional flow, particularly for <br>dispersive mixing. Lower energy consumption results in less viscous dissipation <br>
+<hr>
+<A name=467></a> 7.7Mixing  <b>455</b><br>
+and lower melt temperatures. With lower melt temperatures greater stresses can be <br>generated in the polymer melt, thus improving dispersive mixing. Dispersive mixing <br>will be discussed in Section 7.7.3.<br>
+1.00E+11<br>
+1.00E+10<br>
+1.00E+09<br>
+Simple shear<br>
+1.00E+08<br>
+1.00E+07<br>
+1.00E+06<br>
+1.00E+05<br>
+Energy [J/m^3]<br>
+Uniaxial extension<br>
+1.00E+04<br>
+1.00E+03<br>
+1.00E+02<br>
+1.00E+01<br>
+1.00E+00<br>
+0<br>
+100<br>
+200<br>
+300<br>
+400<br>
+500<br>
+600<br>
+700<br>
+800<br>
+900<br>
+1000<br>
+Area ratio [-]<br>
+<b>Figure 7.131</b>Energy consumption versus area ratio for shear and elongational flow<br>
+In continuous mixers, different fluid elements will invariably experience different <br>amounts of strain, as discussed earlier for the screw extruder. Tadmor and Lidor <br>[206] proposed the use of strain distribution functions (SDF), similar to residence <br>time distribution functions (RTD). The SDF for a continuous mixer f()d is defined <br>as the fraction of exiting flow rate that experienced a strain between  and d. It is <br>also the probability of an entering fluid element to exit with that strain. The cumula-<br>tive SDF, F(), is defined by the following expression:<br>
+ (7.502)<br>
+where 0 is the minimum strain.<br>F() represents the fraction of exiting flow rate with strain less than or equal to . <br>The mean strain of the exiting stream is:<br>
+ (7.503)<br>
+Tadmor and Pinto [207] used the weighted average total strain (WATS) to describe <br>mixing performance in the non-homogeneous flow field of a single screw extruder. <br>This is defined as:<br>
+ (7.504)<br>
+<hr>
+<A name=468></a><b>456</b>  <b>7Functional Process Analysis</b><br>
+where f(t) is the RTD function and (t) the strain undergone by a fluid element at <br>time t. The WATS does not produce an experimentally measurable quantity describ-<br>ing mixing, as discussed by Ottino and Chella [208] in an extensive review of lami-<br>nar mixing of polymeric liquids. Another limitation is that the initial orientation of <br>the minor component is not considered, and changes in orientation are not taken <br>into account. Ottino [209, 210] developed a description of laminar mixing using the <br>mathematical structure of continuum mechanics. This description allows evaluation <br>of the role of initial orientation and the definition of mixing efficiency. This approach <br>was applied to mixing in single screw extruders [211]. The mixing achieved was <br>expressed in terms of mixing cup average area stretch "". This factor was found to <br>depend on down-channel distance, channel width-to-height ratio, helix angle, throttle <br>ratio, and the initial orientation. The effect of helix angle and throttle ratio (ratio of <br>pressure flow to drag flow) is shown in Fig. 7.132.<br>
+[<br>-<br>]<br>
+TS<br>WA<br>
+Area stretch <br>
+Mean strain or <br>
+Throttle ratio  [-]<br>
+Throttle r atio [-]<br>
+<b> <br>Figure  7.132</b>Area stretch versus throttle  <br>
+<b> <br>Figure  7.133</b>Mean strain versus throttle  <br>
+ratio<br>
+ratio<br>
+The mixing in single screw extruders was also studied by Tadmor and Klein [103] <br>who used the mean strain to evaluate the mixing performance of the extruder; their <br>result is shown in Fig. 7.133.<br>By comparing Fig. 7.132 to Fig. 7.133, it is clear that Tadmor's results correspond <br>reasonably well with Ottino's results. The mixing performance improves as the <br>throttle ratio increases. This is to be expected since the output per revolution will <br>decrease, thus the mean residence time will increase with the throttle ratio, while <br>the local shear rates will remain approximately the same. The mixing performance <br>reduces as the helix angle increases from 10 to 30°. As the helix angle increases, the <br>output per revolution increases as well, resulting in a reduced residence time. The <br>residence time seems to play an overriding role because the cross-channel mixing <br>improves with increasing helix angle, but the overall mixing performance reduces. <br>Thus, the reduction in mean residence time overrides the effect of improved cross-<br>channel mixing as the helix angle increases from 10 to 30°. Equation 7.206 (see also <br>
+<hr>
+<A name=469></a> 7.7Mixing  <b>457</b><br>
+Chapter 8) shows that 30° happens to be the optimum helix angle for Newtonian <br>fluids with respect to output. This means that at a helix angle of 30°, the shortest <br>residence time is achieved, provided the depth of the channel is optimum as well.<br>Analyses of laminar mixing in screw extruders generally deal with highly simplified <br>problems. The leakage flow through the flight is usually neglected, as well as the <br>normal velocity components close to the flight flanks. Just these two simplifications <br>constitute a severe limitation on the validity and applicability of results of any mixing <br>analysis. The normal velocity components achieve a reorientation of the minor com-<br>ponent. When the normal velocity components are neglected, the reorientation of <br>the interfacial area cannot be properly accounted for. This is a particularly severe <br>problem in the analysis of multi-flighted mixing sections, such as the Dulmage <br>mixing section; see Section 8.7.2.<br>The laminar mixing action can only be analyzed if the exact flow patterns are known. <br>This can be done reasonably well for simple rectangular flow channels in an extruder <br>screw. However, when mixing sections are incorporated into the extruder screw, <br>the flow patterns generally become quite complex. In one respect, this is desirable <br>because complex velocity profiles tend to improve the mixing effectiveness. How-<br>ever, on the other hand, the mathematical description of the velocity profiles can <br>become very involved. This is particularly true when the minor fluid component has <br>flow properties that are different from the major fluid components, i.e., when the <br>fluid mixture is rheologically non-homogeneous.<br>As a result, quantitative analysis of mixing sections in extruders can be complicated. <br>Therefore, development of mixing devices has been mostly empirical up to about the <br>end of the last millennium. With the advent of more powerful numerical techniques <br>it is now possible to do a complete three-dimensional flow analysis of mixing devices <br>with complex geo metry. The boundary element method has proven to be particularly <br>useful in the analysis of complex mixers. Mixing devices will be discussed in detail <br>in Chapter 8 and numerical techniques and computer simulation in Chapter 12.<br>
+<b>7.7.2Static Mixing Devices</b><br>
+Mixing in extruders occurs not only along the extruder screw but also from the end <br>of the screw to the exit of the die. The flow through the die and possible adapter is a <br>pressure-driven flow where the flow velocities in the center of the channel are high <br>and zero at the wall, see Fig. 7.134.<br>This velocity profile results in a non-uniform shear rate profile with high shear rates <br>at the wall and zero shear rate at the center of the channel; see Fig. 7.134. For a <br>power law fluid in a circular channel the axial velocity profile can be written as [16]; <br>see Fig. 7.110:<br>
+<hr>
+<A name=470></a><b>458</b>  <b>7Functional Process Analysis</b><br>
+Non-Newtonian<br>
+Newtonian<br>
+<b> Figure  7.134</b> <br>
+Velocity and shear rate profiles in <br>
+Velocities<br>
+Shear rates<br>
+ <br>pressure flow through a straight channel<br>
+ (7.505)<br>
+The shear rate profile is determined by taking the first derivative of the velocity <br>with respect to normal distance r. Thus, the shear rate becomes:<br>
+ (7.506)<br>
+The residence time of a fluid element over length L as a function of radial distance is <br>simply:<br>
+ (7.507)<br>
+The shear strain exposure is the shear rate multiplied with the shear exposure time. <br>Thus, the shear strain as a function of radial distance becomes:<br>
+ (7.508)<br>
+From the expression above it is clear that the shear strain distribution in the chan-<br>nel is very non-uniform. The shear strain in the center is zero, while it approaches <br>infinity at the wall, as shown in Fig. 7.135.<br>
+r<br>
+<b> Figure 7.135</b> <br>
+Shear strain distribution in pressure flow through <br>
+Shear strain<br>
+a straight channel<br>
+<hr>
+<A name=471></a> 7.7Mixing  <b>459</b><br>
+The equations above are based on the assumption that the flow is laminar, which <br>will generally be the case for polymer melts due to their high viscosity. If, however, <br>turbulent flow takes place, then even in a simple circular channel efficient mixing <br>can occur. When two fluids of equal density are introduced side by side into an <br>empty pipe with diameter D, then the degree of mixing after length L can be <br>expressed as [260]:<br>
+ (7.509)<br>
+where s is the sample standard derivation, 0 the initial standard derivation, and m <br>
+is a coefficient that depends on the Reynolds number.<br>According to Hiby [289] the mixing quality in most practical mixing operations is <br>sufficient when the coefficient of variation is less than one-hundredth. The coeffi-<br>cient of variation, COV, is the sample standard deviation divided by the average <br>concentration. Thus, the Hiby requirement can be written as:<br>
+ (7.510)<br>
+where s is the sample standard deviation and C the average concentration.<br>Other sources consider the mixing sufficient when the coefficient of variation is less <br>than 0.05 [261]. At a Reynolds number of 8000 the value of m = 0.046 and the pipe <br>length will have to be about 90 D before sufficient mixing is achieved. Various <br>methods can be used to shorten this length as discussed, e.g., by Hartung and Hiby <br>[290] and Fleischmann [291].<br>In regular laminar flow through a straight pipe without mixing elements, the length <br>of the pipe will have to be about 90 D before sufficient mixing is achieved. Clearly, <br>this is much too long for practical applications in plastics extrusion. The mixing <br>action in laminar pressure flow can be substantially improved by incorporating <br>static mixing devices in the channel. These devices split and reorient the flow and, <br>thus, impart improved distributive mixing to the fluid. Because these mixers have <br>no moving parts, they are often referred to as motionless mixers. Some of the advan-<br>tages of static mixing devices are:<br>
+ <br>
+ Can be used for fluids with a wide range of viscosities<br>
+ <br>
+ Continuous mixing device<br>
+ <br>
+ Small space requirement<br>
+ <br>
+ No moving parts, little or no wear, no noise<br>
+ <br>
+ Temperature insensitive<br>
+ <br>
+ Low operational cost<br>
+<hr>
+<A name=472></a><IMG src="Polymer Extrusion by Chris Rauwendaal-472_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-472_22.jpg"><br>
+<b>460</b>  <b>7Functional Process Analysis</b><br>
+<b>7.7.2.1Geometry of Static Mixers<br></b>Static mixing devices usually consist of a combination of identical elements, stacked <br>in series with each element turned ninety degrees relative to the next element. The <br>first static mixer, the Multiflux mixer, was described by Sluijters [243]. This mixer <br>developed at AKZO Corporation of the Netherlands splits the flow in rectangular <br>converging and diverging channels; see Fig. 7.136.<br>
+<b> Figure  7.136</b> <br>
+Channel geo metry in the Multiflux <br>
+mixer<br>
+Top view<br>
+Mid cross section<br>
+Bottom view<br>
+In 1-st element<br>
+Middle 1-st element<br>
+Out 1-st/in 2-nd<br>
+Out 2-nd element<br>
+Out 3-rd element<br>
+Out 4-th element<br>
+<b> Figure  7.137</b> <br>
+The Multiflux static mixer<br>
+Each element of the Multiflux mixer has two channels, as shown in Fig. 7.136. The <br>channels start with a rectangular cross-section and then taper to a square cross-<br>section in the middle of the element. From the square cross-section the channel <br>
+<hr>
+<A name=473></a> 7.7Mixing  <b>461</b><br>
+opens up again to a rectangular cross-section; however, the exit rectangle is per-<br>pendicular to the inlet rectangle. The resulting mixing action is shown in Fig. 7.137. <br>A recent numerical study on the flow and layer distribution in the Multiflux mixer <br>was published by van der Hoeven et al. [283].<br>The ISG mixer has four circular channels in each element, with material from the <br>outside being led to the inside and vice versa; see Fig. 7.138. The flow through this <br>mixer is not very streamlined and the individual channels are quite small, making <br>this mixer unattractive for high-viscosity or thermally sensitive materials.<br>
+<b> Figure  7.138</b> <br>
+The ISG static mixer<br>
+The SMV mixer developed by the Swiss company Sulzer is made up of stacked cor-<br>rugated plates with adjacent plates having opposite orientation. The length of one <br>element is about one diameter; usually several elements are placed in series with <br>one element turned 90° relative to the element next to it. Because of the splitting <br>and reorientation within one element, the mixing action is quite efficient. The mixer <br>geo metry and mixing action is shown in Fig. 7.139. This mixer is primarily used for <br>low-viscosity fluids.<br>
+Inlet 1-st element<br>
+Inlet 2-nd element<br>
+Inlet 3-rd element<br>
+<b> Figure  7.139</b> <br>
+The SMV static mixer<br>
+The SMX mixer is manufactured and sold by Sulzer under license from Bayer. In <br>fact, this mixer used to be called the BMK mixer, which stands for Bayer Kontinuier-<br>lich Mischer (German for Bayer Continuous Mixer). It consists of crossed bars form-<br>ing an approximately 45° angle with the axis of the pipe; see Fig. 7.140.<br>
+<b> Figure 7.140</b> <br>
+The SMX static mixer<br>
+<hr>
+<A name=474></a><b>462</b>  <b>7Functional Process Analysis</b><br>
+This mixer is also described in Chapter 12, where Fig. 12.53 shows a picture of <br>the mixer geo metry, and Fig. 12.54 shows simulated particle tracking through the <br>mixer, while Fig. 12.55 shows the predicted pressure profile. The bars split the flow <br>into layers, which are distributed over the cross-section of the pipe. The degree of <br>mixing increases exponentially with the number of mixing elements. Adjacent ele-<br>ments are turned 90° relative to one another. The mixing action of the SMX element <br>is based on the flow around a tilted bar. When the bar is perpendicular to the flow, <br>the flow splits and recombines behind the bar; see Fig. 7.141 left. However, when <br>the bar is positioned at an angle to the flow, the flow splits but does not recombine <br>behind the bar as a result of secondary flows changing the flow pattern; see Fig. 7.141 <br>right.<br>
+<b>Figure 7.141</b>Flow splitting with straight and oblique bars<br>
+A relatively simple static mixer [245] is the twisted tape mixer developed by Kenics. <br>The geo metry of this mixer is shown in Fig. 7.142. The Kenics mixer is made up of <br>plates that are twisted over 180°, and in some cases over 90°. The next plate is ori-<br>ented at a 90° angle relative to the preceding plate. As a result, the material gets <br>split and reoriented as it flows through the various elements of the mixer. Studies on <br>this static mixer have been published by C.D. Han et al. [215] and others. The advan-<br>tage of the Kenics mixer is its simple design and ease of cleaning.<br>
+<b> Figure 7.142</b> <br>
+The Kenics static mixer<br>
+A static mixer that incorporates the twisted tape concept in a different configuration <br>is the Equalizer from Komax Systems. Each element contains six circular channels <br>with a twisted tape in each channel; see Fig. 7.143. The tape in each element is <br>
+<hr>
+<A name=475></a><IMG src="Polymer Extrusion by Chris Rauwendaal-475_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-475_26.jpg"><br>
+ 7.7Mixing  <b>463</b><br>
+twisted such that a centered additive input will be converted to a pattern with radial <br>spokes. The second mixing element turns the melt stream inside out. This process <br>repeats itself when a number of elements are placed in series.<br>
+<b> Figure  7.143</b> <br>
+The Equalizer from Komax <br>
+Systems<br>
+Another mixer that incorporates the twisted tape feature is the Hi-mixer from the <br>Japanese company Toray. The elements consist of a cylinder with two circular chan-<br>nels and a cone-shaped entrance and exit. Within the circular channels is the twisted <br>tape, twisted over 180°; see Fig. 7.144.<br>
+<b> Figure  7.144</b> <br>
+The Hi-mixer from Toray<br>
+Passing through the mixer, the fluid is split in two smaller flows, split again by the <br>twisted tape and reoriented. The velocity then drops as the fluid enters the chamber <br>between two elements, and the process repeats itself in the next element that is <br>turned 90° relative to the first element.<br>Yet another variation of the twisted tape mixer is the bent and split tape mixer. An <br>example is the Komax mixer shown in Fig. 7.145. It consists of slat steel plates of <br>which the ends are split and bent about 45° in the opposite direction.<br>
+<b> Figure 7.145</b> <br>
+The Komax mixer<br>
+A relatively recent development in static mixers is the Dispersive/Distributive Static <br>Mixer (DDSM) [308]. This mixer is specifically designed to generate strong elonga-<br>tional flow for improved dispersive mixing. The DDSM is an extension of the CRD <br>
+<hr>
+<A name=476></a><b>464</b>  <b>7Functional Process Analysis</b><br>
+mixing technology [299] described in Section 8.7.1. The mixer is also described in <br>Chapter 12, where Fig. 12.56 shows the geo metry of the DDSM, while Fig. 12.57 <br>shows calculated streamlines of particles flowing through the mixer. Figure 12.58 <br>shows the flow number of the particles flowing through the mixer. The DDSM tech-<br>nology has also been applied to breaker plates. The mixing capability of an extruder <br>can be improved with minimum disruption to the process by using a breaker plate <br>that is also a static mixing device.<br>
+<b>7.7.2.2Functional Performance Characteristics<br></b>There are many more static mixing devices used in the polymer processing indus-<br>try; in fact, the number is so large that it is not possible to list them all. Two charac-<br>teristics are of critical importance in application of any static mixing device in actual <br>extrusion operations. The first one, obviously, is the mixing capacity. The second <br>one is the resistance that the static mixer offers to flow, i.e., the pressure drop along <br>the static mixer.<br>
+<b>7.7.2.2.1Mixing</b><br>
+The mixing capacity is clearly related to the number of striations and the striation <br>thickness. Several expressions have been proposed for various static mixers that <br>relate the number of striations to the number of mixing elements, e.g., see [260, <br>263]. The mixing in static mixers can generally be described as ordered plug con-<br>vective mixing. The striation thickness decreases from one mixing element to the <br>next. The reduction in striation thickness can be expressed by:<br>
+ (7.511)<br>
+where n is the number of elements of the static mixer and k is a factor that is deter-<br>mined by the geo metry of the mixer. For the Kenics mixer k = 2, the Komax mixer <br>k = 2, the ISG mixer k = 4, the SMX mixer k = 4, and the SMV mixer k = 8.<br>In experimental studies, however, the striation concept is not often used. This has to <br>do with the difficulty in accurately determining the number of striations beyond the <br>level of coarse mixing. Another measure of mixing is the coefficient of variation <br>(COV), discussed earlier. The COV has been used in several experimental studies to <br>compare the goodness of mixing in various static mixers.<br>The COV is dependent on flow rate and sample size. Therefore, one has to be careful <br>in comparing data from different sources. Allocca [218] compared several static <br>mixers in terms of their COV/length characteristics. For all mixers the COV reduces <br>exponentially with length. However, the rate at which the COV reduces varies con-<br>siderably. Most twisted tape type mixers require a substantial length to accomplish <br>good mixing, while other mixers accomplish the same task in a much shorter dis-<br>tance. If a value of COV = 0.05 represents good mixing, then the minimum length <br>
+<hr>
+<A name=477></a><IMG src="Polymer Extrusion by Chris Rauwendaal-477_1.jpg"><br>
+ 7.7Mixing  <b>465</b><br>
+required for the most efficient static mixer is L = 10 D; see Table 7.3. For the less <br>efficient static mixers the minimum length may have to be L = 30 D or more. Clearly, <br>with such long length one should be concerned about pressure drop, residence time, <br>and chance of degradation.<br>
+<b>7.7.2.2.2Pressure Drop</b><br>
+The other important performance characteristic is the pressure drop. A comparison <br>of various static mixers was reported by Allocca [218]; this is shown in Table 7.3. <br>The experimental set-up that was used to quantify the mixing capability of the vari-<br>ous static mixers is shown in Fig. 7.146.<br>
+Conductivity<br>
+cell<br>
+Recorder<br>
+Tank<br>
+Component 1<br>
+Static<br>
+(tracer)<br>
+mixing<br>
+elements<br>
+Component 2<br>(bulk)<br>
+<b> Figure 7.146</b> <br>
+Metering pumps<br>
+Test set-up for static mixing elements<br>
+The comparison is based on the length required to obtain a COV of 0.05 or less. <br>From this table it becomes clear that good mixing performance is not directly linked <br>to pressure drop. Some relatively inefficient static mixers have low pressure drop, <br>while others have a high pressure drop.<br>The pressure drop in a static mixer with laminar flow can be determined from:<br>
+ (7.512)<br>
+where NSM is a constant depending on the mixer geo metry (see Table 7.3),  is the <br>
+viscosity,   the volumetric flow rate, L the length, and D the diameter. For an empty <br>pipe the value of NSM = 32; the lowest value that is achieved for static mixers is about <br>
+200, while the highest is about 10,000.<br>
+<hr>
+<A name=478></a><b>466</b>  <b>7Functional Process Analysis</b><br>
+<b>Table 7.3</b>Comparison of Various Commercial Static Mixers [218]<br>
+MIXER<br>
+*L/D<br>
+NSM<br>
+Volrel<br>
+Hold-uprel<br>
+Drel<br>
+Lrel<br>
+<b></b>Prel<br>
+Koch SMX<br>
+9<br>
+1237<br>
+1.0<br>
+1.0<br>
+1.0<br>
+1.0<br>
+1.0<br>
+Koch SMXL<br>
+26<br>
+245<br>
+1.8<br>
+1.8<br>
+0.8<br>
+2.4<br>
+0.6<br>
+Koch SMV<br>
+18<br>
+1430<br>
+4.6<br>
+4.5<br>
+1.3<br>
+2.7<br>
+2.3<br>
+Kenics<br>
+29<br>
+220<br>
+1.9<br>
+1.8<br>
+0.8<br>
+2.7<br>
+0.6<br>
+Etoflo HV<br>
+32<br>
+190<br>
+2.0<br>
+2.0<br>
+0.8<br>
+2.7<br>
+0.6<br>
+Komax<br>
+38<br>
+620<br>
+8.9<br>
+8.2<br>
+1.3<br>
+5.4<br>
+2.1<br>
+Lightnin<br>
+100<br>
+290<br>
+29.0<br>
+27.0<br>
+1.4<br>
+15.3<br>
+2.6<br>
+PMR<br>
+320<br>
+500<br>
+511.0<br>
+460.0<br>
+2.4<br>
+86.0<br>
+14.5<br>
+Toray<br>
+13<br>
+1150<br>
+1.9<br>
+0.9<br>
+1.1<br>
+1.6<br>
+1.35<br>
+N-Form<br>
+29<br>
+544<br>
+4.5<br>
+3.8<br>
+1.1<br>
+3.6<br>
+1.4<br>
+Ross ISG<br>
+10<br>
+9600<br>
+9.6<br>
+3.4<br>
+2.1<br>
+2.3<br>
+8.6<br>
+*The L/D ratio refers to the ratio required to achieve COV &lt; 0.05<br>
+<b>7.7.2.2.3Residence Time Distribution</b><br>
+Another performance characteristic of interest is the residence time distribution, <br>RTD, of the mixer. The residence time is a strong function of the distance from the <br>wall. The residence time is short for fluid elements in the center of the channel, but <br>long for elements close to the wall. Data on RTD for various static mixers was pre-<br>sented by Pahl and Muschelknautz [309]; see Fig. 7.147.<br>The data is shown in normalized form with normalized concentration on the vertical <br>axis and normalized residence time on the horizontal axis. It is clear that the RTD <br>for the static mixers shown in Fig. 7.147 is considerably narrower than the RTD in <br>an empty pipe. In terms of RTD, there are no major differences between the various <br>static mixing devices.<br>
+n<br>
+Empty pipe<br>
+SMX (8 elements)<br>
+Kenics (30 elements)<br>
+Hi-Mixer (12 elements)<br>
+Pure plug flow<br>
+Dimensionless concentratio<br>
+Dimensionless residence time<br>
+<b>Figure 7.147</b>RTD data for different static mixers<br>
+<hr>
+<A name=479></a> 7.7Mixing  <b>467</b><br>
+<b>7.7.2.2.4Thermal Homogenization</b><br>
+In many applications of static mixers the ability to reduce melt temperature non-<br>uniformities is important. Various studies have been made on the thermal homoge-<br>nization in static mixers. Nauman [310] introduced the concept of thermal time <br>distribution to characterize the degree of thermal homogeneity. Some models may <br>predict a higher level of thermal homogenization than experimentally observed. <br>This was the case in studies by Cliff and Wilkinson [311] where a traversing ther-<br>mocouple was used to measure radial temperature distribution at the outlet of a <br>20 mm diameter Kenics mixer with nine elements. It was found that significant <br>temperature variations remained at the outlet when two viscoelastic fluids differing <br>only in temperature were introduced at the inlet. Tracer studies showed that a vis-<br>coelastic fluid streamline can be deflected rather than divided by the leading edge <br>of an element. Such behavior, obviously, can reduce the mixing action in a static <br>mixing device.<br>Craig [312] reports on an investigation of the performance of a large diameter static <br>mixer used as a continuous reactor for styrene polymerization. It was shown the <br>mixer behaved adiabatically. This was confirmed by computer simulation using mod-<br>els that had been verified experimentally. Better results were obtained with a static <br>mixer that carries a heat transfer fluid supplied via manifold connections from exter-<br>nal headers. A comparison of different static mixers with respect to thermal homog-<br>enization, pressure drop, and mixing efficiency was published by Mueller [313].<br>
+<b>7.7.2.2.5Mixing Materials with Different Viscosities</b><br>
+The discussion so far has focused on compatible materials of similar viscosity, i.e., <br>simple mixing. In many cases, however, it is necessary to admix low-viscosity addi-<br>tives into a polymer melt. This can involve very high viscosity ratios, as much as ten <br>million to one, and is very difficult for most static mixers. As the viscosity ratio <br>increases, the length of the mixer required also increases. If the length of the mixer <br>is insufficient, the low-viscosity component can separate in the form of splash-out. <br>The splash-out limit, beyond which no further splash-out can be detected, is shown <br>in Fig. 7.148 for SMX type mixing elements for several polymer/additive mixtures.<br>Another difficult mixing job is the mixing of high-viscosity fluid in a low-viscosity <br>matrix. Drop rupture occurs when the drop extension exceeds a critical value. The <br>extension of the drop is determined by the ratio of shear forces to surface forces. <br>Therefore, sufficiently high forces have to be present in the mixer long enough to <br>achieve dispersive mixing. A combination of shear and extensional flow is more <br>effective in reducing drop size than only shear flow. Since some static mixers have <br>elongational flow components, some degree of mixing is possible when high-viscosity <br>components need to be mixed in a lower viscosity matrix and the viscosity ratio is <br>not too large. For large viscosity ratios, clearly, dynamic mixing devices will be much <br>more efficient.<br>
+<hr>
+<A name=480></a><b>468</b>  <b>7Functional Process Analysis</b><br>
+PE/Mineral oil<br>
+PE/Lubricant<br>
+HIPS/Mineral oil<br>
+GPS/Mineral oil<br>
+e (weight %)<br>
+Additiv<br>
+<b> Figure  7.148</b> <br>
+Splash-out limit for SMX mixer for several <br>
+L/D Ratio<br>
+polymer-additive mixtures<br>
+<b>7.7.2.3Miscellaneous Considerations<br></b>In the discussion on residence time distribution it was mentioned that the RTD of a <br>static mixer tends to be narrower than the RTD of a narrow pipe. However, in addi-<br>tion to the RTD the actual residence times are important as well. Static mixers <br>always add extra volume between the extruder and the die. Therefore, when a static <br>mixer is used, the mean residence time will always increase when the entire extru-<br>sion system is considered, i.e., extruder, static mixer, and die. If the static mixer <br>section is long, which it should be to give good mixing, then the increase in mean <br>residence time can be considerable. This should be a concern in thermally less stable <br>polymers.<br>The fact that static mixers have no moving parts means that they have no pressure <br>generating capability. Therefore, static mixing devices always consume pressure <br>and, thus, require the extruder to generate more pressure. This results in reduced <br>extruder efficiency and higher stock temperatures in the extruder. The lack of pres-<br>sure generating capability and the additional residence time in static mixers are <br>problems that are not present in screw mixing sections. These points should be <br>carefully considered when a decision is made to use a dynamic or static mixing <br>device.<br>Other considerations that should play a role in the selection of a static mixing device <br>is the mechanical strength, streamlining, operator friendliness, and price. Clearly, <br>the static mixer should be strong enough to withstand the forces acting on it during <br>use, assembly, and disassembly. Some static mixers have been known to collapse <br>under the pressure of the polymer melt. A streamlined design is necessary to avoid <br>dead spots and to reduce the chance of degradation. Operator friendliness refers to <br>ease of assembly, installation, removal, disassembly, and ability to clean. An opera-<br>tor-friendly design will enhance the change of success in actual production opera-<br>tions. Finally, price is a consideration that cannot be avoided. The comparison in <br>price should not only be between various static mixers but also between a dynamic <br>
+<hr>
+<A name=481></a> 7.7Mixing  <b>469</b><br>
+and a static mixing device. A mixing section on the extruder screw may well be less <br>expensive and achieve better mixing than a static mixer, provided, of course, that an <br>efficient screw mixing section is selected.<br>
+<b>7.7.3Dispersive Mixing</b><br>
+The discussion so far has primarily been concerned with distributive laminar mixing. <br>However, in actual extrusion operations the requirement for good dispersive mixing <br>is often more critical than the distributive mixing. This is particularly true in extru-<br>sion of compounds with pigments or in small gauge extrusion (e.g., low denier fiber <br>spinning, thin film extrusion, etc.). In dispersive mixing, the actual stresses acting <br>on the agglomerates determine whether or not the agglomerate will break down. <br>The breakdown stress will depend on the size, shape, and nature of the agglomerate. <br>The stresses acting on the agglomerate will depend on the flow field and the rheo-<br>logical properties of the fluid.<br>
+<b>7.7.3.1Solid-Liquid Systems<br></b>One of the first engineering analyses of dispersive mixing was made by Bolen and <br>Colwell [220]. They assumed that the agglomerates break when the internal stresses, <br>induced by viscous drag on the particles, exceed a certain threshold value. Bird <br>et al. [221] analyzed the forces acting on a single agglomerate in the form of a rigid <br>dumbbell, consisting of two spheres of radii r1 and r2. The centers of the spheres are <br>
+separated by a distance L, and the dumbbell is located in a homogeneous flow field <br>of an incompressible Newtonian fluid. As a result of the viscous drag acting on the <br>spheres, a force will develop in the connector. The force depends on the drag on each <br>sphere and on the orientation of the dumbbell. Tadmor [222] adopted Bird's approach <br>and extended the analysis to spheres of different radius and to include the effect of <br>Brownian motion.<br>In a steady simple shear flow, the maximum connector force develops when the <br>dumbbell is oriented at a 45° angle to the direction of shear; this force is:<br>
+ (7.513)<br>
+where s is the shear viscosity of the fluid,   the shear rate, L the length of the <br>
+dumbbell, and r1 and r2 the radii of the dumbbell.<br>When the spheres are in contact with each other, Eq. 7.513 reduces to:<br>
+ (7.514)<br>
+From Eqs. 7.513 and 7.514, it can be seen that the force in the connector is directly <br>proportional to the shear stress and to the product of the radii. Thus, in the same <br>
+<hr>
+<A name=482></a><b>470</b>  <b>7Functional Process Analysis</b><br>
+flow field, the force in the connector (breakdown force) reduces as the sizes of the <br>spheres reduce. If the breakdown force of the agglomerate remains about constant, <br>the breakdown will proceed until a certain minimum agglomerate size is reached. <br>Further breakdown will not occur because the flow field will be unable to generate <br>sufficient breakdown force in the agglomerate. For a steady elongational flow, the <br>maximum force in the connector is obtained when the dumbbell is aligned in the <br>direction of flow. When the spheres are in contact with each other, the maximum <br>force is:<br>
+ (7.515)<br>
+where e is the elongational viscosity, e the elongational stress, and  is the rate of <br>
+elongation.<br>Equations 7.514 and 7.515 assume that the spheres do not affect the flow field and <br>that dumbbell interaction can be neglected. By comparing Eqs. 7.514 and 7.515, it <br>can be seen that the breakdown force in elongational flow is twice as large as in <br>simple shear at the same rate of deformation and viscosity. However, the elongational <br>viscosity of polymer melts is greater than the shear viscosity. The elongational vis-<br>cosity at low strain rates is at least three times higher than the shear viscosity, while <br>at higher strain rates the elongational viscosity can be more than three times higher. <br>As a result, the hydrodynamic forces generated in elongational flow are higher than <br>in shear flow and, as a result, dispersive mixing is more efficient.<br>In most commercial mixers used for dispersive mixing, the actual flow patterns in <br>the mixer will be a combination of shear flow and elongational flow. The flow pat-<br>terns are often complex and require advanced numerical techniques to allow accu-<br>rate analysis of flow. Only recently has it become possible to perform a full 3-D <br>analysis of complex mixing devices. This allows a complete engineering approach to <br>the design of mixing devices [299]; see also Chapter 12. In the past, mixing devices <br>were developed based largely on experience, empirical knowledge, and intuition <br>with very little, if any, engineering analysis.<br>From Eqs. 7.514 and 7.515 it can be seen that the breakdown force is directly propor-<br>tional to the viscosity of the matrix. This has some important practical implications. <br>Dispersive mixing should be done at as low a temperature as possible to increase the <br>viscosity and thus the breakdown force. If both a dispersive and a distributive mix-<br>ing element are required in a single screw extruder, the dispersive element should <br>be placed upstream of the distributive element. This placement is more likely to <br>result in a relatively low stock temperature at the inlet of the dispersive mixing ele-<br>ment, while the stock temperature at the inlet of the distributive mixing element will <br>be relatively high as a result of the viscous heat generation in the dispersive mixing <br>element. The low stock temperature in the dispersive mixing element will enhance <br>dispersive mixing, while the high stock temperature in the distributive mixing ele-<br>ment will improve the energy efficiency of the distributive mixing step.<br>
+<hr>
+<A name=483></a> 7.7Mixing  <b>471</b><br>
+The need for high viscosity in dispersive mixing explains why it is often easier to <br>produce a masterbatch of a high filler loading and let it down later, than to produce <br>a compound with relatively low filler loading. The viscosity of the masterbatch will <br>be much higher and, therefore, the dispersive mixing action will be much more <br>effective. Masterbatching is a very common technique in the mixing and compound-<br>ing industry. It should be remembered that in many instances, agglomerate break-<br>down can be achieved more effectively in high speed solid-solid mixing than in liquid-<br>solid mixing [225­227]. The forces that can be transmitted in solid-solid mixing are <br>generally much higher than in liquid-solid mixing, and agglomerate breakdown will <br>occur faster and to a larger extent, resulting in very finely dispersed particles.<br>Dispersive mixing in single screw extruders was studied in detail by Martin [223, <br>224]. An interesting finding of this study was that dispersive mixing is determined <br>not only by the shear stress acting on an agglomerate, but also by the exposure time. <br>It was found that in dispersive mixing of carbon blacks, a certain minimum shear <br>stress exposure time was necessary to accomplish breakdown; this minimum expo-<br>sure time was about 0.2 s. Below this minimum exposure time, no breakdown <br>occurred no matter how high the shear stress. This finding runs counter to common <br>belief that dispersive mixing is only determined by the actual level of stress acting <br>on the agglomerate. However, the minimum stress exposure time is a very impor-<br>tant consideration in the design of dispersive mixing equipment, such as a disper-<br>sive mixing section on an extruder screw. It was also found that the mixing perform-<br>ance of an extruder is very much dependent on the length of the plasticating zone, <br>which depends on the operating conditions. Incorporation of mixing sections can <br>substantially improve an extruder's mixing performance and enable deepening of <br>the metering section as compared to screws without mixing sections.<br>
+<b>7.7.3.2Liquid-Liquid System<br></b>In the previous section we discussed dispersive mixing in solid-liquid systems. <br>Another important type of dispersive mixing is in liquid-liquid systems. This occurs <br>when we mix incompatible or partially incompatible polymer melts. The production <br>of polymer blends is very important in the polymer industry; as a result, liquid- <br>liquid dispersive mixing has received significant attention over the last two or more <br>decades.<br>In the deformation of droplets in immiscible systems the interfacial stress will affect <br>the deformation once the drop size falls below a critical value. The deformation is <br>determined by the ratio of viscous stress to interfacial stress; this ratio is called the <br>Weber or Capillary number. This number can be expressed as:<br>
+ (7.516)<br>
+where  is the viscous stress,  the interfacial tension,  the matrix viscosity,   the <br>shear rate, /R the interfacial stress, and R the radius. When the Capillary number <br>
+<hr>
+<A name=484></a><b>472</b>  <b>7Functional Process Analysis</b><br>
+is large, larger than the critical Capillary number, the viscous stress will dominate <br>the interfacial stress and drops will undergo affine stretching. The critical Capillary <br>number is usually around unity (Cacrit  1). When the Capillary number is small <br>
+(Ca  Cacrit) the interfacial stresses dominate the viscous stresses, which results in <br>
+stable drop sizes.<br>Taylor [64] found that in simple shear flow, a dispersed drop with viscosity ratio p = 1 <br>breaks up when the Ca &gt; 0.5. Breakup seems to occur when the shear stress and the <br>interfacial stress are of the same order of magnitude. The critical Capillary number <br>depends on the type of flow and on the viscosity ratio. In the mixing process two <br>regimes can typically be distinguished:<br>
+A) a viscous stress-dominated regime (Ca &gt;&gt; Cacrit)<br>B) an interfacial stress-dominated regime (Ca  Cacrit)<br>
+In the early stages of mixing, regime A tends to prevail. Typical values for the <br>parameters of the Capillary number are:<br>
+viscosity <br>
+ = 100 Pas<br>
+shear rate <br>
+  = 100 s­1<br>
+interfacial tension <br>
+ = 0.01 N/m<br>
+radius <br>
+R = 0.001 m<br>
+The resulting Capillary number Ca = 1000, which is much larger than the critical <br>Capillary number, and the drops are stretched into fine threads. In the later stages <br>of mixing, regime B will prevail. Assume that the drop size has reduced to R = 1E­6 <br>m, then the Capillary number becomes Ca = 1. Thus, with a small enough drop size, <br>the Capillary number will approach the critical Capillary number (Ca  Cacrit).<br>
+<b>7.7.3.2.1Breakup under Quiescent Conditions</b><br>
+When droplets are suspended in a viscous matrix that is exposed to a shearing and/<br>or elongational flow, the drop will be stretched into long filaments. This stretching <br>does not go on indefinitely; at some point the thread becomes thin enough for inter-<br>facial tension to start playing a role. In other words, the surfaces become active. <br>According to the example above, interfacial tension starts to become important <br>when the thread radius is around 1 micron.<br>The interfacial tension will want to reduce the interface between the two phases, <br>minimizing the surface-to-volume ratio. The smallest surface-to-volume ratio is <br>achieved in a sphere, S/V = 3/R. Thus, an extended liquid thread will tend to break <br>up due to the interfacial tension. The breakup is initiated by small disturbances at <br>the interface, so-called Rayleigh disturbances. These disturbances grow due to the <br>interfacial tension and eventually breakup can occur. The progress of the breakup <br>process is illustrated in Fig. 7.149.<br>
+<hr>
+<A name=485></a> 7.7Mixing  <b>473</b><br>
+<b> Figure 7.149</b> <br>
+Schematic representation of the <br>
+breakup process, top to bottom<br>
+The first theoretical analysis of the breakup of a Newtonian thread in a quiescent <br>Newtonian matrix was performed over a century ago by Rayleigh [284]. The disturb-<br>ances that initiate the breakup process are often referred to as Rayleigh disturb-<br>ances. Rayleigh analyzed only the effect of surface tension, neglecting the viscosi-<br>ties of the two phases. This work was extended by Tomotika [285] by including the <br>effect of viscosity. The analysis considers a sinusoidal liquid cylinder; the radius as <br>a function of axial distance z is:<br>
+ (7.517)<br>
+where  is the wavelength and  the disturbance amplitude.<br>The average radius can be expressed as:<br>
+ (7.518)<br>
+The disturbance amplitude grows exponentially in time:<br>
+ (7.519)<br>
+The growth rate q is given by:<br>
+ (7.520)<br>
+where 0 is the original disturbance amplitude, p the viscosity ratio, and R0 the ini-<br>
+tial thread radius. It can be determined that when the wavelength is greater than <br>the thread circumference, the interfacial area decreases when  increases. Thus, <br>the liquid thread is unstable to distortions with wavelengths greater than the thread <br>circumference. Initially, disturbances of all wavelengths may be present. There is, <br>however, only one disturbance that grows the fastest and will cause breakup. The <br>wavelength of the fastest growing disturbance is the dominant wavelength (m); it <br>
+depends only on the viscosity ratio; see Fig. 7.150.<br>Breakup will occur when the distortion amplitude equals the average thread radius; <br>this occurs when  = 0.8R0. The time necessary to reach this distortion is:<br>
+<hr>
+<A name=486></a><b>474</b>  <b>7Functional Process Analysis</b><br>
+ (7.521)<br>
+Kuhn [69] proposed an estimate of the initial amplitude 0 based on the temperature <br>
+fluctuations due to Brownian motion. The resulting expression shows that the time <br>to break increases with the matrix viscosity and thread radius and reduces with <br>interfacial tension. The radius of the newly formed droplets Rd can be determined <br>
+from the conservation of mass:<br>
+ (7.522)<br>
+When the viscosity ratio p = 1, the dominant wavelength is m = 11.22 R0; see <br>
+Fig. 7.150. The newly formed droplet according to Eq. 7.522 will be Rd = 2.034R0, <br>
+about twice the radius of the original thread. The new drop radius will be a function <br>only of the viscosity ratio; the relationship is shown in Fig. 7.151.<br>
+] 0<br>
+ length [R<br>
+wave<br>
+Dominant <br>
+<b> Figure  7.150</b> <br>
+The dominant wavelength versus <br>
+Log viscosity ratio<br>
+the viscosity ratio<br>
+] 0<br>
+w drop radius [R<br>Ne<br>
+<b> Figure 7.151</b> <br>
+The new drop radius versus <br>
+Log viscosity ratio<br>
+the  viscosity ratio<br>
+<hr>
+<A name=487></a> 7.7Mixing  <b>475</b><br>
+For viscosity ratios typically encountered (1E­3 &lt; p &lt; 1E+2), the radius will be about <br>2 to 2.5 times the original thread radius. In some cases, small drops form together <br>with larger drops; see Fig. 7.149. These satellite drops form in the last stage of the <br>breakup process due to the rapid growth of Rayleigh disturbances on the fine fila-<br>ments in the necked-down regions. Tjahjadi et al. [314] numerically investigated the <br>formation of satellite droplets. They found unique radius distributions of satellite <br>and sub-satellite droplets for several viscosity ratios. As the viscosity ratio increases, <br>the number of satellite droplets decreases and their radius increases.<br>
+<b>7.7.3.2.2Breakup in Flow</b><br>
+The previous discussion focused on the breakup of liquid thread suspended in a <br>quiescent Newtonian fluid. In real mixing operations quiescent conditions will usu-<br>ally not occur, except perhaps for short periods of time. The more important issue, <br>therefore, is how the breakup occurs when the system is subjected to flow. Good <br>reviews on the breakup of liquid threads are available from Acrivos [304], Rallison <br>[305], and Stone [306]. Probably the most extensive experimental study on drop <br>breakup was performed by Grace [286]; data was obtained over an enormous range <br>of viscosity ratios: 10­6 to 10+3! Grace determined the critical Weber (Capillary) <br>number for breakup both in simple shear and in 2-D elongation; the results are re- <br>presented in Fig. 7.152.<br>
+<b> Figure  7.152</b> <br>
+Critical Capillary (Weber) number versus <br>
+the viscosity ratio, after Grace [286]<br>
+The interesting feature of the data from Grace is that the critical Capillary number <br>for shear flow goes to infinity when the viscosity ratio reaches a value of four. This <br>means that it is not possible to break up drops in shear flow when the viscosity of <br>the drop is more than four times higher than the matrix. Such a problem does not <br>exist in elongational flow. Also, the critical Capillary number in elongational flow is <br>lower than in shear flow, particularly at viscosity ratios much below or above unity. <br>Clearly, elongational flow is more efficient in breaking liquid threads into droplets <br>than shear flow. An empirical expression for the critical Capillary number in shear <br>flow was proposed by De Bruijn [287]:<br>
+<hr>
+<A name=488></a><b>476</b>  <b>7Functional Process Analysis</b><br>
+ (7.523)<br>
+Janssen [294] studied the breakup of a viscous thread in stretching flow, both theo-<br>retically and experimentally. In a quiescent matrix one disturbance wavelength is <br>dominant; in an extending matrix the waves are continuously stretched. Different <br>disturbance wavelengths are dominant at different times. As a result, the breakup is <br>postponed in elongational flow compared to quiescent conditions.<br>Janssen found that when the stretching rate is increased, the liquid thread is thinned <br>further before breakup; this leads to smaller drops. A higher viscosity of either the <br>matrix or the drops retards interfacial motions and postpones breakup, leading to <br>smaller drops. At a constant matrix viscosity and low stretching rate, a higher liquid <br>thread viscosity will lead to smaller drops. The results of the theoretical analysis <br>were presented in graphical form; no simple explicit expression was found. The <br>dimensionless drop radius resulting from breakup of a Newtonian liquid thread <br>extending at a uniform rate is shown in Fig. 7.153.<br>
+<b> Figure  7.153</b> <br>
+Dimensionless drop radius <br>
+ <br>versus dimensionless stretch <br>
+rate<br>
+The dimensionless drop radius r* is the actual drop radius divided by the initial <br>amplitude  0. The dimensionless stretching rate e is the actual stretching rate e <br>
+multiplied with the matrix viscosity and the initial amplitude and divided by the <br>interfacial tension; thus e* = me0/. The curves in Fig. 7.153 can be fit quite well <br>
+with an expression of the form:<br>
+ (7.524)<br>
+At low values of the dimensionless stretching rate the drop radius reduces in a <br>power law fashion (i.e., straight line on log-log scale); this portion of the curve is <br>described by the first right-hand side term. At higher values of the dimensionless <br>
+<hr>
+<A name=489></a> 7.7Mixing  <b>477</b><br>
+stretching rate the drop radius reaches an asymptotic plateau; this portion of the <br>curve is described by the second right-hand term. At this point, the drop radius <br>becomes independent of the stretching rate and depends only on the viscosity ratio. <br>Figure 7.153 indicates that the drop radius reduces as the viscosity ratio increases, <br>at least for low values of the dimensionless stretch rate.<br>Rumscheidt and Mason [288] distinguish four classes of deformation and breakup <br>in simple shear flow depending on the viscosity ration p. When p &gt; 1, the deformed <br>drop has rounded ends, while for smaller p values the ends become pointed. When <br>p &lt; 0.1, very small droplets break off and form the sharply pointed ends--this is <br>called <i>tipstreaming.</i> This is caused by gradients in interfacial tension due to convec-<br>
+tion of surfactants along the drop surface. The interfacial tension is lowered at the <br>tip, causing very small droplets to break off.<br>An important parameter in the breakup process is the time required for deforma-<br>tion and breakup. This was measured by Grace [286] under quasi-equilibrium con-<br>ditions. His results are represented in Fig. 7.154.<br>
+irrotational shear<br>
+*  b<br>
+, 0.5t<br>
+Dimensionless time<br>
+ratational shear<br>
+<b> Figure  7.154</b> <br>
+Dimensionless time for drop breakup <br>
+Viscosity ratio<br>
+versus viscosity ratio<br>
+Clearly, the dimensionless breakup time is strongly dependent on the viscosity <br>ratio. The dimensionless breakup time is given by:<br>
+ (7.525)<br>
+Grace [286] found that the breakup time decreases upon exceeding the critical <br>Weber number. Similar experiments by Elemans [307] did not show a decrease in <br>breakup time upon exceeding the critical Weber number. The breakup time for vis-<br>cosity ratios between 0.1 and 1 was found to be around 50 to 100, i.e., tb* = 50­100. <br>
+Experimental work in an opposed-jet device [66] found that with flow occurring a <br>droplet will stretch, while during the no-flow condition, breakup occurs via necking. <br>This is illustrated in Fig. 7.155.<br>
+<hr>
+<A name=490></a><b>478</b>  <b>7Functional Process Analysis</b><br>
+The transient experiment shown in Fig. 7.155 mimics conditions occurring in an <br>extruder. In an extruder, drops are exposed to high rates of deformation as they <br>approach the flight tip, while, after passing through the flight clearance, the rate of <br>deformation is relatively low. A drop is stretched in an elongational flow and after <br>the flow stops, the drop breaks by a necking-in mechanism.<br>
+Flow on<br>
+Flow off<br>
+t=0 sec.<br>
+t=8<br>
+t=3<br>
+t=9<br>
+t=10<br>
+t=7<br>
+<b>Figure 7.155</b>Deformation and breakup of drop in opposed-jet device<br>
+Once the flow has stopped, the interfacial tension drives two competing processes. <br>One is the relaxation back to the original sphere; the other is the development of <br>capillary waves. Capillary waves are like Rayleigh disturbances on a liquid thread. <br>The relaxation of the drop is caused by the pressure difference over the interface of <br>the drop:<br>
+ (7.526)<br>
+where R1 and R2 are the principal radii of curvature. At the ends of the threads <br>
+where the radius is small (R1 = R2 = R), the pressure difference is large (P = 2/R). <br>
+In the middle of the thread the radius approaches infinity (R1  ) and P = /R. As <br>
+a result, this pressure difference flow takes place from the end of the extended drop <br>towards the center. When a disturbance is present at the interface, a pressure differ-<br>ence will develop between the center of the disturbance and the bulb-shaped ends. <br>This pressure difference is also described by Eq. 7.526. If the time scale for the <br>growth of the disturbance is less than the time scale for relaxation, the drop will <br>break as shown in Fig. 7.155. Otherwise, relaxation will occur without breakup.<br>
+<b>7.7.3.2.3Coalescence</b><br>
+The work discussed thus far dealt with deformation and breakup of isolated drops. <br>In real mixing operations the dispersed phase has a large enough volume fraction <br>that interaction between the drops cannot be neglected. Elmendorp [78] found in <br>
+<hr>
+<A name=491></a> 7.7Mixing  <b>479</b><br>
+experiments that coalescence becomes important in the mixing of immiscible sys-<br>tems even at volume fractions of the dispersed phase as low as a few percent. As the <br>volume fraction increases phase inversion will take place. The critical volume frac-<br>tion for phase inversion depends strongly on the viscosity ratio.<br>Theoretical work on coalescence has been done by a number of workers, e.g., Ches-<br>ters [292], Elmendorp [293], and Janssen [294]. Important parameters in the coales-<br>cence are the volume fraction of the dispersed phase and the flow field, because <br>these determine the frequency of collisions, the contact force, and the interaction <br>time. When drops approach each other, the matrix material between the drops has <br>to be removed for coalescence to take place. When the film of matrix material is <br>below a critical value, instabilities rupture the film and the drops coalesce.<br>
+<b>7.7.3.2.4Collision of Drops</b><br>
+The collision frequency of a drop as a function of the shear rate   and the volume <br>fraction of the dispersed phase  can be written as:<br>
+ (7.527)<br>
+It is interesting to note that the collision frequency is independent of the drop <br>radius. On the average a drop collides every time t:<br>
+ (7.528)<br>
+In other words, a drop collides on the average after a total shear strain of  = /8.<br>
+<b>7.7.3.2.5Film Drainage</b><br>
+Initially, the drops approach according to the velocity gradient of the external flow <br>field. The drainage rate ­dh/dt is of the order  R. At a certain separation h0 the <br>
+hydrodynamic interaction becomes significant and the collision starts. The driving <br>force for film drainage is the contact force F that acts during the interaction time tint. <br>
+The drainage rate decreases and the thickness asymptotically decays to zero. When <br>a critical film thickness hcrit is reached, instabilities grow at the interface and film <br>
+rupture occurs; the drops coalesce.<br>The separation distance h0 can be determined from:<br>
+ (7.529)<br>
+The contact force is the Stokes drag force:<br>
+ (7.530)<br>
+<hr>
+<A name=492></a><b>480</b>  <b>7Functional Process Analysis</b><br>
+The critical film thickness can be calculated from:<br>
+ (7.531)<br>
+where A is the Hamaker constant, R the drop radius, and  the interfacial tension.<br>The critical film thickness for rupture is of the order of 50 Å. If the interaction time <br>of the drops is too short to reach the critical film thickness, the drops will not coa-<br>lesce. The drainage of the film is the rate-determining step in coalescence of deform-<br>able drops in polymer blends. Various models have been proposed to describe the <br>film drainage. One model assumes fully mobile interfaces, another model assumes <br>immobile interfaces, and a third model assumes partially mobile interfaces. The <br>mobility of the interfaces is strongly dependent on the presence of impurities, such <br>as surfactants. Surfactants reduce the mobility of the interfaces due to interfacial <br>tension gradients [315].<br>Elmendorp showed [78] that the model based on fully mobile interfaces under- <br>predicts the experimental coalescence time, while the model based on immobile <br>interfaces over-predicts the coalescence time. A model based on partially mobile <br>interfaces was proposed by Chester [316]; the coalescence time in this model is:<br>
+ (7.532)<br>
+where hc is the critical film thickness for rupture, d the viscosity of the dispersed <br>
+phase, and F the contact force. Chester's model is valid for cases where the viscosity <br>ratio is close to unity. When the viscosity ratio is large (p &gt;&gt; 1) the model for immo-<br>bile interfaces is appropriate. When the viscosity ratio is very small (p &lt;&lt; 1) the <br>model for fully mobile interfaces should be used.<br>In polymer blends compatibilizers can reduce the mobility of the interface. Compa-<br>tibilizers often result in a finer morphology; this is due to three factors:<br>1.  Delay of breakup due to lower surface tension, yielding thinner threads and <br>
+smaller droplets.<br>
+2.  Increase in the Weber number enabling the flow to break into smaller droplets <br>
+before the critical Weber number is reached.<br>
+3.  Reduced coalescence.<br>The influence of the contact force F is not intuitively obvious; one would expect a <br>higher contact force to lead to faster coalescence. According to the models with <br>immobile and partially mobile interfaces, the coalescence time increases with the <br>contact force. This is due to the fact that the flattened area between the two deform-<br>able colliding drops increases with F, requiring more film material to be drained <br>over a longer distance.<br>
+<hr>
+<A name=493></a> 7.7Mixing  <b>481</b><br>
+The effect of the contact force has important implications for real mixing operations. <br>It can be expected that coalescence is not likely to occur in regions of high defor-<br>mation rates, due to the high contact forces. However, in these regions deformation <br>of drops leading to breakup will likely take place. Coalescence will take place prefer-<br>entially in regions of low deformation rates and low contact force, while drop defor-<br>mation will be minimal and breakup will occur under semi-quiescent conditions. <br>The probability of coalescence Pc in simple shear with the Chester model can be <br>
+expressed as:<br>
+ (7.533)<br>
+where c is a constant of order unity. According to this equation, coalescence is <br>enhanced by small drop radius R, low viscosity ratio p, and low Capillary number <br>Ca. A low Capillary number occurs at low matrix viscosity, low shear rate, or high <br>interfacial tension.<br>
+<b>7.7.3.2.6Models for Dispersive Mixing of Immiscible Liquids</b><br>
+Janssen [294] developed a model for dispersive mixing using a two-zone model <br>similar to the model used by Manas-Zloczower for dispersive mixing of a polymer <br>melt with an agglomerated filler [317]. The model uses a strong zone (high deforma-<br>tion rate) where affine stretching and thread breakup occurs and a weak zone (low <br>deformation rate) where coalescence and also thread breakup take place. The resi-<br>dence time in the strong zone is short, while the residence time in the weak zone is <br>relatively long. The Janssen model is shown schematically in Fig. 7.156.<br>
+<b>Figure 7.156</b>Schematic representation of the Janssen model<br>
+Some of the results of the Janssen model are quite interesting. It was found that a <br>high viscosity of the dispersed phase promotes a finer dispersion due to the delay of <br>thread breakup and coalescence. In general, lower viscosities of either phase result <br>in coarser morphology. Highly viscous systems cannot be dispersed finer than <br>0.1 micron since coalescence starts to dominate as the drop size reduces much <br>below 1 micron.<br>
+<hr>
+<A name=494></a><b>482</b>  <b>7Functional Process Analysis</b><br>
+The commercial blend Noryl GTE is a practical example of immiscible fluids where <br>a large viscosity ratio leads to a fine morphology. In Noryl GTE a high-viscosity PPE <br>(polyphenylene ether) is dispersed in a much lower viscosity polyamide, viscosity <br>ratio p  20. Simply looking at the critical Weber number for drop breakup (see <br>Fig. 7.152) suggests that a fine dispersion cannot be obtained in shear flow since <br>p &gt;&gt; 1. By using the two-zone model, however, it is predicted that this blend can be <br>dispersed to a length scale of 0.1 micron. This confirms commercial reality: Noryl <br>GTE blends processed on twin screw extruders do indeed achieve fine dispersions <br>with a length scale of around 0.1 micron.<br>A series of articles was published by Utracki et al. [318­322] on the modeling of <br>mixing of immiscible fluids in a twin screw extruder. The fourth paper in the series <br>[321] incorporates several refinements of the earlier model, one of the most impor-<br>tant refinements being the incorporation of the effect of coalescence. The model <br>considers two breakup mechanisms, both based on the micro-rheology. One breakup <br>mechanism is the drop fibrillation and disintegration into fine droplets when the <br>Weber number is greater than four times the critical Weber number. The second <br>mechanism is drop splitting that occurs when the Weber number is below four times <br>the critical Weber number.<br>Utracki et al. [321] found that there is a large difference between computed and <br>experimental values if coalescence effects are not taken into account. By incorporat-<br>ing coalescence into the model, good agreement was obtained between predicted <br>and experimentally determined drop diameters. It was further found that the mor-<br>phology evolution of the PE/PS blend is not very sensitive to output rate or screw <br>speed. The morphology development is, however, strongly dependent on the screw <br>configuration.<br>
+<b>7.7.3.2.7Summary of Liquid-Liquid Dispersive Mixing</b><br>
+In the early stages of the mixing process the length scale of the minor component is <br>such that the Capillary (Weber) number is much larger than the critical Capillary <br>number. In this situation, the mixing is distributive with passive interfaces; inter-<br>facial tension is negligible. The deformation of the drops is affine and occurs as in <br>miscible liquids. Only the total strain is important in describing the mixing process <br>at this point.<br>The affine deformation of the drops causes the drops to extend into long thin threads, <br>which is referred to as fibrillation. This process continues until the local radii <br>become so small that the Weber (Capillary) number starts to approach the critical <br>Weber number. At this point the threads become unstable and disintegrate as a <br>result of interfacial tension-driven processes; the interfaces are now active. The <br>most important mechanisms are the growth of Rayleigh disturbances in the mid-<br>part of the thread, end-pinching, retraction, and necking in the case of relatively <br>short dumbbell-shaped threads.<br>
+<hr>
+<A name=495></a> 7.7Mixing  <b>483</b><br>
+In dispersive mixing the morphology is determined both by the strain rate and the <br>time duration of the strain rate; these are not interchangeable as they are in distri-<br>butive mixing. Therefore, the time scales of the competing processes in the mixing <br>process are quite important. For instance, if the interaction time between two drops <br>is too short, coalescence will not take place. These time scales are determined by the <br>viscosities, the elastic properties, and the interfacial properties.<br>The morphology at the end of mixing is the result of the balance between breakup <br>and coalescence processes. This morphology can change in the operations following <br>the mixing process. The flow through a pelletizing die can further modify the mor-<br>phology, as can cooling in a water bath. Further processing, such as molding or <br>extrusion, can, and most likely will, cause additional changes in the morphology.<br>The theories developed thus far do a reasonable job predicting the behavior of New-<br>tonian fluids. Further work needs to be done on viscoelastic fluids. Work by Milliken <br>and Leal [300], De Bruijn [287], and Janssen [294] showed that the change in the <br>critical Weber number is not too significant when viscoelasticity is introduced to the <br>dispersed phase. The effect of viscoelasticity will be dependent on the time scale of <br>the processes involved; in rapid deformation occurring in very short times, visco-<br>elasticity will have a substantial effect.<br>
+<b>7.7.4Backmixing</b><br>
+Mixing is a critical function in most extrusion operations. One of the most difficult <br>mixing tasks is backmixing. An extrusion operation where good backmixing is very <br>important is when a low percentage color concentrate, CC, is added to a virgin poly-<br>mer. In this case, the initial distance between the CC pellets may be 100 mm or <br>greater. If the final striation thickness needs to be reduced to the micron level, the <br>reduction of the striation thickness needs to be at least five orders of magnitude--this <br>is quite a tough task!<br>This section will analyze how the velocity profiles, axial mixing, and residence time <br>distribution are related. It will be shown why simple conveying screws have poor <br>axial mixing capability. New mixer geometries that are specifically designed to <br>improve backmixing will be discussed.<br>
+<b>7.7.4.1Cross-Sectional Mixing and Axial Mixing<br></b>Most analyses of mixing focus on cross-sectional mixing, e.g., [301]. The cross-sec-<br>tional mixing is determined mostly by the Couette shear rate between the rotating <br>screw and stationary barrel. Typical values of the Couette shear rate in single screw <br>extruders range from 50 to 100 s­1. With a typical residence time in the melt convey-<br>ing zone of about 20 s, the resulting total shear strain ranges from about 1000 to <br>2000 units. This means that the striation thickness in cross-sectional mixing is <br>
+<hr>
+<A name=496></a><b>484</b>  <b>7Functional Process Analysis</b><br>
+reduced by about three orders of magnitude. In many cases, this is not enough to <br>achieve a level of mixing that appears uniform by visual inspections.<br>Axial mixing or backmixing occurs by pressure flow. For pressure flow of a power <br>law fluid between parallel plates, the dimensionless velocity  = v/vmax can be written <br>
+as a function of the dimensionless normal coordinate  = 2y/H as follows:<br>
+ (7.534)<br>
+Figure 7.157 shows the velocity distribution for several values of the power law <br>index.<br>
+Velocity Distribution Power Law Fluid<br>
+1.2<br>
+1.0<br>
+0.8<br>
+0.6<br>
+n=1.0<br>n=0.5<br>n=0.25<br>
+0.4<br>
+Dimensionless Velocity<br>
+0.2<br>
+0.0<br>
+-1.5<br>
+-1.0<br>
+-0.5<br>
+0.0<br>
+0.5<br>
+1.0<br>
+1.5<br>
+Dimensionless Normal Coordinate<br>
+<b>Figure 7.157</b>Velocity profiles for various power law index values<br>
+The velocity profile for a Newtonian fluid (n = 1.0) is a parabola. The shear rate in <br>the center of the channel is zero. As a result, no mixing will take place there. The <br>center region flattens as the power law index reduces. In other words, the velocity <br>profile becomes closer to a plug flow profile as the power law index approaches zero. <br>This means that the low shear rate region expands as the fluid becomes more shear <br>thinning. Thus, the region with poor mixing becomes larger when the power law <br>index reduces.<br>From this simple analysis it becomes clear that the situation for backmixing is sub-<br>stantially more difficult than for cross-sectional mixing. For shear thinning fluids <br>there is a considerable region in the center of the channel where little or no axial <br>mixing takes place.<br>
+<hr>
+<A name=497></a> 7.7Mixing  <b>485</b><br>
+<b>7.7.4.2Residence Time Distribution<br></b>The RTD can be determined from the velocity profiles in the channel. The axial flow <br>in a screw extruder is a pressure flow because there are no axial velocity compo-<br>nents of the screw or barrel. As a first approximation, the axial pressure flow can be <br>considered a flow between parallel plates. The velocity profile for pressure flow of a <br>power law fluid between parallel plates is (see Fig. 7.109):<br>
+ (7.535)<br>
+where W is the width of the channel, H the height of the channel, P the pressure <br>drop over axial length L, n the power law index, and m the consistency index of the <br>fluid.<br>The velocity profile can be written as:<br>
+ (7.536)<br>
+where normal coordinate y ranges from ­H/2 to +H/2.<br>The velocity profile can be written as:<br>
+ (7.537)<br>
+The external RTD function f(t)dt can be determined from:<br>
+ (7.538)<br>
+Coordinate y can be expressed as a function of time by the following substitution:<br>
+ (7.539)<br>
+With this substitution, the external RTD function can be written as:<br>
+ (7.540)<br>
+<hr>
+<A name=498></a><b>486</b>  <b>7Functional Process Analysis</b><br>
+The minimum residence time t0 can be determined from:<br>
+ (7.541)<br>
+The cumulative RTD function F(t) can be found by integrating the external RTD <br>function; it can be expressed as:<br>
+ (7.542)<br>
+For a Newtonian fluid the RTD function becomes:<br>
+ (7.543)<br>
+If we express the RTD as a function of the dimensionless residence time , where  <br>is the actual residence time divided by the mean residence time, we get:<br>
+ (7.544)<br>
+The expression above for a power law fluid has not been published before. With <br>this expression the RTD can plotted at several values of the power law index n; see <br>Fig. 7.158.<br>
+1.0<br>
+0.9<br>
+0.8<br>
+0.7<br>
+0.6<br>
+0.5<br>
+0.4<br>
+0.3<br>
+n=0.25<br>
+Cumulative RTD function<br>
+0.2<br>
+n=1.0<br>n=0.5<br>
+0.1<br>
+0.0<br>
+0.0<br>
+0.5<br>
+1.0<br>
+1.5<br>
+2.0<br>
+2.5<br>
+3.0<br>
+Dimensionless time<br>
+<b>Figure 7.158</b>Residence time distribution curves for several power law index values<br>
+It is clear from Fig. 7.158 that the RTD becomes narrower as the value of the power <br>law index reduces. This means that backmixing reduces as the fluid becomes more <br>shear thinning (lower power law index). Figure 7.158 confirms what we have already <br>
+<hr>
+<A name=499></a> 7.7Mixing  <b>487</b><br>
+seen in the velocity profiles of Fig. 7.157. As the fluid becomes more shear thinning, <br>the velocity profile becomes closer to plug flow and, consequently, the RTD becomes <br>narrower and backmixing more problematic.<br>
+<b>7.7.4.3RTD in Screw Extruders<br></b>Pinto and Tadmor [295] developed expressions for the RTD in single screw extru-<br>ders. The cumulative RTD function can be written as:<br>
+ (7.545)<br>
+The dimensionless time  (time divided by mean residence time) can be expressed <br>as a function of the dimensionless normal coordinate :<br>
+ (7.546)<br>
+The two expressions above were derived for a Newtonian fluid using the flat plate <br>approximation considering both down- and cross-channel velocity components. <br> <br>Figure 7.159 shows the RTD for a single screw extruder as well as for pressure flow <br>of a Newtonian fluid between flat plates.<br>
+1.0<br>
+0.9<br>
+0.8<br>
+0.7<br>
+0.6<br>
+0.5<br>
+Single Screw Extruder<br>
+Flat Plate Newtonian<br>
+0.4<br>
+0.3<br>
+Cumulative RTD function<br>
+0.2<br>
+0.1<br>
+0.0<br>
+0.0<br>
+0.5<br>
+1.0<br>
+1.5<br>
+2.0<br>
+2.5<br>
+3.0<br>
+Dimensionless time<br>
+<b>Figure 7.159</b>RTD of single screw extruder and flat plate<br>
+The single screw RTD is narrower than the flat plate RTD because of the re-circula-<br>tion of the fluid in the screw channel. Fluid spends more time in the lower portion of <br>the channel  = 0 to 2/3 than in the upper portion of the channel  = 2/3 to 1.<br>In order to properly determine the RTD of an extruder we have to consider not only <br>down- and cross-channel velocity components but also normal velocity components <br>that occur at the flight flanks. This will require a numerical analysis: either FDA, <br>FEA, or BEA. Even though the depth of the channel is usually quite small compared <br>to the channel width, the residence time at the flight flanks is substantial because <br>
+<hr>
+<A name=500></a><IMG src="Polymer Extrusion by Chris Rauwendaal-500_1.jpg"><br>
+<b>488</b>  <b>7Functional Process Analysis</b><br>
+the normal velocities are quite small. Joo and Kwon [296] pointed out limitations of <br>the Pinto analysis. As one would expect, the Pinto model under-predicts the residence <br>times relative to a full three-dimensional analysis, particularly with large values of <br>the axial pressure gradient.<br>The Pinto RTD for a single screw extruder is narrower than the RTD for pressure <br>flow between a flat plate for a Newtonian fluid. The effect of shear thinning is to fur-<br>ther narrow the RTD as discussed earlier. These two effects explain why backmixing <br>is such a critical issue in screw extruders.<br>
+<b>7.7.4.4Methods to Improve Backmixing<br></b>A major concern in backmixing is the fluid in the center region of the screw channel <br>where the axial shear strain is zero or close to zero. In a simple conveying screw the <br>fluid in the inner recirculation region will stay within this region until it reaches the <br>end of the screw. When this happens, the material flowing into the die will be poorly <br>mixed.<br>Mixing pins and slots in the screw flights will improve axial mixing because they <br>achieve a short-term splitting and reorientation of the fluid. The effect of mixing <br>pins on backmixing is illustrated in Fig. 7.160.<br>
+<b> Figure  7.160</b> <br>
+Particle tracking results in a mixing section with <br>
+elongational mixing pins<br>
+These results were obtained using a three-dimensional BEM flow analysis. It is clear <br>that one row of mixing pins has limited effect on axial mixing. Backmixing can be <br>improved by varying the spacing between the pins, thus intentionally creating <br>streams of different axial velocities.<br>The challenge in improving axial mixing is to efficiently transfer fluid from the <br>inner re-circulation region to the outer region and vice versa. A simple but effective <br>method of doing this is the inside-out mixer shown in Fig. 7.161.<br>The flight in this mixer is offset so that the material in the center region is cut by the <br>offset flight and then pushed to the screw and barrel surfaces by the normal pres-<br>sure gradients that occur at the flight flank. Results of particle tracking using BEM <br>are shown in Fig. 7.162.<br>
+<hr>
+<A name=501></a><IMG src="Polymer Extrusion by Chris Rauwendaal-501_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-501_2.jpg"><br>
+ 7.7Mixing  <b>489</b><br>
+<b> Figure  7.161</b> <br>
+Solid model of inside-out mixer<br>
+<b> Figure  7.162</b> <br>
+Particle tracking in the inside-out mixer<br>
+The redistribution of the material is shown in Fig. 7.163.<br>
+<b>Figure 7.163</b>Redistribution in the inside-out mixer<br>
+This shows how the fluid from the center region is cut by the offset flight and pushed <br>to the screw surface at the pushing side of the flight and to the barrel surface at the <br>trailing side of the flight. It is clear that a considerable axial distance is necessary to <br>bring about the redistribution of the material.<br>
+<hr>
+<A name=502></a><b>490</b>  <b>7Functional Process Analysis</b><br>
+<b>7.7.4.5Conclusions for Backmixing<br></b>Backmixing is one of the most difficult mixing tasks in screw extruders. This is due <br>to the fact that the axial strain rates in the extrusion process are very low, particu-<br>larly in the center region of the channel. The axial velocity profile is close to plug <br>flow, particularly for strongly shear thinning fluids. Backmixing problems are espe-<br>cially severe when adding a small percentage concentrate to the extruder. When <br>both the polymer and concentrate are in pellet form, the initial striation thickness <br>can be of the order of 100 mm. If a final striation thickness of 1 micron is required, <br>the axial mixing has to achieve a reduction of striation thickness of at least five <br>orders of magnitude.<br>Considering that the axial shear rate in melt conveying is close to zero in the center <br>region of the channel, it is clear that axial mixing will be insufficient unless efficient <br>mixing devices are used. The most efficient way to improve axial mixing is to redis-<br>tribute material from the center of the channel to the outer region of the channel <br>and vice versa. One mixer that aims to achieve such redistribution is the inside-out <br>mixer. Another mixer that was designed specifically to improve axial mixing is the <br>CRD7 mixer; see Fig. 7.164.<br>
+<b> Figure  7.164</b> <br>
+The CRD7 mixer<br>
+Another method of reducing mixing problems with color concentrates is to reduce <br>the initial striation thickness. This can be done by reducing both the natural and <br>color pellet size. Granules are better than pellets and powder is better than granules <br>from a mixing point of view. Smaller particle sizes may lead to other problems <br>though, such as conveying problems and air entrapment. Adding the colorant in <br>liquid form can also reduce the initial striation thickness. This is a main reason <br>liquid colorants are used. However, they can create problems by forming a lubricat-<br>ing layer on the barrel surface and reducing the conveying efficiency of the extruder.<br>
+<hr>
+<A name=503></a> <br>
+Appendix 7.1  <b>491</b><br>
+<b></b><br>
+ <b>Appendix 7.1</b><br>
+<b>Constants of Equation 7.30</b><br>
+ (1)<br>
+ (2)<br>
+ (3)<br>
+ (4)<br>
+ (5)<br>
+ (6)<br>
+ (7)<br>
+ (8)<br>
+ (9)<br>
+ (10)<br>
+ (11)<br>
+ (12)<br>
+ (13)<br>
+ (14)<br>
+ (15)<br>
+ (16)<br>
+<hr>
+<A name=504></a><b>492</b>  <b>7Functional Process Analysis</b><br>
+ (17)<br>
+ (18)<br>
+where e is the effective angle of internal friction and  is the angle between the <br>
+major principal stress and the radial r-axis. Further:<br>
+ (19)<br>
+ (cm = 1 for uniform velocity profile) <br>
+(20)<br>
+ (21)<br>
+where v is the radial velocity component.<br>
+<b></b><br>
+ <b>Appendix 7.2</b><br>
+<b>Constants of the Engstad Equation 7.36<br></b>co is the constant part of the failure function; the unconfined yield strength is written <br>
+as:<br>
+ (1)<br>
+The term Yb is determined from:<br>
+ (2)<br>
+For a passive stress state:<br>
+ (3)<br>
+ (4)<br>
+ (5)<br>
+<hr>
+<A name=505></a> <br>
+Appendix 7.3  <b>493</b><br>
+For an active state:<br>
+ (6)<br>
+ (7)<br>
+ (8)<br>
+<b></b><br>
+ <b>Appendix 7.3</b><br>
+This is a Fortran program to analyze two-dimensional flow of a power law fluid.<br>
+ <br>
+PROGRAM EXP<br>
+ <br>
+REAL H<br>
+ <br>
+DIMENSION C(200), Z(201), TX(201), TY(201, T(201), S(201), SX(201)<br>
+ <br>
+DIMENSION SY(201), VX(201), VY(201), DX(201), DY(201), A(4, 4), B(3)<br>
+ <br>
+DIMENSION GRA (3), DOX(3), SOY(3), SOQ(3)<br>
+ <br>
+READ (6, 20) M, NPRVEL<br>
+20  FORMAT (2I10)<br> <br>
+READ (6, 30) REV, RK, D<br>
+ <br>
+READ (6,30) HMIN, DELH, HMAX<br>
+ <br>
+READ (6,30) GMIN, DELG, GMAX<br>
+ <br>
+READ (6,30) RNMIN, DELRN, RNMAX<br>
+ <br>
+READ (6, 30) PHIMIN, DELPHI, PHIMAX<br>
+ <br>
+READ (6,30) PGMIN, DELPG, PGMAX<br>
+ <br>
+WRITE (7, 29) M, NPRVEL<br>
+29  FORMAT (SI10)<br>30  FORMAT (3F10.4)<br> <br>
+WRITE (7, 31) REV<br>
+31  FORMAT (` SCREW SPEED IN RPM IS', F10.2)<br> <br>
+WRITE (7,32) HMIN<br>
+32  FORMAT (` CHANNEL DEPTH IN CM IS', F10.4)<br> <br>
+WRITE (7,320) DELH<br>
+320  FORMAT (` INCREMENTAL DEPTH IN CM IS', F10.4)<br> <br>
+WRITE (7,321) HMAX<br>
+321  FORMAT (` MAXIMUM DEPTH IN CM IS', F10.4)<br> <br>
+WRITE (7, 33) D<br>
+33  FORMAT (` SCREW DIAMETER IN CM IS', F10.4)<br> <br>
+WRITE (7, 34) RK<br>
+34  FORMAT (` POWER LAW CONST. IN POISE IS', F14.4)<br> <br>
+WRITE (7, 36) GMIN<br>
+36  FORMAT (` INITIAL RED. PRESS. GRADIENT IS', F10.6)<br> <br>
+WRITE (7, 37) DELG<br>
+37  FORMAT (` INCREMENTAL PRESS. GRADIENT IS', F10.6)<br> <br>
+WRITE (7.38) GMAX<br>
+38  FORMAT (` FINAL PRESSURE GRADIENT IS ', F10.6)<br> <br>
+WRITE (7, 39) RNMIN<br>
+<hr>
+<A name=506></a><b>494</b>  <b>7Functional Process Analysis</b><br>
+39  FORMAT (` INITIAL POWER LAW INDEX IS ', F10.6)<br> <br>
+WRITE (7.41) DELRN<br>
+40  FORMAT (` INCREMENTAL POWER LAW INDEX IS ', F10.6)<br> <br>
+WRITE (7, 41) RNMAX<br>
+41  FORMAT (` FINAL POWER LAW INDEX IS ', F10.6)<br> <br>
+WRITE (7, 42) PHIMIN<br>
+42  FORMAT (` INITIAL HELIX ANGLE IN DEGR. IS ', F10.4)<br> <br>
+WRITE (7, 43) DELPHI<br>
+43  FORMAT (` INCREMENTAL HELIX ANGLE IS ', F10.4)<br> <br>
+WRITE (7, 44) PHIMAX<br>
+44  FORMAT (` FINAL HELIX ANGLE IN DEGREES IS ', F10.4)<br> <br>
+WRITE (7, 440) PGMIN<br>
+440  FORMAT (` INITIAL AXIAL PRESS. GRADIENT IN PSI/INCH ', F10.4)<br> <br>
+WRITE (7, 441) DELPG<br>
+441  FORMAT (` INCREMENTAL AXIAL PRESS. GRADIENT IS ', F10.4)<br> <br>
+WRITE (7,444)<br>
+444  FORMAT (` ', ` REDTHRUPUT PRESS-GRAD HELIXANGLE AC-THRUPUT<br> <br>
+DIM', X `THRUPUT DOWNCHA-TX CROSSCH-TY CONSTANTCY')<br>
+ H=HMIN<br> G-GMIN<br> RN=RNMIN<br> PG=PGMIN<br> TXO=10000.0<br> TYO=10000.0<br> CY=10000.0<br> REV-REV/60.0<br> PHIMIN=PHIMIN*17.4533/1000.0<br> DELPHI=DELPHI*17.4533/1000.0<br> <br>
+PHIMAX=PHIMAX*17.4533/1000.0 PHI=PHIMIN<br>
+45  UO=3.14159*D*REV*COS (PHI)<br> <br>
+VO=UO*TAN (PHI)<br>
+ <br>
+IF (PG .EQ. 0) GO TO 48<br>
+ <br>
+CX=27145.67*SIN (PHI) *PG<br>
+C  ABOVE LINE USED WITH CONSTANT AXIAL PRESSURE GRADIENT<br> <br>
+GO TO 49<br>
+48 CX=6.0*RK*G/H*(UO/H)**RN<br>49 NR=0<br>50  DO 70 I=1, ((4*M)+1)<br> <br>
+Z(I)=(FLOAT (I-1))*H/(4.*(FLOAT (M)))<br>
+ <br>
+TX (I)=TXO+(CX*Z (I))<br>
+ <br>
+TY (I)=TYO+(CY*Z (I))<br>
+ <br>
+T (I)=((TX (I)**2)+(TY Ii)**2))**.5<br>
+ <br>
+S (I)=(T (I)/RK)**(1./RN)<br>
+ <br>
+SX (I)=S (I)*TX (I)/T (I)<br>
+ <br>
+SY (I)=S (I)*TY (I)/T (I)<br>
+70 CONTINUE<br> <br>
+VX (1)=0.0<br>
+ <br>
+VY (1)=0.0<br>
+ <br>
+DO 80 I=2, M+1<br>
+ <br>
+K=((FLOAT (I-2))*4)<br>
+ <br>
+DO 90 J=1, 5<br>
+ L=K+J<br> <br>
+DX (J)=SX (L)<br>
+ <br>
+DY (J)=SY (L)<br>
+90 CONTINUE<br> <br>
+CALL SIMP (4, DX, UMX)<br>
+ <br>
+CALL SIMP (4, DY, UMY)<br>
+ <br>
+VX (I)=VX (I-1)+(UMX*H/(4.*(FLOAT (M))))<br>
+ <br>
+VY (I)=VY (I-1)+(UMY*H/(4.*(FLOAT (M))))<br>
+<hr>
+<A name=507></a> <br>
+Appendix 7.3  <b>495</b><br>
+80 CONTINUE<br> <br>
+CALL SIMP (M VX, XOX)<br>
+ <br>
+CALL SIMP (M, VY, YOY)<br>
+ <br>
+QX=XOX*H/(FLOAT (M))<br>
+ <br>
+QY=YOY*H/(FLOAT (M))<br>
+250  DIVX=VX (M+1)-UO<br> <br>
+DIVY=VY (M+1)-VO<br>
+ DIVQ=QY<br> <br>
+IF (NR-1) 180, 185, 185<br>
+180 STVX=DIVX<br> STVY=DIVY<br> STVQ=DIVQ<br>C  WRITE (7, 251) TXO, TYO, CY, STVX, STVY, STVQ<br>251  FORMAT (6E12.3)<br>183  IF ((AMAX1 (STVX, STVY, STVQ))-1E-5) 500, 500, 187<br>187 LAL=1<br> NR=NR+1<br> <br>
+GRA (1)=0.01*TXO<br>
+ <br>
+TXO=TXO+GRA (1)<br>
+ <br>
+GO TO 50<br>
+184  SOX (LAL)=DIVX-STVX<br> <br>
+SOY (LAL)=DIVY-STVY<br>
+ <br>
+SOQ (LAL)=DIVQ-STVQ<br>
+ <br>
+IF (LAL-2) 200, 210, 220<br>
+200 LAL=2<br>201  TXO=TXO-GRA (1)<br>202  GRA (2)=0.01*TYO<br> <br>
+TYO=TYO+GRA (2)<br>
+ <br>
+GO TO 50<br>
+210 LAL=3<br> <br>
+TYO=TYO-GRA (2)<br>
+ <br>
+GRA (3)=0.01*CY<br>
+ <br>
+CY=CY+GRA (3)<br>
+ <br>
+GO TO 50<br>
+220 LAL=1<br> <br>
+CY=CY-GRA (3)<br>
+ <br>
+DO 270 MP=1, 3<br>
+ <br>
+A (1, MP)=SOX (MP)/GRA (MP)<br>
+ <br>
+A (2, MP)=SOY (MP)/GRA (MP)<br>
+ <br>
+A (3, MP)=SOQ (MP)/GRA (MP)<br>
+270 CONTINUE<br> <br>
+B (1)=STVX<br>
+ <br>
+B (2)=STVY<br>
+ <br>
+B (3)=STVQ<br>
+ <br>
+OK=A (2, 1)/A (1,1)<br>
+ <br>
+A (2, 2)=A (2,2)-(A(1, 2)*OK)<br>
+ <br>
+A (2, 3)=A (2, 3)-(A(1, 3)*OK)<br>
+ <br>
+B (2)=B (2)-(B(1)*OK)<br>
+ <br>
+OK=A (3, 1)/A (1, 1)<br>
+ <br>
+A (3, 2)=A (3, 2)-(A (1, 2)*OK)<br>
+ <br>
+A (3, 3)=A (3, 3)-(A (1, 3)*OK)<br>
+ <br>
+B (3)=B (3)-(B(1)*OK)<br>
+ <br>
+OK=A (3, 2)/A (2, 2)<br>
+ <br>
+A (3, 3)=A (3, 3)-(A (2, 3)*OK)<br>
+ <br>
+B (3)=B (3)-(B(2)*OK)<br>
+ <br>
+DLCY=B (3)/A (E, E)<br>
+ <br>
+DLTY=(B (2)-(A (2, 3)*DLCY))/A (2, 2)<br>
+ <br>
+DLTX=B (1)-(A (1, 3)*DLCY)<br>
+ <br>
+DLTX=(DLTX-(A (1, 2)*DLTY))/A (1, 1)<br>
+<hr>
+<A name=508></a><b>496</b>  <b>7Functional Process Analysis</b><br>
+ TXO=TXO+DLTX<br> TYO=TYO+DLTY<br> CY=CY+DLCY<br> NR=0<br> <br>
+GOTO 50<br>
+500 QRED=QX*2.0/(H*UO)<br> <br>
+QREDA=QRED*SIN (2.0*PHI)<br>
+ <br>
+QA=QX*3.14159*D*SIN (PHI)<br>
+ GA=CX*H/(6.0*RK)*(H/UO)**RN<br> PHIDGR=PHI*1000.0/17.4533<br> <br>
+WRITE (7, 501) QREDA, GA, PHIDGR, QA, QRED, TXO, TYO, CY<br>
+501  FORMAT (8E11.4)<br> <br>
+IF (QX .LT. 0) GO TO 560<br>
+ CONTINUE<br> <br>
+IF (NPRVEL .NE. 1) GO TO 556<br>
+ <br>
+WRITE (7, 10000)<br>
+1000 FORMAT  (`1')<br>554  WRITE (7, 555) (Z((4*I)+1), VX (I), VY (I), I=1, M)<br>555  FORMAT (3E15.4)<br>556  IF (G .GE. GMAX) GO TO 560<br> G=G+DELG<br> <br>
+GO TO 45<br>
+557  IF (PG .GE. PGMAX) GO TO 600<br>558 PG=PG+DELPG<br> PHI=PHIMIN<br> H=HMIN<br> <br>
+WRITE (7, 559) PG<br>
+559  FORMAT (` NEW AXIAL PRESSURE GRADIENT IS ', F10.4)<br> <br>
+GO TO 45<br>
+560  IF (RN .GE. RNMAX) GO TO 570<br> RN=RN+DELRN<br> G=GMIN<br> <br>
+WRITE (7, 565) RN<br>
+565  FORMAT (` NEW POWER LAW EXPONENT IS ', F6.3)<br> TXO=10000.0<br> TYO=10000.0<br> CY=10000.0<br> <br>
+GO TO 45<br>
+570  IF (PHI .GE. PHIMAX) GO TO 580<br> PHI=PHI+DELPHI<br> G=GMIN<br> TXO=10000.0<br> TYO=10000.0<br> CY=10000.0<br> <br>
+GO TO 45<br>
+580  IF (H .GE. HMAX) GO TO 558<br> H=H+DELH<br> PHI=PHIMIN<br> <br>
+WRITE (7, 585) H<br>
+585  FORMAT (` NEW CHANNEL DEPTH IS ', F10.4)<br> TXO=10000.0<br> TYO=10000.0<br> <br>
+GO TO 45<br>
+600 STOP<br> END<br> <br>
+SUBROUTINE SIMP (I, C, S)<br>
+ <br>
+DIMENSION C (200)<br>
+ S=0.<br> <br>
+DO 55 L=1, (I-1), 2<br>
+ S=S+(2.*C(L))+4.*C(L+1))<br>
+<hr>
+<A name=509></a> References <br>
+<b>497</b><br>
+55 CONTINUE<br> S=((S+(C(I+1)))-C(1))/3.<br> RETURN<br> END<br> 40<br>SCREW SPEED IN RPM IS 100.00<br>CHANNEL DEPTH IN CM IS 0.2500<br>INCREMENTAL DEPTH IN CM IS 0.0200<br>MAXIMUM DEPTH IN CM IS 0.2500<br>SCREW DIAMETER IN CM IS 6.3500<br>POWER LAW CONST. IN POISE IS 10000.0000<br>INITIAL RED. PRESS. GRADIENT IS 0.300000<br>INCREMENTAL PRESS. GRADIENT IS 0.050000<br>FINAL PRESSURE GRADIENT IS 0.300000<br>INITIAL POWER LAW INDEX IS 1.000000<br>INCREMENTAL POWER LAW INDEX IS 0.200000<br>FINAL POWER LAW INDEX IS 1.000000<br>INITIAL HELIX ANGLE IN DEGR. IS 10.0000<br>INCREMENTAL HELIX ANGLE IS 2.5000<br>FINAL HELIX ANGLE IN DEGREES IS 50.0000<br>INITIAL AXIAL PRESS. GRADIENT IN PSI/INCH 3000.0000<br>INCREMENTAL AXIAL PRESS. GRADIENT IS 2000.0000<br>MAXIMUM AXIAL PRESS. GRADIENT INPSI/INCH 15000.0000<br>
+References<br>1.  S.C. Cowin (Ed.) "Mechanics Applied to the Transport of Bulk Materials," ASME, New <br>
+York (1979)<br>
+2.  S.C. Cowin and M. Satake (Eds.) "Continuum Mechanical and Statistical Approaches in <br>
+the Mechanics of Granular Material," Proceedings U.S.-Japan Seminar, Sendai, Japan, <br>Gakumutsu Bunken Fukyu-Kai (1978)<br>
+3.  C. Orr, Jr., "Particulate Technology," Macmillan, New York (1966)<br>4.  R.L. Brown and J.C. Richards, "Principles of Powder Mechanics," Pergamon Press, <br>
+Oxford (1966)<br>
+5.  J.E. Richards (Ed.) "The Storage and Recovery of Particulate Solids," Inst. Chem. Eng., <br>
+Working Party Rep. (1966)<br>
+6.  K. Wieghardt, Ann. Rev. Fluid Mech., 7, 89­114 (1975)<br>7.  S.B. Savage, J. Fluid Mech., 92, 53­96 (1979)<br>8.  A.W. Jenike, "Gravity Flow of Bulk Solids," Bulletin No. 108 of the Utah Engineering <br>
+Experimental Station, Univ. of Utah, Salt Lake City (1961)<br>
+9.  A.W. Jenike, "Storage and Flow of Solids," Bulletin No. 123 of the Utah Engineering <br>
+Experimental Station, Univ. of Utah, Salt Lake City (1964)<br>
+10.  M. Shahinpoor and J.S.S. Siah, J. Non-Newtonian Fluid Mech., 9, 147­156 (1981)<br>11.  M. Shahinpoor, J. Non-Newtonian Fluid Mech., 12, 31­38 (1983)<br>12.  K.I. Kanatani, Int. J. Eng. Sci., 17, 419­432 (1979)<br>13.  J.T. Jenkins and S.C. Cowin, in [1], p. 79­89<br>14.  R.L. Brown and P.G. Hawksley, Fuel, 26, 171 (1947)<br>
+<hr>
+<A name=510></a><b>498</b>  <b>7Functional Process Analysis</b><br>
+15.  J. Lee, S.C. Cowin, and J.S. Templeton, Trans. Soc. Rheol., 18, 247­269 (1974)<br>16.  P.M. Blair-Fish and P.L. Bransby, Trans. A.S.M.E., J. Eng. Ind., 95, 17­26 (1973)<br>17.  G.C. Gardner, Chem. Eng. Sci., 21, 261­273 (1966)<br>18.  M. Levinson, B. Shmutter, and W.L. Resnick, Powder Technol., 16, 29­43 (1977)<br>19.  L. Bates and D. Kershan, Chem. Eng. Progress, 71, 66­68 (1975)<br>20.  L.M. Connelly, in [1], pp. 35­59<br>21.  R. Butterfield, R.M. Harkness, and K.Z. Andrews, Geotechnique, 8, 308 (1970)<br>22.  H.A. Janssen, Zeitschrift VDI, 1045 (1895)<br>23.  D.M. Walker, Chem. Eng. Sci., 21, 975­997 (1966)<br>24.  J.K. Walters and P.M. Nedderman, Chem. Eng. Sci., 28, 1907­1908 (1973)<br>25.  P.L. Bransby and P.M. Blair-Fish, Chem. Eng. Sci., 29, 1061­1074 (1974)<br>26.  G. Hagen, Berliner Monatsberichte Akad. d. Wiss, 35­42 (1852)<br>27.  R.L. Brown, Nature, 191, 458­461 (1961)<br>28.  J.R. Johanson, Trans. S.M.E., 232, 69­80 (1965)<br>29.  S.B. Savage, Br. J. Appl. Phys., 16, 1885­1888 (1965)<br>30.  S.B. Savage, Ph.D. thesis, McGill Univ., Montreal (1967)<br>31.  S.B. Savage and M. Sayed, in [1], pp. 1­24<br>32.  J.M. Rausch, Ph.D. thesis, Princeton Univ. (1948)<br>33.  J. Bosley, C. Schofield, and C.A. Shook, Trans. Inst. Chem. Eng., 47, T147­T153 (1968)<br>34.  W.N. Sullivan, Ph.D. thesis, California Institute of Technology, Pasadena (1972)<br>35.  D.M. Walker, Powder Technol., 1, 228 (1967)<br>36.  R.K. Eckhoff and P.G. Leversen, Powder Technol., 10, 51 (1974)<br>37.  G. Engstad, Chem. Eng. Sci., 30, 1273­1283 (1975)<br>38.  Y. Lee, Combustion, 32, 20­27 (1960)<br>39.  O. Richmond, Mech. Eng., 85, 46­49 (1963)<br>40.  G.C. Gardner, Chem. Eng. Sci., 18, 35­39 (1963)<br>41.  O. Richmond and H.L. Morrison, in [1], pp. 103­111<br>42.  J.F. Ingen Housz in "Fortschritte beim Extrudieren," Carl Hanser Verlag, Munich (1976)<br>43.  W.H. Darnell and E.A.J. Mol, SPE J., 12, 20 (1956)<br>44.  J.G.A. Lovegrove, Ph.D. thesis, London Univ. (1972)<br>45.  E. Broyer and Z. Tadmor, Polym. Eng. Sci., 12, 12­24 (1972)<br>46.  Z. Tadmor and E. Broyer, Polym. Eng. Sci., 12, 378­386 (1972)<br>47.  H.B. Kessler, R.M. Bonner, P.H. Squires, and C.F.W. Wolf, SPE J., 16, 267 (1960)<br>48.  H. Decker, "Die Spritzmaschine," P. Troester, Hanover, W. Germany (1941)<br>49.  A. Schneiders, Plastverarbeiter, 19, 797­799 (1968)<br>50.  G. Menges, W. Predoehl, R. Hegele, R. Kosel, and W. Elbe, Plastverarbeiter, 20, 79­88 <br>
+and 188­190 (1969)<br>
+<hr>
+<A name=511></a> References <br>
+<b>499</b><br>
+51.  G. Fuchs, Plastverarbeiter, 19, 765­771 (1968) and 20, 237­244 (1969)<br>52.  W.L. Krueger, SPE ANTEC, Boston, 676­678 (1981)<br>53.  R. Rautenbach and E. Goldacker, Kunststoffe, 61, 104­107 (1971)<br>54.  G. Menges and R. Hegele, Plastverarbeiter, 23, 332­338 (1975)<br>55.  B. Franzkoch and G. Menges, SPE ANTEC, Washington, DC, 512­515 (1978)<br>56.  E. Langecker, G.R. Langecker, and W. Fillman, Plastverarbeiter, 28, 531­535 (1977)<br>57.  E. Langecker, German patent application DDS 22 05426, filed February 2, 1972<br>58.  C. Maillefer, U.S. Patent 4,154,535 (also CH 612375, FR 2,385,517, DE 2,813,585, and <br>
+SE 7,80,3477)<br>
+59.  E. Gruenschlosz, SPE ANTEC, New Orleans, 160­165 (1979)<br>60.  B.H. Maddock, SPE ANTEC, New York, 383 (1959)<br>61.  Z. Tadmor, I. Duvdevani, and I. Klein, Polym. Eng. Sci., 7, 198 (1967)<br>62.  Z. Tadmor, Polym. Eng. Sci., 6, 185 (1966)<br>63.  J. Shapiro, Ph.D. thesis, Cambridge Univ., England (1971)<br>64.  J.R. Vermeulen, P.M. Gerson, and W.J. Beek, Chem. Eng. Sci., 26, 1445­1455 (1971)<br>65.  J.R. Vermeulen, P.G. Scargo, and W.J. Beek, Chem. Eng. Sci., 26, 1457­1465 (1971)<br>66.  G. Menges and P. Klenk, Kunststoffe, 57, 590 (1967)<br>67.  P. Klenk, Rheol. Acta., 7, 75­78 (1968)<br>68.  P. Klenk, Plastverarbeiter, 21, 537 (1968)<br>69.  J. Dekker, Kunststoffe, 66, 130 (1976)<br>70.  T.E. Fahey, M.Sc. thesis, Case Western Reserve Univ., September (1981)<br>71.  F.R. Kulas and N.P. Thorshaug, J. Appl. Polym. Sci., 23, 1781­1794 (1979)<br>72.  J.T. Lindt, Polym. Eng. Sci., 16, 284­291 (1976)<br>73.  J.T. Lindt, Polym. Eng. Sci., 21, 1162­1166 (1981)<br>74.  K.A. Gieskes, Internal Report, Twente Univ. of Technology, Mech. Eng. Dept., the Nether-<br>
+lands (1979)<br>
+75.  H.E.H. Meijer, Ph.D. thesis, Twente Univ. of Technology, the Netherlands (1980)<br>76.  J.R. Edmondson, Ph.D. thesis, Univ. of London (1972)<br>77.  I.R. Edmondson and R.T. Fenner, Polymer, 16, 49 (1975)<br>78.  R.T. Fenner, Polymer, 18, 617 (1977)<br>79.  A.P.D. Cox and R.T. Fenner, Polym. Eng. Sci., 20, 562­571 (1980)<br>80.  A.P.D. Cox, J.G. Williams, and D.P. Isherwood, Polym. Eng. Sci., 21, 86­92 (1981)<br>81.  G.M. Gale, Plast. Polym., 6, 183 (1970)<br>82.  G. Mennig, Kunststoffe, 71, 359­362 (1981)<br>83.  R.C. Donovan, Polym. Eng. Sci., 11, 247­257 (1971)<br>84.  R.C. Donovan, Polym. Eng. Sci., 11, 485­491 (1971)<br>85.  J. Shapiro, A.L. Halmos, and J.R.A. Pearson, Polymer, 17, 905­918 (1976)<br>
+<hr>
+<A name=512></a><b>500</b>  <b>7Functional Process Analysis</b><br>
+86.  A.L. Halmos, J.R.A. Pearson, and R. Trottnow, Polymer, 19, 1199­1216 (1978)<br>87.  J.R.A. Pearson, Int. J. Heat Mass Transfer, 19, 405­511 (1976)<br>88.  D.R. Hinrichs and L.U. Lilleleht, Polym. Eng. Sci., 18, 422­426 (1978)<br>89.  D.W. Sundstrom and C.C. Young, Polym. Eng. Sci., 12, 59­63 (1972)<br>89a. D.W. Sundstrom and J.R. Lo, Polym. Eng. Sci., 18, 422­426 (1978)<br>90.  C.I. Chung, Mod. Plast., 45, 178 (1968) and 45, 110 (1968)<br>91.  C.I. Chung, SPE J., 26, 32 (1970)<br>92.  C.I. Chung, Polym. Eng. Sci., 11, 93­98 (1971)<br>93.  C.I. Chung, R.J. Nichols, and G.A. Kruder, Polym. Eng. Sci., 14, 29 (1974)<br>94.  E.M. Mount and C.I. Chung, Polym. Eng. Sci., 18, 711­720 (1978)<br>95.  E.M. Mount, J.G. Watson, and C.I. Chung, Polym. Eng. Sci., 22, 729­737 (1982)<br>96.  D.E. McClelland and C.I. Chung, Polym. Eng. Sci., 23, 100­104 (1983)<br>97.  K.H. Chung and C.I. Chung, Polym. Eng. Sci., 23, 191­196 (1983)<br>98.  N. N., Engineering, 114, 606 (1922) This publications is often attributed to H.S. Rowell <br>
+and D. Finlayson; however, this is disputed by Holmes [99]. Rowell and Finlayson did <br>publish on the subject later in 1928 [100].<br>
+99.  D.B. Holmes, Ph.D. thesis, Delft University of Technology, the Netherlands (1967)<br>100.  H.S. Rowell and D. Finlayson, Engineering, 126, 249 (1928)<br>101.  H.A.A. Helmy, Ph.D. thesis, Univ. of Bradford, England (1975)<br>102.  R.T. Fenner, "Extruder Screw Design," Illiffe, London (1970)<br>103. Z. Tadmor and I. Klein, "Engineering Principles of Plasticating Extrusion," van Nos-<br>
+trand Reinhold Comp., New York (1970)<br>
+104.  J. Nebrensky, J.F.T. Pittman, and J.M. Smith, Polym. Eng. Sci., 13, 209 (1973)<br>105.  R.T. Fenner, Polymer, 18, 617 (1977)<br>106.  Z. Tadmor and C.E. Gogos, "Principles of Polymer Processing," John Wiley &amp; Sons, New <br>
+York (1979)<br>
+107.  Y. Mori and T.K. Matsumoto, Rheol. Acta, 1, 240 (1958)<br>108.  R.E. Colwell and K.R. Nicholls, Ind. Eng. Chem., 51, 841 (1959)<br>109.  W.E. Ball and R.E. Colwell, 43rd Nat'l. Mtg. AIChE, Tulsa, OK, Sept. (1960)<br>110.  D.J. Weeks and W.J. Allen, J. Mech. Eng. Sci., 4, 380 (1962)<br>111.  B.S. Clyde and W.A. Holmes-Walker, Int. Plast. Eng. 2, 338 (1962)<br>112.  F.W. Kroesser and S. Middleman, Polym. Eng. Sci., 5, 231 (1965)<br>113. R.W. Flumerfelt, M.W. Pierick, S.L. Cooper, and R.B. Bird, Ind. Eng. Chem., 8, 354 <br>
+(1969)<br>
+114.  C.J. Rauwendaal, SPE ANTEC, Chicago, 186­199 (1983)<br>115.  H.J. Zamodits, Ph.D. thesis, Univ. of Cambridge, England (1964)<br>116.  B. Yates, Ph.D. thesis, Univ. of Cambridge, England (1968)<br>117.  B. Martin, Ph.D. thesis, London Univ., England (1972)<br>
+<hr>
+<A name=513></a> References <br>
+<b>501</b><br>
+118.  K. Palit, Ph.D. thesis, London Univ., England (1972)<br>119.  N.R. Neelakantan, Ph.D. thesis, Univ. of Wales, England (1974)<br>120.  K.P. Choo, Ph.D. thesis, Univ. of Wales, England (1977)<br>121.  H. Kuehnle, Kunststoffe, 5, 267 (1982)<br>122.  D.E. Hanson, Polym. Eng. Sci., 9, 405 (1967)<br>123.  Z.K. Walczak, J. Appl. Polym. Sci., 17, 153 (1973)<br>124.  M. Rokuda, J. Appl. Polym. Sci., 23, 463 (1979)<br>125.  J.M. Dealy and W.K.W. Tsang, J. Appl. Polym. Sci., 26, 1149 (1981)<br>126.  T.Y. Liu, D.S. Soong, and M.C. Williams, Polym. Eng. Sci., 21, 675 (1981)<br>127.  J.W. White and W. Minoshima, Polym. Eng. Sci., 21, 1113 (1981)<br>128.  B. Maxwell and A. Breckwoldt, J. Rheol., 25, 55 (1981)<br>129.  W.P. Perwadtschuk and V.I. Jankow, Plaste und Kautschuk, 28, 36­44 (1981)<br>130.  J. Boussinesg, J. Mathematique Pures et Appliquees, series 2, 13, 377­424 (1968)<br>131.  W. Meskat, Kunststoffe, 45, 87 (1955)<br>132.  M.L. Booy, Polym. Eng. Sci., 21, 93 (1981)<br>133.  Z. Tadmor, Polym. Eng. Sci., 6, 203 (1966)<br>134.  D.F. Dyer, AIChE J., 15, 823 (1969)<br>135.  R.M. Griffith, Ind. Eng. Chem., 1, 180 (1962)<br>136.  K. Hayashida, Rheol. Acta, 2, 261 (1962)<br>137.  H. Kruger, Kunststoffe, 10, 711 (1963)<br>138.  D.R. Rea and W.R. Schowalter, Trans. Soc. Rheol., 11, 125 (1967)<br>139.  H.J. Zamodits and J.R.A. Pearson, Trans. Soc. Rheol., 13, 357 (1969)<br>140.  K. Palit and R.T. Fenner, AIChE J., 18, 628 (1972)<br>141.  H.H. Winter, Rheol. Acta., 1, 1 (1978)<br>142.  M.L. Hami and J.F.T. Pittman, Polym. Eng. Sci., 20, 339 (1980)<br>143.  K.P. Choo, N.R. Neelakantan, and J.F.T. Pittman, Polym. Eng. Sci., 20, 349 (1980)<br>144.  K.P. Choo, M.L. Hami, and J.F.T. Pittman, Polym. Eng. Sci., 21, 100 (1981)<br>145.  J. Nebrensky, J.F.T. Pittman, and J.M. Smith, Polym. Eng. Sci., 13, 209 (1973)<br>146.  S.M. Dinh and R.C. Armstrong, AIChE J., 28, 294­301 (1982)<br>147.  H.H. Winter, Polym. Eng. Sci., 15, 84­89 (1975)<br>148.  F.N. Cogswell, J. Non-Newtonian Fluid Mech., 2, 373­47 (1977)<br>149.  M.T. Dennison, Trans. J. Plast. Inst., 35, 803­808 (1967)<br>150.  J.J. Benbow and E.R. Howells, Trans. J. Plast. Inst., 30, 240­254 (1960)<br>151.  British Patent 32559/72.<br>152.  J.P. Tordella, in "Rheology," Vol. 4, F.R. Eirich (Ed.), Academic Press, New York (1969) <br>
+Chapter 3<br>
+153.  J.L. White, Appl. Polym. Symp., No. 20, 155 (1973)<br>
+<hr>
+<A name=514></a><b>502</b>  <b>7Functional Process Analysis</b><br>
+154.  J.M. Lupton and J.W. Regester, Polym. Eng. Sci., 5, 235 (1965)<br>155.  L.L. Blyler and A.C. Hart, Polym. Eng. Sci., 10, 193 (1970)<br>156.  S.M. Barnett, Polym. Eng. Sci., 7, 168 (1967)<br>157. E. Boudreaux and J.A. Cuculo, J. Macromol. Sci.--Rev. Macromol. Chem., C16, 39­77 <br>
+(1977­1978)<br>
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+333. J.L. Radovich, "An Experimental Comparison of Heat Removal in Water or Air Cooled <br>
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+<hr>
+<A name=520></a><b>508</b>  <b>7Functional Process Analysis</b><br>
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+<hr>
+<A name=521></a><b>8 Extruder Screw </b><br>
+<b>Design</b><br>
+The single most important mechanical element of a screw extruder is the screw. <br>The proper design of the geometry of the extruder screw is of crucial importance to <br>the proper functioning of the extruder. If material transport instabilities occur as a <br>result of improper screw geometry, even the most sophisticated computerized con-<br>trol system cannot solve the problem. Screw design is often still considered to be <br>more of an art than a science. As a result, misconceptions about certain aspects of <br>screw design still abound today. Since the theory of single screw extrusion is now <br>well-developed (see Chapter 7), the design of screws for single screw extruders can <br>be based on solid engineering principles. Thus, screw design for single screw ex -<br>truders should no longer be an art, but a science based firmly on the principles of <br>polymer processing engineering.<br>Unfortunately, people involved in screw design are not always up-to-date on extru-<br>sion theory. As a result, many extruder screws in use today perform considerably <br>below maximum possible performance, solely because of improper screw design. <br>An example is the still-common use of the square pitch extruder screw. This is a <br>screw with constant pitch with the pitch being equal to the diameter of the screw; <br>this pitch corresponds to a helix angle of 17.66°. It can be demonstrated quite  easily <br>that the square pitch is far from optimum with respect to melting and melt convey-<br>ing for a number of polymers. This fact has been known since the early 1950s, yet <br>most extruder screws in use today still use the constant square pitch design.<br>A factor that may have contributed to the state of affairs in screw design is that there <br>has not been a comprehensive text dealing with screw design. The objective of this <br>chapter is to demonstrate how extrusion theory can be used to properly design <br>extruder screws. Hopefully, this will provide a solid foundation, based on engineer-<br>ing principles, from which better and more effective screw designs can be developed <br>in the future. The principles of screw design are not only important in designing <br>new extruder screws, but also in the analysis of processing problems of an existing <br>extrusion line. It is important to be able to recognize whether a problem is related to <br>poor screw design or to another part of the process. Thus, knowledge of the basic <br>principles of screw design is important to essentially every person involved with <br>extruders.<br>
+<hr>
+<A name=522></a><b>510</b>  <b>8Extruder Screw Design</b><br>
+<b></b><br>
+ <b>8.1Mechanical Considerations</b><br>
+Regardless of the details of the screw geometry, it is important that the screw has <br>sufficient mechanical strength to withstand the stresses imposed by the conveying <br>process in the extruder.<br>
+<b>8.1.1Torsional Strength of the Screw Root</b><br>
+An important requirement for the extruder screw is the ability to transmit the <br>torque required to turn the screw. The most critical area of the screw in this respect <br>is the feed section. In the feed section, the cross-sectional area of the root of the <br>screw is generally the smallest and, thus, the torsional strength the lowest. Also, in <br>the feed section, the entire torque has to be transmitted, while further downstream <br>only a fraction of the total torque has to be transmitted. The torque that is trans-<br>mitted can be determined from the power to the screw, Zscrew, and the screw rpm, N.<br>
+ (8.1)<br>
+where Zscrew is the power to the screw, N the rotational speed of the screw, and C <br>
+a conversion constant.<br>If Z is expressed in horsepower and N in rev/min., the constant C must equal 7120.9 <br>to give a torque expressed in Newton-meter. If Z is expressed in kilowatt, constant C <br>equals 9549.3. This formula is based on the relationship between torque, angular <br>frequency , and power:<br>
+ (8.2)<br>
+where  is expressed in radians/s and N in rpm.<br>From Eq. 8.1, it can be seen that the transmitted torque is directly proportional to <br>the horsepower and inversely proportional to the screw speed. The power to the <br>screw is related to the motor power by:<br>
+ (8.3)<br>
+where motor is the efficiency of the motor and transmission the efficiency of the trans-<br>
+mission.<br>The total efficiency is generally around 0.75. Thus, the power to the screw will be <br>about 75% of the motor power. The stresses in the screw shaft as a result of the <br>torque on the screw are shown in Fig. 8.1.<br>
+<hr>
+<A name=523></a> <br>
+8.1Mechanical Considerations  <b>511</b><br>
+max<br>
+<b> Figure 8.1</b> <br>
+Stresses in the screw shaft<br>
+The shear stress in the shaft resulting from torque T can be expressed as:<br>
+ (8.4)<br>
+where J is the polar moment of inertia.<br>The maximum stress occurs at the circumference (r = R), and the maximum stress <br>is:<br>
+ (8.5)<br>
+In order to avoid failure in the screw shaft, the maximum stress should be less than <br>the allowable stress a of the metal of the screw. The allowable stress of metal ranges <br>
+from about 50 to 100 MPa. Thus, in order to have sufficient torsional strength, the <br>following inequality should hold:<br>
+ (8.6)<br>
+From Eq. 8.6, the maximum channel depth in the feed section can be calculated:<br>
+ (8.7)<br>
+where D is the O.D. of the extruder screw.<br>Consider, as an example, a 150-mm extruder, running at 80 rpm and consuming <br>200 hp at the screw. The torque to the screw, according to Eq. 8.1, is T = 17810 Nm. <br>If a = 100 MPa, then the maximum channel depth in the feed section becomes:<br>
+ (8.8)<br>
+<hr>
+<A name=524></a><b>512</b>  <b>8Extruder Screw Design</b><br>
+It should also be realized that Eqs. 8.6 and 8.7 are valid for solid screws only. If the <br>screw is cored for cooling or heating, the torsional strength of the screw shaft will be <br>reduced. In this case, the polar moment of inertia of the screw shaft becomes:<br>
+ (8.9)<br>
+where Rs is the radius of the root screw and Rc is the radius of the core channel in <br>
+the screw.<br>
+<b>8.1.2Strength of the Screw Flight</b><br>
+The primary loading of the screw flights occurs by the pressure differential P <br>between the leading and trailing side of the flight. In addition, there will be loading <br>from the shear stress acting on the tip of the flight as a result of the leakage flow <br>through the clearance. This situation is shown in Fig. 8.2.<br>
+x(H)<br>
+ P<br>
+y<br>
+H<br>
+x<br>
+ ymax <b> Figure 8.2</b> <br>
+w<br>
+Stresses acting on a screw flight<br>
+As a result of the stresses acting on the screw flight, there will be a distribution of <br>normal stresses in the flight that reaches a maximum value at the screw surface. <br>The torque resulting from this normal stress distribution counteracts the torque <br>resulting from x and P. If the maximum normal stress at the wall is ymax, then a <br>
+torque balance per unit flight length yields:<br>
+ (8.10)<br>
+Thus, the maximum normal stress is:<br>
+ (8.11)<br>
+The cross flight shear stress x acting in the screw flight also reaches a maximum at <br>
+the screw surface. The maximum shear stress max is obtained from a simple force <br>
+balance in the x-direction:<br>
+<hr>
+<A name=525></a> <br>
+8.1Mechanical Considerations  <b>513</b><br>
+ (8.12)<br>
+Thus, the maximum shear stress is:<br>
+ (8.13)<br>
+For the combined shear and normal stress, the following criterion can be used:<br>
+ (8.14)<br>
+From Eqs. 8.11 through 8.14, the ratio of flight height to flight width can be deter-<br>mined such that comb &lt; a. If the shear stress x (H) at the tip of the flight can be <br>
+neglected, the ratio of the flight height to flight width should be:<br>
+ (8.15)<br>
+This relationship is shown in Fig. 8.3.<br>The ratio of flight height to flight width can be thought of as a slenderness ratio. In <br>general, the ratio of the allowable stress to pressure difference, a/P, will be larger <br>
+than 25. Thus, a slenderness ratio of 2 (a common value used in screw design) will <br>yield more than adequate strength if the shear stress at the flight tip is negligible. In <br>some instances, the shear stress at the flight tip cannot be neglected. For instance, <br>a situation where considerable lateral forces are acting on the screw is when there <br>is metal-to-metal contact between screw flight and barrel. In such a case, very large <br>shear stresses can act on the flight tip and these stresses must then be taken into <br>account. However, this is a rather abnormal situation and should be avoided if at all <br>possible.<br>
+4<br>
+3<br>
+x<br>
+) ma<br>
+2<br>
+1<br>
+Slenderness ratio, (H/w<br>
+00<br>
+5<br>
+10<br>
+15<br>
+20<br>
+25<b> Figure 8.3</b> <br>
+a/P<br>
+Slenderness ratio versus a/P<br>
+<hr>
+<A name=526></a><IMG src="Polymer Extrusion by Chris Rauwendaal-526_1.jpg"><br>
+<b>514</b>  <b>8Extruder Screw Design</b><br>
+It should be noted that the stress concentration in the base of the flight can be <br>reduced significantly if the flight width is varied with flight height; see Fig. 8.4.<br>An additional advantage of the flight geometries shown in Fig. 8.4 is that the flight <br>width at the tip can be reduced considerably compared to the standard flight geo-<br>metry without increasing the stresses in the base of the flight. It will be shown later <br>that there are substantial functional benefits that can be derived from the flight <br>geometries as shown in Fig. 8.4; see also Section 7.2.2.<br>
+a<br>
+b<br>
+<b>Figure 8.4</b>Flights with varying flight width<br>
+<b>8.1.3Lateral Deflection of the Screw</b><br>
+There are various causes that will tend to deflect the screw. The most obvious cause <br>is the force of gravity acting on the screw. If the drive support of the screw is consid-<br>ered rigid and the supporting function of the polymer and barrel is neglected, then <br>the sagging of the screw by its own weight can be represented by Fig. 8.5.<br>
+L<br>
+y(L)<br>
+<b> Figure 8.5</b> <br>
+Screw as a cantilever<br>
+The amount of sag at the end of the screw is:<br>
+ (8.16)<br>
+where q is the force per unit length from the weight of the screw, E is the screw <br>elastic modulus, and I is the moment of inertia. If the presence of the screw channel <br>is neglected, then Eq. 8.16 can be written as:<br>
+ (8.17)<br>
+where ag is the gravitational acceleration and  the density of the screw material.<br>
+<hr>
+<A name=527></a> <br>
+8.1Mechanical Considerations  <b>515</b><br>
+Consider a 150-mm extruder screw with a density of 7850 kg/m3 and an elastic <br>modulus of 210 GPa (= 210E9 Pa); the sag for this example is:<br>
+ (8.18)<br>
+The unrestrained sag as a function of screw length is shown in Fig. 8.6.<br>It is clear that the unrestrained sag starts to exceed the standard radial clearance <br>( 0.2 mm) when the L/D ratio exceeds 10. At normal L/D ratios of 20 to 30, the <br>unrestrained sag is about one to two orders of magnitude larger than the standard <br>radial clearance. From these simple considerations, it is obvious that the polymer <br>between the screw and barrel must play a considerable support function to prevent <br>contact between screw and barrel. The supporting force necessary to counteract the <br>sagging by the weight of the screw has to increase very strongly when the L/D is <br>increased.<br>
+5<br>
+4<br>
+L)<br>Y(<br>
+3<br>
+ti<br>on, <br>
+Deflec<br>
+2<br>
+1<br>
+standard clearance<br>
+00<br>
+5<br>
+10<br>
+15<br>
+20<br>
+25<b> Figure 8.6</b> <br>
+Length-to-diameter ratio, L/D<br>
+Unrestrained sag versus screw length<br>
+Another mechanism that can cause lateral movement of the screw is buckling. The <br>collapse force required to cause buckling of a uniform cantilever is:<br>
+ (8.19)<br>
+Thus, the head pressure necessary to cause buckling is: <br>
+ (8.20)<br>
+because the moment of inertia I = <br>
+.<br>
+<hr>
+<A name=528></a><b>516</b>  <b>8Extruder Screw Design</b><br>
+The critical head pressure for buckling as a function of the L/D ratio is shown in <br>Fig. 8.7.<br>
+100<br>
+]a 80<br>[MP<br>
+60<br>
+r buckling <br>fo<br>
+40<br>
+20<br>
+Critical pressure <br>
+020<br>
+25<br>
+30<br>
+35<br>
+40<br>
+45<b> Figure 8.7</b> <br>
+Length-to-diameter ratio, L/D<br>
+Critical head pressure versus L/D ratio<br>
+The elastic modulus is taken as 210 GPa. Considering that normal head pressures <br>range from 20 to 60 MPa, it is clear that the occurrence of buckling is a distinct pos-<br>sibility when the L/D is greater than 20. The equations above do not take into <br>account the weakening of the screw as a result of the screw channel. Fenner and <br>Williams [1] analyzed the effect of channel depth and compression ratio. They found <br>that the critical head pressure for buckling reduces when the channel depth is taken <br>into account. At a ratio of H/D = 0.05 and a compression ratio C = 3, their predicted <br>critical head pressure values are less than half of those predicted by Eq. 8.20 and <br>shown in Fig. 8.7. This simply indicates that buckling is more likely to occur when <br>the L/D is greater than 20. Thus, the polymer will have to support the screw in the <br>center region of the extruder to prevent screw-to-barrel contact as a result of buck-<br>ling. However, the lateral force resulting from buckling Fb is of the order:<br>
+ (8.21)<br>
+where F is the force acting on the end of the screw.<br>Since the ratio /L is about 0.00005, the lateral force resulting from buckling will be <br>quite small.<br>Another possible deflection mechanism is the occurrence of whirling. Whirling <br>occurs when a shaft reaches a critical speed and becomes dynamically unstable <br>with large lateral amplitudes. This phenomenon is due to the resonance frequency <br>when the rotational speed corresponds to the natural frequencies of lateral vibra-<br>tion of the shaft. For uniform beams vibrating in flexure, the natural frequencies <br>can be expressed as:<br>
+<hr>
+<A name=529></a> <br>
+8.1Mechanical Considerations  <b>517</b><br>
+ (8.22)<br>
+where ag is the gravitational acceleration and w1 is the weight per unit length.<br>The value Cn for a beam with one end clamped and one end free is Cn = 0.560. For a <br>
+circular beam, the critical rotational speed becomes (N = 2f):<br>
+ (8.23)<br>
+When E = 210 GPa and  = 7850 kg/m3, the critical rotational speed can be written <br>as:<br>
+ (8.24)<br>
+For a 150-mm extruder with L = 30 D, the critical whirling speed Nw = 33.7 rev/s = <br>
+2022 rpm. It is clear from these numbers that whirling is not likely to occur in nor-<br>mal extrusion operations. Fenner and Williams [1] also calculated the critical rota-<br>tional speed for whirling in extruders; however, they used an incorrect equation for <br>the whirling speed. Instead of using weight per unit length, they used mass per unit <br>length in Eq. 8.22. As a result, their values of the whirling speed are lower by a factor <br>of ag.<br>Another possible cause of lateral deflection of the screw is non-uniform pressure <br>distribution around the circumference of the screw. Figure 8.8 shows a possible <br>pressure distribution that will result in a considerable lateral force on the screw.<br>
+Rv<br>
+d<br>
+<br>
+Rh<br>
+<b> Figure 8.8</b> <br>
+Circumferential pressure distribution that can <br>
+cause screw deflection<br>
+<hr>
+<A name=530></a><b>518</b>  <b>8Extruder Screw Design</b><br>
+The horizontal reaction component can be determined from:<br>
+ (8.25)<br>
+The vertical reaction component can be determined from:<br>
+ (8.26)<br>
+If the pressure is on the average P larger on the left side than on the right side, <br>the horizontal reaction force over length L is:<br>
+ (8.27)<br>
+If D = L = 150 mm and P = 1 MPa ( 145 psi), the reaction force will be 22.5 kN <br>( 5000 lbf.) In reality, the situation is more complicated because the pressure <br> <br>varies both in down-channel and cross-channel directions. However, from this sim-<br>ple example, it is clear that even a small pressure differential of only 1 MPa can <br>cause a significant lateral reaction force on the screw. The reaction force is of such <br>magnitude that it can easily deflect the screw into the barrel and cause substantial <br>wear. This type of pressure-induced deflection is most likely to occur when the pres-<br>sure reaches a sharp maximum or minimum somewhere along the screw. It is also <br>more likely to occur with large helix angles (greater than 20°) as compared to small <br>helix angles (smaller than 20°), and with single-flighted screws as compared to <br> <br>double-flighted  screws.<br>A likely location for sharp pressure peaks is the end of the compression section. <br>Many experimental studies have found that the pressure profile along the length of <br>the extruder reaches a maximum close to the end of the compression section of the <br>screw [77]. If the compression ratio is high, there is a chance of plugging of the solid <br>bed. This can cause local pressure peaks that can deflect the screw into the barrel. <br>It is interesting to note that generally the most severe wear does indeed occur to -<br>wards the end of the compression section. This indicates that quite often the com-<br>pression ratio is too high for the materials being processed, causing sharp pressure <br>peaks towards the end of the compression section.<br>If one considers all the possible causes of lateral screw deflection, gravity, buckling, <br>whirling, and pressure-induced deflection, it seems that pressure-induced deflec-<br>tion is the most likely cause of lateral deflection. The wear caused by pressure-<br>induced deflection is most likely located towards the end of the compression section. <br>This type of wear can be quite dramatic. The author experienced a case where a <br>150-mm extruder was wearing at a rate of about 1 mm (0.040 inch) in about <br>24 hours. The cause turned out to be the installation of a relatively long grooved <br> <br>barrel section without a corresponding change in screw design. A reduction of <br>
+<hr>
+<A name=531></a> <br>
+8.2Optimizing for Output  <b>519</b><br>
+the  length of the grooved barrel section solved the problem; a change in screw <br>design (lower compression ratio) probably would have solved the problem as well. <br>However, this example illustrates that pressure-induced screw deflection can cause <br>severe wear problems. Unfortunately, this type of problem is not easy to diagnose <br>because the exact pressure profile along the length of the extruder is generally not <br>known. Another complicating factor is that this type of wear may take several <br>weeks or even months to become significant. This can make it difficult to relate <br>the wear to the event that triggered it because there can be a considerable time lag <br>or induction time.<br>
+<b></b><br>
+ <b>8.2Optimizing for Output</b><br>
+Generally, the objective of a screw design is to deliver the largest amount of output <br>at acceptable melt quality. Unfortunately, high output and mixing quality are, to <br>some extent, conflicting requirements. As output goes up, residence time goes down <br>and with it the time available to mixing the polymer melt. As a result, the mixing <br>quality goes down when the output increases. However, the mixing quality can gen-<br>erally be restored by incorporating mixing sections, either a mixing section along <br>the screw or a static mixing section.<br>It is also important to realize that all functional zones of the extruder are inter-<br>dependent. It makes little sense to drastically increase the pumping capacity if the <br>melting rate is the real bottleneck in the process. Thus, before designing a new <br>extruder screw to replace an existing screw, one should determine what part of the <br>extruder is limiting the rate. This process will be covered in Chapter 11 on trouble-<br>shooting extruders.<br>
+<b>8.2.1Optimizing for Melt Conveying</b><br>
+The melt conveying theory discussed in Section 7.4 can be used to determine the <br>optimum screw geometry for melt conveying. This optimum geometry will not nor-<br>mally be used in the metering section of the extruder screw. The optimum channel <br>depth for output can be determined from:<br>
+ (8.28)<br>
+The volumetric melt conveying rate for a Newtonian fluid, neglecting the effect of <br>the flight flanks, is given by Eq. 7.198. Considering that the channel width W = ( D <br>sin/p) ­w, down-channel barrel velocity vbz =  D N cos, and down-channel pres-<br>
+<hr>
+<A name=532></a><b>520</b>  <b>8Extruder Screw Design</b><br>
+sure gradient gz = ga sin, where ga is the axial pressure gradient, Eq. 7.198 can be <br>
+written as:<br>
+ (8.29)<br>
+where L is the axial distance over which pressure P is built up; thus, the axial pres-<br>sure gradient ga = P/L.<br>The polymer melt viscosity  will depend on the local shear rate in the screw chan-<br>nel. If it is assumed that the representative shear rate in the channel is the Couette <br>shear rate and that the polymer melt behaves as a power law fluid, which melt vis-<br>cosity can be represented by:<br>
+ (8.30)<br>
+where m is the consistency index and n the power law index; see Eq. 6.23.<br>The optimum channel depth for output rate H*r can now be determined from Eqs. <br>
+8.28 through 8.30.<br>
+ (8.31)<br>
+The optimum depth depends on the diameter, screw speed, power law index, consist-<br>ency index, pressure gradient, and helix angle. The optimum helix angle for output <br>can be determined from:<br>
+ (8.32)<br>
+By using the same equations for the melt conveying rate, the optimum helix angle <br>for output rate *r has to be determined from the following equations:<br>
+ (8.33)<br>
+Equation 8.33 cannot be easily solved in the form it is in. However, the equation can <br>be simplified considerably if it is assumed that w = 0 and by defining a dimension-<br>less down-channel pressure gradient g0z. Equation 8.33 can now be written as:<br>
+ (8.34)<br>
+<hr>
+<A name=533></a> <br>
+8.2Optimizing for Output  <b>521</b><br>
+where:<br>
+ (8.35)<br>
+The optimum helix angle now has to be determined from:<br>
+ (8.36)<br>
+For the Newtonian case (n = 1), the solution becomes:<br>
+ (8.37)<br>
+For the extreme non-Newtonian case (n = 0), the solution is:<br>
+ (8.38)<br>
+Figure 8.9 shows the optimum helix angle as a function of the dimensionless down-<br>channel pressure gradient for n = 1 and n = 0.<br>
+50<br>
+40<br>
+30<br>
+n=0<br>
+n=1<br>
+20<br>
+ angle [degrees]<br>
+10<br>
+<b> Figure 8.9</b><br>
+Optimum helix<br>
+ <br>
+Optimum helix angle versus <br>
+0 0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+ <br>dimensionless down-channel pressure <br>
+Dimensionless downchannel pressure gradient<br>
+gradient<br>
+It can be seen that the two curves are relatively close; thus, the effect of the pseudo-<br>plastic behavior is not very pronounced.<br>It may be more interesting to optimize the channel depth and helix angle simultane-<br>ously. This can be done by inserting Eq. 8.31 into Eq. 8.33. After some calculations, <br>the optimum helix angle can be determined to be:<br>
+ (8.39)<br>
+<hr>
+<A name=534></a><b>522</b>  <b>8Extruder Screw Design</b><br>
+where  is a reduced flight width:<br>
+ (8.39a)<br>
+where p is the number of flights, w the perpendicular flight width, and D the screw <br>O.D.<br>The details of the derivation can be found in an article on screw design of two-stage <br>extruder screws [2]. The optimum helix angle is only a function of the power law <br>index and the reduced flight width. Figure 8.10 shows the optimum helix angle as a <br>function of the power law index at various values of the reduced flight width.<br>
+<b> Figure 8.10</b> <br>
+Optimum helix angle versus power law <br>
+index in simultaneous optimization<br>
+The simplicity of Eq. 8.39 makes it a useful and convenient expression for optimiz-<br>ing the geometry of the melt conveying zone of an extruder. For polymer melts with <br>a power law index in the range of 0.3 to 0.4 and typical values of the flight width, the <br>optimum helix angle is about 22 to 24°.<br>The corresponding optimum channel depth can be found by inserting Eq. 8.39 into <br>Eq. 8.31. The optimum channel depth resulting from simultaneous optimization is <br>simply:<br>
+ (8.40)<br>
+Unfortunately, the optimum channel depth is dependent on many more variables <br>than the optimum helix angle. The latter depends only on the power law index and <br>reduced flight width. In addition to these variables, the optimum channel depth also <br>depends on the screw speed, screw diameter, consistency index, and pressure gra-<br>dient. This means that it is not possible to design a universally optimum screw geo-<br>metry. Thus, one has to determine the most likely operating parameters that the <br>screw is likely to encounter and design for those parameters.<br>
+<hr>
+<A name=535></a> <br>
+8.2Optimizing for Output  <b>523</b><br>
+It should be noted that fundamentally it is not entirely correct to take an expression <br>derived for a Newtonian fluid and insert a power law viscosity form into it. However, <br>if this simplification is not made, the analysis becomes much more complex and <br>analytical solutions much more difficult to obtain, if not impossible. Results of the <br>analytical solutions have been compared to results of numerical computations for a <br>two-dimensional flow of a power law fluid. In most cases, the results are within 10 to <br>20% [2]. It should be noted that the results are exact when the power law index is <br>unity, i.e., for Newtonian fluids. However, if the optimum depth and helix angle for a <br>pseudo-plastic fluid are calculated using expressions valid for Newtonian fluids only, <br>very large errors can result, particularly when the power law index is about one-half <br>or less. It is therefore very important to take the pseudo-plastic behavior into <br>ac count, because the large majority of polymers are strongly non-Newtonian.<br>Instead of optimizing the screw geometry for output, it may be desirable to optimize <br>for pressure-generating capability.<br>The optimum depth H*p and helix angle *p for pressure generation is found by set-<br>
+ting:<br>
+ (8.41)<br>
+An expression for pressure P can be found by rewriting Eq. 8.29. The optimum chan-<br>nel depth is:<br>
+ (8.42)<br>
+The optimum helix angle has to be determined from:<br>
+ (8.43)<br>
+When the depth and helix angle are optimized simultaneously, Eq. 8.42 is inserted <br>into Eq. 8.43. The resulting optimum helix angle for simultaneous optimization of <br>the pressure-generating capacity is:<br>
+ (8.44)<br>
+Comparing this result to the optimum helix angle for output *r, Eq. 8.39, it is clear <br>
+that the two expressions are exactly the same, thus:<br>
+ (8.45)<br>
+<hr>
+<A name=536></a><b>524</b>  <b>8Extruder Screw Design</b><br>
+The optimum channel depth for simultaneous optimization for pressure generation <br>is found by inserting Eq. 8.39 into Eq. 8.37. The resulting expression can be shown <br>to be identical to Eq. 8.35, thus:<br>
+ (8.46)<br>
+Therefore, simultaneous optimization of depth and helix angle for output yields the <br>same results as simultaneous optimization for pressure-generating capability.<br>It should be noted that the expressions for optimum channel depth are only valid <br>when the pressure gradient is positive. When the pressure gradient is negative, the <br>pressure flow will be in the forward direction and the output increases with channel <br>depth without reaching a maximum value. In this situation, the depth of the meter-<br>ing section will be determined by the requirements for complete melting and good <br>mixing.<br>The optimum channel depth and helix angle can also be determined from the melt <br>conveying theory of power law fluids when the flow is considered to be one-dimen-<br>sional; see also Section 7.4.2. By combining Eqs. 7.256 and 7.257, the output can be <br>written as:<br>
+ (8.47)<br>
+where:<br>
+ (8.47a)<br>
+The optimum channel depth can again be determined by taking the partial deriva-<br>tive of output   and setting the result equal to zero. The resulting expression is:<br>
+ (8.48)<br>
+For large positive pressure gradients, Eq. 7.252 should be used to evaluate the first <br>derivative of  with respect to channel depth H. This results in the following expres-<br>sion:<br>
+ (8.49)<br>
+Inserting Eq. 8.49 into Eq. 8.48 yields an expression with only  and s:<br>
+ (8.49a)<br>
+<hr>
+<A name=537></a> <br>
+8.2Optimizing for Output  <b>525</b><br>
+A complete solution of Eq. 8.49(a) may be rather involved; however, it can be seen <br>quite easily that  = 0 is a solution of Eq. 8.49(a). Thus, the channel depth is opti-<br>mized when  = 0, i.e., when the velocity gradient becomes zero at the screw sur-<br>face. The optimum helix angle is obtained by taking the first derivative of output   <br>and setting the result equal to zero. This results in the following expression:<br>
+ (8.49b)<br>
+For simultaneous optimization of channel depth and helix angle,  = 0 and Eq. <br>8.49(b) becomes:<br>
+ (8.49c)<br>
+When  = 0, the first derivative of  with respect to helix angle , determined from <br>Eq. 7.252, becomes:<br>
+ (8.49d)<br>
+Inserting Eq. 8.49(d) into Eq. 8.49(c) yields the solution of the optimum helix angle:<br>
+ (8.49e)<br>
+To compare this result with the result from the modified Newtonian analysis, Eq. <br>8.44, the optimum helix angle can be written as:<br>
+ (8.50)<br>
+Equation 8.50 obviously is different from Eq. 8.44 when the flight width is neglected <br>( = 0).<br>Figure 8.11 compares the results of simultaneous optimization from the modified <br>Newtonian analysis and the one-dimensional power law analysis.<br>It is clear that the one-dimensional power law analysis yields higher values of the <br>optimum helix angle than the modified Newtonian analysis, except when the power <br>law index is unity.<br>Figure 8.12(a) compares the results of simultaneous optimization from the modi-<br>fied Newtonian analysis and the two-dimensional power law analysis, obtained by <br>numerical computations [2].<br>
+<hr>
+<A name=538></a><b>526</b>  <b>8Extruder Screw Design</b><br>
+35<br>
+30<br>
+1D power law analysis<br>
+25<br>
+ angle [degrees]<br>
+Modified Newtonian analysis<br>
+<b>  </b><br>
+<b>Figure 8.11</b> <br>
+20<br>
+Optimum helix angle versus <br>
+power law index, resulting from <br>
+Optimum helix<br>
+modified Newtonian analysis <br>
+15 0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+and one-dimensional power law <br>
+Power law index<br>
+analysis<br>
+40<br>
+10.0<br>
+analytical solution<br>
+numerical solution 2D<br>
+30<br>
+7.5<br>
+]<br>
+mm[<br>
+20<br>
+5.0<br>
+ angle [degrees]<br>
+<b> Figure 8.12(a)</b> <br>
+10<br>
+2.5<br>
+Optimum helix angle versus <br>
+power law index, resulting from <br>
+Optimum helix<br>
+Optimum channel depth <br>
+modified Newtonian analysis <br>
+0<br>
+0<br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+and two-dimensional power law <br>
+Power law index<br>
+analysis<br>
+50<br>
+40<br>
+30<br>
+n=1<br>
+ angle [degrees]<br>
+0.8<br>
+0.6<br>
+20<br>
+0.4<br>
+0.2<br>
+10<br>
+Optimum helix<br>
+<b>  </b><br>
+<b>Figure 8.12(b)</b> <br>
+Optimum helix angle versus <br>
+reduced axial pressure gradient, <br>
+0 0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+resulting from two-dimensional <br>
+Reduced axial pressure gradient<br>
+power law analysis (numerical)<br>
+<hr>
+<A name=539></a> <br>
+8.2Optimizing for Output  <b>527</b><br>
+It can be seen that the values of the optimum helix angle from the modified Newton-<br>ian analysis are about 10 to 20% above those from the two-dimensional power law <br>analysis. This indicates that the results from the modified Newtonian analysis, Eq. 8.44, <br>may be more appropriate than the results from the one-dimensional power law ana-<br>lysis. The use of one-dimensional power law analysis leads to errors when the helix <br>angle is substantially above zero, as discussed in Section 7.4.2. Therefore, one would <br>like to use a two-dimensional power law analysis. However, there are no analytical <br>solutions for this case. From numerical computations, it is possible to develop a plot <br>of optimum helix angle as a function of a dimensionless axial pressure gradient g0a, <br>
+where g0a is:<br>
+ (8.51)<br>
+The plot of the optimum helix angle versus dimensionless axial pressure gradient is <br>shown in Fig. 8.12(b).<br>It should be noted that the optimum helix angle in Fig. 8.12(b) is not the result of <br>simultaneous optimization of channel depth and helix angle, but optimization of the <br>helix angle only. Figure 8.13 shows the optimum helix angle versus dimensionless <br>down-channel pressure gradient as again determined from a two-dimensional analy-<br>sis of a power law fluid.<br>Figure 8.13 can be compared directly to the results of the modified Newtonian analy-<br>sis shown in Fig. 8.9. It is quite clear that the modified Newtonian analysis under-<br>estimates the effect of the non-Newtonian behavior.<br>
+40<br>
+constant axial pressure gradient<br>
+30<br>
+20<br>
+n=0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+ angle [degrees]<br>
+10<br>
+<b> Figure 8.13</b> <br>
+Optimum helix angle versus dimen-<br>
+Optimum helix<br>
+sionless down-channel pressure <br>
+0 0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+ <br>gradient, resulting from two-dimen-<br>
+Dimensionless pressure gradient<br>
+sional power law analysis (numerical)<br>
+<hr>
+<A name=540></a><b>528</b>  <b>8Extruder Screw Design</b><br>
+Another approach to optimization of the screw geometry for melt conveying can be <br>made by using the Newtonian flow rate equation with correction factors for pseudo-<br>plastic behavior; see Eq. 7.291. If the flight width is neglected this equation can be <br>written as:<br>
+ (8.52)<br>
+The optimum helix angle can be determined by taking the first derivative of output <br>
+ with respect to the helix angle and setting the result equal to zero. This results in <br>
+the following expression for the optimum helix angle *:<br>
+ (8.53)<br>
+The solution is:<br>
+ (8.53a)<br>
+where the dimensionless down-channel pressure gradient is given by:<br>
+ (8.53b)<br>
+Figure 8.14(a) shows the optimum helix angle as determined from Eq. 8.53(a) as a <br>function of the dimensionless pressure gradient at various values of the power law <br>index.<br>
+40<br>
+30<br>
+20<br>
+n=0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+ angle [degrees]<br>
+10<br>
+<b> Figure 8.14(a)</b> <br>
+Optimum helix angle versus dimen-<br>
+Optimum helix<br>
+sionless down-channel pressure <br>
+0 0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+ <br>gradient, derived from output of <br>
+Dimensionless pressure gradient<br>
+Eq . 8 .52 (analytical results)<br>
+<hr>
+<A name=541></a> <br>
+8.2Optimizing for Output  <b>529</b><br>
+It can be seen that the optimum helix angle is strongly dependent on the power law <br>index. Comparison with results from the two-dimensional power law analysis, <br>shown in Fig. 8.13, indicates a reasonable agreement when the power law index is <br>larger than one-half (n &gt; 0.5), but relatively large differences when the power law <br>index is smaller than one-half (n &lt; 0.5).<br>If the optimum helix angle is expressed as a function of the axial pressure gradient, <br>the optimum helix angle becomes:<br>
+ (8.54)<br>
+If the reduced axial pressure gradient g0a is defined as:<br>
+ (8.54a)<br>
+then the optimum helix angle can be expressed as:<br>
+ (8.54b)<br>
+Similarly, the optimum channel depth can be determined to be:<br>
+ (8.54c)<br>
+A very simple expression for the optimum helix angle was obtained by Rauwendaal <br>[82]. The optimum helix angle in degrees can be expressed as:<br>
+ (8.54d)<br>
+The optimum helix angle as a function of reduced axial pressure gradient according <br>to Eq. 8.54(b) is shown in Fig. 8.14(b).<br>Figure 8.14(b) can be compared directly to Fig. 8.12(b) showing results from a two-<br>dimensional power law analysis. The agreement between the two sets of results is <br>quite reasonable.<br>Summarizing, it can be concluded that for simultaneous optimization of the channel <br>depth and helix angle, the modified Newtonian analysis yields reasonably accurate <br>results when compared to the two-dimensional power law analysis. The important <br>equations are Eq. 8.39 for the optimum helix angle and Eq. 8.40 for the optimum <br>channel depth. The results of simultaneous optimization from a one-dimensional <br>power law analysis are less accurate than the modified Newtonian analysis.<br>
+<hr>
+<A name=542></a><b>530</b>  <b>8Extruder Screw Design</b><br>
+40<br>
+30<br>
+n=0.2<br>
+0.4<br>
+0.6<br>
+20<br>
+ angle [degrees]<br>
+0.8<br>
+1.0<br>
+10<br>
+Optimum helix 0<br>
+<b> Figure 8.14(b)</b> <br>
+0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+Optimum helix angle versus reduced <br>
+Reduced axial pressure gradient<br>
+axial pressure gradient<br>
+For optimization of just the helix angle or the channel depth, the results from the <br>Newtonian flow rate equation with correction factors for pseudo-plastic behavior, <br>Eq. 8.54, are more accurate than the results from the modified Newtonian analysis, <br>Eq. 8.36. It should again be noted that simultaneous optimization only makes sense <br>for relatively large positive pressure gradients. When the pressure gradient is nega-<br>tive, the output increases monotonically with channel depth. In this case, there is no <br>optimum channel depth.<br>
+<b>8.2.2Optimizing for Plasticating</b><br>
+The geometry for optimum melting performance can be determined from the equa-<br>tions developed in Section 7.3. A convenient relationship to use is the total axial <br>melting length. It is convenient because the effect of varying the helix angle can be <br>evaluated directly, whereas the total down-channel melting length has to be con-<br>verted to axial melting length for a one-to-one comparison. The total axial melting <br>length can be determined from Eq. 7.116:<br>
+ (8.55)<br>
+The initial solid bed width W1 is:<br>
+ (8.56)<br>
+<hr>
+<A name=543></a> <br>
+8.2Optimizing for Output  <b>531</b><br>
+The solid bed velocity can be expressed as:<br>
+ (8.57)<br>
+The term 1 is given by Eq. 7.109(d); it can be written as:<br>
+ (8.58)<br>
+The term A1 in Eq. 8.55 is the compression in axial direction. A1 is related to the <br>
+compression in down-channel direction Az by:<br>
+ (8.59)<br>
+With these equations, the effect of various geometrical variables can be determined. <br>Figure 8.15 shows the total axial melting length as a function of the helix angle at <br>various values of the flight width.<br>The results shown in Fig. 8.15 are predictions for a 50-mm extruder running at a <br>screw speed of 100 rpm with an output of 100 kg/hr.; the barrel temperature is set <br>50°C above the melting point of the polymer. The channel depth in the feed section <br>is 5 mm, the axial compression A1 = 0.008, and the number of flights p = 1. The fol-<br>
+lowing polymer properties were used:<br>  Melt density <br>
+  =  7800 kg/m3<br>
+  Thermal conductivity  km  =  0.25 J/ms °K<br>
+Melt viscosity <br>
+  =  1500 Ns/m2<br>
+Enthalpy change <br>
+H =  4.5E5 J/kg<br>
+5.8D<br>
+th 5.4D<br>
+w=0.2D<br>
+w=0.1D<br>
+w=0<br>
+5.0D<br>
+Axial melting leng<br>
+4.6D 0<br>
+20<br>
+40<br>
+60<br>
+80<br>
+<b> Figure 8.15</b> <br>
+Helix angle [degrees]<br>
+Axial melting length versus helix angle<br>
+These properties are typical of polyethylene; see Table 6.1.<br>
+<hr>
+<A name=544></a><b>532</b>  <b>8Extruder Screw Design</b><br>
+<b>8.2.2.1Effect of Helix Angle<br></b>From Fig. 8.15, it can be seen that the melting length reduces strongly with helix <br>angle at relatively small angles. At larger helix angles, the melting length reduces <br>weakly with helix angle. Very little improvement is obtained by increasing the helix <br>angle beyond 30°. The melting length increases as the flight width increases, par-<br>ticularly at small helix angles. It is clear from Fig. 8.15 that there is no optimum <br>helix angle for which the axial melting length reaches a minimum. The shortest <br>melting length is obtained at a helix angle of 90°; however, the melting length at <br>90° is only about 3% less than the melting length at 30° and about 8% less than the <br>melting length at 17.66° (square pitch). Considering that a helix angle of 90° does <br>not produce any forward drag transport, it makes little sense to apply such an <br>extreme helix angle for the sake of melting.<br>
+<b>8.2.2.2Effect of Multiple Flights<br></b>From Eqs. 8.55 through 8.58, the effect of the number of parallel flights can be <br>evaluated directly. Figure 8.16 shows the axial melting length versus helix angle for <br>a single-flighted, double-flighted, and triple-flighted screw design.<br>
+p=1<br>
+5D<br>
+h<br>
+w=0.1D<br>
+w=0<br>
+4D<br>
+p=2<br>
+w=0.1D<br>
+w=0<br>
+w=0.1D<br>
+Axial melting lengt 3D<br>
+w=0<br>
+p=3<br>
+2D<br>
+<b> Figure  8.16</b> <br>
+0<br>
+20<br>
+40<br>
+60<br>
+80<br>
+Axial melting length versus helix <br>
+Helix angle [degrees]<br>
+angle for 1, 2, and 3 parallel flights<br>
+The predictions are for the same example used earlier in this section. It is clear from <br>Fig. 8.16 that the melting length can be reduced substantially when the number of <br>flights is increased, provided the helix angle is sufficiently large. It can be easily <br>shown that the melting length with p parallel flights L(p) is related to the melting <br>length with one flight L(1) by:<br>
+ (8.60)<br>
+<hr>
+<A name=545></a><IMG src="Polymer Extrusion by Chris Rauwendaal-545_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-545_20.jpg"><br>
+ <br>
+8.2Optimizing for Output  <b>533</b><br>
+Eq. 8.60 is valid for zero compression screws (A1 = 0) when the flight width w is <br>
+negligible; however, it is reasonably accurate even if A1  0. Figure 8.16 shows <br>
+curves with flight width w = 0 and w = 0.1 D for each number of flights. This is <br>shown to demonstrate that the effect of flight width becomes more pronounced as <br>the number of flights is increased. Substantial errors can be made if the flight width <br>is neglected in a multi-flighted screw geometry. Figure 8.16 also shows that the <br>effect of the helix angle on melting length becomes stronger when the number of <br>flights is increased. With three parallel flights, the melting length at 90° is about <br>20% less than at 17.66° (square pitch), while in the single-flighted design the differ-<br>ence is only about 8%.<br>The beneficial effect of multiple flights can be explained by the reduced average <br>melt film; see Fig. 8.17.<br>
+<b> Figure 8.17</b> <br>
+Melting in multi-flighted screw geometry<br>
+As the number of flights is increased, the width of the individual solid beds becomes <br>smaller. As a result, the average melt film thickness reduces, because the melt film <br>thickness increases with the solid bed width. The thinner melt film will improve <br>heat conduction through the melt film and increase the viscous heat generation in <br>the melt film, causing improved melting performance.<br>For optimum melting performance, the most favorable geometry is a multi-flighted <br>design with relatively narrow flights and relatively large helix angle, around 25 to <br>30°. This combination of screw geometry variables is important because a multi-<br>flighted geometry with a small helix angle may actually reduce the melting per-<br>formance.<br>The effect of the helix angle, flight width, and number of flights can also be analyzed <br>in terms of the volumetric efficiency of the extruder screw. This is defined simply as <br>
+<hr>
+<A name=546></a><b>534</b>  <b>8Extruder Screw Design</b><br>
+the channel volume divided by the total volume of channel and flight. The volumet-<br>ric efficiency v can be expressed as:<br>
+ (8.61)<br>
+Figure 8.18 shows the volumetric efficiency as a function of the helix angle at vari-<br>ous numbers of flights.<br>
+<b>Figure 8.18</b>Volumetric efficiency versus helix angle for various numbers of flights<br>
+The volumetric efficiency rises sharply at small helix angles and more slowly at <br>large helix angles. The highest v is obtained when the helix angle is 90°. When the <br>
+number of parallel flights is increased at a small helix angle, the volumetric effi-<br>ciency drops considerably. When  = 17.66° (square pitch), the volumetric effi-<br>ciency is about 85% at p = 1, but it drops to about 50% at p = 4. When  = 30°, the <br>volumetric efficiency is about 95% at p = 1 and drops to about 75% at p = 4. Thus, the <br>adverse effect of multiple flights on volumetric efficiency is much less severe at <br>large helix angles than it is at small helix angles.<br>
+<b>8.2.2.3Effect of Flight Clearance<br></b>The melting rate for a non-zero clearance can be determined from Eq. 7.107; it can <br>be written as:<br>
+ (8.62)<br>
+<hr>
+<A name=547></a> <br>
+8.2Optimizing for Output  <b>535</b><br>
+Equation 8.62 expresses the melting rate per unit length z, where z is the direction <br>
+of the relative velocity between the solid bed and barrel. The melting rate per unit <br>down-channel distance z can be written as:<br>
+ (8.63)<br>
+From Eq. 8.63, it is clear that the melting rate reduces with increasing clearance. <br>Figure 8.19(a) shows the melting rate as a function of radial clearance for the exam-<br>ple used earlier, when  = 17.66° and Ws = ½ D.<br>
+0.06<br>
+]ms 0.04<br>
+0.02<br>
+Melting rate[kg/<br>
+0<br>
+1<br>
+2<br>
+3 <b> Figure 8.19(a)</b> <br>
+Radial clearance [0.001D]<br>
+Melting rate versus radial clearance<br>
+The melting rate drops monotonically with increasing radial clearance. Considering <br>that the standard clearance is 0.001 D; a doubling of the standard clearance causes <br>a reduction in melting rate of about 25%. A tripling of the standard clearance causes <br>a reduction in melting rate of about 35%. It is clear that wear in the plasticating zone <br>of the extruder has a detrimental effect on the melting performance. It is important, <br>therefore, to make sure that the radial clearance in this region is within reasonable <br>limits. Unfortunately, screw and barrel wear often occur in the plasticating zone of <br>the extruder, as discussed in Section 8.1.3. This type of wear will ad versely affect <br>the melting performance and thus reduce overall extruder perform ance. Symptoms <br>of this can be temperature non-uniformities and throughput and pressure fluctua-<br>tions. If these problems occur, it is good practice to check for screw and barrel wear. <br>If the clearance is more than two or three times the standard clearance, the screw <br>and/or barrel should be replaced.<br>Wear in the compression section of the screw is often caused by large compression <br>ratios. This will be discussed next.<br>
+<hr>
+<A name=548></a><b>536</b>  <b>8Extruder Screw Design</b><br>
+<b>8.2.2.4Effect of Compression Ratio<br></b>The compression of the channel depth tends to widen the solid bed, as discussed in <br>Section 7.3. Melting tends to reduce the width of the solid bed. If compression is too <br>rapid, the melting may be insufficient and the solid bed can grow in width. This will <br>generally cause plugging of the channel by the solid bed and should be avoided if at <br>all possible. Plugging will cause output fluctuations, but it may also cause wear in <br>the compression section of the screw, as discussed in Section 8.1.3. It was discussed <br>in Section 7.3 that plugging can be avoided if:<br>
+ (8.64)<br>
+The compression ratio Xc is the channel depth in the feed section divided by the <br>
+channel depth in the metering section. The axial length of the compression section <br>is Lc. Thus, in order to avoid plugging, the length of the compression section should <br>
+obey the following inequality:<br>
+ (8.65)<br>
+From Eq. 8.65, the minimum allowable length of the compression section can be <br>determined if the compression ratio is known. Figure 8.19(b) shows the minimum Lc <br>
+versus the compression ratio for the example used earlier.<br>
+7<br>
+i<br>o<br>at<br>nr<br>sio 5<br>
+ompres<br>
+3<br>
+mc<br>i<br>x<br>mu<br>
+Ma<br>
+<b> Figure 8.19(b)</b> <br>
+1 0<br>
+1D<br>
+2D<br>
+3D<br>
+4D<br>
+5D<br>
+Minimum length of compression <br>
+Length compression section<br>
+ <br>section versus compression ratio<br>
+If a large compression ratio is used, the length of the compression section must be <br>quite long to avoid plugging.<br>The most dangerous combination in design of the compression section is a large <br>compression ratio and a short compression section length. This will lead very easily <br>to plugging. Rapid compression screws should be avoided for this very reason. One <br>of the myths in screw design is that certain polymers, e.g., nylon, require a very <br>
+<hr>
+<A name=549></a> <br>
+8.2Optimizing for Output  <b>537</b><br>
+rapid compression screw in order to extrude properly. Many screws have been <br>designed with a compression length of less than one diameter. Such a screw can <br>only work if the majority of the melting occurs before the compression section. How-<br>ever, this defeats the purpose of having a compression section in the first place. <br>Rapid compression screws do not make much sense from a functional point of view <br>because they are susceptible to surging and wear. It is important to realize that in <br>general polymers do not require a very rapid compression screw in order to extrude <br>properly. The benefits of very rapid compression screws are imaginary and based on <br>a serious misconception. Obviously, a very rapid compression is quite possible in <br>the second stage of a two-stage extruder screw, because essentially all the melting <br>should take place in the first stage; see Section 8.5.2.<br>
+<b>8.2.3Optimizing for Solids Conveying</b><br>
+Optimization of the solids conveying process is very important because solids con-<br>veying is the basis of the entire plasticating extrusion process. If instabilities occur <br>in the solids conveying zone, these instabilities will transmit to the downstream <br>zones and cause fluctuations in output and pressure. It was already discussed in <br>Section 7.2.2.2 that grooved barrel sections provide a powerful tool to improve solids <br>conveying rate and stability. There are, however, some important considerations <br>with respect to screw design to optimize the solids conveying process.<br>
+<b>8.2.3.1Effect of Channel Depth<br></b>It was discussed in Section 7.2.2 that there appears to be an optimum channel depth <br>for which the solids conveying rate reaches a maximum. At low values of the pres-<br>sure increase over the solids conveying section, this optimum channel depth is <br>indeed apparent because this optimum channel depth does not occur when the <br>channel curvature is taken into account. At higher values of the pressure increase, <br>however, there is an actual optimum channel depth even when the channel curva-<br>ture is taken into account. This is shown in Fig. 8.20 for a 114-mm (4.5-in) extruder <br>running at 60 rpm; the coefficient of friction is 0.5 on the barrel and 0.3 on the <br>screw. When the pressure gradient increases, the optimum channel depth decreases.<br>The optimum channel depth can be obtained by taking the first derivative of the <br>solids conveying rate  s with respect to the channel depth H and setting the result <br>
+equal to zero:<br>
+ (8.66)<br>
+Equation 8.66 does not have a simple closed form solution. The optimum channel <br>depth can be evaluated by using a numerical or graphical method. The optimum <br>
+<hr>
+<A name=550></a><b>538</b>  <b>8Extruder Screw Design</b><br>
+channel depth will increase with the coefficient of friction on the barrel. It will <br>decrease with the coefficient of friction on the screw, the number of flights, and the <br>pressure gradient. Unfortunately, the actual coefficients of friction are generally not <br>known with a great degree of precision; thus, accurate prediction of the optimum <br>channel depth is usually not possible. The channel depth in the feed section of <br>screws used in smooth bore extruders is often about 0.15 to 0.20D. In grooved feed <br>extruders the feed section depth is often about 0.1D.<br>
+400<br>
+P1/P0=1<br>
+300<br>
+P1/P0=100<br>
+200<br>
+ing rate [cc/s]<br>
+P<br>
+ey<br>
+1/P0=200<br>
+100<br>
+Conv<br>
+P1/P0=500<br>
+<b> Figure 8.20</b> <br>
+010<br>
+15<br>
+20<br>
+25 Solids conveying rate versus channel <br>
+Channel depth [mm]<br>
+depth<br>
+<b>8.2.3.2Effect of Helix Angle<br></b>The helix angle in the feed section will also have an optimum value for which the <br>solids conveying rate reaches a maximum. This is obvious if one realizes that a zero-<br>degree helix angle results in zero rate and a 90° helix angle also results in zero rate. <br>Thus, somewhere between zero and 90°, the solids conveying rate will reach a max-<br>imum. The optimum helix angle can be determined from:<br>
+ (8.67)<br>
+where:<br>
+ (8.68)<br>
+Equation 8.68 does not have a convenient analytical solution. Thus, the optimum <br>helix angle can be determined by using a numerical or graphical method. Again, the <br>coefficients of friction need to be known to determine the optimum helix angle. <br>Therefore, accurate prediction of the optimum helix angle is usually not possible. In <br>most extruder screws, the helix angle in the feed section ranges from 15 to 25°, <br>with the most common angle being 17.66°.<br>
+<hr>
+<A name=551></a> <br>
+8.2Optimizing for Output  <b>539</b><br>
+<b>8.2.3.3Effect of Number of Flights<br></b>The effect of the number of parallel flights in the feed section has already been dis-<br>cussed in Section 7.2.2. Increasing the number of parallel flights reduces the open <br>cross-sectional area of the channel (see Fig. 8.18) and increases the area of contact <br>between the solid bed and screw. Both of these factors have a negative impact on the <br>solids conveying performance, particularly when the helix angle is relatively small. <br>The volumetric efficiency shown in Fig. 8.18 is the open cross-sectional area relative <br>to the total area between the root of the screw and the barrel.<br>
+<b>8.2.3.4Effect of Flight Clearance<br></b>If the solid polymeric particles are compacted into a solid bed, there will be practi-<br>cally no leak flow through the flight clearance. In many cases, the flight clearance is <br>smaller than the particle size of the polymer. Thus, even if the polymer particles are <br>not fully compacted, the actual radial clearance will generally not be too critical for <br>solids conveying performance.<br>
+<b>8.2.3.5Effect of Flight Geometry<br></b>Dekker [3] studied the effect of various flight geometries on solids conveying per-<br>formance. He proposed that many extrusion instabilities might be due to internal <br>deformation of the solid bed. Internal deformation is more likely to occur when the <br>internal coefficient of friction of the polymer particles is low. Spherical particles <br>tend to have a lower internal coefficient of friction than non-spherical (e.g. cylindri-<br>cal) particles and are, therefore, more susceptible to internal solid bed deformation. <br>This may explain the often observed difference in extrusion behavior between <br>strand pelletized and die-face pelletized material.<br>Dekker [3] compared a trapezoidal flight geometry to a standard rectangular flight <br>geometry. He found that the trapezoidal geometry resulted in higher throughput <br>and more stable extrusion performance, particularly at a high screw speed. The <br>trapezoidal flight geometry is also better from a stress distribution point of view as <br>discussed in Section 8.1.2; see Fig. 8.4(a). It can be expected that the curved flight <br>geometry shown in Fig. 8.4(b) results in similar benefits in terms of solids convey-<br>ing as the trapezoidal flight geometry shown in Fig. 8.4(a); see also Section 7.2.2. <br>Another important benefit of these flight geometries is that the contact area between <br>the solid bed and the screw is reduced. This should have an additional positive effect <br>on solids conveying performance.<br>Spalding et al. [64] studied the effect on the flight flank radius on solids conveying <br>performance. They found that a larger flight flank radius could improve solids con-<br>veying when the level of pressure development in the solids conveying zone is high. <br>Spalding et al. recommend using a flight flank radius of about 1/4 the channel depth <br>of the feed section. However, the basis of this recommendation is not clear from the <br>
+<hr>
+<A name=552></a><b>540</b>  <b>8Extruder Screw Design</b><br>
+data presented in the paper. A flight flank radius of 0.25H is quite small relative to <br>the depth of the channel. A flight flank radius of 0.5 to 1.0H will generally result in <br>better performance than a radius of 0.25H because it provides a gentler transition <br>from the root of the screw to the flight flank. Also, it will create a greater extra normal <br>force at the barrel surface, thus increasing the driving force acting on the solid bed.<br>
+<b></b><br>
+ <b>8.3Optimizing for Power Consumption</b><br>
+In some cases, optimizing for power consumption may be more important than opti-<br>mizing the output. This is the case if the power consumption is excessive, causing <br>high stock temperatures and increasing the chance of degradation. Some polymers <br>have inherent properties that result in high power consumption. A typical example <br>is linear low density polyethylene, LLDPE, and the newer metallocene polyethyl-<br>enes. The cause of high power consumption is generally high polymer melt viscos-<br>ity. The problem can be particularly severe if the viscosity remains high at high <br>shear rates, i.e., if the polymer is not very shear thinning (large power law index). <br>The most difficult polymers with respect to power consumption are low melt index <br>polymers (fractional M.I.) with relatively large power law indici. Such polymers <br>require special attention to the screw design in order to minimize the power con-<br>sumption.<br>The screw geometry that will result in the least amount of power consumption can <br>be determined from the expressions of power consumption developed in Chapter 7. <br>In this section, attention will be focused on the melt-conveying zone. However, <br>other functional zones can be analyzed by the same procedure. Equations for power <br>consumption in melt conveying are given in Section 7.4.1.3. When optimizing for <br>power consumption, one should be concerned about the power consumption at a <br>certain level of throughput. Minimizing power consumption without considering <br>the throughput does not make much sense. The power consumption is optimum <br>when the ratio of power consumption to throughput reaches a minimum. This ratio <br>is the specific energy consumption, SEC. It is the mechanical energy expended per <br>mass of material and is usually expressed in kWhr/kg or hphr/lb. A high level of <br>SEC translates into a large amount of energy expended per mass of polymer; this <br>will result in large temperature increases in the polymer and possibly degradation. <br>Thus, the optimum screw geometry for power consumption is that geometry for <br>which the ratio of power consumption to output reaches a minimum.<br>
+<hr>
+<A name=553></a> <br>
+8.3Optimizing for Power Consumption  <b>541</b><br>
+<b>8.3.1Optimum Helix Angle</b><br>
+The optimum helix angle * for power consumption can be determined by setting:<br>
+ (8.69)<br>
+To evaluate the first derivative of power consumption Z with respect to helix angle , <br>the power consumption has to be written as an explicit function of the helix angle. <br>Using the equation in Section 7.4.1.3 the following expressions can be derived:<br>
+ (8.70)<br>
+where:<br>
+ (8.71a)<br>
+ (8.71b)<br>
+ (8.71c)<br>
+ (8.71d)<br>
+To evaluate the first derivative of output   with respect to the helix angle , the <br>output has to be written as an explicit function of the helix angle. The output can be <br>written as:<br>
+ (8.72)<br>
+where:<br>
+ (8.73a)<br>
+ (8.73b)<br>
+The optimum helix angle now has to be found from the following equation:<br>
+ (8.74)<br>
+<hr>
+<A name=554></a><b>542</b>  <b>8Extruder Screw Design</b><br>
+This equation does not have an obvious simple, analytical solution. One can find the <br>solution either graphically or by using a numerical technique, such as the Newton-<br>Raphson method. Figure 8.21 shows the optimum helix angle as a function of chan-<br>nel depth for a 50-mm extruder, running at 100 rpm.<br>
+40<br>
+P/l = 2E7 [Pa/m]<br>
+30<br>
+20<br>
+ angle [degrees]<br>
+P/l = 8E7 [Pa/m]<br>
+10<br>
+Optimum helix<br>
+0<br>
+<b> Figure 8.21</b> <br>
+0<br>
+2<br>
+4<br>
+6<br>
+8<br>
+10<br>
+Optimum helix angle versus channel <br>
+Channel depth [mm]<br>
+depth<br>
+The flight width is 0.1 D, the radial clearance 0.001 D, and the number of flights is <br>one. The power law index of the polymer melt is 0.5, the consistency index m = 1E4 <br>Pasn, and the melt density 0.8 g/cm3. It can be seen from Fig. 8.21 that the optimum <br>is relatively insensitive to the changes in the channel depth when the pressure gra-<br>dient is moderate. However, at relatively high pressure gradients, the optimum helix <br>angle depends strongly on the channel depth; at large channel depth values, the <br>optimum helix angle becomes quite small.<br>Figure 8.22 shows the actual specific energy consumption, SEC, at the optimum <br>helix angle as a function of channel depth.<br>The SEC at a small channel depth is quite high, but reduces with increasing channel <br>depth. When the pressure gradient is high, the SEC reaches a minimum and starts <br>to increase when the channel depth is further increased. Figure 8.22 thus illustrates <br>a graphical method to determine the optimum channel depth for power consumption <br>when both the channel depth and the helix angle are optimized simultaneously. It <br>can further be seen in Fig. 8.22 that the higher pressure gradient causes a substan-<br>tially higher specific energy consumption. In the normal channel depth range, H = <br>0.01 D--0.05 D, so the difference in SEC is about 50%. This will improve the mixing <br>efficiency of the metering section, but it will also cause more viscous heat genera-<br>tion in the melt and may overheat the polymer.<br>The optimum channel depth H* can be calculated from:<br>
+ (8.75)<br>
+<hr>
+<A name=555></a> <br>
+8.3Optimizing for Power Consumption  <b>543</b><br>
+0.20<br>
+]<br>
+0.15<br>
+r/kg<br>Wh<br>
+0.10<br>
+P/l = 8E7 [Pa/m]<br>
+0.05<br>
+P/l = 2E7 [Pa/m]<br>
+SEC at opt. helix angle [k<br>
+0<br>
+<b> Figure 8.22</b> <br>
+0<br>
+2<br>
+4<br>
+6<br>
+8<br>
+10<br>
+SEC at optimum helix angle versus <br>
+Channel depth [mm]<br>
+channel depth<br>
+This results in another lengthy expression similar to Eq. 8.74. This expression is <br>difficult to solve analytically and is generally solved numerically or graphically, as <br>shown in Fig. 8.22.<br>
+<b>8.3.2Effect of Flight Clearance</b><br>
+The effect of the radial clearance on the optimum helix angle is shown in Fig. 8.23 <br>for two values of the pressure gradient.<br>
+35<br>
+P/l = 2E7 [Pa/m]<br>
+30<br>
+ angle [degrees]<br>
+P/l = 8E7 [Pa/m]<br>
+Optimum helix<br>
+<b> Figure 8.23</b> <br>
+250<br>
+5E-5<br>
+10E-5<br>
+15E-5<br>
+20E-5<br>
+25E-5<br>
+Optimum helix angle versus radial <br>
+Radial clearance  [mm]<br>
+clearance<br>
+It can be seen that the radial clearance has relatively little effect on the optimum <br>helix angle. The corresponding SEC as a function of clearance is shown in Fig. 8.24.<br>It can be seen that there is an optimum value of the clearance * for which the SEC <br>reaches a minimum value. For the large pressure gradient, the optimum clearance <br>*  20E-5 m and for the small pressure gradient, * &gt; 30E-5 m. Thus, the standard <br>
+<hr>
+<A name=556></a><b>544</b>  <b>8Extruder Screw Design</b><br>
+radial clearance is not necessarily the best clearance in the metering section of <br>the extruder. However, as discussed in Section 8.2.2, the radial clearance in the <br>plasticating zone should be as small as possible to enhance the melting capacity. <br>This indicates that a varying clearance along the length of the extruder may not be <br>a bad idea. This can be achieved by varying the barrel inside diameter and/or the <br>screw outside diameter. Unfortunately, wear usually occurs right at a location where <br>it should not happen: in the compression section. If wear would occur in the meter-<br>ing section, it could actually have a beneficial effect.<br>
+0.09<br>
+P/l = 8E7 [Pa/m]<br>
+0.08<br>
+P/l = 2E7 [Pa/m]<br>
+0.07<br>
+ angle [degrees]<br>
+0.06<br>
+Optimum helix<br>
+<b> Figure 8.24</b> <br>
+0.05<br>
+5E-5<br>
+10E-5<br>
+15E-5<br>
+20E-5<br>
+25E-5<br>
+30E-5<br>
+SEC at optimum helix angle versus <br>
+Radial clearance [mm]<br>
+radial clearance<br>
+The optimum radial clearance can be determined from:<br>
+ (8.76)<br>
+Again, the resulting equation cannot be easily solved analytically. Thus, the opti-<br>mum clearance can be found by solving Eq. 8.76 numerically or graphically.<br>
+<b>8.3.3Effect of Flight Width</b><br>
+If the various contributions to power consumption are carefully examined, it can be <br>seen that a substantial portion is consumed in the clearance between the flight tip <br>and the barrel; see, for instance, Eq. 8.71(d). The power consumption in the clear-<br>ance is inversely proportional to the radial clearance and directly proportional to the <br>total flight width pw. However, the viscosity in the clearance will generally be lower <br>than the viscosity in the channel since the polymer melt is pseudo-plastic. Figure <br>8.25 shows how the ratio of power consumption in the clearance Zcl to the total <br>
+power consumption Zt depends on the ratio of flight width w to channel width plus <br>
+flight width W + w for a polymer with a power law index n = 0.5.<br>
+<hr>
+<A name=557></a> <br>
+8.3Optimizing for Power Consumption  <b>545</b><br>
+Figure 8.25 shows that the relative contribution of the power consumption in the <br>clearance increases strongly when the flight width increases. A typical ratio of w/<br>(W + w) is about 0.1; in the example shown in Fig. 8.25 this corresponds to a power <br>consumption in the clearance of about 40% of the total power consumption! The power <br>consumed in the clearance does not serve any useful purpose. It does not aid in trans-<br>porting the polymer forward, but causes a viscous heating of the polymer. Therefore, <br>one would like to make the flight width as narrow as possible to reduce the power <br>consumption in the clearance. The power consumption in the clearance will be more <br>pronounced when the material is more Newtonian in flow behavior, i.e., less shear-<br>thinning. This is shown in Fig. 8.26, where Zcl/Zt is plotted against the power law <br>
+index of the polymer melt. The viscosity in the clearance is determined from:<br>
+ (8.77)<br>
+1.0<br>
+t/Z 0.8<br>
+Z cl<br>
+0.6<br>
+0.4<br>
+r clearance / total power,  0.2<br>
+we<br>Po<br>
+0 0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<b> Figure 8.25</b> <br>
+Relative flight width, w/(w+W)<br>
+Zcl /Ztotal versus w/(W+w)<br>
+The viscosity in the channel is determined from:<br>
+ (8.78)<br>
+The results shown in Figs. 8.25 and 8.26 are for a standard radial clearance of <br>0.001 D.<br>The contribution of the power consumption in the clearance rises dramatically when <br>the power law index increases. When the fluid is Newtonian, around 80% of the total <br>power is consumed in the clearance! Thus, problems with excessive power consump-<br>
+<hr>
+<A name=558></a><b>546</b>  <b>8Extruder Screw Design</b><br>
+tion are more likely to occur when the material of a certain melt index is less shear <br>thinning. This is the main reason behind the extrusion problems encountered with <br>materials such as linear low-density polyethylene, LLDPE [4, 5], and metallocenes.<br>
+1.0<br>
+t/Z 0.8<br>
+Z cl<br>r, <br>we 0.6<br>
+w=0.10D<br>
+0.4<br>
+w=0.05D<br>
+r clearance / total po 0.2<br>
+we<br>Po<br>
+0<br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<b> Figure 8.26</b> <br>
+Power law index<br>
+Zcl/Ztotal versus the power law index<br>
+The most logical way to reduce power consumption in the metering section is to <br>reduce the ratio w/(W + w). This can be achieved in two ways. One is to increase the <br>channel width W by increasing the helix angle, as discussed in Section 8.3.1. The <br>other approach is to reduce the flight width itself. The combination of increasing the <br>helix angle and decreasing the flight width is obviously most effective. This allows <br>a reduction of the w/(W + w) ratio from a typical value of 0.1 down to around 0.03. <br>The minimum flight width is not determined by functional considerations but by <br>mechanical considerations. Functionally, one would like the flight width to be almost <br>infinitely thin. However, there must obviously be sufficient mechanical strength in <br>the flight to withstand the forces acting on it. These mechanical considerations are <br>discussed in detail in Section 8.1.2. The flight width in the metering section wm can <br>
+be considerably narrower than the flight width in the feed section wf because the <br>
+flight height or channel depth is much smaller in the metering section. In order to <br>keep the mechanical stresses in the flight approximately the same, the following <br>rule can be used to determine the flight width in the metering section.<br>
+ (8.79)<br>
+Considering that the flight width in the feed section is generally about 0.1 D, Eq. <br>8.79 can be written as:<br>
+<hr>
+<A name=559></a> <br>
+8.3Optimizing for Power Consumption  <b>547</b><br>
+ (8.80)<br>
+where Xc is the channel depth ratio Hf/Hm.<br>Based on these considerations, a new screw design was recently developed to reduce <br>power consumption in materials like LLDPE; the screw is often referred to as the LL-<br>screw [4, 5].<br>Figure 8.27 shows a standard screw geometry and the LL-screw geometry. Both <br>screws have a 38 mm (1.5 in) diameter and a 24 L/D ratio.<br>
+Standard Extruder Screw<br>
+H<br>
+D=38<br>
+feed section 5D<br>
+compression section 9D<br>
+metering section 10D<br>
+w=5 mm, H=6 mm,   = 17.7°<br>
+w=5 mm, H=2 mm,  = 17.7°<br>
+LL-Extruder Screw<br>
+H<br>
+D=38<br>
+feed section 6D<br>
+compression section 8D<br>
+metering section 10D<br>
+w=4 mm, H=6 mm,    <br>
+ =22.5°  <br>
+  <br>
+w=2.2 mm, H=2 mm,  = 27. 5°<br>
+<b>Figure 8.27</b>The LL-extruder screw versus a standard extruder screw<br>
+Essentially, the two screws differ in the helix angle and the flight width. Thus, the <br>differences in performance between the two screws can be attributed solely to these <br>two geometrical factors. Figure 8.28 shows the predicted output versus hp curves <br>for the two screws shown in Fig. 8.27.<br>
+Extruder output [lbs/hr]<br>
+0<br>
+10<br>
+20<br>
+30<br>
+4<br>
+3<br>
+Standard screw<br>
+2<br>
+r meter section [hp]<br>we<br>
+LL-screw<br>
+1<br>
+P<br>redicted po<br>
+<b> Figure 8.28</b> <br>
+0 0<br>
+5<br>
+10<br>
+Predicted output versus power <br>
+Extruder output [kg/hr]<br>
+consumption<br>
+<hr>
+<A name=560></a><b>548</b>  <b>8Extruder Screw Design</b><br>
+Figure 8.29 shows the actual output versus hp curves for the same two screws.<br>It is clear that the optimized geometry of the LL-screw results in considerably lower <br>power consumption compared to the standard geometry. The drop in power con-<br>sumption is around 35 to 40%! It is also interesting to note that the predicted power <br>consumption, Fig. 8.28, agrees quite well with the actual power consumption, Fig. <br>8.29. The difference in the ordinates is caused by the fact that Fig. 8.28 is predicted <br>power for the metering section only, while Fig. 8.29 is actual total power consump-<br>tion. A patent was issued on the LL-screw [65]; Migrandy Corporation obtained a <br>license to supply extruder screws using this technology.<br>
+Extruder output [lbs/hr]<br>
+0<br>
+10<br>
+20<br>
+30<br>
+8<br>
+6<br>
+Standard screw<br>
+4<br>
+LL-screw<br>
+2<br>
+easured total power [hp]<br>M<br>
+<b> Figure 8.29</b> <br>
+0 0<br>
+5<br>
+10<br>
+Actual output versus power <br>
+Extruder output [kg/hr]<br>
+ <br>consumption<br>
+It should be noted that the benefits of a reduced flight width and increased helix <br>angle are valid for the plasticating zone as well. The power consumption in the <br> <br>plasticating zone of the extruder is also reduced when the flight width is reduced. <br>In Section 8.2.2, it was discussed that increasing the helix angle improves melting <br>performance. Thus, the combination of increased helix angle and reduced flight <br>width should have a beneficial effect not only on the melt conveying zone, but also <br>on the melting zone of the extruder.<br>
+<b></b><br>
+ <b>8.4Single-Flighted Extruder Screws</b><br>
+In the previous sections of this chapter, screw design was analyzed by functional <br>performance. By using the extrusion theory developed in Chapter 7, it was shown <br>how the screw design can be determined quantitatively for optimum performance. <br>In this section, screw design will be approached from another angle. Screw designs <br>in use today will be described and their advantages and disadvantages will be dis-<br>cussed and analyzed.<br>
+<hr>
+<A name=561></a> <br>
+8.4Single-Flighted Extruder Screws  <b>549</b><br>
+<b>8.4.1The Standard Extruder Screw</b><br>
+In many discussions on extrusion, reference is made to a so-called standard or <br> <br>conventional extruder screw. In order to define this term more quantitatively, the <br>general characteristics of the standard extruder will be listed; see also Fig. 8.30:<br>
+ <br>
+ Total length 20­30 D<br>
+ <br>
+ Length of feed section 4­8 D<br>
+ <br>
+ Length of metering section 6­10 D<br>
+ <br>
+ Number of parallel flights 1<br>
+ <br>
+ Flight pitch 1 D (helix angle 17.66°)<br>
+ <br>
+ Flight width 0.1 D<br>
+ <br>
+ Channel depth in feed section 0.15­0.20 D<br>
+ <br>
+ Channel depth ratio 2­4<br>
+Feed section<br>
+Compression<br>
+Metering section<br>
+<b> Figure 8.30</b> <br>
+The standard extruder screw<br>
+These dimensions are approximate, but it is interesting that the majority of the <br>extruder screws in use today have the general characteristics listed above. For pro-<br>file extrusion of PA, PC, and PBTB, Brinkschroeder and Johannaber [46] recommend <br>a channel depth in the feed section of Hf  0.11 (D + 25) and a channel depth in <br>
+the metering section of Hm  0.04 (D + 25), where channel depth H and diameter D <br>
+are expressed in mm. Based on these guidelines, the geometry of a standard extruder <br>screw can be determined easily.<br>Based on the design methodology developed in Sections 8.2 and 8.3, it should be <br>clear that the standard screw design is by no means an optimum screw design. It <br>has developed over the last several decades mostly in an empirical fashion and <br>works reasonably well with many polymers. However, significant improvements in <br>performance can be made by functional optimization using extrusion theory. In this <br>light, it is somewhat surprising that the standard extruder screw is still so popular <br>today. It probably indicates a lack of awareness of the implications of extrusion <br>theo ry on screw design and the improvements that can be realized from functional <br>optimization of the screw geometry. Another interesting note is that several manu-<br>facturers of extruder screws claim to use sophisticated computer programs to opti-<br>mize the screw geometry, but often still end up with a standard square pitch screw. <br>It can be shown from an elementary analysis that the square pitch geometry is not <br>optimum for melting or melt conveying. Thus, if the result of the screw optimization <br>by computer is a square pitch geometry, this indicates that either the computer pro-<br>gram is incorrect or the person using the program is not using it correctly.<br>
+<hr>
+<A name=562></a><IMG src="Polymer Extrusion by Chris Rauwendaal-562_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-562_2.jpg"><br>
+<b>550</b>  <b>8Extruder Screw Design</b><br>
+<b>8.4.2Modifications of the Standard Extruder Screw</b><br>
+There are a large number of modifications of the standard extruder screw in use <br>today. It will not be possible to mention all of them, but an effort will be made to <br>discuss the more significant ones. Figure 8.31 shows the standard screw with an <br>additional flight in the feed section.<br>
+<b>Figure 8.31</b>Standard screw with additional flight in the feed section<br>
+The additional flight is intended to smooth out the pressure fluctuation caused by <br>the flight interrupting the in-flow of material from the feed hopper every revolution <br>of the screw. An additional benefit of the double-flighted geometry is that the forces <br>acting on the screw are balanced; thus, screw deflection is less likely to occur. On <br>the negative side, the additional flight reduces the open cross-sectional channel <br>area and increases the contact area between solid bed and screw. Thus, pressure <br>surges may be reduced, but the actual solids conveying rate will be reduced as well. <br>As a result, a double-flighted feed section in smooth bore extruders often results in <br>reduced performance.<br>Figure 8.32(a) shows a variable pitch extruder screw.<br>
+<b>Figure 8.32(a)</b>Variable pitch extruder screw with increasing pitch<br>
+The varying pitch allows the use of the locally optimum helix angle, i.e., optimum <br>helix angle for solids conveying in the feed section and optimum helix angle for melt <br>conveying in the metering section of the screw. This design is covered by a U.S. pat-<br>ent [6] and is described in a 1980 ANTEC paper [7]. Figure 8.32(b) shows a variable <br>pitch extruder screw as often used for rubber extrusion; see also Section 2.1.4.<br>
+<b> Figure  8.32(b)</b> <br>
+Variable pitch extruder screw <br>
+with reducing pitch<br>
+In this design, the pitch decreases with axial distance as opposed to the screw <br>shown in Fig. 8.32(a). The reducing pitch causes a lateral compression of the mate-<br>rial in the screw channel; as a result, the normal compression from the reducing <br>channel depth can be reduced or eliminated altogether. In fact, many of these vari-<br>able reducing pitch screws maintain the same channel depth along the entire length <br>of the screw.<br>
+<hr>
+<A name=563></a><IMG src="Polymer Extrusion by Chris Rauwendaal-563_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-563_2.jpg"><br>
+ <br>
+8.4Single-Flighted Extruder Screws  <b>551</b><br>
+It should be noted that the variable reducing pitch screw is not a high-performance <br>screw. It is designed primarily to exert minimal shear to the polymer; the L/D ratio <br>is generally quite short, about 10. This screw has been used extensively for rubber <br>extrusion. A smaller than square pitch flight geometry can be beneficial for highly <br>shear thinning polymers. When the power law index is less than 0.2 a smaller than <br>square pitch flight geometry will actually improve melt conveying; see Eq. 8.54(d).<br>A major supplier of LLDPE used to recommend a variable reducing pitch (VRP) <br>screw for extrusion with LLDPE [61]. Considering the approach developed in Sec-<br>tion 8.3, this screw design would seem inappropriate for LLDPE. As discussed in <br>Section 8.3, power consumption can be reduced by increasing the helix angle and <br>flight clearance and by reducing the flight width. The VRP screw recommended for <br>LLDPE does not reduce the flight width and reduces the helix angle. This combina-<br>tion of screw design parameters results in increased power consumption instead of <br>reduced power consumption.<br>The reason, however, that this VRP screw works is that the clearance between flights <br>and barrel is substantially larger than the normal design clearance--about double! <br>This design feature is not much emphasized, however it is the key to the performance <br>of the VRP screw for LLDPE because the power consumption in the flight clearance <br>plays such an overriding role in LLDPE extrusion. Based on the arguments developed <br>in Section 8.3, it is clear that a variable increasing pitch (VIP) screw with a larger <br>flight clearance will be significantly better than the VRP screw. A disadvantage of the <br>larger clearance is reduced melting capacity and reduced heat exchange between the <br>polymer melt and barrel. As a result, the VRP screw with increased flight clearance <br>may not be suitable for extrusion of polymers other than LLDPE.<br>Figure 8.33 shows an extruder screw without a metering section; the so-called zero-<br>meter screw [8].<br>
+<b>Figure 8.33</b>Zero-meter extruder screw<br>
+This screw is more appropriate for a plasticating unit of an injection molding <br>machine. The zero-meter screw is used to reduce the temperature build-up in the <br>material by deepening the depth of the channel in the melt conveying zone of the <br>extruder. The obvious drawback is that the pressure generating capability of the <br>screw will be adversely affected, but this is not a major concern in injection molding <br>applications. In other applications, however, the approach outlined in Section 8.3 is <br>recommended. An extension of the zero-meter screw is the zero-feed zero-meter <br>screw shown in Fig. 8.34.<br>
+<hr>
+<A name=564></a><IMG src="Polymer Extrusion by Chris Rauwendaal-564_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-564_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-564_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-564_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-564_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-564_6.jpg"><br>
+<b>552</b>  <b>8Extruder Screw Design</b><br>
+<b>Figure 8.34</b>Zero-feed zero-meter extruder screw<br>
+This screw essentially consists of only a compression section. This allows a very <br>gradual compression of the material. The screw has been in commercial use for <br>many years and has been successfully used with many polymers, in particular <br>nylon.<br>The exact opposite of the zero-feed zero-meter screw is the very rapid compression <br>screw, shown in Fig. 8.35.<br>
+<b>Figure 8.35</b>Rapid compression screw<br>
+The length of the compression section is generally less than 1 D in these screws. <br>Unfortunately, this screw is often referred to as a nylon screw. This is unfortunate <br>because it implies that nylon should be extruded on a rapid compression screw. <br>However, this is a major misconception in screw design and the success of the zero-<br>feed zero-meter screw with nylon should make that quite clear. Nylon has a rela-<br>tively narrow melting range and turns into a relatively low viscosity melt quite read-<br>ily. However, this does not mean that the compression should be very rapid. The <br>maximum compression can be determined from Eq. 8.64. The relative width of the <br>melting range of the polymer is totally immaterial to the determination of the maxi-<br>mum compression of the extruder screw. The low melt viscosity of nylon will reduce <br>the melting rate, and it indicates that a gradual compression screw will be much <br>more appropriate for nylon than a rapid compression screw. This was conclusively <br>demonstrated as early as 1963 by Bonner [9], who found that the gradual compres-<br>sion screw reduced air entrapment, reduced pressure and output fluctuations, and <br>improved extruder quality.<br>The zero-meter screw is used to reduce the viscous heat generation (power con-<br>sumption) in the melt conveying zone of the extruder by having a relatively deep <br>channel in this portion of the screw, with the depth reducing linearly with distance. <br>Another similar approach is the decompression screw shown in Fig. 8.36.<br>
+<b>Figure 8.36</b>Decompression screw<br>
+<hr>
+<A name=565></a> <br>
+8.5Devolatilizing Extruder Screws  <b>553</b><br>
+The final portion of the screw has a deep channel section following a decompression <br>section. The channel depth is constant over the last screw section. Again, the deeper <br>channel in the final screw section will reduce the pressure generating capability of <br>the screw. A more effective power reduction can be obtained by not only changing <br>the channel depth, but the channel depth, helix angle, flight width, and radial clear-<br>ance in an optimum fashion as discussed in Section 8.3.<br>
+<b></b><br>
+ <b>8.5Devolatilizing Extruder Screws</b><br>
+Devolatilizing extruder screws are used to extract volatiles from the polymer in a <br>continuous fashion. Such extruders have one or more vent ports along the length of <br>the extruder through which volatiles escape. Some of the applications of vented <br>extruders are:<br>
+ <br>
+ Removal of monomers and oligomers in the production of polymers (e.g., PS, <br>
+HDPE, PP).<br>
+ <br>
+ Removal of reaction products of condensation polymerization (e.g., water, metha-<br>
+nol) and oligomers from nylon and polyesters.<br>
+ <br>
+ Removal of air with filled polymers, particularly with glass fiber reinforced poly-<br>
+mers.<br>
+ <br>
+ Removal of residual carrier fluid in emulsion and suspension polymerization (e.g., <br>
+PS, PVC).<br>
+ <br>
+ Removal of water from hygroscopic polymers (e.g., ABS, PMMA, PA, PC, SAN, CA, <br>
+PU, PPO, polysulfone); all polymer particles can have surface moisture left from <br>underwater pelletizing or surface condensation from storage at varying tempera-<br>tures and relative humidity.<br>
+ <br>
+ Removal of solvent and unreacted monomers in solution polymerization (e.g., <br>
+HDPE).<br>
+ <br>
+ Removal of volatile components in compounding of polymers with additives and <br>
+other ingredients.<br>
+Removal of water from hygroscopic polymers is a common use of vented extruders. <br>Most polymers require less than 0.2% moisture in order to properly extrude. In some <br>polymers, this percentage is considerably lower, e.g., PMMA &lt; 0.1%, ABS &lt; 0.1%, <br>CA &lt; 0.05%, PBTB &lt; 0.05%, and PC &lt; 0.02%. Many polymers have an equi librium <br>moisture content at room temperature and 50% R.H. (relative humidity) that is con-<br>siderably higher than the maximum allowable percentage moisture content for <br>extrusion. Some values of the equilibrium moisture content of hygroscopic poly-<br>mers [10] are: ABS 1.5%; PMMA 0.8%; PBTP 0.2%; PC 0.2%; and PA 3%. Such poly-<br>
+<hr>
+<A name=566></a><b>554</b>  <b>8Extruder Screw Design</b><br>
+mers require significant drying or extrusion devolatilization to manufacture good <br>products. In many cases, extrusion devolatilization is preferred over drying.<br>Conventional two-stage extruders can generally reduce the level of volatiles only a <br>fraction of one percent. For example, for PP/xylene with an initial solvent concentra-<br>tion of 0.3 to 1.0%, the amount of solvent removed by single vent extrusion is about <br>50%.<br>
+<b>8.5.1Functional Design Considerations</b><br>
+Figure 8.37 shows a typical two-stage devolatilizing extruder screw.<br>The screw consists of at least five distinct geometrical sections. The first three sec-<br>tions, feed, compression, and metering, are the same as on a conventional screw. <br>After the metering section there is a rapid decompression followed by the extraction <br>section, which, in turn, is followed by a rapid compression and a pump section. Two <br>important functional requirements for good devolatilization are zero pressure in the <br>polymer under the vent port and completely molten polymer under the vent port.<br>
+D<br>
+feed section<br>
+compression section<br>
+metering<br>
+extraction<br>
+pump section<br>
+decompression<br>
+compression<br>
+<b>Figure 8.37</b>Typical two-stage devolatizing extruder screw<br>
+The requirement for zero pressure is made to avoid vent flow, i.e., polymer melt <br>escaping through the vent port. The complete fluxing requirement has several rea-<br>sons. If the polymer is not completely molten in the metering section, there may not <br>be a good seal between the vent port and the feed opening. This will limit the amount <br>of vacuum that can be applied at the vent port. A good vacuum is generally quite <br>important in order to obtain effective devolatilization. Another reason for the com-<br>plete melting requirement has to do with diffusion coefficients. The devolatilization <br>process in extruders is often controlled by diffusion [2]: see also Section 7.6.<br>Diffusion coefficients are very much temperature dependent. When the polymer is <br>below the melting point, diffusion generally occurs at an extremely low rate. The <br>polymer, therefore, should be above the melting point to increase the rate of diffu-<br>sion and with it the devolatilization efficiency. Even when the polymer is in the <br>molten state, the diffusion coefficients can often be increased substantially by <br>increasing the temperature of the polymer melt [11]. Further, when the polymer is <br>in the molten state, surface renewal is possible. This greatly enhances the devola-<br>tilization process. The extent of surface renewal is a strong function of the screw <br>
+<hr>
+<A name=567></a> <br>
+8.5Devolatilizing Extruder Screws  <b>555</b><br>
+design; a multi-flighted, large pitch extraction section will be beneficial to the devo-<br>latilization efficiency. Thus, for the highest devolatilization effectiveness, the poly-<br>mer should be completely molten and at relatively high temperature when it reaches <br>the extrusion section. The complete melting requirement can be worked out by the <br>procedure developed in Section 8.2.2.<br>The requirement for zero pressure can be fulfilled by ensuring that the channel in <br>the extraction section is only partially filled with polymer. There is no chance of <br>pressure build-up, at least in the down-channel direction, when the screw channel is <br>not fully filled. In order to achieve this partial fill, the depth of the extraction section <br>has to be considerably larger than the depth of the metering section, usually at least <br>three times larger, and the transport capacity of the pump section must be larger <br>than the transport capacity of the metering section. In other words, one has to make <br>sure that the polymer can be transported away from the vent port at a rate at least <br>as high as the rate with which it can be supplied to the vent section. If the transport <br>capacity of the pump section is insufficient, the polymer melt will back up in the <br>pump section and eventually escape through the vent port.<br>If the flight pitch is constant and the polymer melt viscosity can be described by <br>Eq. 8.78, the maximum diehead pressure for effective devolatilization can be written <br>as [2]:<br>
+ (8.81)<br>
+This equation was derived by assuming a zero pressure gradient in the metering <br>section and by using the Newtonian throughput-pressure relationship, Eq. 7.198. <br>The optimum channel depth in the pump section H*p can be obtained by setting:<br>
+ (8.82)<br>
+This results in the following expression for the optimum channel depth in the pump <br>section:<br>
+ (8.83)<br>
+The ratio of depth in the pump section to depth in the metering section is often <br>referred to as pump ratio Xp. The optimum pump ratio H*p according to Eq. 8.83 is <br>
+only a function of the power law index; this is shown in Fig. 8.38.<br>
+<hr>
+<A name=568></a><b>556</b>  <b>8Extruder Screw Design</b><br>
+2.0<br>
+1.9<br>
+atio<br>pr 1.8<br>um<br>mp<br>
+1.7<br>
+Optimu<br>
+1.6<br>
+1.5<br>
+<b> Figure 8.38</b> <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Optimum pump ratio versus power <br>
+Power law index<br>
+law index<br>
+The pump ratio should increase when the power law index decreases. The practical <br>lower limit is 1.5 and should be used for polymers with almost Newtonian flow char-<br>acteristics. The upper limit of the pump ratio is 2.0 and should be used for polymers <br>with very strong pseudo-plastic flow behavior.<br>Strictly speaking, one cannot insert a power law melt viscosity in the Newtonian <br>throughput-pressure relationship, as discussed earlier in Section 8.2.1. Thus, Eqs. <br>8.81 and 8.83 are not 100% accurate. However, they are much more accurate than <br>predictions based on pure Newtonian behavior, because the latter can cause sub-<br>stantial errors; see, for instance, Figs. 7.62 through 7.66. The dimensionless maxi-<br>mum pressure is shown as a function of the pump ratio Xp in Fig. 8.39.<br>The dimensionless maximum pressure is the actual maximum pressure divided by <br>the peak maximum pressure for the Newtonian case (Xp = 1.5). The dimensionless <br>
+maximum pressure can be expressed as:<br>
+ (8.84)<br>
+Figure 8.39 shows clearly that the peak maximum pressure is highest for the New-<br>tonian fluid and reduces steadily when the power law index reduces. This indicates <br>that the pressure generating capacity reduces as the fluid becomes more shear thin-<br>ning. At the same time, the optimum pump ratio increases with reducing power law <br>index. From Eq. 8.84, it can be seen that the dimensionless maximum pressure also <br>depends on the average shear rate in the screw channel. Figure 8.40 shows how the <br>dimensionless maximum pressure varies with the pump ratio at several values of <br>the average shear rate.<br>
+<hr>
+<A name=569></a> <br>
+8.5Devolatilizing Extruder Screws  <b>557</b><br>
+<b>Figure 8.39</b>Dimensionless maximum pressure versus pump ratio<br>
+It is evident from Fig. 8.40 that increases in shear rate have a strong effect on the <br>pressure generating capability. This can be seen clearly from the following relation-<br>ship:<br>
+ (8.85)<br>
+where N is the screw speed.<br>
+<b> Figure 8.40</b> <br>
+Dimensionless pressure versus <br>
+pump ratio for n = 0 .5 and <br>
+ <br>various shear rates<br>
+<hr>
+<A name=570></a><b>558</b>  <b>8Extruder Screw Design</b><br>
+In practical terms, this has important implications. It means that when the screw <br>speed is increased, the pressure generating capability increases less than propor-<br>tional to the screw speed.<br>Thus, if the output increases approximately proportional to the screw speed, at some <br>point the diehead pressure can exceed Pmax, causing vent flow. This will happen <br>
+more readily when the material is more shear thinning, i.e., when the power law <br>index is closer to zero.<br>
+<b>8.5.2Various Vented Extruder Screw Designs</b><br>
+There are many different designs of devolatilizing single screw extruders with <br>widely differing devolatilization capacity. Some of the more common ones will be <br>described and discussed next.<br>
+<b>8.5.2.1Conventional Vented Extruder Screw<br></b>The conventional vented extruder screw is shown in Fig. 8.37. In many cases, mix-<br>ing sections are incorporated into the metering section of the screw to improve the <br>homogeneity of the melt entering the extraction section. The volatiles travel with <br>the polymer up to the vent port. This type of venting is referred to as forward devol-<br>atilization. The length of the extraction zone is usually 2 to 5 D. The channel depth <br>in the extraction section is large, particularly if the polymer foams in the extraction <br>section. The channel depth in the extraction section can be as large as 0.4 D on large <br>diameter extruders, 0.3 D on smaller extruders. In order to achieve frequent sur-<br>face renewal, the extraction section is often designed with multiple flights; see <br>Fig. 8.41(a).<br>
+<b>Figure 8.41(a)</b>Extraction section with multiple flights<br>
+For the same reason, the helix angle is often increased from the conventional 17.66° <br>(square pitch) to as high as 40° [12].<br>As discussed in Section 8.5.1, the optimum pump ratio ranges from 1.5 to 2.0. In <br>practice, the pump ratio is often selected in the range from 1.2 to 1.4. However, <br>these lower pump ratios make the extruder more susceptible to vent flow. Some-<br>times vent flow is avoided by starve feeding the extruder. However, if starve feeding <br>
+<hr>
+<A name=571></a> <br>
+8.5Devolatilizing Extruder Screws  <b>559</b><br>
+is necessary to avoid vent flow, it indicates a deficiency in the screw design and it <br>might be better to modify the screw or design a new one. Carley [13] recommends <br>the use of rear valving to adjust the flow rate of the material entering the extraction <br>section. This is shown schematically in Fig. 8.41(b).<br>
+Adjustable<br>
+restriction<br>
+Vent port<br>
+<b>Figure 8.41(b)</b>Rear valving<br>
+Another approach was taken by Heidrich [56], who developed a vented extruder <br>with axial adjustment capability of the screw. This allows variation of a conical gap <br>at the end of the metering section; an example of this feature is shown in Fig. 8.45. <br>In vented extruders without external adjustment capability of the first stage resist-<br>ance, it is generally a good idea to incorporate a pressure consuming mixing ele-<br>ment. This improves melt homogeneity and reduces the pressure of the melt enter-<br>ing the extraction section.<br>
+<b>8.5.2.2Bypass Vented Extruder Screw<br></b>Another method to control the flow rate into the extraction section is to use a bypass <br>system. This system was proposed by Willert of Egan Machinery Company [14]. <br>Maddock and Matzuk [15] discussed the principles of the bypass vented extruder <br>and described actual experiments with different screw geometries. The bypass <br>vented extruder is shown schematically in Fig. 8.42.<br>
+Vent port<br>
+Adjustable restriction<br>
+<b>Figure 8.42</b>Bypass vented extruder system by Egan<br>
+<hr>
+<A name=572></a><IMG src="Polymer Extrusion by Chris Rauwendaal-572_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_48.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_49.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-572_50.jpg"><br>
+<b>560</b>  <b>8Extruder Screw Design</b><br>
+The polymer melt is forced from the metering section into a bypass flow channel by <br>incorporating a multi-flighted screw section with reversed pitch and shallow chan-<br>nels between the metering section and the extraction section. The bypass channel <br>has one or more adjustable restrictions to control the rate of flow into the extraction <br>section. The polymer flows from the bypass channel into the beginning of the extrac-<br>tion section. Maddock and Matzuk concluded that bypass venting allows a wider <br>range of operation, is less susceptible to instabilities, and is less operator-sensitive.<br>A different bypass venting system was described by Anders [16]; it is shown in <br>Fig. 8.43.<br>
+volatiles<br>
+<b> Figure 8.43</b> <br>
+Bypass vented extruder system <br>
+Adjustable restriction<br>
+by Berstorff<br>
+The melt flows from the metering section into a bypass channel, which runs essen-<br>tially parallel to the screw. The melt is forced into the bypass channel by a shallow <br>multi-flighted screw section of reversed pitch located just downstream of the bypass <br>channel inlet. The bypass channel extends about 2 D into the extraction section. The <br>polymer melt flows into the extraction section through a large number of holes. This <br>increases the surface area generation and improves the devolatilization efficiency. <br>The bypass channel has adjustable restrictions to control the flow of material to <br>the extraction section. The devolatilization capability is further enhanced by using <br>a multi-flighted, large pitch design in the extraction section. The bypass system <br>described by Anders is used in multi-stage vented extrusion.<br>
+<b>8.5.2.3Rearward Devolatilization<br></b>Rearward devolatilization is used on melt fed extruders. In rearward devolatiliza-<br>tion, the volatiles are extracted upstream of the feed opening of the extruder. This is <br>shown schematically in Fig. 8.44.<br>
+Rear vent<br>
+Feed<br>
+<b> Figure 8.44</b> <br>
+Rearward devolatilization<br>
+<hr>
+<A name=573></a><IMG src="Polymer Extrusion by Chris Rauwendaal-573_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-573_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-573_3.jpg"><br>
+ <br>
+8.5Devolatilizing Extruder Screws  <b>561</b><br>
+This machine is used in melt fed extrusion. The vent port is generally located at <br>least 1 D from the feed opening to avoid polymer melt getting to the vent port. To <br>improve the devolatilization capability, the melt is often forced into the extruder <br>through numerous small holes; examples of this feature are shown in Figs. 8.45 and <br>8.48.<br>In order to avoid plugging of the vent port, a feedback control mechanism can be <br>incorporated that controls the degree of fill of the extruder. This can be done by <br>measuring the pressure at the beginning of the metering section and by using this <br>reading to adjust the screw speed or feed rate to maintain the same pressure and <br>thus the same degree of fill.<br>
+<b>8.5.2.4Multi-Vent Devolatilization<br></b>Multi-vent devolatilization is used when large amounts of volatiles need to be re -<br>moved from the polymer. In a well-designed system, as much as 15% of volatiles can <br>be removed in one extrusion operation. Figure 8.45 shows a schematic of a multi-<br>stage system used for devolatilization of molten polystyrene.<br>This system incorporates rearward venting, stranding of the melt at the inlet to <br>increase surface area, variable gap before the section vent port by axial screw <br>adjustment, water injection, and bypass venting at the final vent port. Such a system <br>can reduce the monomer level from 15% down to as low as 0.1% in one operation. <br>Such devolatilization performance is quite comparable to that of twin screw devola-<br>tilization systems.<br>
+Feed<br>
+1st vent<br>
+2nd vent<br>
+3rd vent<br>
+Stripping<br>
+agent<br>
+Adjustable gap<br>
+<b>Figure 8.45</b>Efficient multi-stage degassing system<br>
+At high levels of volatiles, the initial devolatilization will be quite rapid because the <br>process will occur primarily by a foam devolatilization; see Section 5.4.2. When the <br>volatile level reduces, the devolatilization will occur by molecular diffusion, reduc-<br>ing the rate of devolatilization considerably. In this situation, the devolatilization <br>can be greatly improved by injecting a stripping agent into the polymer. The strip-<br>ping agent is generally introduced in or right before a mixing section. This causes <br>foaming of the polymer at the extraction section, resulting in much improved devol-<br>atilization. A common stripping agent is water; also used are low boiling organics or <br>
+<hr>
+<A name=574></a><b>562</b>  <b>8Extruder Screw Design</b><br>
+nitrogen. In some cases, the volatile and the stripping agent can form an azeotropic <br>mixture, which boils at a lower temperature than either of the components. An ex -<br>ample is styrene monomers and water [17].<br>A more conventional three-stage extruder screw is shown in Fig. 8.46.<br>
+<b>Figure 8.46</b>Conventional three-stage extruder screw<br>
+This system has two vent ports and is used in applications where a single vent port <br>cannot remove a sufficient amount of volatiles. It is used, for instance, in ABS extru-<br>sion as described by Brozenick and Kruder [18] and with acrylic, polycarbonate, <br>polypropylene, etc., as described by Nichols, Kruder, and Ridenour [19]. This system <br>can remove moisture levels as high as 5 to 7%.<br>
+<b>8.5.2.5Cascade Devolatilization<br></b>In many polymer devolatilization systems, two extruders are arranged in a cascade <br>arrangement. The first extruder is primarily used for solids conveying, plasticating, <br>and mixing. The section extruder is primarily used for melt conveying, i.e., for <br>pumping. The first extruder is often a multi-screw extruder; the second extruder is <br>generally a single screw extruder. Figure 8.47 shows a planetary gear extruder feed-<br>ing a single screw extruder.<br>The venting takes place between the first and second extruder. The system shown in <br>Fig. 8.47 is often used for devolatilization of PVC. The devolatilization effectiveness <br>can be improved by stranding the polymer melt as it enters the second extruder. <br>This is shown in Fig. 8.48.<br>The distance of the strand die to the second extruder is made reasonably long to <br>improve devolatilization. The distance is limited by the fact that the strands cannot <br>cool below the point where the intake of the second extruder is affected.<br>The major advantage of the cascade devolatilizing system is that the control of the <br>output of the first stage to the pressure generating capability of the second stage <br>is much better than in a single extruder devolatilization system. Obviously, the cost <br>will be higher, and the decision for one system or the other must be based on the <br>importance of improved flexibility and controllability.<br>
+<hr>
+<A name=575></a> <br>
+8.5Devolatilizing Extruder Screws  <b>563</b><br>
+<b>Figure 8.47</b>Cascade devolatilization with a planetary gear extruder feeding a single screw <br>
+extruder<br>
+<b> Figure 8.48</b> <br>
+Stranding for improved degassing<br>
+<b>8.5.2.6Venting through the Screw<br></b>An interesting development in devolatilizing extrusion was described by Bernhardt <br>in 1956 [20]. In this extruder, the volatiles are removed through the screw instead <br>of through a vent port in the barrel. The screw has a hollow core connecting with a <br>lateral hole in the extraction section of the screw; see Fig. 8.49(a).<br>The volatiles are withdrawn through a rotary union at the rear of the screw. This <br>venting was tested in practice on acrylic and was found to perform reliably for ex -<br>tensive periods of time. This process is also used for processing PET powder with <br>conventional two-stage screws. This concept has also been applied to barrier screws. <br>
+<hr>
+<A name=576></a><b>564</b>  <b>8Extruder Screw Design</b><br>
+Eastman Kodak Company received a patent (U.S. Patent 6,164,810) on a barrier <br>screw with a vent hole located within about two diameters from the end of the feed <br>section, between a main flight and a barrier flight. The vent hole is located such that <br>there is little chance of polymer plugging the vent hole. The volatiles are vented out <br>to the back of the screw through an axial bore. This screw design can be used to <br>process PET powder into film.<br>
+<b>Figure 8.49(a)</b>Venting through the screw<br>
+In the past barrier screws were used only with PET pellets because pellets are not <br>susceptible to air entrapment. Powders, however, are susceptible to air entrapment <br>and do not process well on conventional barrier screws. The internally vented bar-<br>rier screw developed by Eastman achieves higher throughputs of PET powder. This <br>screw design can also be used for other hygroscopic resins like ABS.<br>An obvious advantage of this approach is that venting can be done on an extruder <br>not equipped with a vent port in the barrel. An equally obvious disadvantage is that <br>plugging of the vent channel in the screw may cause a complete shut-down. The <br>plug may be removed by a blast of high-pressure air into the core of the screw. How-<br>ever, if this does not work, the screw has to be pulled and cleaned. Plugging of the <br>vent channel in the screw, however, may not be as much of a problem as one might <br>think. It should be remembered that the polymer has to adhere to the barrel in order <br>to move forward; however, it does not have to adhere to the screw. Thus, a vent port <br>in the barrel is much more likely to accumulate molten polymer than a vent port in <br>the screw, particularly if the vent port is located close to the trailing flight flank. <br>This type of venting, however, does not seem to have found widespread acceptance.<br>
+<b>8.5.2.7Venting through a Flighted Barrel<br></b>Kearney and Hold [62] proposed a new devolatilizer with helical flights in the barrel <br>and a smooth screw section (rotor); see Fig. 8.49(b). This device is called a rotating <br>drum devolatilizer (RDD).<br>The barrel has multiple helical flights as shown in Fig. 8.49(b). Each turn of the heli-<br>cal channels has an oblong opening following the helical path of the channel. The <br>volatiles are removed through these openings, which are located in the same angular <br>
+<hr>
+<A name=577></a> <br>
+8.5Devolatilizing Extruder Screws  <b>565</b><br>
+position. The vent openings of channels operating at the same vacuum level are cov-<br>ered by a single manifold. Upstream of the vent opening is a replaceable melt barrier, <br>which is used to provide a hydraulic seal between the various stages. The material <br>moves through the RDD by virtue of the contact between the polymer melt and the <br>rotor. Therefore, there is little tendency of the material to accumulate in the vent <br>port. This situation is similar to the conditions existing in venting through the screw <br>described in Section 8.5.2.6. The multiple flights in the housing provide for good <br>mixing and surface renewal, resulting in effective devolatilization. The rotating drum <br>devolatilizer can, in principle, be mounted on the end of an existing extruder. How-<br>ever, full exploitation of its potential benefits will probably require incorporation into <br>new machinery, specially designed to take advantage of the benefits of the RDD.<br>
+<b>Figure 8.49(b)</b>Rotating drum devolatilizer<br>
+<b>8.5.3Vent Port Configuration</b><br>
+As discussed in Section 8.5.2.6, the polymer has to adhere to the barrel in order to <br>be conveyed forward. This means that a vent port in the barrel is likely to pick up <br>molten polymer. It is almost inevitable, simply by the nature of the conveying pro-<br>cess in a screw extruder. For this reason, most vented single screw extruders tend to <br>have a gradual accumulation of polymer in the vent port, requiring periodic clean-<br>ing of the vent port in order to maintain devolatilization efficiency. In order to mini-<br>mize accumulation of polymer in the vent port, it is important that the shape of the <br>vent port be such that there is a minimal chance of material hanging up. For this <br>reason, the leading edge of the vent port is often undercut, with the undercut mak-<br>ing a small angle with the O.D. of the screw. This is shown in Fig. 8.50(a).<br>
+<hr>
+<A name=578></a><IMG src="Polymer Extrusion by Chris Rauwendaal-578_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_48.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_49.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_50.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_51.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_52.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_53.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_54.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_55.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_56.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_57.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_58.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_59.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_60.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_61.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_62.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_63.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_64.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_65.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_66.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_67.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_68.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_69.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_70.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_71.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_72.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_73.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_74.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_75.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-578_76.jpg"><br>
+<b>566</b>  <b>8Extruder Screw Design</b><br>
+<b>Figure 8.50(a)</b>Vent port geometry to avoid vent flow with undercut to minimize vent port <br>
+build-up<br>
+In addition to the undercut, the vent port is often offset from the vertical, again to <br>minimize the chance of polymer scraping off at the leading edge of the vent port. The <br>length of the vent port is usually 0.5 to 1.5 D and the width about 0.25 to 0.75 D.<br>Another vent port geometry is shown in Fig. 8.50(b).<br>
+<b>Figure 8.50(b)</b>Vent port geometry to avoid vent flow (b) with dual openings in the barrel liner<br>
+Again, this geometry minimizes the chance of polymer melt being scraped off at the <br>leading edge of the vent port.<br>Another interesting vent port design is shown in Fig. 8.50(c).<br>
+<b> Figure 8.50(c)</b> <br>
+Combined feed and vent port<br>
+In this design, the feed port is combined with the vent port. The volatiles are <br>removed through the annular space between the feed pipe and the feed port hous-<br>ing. The system is used in conjunction with a vacuum feed hopper as described by <br>
+<hr>
+<A name=579></a> <br>
+8.5Devolatilizing Extruder Screws  <b>567</b><br>
+Franzkoch [21]. Such a system allows devolatilization to occur in the feed hopper <br>section of the extruder. This type of devolatilization is successfully used with pow-<br>ders, particularly when the material in the hopper can be heated. The vacuum feed <br>hopper system and some of the problems associated with it are discussed in Section <br>3.4. Hopper devolatilization of pellets is generally unsuccessful because the larger <br>particle size reduces the surface area and thus the devolatilization effectiveness, as <br>discussed in Section 5.4.1.<br>A very simple vent port that minimizes the chance of build-up in the vent opening is <br>shown in Fig. 8.51(a).<br>
+<b> Figure 8.51(a)</b> <br>
+Tangential vent port geometry<br>
+The tangential vent port is easy to manufacture and eliminates the need to place a <br>deflector plug in the vent port. It can be beneficial to move the vent port from a verti-<br>cal position to a horizontal position or even a position where the opening is in the <br>downward direction. This can be useful when condensate forms inside the vent <br>opening because the condensate will flow away from the screw rather than into it. <br>This is shown in Fig. 8.51(b).<br>
+<b> Figure 8.51(b)</b> <br>
+Vent port with a downward orientation<br>
+<hr>
+<A name=580></a><b>568</b>  <b>8Extruder Screw Design</b><br>
+<b></b><br>
+ <b>8.6Multi-Flighted Extruder Screws</b><br>
+There are a large number of screw geometries using multiple flights. The additional <br>flights can be the same as the main flight; this geometry will be referred to as the <br>conventional multi-flighted screw. In other designs, the additional flight(s) is differ-<br>ent in geometry and function from the main flight; this geometry will be referred to <br>as the barrier flight screw geometry.<br>
+<b>8.6.1The Conventional Multi-Flighted Extruder Screw</b><br>
+The conventional multi-flighted extruder screw has a number of advantages and dis-<br>advantages. The basic geometry is shown in Fig. 8.52.<br>The multi-flighted screw geometry adversely affects the solids conveying and melt <br>conveying rate, as discussed in Chapter 7 and Sections 8.2.1 and 8.2.3. On the other <br>hand, however, the multi-flighted screw geometry can significantly improve melting <br>performance, as discussed in Section 8.2.2, provided the helix angle is sufficiently <br>large. The multi-flighted screw geometry adversely affects power consumption, as <br>discussed in Section 8.3, particularly when the polymer melt is low in melt index <br>and is relatively Newtonian in its flow behavior.<br>
+<b>Figure 8.52</b>Conventional multi-flighted extruder screw<br>
+An advantage of the double-flighted screw geometry is the symmetry of the screw <br>flights. This can reduce the tendency of screw deflection by abrupt changes in pres-<br>sure along the screw channel; this point was discussed in Section 8.1.3. Another <br>possible advantage of a double-flighted screw geometry in the feed section is a more <br>regular intake of material, as discussed in Section 8.4.2. Design considerations of <br>double-flighted extruder screws were discussed by Maddock [22] based on computer <br>simulations. He predicts higher melting rate and reduced power consumption. The <br>latter prediction seems incorrect and must have resulted from improper conside-<br>ration of the power consumption in the flight clearance. However, the prediction of <br>improved melting is correct and this seems to be the main benefit of a multi-flighted <br>screw geometry. Since solids conveying and melt conveying rates are adversely <br>affected by a multi-flighted geometry, it makes sense to incorporate multiple flights <br>only along a particular section of the screw. This should be the screw section where <br>melting will occur. Two possible screw designs are shown in Fig. 8.53.<br>
+<hr>
+<A name=581></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>569</b><br>
+<b>Figure 8.53</b>Multiple-flighted extruder screws for improved melting<br>
+<b>8.6.2Barrier Flight Extruder Screws</b><br>
+Barrier flight extruder screws have been around since the early 1960s. Presently, <br>barrier screws enjoy widespread popularity in the U.S. Every major U.S. extruder <br>manufacturer offers at least one type of barrier flight extruder screw. There are <br>various types of barrier screws and there is little agreement as to which type is bet-<br>ter than the other. Even the advantages and disadvantages of barrier screws com-<br>pared to regular single-flighted screws are not widely known or agreed upon. In this <br>section, a functional analysis will be made of barrier screws based on extrusion <br>theory. From this analysis, the advantages and disadvantages of barrier screws will <br>become clear, also indicating the preferred geometrical configuration for certain <br>applications.<br>The inventor of the barrier flight extruder screw is Maillefer, a pioneer in the field of <br>extrusion. He first applied for a patent in Switzerland on December 31, 1959 [23], <br>and later applied for patents in various other countries. Patents were granted, among <br>other countries, in Germany [24] and in England [25]. Maillefer filed a patent appli-<br>cation in the U.S. on December 20, 1960. However, Maillefer did not obtain a patent <br>in the U.S. because of a particular provision in U.S. patent law. Geyer from Uniroyal <br>filed a patent application for a barrier screw on April 5, 1961, several months after <br>Maillefer's filing date [26].<br>Geyer's patent application describes a barrier flight extruder screw almost identical <br>to the one invented by Maillefer. As a result, an interference procedure developed <br>which resulted in the granting of Geyer's patent and the rejection of Maillefer's <br>claim to the patent. This may seem rather surprising because Maillefer filed his <br> <br>patent application several months before Geyer did. The explanation is that Maillefer, <br>by U.S. patent law, was treated as a foreign national, while Geyer was treated as a <br>U.S. national. If Uniroyal could demonstrate that Geyer conceived of this invention <br>before Maillefer filed his application, which Uniroyal managed to do in court, Uni-<br>royal would be legally entitled to the patent. Maillefer, being considered a foreign <br>national, could not use the priority date based on the conception of the idea.<br>
+<hr>
+<A name=582></a><IMG src="Polymer Extrusion by Chris Rauwendaal-582_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-582_10.jpg"><br>
+<b>570</b>  <b>8Extruder Screw Design</b><br>
+This bit of history explains why seven years elapsed between the filing of Geyer's <br>patent and the date of issue (April 2, 1968). It is an interesting situation because <br>such an interference does not arise very often. It is also interesting because Uniroyal <br>has enforced its patent quite rigorously; various lawsuits have been filed as a result <br>of alleged infringement of the Geyer patent. Uniroyal issued licenses to several com-<br>panies, among others to the old Sterling Extruder Corporation, who used to sell the <br>screw under the name Sterlex High Performance Screw.<br>The principle of all barrier screws is very much the same. At the beginning of the <br>barrier section a barrier flight is introduced into the screw channel. The clearance <br>between the barrier flight and the barrel is generally larger than the clearance <br>between the main flight and the barrel. The barrier clearance is large enough so that <br>polymer melt can flow over the barrier, but it is too small for solid polymer particles <br>to flow over the barrier. As a result, the solid bed will be located at the active side of <br>the barrier flight and the polymer melt mostly at the passive side of the barrier <br>flight. Thus, the barrier flight causes a phase separation, confining the solid bed to <br>one side of the barrier flight while allowing the polymer melt to the other side. This <br>is illustrated in Fig. 8.54.<br>Thus, the barrier screw has a solids channel and a melt channel. In the down-chan-<br>nel direction, the solids channel reduces in cross-sectional area, while the melt <br>channel correspondingly increases in cross-sectional area. At the end of the barrier <br>section, the solids channel reduces to zero, while the melt channel starts to occupy <br>the full channel again.<br>
+<b>Figure 8.54</b>Phase separation in a barrier screw<br>
+This geometry ensures complete melting of the solids because the solids cannot <br>travel beyond the barrier section unless they are able to cross the barrier clearance. <br>This is only possible if the particle has reduced to a size that will allow rapid melt-<br>ing after any possible crossing of the barrier. Another benefit of the barrier design <br>is that all the polymer has to flow through the barrier clearance where it is briefly <br>subjected to relatively high levels of shear. This causes a certain amount of mixing, <br>similar to the mixing in a fluted mixing element; see Section 8.7.1.<br>
+<hr>
+<A name=583></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>571</b><br>
+By the nature of the barrier geometry, the solid bed is confined to a channel that is <br>considerably narrower than the total channel width. This has a possible benefit that <br>the solid bed is less likely to break up, although this has not been conclusively dem-<br>onstrated. On the other hand, it limits the space available to the solid bed. Thus, the <br>screw design will have to be tailored more carefully to the melting profile of the <br>solid bed. The solid bed is more likely to plug the solids channel. Melting must begin <br>considerably before the start of the barrier section in order to allow the introduction <br>of the barrier flight. If this were not the case, plugging would occur immediately.<br>In order to obtain maximum melting efficiency, the solids channel should be filled <br>from flank to flank with solid polymer with only a thin melt film between the solid <br>bed and the barrel. Obviously, this also represents a situation that can easily develop <br>into plugging because the melting has to match the reducing cross-section of the <br>solids channel. In practice, therefore, there is likely to be more polymer melt in the <br>solids channel than just in the melt film. The assumption of the solid bed occupying <br>the full width of the solids channel can be used to determine the maximum possible <br>melting rate. It should be realized, however, that this rate is not likely to be realized <br>in practice. The comparable melting performance of a regular compression screw <br>can be obtained by using the ideal compression A*1:<br>
+ (8.86)<br>
+This results in the following ideal axial melting length for a standard compression <br>screw; see Eqs. 7.116 and 8.55:<br>
+ (8.87a)<br>
+or as down-channel melting length:<br>
+ (8.87b)<br>
+where  is given by Eq. 8.91(a).<br>The following analysis of the melting performance of various barrier screws is based <br>on the analysis developed by Meijer and Ingen Housz [27]. This analysis provides a <br>clear and logical approach to the determination of the melting capacity as a function <br>of the barrier section geometry.<br>
+<hr>
+<A name=584></a><b>572</b>  <b>8Extruder Screw Design</b><br>
+<b>8.6.2.1The Maillefer Screw<br></b>The Maillefer extruder screw is shown in Fig. 8.55.<br>The main feature of this barrier screw is that the helix angle of the barrier flight is <br>larger than the helix angle of the main flight. As a result, there is a continuous <br>reduction in the width of the solids channel and a corresponding increase in the <br>width of the melt channel. This geometry allows a smooth and gradual change of the <br>solids channel as well as the melt channel. A drawback of this geometry is that the <br>solids channel reduces in width. This causes a corresponding reduction in melting <br>rate, since this is directly determined by the width of the solid bed.<br>
+<b> Figure 8.55</b> <br>
+The Maillefer screw<br>
+The melting performance of the Maillefer screw geometry can be analyzed by a <br>down-channel mass balance:<br>
+ (8.88)<br>
+where:<br>
+ (8.88a)<br>
+The right-hand term, the melting rate per unit down-channel distance, is given by <br>Eq. 7.109(c). In the analysis of the melting performance, a complication arises in <br>that the solid bed velocity cannot be assumed constant. This can be appreciated by <br>comparing the solid bed width profile of a standard screw with constant channel <br>depth to the channel width profile of the solids channel in a Maillefer screw. The <br>width of the solid bed Ws in a standard zero-compression screw as a function of <br>
+down-channel distance z can be written as:<br>
+ (8.89)<br>
+where W1 is the channel width and ZT the total melting length. This expression is <br>
+derived based on the assumption that the solid bed velocity vsz is constant along the <br>
+melting zone. This relationship is shown in Fig. 8.56.<br>At the beginning of melting, the width of the solid bed reduces quite rapidly. As <br>melting proceeds, the solid bed width continues to decline but the rate of change <br>reduces with distance. At the final stage of melting, the rate of reduction of solid bed <br>width approaches zero. This varying rate of change in the width of the solid bed is <br>due to the fact that the melting rate reduces as the width of the solid bed reduces.<br>
+<hr>
+<A name=585></a><IMG src="Polymer Extrusion by Chris Rauwendaal-585_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-585_11.jpg"><br>
+ <br>
+8.6Multi-Flighted Extruder Screws  <b>573</b><br>
+W1<br>
+sW<br>
+dth,<br>wi<br>ed<br>i<br>l<br>db<br>So<br>
+<b> Figure 8.56</b> <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Solid bed width profile with constant <br>
+Normalized down channel distance, z/Zt<br>
+solid bed velocity<br>
+The width of the solids channel of the Maillefer screw can be written as:<br>
+ (8.90)<br>
+where  is the difference in helix angle between the main flight and barrier flight <br>and ZT is the total length of the barrier section in the down-channel direction.<br>The relationship becomes clear from examination of a picture of the unwrapped <br>geometry of the Maillefer barrier section; this is shown in Fig. 8.57.<br>
+W1<br>
+Ws<br>
+ZT<br>
+<br>
+<b> Figure  8.57</b> <br>
+Unwrapped Maillefer barrier geometry<br>
+By comparing the channel width profile, Fig. 8.57, to the solid bed width profile in a <br>standard screw with constant solid bed velocity, Fig. 8.56, it is clear that the two <br>profiles are considerably different. If the solid bed is to occupy the full width of the <br>channel W1s in the Maillefer screw, the velocity of the solid bed will have to change <br>
+along the barrier section. This may require substantial deformations in the solid <br>bed, but it will be assumed that the solid bed is capable of undergoing the required <br>
+<hr>
+<A name=586></a><b>574</b>  <b>8Extruder Screw Design</b><br>
+deformations. In reality, the solid bed may not occupy the full width of the channel. <br>However, this is not important at this point since the objective of this exercise is to <br>determine the highest possible melting rate with the Maillefer screw geometry.<br>If it is assumed that the solid bed width equals the channel width W1s, Eq. 8.88 can <br>
+be written as:<br>
+ (8.91)<br>
+where:<br>
+ (8.91a)<br>
+This equation cannot be solved without making some simplifying assumptions. <br> <br>Meijer and Ingen Housz [27] solved this problem by initially taking the term vsz <br>
+constant and evaluating this term by its value at the beginning of melting, i.e., <br>vsz = vszo o. This results in the following expression:<br>
+ (8.92)<br>
+When z = ZT, the left-hand term becomes zero; for the right-hand term to be zero as <br>
+well, the melting length has to be:<br>
+ (8.93)<br>
+where Z*T is the shortest possible melt length in the down-channel direction for a <br>
+standard compression screw; see Eq. 8.87(b).<br>By substituting Eq. 8.93 into Eq. 8.92, an expression is obtained for the solid bed <br>velocity as a function of down-channel distance z:<br>
+ (8.94)<br>
+Equation 8.94 can now be inserted into Eq. 8.91 to obtain a more accurate solution <br>of Eq. 8.91. This can be done by writing 1 as a function of vsz, following Meijer and <br>
+Ingen Housz [27]:<br>
+ (8.95)<br>
+where E is the coefficient relating 1 to vsz.<br>
+<hr>
+<A name=587></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>575</b><br>
+By ignoring higher order terms of E, the solution becomes:<br>
+ (8.96)<br>
+This represents the shortest possible melting length with a Maillefer-type barrier <br>screw. A reasonable maximum value of E = 0.4; this results in the following melting <br>length:<br>
+ (8.97)<br>
+Thus, the best melting length in a Maillefer screw is about 30% longer than an ideal <br>compression screw. For comparison, it is interesting to note that a non-ideal com-<br>pression screw with a channel depth ratio of 4:1 has a total melting length of ZT = <br>
+3/2Z*T, as can be determined from Eq. 7.116. From a theoretical analysis of the <br>
+ <br>melting performance of a Maillefer screw, it can be concluded that the best melting <br>length is about 30% longer than an ideal compression screw and about 10% shorter <br>than a standard compression screw with a compression ratio of four. Considering <br>that the maximum melting performance of the Maillefer screw is only about 15% <br>better than a standard compression (4:1) screw, it can be concluded that the Mail-<br>lefer screw does not offer a significant benefit over a standard compression screw in <br>terms of melting performance, particularly since the actual melting performance of <br>the Maillefer screw will be less than the maximum melting performance.<br>The advantage of the Maillefer screw is primarily the physical separation of the melt <br>pool and the solid bed. As a result, there is less chance of formation of a melt film <br>between the solid bed and the screw and, therefore, there is less chance of solid bed <br>breakup. Thus, the melting process can occur in a more stable fashion but not nec-<br>essarily at a higher rate.<br>
+<b>8.6.2.2The Barr Screw<br></b>The Barr screw is shown in Fig. 8.58.<br>The initial part of the barrier section is the same as the Maillefer screw. However, <br>when the melt channel is sufficiently wide, the barrier flight starts to run parallel to <br>the main flight.<br>
+Conveying direction<br>
+Melt channel<br>
+Solids channel<br>
+<b>Figure 8.58</b>The Barr screw<br>
+<hr>
+<A name=588></a><b>576</b>  <b>8Extruder Screw Design</b><br>
+The cross-sectional area of the solids channel is then reduced by reducing the chan-<br>nel depth while at the same time the channel depth of the melt channel is increas-<br>ing. The advantage of this geometry is that the solids channel width is not continu-<br>ously reducing, as in the Maillefer geometry, but remains constant at a relatively <br>large value. This increases the solid-melt interfacial area and improves the melting <br>performance. It has a few drawbacks as well, however. Since the melt channel is <br> <br>narrow and quite deep in the latter part of the barrier section, the melt conveying <br>efficiency of the melt channel will be less than in the Maillefer screw. Also, at the <br>end of the barrier section the melt channel changes from a narrow, deep channel to <br>a wide, shallow channel over a short length. This rapid change in channel geometry <br>will not be conducive to stable flow conditions. These relatively abrupt changes in <br>channel geometry do not occur in the Maillefer screw.<br>The Barr screw was developed by Barr and Chung when they worked at the old <br> <br>Hartig Plastics Machinery Division of Midland-Ross Corporation. Barr applied for a <br>patent in August 1971, the patent was issued in October 1972 [28]. Hartig used to <br>sell barrier screws covered by this patent under the names "MC3" and MC4" screw. <br>Later when Barr and Chung left Hartig, Chung applied for a patent on a modified <br>Barr screw. The application was filed in July 1975 and the patent issued in January <br>1977 [29]. The modification consists primarily of a difference in the transition from <br>the final melt channel geometry to the metering section. Screws covered by this <br>later patent are sold by Robert Barr, Inc. under the names Barr II, Barr III, and Barr <br>ET screw. In 1982, Uniroyal brought suit against Robert Barr, Inc. for patent infringe-<br>ment. The suit was settled for an undisclosed amount of money.<br>A very similar barrier screw is described by Willert in a patent application dated <br>August 26, 1981 [30]. The difference in the barrier screw is that the main flight <br>becomes a barrier flight at the beginning of the barrier section, while at the same <br>time the main flight branches off at an angle and then starts to run parallel to the <br>barrier flight. The barrier screw design is shown in Fig. 8.59.<br>
+Conveying direction<br>
+Melt channel Solids channel<br>
+<b>Figure 8.59</b>The Lacher/Hsu/Willert barrier screw<br>
+Interestingly enough, Willert's description is essentially identical to the patent of <br>Hsu [32]. This patent was filed in January 1973 and issued January 1975. Screws <br>covered by Hsu's patent used to be sold under the name "Maxmelt Screw" by the <br>Plastics Machinery Division of Hoover Universal. Coincidentally, Hsu used to work <br>at Hartig at the time when the basic concepts of the Barr screw were being devel-<br>
+<hr>
+<A name=589></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>577</b><br>
+oped. Another patent on a barrier screw was obtained by Lacher [57] from NRM <br>Corporation in 1966. This geometry is also very similar to the one described by Hsu <br>[32] and Willert [30]. Both Maillefer and NRM became licensees of Uniroyal.<br>The melting performance of the Barr screw can be analyzed by the same procedure <br>followed for the Maillefer screw. The initial portion of the barrier section can be <br>analyzed just as a Maillefer screw. If zT1 is the length of the initial Maillefer portion <br>
+of the barrier section, the solid bed velocity at z = ZT1 is approximately:<br>
+ (8.98)<br>
+The melting length of the Maillefer portion is:<br>
+ (8.99)<br>
+In the parallel portion of the barrier section, the width of the solids channel is con-<br>stant. The highest melting performance will be reached if the width of the solid bed <br>fills the entire width of the solids channel. If the solid bed velocity is assumed con-<br>stant, the total melting length can be found by using the equations derived from <br>the standard extruder screw; see Eq. 7.116. The total melting length for the parallel <br>barrier portion is simply:<br>
+ (8.100)<br>
+The total melting of the barrier section is the sum of Eqs. 8.99 and 8.100. This <br>results in the following expression for the melting length of the Barr screw:<br>
+ (8.101)<br>
+In Eq. 8.101, the effect of the varying solid bed velocity on 1 has been neglected. <br>
+Since the solid bed varies only in the initial portion of the barrier section, this will <br>not affect the accuracy very much as long as the initial Maillefer section is relatively <br>short relative to the total length of the barrier section.<br>In comparison to the melting length Z*T of the ideal compression screw, the melting <br>
+length of the Barr screw is:<br>
+ (8.102)<br>
+<hr>
+<A name=590></a><b>578</b>  <b>8Extruder Screw Design</b><br>
+A typical ratio of ZT1/ZT is about 0.2. Thus, the shortest possible melting length of <br>
+the Barr screw is about 10% longer than the Z*T of the ideal compression screw and <br>
+about 25% shorter than the melting length of the standard compression (4:1) screw. <br>The Barr screw is slightly more efficient than the Maillefer screw in terms of melt-<br>ing performance, about 15%. Thus, the Barr screw is marginally better than the <br>Maillefer screw in melting performance. However, the Maillefer screw is better in <br>terms of melt conveying performance in the barrier section.<br>
+<b>8.6.2.3The Dray and Lawrence Screw<br></b>The Dray and Lawrence screw is shown in Fig. 8.60.<br>
+<b>Figure 8.60</b>The Dray and Lawrence screw<br>
+The screw geometry is very similar to the Barr screw with the one major difference <br>being an abrupt change in helix angle in the main flight at the point where the barrier <br>flight is introduced. This allows the width of the solids channel to stay just as wide as <br>the full channel width of the feed section. Obviously, this is done in an attempt to <br>maintain the solids channel as wide as possible. However, this also causes an abrupt <br>change in the direction of the solid bed velocity, and this can lead to instabilities.<br>If the effect of the change in helix angle on 1 is neglected, the melting length can <br>
+be shown to be [27]:<br>
+ (8.103)<br>
+where f is the helix angle in the feed section and b the helix angle in the barrier <br>
+section.<br>In this case, however, the down-channel melting length does not provide a good <br>basis for comparison because the helix angle is different along the screw. The total <br>axial melting length can be expressed as:<br>
+ (8.104)<br>
+If typical values are used for b and f, the melting length of the Dray and Lawrence <br>
+screw will about 10 to 20% longer than the ideal compression screw. The melting <br>performance of the Dray and Lawrence screw is thus about the same as the Barr <br>screw and slightly better than the Maillefer screw. A patent on this barrier screw <br>
+<hr>
+<A name=591></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>579</b><br>
+was applied for by Dray of Feed Screw, Inc. and Lawrence of Owens-Illinois, Inc. [31]. <br>The patent was filed in May 1970 and was issued in March 1972. The screw based <br>on this patent was sold by Feed Screw, Inc. under the name "Efficient Screw." At the <br>end of the barrier section, the melt channel has to make a transition to the melt <br>channel of the metering section. Since the solids channel is quite wide, this transi-<br>tion tends to be quite abrupt, even more so than with the Barr screw. Thus, the Dray <br>and Lawrence screw has two abrupt changes in screw channel geometry, one at the <br>beginning and one at the end of the barrier section. This will tend to make the screw <br>more susceptible to surging types of instabilities than the Maillefer screw, which <br>has much more gradual transitions in the screw channel geometry.<br>
+<b>8.6.2.4The Kim Screw<br></b>The Kim screw is basically an improvement of the Dray and Lawrence screw. A pic-<br>ture of the Kim screw is shown in Fig. 8.61.<br>
+<b>Figure 8.61</b>The Kim variable pitch barrier screw (VPB)<br>
+In the Kim screw, the helix angle of the main flight and the barrier flight is changed <br>gradually to obtain a smooth transition from feed section to barrier section. In the <br>Kim screw, the width of the solids channel remains constant as with the Dray and <br>Lawrence screw. A patent was filed in August 1972 and issued February 1975 [33]. <br>The patent was reissued in August 1975 [34] with the number of claims reduced <br>from six to two. The assignee is B.F. Goodrich Company. The screw was licensed to <br>Davis-Standard Division of Crompton &amp; Knowles Corporation and sold under the <br>name "VPB" screw, which stands for variable pitch barrier screw. The VPB screw is <br>covered both by the early Uniroyal patent [26] and the B.F. Goodrich patent [34].<br>Again, the melting performance can be analyzed by the procedure used for the <br>Maillefer screw. Because of the continuously varying helix angle, the analysis is <br>rather involved. Ingen Housz and Meijer [27] found for the total melting length of <br>the Kim screw:<br>
+ (8.105)<br>
+where s is the ratio of the final melt channel width W1m to the initial solids channel <br>
+width W1s:<br>
+ (8.105a)<br>
+<hr>
+<A name=592></a><b>580</b>  <b>8Extruder Screw Design</b><br>
+The value of s in the Kim screw is one, which results in the following value of the <br>
+melting length:<br>
+ (8.106)<br>
+This means that the melting performance of the Kim screw is slightly lower than the <br>Dray and Lawrence screw and the Barr screw. It has an advantage over the Dray and <br>Lawrence screw in that the transition from feed to barrier section occurs more <br>smoothly. However, at the end of the barrier section the same difficulty arises as <br>with the Dray and Lawrence screw.<br>
+<b>8.6.2.5The Ingen Housz Screw<br></b>The Ingen Housz screw combines a barrier geometry with multi-flighted geometry <br>to obtain significant improvements in melting. A picture of the barrier section geo-<br>metry is shown in Fig. 8.62.<br>
+<b> Figure  8.62</b> <br>
+The Ingen Housz screw<br>
+The barrier section geometry is shown with the screw channel unrolled onto a flat <br>plane. The solid bed is divided into several parallel solid channels. The melt is col-<br>lected in several parallel melt channels. It is possible to achieve this multi-flighted <br>geometry by a significant increase in the helix angle. The total melting length can <br>be expressed as [27]:<br>
+ (8.107)<br>
+<hr>
+<A name=593></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>581</b><br>
+With s = 0.25 and p = 3 the total melting length becomes:<br>
+ (8.108)<br>
+This means that with this barrier screw geometry it is possible to obtain a minimum <br>melting length that is shorter than the ideal compression screw. In fact, the multi-<br>flighted barrier screw geometry is the only one that yields significant benefits in <br>terms of melting performance compared to the standard compression screw. A U.S. <br>patent on the Ingen Housz screw was issued August 19, 1980 [35].<br>The Ingen Housz barrier screw has a few drawbacks that may or may not be sig-<br>nificant. At the start of the barrier section, the solid bed is sliced into several nar-<br>rower solid beds. This requires easy deformability of the solid bed. If there is <br> <br>considerable resistance against this deformation in the solid bed, it could lead to <br>instabilities. The melt conveying in the barrier section occurs in deep, narrow <br>channels with a large helix angle. As a consequence, the melt conveying capacity <br>will be poor. This essentially requires the use of a grooved barrel section in the feed <br>section of the extruder to ensure a negative pressure gradient along the barrier <br> <br>section to reach sufficient melt conveying rate. Finally, the transition from the bar-<br>rier section to the metering section will be difficult if the metering section is of <br>conventional geometry. However, if a grooved barrel section is used, the metering <br>section can be deleted altogether since pressure build-up in the melt conveying <br>zone is no longer necessary.<br>Results of extensive experimental tests with the Ingen Housz screw are described <br>by Ingen Housz and Meijer [36]. Tests were run on a 60-mm extruder with LDPE, <br>HDPE, and PP. The high melting capacity of the screw could only be utilized if suffi-<br>cient solids conveying capacity was made available by the use of a grooved barrel <br>section. Outputs as high as 200 kg/hr at 100 rpm were achieved while the total <br>length of the screw was only 16 D. The output was found to be sensitive to the par-<br>ticle size of the polymer. High outputs were achieved with larger particles, while <br>the output with smaller particle size polymer was considerably lower, sometimes as <br>much as 50%. This demonstrates that the output can never be higher than the solids <br>conveying rate in the feed section. Even with the grooved barrel section, the solids <br>conveying rate for some polymers was insufficient to supply the melting section <br>with enough material to utilize the full melting capacity.<br>
+<b>8.6.2.6The CRD Barrier Screw<br></b>The CRD (Chris Rauwendaal Dispersive mixing) barrier screw was developed to <br>enhance the dispersive mixing action when the polymer melt is forced over the bar-<br>rier flight. The unique feature of the CRD barrier screw is that the barrier flight is <br>designed to generate elongational flow as the plastic melt passes over the barrier <br>flight. This can be done by making the pushing flight flank of the barrier flight <br>curved or slanted. This creates a wedge-shaped region between the barrier flight <br>
+<hr>
+<A name=594></a><IMG src="Polymer Extrusion by Chris Rauwendaal-594_1.jpg"><br>
+<b>582</b>  <b>8Extruder Screw Design</b><br>
+and the barrel in which the plastic melt accelerates as it passes over the barrier <br>flight. The acceleration creates the elongational deformation in the plastic melt.<br>Figure 8.63 shows a conventional barrier flight geometry (left) next to a CRD barrier <br>flight geometry on the right. The benefit of the CRD barrier flight geometry is im -<br>proved dispersive mixing and reduced energy dissipation, which results in lower <br>melt temperatures.<br>
+Conventional flight<br>
+CRD flight geometry<br>
+<b>Figure 8.63</b>Standard barrier flight (left) vs . CRD barrier flight (right)<br>
+A drawback of most barrier screws is that their distributive mixing capability is <br>rather poor. As a result, the CRD barrier screw will generally be equipped with a <br>CRD mixing section downstream of the barrier section to improve both dispersive <br>and distributive mixing. CRD mixing sections will be discussed in Section 8.7.1.1. <br>A photograph of a CRD5 mixing section is shown in Fig. 8.64.<br>
+<b>Figure 8.64</b>CRD5 mixing section<br>
+<b>8.6.2.7Summary of Barrier Screws<br></b>The characteristics of the various barrier screws are summarized in Table 8.1. The <br>Maillefer screw has many desirable characteristics despite the fact that its melting <br>performance is not quite as good as the other barrier screws. The Ingen Housz screw <br>clearly has the best melting performance; however, this is at the expense of geo-<br>metrical simplicity.<br>From a functional analysis, a double-flighted Maillefer (DFM) screw would seem to <br>be a good compromise between considerations concerning geometry and output. <br>With a double-flighted geometry, the melting performance can be improved about <br>30% in the best case. This would make the DFM screw more efficient in melting <br>capacity than the Barr screw, the Dray and Lawrence screw, and the Kim screw. In <br>order to minimize the adverse effect of the additional flight, the helix angle of the <br>
+<hr>
+<A name=595></a> <br>
+8.6Multi-Flighted Extruder Screws  <b>583</b><br>
+main flight should be relatively large. However, the helix angle should not be too <br>large in order to maintain good melt conveying capability. A helix angle of about 25° <br>would seem like a reasonable compromise. Figure 8.65 shows a possible configura-<br>tion of the DFM screw.<br>
+<b>Figure 8.65</b>Double-flighted barrier screw<br>
+Extruder manufacturer Davis-Standard introduced a similar double-flighted barrier <br>screw at the 2000 National Plastics Exhibition in Chicago, Illinois. The characteris-<br>tics of the DFM screw are included in Table 8.1.<br>A double-flighted compression screw (4:1 ratio) has a melting capacity only slightly <br>less than the DFM screw (see Table 8.1) and better than the Barr screw, the Dray <br>and Lawrence screw, and the Kim screw. The double-flighted compression screw will <br>be easier to manufacture than any barrier screw. The advantage of barrier screws, <br>that they keep unmelted material from reaching the metering section, can also be <br>obtained by incorporating a fluted mixing section at the beginning of the metering <br>section.<br>
+<b>Table 8.1</b>Characteristics of Various Barrier Extruder Screws<br>
+Transition<br>
+Transition<br>
+Minimum<br>
+Melt<br>
+Ease of<br>
+from feed-<br>
+barrier-<br>
+melting<br>
+conveying<br>
+manufacture<br>
+barrier<br>
+meter<br>
+length LT*<br>
+capacity<br>
+Maillefer<br>
+Smooth<br>
+Smooth<br>
+1.3L<br>
+Good<br>
+Good<br>
+Barr<br>
+Smooth<br>
+Abrupt<br>
+1.1L<br>
+Fair<br>
+Fair<br>
+DL<br>
+Abrupt<br>
+Abrupt<br>
+1.1L<br>
+Fair<br>
+Fair<br>
+Kim<br>
+Smooth<br>
+Abrupt<br>
+1.2L<br>
+Fair<br>
+Difficult<br>
+Ingen Housz<br>
+Abrupt<br>
+Abrupt<br>
+0.65L<br>
+Poor<br>
+Difficult<br>
+DFM<br>
+Smooth<br>
+Smooth<br>
+0.9L<br>
+Good<br>
+Good<br>
+Compression (4:1) <br>
+ <br>
+ <br>
+ <br>
+ <br>
+ <br>
+single-flighted<br>
+Smooth<br>
+Smooth<br>
+1.5L<br>
+Good<br>
+Excellent<br>
+Compression (4:1) <br>
+ <br>
+ <br>
+ <br>
+ <br>
+ <br>
+double-flighted<br>
+Smooth<br>
+Smooth<br>
+1.1L<br>
+Good<br>
+Excellent<br>
+LT* minimum melting length in regular compression screw<br>
+Table 8.2 summarizes advantages and disadvantages of barrier screws.<br>Barrier screws became popular because they generally replaced simple conveying <br>screws and were able to improve performance. However, when comparing barrier <br>screws to well-designed non-barrier screws with mixing sections, the barrier screw <br>generally does not perform as well and is more expensive. In a barrier screw, there <br>
+<hr>
+<A name=596></a><b>584</b>  <b>8Extruder Screw Design</b><br>
+is less space available for the solid bed; as a result, they are inherently more suscep-<br>tible to plugging of the solid bed, which leads to surging. One situation where a bar-<br>rier screw can offer an advantage is in deep-flighted large diameter screws. In such <br>screws, melting will not be efficient, and a barrier geometry can force the solid bed <br>close to the barrel to improve melting compared to the performance of a non-barrier <br>type screw.<br>
+<b>Table 8.2</b>Summary of Barrier Screw Characteristics<br>
+Advantages of barrier screw<br>
+Disadvantages of barrier screws<br>
+More stable operation than simple  Not better than well designed non-barrier screws with good <br>
+conveying screw<br>
+ <br>mixing  sections<br>
+Some dispersive mixing as melt <br>
+More expensive than non-barrier screws, particularly with OEMs*<br>
+flows over the barrier flight<br>Little chance of unmelted material <br>
+Inherently more susceptible to plugging because less space for <br>
+traveling beyond the barrier section solid bed<br>Widely available in the polymer <br>
+Barrier screws have to be carefully tailored to melting character-<br>
+extrusion industry<br>
+istics of the polymer; as a result, barrier screws are not good <br>
+general-purpose screws<br>
+*OEM is Original Equipment Manufacturer<br>
+<b></b><br>
+ <b>8.7Mixing Screws</b><br>
+The mixing capacity of standard extruder screws is limited as discussed in Section <br>7.7. As a result, many modifications have been made to the standard extruder screw, <br>in an effort to improve the mixing capacity. The number of mixing elements that <br>have been used on extruder screws is very large. Therefore, it is not possible to dis-<br>cuss all mixing screws used in the industry. This discussion will be limited to the <br>more common and important types of mixing elements. Before selecting a mixing <br>element, it is important to determine whether distributive or dispersive mixing is <br>required; see Section 7.7. Therefore, the mixing sections will be divided into dis-<br>tributive and dispersive mixing sections to indicate their preferred application.<br>
+<b>8.7.1Dispersive Mixing Elements</b><br>
+Dispersive mixing elements are used when agglomerates or droplets, such as gels, <br>need to be broken down. This is particularly important in small or thin gauge extru-<br>sion, e.g., fiber spinning, thin film extrusion. The most common dispersive mixing <br>section is the fluted or splined mixing section. In this mixing section, one or more <br>barrier flights are placed along the screw such that the material has to flow over the <br>barrier flight(s). In the barrier clearance the material is subjected to a high shear <br>
+<hr>
+<A name=597></a> <br>
+8.7Mixing Screws  <b>585</b><br>
+rate; the corresponding shear stress should be large enough to break down the par-<br>ticles in the polymer melt. A well-known fluted mixing section is the Union Carbide <br>(UC) mixing section invented by LeRoy [37]; see Fig. 8.66(a).<br>
+Outlet channel<br>
+Inlet<br>channel<br>
+Barrier flight<br>
+Undercut<br>
+Main flight<br>
+<b>Figure 8.66(a)</b>The LeRoy mixing section (also called Maddock or UC mixing section)<br>
+Maddock from Union Carbide published results of experiments with this mixing <br>section [38]; since then the mixing section is often referred to as the Maddock mix-<br>ing section.<br>The UC mixing section has longitudinal splines, i.e., a barrier flight helix angle of <br>90°. All material has to flow over the barrier flight because the inlet channel is closed <br>at the end of the mixing section. Thus, all of the material is forced over the barrier <br>flight, yielding a uniform dispersive mixing action. A drawback of this mixing sec-<br>tion is that it is pressure consuming, i.e., it reduces the output of the extruder. Also, <br>the longitudinal geometry with constant channel depth results in stagnating regions. <br>Thus, the design will be less suitable with materials of limited thermal stability.<br>A recent version of the LeRoy/Maddock mixer is the BT mixer developed by Luker <br>[102, 103]; see Fig. 8.66(b). In this fluted mixer, the inlet channels are open at the <br>beginning as well as at the end of the mixer. The purpose of this arrangement is to <br>allow material leaving the exit channels to flow back into the inlet channels. Poten-<br>tially, this allows the material to experience more than one exposure to the high <br>stress regions of the mixer. The drawback of this arrangement is that it is possible <br>for the material to flow through the inlet channel without flowing over the barrier <br>flight.<br>
+Outlet channel<br>
+Inlet<br>
+channel<br>
+Barrier flight<br>
+Undercut<br>
+Main flight<br>
+<b>Figure 8.66(b)</b>The BT mixer<br>
+<hr>
+<A name=598></a><b>586</b>  <b>8Extruder Screw Design</b><br>
+The open inlet channel eliminates the advantage of regular fluted mixers where all <br>of the incoming material is forced over a barrier flight. It is claimed that the BT <br>mixer generates elongational flow; however, it is not clear how this elongational flow <br>is generated. The best way to determine if elongational flow occurs in a mixer is to <br>perform a full 3-D flow analysis. Full 3-D analysis of the conventional LeRoy mixer <br>indicates that elongational flow does indeed occur in the mixer; see Section 12.4.3.4. <br>However, the elongational flow occurs in a region with low strain rate (and thus low <br>stresses). Unfortunately, elongational flow at low strain rate is ineffective for disper-<br>sive mixing because high stresses are required to rupture the agglomerates or drop-<br>lets. There has been no report of a full 3-D flow analysis of the BT mixer and no <br>quantitative assessment of the elongational flow in the mixer.<br>A more effective way to generate multiple high stress exposures is to use a fluted <br>mixer with one or more intermediate flutes between each inlet and outlet flute. The <br>intermediate flutes are normally closed at both ends so that all incoming material is <br>forced over the barrier flight of the flute. An example is the CRD fluted mixer shown <br>in Fig. 8.92, which has four intermediate flutes between each inlet and outlet flute. <br>As a result, all material passing through the mixer is exposed to four high stress <br>exposures as shown by the arrows in Fig. 8.92. The helical orientation of the flutes <br>reduces the pressure drop and improves dispersive mixing. Further, the barrier <br>flight has a wedge-shaped pushing flight flank to generate strong elongational flow <br>for effective dispersive mixing. Different mixing flight geometries are shown in <br>Fig. 8.82; another version of the CRD fluted mixer is shown in Fig. 8.66(c). The CRD <br>mixer is covered by two U.S. patents [79, 80] and several international patents.<br>
+Inlet channel<br>
+Material Transport Direction<br>
+Outlet channel<br>
+A<br>
+D<br>
+<b>Section A-A</b><br>
+A<br>
+L<br>
+Main flight<br>
+Barrier flight<br>
+Undercut<br>
+R<br>
+R<br>
+<br>
+Outlet channel<br>
+Inlet channel<br>
+Tangential pushing barrier flight flank<br>
+tangential with inlet channel radius R<br>
+Unrolled view of mixer<br>
+<b>Figure 8.66(c)</b>The CRD fluted mixer<br>
+<hr>
+<A name=599></a><IMG src="Polymer Extrusion by Chris Rauwendaal-599_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-599_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-599_3.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>587</b><br>
+Another fluted mixing section is the Egan mixing section invented by Gregory and <br>Street [47]; see Fig. 8.67.<br>
+Inlet flute<br>
+Outlet flute<br>
+<b> Figure 8.67</b> <br>
+Main flight<br>
+Barrier flight<br>
+The Egan fluted mixing section<br>
+In this mixing section, the splines run in a helical direction, i.e., the barrier flight <br>helix angle is less than 90°. The advantage of the helical splines is the fact that this <br>enables forward drag transport in the inlet and outlet channel. As a result, the fluted <br>mixing section with helical flutes will consume less pressure than the fluted mixing <br>section with longitudinal flutes; see Fig. 8.75. Thus, the helically fluted mixing will <br>reduce the extruder output to a lesser extent. In fact, if the mixing section is properly <br>designed it can even generate pressure, causing an improvement in extruder output. <br>Another feature of the Egan mixing section is a gradual reduction of the depth of the <br>inlet channel, leading to zero depth at the end of the mixing section. This channel <br>depth profile is reversed in the outlet channel. The channel depth taper reduces the <br>chance of hang-up of material, and thus reduces the chance of degradation.<br>A similar mixing device was later patented by Gregory [48]. The difference between <br>this mixing device and the Egan mixing section is a constant depth in inlet and out-<br>let channel and a concave channel geometry. Another similar mixing section was <br>patented by Dray [49]; see Fig. 8.68.<br>
+<b> Figure 8.68</b> <br>
+The Dray fluted mixing section<br>
+The main difference in this mixing section is that the outlet channel is open at the <br>start of the mixing section. Thus, not all material is forced over the barrier clear-<br>ance. Therefore, this mixing device will not result in a uniform shear history of the <br>material. An extreme form of a fluted mixing section is the annular blister ring; see <br>Fig. 8.69.<br>
+<b> Figure 8.69</b> <br>
+Blister ring<br>
+Blister ring mixing section<br>
+The annular blister ring is simply a smooth cylindrical screw section with a small <br>radial clearance. All the material has to pass through this clearance to exit from the <br>extruder. Since no positive drag transport takes place over the barrier clearance, the <br>
+<hr>
+<A name=600></a><b>588</b>  <b>8Extruder Screw Design</b><br>
+pressure drop over the blister ring will be high compared to other fluted mixing sec-<br>tions. A blister ring with a small barrier clearance will generally cause a significant <br>reduction in output. The pressure drop over the blister ring can be written by using <br>the expressions in Table 7.1:<br>
+ (8.109)<br>
+This expression is for a power law fluid and is valid if the effect of the screw rotation <br>on the melt viscosity can be neglected. In reality, however, the effective viscosity <br>in the clearance will be reduced as a result of the rotation of the screw. The shear <br>stress is composed of a shear stress in the tangential direction and a shear stress in <br>the axial direction. The tangential shear stress can be determined by evaluating the <br>stress at the center of the channel where the axial shear stress is zero. This yields <br>the following expression for the shear stress in the tangential direction:<br>
+ (8.110)<br>
+The tangential shear stress is constant over the depth of the channel. The axial <br>shear stress can be related to the axial pressure gradient by a simple force balance; <br>this yields:<br>
+ (8.111)<br>
+The total shear stress is obtained by vectorial addition of the axial and tangential <br>shear stress:<br>
+ (8.112)<br>
+The axial velocity gradient can be determined from:<br>
+ (8.113)<br>
+The axial velocity is obtained by integration of the axial velocity gradient:<br>
+ (8.114)<br>
+And the volumetric flow rate is obtained by integration of the axial velocity:<br>
+ (8.115)<br>
+<hr>
+<A name=601></a> <br>
+8.7Mixing Screws  <b>589</b><br>
+Closed form solutions of Eqs. 8.114 and 8.115 are only possible for a few specific <br>values of the power law index n, namely those for which (1--n)/2n is an integer. <br>Worth [50] derived a solution for a power law index value n = 1/3. This is a useful <br>case because many of the high-volume commodity polymers have a power law index <br>close to 1/3; see also Table 6.1. When n = 1/3, the throughput as a function of pres-<br>sure can be written as:<br>
+ (8.116)<br>
+The pressure drop now has to be found by solving a cubic equation. This solution <br>can be written as:<br>
+ (8.117)<br>
+where:<br>
+ (8.118a)<br>
+ (8.118b)<br>
+ (8.118c)<br>
+A comparison of the pressure drop predicted with Eq. 8.118 to the P predicted with <br>Eq. 8.109 is shown in Fig. 8.70.<br>
+400<br>
+Eq'n 8-118<br>
+300<br>
+Eq'n 8-109<br>
+Eq'n 8-118<br>
+Eq'n 8-109<br>
+200<br>
+ = 0.508 mm<br>
+Throughput [cc/sec]<br>
+ = 0.254 mm<br>
+100<br>
+0 0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50 <b> Figure 8.70</b> <br>
+Pressure drop [MPa]<br>
+Pressure drop over blister ring<br>
+<hr>
+<A name=602></a><b>590</b>  <b>8Extruder Screw Design</b><br>
+The prediction is for a 114-mm (4.5 in) extruder running at 100 rpm with a blister <br>length of 12.7 mm (0.5 in) and a flow rate of 131 cm3/s (8 in3/s). The reduction of the <br>pressure drop as a result of the screw rotation is about 15%. This indicates that the <br>simple Eq. 8.109 gives a reasonably accurate prediction of the pressure drop.<br>The pressure drop in a fluted mixing section can be calculated for a Newtonian fluid. <br>The first theoretical analysis was performed by Tadmor and Klein [51]. Their final <br>equation for the pressure drop contains five dimensionless numbers, which makes <br>determination of the effect of certain design variables rather indirect. A non-iso-<br>thermal and non-Newtonian analysis was performed by Lindt et al. [52]. This analy-<br>sis requires numerical techniques to solve the equations. Therefore, this analysis <br>can only be used if one develops the computer software to perform the calculations. <br>A simpler analysis was made by the author [53], leading to closed form analytical <br>solutions from which the effect of the most important design variables can be easily <br>evaluated.<br>To determine the pressure drop as a function of flow rate, one pair of inlet and outlet <br>channels will be examined in detail; see Fig. 8.71.<br>
+<b>.</b>V<br>
+x<br>
+i<br>
+z<br>
+Screw axis<br>
+<br>
+wcl<br>
+<b>.</b><br>
+<b> Figure 8.71</b> <br>
+Vo<br>
+Inlet and outlet of a fluted mixer<br>
+The volumetric flow rate at the entrance to the inlet channel is  i(o), where the sub-<br>
+script i refers to the inlet channel.  i(o) is the total volumetric output of the extruder <br>
+divided by the number of inlet channels. The flow rate through the inlet channel <br>decreases in the down-channel direction as a result of leakage over the barrier flight. <br>At the same time there is a corresponding increase in the flow rate through the out-<br>let channel. If the fluid is considered Newtonian and isothermal, the flow rate in the <br>inlet channel as a function of down-channel distance z can be written as:<br>
+ (8.119)<br>
+where  is the radial clearance of the barrier flight.<br>
+<hr>
+<A name=603></a> <br>
+8.7Mixing Screws  <b>591</b><br>
+Equation 8.119 is a simplified form of the equations presented in Section 7.4.1.2. <br>The radial clearance of the non-barrier flight is taken to be zero. The initial pressure <br>gradient G1 in the inlet channel can be obtained from:<br>
+ (8.120)<br>
+The flow rate in the exit or outlet channel  e(z) can be written as:<br>
+ (8.121)<br>
+The subscript e refers to the exit channel. Considering that the flow rate at the exit <br>of the outlet channel  e(zm) equals the flow rate at the entry to the inlet channel <br>
+i(o), the pressure gradient at the exit of the outer channel G2 can be written as:<br>
+ (8.122)<br>
+The leakage flow from the inlet channel to the outlet channel  1 is a combination of <br>
+drag flow and pressure flow. The leakage flow per unit down-channel distance  1 <br>
+can be written as:<br>
+ (8.123)<br>
+where wc1 is the perpendicular barrier flight width and c1 the polymer melt visco-<br>
+sity in the clearance.<br>From a mass balance, it follows that the local flow rate in the exit channel equals the <br>total flow rate minus the local flow rate in the inlet channel:<br>
+ (8.124)<br>
+This equation leads to the following relationship between the two pressure gradients:<br>
+ (8.125)<br>
+Equation 8.125 is valid when the local channel width W and channel depth H of the <br>inlet channel are the same as those of the outlet channel. If the dimensions of both <br>channels do not change with down-channel distance, the sum of the pressure gra-<br>dients will be constant along the length of the mixing section. In this case, Eq. 8.125 <br>can be written as:<br>
+ (8.126)<br>
+<hr>
+<A name=604></a><b>592</b>  <b>8Extruder Screw Design</b><br>
+Thus, the pressure in the exit channel can be related to the pressure in the inlet <br>channel by:<br>
+ (8.127)<br>
+where:<br>
+ (8.127a)<br>
+Another important relationship can be obtained by considering that the local change <br>in flow rate of the inlet channel over an incremental increase in down-channel dis-<br>tance equals the local leakage flow:<br>
+ (8.128)<br>
+If the channel dimensions do not change in the down-channel direction, Eq. 8.128 <br>can be written as:<br>
+ (8.129)<br>
+With Eq. 8.127, the differential equation can be written as:<br>
+ (8.130)<br>
+where:<br>
+ (8.130a)<br>
+Equation 8.130 is a non-homogeneous equation of the second order. The solution to <br>the homogeneous equation is:<br>
+ (8.131)<br>
+where:<br>
+ (8.131a)<br>
+An obvious particular solution is:<br>
+ (8.132)<br>
+<hr>
+<A name=605></a> <br>
+8.7Mixing Screws  <b>593</b><br>
+The general solution is the sum of the solution to the homogeneous equation plus <br>the particular solution. Thus, the pressure profile in the inlet channel can be de -<br>scribed by:<br>
+ (8.133)<br>
+With Eq. 8.127, the pressure profile in the outlet channel can be written as:<br>
+ (8.134)<br>
+Two boundary conditions are necessary to evaluate constants C1 and C2. The follow-<br>
+ing boundary conditions can be used:<br>
+ (8.135)<br>
+This results in the following expressions for C1 and C2:<br>
+ (8.136)<br>
+ (8.137)<br>
+Figure 8.72 shows the pressure profile in the inlet channel and outlet channel for <br>a 114-mm (4.5-in) extruder running at 100 rpm with a throughput of 164 cm3/s <br>(10 in3/s).<br>
+40<br>
+P-in<br>
+30<br>
+]<br>
+20<br>
+Pressure [MPa 10<br>
+0<br>
+P-out<br>
+<b> Figure 8.72</b> <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Pressure profiles in inlet <br>
+Dimensionless axial distance<br>
+and outlet channels<br>
+<hr>
+<A name=606></a><b>594</b>  <b>8Extruder Screw Design</b><br>
+The length of the mixing section is 2 D, the barrier clearance is 0.5 mm (0.020 in), <br>the helix angle is 45°, and the number of inlet channels is 3. The local viscosity is <br>evaluated with the power law equation by using the Couette shear rate; the consist-<br>ency index is 13,800 Pasn (2 psisn) and the power law index is 0.5. The pressure in <br>the inlet channel reduces initially but later starts to increase again. This indicates <br>some degree of pressure generating capability of the inlet channel. In the outlet <br>channel, the pressure rises initially and drops in the later portion of the channel. <br>Thus, both the inlet channel and outlet channel have some pressure generating <br>capability. This is primarily achieved by the helical orientation of the flutes.<br>The importance of the helix angle is shown in Fig. 8.73.<br>
+40<br>
+P-in<br>
+30<br>
+]<br>
+90° helix angle<br>
+20<br>
+Pressure [MPa 10 P-in<br>
+P-out<br>
+50° helix angle<br>
+0<br>
+<b> Figure  8.73</b> <br>
+0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Pressure profiles with two <br>
+Dimensionless axial distance<br>
+helix angles<br>
+Two sets of pressure profiles are shown, one for a mixing section with a 90° helix <br>angle and one for a mixing section with a 50° helix angle. In this case, the barrier <br>clearance is 0.635 mm (0.025 in) and the throughput is 131 cm3/s (8 in3/s). The <br>profiles are determined such that the final pressure has the same value (5 MPa = <br>725 psi). It is evident that the helix angle has a strong effect on the pressure pro-<br>files and the total pressure drop. There is a significant pressure generation in the <br>inlet channel and outlet channel of the helical mixing section, resulting in a rela-<br>tively small total pressure drop. On the other hand, there is no pressure generating <br>capa city in the axially oriented mixing section as evidenced by the monotonic drop <br>in pressure in both the inlet and outlet channel. This results in a rather large total <br>pressure drop, about three times as high as the helically oriented mixing section!<br>The total pressure drop over the mixing section Pm is simply:<br>
+ (8.138)<br>
+<hr>
+<A name=607></a> <br>
+8.7Mixing Screws  <b>595</b><br>
+With Eqs. 8.133 through 8.137, this results in the following expression for the pres-<br>sure drop over the mixing section:<br>
+ (8.139)<br>
+where:<br>
+ (8.139a)<br>
+When B2zm ranges between 0 and 1, the following approximation can be made:<br>
+ (8.140)<br>
+The total pressure drop can now be written as the sum of two terms:<br>
+ (8.141)<br>
+The pressure drop in the clearance Pcl is given by the first two terms on the right-<br>
+hand side of Eq. 8.139. The pressure drop inlet and outlet channel Pch is the last <br>
+term of Eq. 8.139.<br>The first term Pcl is the pressure drop caused by the clearance. When B2zm is less <br>
+than unity, Eq. 8.140 can be used to express Pcl as:<br>
+ (8.142a)<br>
+When B2zm is larger than unity, Pcl can be expressed as:<br>
+ (8.142b)<br>
+The second term Pch is the pressure drop in the inlet and outlet channel; this can <br>
+be written as:<br>
+ (8.143)<br>
+The first term Pcl is inversely proportional to the cube of the barrier clearance. <br>
+Thus, when the barrier clearance is small, the pressure drop over the clearance will <br>increase very rapidly and will be the major component of the total pressure drop. <br>The pressure drop over the clearance Pcl can be made zero by making sure that the <br>
+<hr>
+<A name=608></a><b>596</b>  <b>8Extruder Screw Design</b><br>
+drag flow rate over the barrier clearance equals the flow rate at the entrance to the <br>inlet channel. Thus, the mixing section should be designed such that:<br>
+ (8.144)<br>
+If the flow rate through the mixing section is assumed to be about two-thirds of the <br>drag flow rate of the preceding screw section and the axial length is about two dia-<br>meters (Lm  D), then Eq. 8.144 can be simplified to the following form:<br>
+ (8.145)<br>
+D is the screw diameter, p is the number of inlet channels, and the constant C in <br>many cases is about 0.01. When the pressure drop over the clearance is zero, the <br>total pressure drop becomes simply:<br>
+ (8.146)<br>
+The pressure drop over the channel reaches its maximum value when the helix <br>angle is 90°. Thus, this corresponds to the most unfavorable geometry because it <br>will result in the largest drop in output. If the channel depth and helix angle are <br>optimized simultaneously, the pressure drop in the channel will reach a minimum <br>when the helix angle is 52.24° [63]. The corresponding optimum channel depth is <br>H* = 0.314 /(FpD2N).<br>The various factors that influence the pressure drop over the mixing section can <br>now be easily analyzed. The pressure drop increases proportionally with the flow <br>rate through the mixing section, as shown in Fig. 8.74.<br>
+<b> Figure 8.74</b> <br>
+Pressure drop versus flow rate<br>
+<hr>
+<A name=609></a> <br>
+8.7Mixing Screws  <b>597</b><br>
+Therefore, the design of the mixing section has to be matched to the preceding screw <br>section in order to avoid excessive pressure drop. The effect of the helix angle is <br>shown in Fig. 8.75.<br>
+25<br>
+] 20<br>
+15<br>
+Pressure drop [MPa 10<br>
+5<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+Helix angle [degrees]<br>
+<b>Figure 8.75</b>Pressure drop versus helix angle<br>
+As discussed earlier, the helix angle has a strong effect on the pressure drop. The <br>minimum pressure drop occurs at a helix angle between 50 and 60°. Below a helix <br>angle of 50° and above 60° the pressure drop increases quite rapidly. Thus, the <br>proper value of the helix angle is around 50 to 60°. The optimum helix angle for <br>shear thinning fluids is less than 50°.<br>The effect of the barrier flight width is shown in Fig. 8.76.<br>
+40<br>
+]a 20<br>[MP<br>
+0<br>
+Pressure drop  -200<br>
+0.05D<br>
+0.10D<b> Figure 8.76</b> <br>
+Barrier flight width<br>
+Pressure drop versus barrier flight width<br>
+The pressure drop increases in an approximately proportional fashion with the bar-<br>rier flight width. This is true if the pressure drop over the clearance is positive (Pcl <br>
+&gt; 0). When the pressure drop over the clearance is made zero (Pcl = 0), the width of <br>
+<hr>
+<A name=610></a><b>598</b>  <b>8Extruder Screw Design</b><br>
+the barrier flight no longer affects the total pressure drop (see Eq. 8.146). In all <br>cases, however, the width of the barrier flight will strongly influence the power con-<br>sumption and the viscous heat generation in the material.<br>The effect of the barrier clearance is shown in Fig. 8.77.<br>
+40<br>
+] 30<br>
+20<br>
+Pressure drop [MPa 10<br>
+0<br>
+0.25<br>
+0.50<br>
+0.75<br>
+1.00<br>
+Barrier clearance [mm]<br>
+<b>Figure 8.77</b>Pressure drop versus barrier flight clearance<br>
+When the clearance is less than about 1/2 mm (0.020 in), the pressure drop in -<br>creases quite dramatically. When the clearance is larger than about 3/4 mm <br>(0.030 in), the effect of the clearance becomes quite small. In fact, when the clear-<br>ance is larger than 1 mm (0.040 in), the pressure drop starts to increase because of <br>the reduced drag flow in the inlet channel. It should be noted, however, that changes <br>in the barrier flight width and barrier clearance directly affect the dispersive mix-<br>ing capability of the mixing section.<br>Dispersion of agglomerates or gels requires the application of a certain minimum <br>stress to break down the particles. The minimum stress level depends on the nature <br>of the particle as discussed by Martin [54] and Tadmor et al. [55]. For carbon black, <br>the critical stress level as determined by Martin [54] was found to be around 60 kPa <br>(9 psi). In addition to a minimum stress, there is also a minimum high stress expo-<br>sure time as discussed by Martin [54]. When the duration of high stress is below a <br>minimum exposure time, no dispersion will occur even at very high stress levels. <br>For carbon black, Martin [54] found the minimum exposure time to be about 0.2 s. <br>This means that the width of the barrier flight should be large enough so that the <br>residence time of the polymer in the clearance exceeds the minimum exposure time <br>tmin. Therefore, the width of the barrier clearance should be:<br>
+ (8.147)<br>
+where N is expressed in revolutions per minute.<br>
+<hr>
+<A name=611></a> <br>
+8.7Mixing Screws  <b>599</b><br>
+If the critical stress level is min, the barrier clearance should be:<br>
+ (8.148)<br>
+Thus, the barrier flight width wcl and the barrier clearance  have to be designed for <br>
+both pressure drop and dispersive mixing capacity.<br>The effect of the degree of non-Newtonian behavior is shown in Fig. 8.78.<br>
+100<br>
+80<br>
+] 60<br>
+40<br>
+Pressure drop [MPa 20<br>
+0<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+Power law index<br>
+<b>Figure 8.78</b>Pressure drop versus power law index<br>
+The effect of the pseudo-plasticity is determined by evaluating the local viscosity <br>with the power law equation:<br>
+ (8.149)<br>
+where m is the consistency index,   the local shear rate, and n the power law index <br>(see also Eq. 6.23). If the local shear rate is approximated by the Couette shear rate, <br>the viscosity in the clearance becomes:<br>
+ (8.150)<br>
+Similarly, the viscosity in the channel:<br>
+ (8.151)<br>
+<hr>
+<A name=612></a><IMG src="Polymer Extrusion by Chris Rauwendaal-612_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-612_19.jpg"><br>
+<b>600</b>  <b>8Extruder Screw Design</b><br>
+As the power law index increases, the pressure drop increases quite substantially. <br>Therefore, materials with relatively Newtonian flow characteristics can be expected <br>to cause higher pressure drops than strongly non-Newtonian materials. For exam-<br>ple, a material like LLDPE will give a much higher pressure drop than regular LDPE <br>of the same melt index because of the larger power law index of LLDPE [4, 5].<br>The effect of the axial length of the mixing section is shown in Fig. 8.79(a).<br>
+40<br>
+]<br>
+20<br>
+0<br>
+Pressure drop [MPa<br>
+0<br>
+1D<br>
+2D<br>
+3D<br>
+4D<b> Figure 8.79(a)</b> <br>
+Axial length<br>
+Pressure drop versus axial length<br>
+The pressure drop reduces substantially when the axial length is increased. Axial <br>lengths of less than 2 D generally create excessive pressure drops.<br>Finally, the effect of the number of inlet channels is shown in Fig. 8.79(b).<br>
+100<br>
+80<br>
+] 60<br>
+40<br>
+Pressure drop [MPa 20<br>
+0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+6<br>
+Number of inlet channels<br>
+<b>Figure 8.79(b)</b>Pressure drop versus the number of inlet channels<br>
+<hr>
+<A name=613></a><IMG src="Polymer Extrusion by Chris Rauwendaal-613_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-613_23.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>601</b><br>
+The pressure drop initially reduces with the number of inlet channels but later <br>increases. Thus, there is an optimum number of inlet channels that results in the <br>lowest pressure drop across the mixing section. The optimum number of inlet chan-<br>nels is generally about three or four.<br>The most important design features for a fluted mixing section can be summarized <br>as follows. The helix angle should be about 50 to 60°, the clearance should not be <br>smaller than about 1/2 mm (0.020 in), and the axial length should not be less than <br>2 D.<br>Further, the number of inlet channels should be three or four. The chance of hold-up <br>of material can be reduced by tapering the channel depth. The chance of hold-up can <br>be further reduced by tapering the channel width. This leads to the geometry shown <br>in Fig. 8.80.<br>
+<b> Figure 8.80</b> <br>
+Main flight, no undercut<br>
+Barrier flight with undercut<br>
+Z-shaped fluted mixing section<br>
+This geometry has the additional advantage that the pressure drop at the inlet to the <br>mixing section is substantially reduced. This entrance pressure drop has not been <br>taken into account in the analysis. However, it is obvious that the entrance pressure <br>drop in conventional fluted mixing sections can be substantial because the material <br>is forced from the wide screw channel into a number of narrow inlet channels.<br>Obviously, all barrier-type extruder screws (see Section 8.6.2) impart some degree <br>of dispersive mixing to the polymer because all the polymer has to flow over the bar-<br>rier flight to leave the extruder. Thus, every polymer element is exposed to a brief <br>but relatively intensive shearing in the barrier clearance. Simulation of fluted mix-<br>ers is discussed in Section 12.4.3.4; see also Fig. 12.43 and Fig. 12.44.<br>Dispersive mixing also occurs in the double wave mixing screw; see Fig. 8.81.<br>
+Unrolled channel<br>
+<b> Figure 8.81</b> <br>
+The double wave screw<br>
+<hr>
+<A name=614></a><b>602</b>  <b>8Extruder Screw Design</b><br>
+This screw was developed by Kruder of HPM [39] and is basically an extension of <br>the single channel wave screw [40]. Polymer is forced over the center barrier flight <br>by a cyclic variation of the channel depth. When one channel is increasing in depth, <br>the other is reducing. When the first channel reaches its maximum depth, the other <br>channel reaches its minimum depth. Then the first channel starts to reduce in depth <br>and the other channel starts to increase in depth. This process is repeated many <br>times. This screw design improves mixing performance, but the screw is relatively <br>expensive to manufacture.<br>
+<b>8.7.1.1The CRD Mixer<br></b>As discussed in Section 7.7.3, dispersive mixing in shear flow is substantially less <br>efficient than in elongational flow. Important requirements for dispersive mixing <br>elements were formulated by Rauwendaal [66]; they are:<br>1.  The mixing section should have a high stress region, HSR, where the material is <br>
+subjected to high, preferably elongational, stresses to break down agglomerates <br>and droplets.<br>
+2.  The HSR should be designed such that the exposure to high shear stresses occurs <br>
+only for a short time to avoid excessive power consumption and melt temperature <br>rise.<br>
+3.  All fluid elements should experience the same high stress level multiple times to <br>
+achieve uniform and efficient mixing.<br>
+If we analyze current dispersive mixers based on these requirements, we find that <br>most current dispersive mixers only meet these requirements partially.<br>The most commonly used dispersive mixer in single screw extruders is the LeRoy <br>mixer, popularized by Maddock. There are several versions of the fluted mixing sec-<br>tion [66] commercially available, with the helical LeRoy being a popular mixing <br>section because of its low pressure drop and good streamlining. Like the LeRoy <br>mixer, most current dispersive mixers rely on shear stresses to achieve breakdown <br>of the agglomerates. However, because elongational flow has open streamlines and <br>generates higher stresses, it is more effective in breaking down agglomerates and <br>droplets [67]. Therefore, elongational stresses are preferred in a dispersive mixer. <br>A new dispersive mixer based on the generation of elongational flow was developed <br>at the NRC in Montreal, Canada [68]. This extensional flow mixer (EFM) is placed at <br>the discharge end of an extruder, and the flow through the mixer is pressure driven <br>because the EFM is a static mixer.<br>In most current (shear flow) dispersive mixers, the material passes through the <br>HSR only once, thus severely limiting the level of dispersion that can be achieved. <br>To achieve a fine level of dispersion it is generally necessary for the agglomerates <br>or droplets to be broken down several times. Therefore, a single pass through a high <br>stress region is not sufficient in most cases and multiple passes through a high <br>
+<hr>
+<A name=615></a> <br>
+8.7Mixing Screws  <b>603</b><br>
+stress region are critical. If the agglomerate is of the order of 1000 m and needs <br>to be reduced to the 1 m level, it will take about 10 rupture events if we assume <br>that each rupture event reduces the agglomerate size by 50%. It should be noted <br>that drop breakup does not always reduce the drop size by 50%. The most efficient <br>mechanism for dispersing liquids is to deform droplets into extended threads at <br>high capillary number and let them disintegrate into smaller droplets. The droplets <br>that form can be much smaller than the initial droplet size--formation of over 10,000 <br>droplets from a single drop has been reported.<br>If each pass through a high stress region produces one rupture, then it becomes <br>clear that a dispersive mixer that exposes the polymer melt to only one high stress <br>exposure is not likely to achieve a fine level of dispersion. This is an important rea-<br>son why current dispersive mixers in single screw extruders generally do not work <br>well. The lack of strong elongational flow and multiple passes through the HSRs <br>explain why current dispersive mixers for single screw extruders have limited dis-<br>persive mixing capability.<br>With the requirements formulated above, new geometries have been developed that <br>substantially improve dispersive mixing; these mixers are protected by U.S. and <br>international patents [79, 80]. These mixers, called CRD mixers, can be incorpo-<br>rated along the extruder screw. As stated earlier, the key to the enhanced mixing <br>efficiency is the generation of elongational flow in the high stress regions and <br>achieving multiple passes of all fluid elements through the HSRs.<br>Elongational flow is not easily achieved in screw extruders. It is generated most effi-<br>ciently by modifying the leading flight flanks of a mixing section such that the space <br>between the flank and the barrel becomes wedge shaped. Such geometries create <br>lobal mixing and are used in twin screw extruders [69]. This can be done by either <br>slanting the leading flight flank or by using a curved flight flank geometry as shown <br>in Fig. 8.82.<br>
+<b> Figure 8.82</b> <br>
+Flight geometries to create elongational flow <br>
+Flat-slanted pushing<br>
+Curved-slanted<br>
+(the arrows indicate the movement of the screw flight <br>
+flight flank<br>
+pushing flight flank<br>
+relative to the barrel)<br>
+Multiple passes through the HSRs can be achieved by using a multi-flighted geo-<br>metry combined with a generous flight clearance. A possible geometry is shown in <br>Fig. 8.83.<br>In order to achieve multiple passes through the HSRs, they should be designed such <br>that significant flow takes place through them. This issue was studied by Tadmor <br>and Manas-Zloczower [70]. Substantial flow through the HSR can be achieved by <br>
+<hr>
+<A name=616></a><IMG src="Polymer Extrusion by Chris Rauwendaal-616_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-616_28.jpg"><br>
+<b>604</b>  <b>8Extruder Screw Design</b><br>
+increasing the flight clearance. However, this is only part of the story because with-<br>out randomization of the polymer melt, increasing the flight clearance will only <br>result in mixing of the outer re-circulating region [66]. Another problem with a <br>large flight clearance is that it leaves a thick stagnant layer of polymer melt on the <br>barrel surface. Using at least one wiping flight in addition to the mixing flights can <br>circumvent this problem. One version of the CRD mixer is shown in Fig. 8.83.<br>
+Wiping flight<br>
+Wiping flight<br>
+Mixing flight<br>
+Mixing flight<br>
+ing flight<br>
+<b>Figure 8.83</b>Six-flighted CRD mixer with two continuous wiping flights and four slotted mixing <br>
+flights<br>
+Instead of incorporating separate wiping and conveying flights, it is possible to use <br>one or more flights that incorporate wiping and mixing segments along their length. <br>A possible geometry is shown in Fig. 8.84. Complete barrel wiping can be achieved <br>by making sure that at least one wiping flight segment is present at every axial posi-<br>tion along the mixer.<br>
+<b> Figure 8.84</b> <br>
+Four-flighted CRD mixer with wiping flight segments <br>
+Transport direction<br>
+ <br>followed by three mixing flight segments<br>
+By intentionally incorporating distributive mixing in the dispersive mixer, randomi-<br>zation of the fluid elements can be achieved. This gives each fluid element equal <br>chance to experience the dispersive lobal mixing action. Without this, only fluid <br>elements in the outer re-circulating region (shell) would participate in the disper-<br>sive mixing process [66, 71]. Slotted flight geometries have proven to be quite effec-<br>tive for distributive mixing [72]. The helix angle of the mixing flights can be posi-<br>tive, negative, and even zero. It is possible to use elements with 90° helix angle and <br>stagger the elements to achieve forward or rearward conveying, similar to kneading <br>blocks in co-rotating twin screw extruders. The difference is that the dispersion <br>disks can be designed to achieve maximum dispersion without the geometric con-<br>
+<hr>
+<A name=617></a><IMG src="Polymer Extrusion by Chris Rauwendaal-617_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-617_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-617_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-617_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-617_5.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>605</b><br>
+straints associated with self-wiping action [66]. An example of a collection of stag-<br>gered dispersion disks is shown in Fig. 8.85.<br>
+<b> Figure 8.85</b> <br>
+Staggered dispersion disks<br>
+The mixer should be designed such that all fluid elements are exposed to a mini-<br>mum number of passes through the high stress region. This requires a high enough <br>flow rate through the high stress regions and efficient distributive mixing. Deter-<br>mination of the appropriate clearance of the mixing flights is discussed in the next <br>section.<br>
+<b>8.7.1.1.1Determining Flight Clearance with Passage Distribution Function</b><br>
+According to Tadmor and Manas-Zloczower [70] the passage distribution function <br>can be written as:<br>
+ (8.152)<br>
+where k is the number of passes through the clearance, and the dimensionless time <br> = tr/  is the ratio of the residence time tr and the mean residence time   of the <br>
+control volume. The residence time for a Newtonian fluid can be approximated as <br>follows:<br>
+ (8.153)<br>
+where z is the helical length of the screw section considered, vbz the down-channel <br>
+barrel velocity, and r the throttle ratio (pressure flow rate divided by drag flow rate). <br>The mean residence time is the ratio of the control volume WHz to the volumetric <br>leakage flow rate over the flight; it can be determined from:<br>
+ (8.154)<br>
+<hr>
+<A name=618></a><b>606</b>  <b>8Extruder Screw Design</b><br>
+where W is the channel width, H the channel depth,  the radial flight clearance, vbx <br>
+the cross-channel barrel velocity, and wf the flight width. The dimensionless time <br>
+can be written as:<br>
+ (8.155)<br>
+where L is the axial length corresponding to down-channel distance z. The fraction <br>of the fluid experiencing zero passes through the clearance is:<br>
+ (8.156)<br>
+The G0 fraction is a very important characteristic of an open design type mixer. It <br>
+should be low to ensure that most of the fluid experiences at least one or more <br>passes through the clearance. In single screw extruders with a simple conveying <br>screw the  value is typically about 0.1. This corresponds to a G0 fraction of around <br>
+0.9. In this case, most of the fluid passes through the extruder without ever passing <br>through the clearance. The passage distribution function for this case is shown in <br>Fig. 8.86(a).<br>
+<b> Figure 8.86(a)</b> <br>
+The passage distribution <br>
+function for  = 0 .1<br>
+We can use the expressions above to determine the minimum  value that will yield <br>a G0 less than 0.01, meaning that less than 1% of the fluid will not pass through the <br>
+clearance at all. This is achieved when the dimensionless time  &gt; 4.6. For certain <br>values of L, H, W, r, , and wf we can then determine how large the flight clearance <br>
+ has to be to make  &gt; 4.6 or G0 &lt; 0.01. The passage distribution function for  = 4.6 <br>
+is shown in Fig. 8.86(b).<br>The distribution at  = 4.6 is quite different from that at  = 0.1. With  = 4.6, the G0 <br>
+fraction is quite low and most of the fluid experiences four passes through the HSR. <br>The value G4 is about 0.19; this means that about 19% of the fluid passes through <br>
+the HSR four times.<br>
+<hr>
+<A name=619></a> <br>
+8.7Mixing Screws  <b>607</b><br>
+<b>Figure 8.86(b)</b>The passage distribution function for  = 4 .6<br>
+When L = 3W, r = 0, and  = 17.67°, the ratio of /H has to be about 0.8 to achieve a <br>G0 &lt; 0.01. Clearly, with such a high ratio of /H it will be almost impossible to create <br>
+large stresses in the clearance and to accomplish effective dispersive mixing. From <br>Eq. 8.155 it is clear what geometric variables we have to change to achieve a low G0 <br>
+fraction at a small clearance. We can do this by:<br>1.  increasing L, the length of the mixing section<br>2. increasing , the helix angle<br>3.  reducing wf, the width of the flight<br>4.  increasing the number of flights. Increasing the number of flights reduces the <br>
+channel width, W.<br>
+If we increase the helix angle from 17.67° to 60° with the other values being the <br>same, the /H ratio has to be about 0.35 or greater for the G0 fraction to be less than <br>
+0.01. This value is still rather large, but substantially better than 0.8. The /H ratio <br>can be further reduced by increasing the length of the mixing section, reducing the <br>flight width, or by increasing the helix angle even more. The point is that this proce-<br>dure allows a first order determination of the design variables. Further refinement <br>of the initial values can be obtained from computer simulation.<br>
+<b>8.7.1.1.2Determining Flight Clearance with Respect to Stress Level</b><br>
+Another important requirement for dispersive mixing is that the stresses generated <br>in the HSR are high enough to achieve rupture of the agglomerate or droplet. The <br>highest shear stresses occur in the region where the mixing flight has the smallest <br>flight clearance. The shear rate at this point can be expressed as:<br>
+ (8.157)<br>
+<hr>
+<A name=620></a><b>608</b>  <b>8Extruder Screw Design</b><br>
+If the shear viscosity of the polymer melt is s, the maximum shear stress can be <br>
+written as:<br>
+ (8.158)<br>
+If the critical shear stress required for rupture is crit, the maximum flight clearance <br>
+that can achieve dispersion can be expressed as:<br>
+ (8.159)<br>
+If the viscosity is s = 500 Pas, the screw diameter D = 120 mm, the screw speed <br>
+N = 1.5 rev/s, and the critical shear stress crit = 140,000 Pa, then the maximum <br>
+clearance of the mixing flight is max = 2 mm.<br>
+<b>8.7.1.1.3Determining the Proper Flight Flank Geometry</b><br>
+As explained earlier, for efficient dispersion it is more important to achieve elon-<br>gational stresses than shear stresses. Elongational stresses are generated in the <br>wedge-shaped region between the pushing flight flank and the barrel. The elonga-<br>tion rate in the wedge can be obtained by using a procedure suggested by Cogswell <br>[73]. The average stretch rate close to the entrance of the flight clearance can thus <br>be written as:<br>
+ (8.160)<br>
+where  is the wedge angle between the pushing flight flank and the barrel surface, <br>and rd is the throttle ratio (pressure flow rate divided by drag flow rate).<br>The elongational stress can thus be expressed as:<br>
+ (8.161)<br>
+where e is the elongational viscosity.<br>If the critical elongational stress required for rupture is crit, the following inequal-<br>
+ity must be satisfied for dispersion to occur:<br>
+ (8.162)<br>
+From this expression the critical parameters for the flight flank geometry can be <br>determined. Unfortunately, the expressions above are valid only for small values of <br>the wedge angle . As a result, these expressions have limited usefulness. If we <br>
+<hr>
+<A name=621></a> <br>
+8.7Mixing Screws  <b>609</b><br>
+assume that the drag flow in the channel is forced through the flight clearance, the <br>average stretch rate can be approximated by:<br>
+ (8.163)<br>
+With this expression, we can determine a maximum flight clearance for dispersive <br>mixing based on the requirement that the elongational stress must be greater than <br>the critical elongational stress. This leads to the following expression:<br>
+ (8.164)<br>
+If the diameter D = 120 mm, the screw speed N = 1.5 rev/s, the elongational viscos-<br>ity e = 1,500 Pas,  = 30°, and the critical elongational stress crit = 100,000 Pa, <br>
+then the maximum flight clearance is max = 2.45 mm. The expressions above can be <br>
+used for a first order approximation of the critical geometrical parameters of the <br>mixer. For accurate determination, numerical techniques are necessary to capture <br>the complexity of actual flow.<br>
+<b>8.7.1.1.4Slot Geometry</b><br>
+The slots in the mixing section can be used to achieve efficient distributive mixing, <br>similar to mixing in a Saxton mixing section; see Section 8.7.2. The slot geometry <br>used in most distributive mixers is a straight slot. For dispersive mixing, however, <br>it is better to use a tapered slot because this will create additional elongational flow <br>as material passes through the slot. The geometry of the slot can be made such <br>that the flight maintains full wiping capability. The geometry of the slotted flight is <br>shown in Fig. 8.87.<br>
+<b> Figure 8.87</b> <br>
+Flight geometry with tapered slot<br>
+The flight maintains complete wiping capability when the axial component of the <br>pushing slot flank, Lf2, is greater than the axial slot width, Ls. This is the case when:<br>
+ (8.165)<br>
+<hr>
+<A name=622></a><b>610</b>  <b>8Extruder Screw Design</b><br>
+When the flight helix angle f = 45°, the inequality simplifies to:<br>
+ (8.166)<br>
+Figure 8.88 shows the smallest values of the slot flank angle for which the inequal-<br>ity above is satisfied.<br>As Fig. 8.88 indicates, the slot flank angle must be increased as the ratio of flight <br>width to slot width decreases. When the slot width is twice the flight width, the slot <br>flank angle has to be 90° to maintain full wiping. As a result, this will be the small-<br>est value of the width ratio that will be practical. The preferred range of the flight to <br>slot width ratio is from 1:1 to 3:1.<br>
+<b> Figure 8.88</b> <br>
+Minimum slot flank angle for flight <br>
+helix angle of 45°<br>
+<b>8.7.1.1.5Computer Simulation</b><br>
+The analytical approach to mixer design has some severe limitations because of the <br>difficulties in analyzing flow in a complicated mixer geometry. A better approach to <br>analyze complicated mixers is to use mathematical modeling and computer simula-<br>tion. One simulation tool that lends itself well to the analysis of complicated mixer <br>geometries is the boundary element method (BEM). This method allows a determi-<br>nation of the optimum value of the flight clearance, flight flank geometry, and spac-<br>ing of the slots to achieve the proper combination of dispersive and distributive <br>mixing action. Recently, a three-dimensional BEM package was developed at the <br>University of Wisconsin in Madison [74] and commercialized by The Madison Group <br>[75].<br>To help determine the flight flank geometry and clearance, a two-dimensional BEM <br>analysis was initially performed. To evaluate the strength of the elongational flow <br>versus the shear flow, the flow number [76] was analyzed. The flow number is the <br>ratio of the magnitude of the rate of deformation tensor   to the sum of   + , where <br> is the magnitude of the vorticity tensor.<br>
+ (8.167)<br>
+<hr>
+<A name=623></a> <br>
+8.7Mixing Screws  <b>611</b><br>
+When   = 1.0 the flow is pure elongational flow,  = 0.5 simple shear flow, and <br> = 0.0 pure rotational flow. A high value of the flow number is desired for effective <br>mixing. Greater hydrodynamic forces are generated in elongational flow as dis-<br>cussed in Section 7.7.3. Also, elongational flow can disperse high viscosity droplets <br>such as gels while shear flow is incapable of dispersing gels.<br>Using the BEM simulation, the flow number and forces at any point in the mixer can <br>be computed. Moreover, particles can be tracked through the mixer to determine <br>streamlines and detect possible stagnant regions. Figure 8.89(a) shows the calcu-<br>lated streamlines in the mixing section.<br>
+<b> Figure 8.89(a)</b> <br>
+Predicted streamlines in CRD mixer shown in Fig . 8 .83<br>
+Here, at every time step, the strain rates and flow numbers are calculated. Figure <br>8.89(b) shows the flow number of a particle as it flows through the system.<br>
+<b> Figure 8.89(b)</b> <br>
+Flow number versus time for <br>
+a point traveling through the <br>
+nip region of the mixer shown <br>
+in Fig . 8 .82<br>
+Flow numbers are achieved as high as 0.95, indicating that strong elongational flow <br>can be generated in the new mixers. Similarly, Fig. 8.90 shows the magnitude of the <br>rate of deformation tensor of the particle as it flows through the system.<br>As the particle approaches the flight, it "feels" an increase in the elongational flow. <br>While passing over the top of the flight, the elongational flow switches to shear flow, <br>
+<hr>
+<A name=624></a><IMG src="Polymer Extrusion by Chris Rauwendaal-624_1.jpg"><br>
+<b>612</b>  <b>8Extruder Screw Design</b><br>
+but at the same time the magnitude of the rate of deformation tensor increases. This <br>effect will increase the mixing capability of the system.<br>
+<b> Figure 8.90</b> <br>
+Strain rate versus time for points <br>
+traveling through the nip<br>
+One of the goals of this mixing section is to provide improved distributive mixing as <br>well as dispersive mixing. Introducing grooves in the modified flight will increase <br>the distributive mixing and at the same time allow the re-circulation areas shown in <br>Fig. 8.89(a) to be broken up. To calculate the splitting of the material (distributive <br>mixing effect) as it flows through the mixer, a three-dimensional BEM analysis was <br>performed. Figure 8.91 shows how a grouping of particles flows through a region of <br>the mixer.<br>
+<b> Figure 8.91</b> <br>
+Tracking of multiple points in 3-D simulation<br>
+<hr>
+<A name=625></a><IMG src="Polymer Extrusion by Chris Rauwendaal-625_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-625_9.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>613</b><br>
+As expected, some particles flow over the modified flight while others flow through <br>the groove. Again, this effect will increase the distributive and dispersive mixing <br>capability of the mixer. Simulation of the CRD mixer is discussed further in Section <br>12.4.3.6 (see Figs. 12.48 to 12.50).<br>One of the findings of the BEM simulations was that the number of passes through <br>the mixing clearance reduces as the pressure gradient along the mixer reduces. <br>This effect was also observed in mixing experiments when tests were performed at <br>low discharge pressure. In extrusion experiments [77], it was found that the mixing <br>quality reduces when the discharge pressure is low, less than 5 MPa. Obviously, this <br>problem is inherent in any open mixer design. It can be avoided by adopting a closed <br>mixer design, such as the fluted mixer. Figure 8.92(a) shows the geometry of a CRD <br>fluted mixer that achieves four passes through the mixing clearance.<br>
+<i><b>Detail</b></i><br>
+Main flight<br>
+Mixing flight<br>
+1<br>
+2<br>
+3<br>
+4<br>
+<b>Figure 8.92(a)</b>A fluted CRD mixer<br>
+The advantage of this geometry is that all fluid elements are exposed to four pas-<br>sages of the mixing clearance regardless of the discharge pressure. The disadvan-<br>tage is that the distributive mixing capability is reduced relative to the open mixer <br>and extruder output will tend to be lower.<br>
+<hr>
+<A name=626></a><b>614</b>  <b>8Extruder Screw Design</b><br>
+Another method of making sure that all fluid elements are exposed to the elonga-<br>tional mixing action is to use rings of elongational mixing pins (EMP). The pins have <br>the shape of an elongated polygon as shown in Fig. 8.92(b). The rings take up very <br>little space and provide many splitting and reorientation events in addition to the <br>elongational mixing action. The axial EMP shown in Fig. 8.92(b) obviously has no <br>forward pumping capability. This can be a benefit in some applications such as <br>grooved feed extruders.<br>
+<b> Figure 8.92(b)</b> <br>
+A CRD-EMP mixer<br>
+<b>8.7.1.1.6Applications of the CRD Mixer</b><br>
+The CRD mixer has been commercially available since late 1998; as of early 2013 <br>there are over 2000 CRD mixing screws in operation. The first CRD screw was used <br>in a foamed profile extrusion operation, resulting in improved product quality and <br>process stability. This company now has 50 extrusion lines running with CRD mix-<br>ing screws. Foamed plastic extrusion is one of the most critical operations with <br>regard to mixing and melt temperature control. Since the CRD mixer can improve <br>mixing without increasing viscous dissipation it is well suited for foamed polymer <br>extrusion.<br>The second application of the CRD mixer was in the production of color concentrates <br>(CC) on a single screw compounding extruder. The machine was a two-stage extruder <br>and two CRD mixers were used at the end of the first and second stages. The CC <br>quality was improved to the point that dispersing agents could be eliminated from <br>the compounds run on this extruder. CRD mixers are used in single screw extrud-<br>
+<hr>
+<A name=627></a><IMG src="Polymer Extrusion by Chris Rauwendaal-627_1.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>615</b><br>
+ers, twin screw extruders, injection molding machines, and blow molding machines. <br>Other applications of the CRD are post-consumer reclaim with filler, medical appli-<br>cations, heat shrinkable tubing, blown film extrusion, profile extrusion, fiber spin-<br>ning, sheet extrusion, and reactive extrusion.<br>CRD mixers are used in molding operations. Injection molding screws have been <br>equipped with regular CRD mixers, and a new non-return valve (NRV) has been <br>developed that incorporates CRD mixing elements within the NRV [81]. This CRD-<br>NRV is a slide ring NRV with three EMP rings along the length of the valve as shown <br>in Fig. 8.93.<br>
+<b> Figure 8.93</b> <br>
+CRD non-return valve for injection molding<br>
+The first EMP ring is placed at the start of the NRV. The second EMP ring is machined <br>into the internal diameter of the ring. The last EMP ring is machined into the conical <br>tip of the NRV in the form of conical holes. The CRD-NRV fits within the same space <br>as a regular NRV and thus allows simple installation.<br>There are now several twin screw extruders that use CRD mixing elements to en -<br>hance the distributive and dispersive mixing action. This is not surprising for non-<br>intermeshing twin screw extruders, but it is for intermeshing twin screw extruders <br>because it is generally believed that conventional kneading disks provide good dis-<br>persive mixing.<br>
+<b>8.7.1.1.7Conclusions</b><br>
+The CRD mixer technology allows single screw extruders to achieve dispersive mix-<br>ing as good as that of intermeshing twin screw extruders; this was confirmed by <br>mixing experiments [77]. This finding contradicts traditional thinking about mixing <br>in single screw extruders [78]. The new mixer technology allows single screw ex -<br>truders to be used in applications where thus far only twin screw extruders could be <br>considered. Thus, the use of single screw extruders can be broadened significantly. <br>The CRD mixers can be incorporated into existing or new extruder screws, making <br>implementation simple and inexpensive. Mixers that are mounted downstream of <br>extruders are more difficult to install and more expensive.<br>The new mixers can improve mixing not only in single screw extruders, but also in <br>non-intermeshing twin screw extruders. Current tangential extruder have limited <br>dispersive mixing capability. Using the new mixer technology can improve the dis-<br>persive mixing capability of these extruders. Some aspects of the mixing technology <br>can even be applied to intermeshing twin screw extruders to improve dispersive <br>
+<hr>
+<A name=628></a><b>616</b>  <b>8Extruder Screw Design</b><br>
+mixing. In fact, currently several intermeshing twin screw extruders are using CRD <br>type mixing elements. Also, internal mixers can benefit from this new mixer tech-<br>nology; this applies to both batch and continuous internal mixers. In internal mixers <br>the empirical sigma type mixing rotor can be replaced with a more efficient CRD <br>type rotor designed from sound engineering principles.<br>The boundary element method is a useful tool in the development and design of <br>mixing sections with complex geometry. BEM provides a tool that allows a quantita-<br>tive approach to the design of mixing devices. The BEM results of the new mixers <br>indicate that strong elongational flow can indeed be generated by the wedge-shaped <br>geometry of the mixing flights. Also, multiple passes through the HSRs can be <br>achieved, provided that the mixing flight clearance is properly dimensioned.<br>Multi-flighted CRD mixers expose all the material to multiple high stress events to <br>achieve a fine dispersion. Conventional mixers, like the Maddock, expose the mate-<br>rial to only one high stress event, thus limiting the mixing efficiency. The CRD mixer <br>is designed with elongational mixing action and forward pumping capability, allow-<br>ing effective mixing without increasing power consumption or melt temperatures. <br>As a result, the CRD mixer can be made quite long with typical lengths of 6 D. Some <br>of the important benefits of the elongational mixing action are lower melt tempe-<br>ratures, less melt temperature fluctuation, reduced die lip buildup, and the ability to <br>disperse gels--shear based mixers cannot disperse gels.<br>
+<b>8.7.1.2Mixers to Break Up the Solid Bed<br></b>In extrusion visualization experiments it is frequently observed that large chunks <br>break off from the solid bed and travel far down the length of the screw [89]. There <br>are two basic methods to deal with this problem. One is to use some type of a barrier <br>device to keep the large clusters of pellets from traveling to the end of the screw. <br>This can be done with a barrier screw or with a fluted mixing section. The drawback <br>of this method is that there is the risk of choking the polymer flow when too much <br>unmelted material accumulates at the end of the barrier section. Another method is <br>to break up the large clusters into much smaller clusters or even pellets. The advan-<br>tage of this approach is that it actually improves melting as discussed in Section <br>7.3.1.2.<br>The CRD mixer described above is normally designed to break up agglomerates into <br>much smaller aggregates or even individual particles; the final size is typically at <br>the micron or submicron level. However, the same principle can be applied to break <br>up large clusters of unmelted polymer particles into smaller clusters or individual <br>particles (usually pellets) at the millimeter level. A number of variations of the CRD <br>mixer have been developed with the specific objective to break up clusters of un -<br>melted particles--these mixers are called the Cluster BusterTM or CB mixer [89]. An <br>example of the CB mixer is shown in Fig. 8.94.<br>
+<hr>
+<A name=629></a><IMG src="Polymer Extrusion by Chris Rauwendaal-629_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-629_24.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>617</b><br>
+Tra nsport direction<br>
+L<br>
+flight clearancew<br>
+SIDE VIEW OF MIXER<br>
+land<br>
+undercut<br>
+wiping flight<br>
+mixing flight <br>
+f<br>
+R<br>
+m<br>
+SECTION B-B<br>
+B<br>
+Wflight<br>
+B<br>
+f<br>
+R<br>
+R<br>
+D<br>
+A<br>
+SECTION A-A<br>
+A<br>
+UNROLLED VIEW OF MIXER<br>
+<b>Figure 8.94</b>The Cluster Buster mixer<br>
+Experiments with the CB mixer have shown significant improvement in extrudate <br>quality and process stability, allowing higher throughput rates to be achieved. The <br>CB mixer makes it possible to generate a dispersed solids melting (DSM) mecha-<br>nism toward the end of the melting zone in single screw extruders. Since DSM melt-<br>ing is significantly more effective than the conventional contiguous solids melting, <br>the CB mixer accelerates melting and improves the melt quality produced by the <br>screw.<br>Other mixing screws have been developed in the past to disrupt the solid bed and <br>mix unmelted with melted material. The double wave screw shown in Fig. 8.80 <br>breaks up the solid bed and mixes the material by forcing a cross-channel flow by <br>the cyclic variation in channel depth. The principle of the double wave screw was <br>used by Barr in his energy transfer (ET) screw [90]. The ET section is basically a <br>double wave section with occasional undercuts in both flights to force a cross-chan-<br>nel mixing between the two channels. Modeling of the ET mixer is discussed in Sec-<br>tion 12.4.3.2; see also Figs. 12.23 to 12.25.<br>
+<hr>
+<A name=630></a><b>618</b>  <b>8Extruder Screw Design</b><br>
+<b>8.7.1.3Summary of Dispersive Mixers<br></b>Table 8.3 lists important attributes of dispersive mixers. The most important issues <br>for the dispersive mixing capability are the type of flow and the number of passes <br>through the high stress regions.<br>
+<b>Table 8.3</b>Comparison of Dispersive Mixers for Single Screw Extruders<br>
+Mixer<br>
+Pressure Dead Barrel Cost of Number of Distributive<br>
+Type of<br>
+drop<br>
+spots wiped<br>
+mixer<br>
+passes<br>
+mixing<br>
+flow<br>
+Blister<br>
+High<br>
+Some<br>
+No<br>
+Low<br>
+1<br>
+Poor<br>
+Shear<br>
+Egan<br>
+Fair<br>
+No<br>
+Yes<br>
+Fair<br>
+1<br>
+Fair<br>
+Shear<br>
+LeRoy/Maddock<br>
+Fair<br>
+Yes<br>
+Yes<br>
+Fair<br>
+Fair<br>
+Shear<br>
+Fluted CRD<br>
+Low<br>
+No<br>
+Yes<br>
+Fair<br>
+&gt;1<br>
+Fair<br>
+Elongation<br>
+Zorro<br>
+Low<br>
+No<br>
+Yes<br>
+Fair<br>
+1<br>
+Fair<br>
+Shear<br>
+Double wave<br>
+Low<br>
+No<br>
+Yes<br>
+Med.<br>
+&gt;1<br>
+Fair<br>
+Shear<br>
+Energy transfer<br>
+Low<br>
+No<br>
+Yes<br>
+Med.<br>
+&gt;1<br>
+Fair<br>
+Shear<br>
+Helical LeRoy<br>
+Low<br>
+No<br>
+Yes<br>
+Fair<br>
+1<br>
+Fair<br>
+Shear<br>
+Planetary gear<br>
+Fair<br>
+No<br>
+Yes<br>
+High<br>
+&gt;1<br>
+Excellent<br>
+Shear<br>
+CRD mixer<br>
+Low<br>
+No<br>
+Yes<br>
+Fair<br>
+&gt;1<br>
+Good<br>
+Elongation<br>
+CB mixer<br>
+Low<br>
+No<br>
+Yes<br>
+Fair<br>
+&gt;1<br>
+Good<br>
+Elongation<br>
+The blister ring has few attractive attributes. One advantage of the blister ring is <br>that it takes up little space. The differences between the fluted mixers (Egan, LeRoy, <br>fluted CRD, Zorro, and helical LeRoy) are relatively small with the exception that the <br>fluted CRD can generate elongational flow. Both the planetary gear mixer and the <br>CRD achieve multiple passes through the high stress regions with effective flow <br>splitting and, as a result, have good dispersive and distributive mixing capability. <br>The planetary gear mixer, however, is quite expensive. The double wave screw and <br>the energy transfer screw are designed to mix unmelted and melted material by <br>forcing cross-channel flow in a double-flighted section with varying channel depth. <br>This geometry is difficult to manufacture; as a result, the cost of these mixers is <br>relatively high.<br>The CRD and CB mixers are the only dynamic mixers specifically designed to create <br>strong elongational flow. This allows more effective dispersive mixing with lower <br>viscous dissipation. Elongational mixers are the only mixers that have the capability <br>of dispersing gels. The CB mixer is specifically designed to break up clusters of <br>unmelted particles to generate dispersed solids melting. This accelerates the melt-<br>ing process and improves melt quality. Since the CRD and CB mixers achieve flow <br>splitting and reorientation they also have effective distributive mixing capability.<br>
+<hr>
+<A name=631></a><IMG src="Polymer Extrusion by Chris Rauwendaal-631_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-631_2.jpg"><br>
+ <br>
+8.7Mixing Screws  <b>619</b><br>
+<b>8.7.2Distributive Mixing Elements</b><br>
+Distributive mixing is needed where different polymers are blended together with <br>the viscosities reasonably close together. Distributive mixing is easier to achieve <br>than dispersive mixing. Essentially any disruption of the velocity profiles in the <br>screw channel will cause distributive mixing. A common distributive mixing ele-<br>ment is the pin mixing section; see Fig. 8.95.<br>
+<b> Figure 8.95</b> <br>
+The pin mixing section<br>
+The pins cause disturbances in the velocity profile and thus cause mixing. Many dif-<br>ferent patterns have been used to place the pins; however, there is little agreement <br>as to what pattern is most effective.<br>Another well-known mixing element is the Dulmage mixing section shown in <br>Fig. 8.96.<br>
+<b> Figure 8.96</b> <br>
+The Dulmage mixing section<br>
+The Dulmage mixer is a multi-flighted mixer with circumferential grooves machined <br>into the flights to create a number of slots in the flights. The polymer is divided into <br>many narrow channels combined, divided again, etc. This design was patented al -<br>most 30 years ago with Dow Chemical as the assignee [41]. The drawback of the <br>circumferential grooves is that the barrel is not completely wiped by the screw. This <br>can create problems with stagnation and reduced heat transfer between the poly-<br>mer melt and the barrel. A relatively similar mixing section is the Saxton mixing <br>section, shown in Fig. 8.97.<br>
+<b> Figure 8.97</b> <br>
+The Saxton mixing section<br>
+The difference between the Saxton and the Dulmage mixers is that the slots in the <br>Saxton mixer are machined in the helical rather than the circumferential direction. <br>The advantage of this geometry over the Dulmage is that the Saxton mixer com-<br>pletely wipes the barrel surface. Therefore, there is less chance of stagnation and <br>
+<hr>
+<A name=632></a><b>620</b>  <b>8Extruder Screw Design</b><br>
+better heat transfer between the polymer melt and the barrel. This design was pat-<br>ented in 1961 with the assignee being E.I. DuPont de Nemours and Company [42]. <br>Pineapple-shaped mixing sections are also used for distributive mixing; see Fig. 8.98.<br>
+<b> Figure 8.98</b>The pineapple mixing section<br>
+The pineapple mixer is basically a form of a Saxton type mixer with enough slots <br>machined into the flights that the flight segments become diamond shaped. Mixing <br>in pineapple mixers was studied in detail by Rios et al. [99, 100] both experimen-<br>tally and through numerical simulation of flow using the boundary element method. <br>This study is discussed in detail in Section 12.4.3.4; see also Figs. 12.37 to 12.42.<br>A simple screw geometry with slots machined into the flights is shown in Fig. 8.99.<br>
+<b> Figure 8.99</b> <br>
+The slotted extruder screw<br>
+A Swedish company, Axon, has patented such a design in many European countries <br>[43]. Many other slotted mixing sections have been developed over the years.<br>A new concept in mixing screws was developed by J. Fogarty [98]. This screw, the <br>Turbo-ScrewTM, has rectangular openings (windows) machined into the screw flights <br>to enhance mixing and heat transfer. This patented [101] screw geometry requires <br>considerable flight height; therefore, this design can be applied to extrusion opera-<br>tions where deep flighted screws are necessary. One example of such an operation is <br>foamed plastic extrusion where deep flighted screws are used in the secondary <br>extruder to cool down the polymer melt. The Turbo-Screw has been used extensively <br>in foamed plastic operations and has allowed significant increases in throughput as <br>a result of its improved mixing and heat transfer capability. The Turbo-Screw is dis-<br>cussed in more detail in Section 12.4.3.5; see also Figs. 12.45 to 12.47.<br>Another distributive mixing section is the cavity transfer mixing (CTM) section <br>shown in Fig. 8.100.<br>
+<b> Figure 8.100</b> <br>
+The cavity transfer mixer<br>
+<hr>
+<A name=633></a> <br>
+8.7Mixing Screws  <b>621</b><br>
+This mixing section was developed by Gale at RAPRA. It was licensed in the U.S. <br>by David-Standard. It is interesting to note that the cross-cavity mixer concept was <br>described in a patent as early as 1961 [45]. The CTM mixer has cavities both in the <br>rotor and barrel housing; the combination of shearing and reorientation appears to <br>give effective distributive mixing. A mixer similar to the CTM mixer was developed <br>earlier by Barmag in Remscheid, Germany [58]. The mixer reportedly performs both <br>dispersive and distributive mixing with the ability to reduce the particle or drop <br>size of additives down to the micrometer (10-6 m) range. Reifenhauser in Troisdorf, <br>Germany, also has a similar mixer called the Staromix; it has ellipsoidal cavities in <br>the axial direction rather than hemispherical cavities. Another German extruder <br>manufacturer, Paul Kiefel Extrusionstechnik, offers their CT mixer, which uses ellip-<br>soidal cavities in the helical direction. At this point in time there are several mixers <br>with characteristics very similar to the CTM; since the basic RAPRA patent has <br>expired, companies can freely use the CTM or modifications of it.<br>One of the drawbacks of the CTM and similar mixers is that the barrel is not com-<br>pletely wiped by the screw (or rotor). As a result, the CTM is difficult to clean and <br>there is a possibility of stagnation. Therefore, the CTM is not attractive for short runs <br>with frequent material changes because the changeover time can be quite long--it <br>can take two to three hours to clean a CTM. In addition, the CTM is quite expensive <br>and has no pressure generating capability. Because of these disadvantages the CTM <br>is not used as widely as one might expect based on its good mixing characteristics.<br>
+<b>8.7.2.1Ring or Sleeve Mixers<br></b>An interesting mixing device with features similar to the CTM was developed at <br>Twente University in the Netherlands by Semmekrot and patented in several coun-<br>tries [83]. This mixer is called the Twente Mixing Ring or TMR. The TMR consists of <br>a screw with hemispherical cavities in the screw like the CTM. However, it does not <br>have cavities in the barrel. Instead, the CTM uses an annular ring between the screw <br>and the barrel with circular holes machined through the ring in the radial direction <br>[84, 85]. Figure 8.101 shows a picture of the TMR.<br>
+Stationary  barrel<br>
+Sleeve with holes,<br>
+Mixing element,<br>
+rotating at N2<br>
+rotating at N1<br>
+<b> Figure 8.101</b> <br>
+Path of polymer<br>
+The Twente Mixing Ring (TMR)<br>
+<hr>
+<A name=634></a><IMG src="Polymer Extrusion by Chris Rauwendaal-634_1.jpg"><br>
+<b>622</b>  <b>8Extruder Screw Design</b><br>
+The ring in the TMR rotates with the screw by the dragging action of the screw but <br>at a lower rotational speed. As a result, there is a relative motion between the cavi-<br>ties in the screw and the holes in the ring, which results in a mixing action similar <br>to the CTM. An important advantage of the TMR over the CTM is that it does not <br>have cavities in a stationary barrel. This improves the self-cleaning action of the <br>mixer and makes installation significantly easier. The CTM requires a special barrel <br>section and an extension of the screw; this is both costly and makes installation dif-<br>ficult. The TMR fits within the normal length of an extruder or injection molding <br>machine. This makes the TMR less expensive and easier to install. The TMR is also <br>used in injection molding as part of the non-return valve. Figure 8.102 shows a pic-<br>ture of a TMR non-return valve.<br>
+<b> Figure 8.102</b> <br>
+TMR non-return valve for injection molding<br>
+The TMR non-return valve combines the mixing action with the valve action, similar <br>to the CRD non-return valve shown in Fig. 8.93. The TMR started a new class of mix-<br>ers called "ring" mixers, also called "sleeve" mixers. The design of TMR is patented <br>[83]; however, other companies have found ways to get around this patent and even <br>obtain patents on their own ring mixer. An example is the Fluxion mixer by Robert <br>Barr [86]. This mixer differs from the CTM in that the sleeve has circumferential <br>rings on the outside surface. Obviously, this is a potential source of stagnation and, <br>therefore, not attractive from a functional point of view. However, this feature made <br>it possible to get around the TMR patent. The Fluxion mixer has been tested at Dow <br>Chemical [87, 88].<br>
+<b>8.7.2.2Variable Depth Mixers<br></b>In variable depth mixers, the channel depth of the mixer is varied to obtain improved <br>mixing. An example of a variable depth mixer is the double wave screw shown in <br>Fig. 8.81 and the energy transfer screw; see Fig. 12.23. Another example is the Pul-<br>sar mixing section shown in Fig. 8.103.<br>
+<b> Figure 8.103</b> <br>
+The Pulsar mixing section<br>
+In the Pulsar mixer a helical groove is machined into the root of the screw; the helix <br>angle of the groove is greater than the helix angle of the flight. As a result, some <br>
+<hr>
+<A name=635></a> <br>
+8.7Mixing Screws  <b>623</b><br>
+cross-channel mixing is induced by the groove. The Strata-blend mixer is another <br>variable depth mixer; the mixer is shown in Fig. 8.104.<br>
+<b> Figure 8.104</b> <br>
+The Strata-blend mixer<br>
+Three grooves are machined into the root of the screw in the Strata-blend mixer. The <br>grooves have the same helix angle as the flight, and the grooves are not continuous. <br>This forces material to flow from one groove to the next. Most variable depth mixers <br>have little flow splitting and reorientation; therefore, their distributive mixing cap-<br>ability tends to be limited.<br>
+<b>8.7.2.3Summary of Distributive Mixers<br></b>Based on the important characteristics of mixers we can compile the various distri-<br>butive mixers and list how they perform with respect to different criteria; this is <br>shown in Table 8.4.<br>
+<b>Table 8.4</b>Comparison of Various Distributive Mixers<br>
+Mixers<br>
+Pressure Dead<br>
+Barrel<br>
+Operator<br>
+Mixer<br>
+Disp. Shear Splitting,<br>
+drop<br>
+spots<br>
+wiped<br>
+friendly<br>
+cost<br>
+mixing strain<br>
+reorient-<br>
+ing<br>
+Pins<br>
+High<br>
+Yes<br>
+Partial<br>
+Good<br>
+Low<br>
+No<br>
+Low<br>
+Fair<br>
+Dulmage<br>
+Low<br>
+No<br>
+Partial<br>
+Good<br>
+Fair<br>
+No<br>
+High<br>
+Good<br>
+Saxton<br>
+Low<br>
+No<br>
+Yes<br>
+Good<br>
+Fair<br>
+No<br>
+High<br>
+Good<br>
+CRD<br>
+Low<br>
+No<br>
+Yes<br>
+Good<br>
+Fair<br>
+Yes<br>
+High<br>
+Good<br>
+CTM<br>
+High<br>
+Yes<br>
+Yes<br>
+Bad<br>
+High<br>
+Some<br>
+High<br>
+Good<br>
+TMR<br>
+High<br>
+Yes<br>
+Yes<br>
+Fair<br>
+Medium<br>
+Some<br>
+High<br>
+Good<br>
+Axon<br>
+Low<br>
+No<br>
+Yes<br>
+Good<br>
+Low<br>
+No<br>
+High<br>
+Low<br>
+Double <br>
+Low<br>
+No<br>
+Yes<br>
+Good<br>
+High<br>
+Some<br>
+High<br>
+Low<br>
+wave<br>Pulsar<br>
+Low<br>
+No<br>
+Yes<br>
+Good<br>
+Fair<br>
+No<br>
+Fair<br>
+Low<br>
+Strata-blend<br>
+Low<br>
+Yes<br>
+Yes<br>
+Good<br>
+Fair<br>
+No<br>
+Fair<br>
+Low<br>
+The last column in Table 8.4 is the most important when it comes to distributive <br>mixing effectiveness. The Dulmage, Saxton, CTM, and TMR all do very well in this <br>category. The Saxton mixer combines good mixing with low cost, good streamlining, <br>ease of use, and low pressure drop. The CRD has characteristics similar to the Sax-<br>ton mixer with the difference that the CRD is also capable of dispersive mixing.<br>It should be remembered that static mixing devices can also be quite effective in <br>distributive mixing capacity; see Section 7.7.2. Thus, if distributive mixing is re -<br>quired, one should consider application of a static mixing device.<br>
+<hr>
+<A name=636></a><b>624</b>  <b>8Extruder Screw Design</b><br>
+<b></b><br>
+ <b>8.8Efficient Extrusion of Medical Devices</b><br>
+<b>8.8.1Introduction</b><br>
+Medical extrusion can present special challenges in terms of product size, dimen-<br>sional control, physical properties, and others. Automation of the extrusion line is <br>critical to achieve high levels of process stability and reproducibility. Polymer degra-<br>dation can be a significant concern in medical extrusion. Degradation is affected by <br>the stresses and temperatures that occur in extrusion; both depend strongly on the <br>screw geometry.<br>Melt temperatures can vary significantly in extrusion. Therefore, screws with mix-<br>ing elements are required to achieve good melt temperature uniformity. Barrier <br>screws can achieve relatively uniform melt temperatures but can result in large melt <br>pressure variation. Barrier screws also tend to result in significant reduction in <br>molecular weight (MW). This is likely due to the high shear stresses that occur when <br>the polymer melt flows over the barrier flight. The same problem occurs in fluted <br>mixing sections. This explains why screws with fluted mixing sections (e.g., the <br>LeRoy-Maddock mixer) tend to result in significant MW reduction as well.<br>Mixing screws based in elongational mixing devices and without barrier flights <br>result in minimal MW reduction. These screws have little melt temperature and <br>pressure variation, can disperse gels, and achieve a high quality product with mini-<br>mal dimensional variation. As a result, such extruder screws are attractive in medi-<br>cal extrusion operations where MW reduction has to be minimized and dimensional <br>control and product quality maximized.<br>In the extrusion of medical devices, there are special requirements that go beyond <br>those that apply to the extrusion of non-medical products [106­108, 134]. Figure <br>8.105 illustrates the multiple requirements that apply to the extrusion of medical <br>products.<br>
+Efficient<br>
+Total line<br>
+machinery<br>
+Preventive<br>
+control<br>
+maintenance<br>
+Instrumentation<br>
+and control<br>
+Quality<br>
+<b>Efficient</b><br>
+materials<br>
+Data acquisition<br>
+<b>Extrusion</b><br>
+Trained and motivated<br>
+system<br>
+work force<br>
+Design of<br>
+Efficient<br>
+experiments<br>
+troubleshooting<br>
+Statistical<br>
+process<br>
+Ef icient<br>
+control<br>
+Good manufacturing<br>
+change-over<br>
+practices (GMP)<br>
+<b>Figure 8.105</b>There are multiple requirements for efficient medical extrusion<br>
+<hr>
+<A name=637></a> <br>
+8.8Efficient Extrusion of Medical Devices  <b>625</b><br>
+Essentially all of these requirements apply to all extrusion operations. However, too <br>often, a number of these requirements are disregarded; this is unacceptable in med-<br>ical extrusion.<br>
+<b>8.8.2Good Manufacturing Practices in Medical Extrusion</b><br>
+Good manufacturing practices (GMP) are essential in medical extrusion and involve <br>both a high level of sanitation and process reproducibility. Process reproducibility <br>can be quantified with statistical techniques that have been developed in the field <br>of statistical process control (SPC). Therefore, SPC is a necessary requirement in <br>medical extrusion [107]. GMP also requires full documentation and traceability. <br>With respect to medical extruder machine design, the following aspects are of great <br>importance:<br>
+ <br>
+ Polished, detailed design for all components with respect to cleaning<br>
+ <br>
+ Stainless steel machine frame and barrel cover<br>
+ <br>
+ Ground and polished welding seams<br>
+ <br>
+ Complete documentation and calibration of all process parameters<br>
+ <br>
+ Quick-release couplings for cooling and heating system<br>
+ <br>
+ FDA-approved gear oils and lubricants<br>
+ <br>
+ Mercury-free pressure sensors<br>
+ <br>
+ Contact surfaces made out of stainless steel or nickel-based alloys<br>
+ <br>
+ FDA-approved paints for parts that cannot be made out of stainless steel<br>
+ <br>
+ Validated programmable logic control (PLC) and computer-based control<br>
+<b>8.8.3Automation of the Medical Extrusion Process</b><br>
+The extrusion process allows a high degree of automation. In fact, once the extruder <br>has reached steady-state operation, operator intervention is no longer necessary if <br>complete line control is effectively used. These types of operations offer multiple <br>advantages, such as:<br>
+ <br>
+ Several extrusion lines can be handled by a single operator ("lights-off operation")<br>
+ <br>
+ Product dimensions, such as diameter and wall thickness, are controlled auto-<br>
+matically<br>
+ <br>
+ Product variability is minimized with full line control<br>
+ <br>
+ Electronic data acquisition systems (DAS) allow real-time monitoring of all pro-<br>
+cess variables and pertinent product dimensions<br>
+ <br>
+ Pertinent process and product data can be processed by the DAS to yield process <br>
+control charts and process capability; SPC information is available real-time<br>
+<hr>
+<A name=638></a><b>626</b>  <b>8Extruder Screw Design</b><br>
+ <br>
+ On-line SPC allows instant detection of out-of-spec conditions or out-of-control <br>
+conditions; this allows immediate corrective action, minimizing scrap<br>
+<b>8.8.4Minimizing Polymer Degradation</b><br>
+One of the critical aspects of polymer extrusion is degradation of the polymer. Deg-<br>radation reduces physical properties with or without discoloration of the product; <br>in some cases, gels are produced. There are a number of degradation mechanisms <br>[110], the main ones being thermal, mechanical, biological, and radiation. Degrada-<br>tion in extrusion is affected by temperatures, mechanical stresses, and residence <br>times. The extruder should be designed to minimize all three of these factors.<br>Degradation most often results in the reduction of the molecular weight (MW) of the <br>polymer and the generation of monomers and oligomers. This leads to lower physi-<br>cal properties; this, in turn, can lead to out-of-spec conditions for the extruded pro-<br>duct.<br>The melt temperatures and stresses to which the polymer is exposed in the extruder <br>are strongly influenced by the geometry of the extruder screw. The functions of the <br>extruder screw are<br>
+ <br>
+ Conveying<br>
+ <br>
+ Pressure build-up<br>
+ <br>
+ Heating and melting<br>
+ <br>
+ Mixing, both distributive and dispersive<br>
+ <br>
+ Degassing (in vented extruders)<br>Currently, in most medical extrusion operations, the mixing of screws is used. This <br>is necessary to make high-quality extruded medical products. Simple conveying <br>screws (without mixing elements) are rarely used anymore because they lead to <br>poor homogeneity in the product, as well as dimensional variation.<br>
+<b>8.8.5Melt Temperatures Inside the Extruder</b><br>
+Melt temperatures in the extruder tend to be highly nonuniform because of the low <br>thermal conductivity of polymers. Therefore, it is more efficient to heat the polymer <br>by viscous heat generation than by heat from the barrel heaters. The actual melt <br>temperatures inside the machine can be quite different from the barrel temperature. <br>Also, temperature peaks within the machine are often much higher than the bulk <br>average melt temperature.<br>
+<hr>
+<A name=639></a><IMG src="Polymer Extrusion by Chris Rauwendaal-639_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_48.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_49.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_50.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_51.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_52.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_53.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_54.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_55.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_56.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_57.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_58.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_59.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_60.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_61.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_62.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_63.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_64.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_65.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_66.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_67.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_68.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_69.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_70.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_71.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_72.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_73.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_74.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_75.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_76.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-639_77.jpg"><br>
+ <br>
+8.8Efficient Extrusion of Medical Devices  <b>627</b><br>
+Melt temperatures inside the extruder are difficult to measure because one cannot <br>use an immersion melt temperature sensor along the barrel because the probe will <br>be sheared off by the screw flight. Downstream of the screw, the melt temperature is <br>normally measured with an immersion probe; these can be made with adjustable <br>depth of the probe; see Fig. 8.106.<br>
+<b>Figure 8.106</b>Immersion melt temperature probe with adjustable depth<br>
+Steady state melt temperatures can be calculated using finite element analysis <br>[111­113]. Figure 8.107 shows the melt temperature distribution in a cross-section <br>of the screw channel in the metering section. Figure 8.107 shows that significant <br>temperature changes can occur across the width and depth of the channel. The <br>greatest temperature gradients occur across the depth of the channel.<br>
+175.0<br>
+189.9<br>
+204.7<br>
+219.6<br>
+234.4<br>
+249.3<br>
+264.1<br>
+279.0<br>
+293.8<br>
+182.4<br>
+197.3<br>
+212.1<br>
+227.0<br>
+241.8<br>
+256.7<br>
+271.6<br>
+286.4<br>
+<b>Figure 8.107</b>Melt temperature distribution in a 63 .5-mm single screw extruder<br>
+In reality, the temperature distribution is dynamic; in other words, melt temperature <br>changes with time. These changes can be significant, but short-term (0­10 seconds) <br>temperature changes cannot be measured with a conventional melt tempe rature sen-<br>sor because the thermal mass of the probe is too large. Infrared melt temperature <br>measurement allows detection of rapid (millisecond range) melt temperature fluc-<br>tuation [114­118].<br>
+<b>8.8.6Melt Temperatures and Screw Design</b><br>
+Short-term melt temperature changes can be measured with fast-response thermo-<br>couples. A number of studies using a fast-response thermocouple mesh were con-<br>ducted at Polymer IRC, School of Engineering, Design, and Technology at the Uni-<br>versity of Bradford, England [121­125]. In these studies, dynamic melt temperatures <br>
+<hr>
+<A name=640></a><b>628</b>  <b>8Extruder Screw Design</b><br>
+were observed on a 63.5 mm single screw extruder running a high-density polyethy-<br>lene at different screw speeds; three screw geometries are shown in Fig. 8.108.<br>
+4D/10.53 mm<br>
+10D/tapered<br>
+10D/3.46 mm<br>
+12D/10.53 mm<br>
+2D<br>
+10D/3.5 mm<br>
+taper<br>
+5.5D/12.19 mm<br>
+12.9D/barrier section<br>
+5.6D/4.83 mm<br>
+fluted mixer<br>
+<b>Figure 8.108</b>Three screw geometries tested<br>
+In Fig. 8.108, the first screw (top) is a simple conveying screw with gradual com-<br>pression (10 D length). The second screw (middle) is a simple conveying screw with <br>rapid compression (2 D length); the third screw (bottom) is a barrier screw with a <br>long barrier section (12.9 D) and a fluted mixing section (3 D) in the metering sec-<br>tion of the screw.<br>In these studies, it was found that significant melt temperature variations occur, as <br>much as 45°C over 5 to 10 seconds. Figure 8.109 shows these melt temperature <br>variations for three screw geometries at three barrel temperature profiles and at two <br>screw speeds.<br>
+50<br>
+screw speed 50 rpm<br>
+45<br>
+screw speed 90 rpm<br>
+40<br>
+35<br>
+30<br>
+25<br>
+20<br>
+15<br>
+10<br>
+5<br>
+0<br>
+220<br>
+200<br>
+180<br>
+220<br>
+200<br>
+180<br>
+220<br>
+200<br>
+180<br>
+gradual compression<br>
+rapid compression<br>
+barrier screw<br>
+<b>Figure 8.109</b>Melt temperature variations for three screw geometries<br>
+<hr>
+<A name=641></a> <br>
+8.8Efficient Extrusion of Medical Devices  <b>629</b><br>
+It is interesting to note that the melt temperature variation for the simple conveying <br>screws is quite large. The melt temperature variation at a screw speed of 90 rpm is <br>greater than at 50 rpm--this is true for all three screws. The melt temperature <br>var iation at 90 rpm for the simple conveying screws is about an order of magnitude <br>higher than for the barrier screw.<br>In simple conveying screws, the helical screw channel extends over the entire length <br>of the screw without interruptions, slots, or barriers. As a result, these screws have <br>poor melting and mixing characteristics. These screws are susceptible to unmelt; <br>this is a condition where unmelted particles reach the discharge end of the extruder.<br>It is well known [126] that simple conveying screws have very limited melting and <br>mixing capability. Therefore, it is not surprising that these screws generate melt <br>temperature variations. However, the magnitude of these changes is surprising. In <br>the extrusion industry it is often assumed that the melt temperature changes are <br>quite small (less than 2 to 3°C). The data in Fig. 8.109 show that the actual melt <br>temperature variations can be higher by at least an order of magnitude.<br>High melt temperature variations do not necessarily result in large pressure fluctua-<br>tions. This is evident in Fig. 8.110 where the melt pressure variation is shown for <br>the same three screws at the same screw speeds and barrel temperature profiles.<br>
+3.5<br>
+3.0<br>
+2.5<br>
+screw speed 50 rpm<br>
+screw speed 90 rpm<br>
+2.0<br>
+1.5<br>
+1.0<br>
+0.5<br>
+0<br>
+220<br>
+200<br>
+180<br>
+220<br>
+200<br>
+180<br>
+220<br>
+200<br>
+180<br>
+gradual compression<br>
+rapid compression<br>
+barrier screw<br>
+<b>Figure 8.110</b>Melt pressure variations for three screw geometries<br>
+<hr>
+<A name=642></a><b>630</b>  <b>8Extruder Screw Design</b><br>
+The 220 barrel temperature refers to the temperature of the last barrel temperature <br>zone; the actual profile is 150-185-205-220. For the 200 barrel temperature the <br>actual profile is 140-170-185-200 and for the 180 barrel temperature the actual pro-<br>file is 130-155-165-180. For the simple conveying screws, the barrel temperature <br>profile (BTP) has little effect on the pressure variation, and the pressure variation is <br>quite small. For the barrier screw, the BTP has a significant effect on the pressure <br>variation and the pressure variation is much higher--as much as an order of mag-<br>nitude.<br>The small melt temperature variation with the barrier screw is a distinct advantage; <br>however, the large melt pressure variation with the barrier screw is a distinct dis-<br>advantage. This large pressure variation is likely due to plugging; this is a condition <br>where the solid bed does not melt fast enough to accommodate the reduction in the <br>size of the solids channel. Barrier screws with a long barrier section (over 10 D) are <br>particularly susceptible to this problem [110, 133].<br>
+<b>8.8.7Molecular Degradation and Screw Design</b><br>
+In the extrusion of medical devices, molecular degradation is often a critical issue. <br>This degradation of the MW can be strongly influenced by the geometry of the ex -<br>truder screw. Unfortunately, little information on this problem is available in the <br>open literature. Paakinaho et al. [127] published results of a study on the MW re -<br>duction along the length of a single screw extruder for three polylactic acid (PLA) <br>resins; see Fig. 8.111.<br>PLA22 is a low MW resin, PLA48 a medium MW resin, and PLA63 a high MW resin. <br>Figure 8.111 shows that PLA22 has little or no MW reduction along the length of the <br>extruder, while PLA48 has a moderate MW reduction, about 25%, while PLA63 has <br>significant MW reduction, over 50%. In fact, the final MW of PLA63 at the discharge <br>is actually lower than that of PLA48. This indicates that for this extrusion operation, <br>there is no benefit to using the high MW PLA63 because its MW at the discharge <br>end of the extruder is lower than that of the medium MW PLA48.<br>Figure 8.112 shows the effects of screw speed and extruder length on the MW at the <br>discharge of a 0.75-inch extruder in the extrusion of PLLA polymer. The results <br>demonstrate that lower screw speed results in lower MW. This is likely caused by <br>longer residence time at lower screw speed. Figure 8.112 also shows that the longer <br>extruder (L/D = 25:1) results in a lower MW than the shorter extruder (L/D = 20:1).<br>Finally, Fig. 8.112 shows that using a nitrogen blanket at the feed opening results <br>in a slight increase in MW compared to standard air at the feed opening. These data <br>were generated by a medical company in a study to increase their understanding on <br>how to best retain PLLA molecular weight and the physical properties that depend <br>upon the MW.<br>
+<hr>
+<A name=643></a> <br>
+8.8Efficient Extrusion of Medical Devices  <b>631</b><br>
+0.4<br>
+Feed section<br>
+Transition section<br>
+Metering section<br>
+PLA63<br>
+0.3<br>
+PLA48<br>
+0.2<br>
+PLA22<br>
+0.1<br>
+0.0<br>
+5D<br>
+10D<br>
+15D<br>
+20D<br>
+25D<br>
+Distance along screw<br>
+<b>Figure 8.111</b>MW reduction along extruder screw for three PLA resins<br>
+L/D 25:1<br>
+L/D 20:1<br>
+nitrogen<br>
+50K<br>
+40K<br>
+30K<br>
+20K<br>
+10K<br>
+<b> Figure 8.112</b> <br>
+Molecular weight changes <br>
+0<br>
+by screw speed, screw <br>
+10 rpm<br>
+20 rpm<br>
+30 rpm<br>
+length, and atmosphere<br>
+The effect of screw geometry on MW is shown in Fig. 8.113. In this study, a 0.75-<br>inch extruder was used with a thermoplastic elastomer (TPE). Three screw geome-<br>tries were tested: a barrier screw, a CRD mixing screw, and a Maddock mixing screw. <br>
+<hr>
+<A name=644></a><b>632</b>  <b>8Extruder Screw Design</b><br>
+These studies were conducted by a large medical device company in the develop-<br>ment of implantable devices. The MW results are critical because the polymer is <br>used for implantable CRM lead wire insulation, which is susceptible to hydrolytic <br>and metal ion degradation in the body.<br>
+150000<br>
+140000<br>
+<b>t<br>o<br>n<br>]</b><br>
+lower spec limit<br>
+<b>al </b>130000<br>
+<b> [<br>D<br>W<br>e M<br>ag </b>120000<br><b>ver<br>A</b><br>
+110000<br>
+100000<br>
+Barrier<br>
+CRD<br>
+CRD<br>
+Barrier<br>
+Barrier<br>
+CRD<br>
+Maddock<br>
+<b>Extruder Screw Design</b><br>
+<b>Figure 8.113</b>Molecular weight for three extruder screws<br>
+In this particular application, the molecular weight of the extruded product had <br>to be above a certain minimum level. This turned out to be a significant challenge <br>because the early extrusion trials did not achieve the minimum MW level in the <br>TPE. As a result, a special screw was designed for this application to minimize the <br>stresses that the polymer melt is exposed to in the extrusion process--this is the <br>CRD mixing screw.<br>The barrier screw geometry is shown in Fig. 8.108. The Maddock screw has a fluted <br>mixing element similar to the one shown in the bottom of Fig. 8.108. The CRD screw <br>is a screw with a distributive mixing element that is based on elongational flow <br>rather than shear flow [128­131].<br>Figure 8.113 shows that the barrier screw results in the lowest MW. The Maddock <br>mixing screw resulted in a slightly higher MW. The barrier screw and Maddock <br>screw were not able to produce MW values above the specification level. In barrier <br>screws and fluted mixers, the polymer melt is forced over a barrier flight. As a result, <br>all the melt is exposed to the high shear stresses that occur between the crest of <br>the barrier flight and the extruder barrel. These shear stresses can be high enough <br>to result in a significant reduction of the MW.<br>Figure 8.114 shows more results of the MW for the different extruder screws. Each <br>screw was run at three different temperature profiles. <i>Low</i> indicates barrel tempera-<br>
+<hr>
+<A name=645></a><IMG src="Polymer Extrusion by Chris Rauwendaal-645_1.jpg"><br>
+ <br>
+8.8Efficient Extrusion of Medical Devices  <b>633</b><br>
+tures of 375, 385, and 395°F. <i>Medium</i> indicates barrel temperature of 395, 420, and <br>
+415°F. <i>High</i> indicates barrel temperatures of 400, 420, and 420°F.<br>
+<b>Figure 8.114</b>Molecular weight for four screw and three barrel temperatures<br>
+The MW results for the Maddock screw, general-purpose screw, and barrier screw <br>do not differ markedly; for each barrel temperature profile, the CRD screw achieved <br>the best MW results, resulting in the highest MW values. In all trials with the CRD <br>screw the MW values were above the specification limit. As a result, this screw was <br>selected for the production process. Figure 8.115 shows a photograph of a slotted <br>CRD mixer.<br>
+<b>Figure 8.115</b>Example of CRD mixing element<br>
+<hr>
+<A name=646></a><b>634</b>  <b>8Extruder Screw Design</b><br>
+This CRD mixer uses multiple flights with right-handed orientation. As a result, this <br>mixer has forward pumping capability. The mixing action is achieved by machining <br>slots into the flights. The slots are tapered to generate elongational flow in the slots.<br>Elongational flow results in less viscous heating than shear flow. Therefore, elonga-<br>tional mixing devices offer the following benefits:<br>
+ <br>
+ Lower melt temperatures<br>
+ <br>
+ Fewer gels created<br>
+ <br>
+ Higher viscosity<br>
+ <br>
+ Lower motor load<br>
+ <br>
+ Better dispersion<br>
+ <br>
+ Gels can be dispersed in elongational flow<br>
+ <br>
+ Less degradation<br>Gels are generally crosslinked droplets that can be created in the polymerization <br>process as well as in the extrusion process [110]. However, gels cause frequent <br>problems in medical tubing extrusion where wall thicknesses tend to be very small. <br>Most gels cannot be dispersed in shear flow because the viscosity of the gel is much <br>greater (one or two orders of magnitude) than the matrix viscosity. Grace [132] stud-<br>ied drop breakup in shear and elongational flow. He found that drop breakup in <br>shear flow is not possible when the viscosity of the drop is four times the matrix <br>viscosity or higher. In elongational flow, drop breakup is possible even at very high <br>viscosity ratios (as high as 1000:1).<br>
+<b>8.8.8Conclusions</b><br>
+Efficient extrusion of medical devices is a multi-faceted problem. Only some of the <br>critical issues are discussed here. Melt temperatures in extrusion can be highly <br>non-uniform, particularly in simple conveying screws. Therefore, screws with mix-<br>ing devices are required to produce acceptable melt temperature uniformity. Barrier <br>screws can achieve more uniform melt temperatures than simple conveying screws, <br>but they can have greater melt pressure variation. Pressure variation results in <br>dimensional variation of the extruded product. Therefore, pressure variation in <br>extrusion needs to be kept as low as possible.<br>High stresses and temperatures in the extrusion process result in molecular weight <br>reduction. This reduction is dependent on a number of factors, screw design being <br>one of the most significant issues. Barrier screws tend to result in significant MW <br>reduction. This is likely due to the high shear stresses that occur when the polymer <br>melt flows over the barrier flight. The same problem occurs in fluted mixing sec-<br>tions. This explains why screws with fluted mixing sections (e.g., the LeRoy-Mad-<br>dock mixer) tend to result in significant MW reduction as well.<br>
+<hr>
+<A name=647></a> 8.9Scale-Up  <b>635</b><br>
+Mixing screws based in elongational mixing devices and without barrier flights <br>result in minimal MW reduction. These screws have little melt temperature and <br>pressure variation, can disperse gels, and achieve a high-quality product with mini-<br>mal dimensional variation. As a result, such extruder screws are attractive in medi-<br>cal extrusion operations when MW reduction must be minimized while dimensional <br>control and product quality are maximized.<br>
+<b></b><br>
+ <b>8.9Scale-Up</b><br>
+One of the first articles on scale-up was written by Carley and McKelvey [59]. By <br>analyzing the melt pumping function, they showed that output and power consump-<br>tion increase by the diameter ratio cubed if channel depth and width are increased <br>in proportion to the diameter ratio and if the screw speed is kept constant. This <br>analysis is only applicable to melt fed extruders where the polymer melt exhibits <br>Newtonian flow characteristics. In plasticating extruders, one has to be concerned <br>with solids conveying, melting, pumping, mixing, and the temperature profiles in <br>the polymer melt. The actual scale-up factors will generally be a compromise be -<br>tween the various functional requirements because different functional require-<br>ments often result in conflicting scale-up factors, e.g., mixing and heat transfer.<br>
+<b>8.9.1Common Scale-Up Factors</b><br>
+The most commonly used scale-up method maintains constant shear rate by increas-<br>ing the channel depth proportional to the square root of the diameter ratio and by <br>reducing the screw speed by the square root of the diameter ratio [60]. The resulting <br>scaling factors for output, residence time, melting capacity, power consumption, and <br>specific energy consumption are shown in Table 8.5.<br>The pumping capacity can be checked by using Eq. 7.291. The drag flow rate can be <br>written as:<br>
+ (8.168)<br>
+If the helix angle  is constant, W   D, H   D, and N   1D, the drag flow rate ratio <br>becomes:<br>
+ (8.169)<br>
+<hr>
+<A name=648></a><b>636</b>  <b>8Extruder Screw Design</b><br>
+The pressure flow rate can be written as:<br>
+ (8.170)<br>
+The pressure flow ratio becomes:<br>
+ (8.171)<br>
+In order for the pressure flow rate to increase by the same rate as the drag flow, the <br>helical length for pressure build-up has to be:<br>
+ (8.172)<br>
+This means that the axial length of the metering section can be increased by the <br>square root of the diameter ratio. Thus, the L/D of the metering section can be <br>reduced by the square root of the diameter ratio.<br>The melting capacity can be evaluated by examining Eq. 7.166. The melting rate  p <br>
+can be written as:<br>
+ (8.173)<br>
+Thus, the ratio of melting capacity becomes:<br>
+ (8.174)<br>
+With vb   D, z2  D, and Ws2   D, the ratio becomes:<br>
+ (8.175)<br>
+This indicates that the increase in melting rate does not keep up with the increase <br>in output. This problem can be alleviated by increasing the length of the melting <br>section more than a simple proportional increase. In order to keep the increase in <br>melting rate the same as the increase in total throughput, the length of the melting <br>section should increase by z2   D1.25. Thus, the L/D of the melting section will <br>
+increase by L/Dmelt   D0.25. In order to match the melting capacity and the pumping <br>
+capacity, the L/D of the melting section has to be increased and the L/D of the <br>pumping section reduced.<br>
+<hr>
+<A name=649></a> 8.9Scale-Up  <b>637</b><br>
+The solids conveying rate can be evaluated by using Eqs. 7.46 and 7.48:<br>
+ (8.176)<br>
+If the solids conveying angle is considered to be relatively constant, the solids con-<br>veying ratio can be written as:<br>
+ (8.177)<br>
+Thus, the solids conveying rate increases at the same rate as the pumping capacity.<br>The screw power consumption can be evaluated by considering that the power con-<br>sumption is roughly determined by the barrel surface area A, the shear stress acting <br>at the barrel surface area , and the barrel velocity vb:<br>
+ (8.178)<br>
+Considering that A = DL and that the shear stress will be constant because the <br>shear rate is constant as long as the polymer melt viscosity remains constant, the <br>power consumption becomes:<br>
+ (8.179)<br>
+If L   D and N   1/D, the power consumption becomes:<br>
+ (8.180)<br>
+This is not a good situation because the power consumption increases more rapidly <br>than the output, causing an increase in specific energy consumption and thus melt <br>temperature. If the length L is reduced less than proportional to D, then the increase <br>in power consumption can be reduced. In order to match the increase in power con-<br>sumption to the increase in output, the total length L should increase as:<br>
+ (8.181)<br>
+This would mean a reduction in the total L/D by D. For the pumping section this is <br>possible; however, it is not possible for the melting section. The other alternative is <br>to reduce the screw speed by more than the square root of the diameter ratio.<br>The effect of the common scale-up factors on extruder performance is presented in <br>tabular form in Table 8.5.<br>The scale-up factors listed in Table 8.5 are the exponents of the diameter ratio. For <br>instance, if the exponent for channel depth is 0.5, then the channel depth of the <br>large extruder H2 is related to the channel depth of the small extruder H1 by the fol-<br>
+<hr>
+<A name=650></a><b>638</b>  <b>8Extruder Screw Design</b><br>
+lowing relationship: H2 = H1 (D2/D1)0.5; D2 is the diameter of the large extruder and <br>
+D1 of the small extruder.<br>
+<b>Table 8.5</b>Common Scale-Up Factors<br>
+Common scale-up<br>
+Scale-up for heat transfer<br>
+Scale-up for mixing<br>
+Channel depth<br>
+0.5<br>
+0.5<br>
+1<br>
+Screw speed<br>
+­0.5<br>
+­1.0<br>
+0<br>
+Output<br>
+2.0<br>
+1.5<br>
+3<br>
+Shear rate<br>
+0.0<br>
+­0.5<br>
+0<br>
+Tip speed<br>
+0.5<br>
+0.0<br>
+1<br>
+Residence time<br>
+0.5<br>
+1.0<br>
+0<br>
+Melting rate<br>
+1.75<br>
+1.5<br>
+2<br>
+Solids conveying<br>
+2.0<br>
+1.5<br>
+3<br>
+Screw power<br>
+2.5<br>
+1.5<br>
+3<br>
+Specific energy<br>
+0.5<br>
+0.0<br>
+0<br>
+<b>8.9.2Scale-Up for Heat Transfer</b><br>
+In Section 5.3.3, heat transfer was analyzed in a Newtonian fluid between two plates, <br>one stationary at temperature T0 and one moving at velocity v and at temperature T1. <br>
+When conduction, convection, and dissipation all play a role of importance, the tem-<br>perature profile is described by Eq. 5.69. Two dimensionless numbers determine <br>the temperature distribution, the Graetz number, and the Brinkman number. If <br>these numbers remain the same in scale-up, the temperature profile in the polymer <br>melt will also remain the same. A constant Graetz number requires that:<br>
+ (8.182)<br>
+It can be assumed that the thermal diffusivity () is constant. Considering that <br>v = DN and the L/D ratio is usually constant, Eq. 8.182 can be written as:<br>
+ (8.183)<br>
+A constant Brinkman number requires that:<br>
+ (8.184)<br>
+If it can be assumed that the viscosity , the thermal conductivity k, and the imposed <br>temperature difference T are constant, Eq. 8.184 becomes:<br>
+ (8.185)<br>
+<hr>
+<A name=651></a> 8.9Scale-Up  <b>639</b><br>
+This means that the circumferential speed (v = DN) has to be constant. With <br>Eqs. 8.183 and 8.184 the channel depth and screw speed can be expressed as a <br>function of diameter:<br>
+ (8.185a)<br>
+ (8.185b)<br>
+The effect of these scale-up factors on extruder performance is shown in Table 8.5. <br>It can be seen that there is a good match between pumping rate, melting rate, and <br>solids conveying rate. Further, the specific energy consumption remains constant, <br>thus the melt temperature level should be about the same in scale-up. The main <br>disadvantage of this approach is that the output will be considerably lower than the <br>common scale-up factors; see Table 8.5. For instance, if D1 = 50 mm and  1 = 100 <br>
+kg/hr, then for D2 = 150 mm, the common scale-up factor will give  2 = 900 kg/hr, <br>
+whereas the scale-up for heat transfer will give  2 = 520 kg/hr. In practice, there-<br>
+fore, the screw speed will be increased until the melt temperature almost reaches <br>the maximum acceptable temperature. The residence time increases faster than <br>with the common scale-up factors.<br>
+<b>8.9.3Scale-Up for Mixing</b><br>
+If it is assumed that the two most important parameters in mixing are shear rate <br>and residence time, then scale-up rules can be derived that will keep these para-<br>meters constant. The shear rate is approximately:<br>
+ (8.186)<br>
+The residence time is:<br>
+ (8.187)<br>
+If L/D is constant, then Eq. 8.187 requires that the screw speed be constant. From <br>Eq. 8.186, it can be seen that with constant N the ratio of diameter to channel depth <br>also must be constant. Thus, the scale-up factors for mixing become:<br>
+ (8.187a)<br>
+ (8.187b)<br>
+The effect of these geometric scale-up factors on extruder performance is shown in <br>Table 8.5. The main problem with this scale-up approach is that the output increases <br>much faster than the melting capacity. This approach, therefore, will not work <br>
+<hr>
+<A name=652></a><b>640</b>  <b>8Extruder Screw Design</b><br>
+unless special design changes are made in the melting section of the screw to sub-<br>stantially enhance the melting capacity; see Section 8.2.2. The advantage of this <br>scale-up approach is that very high outputs are obtained and the specific energy <br>consumption remains constant.<br>A comparison of the effect on output of the different scale-up strategies is shown <br>in Table 8.6. If the diameter of the small extruder is 50 mm (2 in) and the output <br>100 kg/hr (220 lbs/hr), the output for a 150-mm (6 in) extruder will be as shown in <br>Table 8.6.<br>From Table 8.6, it can be seen that scale-up for mixing results in very high output <br>values. Summarizing, it can be stated that scale-up for heat transfer will result in <br>matched solids conveying, melting, and pumping in addition to constant specific <br>energy consumption. Scale-up for heat transfer, therefore, will result in good <br>extruder performance and melt temperature control. However, outputs according to <br>the scale-up for heat transfer are rather low. Outputs can be increased by increasing <br>the screw speed by more than N   1/D; however, this will result in an increase in <br>specific energy and insufficient melting capacity. Thus, the melt quality will deterio-<br>rate. In practice, one would increase the screw speed to just below where the melt <br>quality becomes unacceptable.<br>
+<b>Table 8.6</b>Output According to Various Scale-Up Rules<br>
+Scale-up method<br>
+Throughput large extruder (150 mm)<br>
+Common scale-up factors<br>
+900 kg/hr<br>
+Scale-up for heat transfer<br>
+520 kg/hr<br>
+Scale-up for mixing (geometric)<br>
+2700 kg/hr<br>
+Geometric scale-up has many attractive features. The main drawback is that the <br>melting capacity does not increase as fast as the solids conveying and melt convey-<br>ing capacity. This can create melting problems on the larger machine unless special <br>measures are taken to improve melting. One simple way of doing this is to increase <br>the length of the extruder. The melting problem is less likely to be a problem with <br>amorphous polymers than with semi-crystalline polymers because the enthalpy rise <br>with amorphous polymer tends to be significantly lower than with semi-crystalline <br>polymers; see Section 6.3.4.<br>
+<b>8.9.4Comparison of Various Scale-Up Methods</b><br>
+Rauwendaal [91] compared a number of existing scale-up methods and proposed <br>two new methods. The different scale-up methods are compared by how the three <br>primary variables are changed in the scale-up. The primary variables are channel <br>depth, length, and screw speed. The resulting performance of the extruder can be <br>
+<hr>
+<A name=653></a> 8.9Scale-Up  <b>641</b><br>
+expressed as a function of the exponents of the primary variables. This is shown in <br>Table 8.7 with the primary variables listed at the top.<br>
+<b>Table 8.7</b>Basic Relationships and Three Scale-Up Methods<br>
+Relations<br>
+I<br>
+II<br>
+III<br>
+Channel depth<br>
+h<br>
+1<br>
+0.5<br>
+(1+n)/(1+3n)<br>
+Axial length<br>
+l<br>
+1<br>
+1<br>
+1<br>
+Screw speed<br>
+v<br>
+0<br>
+­0.5<br>
+­(2+2n)/(1+3n)<br>
+Shear rate<br>
+1+v­h<br>
+0<br>
+0<br>
+­2/(1+3n)<br>
+Pumping rate<br>
+h+2+v<br>
+3<br>
+2<br>
+(1+5n)/(1+3n)<br>
+Melting1)<br>
+1+0.5v+lt<br>
+2<br>
+1.75<br>
+(1+5n)/(1+3n)<br>
+Melting2)<br>
+2+v+0.5nv+lt<br>
+3<br>
+2.5­0.25n<br>
+(­n2+6n+1)/(1+3n)<br>
+Solids conveying<br>
+2+h+v<br>
+3<br>
+2<br>
+(1+5n)/(1+3n)<br>
+Residence time<br>
+­1­v+l<br>
+0<br>
+0.5<br>
+(2+2n)/(1+3n)<br>
+Shear strain<br>
+1­h<br>
+0<br>
+0.5<br>
+2n/(1+3n)<br>
+Power consumption<br>
+2+n+l+nv+v-nh<br>
+3<br>
+2.5<br>
+(1+5n)/(1+3n)<br>
+Specific energy<br>
+1­h+n+nv­nh<br>
+0<br>
+0.5<br>
+0<br>
+Area/throughput<br>
+­1­h+l­v<br>
+­1<br>
+0<br>
+(1+n)/(1+3n)<br>
+I  scale-up proposed by Carley and McKelvey [92]<br>
+II  scale-up proposed by Maddock [93]<br>
+III  scale-up proposed by Pearson [94]<br>1)  at low Brinkman number<br>2)  at high Brinkman number<br>
+The scale-up proposed by Carley and McKelvey [92] is the same as the scale-up for <br>mixing also called geometrical scale-up, discussed in Section 8.8.3. The scale-up <br>proposed by Maddock [93] is the same as the common scale-up discussed in Section <br>8.8.1. The scale-up proposed by Pearson is the most comprehensive and consistent. <br>There is good balance between solids conveying, melting, and melt conveying; fur-<br>ther, the specific energy consumption is constant. A drawback of the Pearson scale-<br>up is that the output increase is rather low; this makes the scale-up unattractive in <br>practice. More scale-up methods are listed in Table 8.8.<br>The scale-up by Fenner and Yi [95] suffers from the fact that the specific energy <br>consumption increases a large amount. This is generally detrimental in scale-up. <br>Potente and Fischer [96] developed scale-up rules for both conventional and feed <br>controlled (grooved feed) extruders. The features of this scale-up are not attractive. <br>The solids conveying rate does not match the melting or melt conveying rate, the <br>throughput increase is rather low, the residence time increases considerably, and <br>the specific energy increases for shear thinning polymers (n &lt; 1).<br>Rauwendaal [97] proposed two new scale-up methods that result in constant me -<br>chanical specific energy consumption and high throughput rates. The first one keeps <br>the specific surface area constant. This scale-up should work well for high values of <br>
+<hr>
+<A name=654></a><b>642</b>  <b>8Extruder Screw Design</b><br>
+the Brinkman number; at low values of the Brinkman number the melting rate may <br>be insufficient. The second scale-up method keeps the melting rate at a low Brink-<br>man number equal to the pumping rate and, thus, should be useful in cases where <br>the first scale-up method cannot be used.<br>
+<b>Table 8.8</b>Comparison of Several Scale-Up Methods<br>
+IV<br>
+V<br>
+VI<br>
+VII<br>
+Channel depth<br>
+0.3<br>
+0.7<br>
+(1+n)/(1+2n)<br>
+1/(2n)<br>
+Axial length<br>
+1<br>
+1<br>
+1<br>
+(1+n)/(2n)<br>
+Screw speed<br>
+­0.3<br>
+­0.6<br>
+­(1+n)/(1+2n)<br>
+­1<br>
+Shear rate<br>
+0.4<br>
+­0.3<br>
+­1/(1+2n)<br>
+­1/(2n)<br>
+Pumping rate<br>
+2<br>
+2.1<br>
+2<br>
+(1+2n)/(2n)<br>
+Melting1)<br>
+1.85<br>
+1.7<br>
+(3+7n)/(2+4n)<br>
+(1+2n)/(2n)<br>
+Melting2)<br>
+2.7­0.15n<br>
+2.4­0.3n<br>
+(­n2+9n+6)/(2+4n)<br>
+(­n2+3n+1)/(2n)<br>
+Solids conveying<br>
+2<br>
+2.1<br>
+2<br>
+(1+2n)/(2n)<br>
+Residence time<br>
+0.3<br>
+0.6<br>
+(1+n)/(1+2n)<br>
+(1+n)/(2n)<br>
+Shear strain<br>
+0.7<br>
+0.3<br>
+n/(1+2n)<br>
+0.5<br>
+Power consumption<br>
+2.7+0.4n<br>
+2.4­0.3n<br>
+2<br>
+(1+2n)/(2n)<br>
+Specific energy<br>
+0.7+0.4n<br>
+0.3­0.3n<br>
+0<br>
+0<br>
+Area/throughput<br>
+0<br>
+­0.1<br>
+0<br>
+0.5<br>
+IV scale-up proposed by Fenner and Yi [95]<br>
+V  scale-up proposed by Fischer and Potente [96]<br>
+VI scale-up proposed by Rauwendaal [97]<br>
+VII scale-up proposed by Rauwendaal [97]<br>1)  at low Brinkman number<br>2)  at high Brinkman number<br>
+<b></b><br>
+ <b>8.10Rebuilding Worn Screws and Barrels</b><br>
+In a correctly designed extruder, the majority of the wear should be concentrated on <br>the screw because the screw can be replaced and rebuilt more easily than the bar-<br>rel. In fact, the rebuilding of extruder screws has become so common that the <br>rebuilding business has become a major segment of the extrusion industry. There <br>are more than 70 companies in the U.S. involved in the rebuilding of extrusion <br>equipment. For a number of these companies, screw rebuilding constitutes the <br>major part of their business.<br>One reason for the popularity of screw rebuilding is the fact that rebuilding is usu-<br>ally considerably less expensive than replacement with a new screw. Rebuilding is <br>usually done with hardfacing materials. With the proper choice of hardfacing ma -<br>terial, the rebuilt screw can be better than the original screw. It usually makes no <br>
+<hr>
+<A name=655></a> <br>
+8.10Rebuilding Worn Screws and Barrels  <b>643</b><br>
+economic sense to rebuild small extruder screws (diameter less than 40 mm) be -<br>cause the cost of rebuilding may be the same (or higher) as the manufacture of a <br>new screw. Also, applying hardfacing to a worn, small-diameter screw is difficult <br>and the results are often less than satisfactory. However, larger diameter screws <br>can be hardfaced without much trouble and generally can be rebuilt numerous <br>times.<br>Properties of several hardfacing materials are listed in Table 8.9 [104].<br>
+<b>Table 8.9</b>Properties of Hardfacing Materials<br>
+Product<br>
+Base<br>
+Hardness Cracking<br>
+%<br>
+%<br>
+%<br>
+%<br>
+Cost/lb<br>
+material<br>
+Rc<br>
+Tendency Carbon Chro- Tungs- Boron<br>
+[$]<br>
+mium<br>
+ten<br>
+Stellite 1<br>
+Cobalt<br>
+48­54<br>
+High<br>
+2.5<br>
+30.0<br>
+12<br>
+­<br>
+25­40<br>
+Stellite 6<br>
+Cobalt<br>
+37­42<br>
+Medium<br>
+1.1<br>
+28.0<br>
+4<br>
+­<br>
+25­40<br>
+Stellite 12<br>
+Cobalt<br>
+41­47<br>
+Medium<br>
+1.4<br>
+29.0<br>
+8<br>
+­<br>
+25­40<br>
+Colmonoy 5<br>
+Nickel<br>
+45­50<br>
+Medium<br>
+0.65<br>
+11.5<br>
+­<br>
+2.5<br>
+15­25<br>
+Colmonoy 56<br>
+Nickel<br>
+50­55<br>
+High<br>
+0.70<br>
+12.5<br>
+­<br>
+2.7<br>
+15­25<br>
+Colmonoy 6<br>
+Nickel<br>
+56­61<br>
+High<br>
+0.75<br>
+13.5<br>
+­<br>
+3.0<br>
+15­25<br>
+Colmonoy 83<br>
+Nickel<br>
+50­55<br>
+High<br>
+2.0<br>
+20.0<br>
+34<br>
+1.0<br>
+40­50<br>
+N-45<br>
+Nickel<br>
+30­40<br>
+Medium<br>
+0.3<br>
+11.0<br>
+­<br>
+2.2<br>
+15­25<br>
+N-50<br>
+Nickel<br>
+40­45<br>
+Medium<br>
+0.4<br>
+12.0<br>
+­<br>
+2.4<br>
+15­25<br>
+N-56<br>
+Nickel<br>
+45­50<br>
+High<br>
+0.6<br>
+13.5<br>
+­<br>
+2.8<br>
+15­25<br>
+The steps involved in rebuilding a screw are [104]:<br>1. The screw is set up in a lathe and a center is found. At this time the screw is <br>
+checked for straightness and concentricity;<br>
+2. The screw is polished and prepped for stripping off the existing chrome;<br>3. The entire screw is submerged in an acid bath to remove the chrome plating;<br>4. The screw then moves to the grinder where it is ground undersize;<br>5.  The screw flights are welded with a hardfacing material such as Colmonoy 56 or <br>
+Stellite 12;<br>
+6. The screw goes back to the grinder for rough grind after welding. The screw is <br>
+also checked for straightness;<br>
+7. The flight grinder is used to trim the sides of the flight;<br>8. The screw goes to the polishing booth for a rough polish;<br>9. The screw is inspected and buffed for chrome plating if needed;<br>10. Chrome plating is applied to the entire root and bearing surface;<br>11. The screw is buffed after chrome;<br>12. The screw is ground to the final O.D. specification;<br>
+<hr>
+<A name=656></a><b>644</b>  <b>8Extruder Screw Design</b><br>
+13. Final polish and buff as needed;<br>14. Grind front surface, size register and board the O.D.;<br>15. Final inspection.<br>
+<b>8.10.1Application of Hardfacing Materials</b><br>
+There are four commonly used hardfacing techniques in the industry [105]. They <br>are oxyacetylene, tungsten inert gas (TIG), plasma transfer arc (PTA), and metal <br>inert gas (MIG). Each method has certain advantages and disadvantages that will be <br>discussed next. Sometimes a layer of stainless steel is applied on the flight before <br>applying the hardfacing material. This can be done to control the dilution of the <br>hardfacing material with the screw base material, to improve the bond, and to <br>reduce the cracking of the hardfacing.<br>
+<b>8.10.1.1Oxyacetylene Welding<br></b>In this process an intense flame is produced by burning a controlled mixture of oxy-<br>gen and acetylene gas; see Fig. 8.116.<br>
+<b> Figure 8.116</b> <br>
+Oxyacetylene welding<br>
+The gases are drawn from separate sources through pressure regulators and intro-<br>duced into a torch for mixing. The gases exit the welding nozzle where they are <br>ignited. The flame intensity depends on the flow rate of the gases, the gas mixture <br>ratio, the properties of the fuel gas selected, and the type of nozzle used.<br>Welds are formed from the weld puddle created through contact of the flame, the <br>work piece, and the welding rod. Oxyacetylene welding requires a high degree of <br>skill to obtain high-quality deposits and the process is slow. The benefit of oxyace-<br>tylene welding is that it provides the least base metal dilution of any method. A one-<br>layer deposit is usually sufficient to reach the desired hardness.<br>
+<hr>
+<A name=657></a> <br>
+8.10Rebuilding Worn Screws and Barrels  <b>645</b><br>
+<b>8.10.1.2Tungsten Inert Gas Welding<br></b>TIG welding is an arc fusion welding process in which intense heat is produced by <br>an electric arc between a non-consumable, torch-held tungsten electrode and a work <br>piece; see Fig. 8.117.<br>
+<b> Figure 8.117</b> <br>
+Tungsten inert gas welding process<br>
+An inert shielding gas, generally argon, is introduced through the torch to protect <br>the weld zone from atmospheric contamination. TIG welding is the method most <br>commonly used in the manufacturing and rebuilding of extruder screws. The local-<br>ized, intense heat of TIG results in some base metal dilution. As a result, it may be <br>necessary to apply a second layer to achieve full hardness of the hardfacing mate-<br>rial.<br>
+<b>8.10.1.3Plasma Transfer Arc Welding<br></b>A PTA torch consists of an electrode in the center surrounded by a double-walled <br>tube that carries the powdered metal. Argon gas passes through this annulus while <br>metal powder is metered through the holes in the inside wall of the tube. Both exit <br>onto the work piece through an arc struck between the electrode and the work piece; <br>see Fig. 8.118. Argon gas is circulated around the welding zone to provide a shield <br>around the arc region.<br>
+<b> Figure 8.118</b> <br>
+Plasma transfer arc welding process<br>
+<hr>
+<A name=658></a><b>646</b>  <b>8Extruder Screw Design</b><br>
+<b>8.10.1.4Metal Inert Gas Welding<br></b>In the MIG welding process, an electric arc is established between the work piece <br>and a wire electrode. The electrode is continuously fed by a wire feeder through a <br>torch. The arc continuously melts to form the weld puddle. An appropriate gas or <br>gas mixture shields the weld area from atmospheric contamination. The MIG pro-<br>cess has advantages of high deposition rates, faster speed, and excellent weld <br> <br>quality. A drawback of MIG is that base metal dilution is more than other processes. <br>As a result, a second layer of weld may have to be applied to achieve the desired <br>hardness.<br>
+<b>8.10.1.5Laser Hardfacing<br></b>Another method of applying hardfacing is laser hardfacing on the flight lands. The <br>common hardfacing materials listed in Table 8.9 can be applied by laser hardfac ing <br>along with tungsten carbide composites. The tungsten carbide particle can be spher-<br>ical or angular in shape. Important benefits of the laser process are the low heat <br>input, the low dilution of the deposited alloy, the large variety of hardfacing compo-<br>sitions (powder fed), and the overlays are metallurgically bonded and impervious.<br>These characteristics lead to high quality overlays without cracks, minimal porosity, <br>and high hardness values. For instance, Colmonoy 56 PTA powder can be laser <br>deposited with resulting hardness values in the low 60s Rc. The final thickness that <br>can be achieved for crack sensitive materials ranges from 0.4 to 1.5 mm. The final <br>thickness achievable for less crack sensitive materials like Stellite 6 is not limited <br>because multi-layer deposits can be applied. This method can be used on new <br>screws; the use in rebuilding screws is evaluated on a case-by-case basis. See Table <br>8.10 for a comparision of different welding methods.<br>
+<b>Table 8.10</b>Comparison of Different Welding Methods<br>
+Method<br>
+Speed of<br>
+Metal<br>
+Integrity<br>
+Ease of<br>
+application<br>
+dilution<br>
+of the weld<br>
+automation<br>
+Oxyacetylene<br>
+Poor<br>
+Good<br>
+Fair<br>
+Poor<br>
+TIG<br>
+Good<br>
+Fair<br>
+Good<br>
+Fair<br>
+PTA<br>
+Fair<br>
+Fair<br>
+Good<br>
+Good<br>
+MIG<br>
+Excellent<br>
+Poor<br>
+Good<br>
+Excellent<br>
+Laser<br>
+Excellent<br>
+Excellent<br>
+Excellent<br>
+Excellent<br>
+<b>8.10.2Rebuilding of Extruder Barrels</b><br>
+Rebuilding barrels is usually considerably more difficult than rebuilding screws. If <br>the barrel wear does not exceed about 0.5 mm, the whole barrel can be honed to a <br>larger diameter and an oversized screw can be placed in the machine. The obvious <br>
+<hr>
+<A name=659></a> References <br>
+<b>647</b><br>
+disadvantage of this procedure is that non-standard barrel and screw dimensions <br>result. Thus, screws from other machines can no longer be used in the non-standard <br>extruder. If barrel wear occurs near the end of the barrel, a sleeve can be placed in <br>the barrel. In most cases, however, the barrel wear is such that replacement of the <br>barrel makes more sense than sleeving or increasing I.D. by honing.<br>
+References<br>1.  R.T. Fenner and J.G. Williams, Polym. Eng. Sci., 11, 474­483 (1971)<br>2.  C.J. Rauwendaal, SPE ANTEC, Chicago, 186­299 (1983)<br>3.  J. Dekker, Polytechnisch Tijdschrift (Dutch), 31, 742­746 (1976)<br>4.  C.J. Rauwendaal, SPE ANTEC, Chicago, 151­154 (1983)<br>5.  C.J. Rauwendaal, Plastics Technology, August, 61­63 (1983)<br>6.  C.J. Rauwendaal, U.S. Patent 4,129,386, "Extruder Screw to Increase Throughput"<br>7.  C.J. Rauwendaal, SPE ANTEC, New York, 110­113 (1980)<br>8.  B. Miller, Plastics World, March, 34­38 (1982)<br>9.  R.M. Bonner, 89 October, 1069­1073 (1963)<br>10.  W. Backhoff, R. von Hooren, and F. Johannaber, Kunststoffe, 6, 307 (1977)<br>11.  J.L. Duda, J.S. Vrentas, S.T. Ju, and H.T. Liu, AIChE J., 28, 279 (1982)<br>12.  D. Anders, in "Entgasen von Kunststoffen," VDI-Verlag, Duesseldorf (1980)<br>13.  J.F. Carley, SPE J., 24, 36­41 (1968)<br>14.  W.H. Willert, paper given at Newark section of the SPE on February 8 (1961)<br>15.  B.H. Maddock and P.P. Matzuk, SPE J., 18, 405­408 (1962)<br>16.  D. Anders, Kunststoffe, 66, 250­257 (1976)<br>17.  H. Werner and J. Curry, SPE ANTEC, Boston, 623­626 (1981)<br>18.  N.J. Brozenick and G.A. Kruder, SPE ANTEC, San Francisco, 176­181 (1974)<br>19.  R.J. Nichols, G.A. Kruder, and R.E. Ridenour, SPE ANTEC, Atlantic City, 361­363 <br>
+(1976)<br>
+20.  D. Bernhardt, SPE J., 12, 40­57 (1956)<br>21.  B. Franzkoch, in "Entgasen von Kunststoffen," VDI-Verlag, Duesseldorf (1980)<br>22.  B.H. Maddock, SPE ANTEC, San Francisco, 247­251 (1974)<br>23.  Ch. Maillefer, Swiss Patent 363,149<br>24.  Ch. Maillefer, German Patent 1,207,074<br>25.  Ch. Maillefer, British Patent 964,428<br>26.  P. Geyer, U.S. Patent 3,375,549<br>27.  J.F. Ingen Housz and H.E.H. Meijer, Polym. Eng. Sci., 21, 352­359 (1981)<br>28.  R. Barr, U.S. Patent 3,698,541<br>29.  C.I. Chung, U.S. Patent 4,000,884<br>
+<hr>
+<A name=660></a><b>648</b>  <b>8Extruder Screw Design</b><br>
+30.  W.H. Willert, European Patent Application 34,505, August 26, 1981<br>31.  R.F. Dray and D.L. Lawrence, U.S. Patent 3,650,652<br>32.  J.S. Hsu, U.S. Patent 3,858,856<br>33.  D.T. Kim, U.S. Patent 3,867,079<br>34.  D.T. Kim, U.S. Patent 3,897,938<br>35.  J.F. Ingen Housz, U.S. Patent 4,218,146<br>36.  J.F. Ingen Housz and H.E.H. Meijer, Polym. Eng. Sci., 21, 1156­1161 (1981)<br>37.  G. LeRoy, U.S. Patent 3,486,192<br>38.  B.H. Maddock, SPE J., July, 23­29 (1967)<br>39.  G. Kruder, U.S. Patent 4,173,417<br>40.  G. Kruder, U.S. Patent 3,870,284<br>41.  F.E. Dulmage, U.S. Patent 2,753,595<br>42.  R.L. Saxton, U.S. Patent 3,006,029<br>43.  German Patent 2,026,834<br>44.  G.M. Gale, SPE ANTEC, Chicago, 109­112 (1983)<br>45.  British Patent 930,339<br>46.  F.J. Brinkschroeder and F. Johannaber, Kunststoffe, 71, 138­143 (1981)<br>47.  R.B. Gregory and L.F. Street, U.S. Patent 3,411,179<br>48.  R.B. Gregory, U.S. Patent 3,788,614<br>49.  R.G. Dray, U.S. Patent 3,788,612<br>50.  R.A. Worth, Polym. Eng. Sci., 19, 198­202 (1979)<br>51.  Z. Tadmor and I. Klein, Polym. Eng. Sci., 13, 382 (1973)<br>52.  B. Elbirli, J.T. Lindt, S.R. Gottgetreu, and S.M. Baba, SPE ANTEC, Chicago, 104 (1983)<br>53.  C.J. Rauwendaal, SPE ANTEC, New Orleans, 59­63 (1984)<br>54.  G. Martin, Industrie-Anzeiger, 14, 2651 (1971)<br>55.  I. Manas-Zloczower, A. Nir, and Z. Tadmor, Rubber Chem. Technol., 55, 1250 (1983)<br>56.  P. Heidrich, German Patent 1,145,787 (1959)<br>57.  F.K. Lacher, U.S. Patent 3,271,819<br>58.  M.H. Pahl, "Dispersives Mischen mit Dynamischen Mischern," VDI-Verlag, Duessel-<br>
+dorf, 177­196 (1978)<br>
+59.  D.F. Carley and J.M. McKelvey, Ind. Eng. Chem., 45, 985 (1953)<br>60.  C.I. Chung, Polym. Eng. Sci., 24, 626­632 (1984)<br>61.  J.C. Miller, Tappi J., 67, 64­67, June (1984)<br>62.  D. Kearney and P. Hold, SPE ANTEC, Washington, DC, 17­22 (1985)<br>63.  C.J. Rauwendaal, Polymer Extrusion III Conference, London, 7/1­7/16, September <br>
+11­13 (1985)<br>
+<hr>
+<A name=661></a> References <br>
+<b>649</b><br>
+64.  M.A. Spalding, J. Dooley, and K.S. Hyun, "The Effect of Flight Radius Size on the Per-<br>
+formance of Single-Screw Extruders," 57th SPE ANTEC, 190­194 (1999)<br>
+65.  C.J. Rauwendaal, U.S. Patent 4,798,473, "Extruder Screw to Reduce Energy Use"<br>66.  C.J. Rauwendaal, "Polymer Mixing, A Self-Study Guide," Carl Hanser Verlag, Munich <br>
+(1998)<br>
+67.  L.A. Utracki, "Mixing in Extensional Flow," 14th Annual Meeting Polymer Processing <br>
+Society, Yokohama, Japan, June 8­12 (1998)<br>
+68.  X.Q. Nguyen and L.A. Utracki, U.S. Patent 5,451,106<br>69.  W. Thiele, Polyblends `95-RETEC, Montreal, October 19­20 (1995)<br>70.  Z. Tadmor and I. Manas-Zloczower, Adv. Polym. Technol., 3, no. 3, 213­221 (1983)<br>71.  T.H. Kwon, J.W. Joo, and S.J. Kim, "Kinematics and Deformation Characteristics as a <br>
+Mixing Measure in the Screw Extrusion Process," Polym. Eng. Sci., 34, no. 3, 174­189 <br>(1994)<br>
+72.  C.J. Rauwendaal in "Mixing and Compounding of Polymers, Theory and Practice," <br>
+I. Manas-Zloczower and Z. Tadmor (Eds.), Carl Hanser Verlag, Munich (1994)<br>
+73.  F.N. Cogswell, J. Non-Newtonian Fluid Mech., 4, 23 (1978)<br>74.  P.J. Gramann, L. Stradins, and T.A. Osswald, Int. Polym. Proc. 8, 287 (1993)<br>75. BEM<i>flow,</i> Boundary Element Fluid and Heat Transfer Simulation Program, (c)1996, <br>
+The Madison Group: PPRC<br>
+76.  J. Cheng and I. Manas-Zloczower, Polym. Eng. Sci., 29, 11 (1989)<br>77.  C.J. Rauwendaal, M. del Pilar Noriega, A. Rios, T. Osswald, P. Gramann, B. Davis, and <br>
+O. Estrada, "Experimental Study of New Dispersive Mixer," SPE ANTEC, New York <br>(1999)<br>
+78.  A. Gale, "Compounding in Single-Screw Extruders," Adv. Polym. Technol., 16, no. 4, <br>
+251­262 (1997)<br>
+79.  C.J. Rauwendaal, U.S. Patent 5,932,159, "Screw Extruder with Improved Dispersive <br>
+Mixing," August 3 (1999)<br>
+80.  C. Rauwendaal, P. Gramann, B. Davis, and T. Osswald, U.S. Patent 6,136,246, "Screw <br>
+Extruder with Improved Dispersive Mixing Elements," October 24 (2000)<br>
+81.  C.J. Rauwendaal, "Non-Return Valve with Distributive and Dispersive Mixing Capabil-<br>
+ity," 58th SPE ANTEC, Orlando, FL, 638­641 (2000)<br>
+82.  C.J. Rauwendaal, "The ABCs of Extruder Screw Design," Adv. Polym. Technol. 9, no. 4, <br>
+301­308 (1989)<br>
+83.  G. Semmekrot, U.S. Patent 5,013,233 (1991)<br>84.  M. Esseghir et al., Adv. Polym. Techno., 17, 1 (1998)<br>85.  A.J. Ingen Housz and S.A. Norden, Int. Polym. Process., 10, 120 (1995)<br>86.  R. Barr, U.S. Patent 5,988,866 (1998)<br>87.  J.A. Myers, R.A. Barr, M.A. Spalding, and K.R. Hughes, SPE ANTEC Tech. Papers, 45, <br>
+157 (1999)<br>
+<hr>
+<A name=662></a><b>650</b>  <b>8Extruder Screw Design</b><br>
+88.  B.A. Salamon, M.A. Spalding, J.R. Powers, M. Serrano, W.C. Sumner, S.A. Somers, and <br>
+R.B. Peters, SPE ANTEC Tech. Papers, 46, 479­483 (2000)<br>
+89.  C. Bos and C.J. Rauwendaal, internal research report (2001)<br>90.  R.A. Barr, U.S. Patent 4,405,239<br>91. C.J. Rauwendaal, "Scale-Up of Single Screw Extruders," Polym. Eng. Sci., 27, no. 14, <br>
+1059­1068 (1989)<br>
+92.  J.F. Carley and J.M. McKelvey, Ind. Eng. Chem., 45, 985 (1953)<br>93.  B.H. Maddock, SPE J., 15, 983 (1959)<br>94.  J.R.A. Pearson, Plast. Rubber Process., Sept., 113 (1976)<br>95.  R.T. Fenner and B. Yi, Plast. Rubber Process., Sept., 119 (1976)<br>96.  E. Fischer and H. Potente, Kunststoffe, 67, 242 (1977)<br>97.  C.J. Rauwendaal, Polym. Eng. Sci., 27, no. 14, 1059­1068 (1987)<br>98.  J. Fogarty, C.J. Rauwendaal, D. Fogarty, and A. Rios, "Turbo-Screw, New Screw Design <br>
+for Foam Extrusion," SPE ANTEC Tech. Papers (2001)<br>
+99. A. Rios, P. Gramann, T. Osswald, M. Noriega, "Experimental and Numerical Study of <br>
+Rhomboidal Mixing Sections," Int. Polym. Process., 9, no. 1, 12­19 (2000)<br>
+100.  P. Gramann, M. Noriega, A. Rios, T. Osswald, "Understanding a Rhomboid Distributive <br>
+Mixing Head Using Computer Modeling and Flow Visualization Techniques," SPE <br>ANTEC Tech. Papers (1997)<br>
+101.  J. Fogarty, U.S. Patent 6,015,227, "Thermoplastic Foam Extrusion Screw with Circula-<br>
+tion Channels," (1998)<br>
+102.  K. Luker, "Laboratory Tools for Compounders," Continuous Compounding Conference, <br>
+Beachwood, OH, November 14­15 (2000)<br>
+103. K. Luker, "Recent Laboratory and R&amp;D Developments in Wood-Plastic Composites," <br>
+Wood-Plastic Conference, Baltimore, MD, December 5­6 (2000)<br>
+104. "Plasticating Components 2000," a publication from Spirex Corporation, Youngstown, <br>
+Ohio (2000)<br>
+105.  V. Anand and K. Das, "The Selection of Screw Base and Hard Facing Materials," publica-<br>
+tion CEC SSB 493.01 from Canterbury Engineering Company, Inc., Champlee, Georgia<br>
+106.  C. Rauwendaal, "How to Get Peak Performance &amp; Efficiency Out of Your Extrusion Line," <br>
+Plast. Technol. Mag., June, 33­35 (2010)<br>
+107. C. Rauwendaal, "Boosting Extruder Productivity--Optimize Product Changeover and <br>
+Purging," Plast. Technol. Mag., Sept. (2010)<br>
+108. C. Rauwendaal, "Boosting Extruder Productivity--Trim Your Material &amp; Energy Cost," <br>
+Plast. Technol. Mag., Nov., 26­28 (2010)<br>
+109. C. Rauwendaal, "Statistical Process Control in Extrusion and Injection Molding," Carl <br>
+Hanser Verlag, Munich (2000)<br>
+110.  M. del Pilar Noriega E., and C. Rauwendaal, "Troubleshooting the Extrusion Process," <br>
+2nd ed., Carl Hanser Verlag, Munich (2010)<br>
+<hr>
+<A name=663></a> References <br>
+<b>651</b><br>
+111. C. Rauwendaal, "Leakage Flow in Screw Extruders," Doctoral Thesis, Twente Univer-<br>
+sity of Technology, Department of Mechanical Engineering­Polymer Processing, The <br>Netherlands (1988)<br>
+112. J. Anderson and C. Rauwendaal, "Finite Element Analysis of Flow in Extruders," 52nd <br>
+SPE ANTEC, 298­305, San Francisco, CA (1994)<br>
+113.  C. Rauwendaal, "Modelling and Simulation of Plasticating Single Screw Extrusion, State <br>
+of the Art and Remaining Challenges," softEXTRUSION Workshop, Alvor, Portugal, Oct. <br>15 (2004)<br>
+114.  X. Shen, R. Malloy, J. Pacini, "An Experimental Evaluation of Melt Temperature Sensors <br>
+for Thermoplastic Extrusion," SPE ANTEC Tech. Papers, 918­926 (1992)<br>
+115.  K.D. Sabota, D.R. Lawson, and J.S. Huizinga, "Advanced Temperature Measurements in <br>
+Polymer Extrusion," SPE ANTEC Tech. Papers, 2832­2842 (1995)<br>
+116.  W. Obendrauf, G.R. Langecker, and W. Friesenbichler, "Temperature Measuring in Plas-<br>
+tics Processing with Infrared Radiation Thermometers," Int. Polym. Process., 13, no. 1, <br>71­77 (1998)<br>
+117. C. Maier, "Infrared Temperature Measurement of Polymers," Polym. Eng. Sci., 36, no. <br>
+11, 1502­1512 (1996)<br>
+118. A. Bendada, M. Lamontagne, "A New Infrared Pyrometer for Polymer Temperature <br>
+Measurement during Extrusion Molding," Infrared Phys. Technol., 46, 11­15 (2004)<br>
+119.  W. Schlaffer and H. Janeschitz-Kriegl, "Measurements of Radial Temperature Profiles in <br>
+a Single-Screw Extruder," Plast. &amp; Polym., 193­199 (1971)<br>
+120. I. Bruker, C. Miaw, A. Hasson, and G. Balch, "Numerical Analysis of the Temperature <br>
+Profile in the Melt Conveying Section of a Single Screw Extruder: Comparison with <br>Experimental Data," Polym. Eng. Sci., 27, no. 7, 504­509 (1987)<br>
+121. E.C. Brown, A.L. Kelly, and P.D. Coates, "Melt Temperature Field Measurement in <br>
+ <br>Single Screw Extrusion using Thermocouple Meshes," Rev. Sci. Instrum., 75, no. 11, <br>4742­748 (2004)<br>
+122.  A.L. Kelly, E.C. Brown, and P.D. Coates, The Effect of Screw Geometry on Melt Tempera-<br>
+ture Profile in Single Screw Extrusion, Polym. Eng. Sci., 46, no. 12, 1706­1714 (2006)<br>
+123.  A.L. Kelly, E.C. Brown, K. Howell, and P.D. Coates, "Melt Temperature Field Measure-<br>
+ment in Extrusion Using Thermocouple Meshes," Plast. Rubber Comp., 37, 151­157 <br>(2008)<br>
+124.  E.C. Brown, P. Olley, and P.D. Coates, "In-line Melt Temperature Measurement during <br>
+Real Time Ultrasound Monitoring of Single Screw Extrusion," Plast. Rubber Comp., 29, <br>3­13 (2000)<br>
+125.  A.L. Kelly et al., "Thermal Optimisation of Polymer Extrusion using In-process Monitor-<br>
+ing Techniques," paper submitted for International Conference on Sustainable Thermal <br>Energy Management, Oct. 25­26, Newcastle upon Tyne, UK (2011)<br>
+126.  C. Rauwendaal, "Polymer Extrusion," 4th ed., Carl Hanser Verlag, Munich (2001)<br>127. K. Paakinaho et al., "Melt Spinning of Poly (L/D) Lactide 96/4: Effects of Molecular <br>
+Weight and Melt Processing on Hydrolytic Degradation," Polym. Degrad. Stab., 94, <br>438­442 (2009)<br>
+<hr>
+<A name=664></a><b>652</b>  <b>8Extruder Screw Design</b><br>
+128.  T. Osswald, P. Gramann, B. Davis, and C. Rauwendaal, "A New Dispersive Mixer for Sin-<br>
+gle Screw Extruders," 56th SPE ANTEC, Atlanta, GA, 277­283 (1998)<br>
+129.  T. Osswald, P. Gramann, B. Davis, M. del Pilar Noriega, O. Estrada, and C. Rauwendaal, <br>
+"Experimental Study of a New Dispersive Mixer," 57th SPE ANTEC, New York, 167­176 <br>(1999); also the annual meeting of the Polymer Processing Society in Den Bosch, The <br>Netherlands (1999)<br>
+130. G. Ponzielli and C. Rauwendaal, "Performance Characteristics of Elongational Mixing <br>
+Screws," SPE ANTEC, Chicago, IL, (2004)<br>
+131. C. Rauwendaal, "New Developments in Extruder Screw Design," Plast. Technol. Asia, <br>
+July, 14­17 (2006)<br>
+132. H.P. Grace, "Dispersion Phenomena in High Viscosity Immiscible Fluid Systems and <br>
+Application of Static Mixers as Dispersion Devices in Such Systems," Chem. Eng. Com-<br>mun., 14, 225­277 (1982)<br>
+133.  C. Rauwendaal, "Recent Advances in Barrier Screw Design," Plast. Addit. Compd., Sept./<br>
+Oct., 2­5 (2005)<br>
+134.  C. Rauwendaal, "Efficient Extrusion of Medical Devices," presentation at conference on <br>
+Plastics in Medical Products, organized by Plastics News, April 11­13, Huron, Ohio <br>(2011)<br>
+<hr>
+<A name=665></a><b>9 Die Design</b><br>
+Die design is one aspect of extrusion engineering that has remained more of an art <br>than any other aspect. The obvious reason is that it is difficult to determine the opti-<br>mum flow channel geometry from engineering calculations. Realistic analysis of <br>flow through dies in many cases requires computation of three-dimensional velocity <br>profiles inside the die. This can be a challenge for simple Newtonian fluids. For vis-<br>coelastic fluids the computational complexity is greater still. If temperature effects <br>are considered as well, the problem gets even more complicated. Even if the flow <br>inside the die can be modeled accurately, it is equally important to predict what hap-<br>pens to the extruded polymer once it leaves the die. At this point effects like extru-<br>date swell, drawdown, cooling, and relaxation start affecting the actual size and <br>shape of the extrudate. Describing all these events quantitatively with good accu-<br>racy and within a reasonable time frame will remain an engineering challenge for <br>some time to come.<br>Accurate description of flow of the polymer melt through the die requires know-<br>ledge of the viscoelastic behavior of the polymer melt. The polymer melt can no <br>longer be considered a purely viscous fluid because elastic effects in the die region <br>can be significant. Unfortunately, there are no simple constitutive equations that <br>adequately describe the flow behavior of polymer melt over a wide range of flow con-<br>ditions. Thus, a simple die flow analysis is generally very approximate, while more <br>accurate die flow analyses tend to be quite complicated.<br>Finite element analysis (FEM) has become a popular method for numerical simula-<br>tion of flow through dies. One of the benefits of FEM is that it can handle non-linear <br>fluids well. A newer numerical technique gaining popularity is boundary element <br>analysis (BEM) Three-dimensional flow analysis with BEM can handle complex flow <br>geometries well; however, BEM at this point is not as good as FEM in handling non-<br>linear fluids. Less detailed analyses often use control volume analysis to reduce the <br>computational effort. The different numerical techniques will be discussed in more <br>detail in Chapter 12.<br>Michaeli's book on extrusion dies [1] contains some information on the use of FEM <br>in die design. The book by Crochet, Davies, and Walters [30] contains general infor-<br>mation on the use of FEM in the analysis of non-Newtonian flow. More recent books <br>on computer simulation in polymer processing are the books by Tucker [38] and <br>O'Brien [39]. The last book deals specifically with modeling for extrusion.<br>
+<hr>
+<A name=666></a><b>654</b>  <b>9Die Design</b><br>
+<b></b><br>
+ <b>9.1Basic Considerations</b><br>
+The objective of an extrusion die is to distribute the polymer melt in the flow chan-<br>nel such that the material exits from the die with a uniform velocity. The actual <br>distribution will be determined by the flow properties of the polymer, the flow chan-<br>nel geometry, the flow rate through the die, and the temperature field in the die. If <br>the flow channel geometry is optimized for one polymer under one set of conditions, <br>a simple change in flow rate or in temperature can make the geometry non-opti-<br>mum. Except for circular dies, it is essentially impossible to obtain a flow channel <br>geometry that can be used, as such, for a wide range of polymers and for a wide <br>range of operating conditions. For this reason, one often incorporates adjustment <br>capabilities into the die by which the distribution can be changed externally while <br>the extruder is running. The flow distribution is generally changed in two ways: <br>i) by changing the flow channel geometry by means of choker bars, restrictor bars, <br>valves, etc., and ii) by changing the local die temperature. Mechanical adjustment <br>capabilities complicate the design of the die but enhance its flexibility and controll-<br>ability.<br>Some general rules that are useful in die design are:<br>
+ <br>
+ No dead spots in the flow channel<br>
+ <br>
+ Steady increase in velocity along the flow channel<br>
+ <br>
+ Assembly and disassembly should be easy<br>
+ <br>
+ Land length about 10X land clearance<br>
+ <br>
+ Avoid abrupt changes in flow channel geometry<br>
+ <br>
+ Use small approach angles<br>In die design, problems often occur because the product designer has little or no <br>appreciation for the implications of the product design details on the ease or diffi-<br>culty of extrusion. In many cases, small design changes can drastically improve the <br>extrudability of the product. Some basic guidelines in profile design to minimize <br>extrusion problems are:<br>
+ <br>
+ Use generous internal and external radii on all corners; the smallest possible <br>
+radius is about 0.5 mm<br>
+ <br>
+ Maintain uniform wall thickness (important!)<br>
+ <br>
+ Avoid very thick walls<br>
+ <br>
+ Make interior walls thinner than exterior walls for cooling<br>
+ <br>
+ Minimize the use of hollow sections<br>Figure 9.1 illustrates applications of these guidelines to different profiles and flow <br>channel geometries.<br>
+<hr>
+<A name=667></a> <br>
+9.1Basic Considerations  <b>655</b><br>
+<b>9.1.1Balancing the Die by Adjusting the Land Length</b><br>
+Mechanical adjustment of the die flow channel can be done in two basic ways. The <br>length of the channel can be adjusted to make sure the average flow velocity is uni-<br>form. The other method is to adjust the height of the channel. In this section we will <br>discuss how to achieve uniform flow by adjustment of the land length.<br>
+<b>Figure 9.1</b>Examples of application of die design guidelines<br>
+In this analysis we will assume that the width of the flow channel is large relative to <br>the height of the channel. In this case the flow rate for a power law fluid can be ex -<br>pressed as:<br>
+ (9.1)<br>
+The geometrical factors are the channel width W, the channel height H, and the <br>channel length L. The important flow properties are the power law index n and the <br>consistency index m. Equation 9.1 represents the volumetric flow rate through the <br>die flow channel. The average flow velocity can be obtained by dividing the volumet-<br>ric flow rate by the cross-sectional area of the channel. This results in the following <br>expression for the average velocity:<br>
+ (9.2)<br>
+<hr>
+<A name=668></a><b>656</b>  <b>9Die Design</b><br>
+This expression shows that the average velocity is no longer dependent on the width <br>of the channel. This is true as long as the channel width is much greater than the <br>height (W &gt;&gt; H). If the extruded profile has sections with different thickness, the <br>channel height will have to be different in different sections of the die flow channel. <br>If the land length is the same in all sections of the die, differences in channel height <br>will result in large differences in average velocity. This can be quantified by the fol-<br>lowing expression showing the ratio of average velocities v1/v2 as a function of the <br>
+height ratio H1/H2.<br>
+ (9.3)<br>
+Figure 9.2 shows how the velocity ratio changes with the thickness ratio at several <br>values of the power law index.<br>From Fig. 9.2 it is clear that even small differences in thickness can result in large <br>velocity differences, particularly for small values of the power law index. Even with <br>thickness differences of 50% the velocity difference can be as high as 5:1 up to 10:1 <br>for power law index values between 0.2 and 0.4. Most high-volume polymers have <br>power law index values in this range. Velocity differences of 10:1 will cause severe <br>distortion in the extrudate, which means that balancing the flow will be absolutely <br>necessary.<br>
+<b>Figure 9.2</b>Velocity ratio versus thickness ratio for several values of the power law index<br>
+<hr>
+<A name=669></a> <br>
+9.1Basic Considerations  <b>657</b><br>
+Equation 9.2 can be used to determine how the land length can be changed to main-<br>tain uniform average velocity. In order for the average velocity in the channel sec-<br>tion with height H1 to be the same as in the section with height H2, the land length <br>
+ratio has to be:<br>
+ (9.4)<br>
+Figure 9.3 shows how the land length ratio changes with height ratio at several val-<br>ues of the power law index.<br>
+<b>Figure 9.3</b>Land length ratio versus height ratio at several values of the power law index<br>
+When the height ratio is large the land length ratio has to become very large to <br>maintain uniform flow, particularly for large values of the power law index. When <br>the land length ratio has to be greater than about 5:1 it may no longer be practical to <br>adjust by land length. For a polymer with a power law index of n = 0.4, this means <br>that the height ratio should be no greater than about 3:1.<br>One of the problems with balancing by land length is that it induces transverse <br>pressure differences along the length of the flow channel. Pressure differences <br>across the channel will create cross flow, and this will make the effect of balancing <br>by land length unpredictable unless a three-dimensional flow analysis is used. One <br>way to avoid cross flow is to place partitions between the sections with different <br>thickness of the die so that cross flow between the different sections is not possible. <br>In effect, this creates separate die flow channels; an example is shown in Fig. 9.4.<br>
+<hr>
+<A name=670></a><b>658</b>  <b>9Die Design</b><br>
+Partition<br>
+<b> Figure 9.4</b> <br>
+Example of partitions in a die flow channel<br>
+The partitions can be terminated just before the exit of the die to allow the separate <br>streams to merge. This approach allows greater control of the flow through the die. <br>The drawback with partitions is that they can create knit lines between the various <br>partitions. Knit lines will be discussed more in Section 11.3.7.5.1. Partitions between <br>sections of the die with different thickness can be considered as a way of balancing <br>by channel width.<br>The land length is usually adjusted by changing the land length at the entry to the <br>land; this is called "back relieving" the land. An example of back relieving is shown <br>in Fig. 9.5(a).<br>
+<b> Figure 9.5(a)</b> <br>
+Example of back relieving a profile die<br>
+It is also possible to change the land length at the end of the land, at the die exit. <br>This is called front relieving of the die; an example of front relieving is shown in Fig. <br>9.5(b).<br>
+<hr>
+<A name=671></a> <br>
+9.1Basic Considerations  <b>659</b><br>
+<b> Figure 9.5(b)</b> <br>
+Example of front relieving a profile die<br>
+Obviously, front relieving results in a die exit surface that is no longer flat. This <br>makes it difficult to clean the die surface; as a result, front relieving of dies is prac-<br>ticed less than back relieving.<br>
+<b>9.1.2Balancing by Channel Height</b><br>
+As discussed in the previous section, balancing by land length does not always lead <br>to satisfactory results. The other method is to balance by channel height. An ex ample <br>is shown in Fig. 9.6.<br>
+9<br>
+3<br>
+<b> Figure 9.6</b> <br>
+U-shaped profile with circular sections<br>
+In Fig. 9.6 there is a thin wall connected to larger diameter circular sections. With-<br>out balancing, the flow through the circular section will be substantially greater <br>than the slit section. The average velocity in the circular section will be:<br>
+ (9.5)<br>
+As a result, the ratio of the average velocity in the circular section and the slit sec-<br>tion will be:<br>
+ (9.6)<br>
+This relationship is shown in Fig. 9.7.<br>
+<hr>
+<A name=672></a><b>660</b>  <b>9Die Design</b><br>
+<b>Figure 9.7</b>Velocity ratio circle-to-slit versus the diameter-to-wall-thickness ratio<br>
+The average flow velocity in the thin section is given by Eq. 9.2. With the wall <br>w = 3 mm and the diameter of the circular section D = 9 mm the velocity ratio will <br>be 3.375 when the power law index is 1. When the power law index is 0.4, the velo-<br>city ratio will be 6.764. This will clearly cause problems; therefore, balancing will <br>be required. The balancing can be done by land length. This will require the follow-<br>ing ratio of land length:<br>
+ (9.7)<br>
+This relationship is shown in Fig. 9.8.<br>
+<b>Figure 9.8</b>Ratio of Lcircle/Lslit versus the ratio D/H at several values of the power law index<br>
+<hr>
+<A name=673></a> <br>
+9.1Basic Considerations  <b>661</b><br>
+When D/H = 3, the land length of the circular section has to be 4.3 times longer than <br>the land of the slit. This is quite a large difference and will likely result in cross flow. <br>In this case, a better method of balancing the flow will be to place cylindrical pins in <br>the circular section of the die as shown in Fig. 9.9.<br>
+3<br>
+9<br>
+3<br>
+<b> Figure 9.9</b> <br>
+Example of balancing by adjusting the channel height<br>
+If the circular sections of the profile need to be solid the cylindrical pins can be ter-<br>minated before the exit of the die so that the polymer melt can fill the entire cylind-<br>rical section.<br>Another example of balancing by channel height is the profile shown in Fig. 9.10.<br>
+<b> Figure 9.10</b> <br>
+Example of profile with large difference in channel <br>
+height<br>
+This profile will be difficult to extrude because the wall thickness differs by a factor <br>of three; this will result in large velocity differences. One way to solve this problem <br>is to make the profile hollow as shown in Fig. 9.1. However, if the profile needs to be <br>solid, a different approach will be required. One possible approach is to place parti-<br>tions in the thick section of the die so that the thickness of the individual sections <br>will be approximately the same. This is shown in Fig. 9.11.<br>
+<b> Figure 9.11</b> <br>
+Example of adjusting channel height by <br>
+ <br>partitioning the flow channel<br>
+<hr>
+<A name=674></a><b>662</b>  <b>9Die Design</b><br>
+Obviously, there are other ways that the die can be partitioned to balance the flow, <br>but the method shown in Fig. 9.11 can work quite well to avoid large velocity differ-<br>ences in profile with thin and thick sections.<br>
+<b>9.1.3Other Methods of Die Balancing</b><br>
+Balancing problems frequently occur when the extruded product has differences in <br>wall thickness. However, even without differences in wall thickness there can be <br>distortion in the extruded product. An example is a simple rectangular or square <br>profile; the velocity distribution on a square channel is shown in Fig. 7.105. Clearly, <br>the velocities in the corner are less than they are along the middle of the wall. As a <br>result, there will be more drawdown at the corners than at the mid-sections of the <br>wall. Obviously, this problem is inherent to shapes with corners, particularly cor-<br>ners with a small radius and corners smaller than right-angle corners.<br>For this reason, the easiest shapes to extrude are circular and annular shapes. The <br>lower velocities in the corners of a square or rectangular profile can be increased by <br>reducing the land length in the corners. This can be done several ways as shown in <br>Fig. 9.12.<br>
+<b>Figure 9.12</b>Examples of back relieving to increase velocities in corner<br>
+The examples shown in Fig. 9.12 are methods for local balancing of the land length. <br>Another method of adjusting the shape of the die is to incorporate moveable ele-<br>ments in the channel such as choker bars, flex lips, and membranes. These elements <br>are frequently used in film and sheet dies. However, they can be used in other types <br>of dies as well.<br>
+<hr>
+<A name=675></a> <br>
+9.2Film and Sheet Dies  <b>663</b><br>
+<b></b><br>
+ <b>9.2Film and Sheet Dies</b><br>
+Dies for flat film are essentially the same as dies for sheet extrusion. The difference <br>between sheet and film is primarily the thickness. Webs with a thickness of less <br>than 0.5 mm are generally referred to as film, while webs with a thickness of more <br>than 0.5 mm are generally referred to as sheet. Three distribution channel geo-<br>metries used in sheet dies are shown in Fig. 9.13.<br>Fig. 9.13(a) shows the T-die. This flow channel geometry is simple and easy to <br>machine. However, the distribution of the polymer melt is not very uniform and the <br>flow channel geometry is not well streamlined. Thus, this die is not suitable for high-<br>viscosity polymers with limited thermal stability. The T-die is used in extrusion <br>coating applications. Analyses of the flow in a T-die have been made by Weeks [3, 4], <br>Ito [5], and Pearson [6].<br>
+<b> Figure 9.13(a)</b> <br>
+The T-die<br>
+The fishtail die is shown in Fig. 9.13(b). This die results in a more uniform melt <br> <br>distribution than the T-die; however, a completely uniform distribution is still diffi-<br>cult to obtain with this geometry. Analyses of the flow in fishtail dies have been <br>made by Ito [7] and Chejfec [8].<br>
+<b> Figure 9.13(b)</b> <br>
+The fishtail die<br>
+<hr>
+<A name=676></a><b>664</b>  <b>9Die Design</b><br>
+Figure 9.13(c) shows the coat hanger die.This is a die geometry commonly used in <br>sheet extrusion. The geometry of the coat hanger section can be designed to give a <br>very uniform distribution of the polymer melt. Obviously, the coat hanger die is <br>more difficult to machine and, therefore, more expensive than the T-die and fishtail <br>die. Analyses of the flow in coat hanger dies have been made by Ito [9, 10], Wortberg <br>[11], Goermar [12, 13], Chung [14], Klein [15], Schoenewald [16], Vergnes [17], <br> <br>Matsubara [18, 19], and many more. Goermar used the sinh law (see Eq. 6.32) as the <br>constitutive equation for the polymer melt. He obtains a remarkably simple expres-<br>sion for the geometry of the coat hanger section; see Fig. 9.13(c). For the manifold <br>radius as a function of distance, Goermar derived the following expression:<br>
+ (9.8a)<br>
+and for the land length:<br>
+ (9.8b)<br>
+where x is the distance from the edge of the die, b the half width, R0 the manifold <br>
+radius in the center, and L0 the preland length in the center.<br>
+<b> Figure 9.13(c)</b> <br>
+The coat hanger die<br>
+<b>9.2.1Flow Adjustment in Sheet and Film Dies</b><br>
+Figure 9.14 shows two commonly used techniques to change the flow channel geo-<br>metry in sheet dies.<br>The first is the flex lip adjustment. A number of bolts along the width of the die <br>allow local closing of the final land gap. This allows fine adjustments of the extru-<br>date thickness at discrete points. The gap can be adjusted by as much as 1 mm or <br>more if the flex lip is properly designed. The choker bar is not used as often as the <br>
+<hr>
+<A name=677></a><IMG src="Polymer Extrusion by Chris Rauwendaal-677_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-677_9.jpg"><br>
+ <br>
+9.2Film and Sheet Dies  <b>665</b><br>
+flex lip. The choker bar adjustment works in a similar fashion as the flex lip adjust-<br>ment. The choker bar can be locally deformed by a number of bolts located along the <br>width of the die. The deformation of the choker bar causes a change in the height of <br>the flow channel and, thus, allows an adjustment of the flow distribution in the die. <br>A third adjustment possibility, not shown in Fig. 9.14, is the die temperature. Local <br>heating or cooling of certain die sections enhance or restrict flow; this is another <br>means of flow distribution adjustment. Temperature adjustment will be more effec-<br>tive with polymers whose viscosity is quite sensitive to temperature; this includes <br>most amorphous polymers (see also Table 6.1).<br>
+Choker bar<br>
+Flex lip adjustment<br>
+<b> Figure 9.14</b> <br>
+Methods to change the flow channel geometry in sheet <br>
+dies<br>
+Figure 9.15 also shows a feature of the sheet die that is useful when a large thick-<br>ness range is necessary.<br>
+Choker bar<br>
+Flex lip adjustment<br>
+<b> Figure 9.15</b> <br>
+Removable lower lip<br>
+Sheet die with removable lower lip<br>
+By incorporating a removable lower die lip, the die can be used, for instance, for <br>1-mm sheet extrusion and for 4-mm sheet extrusion by changing only the lower die <br>lip. Without the removable die lip, another sheet die would have to be used because <br>the flex lip can only adjust over a limited range. In some automated extrusion lines, <br>the gap of the sheet die is adjusted automatically. This is done with heat expandable <br>die lip bolts [20], first developed by Welex in the early 1970s. A similar type of sheet <br>die using thermal bolts is now also offered by Egan Machinery Company and a num-<br>ber of other companies.<br>
+<hr>
+<A name=678></a><b>666</b>  <b>9Die Design</b><br>
+Another concept is used by Harrel, Inc. Their sheet die has die temperature adjust-<br>ment capability at various locations along the width of the die. By raising or lower-<br>ing the temperature automatically, the sheet thickness can be controlled without <br>changing the die lip gap. An interesting automatically adjustable sheet die was <br>developed by Hexco. This sheet die uses a PC servo positioner to hydraulically con-<br>trol each die bolt. This design is said to reduce response time from minutes, which <br>is typical in thermally adjusted sheet dies, down to a few seconds. Hexco went out of <br>business in 1983; as result, this particular die is no longer available. However, simi-<br>lar systems have been developed by other manufactures. For instance, Japan Steel <br>Works has developed an automatic die gap adjustment system that uses a servo <br>motor that can travel along the width of the die to adjust any die bolt a certain <br>amount as determined by an automatic sheet thickness measurement downstream <br>of the die. The advantage of a mechanical adjustment as opposed to heat expandable <br>bolts is that the thickness adjustment occurs more rapidly.<br>An elegant approach to thickness adjustment in sheet and film dies was developed <br>by Gross [46]. Gross developed a system with a flexible membrane inside the die <br>to allow rather simple thickness adjustment using low force actuators to create a <br>smoothly curved surface without sharp angles or dead spots. This system is sche-<br>matically shown in Fig. 9.16. The membrane die has been licensed by several manu-<br>facturers of extrusion dies worldwide [46].<br>
+<b> Figure 9.16</b> <br>
+Schematic of coextrusion membrane die<br>
+A drawback of the conventional coat hanger die with the teardrop-shape distribution <br>channel is the fact that the distribution changes when the power law index of the <br>material changes. Thus, the distribution will change when a change in polymer is <br>made and also when the output is changed because the power law index is generally <br>somewhat dependent on shear rate. Therefore, flex lips and choker bars are gener-<br>ally used to compensate for the imperfect distribution in the die. A modified coat <br>hanger geometry was proposed by Winter and Fritz [31] that eliminates the problem <br>of the power law index dependence of the distribution.<br>
+<hr>
+<A name=679></a> <br>
+9.2Film and Sheet Dies  <b>667</b><br>
+<b>9.2.2The Horseshoe Die</b><br>
+The main feature of the new coat hanger geometry is a slit-shaped distribution in <br>the shape of a horseshoe channel, see Fig. 9.17.<br>If the ratio of channel width W to channel depth H is larger than about 10, the shape <br>factor becomes independent of the power law index. For a manifold with constant <br>channel width W, the distribution channel geometry is given by (see Fig. 9.17):<br>
+ (9.9a)<br>
+and:<br>
+ (9.9b)<br>
+<b> Figure 9.17</b> <br>
+Geometry of the horseshoe <br>
+manifold<br>
+For a manifold with constant aspect ratio, W/H = a, the depth profile is given by:<br>
+ (9.9c)<br>
+The corresponding contour line is given by:<br>
+ (9.9d)<br>
+where:<br>
+ (9.9e)<br>
+<hr>
+<A name=680></a><b>668</b>  <b>9Die Design</b><br>
+The integration constant C is determined by setting H(x) = h at y = 0. The front factor <br>is defined as:<br>
+ (9.9f)<br>
+The shape factor F for a rectangular flow channel is given by Eqs. 7.219 and 7.221. <br>If the shape factor is taken as unity, the contour line becomes:<br>
+ (9.9g)<br>
+where:<br>
+ (9.9h)<br>
+The new coat hanger geometry can be applied to flat sheet dies as well as annular <br>dies. Winter [31] reports on applications with blow molding dies with a circum-<br>ferential thickness distribution of the parison between 5 and 8%. Very good results <br>were also obtained on a flat sheet die of 0.25 m width and a flat profile die with a <br>width of 2 m. A drawback of the horseshoe manifold is that the pre-land is quite <br>long. This makes the die more susceptible to clamshelling (the separation of the die <br>lips in the center region of the die due to high pressure of the polymer melt inside <br>the die). As a result, this manifold is less suitable for very wide dies when high <br>internal pressures occur inside the die.<br>
+<b></b><br>
+ <b>9.3Pipe and Tubing Dies</b><br>
+The difference between pipe and tubing is mainly determined by size. Small dia-<br>meter products (less than 10 mm) are generally referred to as tubing, while large <br>products are generally referred to as pipe. Annular products can be extruded on in-<br>line dies and crosshead dies. In the crosshead die the polymer melt makes a turn as <br>it flows through the die; an example of a crosshead die Fig. 9.18.<br>The direction of the inlet flow is perpendicular to the outlet flow. The polymer melt <br>makes a 90° turn and splits at the same time over the core tube. The polymer melt <br>recombines below the core tube; this is where a weld line will form. After the 90° <br>turn, the polymer melt flows through the annular flow channel where it adopts more <br>or less the shape of the final land region.<br>
+<hr>
+<A name=681></a><IMG src="Polymer Extrusion by Chris Rauwendaal-681_1.jpg"><br>
+ <br>
+9.3Pipe and Tubing Dies  <b>669</b><br>
+<b>Figure 9.18</b>Example of a crosshead die<br>
+Weld lines are usually unavoidable in hollow extruded products. The polymer has <br>to be given sufficient opportunity to "heal" along the knit lines. This healing process <br>is essentially a reentanglement of the polymer molecules. Important parameters in <br>this process are time, temperature, and pressure. Analyses of the healing process <br>have been made by Prager and Tirrell [21] and Wool et al. [22­24]. The problem <br>of weld lines also occurs in injection molding, as discussed, for instance, by Mal-<br>guarnera and Manisali [25]. The healing time reduces with temperature but in -<br>creases with molecular weight. In practical terms, this means that the point of weld <br>line formation has to be a reasonably large distance upstream of the die exit to en -<br>able the polymer to heal sufficiently.<br>The crosshead die is also used for wire coating. In wire coating, a conductor passes <br>through the hollow center of the core tube and becomes coated with polymer melt <br>close to the die exit. The conductor may be a bare conductor or it may already have <br>been coated with one or more layers of polymer. In wire coating, one distinguishes <br>between high-pressure extrusion and low-pressure extrusion; see Fig. 9.19.<br>In high-pressure extrusion, Fig. 9.19, the polymer melt meets the conductor before <br>the die exit. This allows for good contact between the conductor and the polymer. In <br>low-pressure extrusion, Fig. 9.19, the polymer melt meets the conductor after the <br>die exit. The polymer is tubed down over the conductor. Low-pressure extrusion is <br>used when good contact between the wire and the polymer is not essential, for <br>instance, when a loose jacket needs to be extruded over a coated wire.<br>
+<hr>
+<A name=682></a><b>670</b>  <b>9Die Design</b><br>
+<b>Figure 9.19</b>High- and low-pressure wire coating<br>
+In most crosshead dies, the location of the die relative to the tip can be adjusted by <br>means of centering bolts; see Fig. 9.18. This allows adjustment of the wall thickness <br>distribution and concentricity. In some extrusion lines with in-line wall thickness <br>measurement, an automatic wall thickness control is obtained by using the signal <br>from the wall thickness probe to automatically adjust the position of the die. In <br>some cases, a slight internal air pressure is applied through the center of the core <br>tube in order to maintain the I.D. of the tubing or to prevent collapse of the tubing. <br>This is particularly useful in extrusion of tubing with very small I.D., 0.1 mm or <br>less. In wire coating, sometimes a vacuum is applied to the center of the core tube to <br>prevent air being dragged along with the conductor; this can cause imperfect con-<br>tact between the polymer and the conductor.<br>Pipes are often extruded with in-line pipe dies; see Fig. 9.20.<br>
+<b>Figure 9.20</b>In-line pipe die<br>
+In these dies, the center line of the die is in line with the center line of the extruder. <br>The central torpedo is supported by a number of spider legs, usually three or more. <br>The spider legs are relatively thin and streamlined to minimize the disruption of the <br>velocity profile. Of course, as the polymer recombines after the spider leg, a weld <br>line will form. Thus, the location of the spider support should be far enough from <br>the die exit to enable the polymer to heal. The location of the die is generally adjust-<br>able relative to the pin, just as in the crosshead die.<br>
+<hr>
+<A name=683></a> <br>
+9.3Pipe and Tubing Dies  <b>671</b><br>
+A manifold geometry that largely eliminates weld lines is the spiral mandrel die. <br>These dies were originally developed for blown film extrusion; see Section 9.4. How-<br>ever, it became clear that spiral mandrel dies are equally (if not more) beneficial in <br>pipe and tubing extrusion. Conventional tubing and pipe dies create a weld line <br>(actually a weld region) that runs along the length of the extruded product and <br>extends from the I.D. to the O.D. of the tube. Such a weld line reduces the hoop <br>strength of a tube because the weld line (weld region) has the least favorable orien-<br>tation relative to the stresses in the tube caused by internal pressure.<br>On the other hand, a spiral mandrel manifold creates flow in the helical direction, <br>resulting in some degree in circumferential orientation. This combined with the <br>near absence of weld lines make spiral mandrel dies capable of producing a tube or <br>pipe with better mechanical properties simply by modifying the flow inside the die.<br>Another method of improving circumferential orientation in tubing and pipe dies is <br>by inducing relative motion between the tip and the die. This can be done by rotat-<br>ing the tip relative to a stationary die or by rotating the die relative to a stationary <br>tip. It is even possible to rotate both the tip and the die separately and in different <br>directions. With rotation of the tip and/or die there is a greater degree of control <br>over the orientation of the extruded tube than with a spiral mandrel section. Ob -<br>viously, this is at the expense of increased mechanical complexity.<br>An example of a rotating tubing die is shown in Fig. 9.21.<br>
+<b> Figure 9.21</b> <br>
+Tubing crosshead with capability to <br>
+rotate the tip<br>
+<b>9.3.1Tooling Design for Tubing</b><br>
+Tubing is used in many industries for the transport of fluids. One of the more interest-<br>ing applications is heat shrinkable tubing, which is made by extrusion, followed by <br>crosslinking and expansion. Another important use is in the medical industry as cath-<br>eter tubing. One of the more sophisticated medical devices based on polymeric tubing <br>is the balloon catheter used for percutaneous transluminal coronary angioplasty.<br>
+<hr>
+<A name=684></a><IMG src="Polymer Extrusion by Chris Rauwendaal-684_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-684_32.jpg"><br>
+<b>672</b>  <b>9Die Design</b><br>
+The requirements for medical tubing with respect to dimensional tolerances and <br>overall quality are stricter than almost any other application. This, coupled with the <br>small tubing sizes typically produced, presents major challenges to the producers of <br>medical tubing. The requirements are often so severe that off-the-shelf extrusion <br>equipment may not do the job well enough. As a result, a number of producers of <br>medical tubing have developed some of their own machinery.<br>
+<b>9.3.1.1Definitions of Various Draw Ratios<br></b>Tooling design for tubing is a critical issue in tubing extrusion but there is limited <br>useful information available. Important issues in the design of tubing tooling are the <br>various draw ratios that define the tooling and the extrusion process. The dimen-<br>sions of the tip (mandrel) and die are determined by the drawdown in the extrusion <br>process. There are various draw ratios in tubing extrusion that describe how the <br>tubing is drawn down at the exit of the die. The diameter draw ratio (DDR) is the <br>average diameter of the tip and die divided by the average diameter of the tubing.<br>
+ (9.10)<br>
+where Dt is the tip diameter, Dd is the die diameter, Do is the tubing outside diameter, <br>
+and Di the tubing inside diameter; see Fig. 9.22.<br>
+Die<br>
+D<br>
+D<br>
+d<br>
+Dt<br>
+o<br>
+Di<br>
+Tip<br>
+<b> Figure 9.22</b> <br>
+Definition of Dd, Dt, Do, and Di<br>
+Another important parameter is the wall draw ratio (WDR). This is the gap between <br>the tip and die divided by the wall thickness of the tubing.<br>
+ (9.11)<br>
+A third measure is the area draw ratio (ADR). This is the cross-sectional area be -<br>tween the tip and die divided by the tubing cross-sectional area.<br>
+ (9.12)<br>
+<hr>
+<A name=685></a> <br>
+9.3Pipe and Tubing Dies  <b>673</b><br>
+It should be noted that sometimes the area draw ratio is represented by the letters <br>DDR as an acronym for drawdown ratio. Obviously, this can cause confusion with <br>the diameter draw ratio. Often the area draw ratio is called simply the draw ratio. It <br>is important therefore to check what exactly is meant when the term draw ratio is <br>used. The ADR can be calculated from the WDR and the DDR as follows:<br>
+ (9.13)<br>
+Thus, the wall and diameter draw ratios determine the area draw ratio in the extru-<br>sion process. A high ADR increases orientation and the chance of pinholes and break-<br>aways. A low ADR reduces orientation and increases the chance of melt fracture.<br>A fourth measure of drawdown is the draw ratio balance or DRB. This is the dia-<br>meter ratio of the die and tip divided by the diameter ratio of the tubing.<br>
+ (9.14)<br>
+The draw ratio balance should be equal to or larger than one: DRB  1. Yet another <br>parameter that is used is the sizing ratio or SR. This is the wall draw ratio divided <br>by the diameter draw ratio.<br>
+ (9.15)<br>
+A balanced draw occurs when the sizing ratio ranges from 1.0 to 1.3. When the SR <br>is larger than 1.3, there is a danger of getting tear holes in the tubing. Low SR values <br>can cause instabilities in the sizing of the tubing. Rubbers and high molecular <br>weight polymers can be run with low SR values. Low viscosity polymers should be <br>run with high SR values. High SR values will increase orientation and the chance of <br>breakaways. High SR values will require higher internal and/or lower external air <br>pressure to obtain tubing size.<br>Different polymers can be drawn down by varying amounts. Fluoropolymers, such <br>as FEP, PFA, and ETFE, can be drawn down a great deal with ADR values of 10 to 100 <br>and higher. With special extrusion techniques, PFA can have an ADR of over 250:1. <br>Polyethylenes have a medium ability to be drawn down, polyurethanes and poly-<br>vinyls low to medium.<br>
+<b>9.3.1.2Land Length<br></b>In addition to the tip and die diameter, the land length and the cone angle are impor-<br>tant design parameters. In many situations a long land length is desired because a <br>long land tends to:<br>
+ <br>
+ Reduce tip and die drool<br>
+ <br>
+ Increase orientation<br>
+<hr>
+<A name=686></a><b>674</b>  <b>9Die Design</b><br>
+ <br>
+ Reduce the chance of pinholes<br>
+ <br>
+ Reduce the swelling of the extrudate (die swell)<br>
+ <br>
+ Improve shape definition<br>The main drawback of a long land length is increased diehead pressure. Because <br>the land region usually has the highest restriction to flow, a longer land can in -<br>crease pressure substantially. Another drawback of a long land length is that a <br>long tip is more susceptible to mechanical deformation; the tip can bend more <br> <br>easily. This is a particular concern in small diameter tubing. Typical rules for the <br>land length are:<br>
+ <br>
+ Land length divided by gap between tip and die (L/H) from 10:1 to 20:1<br>
+ <br>
+ Land length divided by the diameter of the tip (L/Dt) from 10:1 to 25:1<br>The gap between the tip and the die, H, is half the die diameter minus half the tip <br>diameter, or H = 0.5Dd­0.5Dt. The land length values that follow from these rules <br>
+often result in excessive pressures with dealing with high viscosity materials. In <br>many cases, therefore, the pressure drop will determine what land length is practi-<br>cal.<br>A special extrusion technique that is occasionally used is the extended mandrel. <br>This is particularly used for thin wall tubing. In this technique the tip extends <br>beyond the die by a considerable distance. The purpose of the extended mandrel is <br>to obtain better shaper definition. Another use of the extended mandrel is to provide <br>localized heating of the tip using an induction coil at the die exit. This is a variation <br>of the G-Process discussed under special features. It is a suitable method to elimi-<br>nate internal melt fracture or internal die drool. The mandrel extension should be <br>made of a ferro-magnetic material to obtain an efficient temperature increase under <br>the influence of the alternating magnetic field of the induction coil.<br>
+<b>9.3.1.3Taper Angles<br></b>The taper angle of the tip and die depends largely on whether the tooling is self-<br>centering or adjustable; see Fig. 9.23.<br>The taper angle used in self-centering tooling typically ranges from 30° to 40°, in <br>adjustable tooling from 8° to 15°. The term self-centering tooling is not totally cor-<br>rect because it does not always center itself; a more appropriate term is non-adjust-<br>able tooling. Relatively few studies have been published on the influence of die entry <br>angle on the extrudate quality. Han [32] found that the entry angle affects melt <br>fracture in certain polymers such as LDPE. When the entry angle is as large as <br>120°, melt fracture occurs in LDPE. At smaller entry angles melt fracture does not <br>occur. In other polymers, such as HDPE, the entry angle has no effect on the extru-<br>date distortion.<br>
+<hr>
+<A name=687></a><IMG src="Polymer Extrusion by Chris Rauwendaal-687_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-687_19.jpg"><br>
+ <br>
+9.3Pipe and Tubing Dies  <b>675</b><br>
+Flow splitter<br>
+(helicoid)<br>
+Tip<br>
+Die<br>
+Die holder<br>
+Core tube<br>
+<b>Figure 9.23</b>Example of self-centered crosshead die<br>
+Usually the taper angle of the die is slightly larger than the tip. It is also possible to <br>make the taper angle of the tip equal to the die. It is important to have a gradual <br>reduction in the cross-sectional area of the flow channel. Since the average diameter <br>of the flow channel reduces, the area will reduce even if the taper angles are the <br>same. Powell [33] describes design procedures to determine the length and included <br>angle of converging sections such that the critical tensile deformation rate is not <br>exceeded. In converging flow the tensile stress increases in the flow direction and <br>reaches a maximum value at the narrow end of the taper. The average tensile defor-<br>mation rate at the narrow end of a tapered wedge with included angle of 2 is given <br>by:<br>
+ (9.16)<br>
+where  --is the shear rate in the small end of the wedge.<br>In a rectangular geometry this shear rate can be calculated from:<br>
+ (9.17)<br>
+Typical values of the critical tensile deformation rate range from about 1 to 100 s­1, <br>depending on the type of polymer. When a polymer melt flows from a large to a <br>small channel, it forms a natural streamline angle. When the taper angle of the tool-<br>ing is larger than the natural streamline angle, dead spots will form. This can be <br>avoided by making the half-angle of the entry to the die equal to or smaller than the <br>natural half-angle.<br>
+<hr>
+<A name=688></a><b>676</b>  <b>9Die Design</b><br>
+<b>9.3.1.4Special Features<br></b>In some cases a slight taper is used in the land region. Haas and Skewis [34] reported <br>that a slight taper reduces extrudate roughness. For high draw ratios, DuPont re -<br>commends [36, 37] to radius the face of the tip with a radius of 0.1 of the tip dia-<br>meter. In some cases, a small chamfer is used on the I.D. of the die, e.g., 0.8 mm by <br>45°. In order to obtain higher and/or faster drawdown, a cone heater can be used <br>according to Blair [37]. This is a heater at the exit of the die used to prevent the cone <br>of polymer from cooling too rapidly. The cone heater primarily heats the air sur-<br>rounding the cone.<br>With materials that are susceptible to melt fracture, the exit region of the tooling <br>can be heated to temperatures considerably higher than the rest of the extruder. <br>One approach is to use an induction heater at the die exit to heat just the region sur-<br>rounding the die orifice. This is called the G-Process by DuPont [35, 36]. In this <br>process a high-frequency heater is used. The heat is generated due to the metals <br>resistance to the flow of electrons and to the hysteresis losses occurring during <br>rapid magnetization and demagnetization.<br>
+<b></b><br>
+ <b>9.4Blown Film Dies</b><br>
+The most common die used in blown film extrusion is the spiral mandrel die. In this <br>die, the polymer is divided into a number of spiraling channels with the depth of the <br>channels reducing in the direction of flow. The popularity of the spiral mandrel die <br>is due to its relatively low pressure requirement and its excellent melt distribution <br>characteristics. Spiral mandrel dies can be used with a wide range of materials over <br>a wide range of operating conditions.<br>A simpler die is the conventional crosshead die; see Fig. 9.24. This design is more <br>susceptible to weld lines; however, with the correct design good blown film can be <br>produced.<br>The distribution characteristics of conventional crosshead dies may not be good <br>enough for application in blown film extrusion, where wall thicknesses are gener-<br>ally quite small (the typical range is 0.005 mm to 0.25 mm). Spiral mandrel dies can <br>achieve good flow distribution and largely eliminate weld lines. As a result, spiral <br>mandrel dies are widely used in blown film extrusion.<br>A simplified picture of a spiral mandrel die is shown in Fig. 9.25.<br>The incoming polymer melt stream is divided into separate feed ports. Each feed <br>port feeds the polymer into a spiral groove machined into the mandrel. The cross-<br>sectional area of the groove decreases with distance, while the gap between the <br>mandrel and the die increases towards the die exit. This multiplicity of flow chan-<br>
+<hr>
+<A name=689></a> <br>
+9.4Blown Film Dies  <b>677</b><br>
+nels results in a smearing or layering of polymer melt from the various feed ports, <br>yielding a good distribution of the polymer melt exiting from the die. It is obvious <br>that local gap adjustment is not possible as it is with flat sheet dies. As a result, the <br>wall thickness uniformity with spiral mandrel dies is generally not as good as with <br>flat sheet dies. The latter can generally achieve a wall thickness uniformity of about <br>±5%, while the blown film die achieves a wall thickness uniformity of about ±10%. <br>For this reason, the die is generally made to rotate to evenly distribute the wall <br>thickness non-uniformities. If this were not done, the final roll of product would <br>show very noticeable variations in diameter.<br>
+<b> Figure 9.24</b> <br>
+Example of simple crosshead die for blown film<br>
+<b> Figure 9.25</b> <br>
+Example of a spiral mandrel die<br>
+Analyses of flow in spiral mandrel dies have been made by Ast [26], Wortberg and <br>Schmidtz [27], Proctor [28], Predoehl [29], Menges et al. [40], Rauwendaal [41], and <br>others. Proctor showed the effect of the spiral runout angle, the spiral helix angle, <br>and the taper angle of the annular channel; he assumes that the pressure gradient <br>is constant. Rauwendaal removed this simplifying assumption and studied the effect <br>of die design variables on flow distribution.<br>
+<hr>
+<A name=690></a><b>678</b>  <b>9Die Design</b><br>
+Rauwendaal [41] used the following assumptions to analyze the flow in the spiral <br>mandrel die:<br>
+ <br>
+ The polymer melt behaves as a temperature-independent power law fluid.<br>
+ <br>
+ The curvature of the mandrel can be neglected.<br>
+ <br>
+ Zero pressure gradient in tangential direction.<br>
+ <br>
+ The flow in the spiral channel can be approximated by pressure flow in a rectan-<br>
+gular channel of width W and height H by using a shape factor.<br>
+ <br>
+ The leakage flow in the flight clearance can be approximated by a pressure flow in <br>
+a rectangular channel with height .<br>
+ <br>
+ The leakage flow does not affect the flow in the spiral channel.<br>The flow will be analyzed by unrolling the spiral mandrel onto a flat plane as shown <br>in Fig. 9.26.<br>
+z = zm,  l = lm<br>
+<br>
+b<br>
+w<br>
+B<br>
+W<br>
+<br>
+l<br>
+H<br>
+z<br>
+0<br>
+<br>
+zc<br>
+<b>Figure 9.26</b>Unrolled spiral mandrel geometry<br>
+The flow in the spiral mandrel channel can be expressed as:<br>
+ (9.18)<br>
+where gz is the helical down-channel pressure gradient, Fp the shape factor for the <br>
+spiral channel, and s the reciprocal power law index (s = 1/n). For small values of <br>the height-to-width ratio (H/W &lt; 0.5) the shape factor can be taken to be constant, <br>Fp = 0.45. The leakage flow per unit tangential distance can be expressed as:<br>
+ (9.19)<br>
+where gl is the axial pressure gradient.<br>
+<hr>
+<A name=691></a> <br>
+9.4Blown Film Dies  <b>679</b><br>
+The pressure gradient in the down-channel direction can be related to the axial <br>pressure gradient by:<br>
+ (9.20)<br>
+Within the first zc of helical distance the flow in the spiral mandrel channel is not <br>
+affected by the leakage flow from a channel below it. For this section of the die the <br>following mass balance is valid, assuming the melt density to be constant:<br>
+ (9.21)<br>
+Beyond the first zc of helical distance the flow in the spiral channel is affected by <br>
+the leakage flow from the channel below it. For this section of the die the following <br>mass balance is valid:<br>
+ (9.22)<br>
+With these equations, the flow distribution can be calculated in a straightforward <br>fashion. This calculation proceeds in a step-wise fashion. The step size has to be <br>small enough to make sure that the ratio of  (z + z)/ (z) is close to unity.<br>
+<b>9.4.1The Spiral Mandrel Geometry</b><br>
+The important design parameters are:<br>N  the number of grooves<br>  the groove helix angle<br>  the spiral runout angle<br>  the taper angle of the annular channel<br>W  the perpendicular groove width<br>w  the perpendicular flight width<br>H0  the initial groove depth<br>D  the mandrel diameter<br>0  the initial flight clearance<br>lm  the axial groove length<br>The axial groove width B is related to the perpendicular groove width W by the fol-<br>lowing relationship:<br>
+ (9.23)<br>
+<hr>
+<A name=692></a><b>680</b>  <b>9Die Design</b><br>
+A similar relationship is valid for the axial flight width. The tangential flight and <br>channel width, W/sin and w/sin, are related to the mandrel diameter D and the <br>number of flights N by the following expression:<br>
+ (9.24)<br>
+The groove depth can be expressed as a function of helical distance z by the follow-<br>ing relationship:<br>
+ (9.25)<br>
+The flight clearance can be expressed as a function of z by:<br>
+ (9.26)<br>
+With these expressions a complete analysis of the flow process in the spiral mandrel <br>section can be made.<br>
+<b>9.4.2Effect of Die Geometry on Flow Distribution</b><br>
+A standard die geometry was chosen with a diameter of 250 mm with 10 grooves, <br>a helix angle of 20°, and an axial length of 250 mm. Further, the initial channel <br>depth was 19 mm and the taper angle of the annular channel 2°. The total flow rate <br>was taken as 82 cm3/s. The consistency index of the polymer melt is 0.145 Pasn <br>and the power law index n = 0.5.<br>The development of the leakage flow along the length of the die is shown in Fig. <br>9.27.<br>
+<b> Figure 9.27</b> <br>
+Development of leakage flow <br>
+along the length of the die<br>
+<hr>
+<A name=693></a> <br>
+9.4Blown Film Dies  <b>681</b><br>
+Figure 9.27 shows the leakage flow per tangential distance against the dimension-<br>less helical distance along the spiral channel. The leakage flow increases rapidly at <br>about 1/3 the channel length and reaches its final distribution at about 2/3 the <br>channel length. The corresponding flow rate in the helical channel is shown in Fig. <br>9.28.<br>
+<b> Figure 9.28</b> <br>
+Flow rate in the helical channel <br>
+versus reduced length<br>
+In the midsection of the channel the spiral flow drops quickly and reaches almost <br>zero value at about 2/3 the channel length; this is where the leakage flow reaches <br>its final distribution.<br>The pressure profile along the channel is shown in Fig. 9.29.<br>
+<b> Figure 9.29</b> <br>
+Pressure profile along the length of <br>
+the channel<br>
+The pressure gradient reduces at first, reaches a minimum at about the center <br>region, and then increases to a relatively low value. It is clear that the pressure gra-<br>
+<hr>
+<A name=694></a><b>682</b>  <b>9Die Design</b><br>
+dient is by no means constant; there is about a 5X difference between the high and <br>low value. Therefore, the pressure gradient should not be assumed to be constant <br>along the channel.<br>The effect of the number of grooves is shown in Fig. 9.30.<br>
+<b> Figure 9.30</b> <br>
+Effect of the number of grooves on flow <br>
+distribution<br>
+The effect of the number of grooves on flow distribution is quite strong. With five <br>grooves the distribution is poor, with ten grooves acceptable, and with 15 grooves <br>very good. An additional benefit of more grooves is reduced pressure drop. There-<br>fore, increasing the number of grooves has two important benefits.<br>Another important design variable is the taper angle of the annular channel. This <br>effect of the taper angle on the distribution is shown in Fig. 9.31.<br>
+<b> Figure 9.31</b> <br>
+Effect of taper angle on flow distribution<br>
+The best distribution is obtained when the taper angle is 2°. Taper angles of 1° and <br>3° result in a less uniform distribution.<br>A third design variable with an interesting effect is the initial clearance. Figure 9.32 <br>shows how the flow distribution is affected by the initial clearance.<br>A significant improvement in flow distribution can be obtained by using a non-zero <br>initial clearance. However, if the value is taken too large the distribution becomes <br>more non-uniform again. In this example there is an optimum value between 0 and <br>5 mm. A further benefit of the increased initial clearance is reduced pressure drop.<br>
+<hr>
+<A name=695></a> <br>
+9.4Blown Film Dies  <b>683</b><br>
+<b> Figure 9.32</b> <br>
+Effect of initial clearance on flow <br>
+ <br>distribution<br>
+Another important design variable is the groove helix angle. Figure 9.33 shows how <br>the helix angle affects the flow distribution.<br>
+<b> Figure 9.33</b> <br>
+Effect of helix angle on flow distribution<br>
+The helix angle has a strong effect on the flow distribution. At a helix angle of 15° <br>a very good flow distribution is obtained, while the distribution gets progressively <br>worse when the helix angle is increased. A disadvantage of reduced helix angle is <br>increased pressure drop. However, the increase in pressure drop is rather small.<br>The initial groove depth is another important design variable. The effect of groove <br>depth is shown in Fig. 9.34.<br>
+<b> Figure 9.34</b> <br>
+Effect of initial groove depth on flow <br>
+ <br>distribution<br>
+<hr>
+<A name=696></a><b>684</b>  <b>9Die Design</b><br>
+The flow distribution improves as the initial groove depth is increased. By increas-<br>ing the groove depth from 12.5 to 25 mm the distribution uniformity improves about <br>40%. Finally, the effect of reducing groove width was studied from 19 mm initial <br>width to 10 mm final width. Within this range of width values the reduction of <br>groove width did not improve the flow distribution.<br>
+<b>9.4.3Summary of Spiral Mandrel Die Design Variables</b><br>
+Three design variables have a strong effect on the flow distribution in spiral man-<br>drel dies. These are the number of grooves, the initial clearance, and the groove <br>helix angle. Increasing the number of grooves improves distribution and reduces <br>pressure drop. Typical values of the number of grooves range from 1 to 2 per inch <br>(25 mm) of die diameter. A small non-zero clearance improves the flow distribution <br>and reduces pressure drop. There is an optimum initial clearance beyond which the <br>flow distribution exhibits more variation. Small groove helix angles improve the <br>flow distribution; however, this increases pressure drop.<br>The optimum taper angle was found to be between 1° and 3°. The flow distribution <br>uniformity improves with initial groove depth, while the pressure drop reduces at <br>the same time. A gradually reducing groove width does not result in improved flow <br>distribution.<br>Obviously, the actual flow variation in spiral mandrel dies can be greater than the <br>values predicted. Other variations in flow distribution can occur because of consist-<br>ency variations in the polymer melt due to temperature non-uniformities and/or <br>insufficient mixing. Flow variations can also occur due to mechanical variations in <br>the dimensions of the die. Further, elastic effects can affect the flow distribution. As <br>a result, good quality film requires not only a good die design but also an extruder <br>with good melting and mixing capability, consistent resin properties, uniform film <br>cooling, constant tension, etc.<br>
+<b></b><br>
+ <b>9.5Profile Extrusion Dies</b><br>
+Apart from rectangular and annular extrudates, there is a tremendous variety of ex -<br>truded profiles with other shapes. Profile dies usually describe dies used to produce <br>shapes other than rectangular or annular. Profile extrusion is often the most diffi-<br>cult type of extrusion. This is, to a large extent, due to the fact that it is very difficult <br>to accurately predict the required geometry of the flow channel that will yield an <br>extruded product of proper shape and dimensions.<br>
+<hr>
+<A name=697></a> <br>
+9.5Profile Extrusion Dies  <b>685</b><br>
+In profile extrusion, there are two extremes in die design. One extreme is the plate <br>die shown in Fig. 9.35.<br>
+<b> Figure 9.35</b> <br>
+Plate die<br>
+In this design, a plate with the required opening is placed abruptly at the end of the <br>die flow channel with a minimum amount of streamlining. This type of die is simple, <br>easy to make, and easy to modify. However, there is a large dead flow region, and <br>degradation is a definite concern with polymers with limited thermal stability. This <br>type of die, therefore, should be used with relatively stable polymers and preferably <br>only for short times.<br>The other extreme is the highly streamlined die shown in Fig. 9.36.<br>
+<b> Figure 9.36</b> <br>
+Fully streamlined profile die<br>
+In this die, there is a gradual transition from the geometry of the die inlet channel to <br>the geometry of the die outlet channel. Obviously, this die is more complex, more <br>difficult to manufacture, and more difficult to modify. Good streamlining is impor-<br>tant if the polymer is susceptible to degradation. A well-streamlined profile die is <br>more likely to be used for a long extrusion run because the manufacturing cost can <br>be spread over a larger amount of product and because degradation becomes more <br>of a problem in longer runs. Obviously, there are an infinite number of intermediate <br>die designs between the totally non-streamlined plate die and the fully streamlined <br>profile die.<br>
+<hr>
+<A name=698></a><b>686</b>  <b>9Die Design</b><br>
+<b></b><br>
+ <b>9.6Coextrusion</b><br>
+Coextrusion is the simultaneous extrusion of two or more polymers through a single <br>die where the polymers are joined together such that they form distinct, well-bonded <br>layers forming a single extruded product. Coextrusion has been applied in film, <br>sheet, tubing, blown film, wire coating, and profile extrusion.<br>Advantages of coextrusion are better bonds between layers, reduced materials and <br>processing costs, improved properties, and reduced tendency for pinholes, delami-<br>nation, and air entrapment between the layers. Another advantage of coextrusion is <br>that it is often possible to reuse scrap material and locate it in an inside layer of the <br>extruded product so that it does not affect the appearance of the product. An obvious <br>disadvantage of coextrusion is that the tooling is more difficult to design and manu-<br>facture and, therefore, more expensive. Further, it requires at least two extruders, <br>and it takes more operational skill to run a coextrusion line.<br>Coextrusion is used in a variety of packaging applications to obtain the required <br>combination of properties, for instance, good moisture resistance, gas barrier prop-<br>erties, reduced costs, tear strength, etc. A combination of polyethylene/nylon/poly-<br>ethylene is popular in sterile-packaged disposables. A combination of LDPE/HDPE <br>is used for shrink film and shopping bags to obtain a balance of rigidity and low <br>cost. PS/foamed-PS coextrusion is used in production of egg cartons and meat trays. <br>In sheet extrusion, the combination ABS/polystyrene is used for refrigerator door <br>liners and margarine tubs. The ABS is applied for chemical resistance and the poly-<br>styrene for economy. Essentially the number of applications is infinite and certainly <br>the number of coextruded products will continue to rise.<br>There are basically three different techniques for coextrusion. The first employs <br>feed block dies where the various melt streams are combined in a relatively small <br>cross-section before entering the die. The advantage of this system is simplicity and <br>low cost. Existing dies can be used with little or no modification. Disadvantages are <br>that the flow properties of the different polymers have to be quite close to avoid <br>interface distortion. There is no individual thickness control of the various layers, <br>only an overall thickness control. Figure 9.37 shows a schematic of a feed block <br>sheet die.<br>The second coextrusion technique uses multi-manifold internal combining dies. The <br>different melt streams enter the die separately and join just inside the final die ori-<br>fice. The advantage of this system is that polymers with large differences in flow <br>properties can be combined with minimum interface distortion. Individual thick-<br>ness control of the different layers is possible; this enables a higher degree of layer <br>uniformity. Disadvantages are complex die design, higher cost, and limited number <br>of layers that can be combined.<br>
+<hr>
+<A name=699></a> 9.6Coextrusion  <b>687</b><br>
+<b> Figure 9.37</b> <br>
+Schematic of feed block system<br>
+Figure 9.38 shows a multi-manifold blown film coextrusion die enabling extrusion <br>of three different polymers.<br>
+<b> Figure 9.38</b> <br>
+Multi-manifold blown film die, <br>
+three layers<br>
+Coextrusion is practiced on a wide scale in blown film. There are many five-layer <br>blown film coextrusion dies used in the industry; five-layer films are now consid-<br>ered a commodity [41]. Even seven-layer dies are not unusual. Some coextrusion <br>dies use as many as 8 to 10 layers. Most of these multi-layer dies are used in high-<br>barrier packaging for food. Conventional blown film coextrusion dies have a con-<br>centric arrangement of spiral mandrel manifolds. In some cases conical spiral man-<br>drel sections are used, while in other cases the spiral mandrel section is machined <br>into a flat horizontal surface. The latter arrangement is referred to as a "pancake" <br>coextrusion die, because the different sections of the die are stacked like pancakes. <br>A schematic of a pancake coextrusion system is shown in Fig. 9.39.<br>
+<hr>
+<A name=700></a><IMG src="Polymer Extrusion by Chris Rauwendaal-700_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-700_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-700_3.jpg"><br>
+<b>688</b>  <b>9Die Design</b><br>
+<b> Figure 9.39</b> <br>
+Example of a "pancake" coextrusion system<br>
+The advantage of the pancake system is that many modules (disks) can be stacked <br>together in a more or less modular fashion, allowing for as many as ten layers to be <br>produced. Another advantage is the relatively compact design that keeps the space <br>requirements to a minimum.<br>Figure 9.40 shows a multi-manifold sheet die for two-layer coextrusion.<br>
+Choker bar adjustment nut<br>
+Flex lip<br>
+Choker bar<br>
+Upper manifold<br>
+adjustment bolt<br>
+Flex lip<br>
+<b>Plastic A</b><br>
+<b>Plastic B</b><br>
+<b> Figure 9.40</b> <br>
+Multi-manifold sheet die, <br>
+Lower manifold<br>
+two layers<br>
+The upper layer can be adjusted with a choker bar, while the final combined layer <br>thickness can be adjusted with the flex lip.<br>Coextrusion of more than three layers is difficult in a sheet die because the die geo-<br>metry becomes quite complex. Figure 9.41 shows another design of a triple-layer <br>coextrusion sheet die.<br>In this design, the layer thickness can be adjusted by die vanes. The internal vane <br>adjustment allows a greater degree of adjustment in overall actual layer thickness; <br>however, it does not enable local thickness adjustment as is possible with the choker <br>bar.<br>The third coextrusion technique uses multi-manifold external-combining dies, <br>which have completely separate manifolds for the different melt streams as well as <br>distinct orifices through which the streams leave the die separately, joining just <br>beyond the die exit. This technique is also referred to as multiple lip coextrusion. <br>The layers are combined after exiting while still molten and just downstream of the <br>die. For flat film dies, pressure rolls are used to force the layers together. In blown <br>
+<hr>
+<A name=701></a> 9.6Coextrusion  <b>689</b><br>
+film extrusion, air pressure inside the expanding bubble provides the necessary <br>pressure for combining the layers. This technique is more expensive than the feed <br>block technique; however, gage control of individual layers is more accurate, pin-<br>holes are eliminated, and the system is easier to start up.<br>One interesting benefit that can be obtained with coextrusion is a more uniform <br>temperature distribution in the material. This can be realized by coextruding a thin, <br>low-viscosity outer layer over a high-viscosity inner layer. The highest shear rate <br>and heat generation normally occurs at the wall. By having a low-viscosity material <br>at the location of maximum shear rate, the heat generation is reduced at this point <br>and a more even temperature profile is obtained. This is a useful technique for ther-<br>mally unstable polymers. As discussed before in Section 7.5.3, this technique can <br>also avoid the occurrence of shark skin and melt fracture type flow instabilities.<br>
+A<br>
+B<br>
+A<br>
+<b> Figure 9.41</b> <br>
+Vane<br>
+Three-layer coextrusion sheet die with vanes<br>
+The feed block system and the multi-manifold system are compared in Table 9.1.<br>
+<b>Table 9.1</b>Comparison of Feed Block and Multi-Manifold Systems<br>
+Feed block system<br>
+Multi-manifold system<br>
+Low cost<br>
+High cost<br>
+Many layers can be combined<br>
+Limited number of layers<br>
+Simple design<br>
+Complex design<br>
+Viscosities have to be closely matched<br>
+Viscosities can vary considerably<br>
+Limited number of polymers can be combined<br>
+Many different polymers can be combined<br>
+Limited layer uniformity<br>
+Good layer uniformity<br>
+Different number of layers in same die<br>
+Number of layers is fixed by die design<br>
+Suitable for sheet and flat film<br>
+Suitable for many different shapes<br>
+It is possible to combine a feed block with a multi-manifold die to obtain a highly <br>versatile coextrusion system that can handle many layers and polymers with large <br>differences in flow characteristics.<br>
+<hr>
+<A name=702></a><b>690</b>  <b>9Die Design</b><br>
+<b>9.6.1Interface Distortion</b><br>
+One of the challenges in coextrusion is to control the uniformity of the individual <br>layers. It is well known that when two fluids with different viscosity flow side by <br>side the interface will distort because the fluid with the lowest viscosity has a ten-<br>dency to flow to the high shear rate regions. As a result, the low-viscosity material <br>will tend to encapsulate the high-viscosity material. This situation is illustrated in <br>Fig. 9.42.<br>The extent of the interface distortion will depend on the length of the flow channel. <br>If the channel is long enough the high-viscosity fluid will be completely encapsu-<br>lated as shown in Fig. 9.42. If the channel is short the interface distortion will show <br>an intermediate configuration as shown in time t1 or t2 in Fig. 9.42. The process of <br>
+viscous encapsulation was studied by Gifford, as discussed in Section 12.4.2; see <br>Fig. 12.18.<br>The distortion shown in Fig. 9.42 is driven by viscosity differences. However, even <br>when coextruding polymers with exactly the same viscosity, interfacial distortion <br>has been observed [43].<br>
+<b> Figure 9.42</b> <br>
+Illustration of encapsulation <br>
+due to viscosity differences<br>
+Clearly, there are other mechanisms by which interfacial distortion occurs. Dooley <br>and Hughes [43] performed careful experiments to illustrate the extent of interface <br>distortion in coextrusion of polymers with the same flow characteristics. In their <br>analysis they attribute the interface distortion to normal stress differences within <br>the polymer melt, and they used a finite element program capable of handling visco-<br>elastic fluids to predict the distortion within the fluid. This issue is discussed fur-<br>ther in Section 12.4.2; see Figs. 12.19 to 12.21.<br>Svabik, Samsonkova, and Perdikoulias [45] proposed another explanation for the <br>interface distortion in coextrusion of fluids with equal viscosity. They performed <br>three-dimensional flow analysis of coextrusion flow and found that even in coextru-<br>sion with Newtonian fluids with equal viscosity layer distortion takes place. Obvi-<br>ously, this type of distortion cannot be caused by normal stress differences since <br>these do not occur in Newtonian fluids. Also, with the viscosities being equal the <br>distortion cannot be caused by viscosity differences. The predicted layer distortion <br>is schematically illustrated in Fig. 9.43. The authors call the distortion resulting <br>from purely viscous flow geometrical encapsulation.<br>
+<hr>
+<A name=703></a><IMG src="Polymer Extrusion by Chris Rauwendaal-703_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-703_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-703_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-703_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-703_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-703_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-703_7.jpg"><br>
+ 9.6Coextrusion  <b>691</b><br>
+<b>Figure 9.43</b>Illustration of geometrical encapsulation<br>
+Geometric encapsulation is caused by the parabolic velocity profiles in the die flow <br>channel. Because the velocities are highest in the center region of the channel the <br>layer thickness increases in this part of the channel, while the thickness of the out-<br>side layers reduces in the center region of the channel. At the side walls the inner <br>layer thickness reduces while the outside layer thickness increases. The distortion <br>caused by geometric encapsulation becomes more severe as the fluid becomes shear <br>thinning.<br>It appears that there are several mechanisms for interface distortion. One is distor-<br>tion caused by viscosity differences (viscous encapsulation), another is caused by <br>normal stress differences in the fluid (elastic encapsulation), and a third is caused <br>by normal velocity differences within the fluid (geometrical encapsulation).  Ob -<br>viously, the distortion will increase when viscosity differences are large and when <br>normal stress differences play a significant role.<br>There are two approaches to minimizing the interface distortion. One is to reduce <br>the length over which the different melt streams flow together. This is done in multi-<br>manifold dies where the different melt streams are combined just before the exit of <br>the die. Another approach is to modify the initial configuration of the layers in such <br>a way that the final layer configuration is the one desired. This approach is called <br>profiling, and this is a method frequently used in feed block coextrusion systems to <br>achieve uniform layer distribution at the exit of the die. This principle is illustrated <br>in Fig. 9.44.<br>
+<b>Figure 9.44</b>Illustration of profiling in coextrusion<br>
+<hr>
+<A name=704></a><b>692</b>  <b>9Die Design</b><br>
+Profiling can be used with feed block coextrusion systems and also with vane-type <br>coextrusion dies as shown in Fig. 9.41. With 3-D flow simulation the rearrangement <br>of the layers can be predicted. Particles defining a flat interface can be tracked <br>upstream to the input plane to determine the appropriate initial layer geometry. <br>Examples of this approach are discussed by Perdikoulias et al. [45].<br>
+<b></b><br>
+ <b>9.7Calibrators</b><br>
+In pipe extrusion, the extrusion die is often followed by a sizing die, where the <br>actual dimensions of the pipe are primarily determined. Such a device is generally <br>referred to as a calibrator. Calibrators are required if the extrudate emerging from <br>the die has insufficient melt strength to maintain the required shape. The calibrator <br>is in close contact with the polymer melt and cools the extrudate. When the extru-<br>date leaves the calibrator, it has sufficient strength to be pulled through a haul-off <br>device such as a catapuller without significant deformation of the extruded product.<br>In a vacuum calibrator, a vacuum is applied to ensure good contact between the <br>calibrator and the extrudate and to prevent collapse of the extrudate. The use of a <br>vacuum calibrator is generally easier than the use of positive air pressure within <br>the extrudate because a vacuum is easier to maintain at a constant level. Positive <br>internal air pressure tends to vary with the length of the extrudate and it is difficult <br>to maintain when the extrudate has to be cut into discrete lengths. Calibrators are <br>useful when good, accurate shape control is important. When the requirements for <br>shape control are less stringent, the extrudate shape is often maintained by support <br>brackets placed downstream of the extrusion die. In fact, the extrudate shape can <br>be modified substantially by the support brackets. This can be useful because it <br>allows modification of the shape of the extrudate without changing the die, but only <br>by changing the shape of the support brackets.<br>As discussed by Michaeli [1], there are five types of calibrators:<br>1.  Slide calibrators<br>2.  External calibrators with internal air pressure<br>3.  External vacuum calibrators<br>4.  Internal calibrators<br>5.  Precision profile pultrusion (Technoform process)<br>Slide calibrators are used for simple and open profiles. The extrudate is more or less <br>in contact with cooled plates, and the profile is pulled through the calibrator, caus-<br>ing some amount of drawdown.<br>
+<hr>
+<A name=705></a><IMG src="Polymer Extrusion by Chris Rauwendaal-705_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-705_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-705_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-705_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-705_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-705_6.jpg"><br>
+ 9.7Calibrators  <b>693</b><br>
+A schematic of a water-cooled slide calibrator is shown in Fig. 9.45.<br>
+<b> Figure 9.45</b> <br>
+Example of profile calibration system<br>
+An example of an external calibrator with internal air pressure is shown in Fig. 9.46.<br>
+<b>Figure 9.46</b>External calibrator with internal air pressure<br>
+The air pressure is maintained by a sealing mandrel located inside the extrudate <br>downstream of the calibrator. The mandrel is attached to the tip by a cable to fix its <br>position. This type of calibration is used in larger diameter pipe for PVC (D &gt; 350 mm) <br>and PE (D &gt; 100 mm). An example of an external vacuum calibrator is shown in <br>Fig. 9.47.<br>
+<b>Figure 9.47</b>Calibrator with external vacuum<br>
+This internal calibrator modifies the annular shape of the extrudate emerging from <br>the die into a more or less triangular shape. This allows shape modification just by <br>
+<hr>
+<A name=706></a><b>694</b>  <b>9Die Design</b><br>
+the calibrator. However, internal calibration is not used very often. In precision pro-<br>file pultrusion, a small amount of polymer is allowed to accumulate between the die <br>and the calibrator. The accumulation is controlled by a sensor through adjustment <br>of the haul-off speed. The extrudate is pulled through a short, intensely cooled cali-<br>brator followed by a water bath. This type of calibration is referred to as the Techno-<br>form process; it was developed by Reifenhäuser KG.<br>An example of a dry vacuum calibration system is shown in Fig. 9.48<br>
+<b>Figure 9.48</b>Example of dry vacuum calibration system<br>
+References<br>1.  W. Michaeli, "Extrusionswerkzeuge fuer Kunststoffe," Carl Hanser Verlag, Munich <br>
+(1979), in German<br>
+2.  W. Michaeli, "Extrusion Dies," Hanser Gardner Publications, Cincinnati, OH (1984)<br>3.  D.J. Weeks, Br. Plast., 31, 156­160 (1958)<br>4.  D.J. Weeks, Br. Plast., 31, 201­205 (1958)<br>5.  J.M. McKelvey and K. Ito, Polym. Eng. Sci., 11, 258­263 (1971)<br>6.  J.R.A. Pearson, Trans. Plast. Inst., 32, 239­244 (1964)<br>7.  K. Ito, Jpn. Plast., 4, 27­30 (1970)<br>8.  M.B. Cheijfec, Plast. Massy, 12, 31­33 (1973)<br>9.  K. Ito, Jpn. Plast., 2, 35­37 (1968)<br>10.  K. Ito, Jpn. Plast., 3, 32­34 (1969)<br>11.  J. Wortberg, Ph. D. thesis, RWTH Aachen, Germany (1978). Also in: "Berechnen von <br>
+Extrudierwerkzeugen," VDI-Verlag GmbH, Duesseldorf, Germany (1978)<br>
+12.  P. Fischer, E.H. Goermar, M. Herner, and U. Kosel, Kunststoffe, 61, 342­355 (1971)<br>13.  E.H. Goermar, Ph.D. thesis, RWTH Aachen, Germany (1968)<br>14.  C.I. Chung and D.T. Lohkamp, SPE ANTEC, Atlanta, 363­365 (1975)<br>15.  I. Klein and R. Klein, SPE J., 29, 33­37 (1973)<br>16.  H. Schoenewald, Kunststoffe, 68, 238­243 (1978)<br>17.  B. Vergnes, P. Saillard, and B. Plantamura, Kunststoffe, 70, 750­752 (1980)<br>
+<hr>
+<A name=707></a> References <br>
+<b>695</b><br>
+18.  Y. Matsubara, Polym. Eng. Sci., 19, 169­172 (1979)<br>19.  Y. Matsubara, Polym. Eng. Sci., 20, 716­719 (1980)<br>20.  J. Sneller, Mod. Plast., 48­52 (1983)<br>21.  S. Prager and M. Tirrell, J. Chem. Phys., 75, 5194­5198 (1981)<br>22.  Y.H. Kim and R.P. Wool, Macromolecules, 16, 1115 (1983)<br>23.  R.P. Wool and K.M. O'Connor, J. Polym. Sci., Letters Ed., 20 (1982)<br>24.  R.P. Wool and K.M. O'Connor, J. Appl. Phys., 52 (1981)<br>25.  S.C. Malguarnera and A. Manisali, SPE ANTEC, New York, 124­128 (1980)<br>26.  W. Ast, Kunststoffe, 66, 186­192 (1976)<br>27.  J. Wortberg and K.P. Schmidtz, Kunststoffe, 72, 198­205 (1982)<br>28.  B. Proctor, SPE ANTEC, Washington, D.C. , 211­218 (1971)<br>29.  W. Predoehl, "Technologie Extrudierter Kunststoffolien," VDI-Verlag GmbH, Duessel-<br>
+dorf, Germany (1979)<br>
+30.  M.J. Crochet, A.R. Davies, and K. Walters, "Numerical Simulation of Non-Newtonian <br>
+Flow," Elsevier, Amsterdam, The Netherlands (1984)<br>
+31.  H.H. Winter and H.G. Fritz, SPE ANTEC, New Orleans, 49­52 (1984)<br>32.  C.D. Han, "Influence of the Die Entry Angle on the Entrance Pressure Drop, Recover-<br>
+able Elastic Energy, and the Onset of Flow Instability in Polymer Melt Flow," J. Appl. <br>Polym. Sci., 17, 1403­1413 (1973)<br>
+33.  P.C. Powell, "Design of Extruder Dies Using Thermoplastics Melt Properties Data," <br>
+Polym. Eng. Sci., 14, 298­307 (1974)<br>
+34.  K.U. Haas and F.H. Skewis, "The Wire Coating Process: Die Design and Polymer Flow <br>
+Characteristics," SPE ANTEC, 8­11 (1974)<br>
+35.  "Extrusion Guide for Melt Processable Fluoropolymers," DuPont Technical Literature<br>36.  J.A. Blair, "Teflon FEP Fluorocarbon Resin, Techniques for Processing by Melt Extru-<br>
+sion," DuPont Technical Literature, TR 108<br>
+37.  J.A. Blair, "Methods for Increasing Extrusion Rates of Teflon FEP Fluorocarbon Resins," <br>
+DuPont Technical literature, TR 108<br>
+38.  C.L. Tucker III, "Fundamentals of Computer Modeling for Polymer Processing," Hanser <br>
+Publishers, Munich (1989)<br>
+39.  K. O'Brien, "Applications of Computer Modeling for Extrusion and Other Continuous <br>
+Polymer Processes," Carl Hanser Publishers, Munich (1992)<br>
+40.  G. Menges, A. Mayer, T. Bartilla, and J. Wortberg, "A New Concept for the Design of <br>
+ <br>Spiral Mandrel Dies," Adv. Polym. Technol., 4, no. 2, 177­185 (1984)<br>
+41.  C.J. Rauwendaal, "Flow Distribution in Spiral Mandrel Dies," Polym. Eng. Sci., 27, <br>
+186­191 (1987)<br>
+42.  P.A. Toensmeier, "High-Value Niche Grows in Multi-Layer Die Design," Mod. Plast., <br>
+Dec., 52­54 (2000)<br>
+43.  J. Dooley and K. Hughes, "Analyzing the Flow Through Dies Containing Different Chan-<br>
+nel Geometries," SPE ANTEC (1996)<br>
+<hr>
+<A name=708></a><b>696</b>  <b>9Die Design</b><br>
+44.  B.L. Koziey, J. Vlachopoulos, J. Vlcek, and J. Svabik, "Profile Die Design by Pressure <br>
+Balancing and Cross Flow Minimization," SPE ANTEC (1996)<br>
+45.  J. Svabik, P. Samsonkova, and J. Perdikoulias, Proceedings of Vinyltech, Mississauga, <br>
+Canada, 111 (1999)<br>
+46.  H. Gross, W. Michaeli, F. Pöhler, and J. Ullrich, "(Membran Statt Staubalken) Membrane <br>
+instead of Restrictor Bar," Kunststoffe, Oct., 1352­1358 (1994)<br>
+47.  J. Callari, "Flow Tuner, Precise Block the Latest in Flat Dies," Plast. World, Oct., 16 <br>
+(1996)<br>
+<hr>
+<A name=709></a><b> 10 Twin Screw </b><br>
+<b>Extruders</b><br>
+<b></b><br>
+ <b>10.1Introduction</b><br>
+The first twin screw extruders for polymer processing were developed in the late <br>1930s in Italy. Roberto Colombo developed the co-rotating twin screw extruder, and <br>Carlo Pasquetti developed the counter-rotating twin screw extruder. Early twin <br>screw extruders had a number of mechanical problems. The most important limi-<br>tation was the thrust bearing design. Because of the limited space, it is difficult to <br>design a thrust bearing with good axial and radial load capability. The early thrust <br>bearings were not strong enough to give the twin screw extruders good mechanical <br>reliability. In the late 1960s, special thrust bearings were developed especially for <br>application in twin screw extruders. Since that time, the mechanical reliability of <br>twin screw extruders has been comparable to that of single screw extruders. How-<br>ever, twin screw extruders generally still do not have as high a thrust bearing rating <br>as single screw extruders.<br>Twin screw extruders have established a solid position in the polymer processing <br>industry. The two main areas of application for twin screw extruders are profile <br>extrusion of thermally sensitive materials (e.g., RPVC) and specialty polymer pro-<br>cessing operations, such as compounding, devolatilization, chemical reactions, etc. <br>Twin screw extruders used in profile extrusion have a closely fitting flight and chan-<br>nel profile and operate at relatively low screw speeds, in the range of about 20 rpm. <br>These machines offer several advantages over single screw extruders. Better feeding <br>and more positive conveying characteristics allow the machine to process hard-to-<br>feed materials (powders, slippery materials, etc.) and yield short residence times <br>and a narrow residence time distribution (RTD). Better mixing and larger heat trans-<br>fer area allow good control of the stock temperatures. Good control over residence <br>times and stock temperatures obviously are key elements in the profile extrusion of <br>thermally sensitive materials. Most twin screw extruders used in profile extrusion <br>are closely intermeshing and counter-rotating, although a few co-rotating twin screw <br>extruders are used.<br>Specialty polymer processing operations are performed on a number of twin screw <br>extruders with a variety of designs. An overview of the different types of twin screw <br>extruders is shown in Table 2.2, Chapter 2. High speed intermeshing co-rotating <br>
+<hr>
+<A name=710></a><b>698</b>  <b> 10Twin Screw Extruders</b><br>
+extruders are used in compounding and devolatilization. Co-rotating twin screw <br>extruders are also used as chemical reactors. Co-rotating twin screw extruders used <br>in compounding often operate at high speeds, with typical screw speeds ranging <br>from 300 to 600 rpm. Very high speed co-rotating twin screw extruders are available <br>that can run at speeds as high as 1200 to 1400 rpm. Obviously, not all compounds <br>can be processed at screw speeds this high.<br>Non-intermeshing extruders are used for mixing, chemical reactions, and devolatili-<br>zation. The conveying mechanism in non-intermeshing extruders is considerably <br>different from that in intermeshing extruders; it is closer to the conveying mechanism <br>in a single screw extruder, although there are substantial differences. As a result, <br>non-intermeshing twin screw extruders do not have positive conveying characteris-<br>tics. However, it should be realized that positive conveying characteristics generally <br>result in poor axial mixing capability. Thus, if axial mixing is required, positive <br>conveying characteristics can be a disadvantage. Table 10.1 compares high speed to <br>low speed twin screw extruders.<br>
+<b>Table 10.1</b>Comparison of High Speed to Low Speed Twin Screw Extruders<br>
+High speed TSE<br>
+Low speed TSE<br>
+Primarily used in compounding<br>
+Primarily used in profile extrusion<br>
+Screw speeds from 200 to 1400 rpm<br>
+Screw speeds from 10­40 rpm<br>
+Always operated by starve feeding, flood feeding  Can be operated by starve feeding, in some cases <br>
+not possible<br>
+flood feeding is possible<br>
+Operates at a low degree of fill, around 20­40% <br>
+Operates at a high degree of fill<br>
+typically<br>Good mixing characteristics in most cases<br>
+Poor mixing capability in most cases<br>
+Fair conveying characteristics, limited pressure <br>
+Good conveying characteristics, good pressure <br>
+generating capability<br>
+generating capability<br>
+Fair output stability<br>
+Good output stability<br>
+Pressures in the extruder are generally low<br>
+Pressures can be relatively high<br>
+Machines generally have long L/D ratio, typically  Machines have short L/D ratio, typically less than <br>
+over 30:1<br>
+30:1<br>
+Sequential feeding commonly used<br>
+Sequential feeding not common<br>
+Modular screws and barrel commonly used<br>
+Most extruders use non-modular screws and barrel<br>
+High price, much higher than single screw <br>
+Low price, closer to the price of single screw <br>
+extruders of same diameter<br>
+extruders<br>
+Parallel screws are used in high speed extruders,  Small diameter extruders can use conical screws, <br>
+conical screws are not used<br>
+large diameter extruders are all parallel<br>
+<hr>
+<A name=711></a> <br>
+10.2Twin versus Single Screw Extruder  <b>699</b><br>
+<b></b><br>
+ <b>10.2Twin versus Single Screw Extruder</b><br>
+The characteristics of twin screw extruders may be better appreciated by consider-<br>ing the fundamental differences between single and twin screw extruders. One <br>major difference is the type of transport that takes place in the extruder. Material <br>transport in a single screw extruder is a drag-induced type of transport: frictional <br>drag in the solids conveying zone and viscous drag in the melt conveying zone. <br>Therefore, the conveying behavior is to a large extent determined by the frictional <br>properties of the solid material and the viscous properties of the molten material. <br>There are many materials with unfavorable frictional properties, which cannot be <br>fed into single screw extruders without experiencing feed problems. On the other <br>hand, the transport in an intermeshing twin screw extruder is to some extent <br>a positive displacement type of transport. The degree of positive displacement <br>depends on how well the flight of one screw closes the opposing channel of the other <br>screw. The most positive displacement is obtained in a closely intermeshing, counter-<br>rotating geometry. For instance, a gear pump can be considered to be a counter-<br>rotating twin screw extruder with the helix angle of the screw flights being 90° or <br>close to it. However, even a gear pump is not a pure positive displacement device <br>because the machine cannot be designed with zero clearances. Thus, leakage flows <br>will reduce the degree of positive conveying that can be achieved in a twin screw <br>extruder.<br>Another major difference between the single and twin screw extruder is the velocity <br>patterns in the machine. The velocity profiles in single screw extruders are well <br>defined and fairly easy to describe; see Section 7.4. The situation in twin screw <br>extruders in more complicated. The velocity profiles in twin screw extruders are <br>complex and more difficult to describe. A number of workers have analyzed the flow <br>patterns by neglecting the flow in the intermeshing region [1­5]. However, the mix-<br>ing characteristics and the overall behavior of the machine is primarily determined <br>by the leakage flows occurring in the intermeshing region. Thus, results from analy-<br>ses that do not consider the flow in the intermeshing region have limited practical <br>applicability. On the other hand, analyses that attempt to accurately describe the <br>flow in the intermeshing region can easily become very complex [6, 7].<br>The complex flow patterns in twin screw extruders have several advantages, such as <br>good mixing, good heat transfer, large melting capacity, good devolatilization capacity, <br>and good control over stock temperatures. One disadvantage of the complex flow <br>patterns is that they are difficult to describe. The theory of twin screw extruders is <br>not as well developed as the theory of single screw extruders. As a result, it is diffi-<br>cult to predict the performance of a twin screw extruder based on extruder geome-<br>try, polymer properties, and processing conditions. Conversely, it is equally difficult <br>to predict the proper screw geometry when a certain performance is required in a <br>particular application. This situation has led to twin screw extruders of modular <br>
+<hr>
+<A name=712></a><b>700</b>  <b> 10Twin Screw Extruders</b><br>
+design. These machines have removable screw and barrel elements. The screw <br>design can be altered by changing the sequence of the screw elements along the <br>shaft. In this way, an almost infinite number of screw geometries can be put together. <br>The modular design, therefore, creates excellent flexibility and allows careful opti-<br>mization of screw and barrel geometry to each particular application. Unfortunately, <br>modular screws and barrels also increase the cost of the extruder a great deal. A <br>number of modular screw elements for a co-rotating extruder are shown in Fig. 10.1.<br>
+<b> Figure 10.1</b> <br>
+Screw elements for co-rotating twin <br>
+screw extruder<br>
+Modular screw elements for a counter-rotating extruder are shown in Fig. 10.2.<br>
+<b> Figure 10.2</b> <br>
+Screw elements for counter-<br>
+rotating twin screw extruder<br>
+Table 10.2 presents a comparison of characteristics of twin and single screw ex- <br>truders. The comparison is based on intermeshing twin screw extruders.<br>
+<b>Table 10.2</b>Comparison of Twin Screw and Single Screw Extruder<br>
+Twin Screw Extruder (TSE)<br>
+Single Screw Extruder (SSE)<br>
+Used in profile, compounding, and reactive <br>
+Used in simple profile extrusion and coextrusion<br>
+extrusion<br>Often used with modular design of screw and <br>
+Modular design of screw and barrel is rarely used--<br>
+barrel--great flexibility<br>
+less flexibility<br>
+Prediction of extruder performance is often <br>
+Prediction of extruder performance less difficult <br>
+ <br>difficult<br>
+than for twin screw extruder<br>
+<hr>
+<A name=713></a><IMG src="Polymer Extrusion by Chris Rauwendaal-713_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-713_17.jpg"><br>
+ <br>
+10.3Intermeshing Co-Rotating Extruders  <b>701</b><br>
+Twin Screw Extruder (TSE)<br>
+Single Screw Extruder (SSE)<br>
+Good feeding, can handle pellets, powder, liquids Fair feeding, slippery additives tend to give <br>
+ <br>problems<br>
+Good melting; dispersed solids melting <br>
+Fair melting; contiguous solids melting mechanism<br>
+ <br>mechanism<br>Good distributive mixing with effective mixing <br>
+Good distributive mixing with effective mixing <br>
+elements<br>
+ <br>elements<br>
+Good dispersive mixing with effective mixing <br>
+Good dispersive mixing with effective mixing <br>
+ <br>elements<br>
+ <br>elements<br>
+Good degassing<br>
+Fair degassing<br>
+Intermeshing TSE can have completely self- <br>
+Not self-wiping: barrel is wiped but screw root and <br>
+wiping characteristics<br>
+flight flanks are not<br>
+Modular TSE is very expensive<br>
+SSE is relatively inexpensive<br>
+Co-rotating TSE can run at very high screw <br>
+SSE usually run between 10­150 rpm; high screw <br>
+speed, up to 1400 rpm<br>
+speeds possible but not often used<br>
+<b></b><br>
+ <b>10.3Intermeshing Co-Rotating Extruders</b><br>
+There are two types of intermeshing co-rotating extruders: the low speed extruder <br>and the high speed extruder. The two machines are different in design, in operating <br>characteristics, and in areas of application. The low speed co-rotating twin screw <br>extruder is primarily used in profile extrusion, while the high speed extruder is <br>primarily used in compounding.<br>
+<b>10.3.1Closely Intermeshing Extruders</b><br>
+The low speed extruder has closely intermeshing screw geometry where the flight <br>profile fits closely into the channel profile, i.e., a conjugated screw profile. A typical <br>screw geometry of the closely intermeshing co-rotating (CICO) twin screw extruder <br>is shown in Fig. 10.3.<br>
+A<br>
+<b> Figure 10.3</b> <br>
+A<br>
+Section A-A<br>
+Screw geometry of a CICO extruder<br>
+<hr>
+<A name=714></a><IMG src="Polymer Extrusion by Chris Rauwendaal-714_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-714_20.jpg"><br>
+<b>702</b>  <b> 10Twin Screw Extruders</b><br>
+The conjugated screw profile shown in Fig. 10.3 appears to form a good seal between <br>the two screws. However, a cross-section through the intermeshing region, shown in <br>Fig. 10.4, reveals the presence of relatively large openings between the channels of <br>the two screws.<br>Therefore, the conveying characteristics of the CICO extruder are not as positive as <br>those of a closely intermeshing counter-rotating extruder (CICT); see also Figs. <br>10.27 and 10.28.<br>
+B<br>
+B<br>
+Area II<br>
+Area I<br>
+<b> Figure 10.4</b> <br>
+Cross-section through the intermeshing region <br>
+Section B-B<br>
+of a CICO extruder<br>
+The co-rotating twin screw extruder has a sliding type of intermeshing as shown in <br>Fig. 10.5.<br>
+<b> Figure 10.5</b> <br>
+Sliding type of intermeshing in co-rotating <br>
+twin screw extruders<br>
+The screw velocities in the intermeshing region are in opposite directions. There-<br>fore, material entering the intermeshing region will have little tendency to move <br>through the entire intermeshing region unless the flight flank clearance is quite <br>large; this situation is shown in Fig. 10.6(a).<br>
+<b> Figure 10.6(a)</b> <br>
+Intermeshing region with a large flank clearance<br>
+Because of the relatively large open areas between the channels, material entering <br>the intermeshing region will tend to flow into the channel of the adjacent screw. The <br>material will move in an open figure-eight pattern, as shown in Fig. 10.6(b), while at <br>the same time moving in the axial direction.<br>
+<hr>
+<A name=715></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>703</b><br>
+<b> Figure 10.6(b)</b> <br>
+Movement of material in open figure-eight pattern<br>
+The material close to the passive flight flank cannot flow into the channel of the <br>adjacent screw because it is obstructed by the flight of the adjacent screw. The mate-<br>rial, therefore, will undergo a circulatory flow as shown in Fig. 10.7.<br>
+<b> Figure 10.7</b> <br>
+Circulatory flow at the passive flight flank<br>
+This material fraction will move forward at axial velocity va:<br>
+ (10.1)<br>
+The obstructed material fraction will contribute to the positive conveying character-<br>istics of the extruder. If the obstructed area (Area I in Fig. 10.4) is large relative to <br>the open area (Area II in Fig. 10.4), then the conveying characteristics will be quite <br>positive. If the open area is large relative to the obstructed area, then the positive <br>conveying characteristics will be considerably reduced, resulting in a wide resi-<br>dence time distribution (RTD) and a more pressure-dependent throughput. CICO <br>extruders have relatively positive conveying characteristics because their screw <br>geometry is such that the open area is small relative to the obstructed area.<br>The sliding type of intermeshing will result in high pressure regions at the point <br>where the material enters the intermeshing region; this is shown in Fig. 10.8.<br>
+High pressure region<br>
+<b> Figure 10.8</b> <br>
+High pressure regions at the entrance <br>
+High pressure region<br>
+to the intermeshing region<br>
+<hr>
+<A name=716></a><IMG src="Polymer Extrusion by Chris Rauwendaal-716_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-716_23.jpg"><br>
+<b>704</b>  <b> 10Twin Screw Extruders</b><br>
+The pressure build-up occurs primarily because of the reduction in cross-sectional <br>area of the flow channel as the material enters the intermeshing region. Pressure <br>build-up also occurs because of the change in flow direction that occurs as the mate-<br>rial enters the intermeshing region. Obviously, the pressure build-up will be most <br>severe when the open area is small compared to the obstructed area, which is the <br>case in CICO twin screw extruders. These high pressure regions will result in lateral <br>forces on the screws trying to push the screws apart. These separating forces will <br>increase with screw speed. Clearly, the separating forces should not be as large as to <br>cause contact between the screws and barrel, since this will result in severe wear. <br>Therefore, CICO extruders have to run at low speed in order to avoid large pressure <br>peaks in the intermeshing region.<br>
+<b>10.3.2Self-Wiping Extruders</b><br>
+High speed co-rotating extruders have a closely matching flight profile, as shown in <br>Fig. 10.9.<br>
+<b> Figure 10.9</b> <br>
+Flight geometry in CSCO extruders<br>
+There is considerable openness from one channel to the adjacent channel. This is <br>obvious both from the top view of the screws shown in Fig. 10.9 as well as from the <br>cross-section through the intermeshing region, shown in Fig. 10.10.<br>
+B<br>
+B<br>
+Area I<br>
+Area II<br>
+<b> Figure 10.10</b> <br>
+Cross-section through the intermeshing region <br>
+Section B-B<br>
+in CSCO extruders<br>
+Thus, the open area II is large relative to the obstructed area I. Therefore, there is <br>relatively little tendency for large pressure peaks to form in the intermeshing region. <br>The screws can therefore be designed with relatively small clearances between the <br>
+<hr>
+<A name=717></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>705</b><br>
+two screws; the screws are then closely self-wiping. Twin screw extruders of this <br>design are generally referred to as closely self-wiping co-rotating extruders (CSCO).<br>Since the tendency to develop large pressure peaks in the intermeshing region is <br>quite small with CSCO extruders, they can run at high speeds, as high as 1400 rpm. <br>This is made possible by the relatively large open area in the intermeshing region. <br>However, this geometrical characteristic also results in a relatively non-positive <br> <br>conveying characteristic with a corresponding wide RTD and pressure-sensitive <br>throughput. These machines, therefore, are not well suited for direct profile extru-<br>sion. A large fraction of the material will follow the figure-eight flow pattern dis-<br>cussed earlier. This fraction in the CSCO extruder will be considerably larger than <br>in the CICO extruder. The progression of the material in one channel is shown in <br>Fig. 10.11 for a double-flighted geometry.<br>
+<b> Figure 10.11</b> <br>
+Transport in a double-flighted CSCO extruder<br>
+Note that the material is displaced by an axial distance of three times the pitch <br>when it reenters the screw. Thus, in a double-flighted geometry, there are three <br>more or less independent down-channel flows. When the number of parallel flights <br>is p, the number of independent down-channel flows ni is:<br>
+ (10.2)<br>
+<b>10.3.2.1Geometry of Self-Wiping Extruders<br></b>The flight and channel geometry of self-wiping co-rotating twin screw extruders is <br>determined by the screw diameter, centerline distance, helix angle, and the number <br>of parallel flights. When these geometrical parameters are selected, the cross-sec-<br>tion geometry is fixed and can be determined from kinematic principles as described <br>in detail by Booy [8]. If t is the tip angle, i the angle of intermesh, p the number of <br>
+parallel flights, and D the screw diameter, then the centerline distance Lc can be <br>
+determined from:<br>
+ (10.3)<br>
+Thus, when the diameter, centerline distance, and number of flights are selected, <br>the tip angle and thus the flight width are fixed by Eq. 10.3. The angles t and i are <br>
+angles in the plane perpendicular to the screw axis; see also Fig. 10.12.<br>
+<hr>
+<A name=718></a><b>706</b>  <b> 10Twin Screw Extruders</b><br>
+<b>t</b><br>
+2<b>i</b><br>
+<b> Figure 10.12</b> <br>
+Geometry of self-wiping twin screw <br>
+<b>LC</b><br>
+extruders<br>
+The angle of intermesh i is related to the tip angle by:<br>
+ (10.4)<br>
+Thus, the centerline distance can be expressed simply as:<br>
+ (10.5)<br>
+When the angle of intermesh is zero, the centerline distance simply equals the <br>screw diameter. The maximum channel depth at the root of the screw is given by:<br>
+ (10.6)<br>
+For non-zero values of the tip and root angle, the centerline distance has to increase <br>with the tip angle (see Fig. 10.13[A]), with a corresponding reduction in the <br>intermesh angle and channel depth. Fig. 10.13(B) shows a graph of the ratio of <br> <br>channel depth to screw diameter plotted against the tip angle for three values of the <br>number of flights.<br>Once the tip angle and the number of flights are selected, the centerline distance <br>and the root diameter are fixed in a self-wiping co-rotating extruder. This imposes a <br>considerable design constraint on co-rotating twin screw extruders.<br>One of the implications of this constraint is that the centerline distance for triple-<br>flighted or trilobal screws, p = 3, has to be quite large. The smallest centerline dis-<br>tance for a triple-flighted geometry is CL/D = 0.53 (= 0.866); see Eq. 10.3 and the <br>
+maximum channel depth 0.134 D. As a result, the channel depth of the screw <br>becomes relatively small and so does the channel volume. Consequently, the <br>throughput capability of triple-flighted screws is limited.<br>
+<hr>
+<A name=719></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>707</b><br>
+A<br>
+1<br>
+Triple flighted<br>
+0.9<br>
+Double flighted<br>
+0.8<br>
+<b>]</b><br>
+<b>CL/D [- </b>0.7<br>
+0.6<br>
+Single flighted<br>
+0.5<br>
+0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+<b>Tip angle [degrees]</b><br>
+B<br>
+0.5<br>
+0.45<br>
+Single flighted<br>
+0.4<br>
+0.35<br>
+0.3<br>
+0.25<br>
+0.2<br>
+<b>Channel depth [H/D] </b>0.15<br>
+0.1<br>
+Double flighted<br>
+0.05<br>
+Triple flighted<br>
+0<br>
+0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+<b>Tip angle [degrees]</b><br>
+<b>Figure 10.13(A­B)</b>Channel depth versus tip angle for self-wiping co-rotating twin screws<br>
+The centerline distance can be made considerably shorter in double-flighted, bilobal <br>screws (p = 2). In this case the smallest centerline distance is CL/D = 0.52 (= 0.707) <br>
+and the maximum channel depth 0.293 D. Thus, the channel depth and channel <br>volume can be made substantially larger, which results in improved throughput <br>capability. These considerations have led several twin screw extruder manufacturers <br>to switch from triple-flighted co-rotating extruders to double-flighted extruders.<br>The free volume of self-wiping co-rotating extruders can be expressed as CR3(L/D). <br>For single-flighted screws the constant C = 7.0, for double-flighted screws C = 6.8, <br>and for triple-flighted screws C = 3.8. Clearly, the free volume reduces a great deal <br>when going from a double-flighted to a triple-flighted geometry.<br>Because the leading flight flank of one screw sweeps both the leading and trailing <br>flight flank of the other screw, the geometry of the leading flight flank has to be the <br>
+<hr>
+<A name=720></a><b>708</b>  <b> 10Twin Screw Extruders</b><br>
+same as that of the trailing flight flank. This is not the case for counter-rotating twin <br>screw extruders.<br>The construction of the screw geometry for co-rotating extruders was described in <br>detail by Booy [8]. Figure 10.14 shows the steps involved in constructing the exact <br>geometry of the screws.<br>
+<b> Figure 10.14</b> <br>
+Construction of the geometry of <br>
+self-wiping co-rotating screws<br>
+The steps in the construction are as follows:<br>1.  Draw line AB such that AB equals the centerline distance; this defines the <br>
+intermesh angle 2i<br>
+2.  Locate point D such that angle COD equals /n, where n is number of flights<br>3.  Point P is located on the circle midway between points B and D<br>4.  Make PD equal to QD; this defines one screw tip; the other tip is Q'P'<br>5.  Center Mp of flank curve through P lies on a circle at distance Lc from P<br>6.  Construct flank curve PR, then add other flank curves<br>7.  One screw cross-section is now completely defined<br>The channel depth reduces along the flight flank. If circumferential angle  starts at <br>the beginning of the flight flank the channel depth as a function of angle  can be <br>written as:<br>
+ (10.7)<br>
+Figure 10.15 illustrates how the channel depth varies with circumferential angle .<br>The axial coordinate 1 is related to the circumferential angle  by:<br>
+ (10.8)<br>
+where  is the helix angle of the screw flight.<br>
+<hr>
+<A name=721></a><IMG src="Polymer Extrusion by Chris Rauwendaal-721_1.jpg"><br>
+ <br>
+10.3Intermeshing Co-Rotating Extruders  <b>709</b><br>
+H()<br>
+<b> Figure 10.15</b> <br>
+Channel depth as a function of angle <br>
+Thus, the channel depth profile as a function of axial distance 1 is:<br>
+ (10.9)<br>
+The cross-channel coordinate x is related to the circumferential angle  by:<br>
+ (10.10)<br>
+Thus, the channel depth as a function of cross-channel distance is:<br>
+ (10.11)<br>
+When the coordinate x is zero, the channel depth reaches its maximum value; see <br>Eq. 10.6. The maximum channel depth is maintained over a circumferential angular <br>distance of t, which corresponds to a cross-channel distance 0.5 Dtsin.<br>In CSCO extruders, a major portion of the material entering the intermeshing region <br>in a channel of one screw will transfer to an adjacent channel of the other screw. <br>This is shown in Fig. 10.16 where the material transport in one channel is shown in <br>a cross-section perpendicular to the screw axes.<br>Just before the intermeshing region, the flow channel area is determined by the area <br>between the screw and the barrel, i.e., the screw channel area A1. In the intermesh-<br>
+ing region itself, the flow channel area is determined by the area between the two <br>screws and the barrel. Initially, the flow channel area increases to a maximum value <br>and then it reduces back to area A1 at the end of the intermeshing region. This <br>
+action causes a relatively effective transfer of material from one screw to the other <br>and vice versa if the flight width is small relative to the width of the channel--this <br>is the case in CSCO extruders. As the flight width increases, the interscrew material <br>transfer becomes more restricted, resulting in increased circulatory flow at the <br>entrance to the intermeshing region and increased pressure build-up at this point. <br>
+<hr>
+<A name=722></a><b>710</b>  <b> 10Twin Screw Extruders</b><br>
+In fact, when the flight width becomes sufficiently large, the characteristics of the <br>extruder will change to those of a CICO extruder; see Fig. 10.17.<br>
+Area A1<br>
+<b>Figure 10.16</b>Material transfer in a CSCO extruder<br>
+<b> Figure 10.17</b> <br>
+Co-rotating extruder with wide flights<br>
+The cross-sectional area of the barrel is:<br>
+ (10.12)<br>
+<hr>
+<A name=723></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>711</b><br>
+The cross-sectional area of one screw is:<br>
+ (10.13)<br>
+The open cross-sectional area between barrel and screw is simply:<br>
+ (10.14)<br>
+This can be written as:<br>
+ (10.15)<br>
+Thus, at a certain diameter, the open area is primarily a function of the number of <br>parallel flights p and the intermeshing angle i; see Fig. 10.18.<br>
+<b> Figure 10.18</b> <br>
+Open area versus intermesh angle with <br>
+double- and triple-flighted screws<br>
+The open volume is obtained by simply multiplying the open area with axial screw <br>length L:<br>
+ (10.16)<br>
+The surface area of the screw is obtained by multiplying the periphery of the screw <br>with axial length L:<br>
+ (10.17)<br>
+The surface area of the barrel is:<br>
+ (10.18)<br>
+<hr>
+<A name=724></a><b>712</b>  <b> 10Twin Screw Extruders</b><br>
+The channel area formed between the screw and barrel, screw channel area A1 (see <br>
+Fig. 10.16) is:<br>
+ (10.19)<br>
+<b> Figure 10.19</b> <br>
+Angle  from leading tip to start of intermeshing <br>
+region<br>
+By using Eq. 10.13 for As, Eq. 10.19 can be written as:<br>
+ (10.20)<br>
+When the leading tip of a screw channel enters the intermeshing region, the flow <br>channel area increases because of the contribution of the adjacent screw; see Fig. <br>10.16. The flow channel area reaches a maximum Amax, when the leading tip reaches <br>
+the end of the intermeshing region. The flow channel area then reduces again and <br>becomes area A1 when the trailing tip of the screw channel enters the intermeshing <br>
+region. The open cross-sectional area A0 is thus formed by 2p ­1 channels, two of <br>
+which are in the intermeshing region, Atop and Abot. When Atop is increasing, Abot is <br>
+decreasing and vice versa. The total area of intermesh Aint is the sum of Atop and Abot; <br>
+this is a constant. Area Aint is obtained from:<br>
+ (10.21)<br>
+The maximum value of Atop and Abot is simply:<br>
+ (10.22)<br>
+If angle  is the angle from the leading tip to the start of the intermeshing region <br>(see Fig. 10.19), then the area Atop can be expressed as a function of angle .<br>This relationship is shown graphically in Fig. 10.20 for a geometry with p = 3 and <br>t = 0.<br>Area Atop increases from A1 = 0.0856D2 to reach a maximum of Amax = 0.1259D2, an <br>
+increase of 47%. At the same time, area Abot reduces from the maximum Amax down <br>
+to A1. Thus, when area Atop is decompressed, area Abot is compressed and vice versa. <br>
+This action will promote leakage flow through the nip when the degree of fill is high <br>and will cause an alternating dynamic pressure field in area Atop and Abot.<br>
+<hr>
+<A name=725></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>713</b><br>
+<b> Figure 10.20</b> <br>
+Atop as a function of angle <br>
+<b>10.3.2.2Conveying in Self-Wiping Extruders<br></b>The conveying process in single screw extruders is generally analyzed by using the <br>flat plate model; see Chapter 7. A similar analysis can be used in CSCO extruders <br>when both screws are rolled onto a flat plate as shown in Fig. 10.21.<br>
+<b>Figure 10.21</b>Flat plate model in CSCO extruders<br>
+The down-channel length of each flat screw segment is S/sin, where S is the pitch <br>of the screw flight and  the helix angle. The screws are offset in the cross-channel <br>direction by a distance Xo, where Xo is often taken to be equal to the cross-channel <br>
+flight width. Obviously, this model is a severe simplification of the actual conveying <br>process because it cannot accurately represent the interscrew material transfer in <br>the intermeshing region. However, the advantage of the flat plate model is that one <br>can follow a very similar approach to the one used in single screw extruders.<br>In most cases, the CSCO extruder will be starve fed. Thus, the output from the <br>extruder is determined by the device feeding the extruder and not by the extruder <br>
+<hr>
+<A name=726></a><b>714</b>  <b> 10Twin Screw Extruders</b><br>
+itself. This means that the screw channels are partially filled with material over a <br>considerable length of the extruder. Only certain sections of the screw will be fully <br>filled. The last section of the screw is generally completely filled because it must <br>generate the required diehead pressure. Other sections of the screw that will be <br>completely filled are screw sections with neutral or reversing screw elements.<br>The fully filled length of the machine will increase with diehead pressure. The reason <br>that the fully filled length of the machine is determined by the pressure-generating <br>requirement is that pressure can only be generated when the channel is completely <br>filled with material. If the screw channel is only partially filled with material, no <br>pressure can be generated in the down-channel direction. Local sections of the screw <br>can be fully filled if a restrictive screw element is placed along the screw, such as a <br>reversed-flighted screw element or a kneading block.<br>
+<b>10.3.2.2.1Partially Filled Screws</b><br>
+Following Werner [9], the degree of fill  is defined as the ratio of filled channel area <br>A to total area:<br>
+ (10.23)<br>
+This is shown graphically in Fig. 10.22, where the areas are determined perpendi-<br>cular to the screw flights.<br>
+x<br>
+x1<br>
+<b> Figure 10.22</b> <br>
+Partially filled screw channel<br>
+The total cross-channel area A is simply:<br>
+ (10.24)<br>
+where A1 is the channel area in a plane perpendicular to the screw axis given by <br>
+Eq. 10.20.<br>Area A is related to the filled cross-channel distance x by:<br>
+ (10.25)<br>
+where x1 = 0.25 tDsin.<br>
+<hr>
+<A name=727></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>715</b><br>
+A typical relationship between the degree of fill and a normalized cross-channel <br>distance is shown in Fig. 10.23.<br>
+<b> Figure 10.23</b> <br>
+Degree of fill versus normalized cross- <br>
+channel distance<br>
+The velocity profiles can be determined if the following assumptions are made:<br>1.  The fluid is Newtonian<br>2.  The flow is steady and fully developed<br>3.  The flow is isothermal<br>4.  No slip at the wall<br>5.  Body and inertia forces are negligible<br>6.  Channel curvature in the down-channel direction is negligible<br>The equation of motion in the down-channel direction can be written as:<br>
+ (10.26)<br>
+The solution to this equation for a rectangular channel is given in Section 7.4. <br>However, the channel of a CSCO extruder does not have a rectangular shape but <br>resembles more the segment of a circle. Since the channel depth is a rather lengthy <br>function of the cross-channel distance (see Eq. 10.11), a simple analytical solution to <br>Eq. 10.26 is not very likely to be found. Thus, one has to resort to numerical tech-<br>niques to solve Eq. 10.26. If it is assumed that:<br>
+ (10.27)<br>
+then the down-channel velocity profile becomes simply:<br>
+ (10.28)<br>
+<hr>
+<A name=728></a><b>716</b>  <b> 10Twin Screw Extruders</b><br>
+The equation of motion in the cross-channel direction can be written as:<br>
+ (10.29)<br>
+If it is assumed that:<br>
+ (10.30)<br>
+then the cross-channel velocity profile can be written as:<br>
+ (10.31)<br>
+where y is measured from the barrel surface in the negative direction.<br>The cross-channel pressure gradient can be determined from the condition that the <br>net flow in the x-direction is zero, i.e., leakage flow is neglected. Thus, the cross-<br>channel pressure gradient is:<br>
+ (10.32)<br>
+and the resulting cross-channel velocity profile becomes:<br>
+ (10.33)<br>
+If the leakage flow over the flights is neglected and also the forced positive con-<br>veying of the obstructed material fraction (see Eq. 10.1), then the volumetric flow <br>rate can be expressed as:<br>
+ (10.34)<br>
+If it is assumed that the down-channel velocity profile can be approximated with <br>Eq. 10.28, then the volumetric throughput is approximately:<br>
+ (10.35)<br>
+By using Eq. 10.23, the equation can be written as:<br>
+ (10.36)<br>
+<hr>
+<A name=729></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>717</b><br>
+Thus, the throughput according to Eq. 10.36 is directly proportional to the degree of <br>fill  and the screw speed N. The power consumption in the screw channel can be <br>determined from:<br>
+ (10.37)<br>
+The power consumption in the flight clearance can be written as:<br>
+ (10.38)<br>
+The total power consumption per unit down-channel length is:<br>
+ (10.39)<br>
+The specific energy consumption (SEC) over axial length L is:<br>
+ (10.40)<br>
+Another approach to the analysis of the conveying process of partially filled screws <br>was proposed by Booy [29]. By assuming that the bank of material at the leading <br>flight edge is approximately symmetrical with respect to the bisectrix (see Fig. <br>10.24), and that the screw flank contacting the bank of material is reasonably flat, <br>the velocity distribution in the bank can be analyzed as follows. The motion of the <br>bank of material can be considered as a superposition of the motions of Fig. 10.24(b) <br>and (c).<br>
+<b>Figure 10.24(a­c)</b>Partially filled screw channel<br>
+The velocity distribution in Fig. 10.24(c) will be such that no net flow occurs in the <br>down-channel direction. The velocity distribution in Fig. 10.24(b) will cause a down-<br>channel flow at a uniform velocity of vbz/2. The average total velocity of the material <br>
+in the bank is the resultant of vbx and vbz/2, as shown in Figs. 10.24. The axial <br>
+velocity component is:<br>
+ (10.41)<br>
+<hr>
+<A name=730></a><IMG src="Polymer Extrusion by Chris Rauwendaal-730_1.jpg"><br>
+<b>718</b>  <b> 10Twin Screw Extruders</b><br>
+When the flight clearances are neglected, this approach yields the same expression <br>for throughput as the one derived earlier; see Eqs. 10.35 and 10.36. When the flight <br>clearance is not neglected, it can be assumed that the thin layers smeared out on the <br>screw and barrel surfaces are mostly stagnant and about half the thickness of the <br>clearance. With the expressions for the screw surface area ASS, barrel surface area <br>
+ASb, and open area A0, the degree of fill resulting from non-zero clearances can be <br>
+expressed as:<br>
+ (10.42)<br>
+where c is the clearance between the screws and  the clearance between screw <br>
+and barrel.<br>Open area A0 is given by Eq. 10.15. The rest of the material moves at an axial velocity <br>
+va and corresponds to a degree of fill 0. The cross-section AL through the moving <br>
+material is:<br>
+ (10.43)<br>
+The flow rate is then:<br>
+ (10.44)<br>
+The total degree of fill now becomes:<br>
+ (10.45)<br>
+When the extruder is run empty, the remaining degree of fill will be 1 when the <br>
+effect of gravity is negligible.<br>
+<b>10.3.2.2.2Fully Filled Screws</b><br>
+Booy [29] analyzed the pumping performance of fully filled co-rotating twin screw <br>extruders by distinguishing two flow regimes. One flow regime (I) is bounded by <br>both screw and barrel surface, and one flow regime (II) is bounded primarily by the <br>two screw surfaces in the intermeshing region; see Fig. 10.25.<br>
+I<br>
+I<br>
+II<br>
+I<br>
+I<br>
+<b> Figure 10.25</b> <br>
+Flow regimens in twin screw extruder<br>
+<hr>
+<A name=731></a> <br>
+10.3Intermeshing Co-Rotating Extruders  <b>719</b><br>
+The flow in regime I is analyzed by unwrapping screws as shown in Fig. 10.26.<br>
+<b> Figure 10.26</b> <br>
+Unwrapped twin screw geometry<br>
+The drag flow rate is written as:<br>
+ (10.46)<br>
+The pressure flow rate is written as:<br>
+ (10.47)<br>
+Equations 10.46 and 10.47 can be applied only when the tip angle t is small. <br>
+Regime II is assumed not to contribute to the pressure generation and is further <br>assumed to move forward at a rate of one lead per revolution. If the cross-sectional <br>area of regime II is Aa, then the flow rate through this domain is:<br>
+ (10.48)<br>
+The total flow rate then becomes:<br>
+ (10.49)<br>
+When the width W of the channel is large relative to the depth Hmax of the channel, <br>
+the shape factor for drag can be approximated by [29]:<br>
+ (10.50a)<br>
+Similarly, the shape factor for drag can be approximated by:<br>
+ (10.50b)<br>
+<hr>
+<A name=732></a><IMG src="Polymer Extrusion by Chris Rauwendaal-732_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-732_45.jpg"><br>
+<b>720</b>  <b> 10Twin Screw Extruders</b><br>
+Expressions for the limiting shape factors when the width of the channel is small <br>relative to the depth (W &lt;&lt; Hmax) are given by Booy [29]. However, this type of chan-<br>
+nel geometry is generally not encountered in commercial twin screw systems. <br>Numerical simulation of the flow and heat transfer in twin screw extruders is <br> <br>covered in Chapter 12. Section 12.3.2 discusses 2-D analysis of twin screws, and <br>Section 12.4.3.3 deals with 3-D analysis of flow and heat transfer in twin screw <br>extruders. Since 2000, major advances have been made in the numerical methods <br>used to simulate twin screw extruders. The boundary element method now allows <br>full 3-D analysis of flow in TSEs. A significant advance in the finite element method <br>is the mesh superposition technique that allows analysis of complicated geometries <br>with relative ease. This is discussed in more detail in Chapter 12.<br>
+<b></b><br>
+ <b>10.4 Intermeshing  Counter-Rotating </b><br>
+<b>Extruders</b><br>
+A typical screw geometry of a closely intermeshing counter-rotating (CICT) twin <br>screw extruder is shown in Fig. 10.27.<br>
+A<br>
+<b> Figure 10.27</b> <br>
+A<br>
+Section A-A<br>
+Geometry of a CICT extruder<br>
+A cross-section through the intermeshing region (see Fig. 10.28) shows that the <br>openings between the channels of the two screws are quite small.<br>
+A<br>
+B<br>
+B<br>
+A<br>
+Section A-A<br>
+<b> Figure 10.28</b> <br>
+Cross-section through the inter-<br>
+Section B-B<br>
+meshing region in a CICT extruder<br>
+<hr>
+<A name=733></a> <br>
+10.4Intermeshing Counter-Rotating Extruders  <b>721</b><br>
+These openings are considerably smaller than in intermeshing co-rotating extruders; <br>see also Figs. 10.3 and 10.4. As a result, CICT extruders can achieve relatively posi-<br>tive conveying characteristics.<br>The counter-rotating twin screw extruder has a milling type of intermeshing as <br>shown in Fig. 10.29.<br>
+<b> Figure 10.29</b> <br>
+Milling type of intermeshing <br>
+in counter-rotating extruders<br>
+The screw velocities in the intermeshing region are in the same direction. There-<br>fore, the material entering the intermeshing region will have a strong tendency to <br>flow through the intermeshing region. If the clearances between the two screws are <br>rather small, the flow through the intermeshing region will be quite small. This will <br>result in a bank of material accumulating at the entry of the intermeshing region. <br>The material drawn into the nip will exert considerable pressure on the two screws. <br>This can cause deflection of the screws. Therefore, CICT extruders generally run at <br>low speed to avoid excessive pressures developing in the intermeshing region. By <br>designing the screws with larger clearances, the allowable screw speeds can be <br>increased; however, this is at the expense of the positive conveying characteristics. <br>Thus, the maximum allowable screw speed on a CICT extruder is often a good indi-<br>cation of the conveying characteristics of the machine. A low maximum screw speed <br>(about 20 to 40 rpm) indicates a machine with positive conveying characteristics <br>with the most likely area of application being profile extrusion. A high maximum <br>screw speed (about 100 to 200 rpm or higher) indicates a machine with less positive <br>conveying characteristics with likely areas of application being compounding, con-<br>tinuous chemical reactions, and other specialty polymer processing operations.<br>The theoretical maximum output of a CICT extruder is:<br>
+ (10.51)<br>
+where p is the number of parallel flights, N the screw speed, and V the volume of the <br>C-shaped chamber.<br>In Eq. 10.51, it is assumed that the screw channels are fully filled with material and <br>that there is no leakage of material. Equation 10.51 was first proposed by Schenkel <br>[10]. It was found that the actual throughput of CICT extruders is usually consider-<br>ably below  max. Doboczky [11, 12] and Klenk [13, 14] introduced correction factors <br>
+<hr>
+<A name=734></a><IMG src="Polymer Extrusion by Chris Rauwendaal-734_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-734_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-734_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-734_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-734_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-734_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-734_7.jpg"><br>
+<b>722</b>  <b> 10Twin Screw Extruders</b><br>
+to bring the predicted throughput values in line with actual throughput values. <br>However, these correction factors were mostly empirical and, thus, of limited useful-<br>ness. Janssen [15] performed a detailed analysis of the leakage flows in counter-<br>rotating extruders. He distinguished four kinds of leakage; see Fig. 10.30.<br>1.  Leakage through the clearance between the screw flight and barrel,  f<br>2.  Calender leakage,  c, between the root of the screw and the tip of the flight<br>3.  Interscrew leakage through the gap between the flight flanks (the tetrahedron <br>
+gap),  t, in the radial direction<br>
+4.  Leakage through the side gap,  s, in the tangential direction<br>
+<b> Figure 10.30</b> <br>
+Leakage flows in CICT extruder<br>
+The volume of the C-shaped chamber is approximately:<br>
+ (10.52)<br>
+The geometry of a C-shaped chamber is shown in Fig. 10.31.<br>
+<b> Figure 10.31</b> <br>
+Geometry of a C-shaped chamber<br>
+For screws with straight flight flanks, the flight width is:<br>
+ (10.53)<br>
+where  is the flight flank angle.<br>Thus, the mean flight width wm is:<br>
+ (10.54)<br>
+<hr>
+<A name=735></a> <br>
+10.4Intermeshing Counter-Rotating Extruders  <b>723</b><br>
+and the mean channel width is:<br>
+ (10.55)<br>
+The axial pressure gradient in the screw channel, using the same assumptions as in <br>Section 10.3.2.2, becomes:<br>
+ (10.56)<br>
+The axial velocity va is given by Eq. 10.1. The derivation of the axial pressure gradient <br>
+is essentially the same as the derivation of the cross-channel pressure gradient <br>given in Eq. 10.32.<br>
+<b> Figure 10.32</b> <br>
+Pressure profile in CICT extruder<br>
+If it is assumed that the diehead pressure Pd is built up uniformly along the filled <br>
+length of the extruder Lf, the drag-induced axial channel pressure (Eq. 10.56) can be <br>
+superimposed on the linear pressure profile, as shown in Fig. 10.32.<br>The pressure drop PB over the axial channel width B becomes:<br>
+ (10.57)<br>
+The axial drop over a screw flight Pf is:<br>
+ (10.58)<br>
+where P is the pressure drop per chamber as a result of the diehead pressure.<br>
+<hr>
+<A name=736></a><b>724</b>  <b> 10Twin Screw Extruders</b><br>
+The leakage flow over the flight of a C-shaped chamber can be written as:<br>
+ (10.59)<br>
+The pressure drop across the calender gap Pc, according to Janssen [15], can be <br>
+written as:<br>
+ (10.60)<br>
+where c is the calender gap; see Fig. 10.33.<br>
+<b> Figure 10.33</b> <br>
+Calender gap geometry<br>
+Pressure drop Pc must equal the tangential pressure drop over a chamber plus the <br>
+contribution of the diehead pressure. In order to determine the calender leakage <br>flow  c, the tangential pressure drop over a chamber must be known. If  tan is the <br>
+tangential flow in the C-shaped chamber, the tangential pressure gradient P/ <br>
+after Janssen [15] can be determined from the following relationship:<br>
+ (10.61)<br>
+where:<br>
+ (10.62)<br>
+ (10.63)<br>
+<hr>
+<A name=737></a> <br>
+10.4Intermeshing Counter-Rotating Extruders  <b>725</b><br>
+ (10.64)<br>
+ (10.65)<br>
+Functions I0 and I1 are modified Bessel functions of the first kind and zero and first <br>
+order; K0 and K1 are modified Bessel functions of the second kind and zero and first <br>
+order. A correction factor is required, which has to be subtracted from the through-<br>put. This correction factor Fc is the channel volume displaced in one revolution mul-<br>
+tiplied with the rotational speed:<br>
+ (10.66)<br>
+Equations 10.61 through 10.66 allow the determination of the tangential pressure <br>gradient, which, in turn, allows the determination of the leakage flow through the <br>calender gap.<br>Leakage through the tetrahedron gap causes interscrew material transfer. In fact, it <br>is the only leakage flow that causes interscrew transfer. The tetrahedron gap <br>increases when the flight flank angle is increased. Janssen, Mulders, and Smith [16] <br>developed an empirical formula for the leakage flow through the tetrahedron gap:<br>
+ (10.67)<br>
+where f is the clearance between the flight flanks:<br>
+ (10.68)<br>
+The drag component  sd of the side leakage is:<br>
+ (10.69)<br>
+The pressure component  sp of the side leakage flow is approximately:<br>
+ (10.70)<br>
+where:<br>
+ (10.71)<br>
+<hr>
+<A name=738></a><b>726</b>  <b> 10Twin Screw Extruders</b><br>
+and:<br>
+ (10.72)<br>
+The calender leakage and side leakage can be combined as:<br>
+ (10.73)<br>
+The pressure generation in the C-shaped chamber can be determined from:<br>
+ <br>
+(10.74)<br>
+The drag-induced pressure generation in the C-shaped chamber plus the pressure <br>rise due to the diehead pressure should equal the pressure drop through the calen-<br>der gap:<br>
+ <br>
+(10.75)<br>
+where P is the pressure drop per chamber as a result of the diehead pressure.<br>Since the combined calender and side leakage flow equals the tangential flow in the <br>chamber, this flow can be expressed as:<br>
+ (10.76)<br>
+The total output of the extruder can be determined from:<br>
+ (10.77)<br>
+This relationship allows the determination of the throughput pressure relationship <br>for the filled length of a CICT extruder as a function of viscosity and machine geo-<br>metry. Experimental verification with Newtonian fluids [15] has shown this relation-<br>ship to be accurate to about 5 to 10%. Figure 10.34 shows the dimensionless output <br>
+0 as a function of flight flank angle when the dimensionless pressure drop P0 = <br>
+1E4.<br>The dimensionless output is:<br>
+ (10.78)<br>
+Figure 10.34 is valid for a CICT extruder with the following dimensions: p = 1, D = <br>70 mm, S = 20 mm, H = 10 mm,  = 0.1 mm, c = 0.2 mm, and f = 0. Increasing the <br>
+flight flank angle causes a significant drop in output. Thus, increased flight flank <br>angle strongly reduces the positive conveying characteristics. Figure 10.35 shows <br>
+<hr>
+<A name=739></a> <br>
+10.4Intermeshing Counter-Rotating Extruders  <b>727</b><br>
+the dimensionless output as a function of the calender clearance when P0 = 1E4 <br>and  = 6°; all other dimensions are as in Fig. 10.34.<br>
+1.0<br>
+0.8<br>
+0.6<br>
+0.4<br>
+Po= 1E4<br>
+Dimensionless throughput 0.2<br>
+0<br>
+<b> Figure 10.34</b> <br>
+0<br>
+2<br>
+4<br>
+6<br>
+8<br>
+10<br>
+12<br>
+Dimensionless output <br>
+Flight flank angle [degrees]<br>
+ <br>versus the flight flank angle<br>
+<b> Figure 10.35</b> <br>
+Dimensionless output versus <br>
+the calender gap<br>
+Clearly, the effect of calender clearance is similar to the effect of flight flank angle.<br>The effect of the radial flight clearance is shown in Fig. 10.36 when P0 = 1E4.<br>
+<hr>
+<A name=740></a><b>728</b>  <b> 10Twin Screw Extruders</b><br>
+<b> Figure 10.36</b> <br>
+Dimensionless output versus <br>
+radial flight clearance<br>
+The output starts to drop off rapidly when the flight clearance becomes larger <br>than 0.005 D. The radial flight clearance normally ranges from 0.001 to 0.002 D. <br>Figure 10.37 shows the effect of side clearance when P0 = 1E4. Increased side <br>clearance causes a strong reduction in output.<br>
+<b> Figure  10.37</b> <br>
+Dimensionless output versus <br>
+side clearance<br>
+The effect of flight pitch is shown in Fig. 10.38.<br>When the pitch is larger than 1/4 D, the dimensionless output is relatively insensi-<br>tive to the pitch. However, the dimensionless output reduces strongly when the <br>pitch becomes less than 1/4 D.<br>
+<hr>
+<A name=741></a> <br>
+10.4Intermeshing Counter-Rotating Extruders  <b>729</b><br>
+<b> Figure  10.38</b> <br>
+Dimensionless output <br>
+ <br>versus flight pitch<br>
+The effect of the channel depth is shown in Fig. 10.39 for two values of the pressure <br>differential, P0 = 2E4 and P0 = 5E4.<br>
+1.0<br>
+0.8<br>
+Po=2E4<br>
+0.6<br>
+Po=5E4<br>
+0.4<br>
+Dimensionless throughput 0.2<br>
+0<br>
+<b> Figure 10.39</b> <br>
+0<br>
+0.10<br>
+0.20<br>
+0.30<br>
+Dimensionless output <br>
+Channel depth [D]<br>
+ <br>versus channel depth<br>
+At each pressure gradient, there is an optimum channel depth for which the output <br>reaches a maximum value. This is similar to the situation in single screw extruders. <br>The optimum channel depth in these examples ranges from about 0.05 to 0.010 D. <br>As the pressure gradient increases, the optimum channel depth decreases.<br>
+<hr>
+<A name=742></a><IMG src="Polymer Extrusion by Chris Rauwendaal-742_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-742_16.jpg"><br>
+<b>730</b>  <b> 10Twin Screw Extruders</b><br>
+<b></b><br>
+ <b>10.5Non-Intermeshing Twin Screw Extruders</b><br>
+Non-intermeshing twin screw extruders are double screw machines where the cen-<br>terline distance between the screws is larger than the sum of the radii of the two <br>screws. Commercial examples of non-intermeshing twin screw extruders are coun-<br>ter-rotating (NOCT extruders). The conveying in NOCT extruders is similar to that in <br>a single screw extruder. The main difference is the fact that there is a possibility of <br>exchange of material from one screw to another. If the apex area (see Fig. 10.40) is <br>zero, the NOCT extruder behaves as two single screw extruders.<br>
+A<br>
+a<br>
+Apex<br>
+area<br>
+<b> Figure 10.40</b> <br>
+A<br>
+Section A-A<br>
+NOCT extruder geometry<br>
+Because of the non-zero apex area, the output of the NOCT extruder will be less than <br>twice the output of a single screw extruder with the same screw diameter.<br>The NOCT extruder has less positive conveying characteristics than a single screw <br>extruder. As a result, however, it has better backmixing characteristics than a single <br>screw extruder. Therefore, the NOCT extruder is primarily used in blending opera-<br>tions, devolatilization, chemical reactions, etc. The particular conveying characteris-<br>tics of the NOCT extruder make it undesirable for profile extrusion. In one commer-<br>cial example of a NOCT extruder, the screws are of different length such that the <br>last section of the extruder has a single screw discharge. This design is shown in <br>Fig. 10.41.<br>
+<b>Figure 10.41</b>NOCT extruder with unequal screw lengths<br>
+<hr>
+<A name=743></a> <br>
+10.5Non-Intermeshing Twin Screw Extruders  <b>731</b><br>
+Two advantages of this configuration are improved pumping characteristics and a <br>thrust load on one screw only. The thrust load on the secondary (short) screw is <br>very small. Thus, the thrust bearing design is greatly facilitated. A disadvantage of <br>this construction is a non-symmetrical conveying process with a chance of hang-up <br>of material in the transition region.<br>The first theoretical study of the conveying process in non-intermeshing twin screw <br>extruders was made by Kaplan and Tadmor [17]. They simplified the actual geome-<br>try (see Fig. 10.42) to a flat plate model.<br>
+<b> Figure 10.42</b> <br>
+Actual geometry of NOCT extruder<br>
+The flat plate model involves three plates: the two outside plates representing the <br>screw surface and the middle plate representing the barrel; see Fig. 10.43.<br>
+<b>Figure 10.43</b>Flat plate approximation of NOCT extruder<br>
+The center plate has slots that run perpendicular to the circumferential velocity vb. <br>
+The tangential slot width is:<br>
+ (10.79)<br>
+where a is the apex angle; see Fig. 10.40.<br>The tangential distance between the slots is:<br>
+ (10.80)<br>
+<hr>
+<A name=744></a><b>732</b>  <b> 10Twin Screw Extruders</b><br>
+This model yields the following output-pressure relationship for Newtonian fluids:<br>
+ (10.81)<br>
+where:<br>
+ (10.82)<br>
+ (10.83)<br>
+ (10.84)<br>
+Equation 10.81 gives the output for one screw. Thus, the total output is twice the <br>value from Eq. 10.81. The factor f is the ratio of uninterrupted barrel circumference <br>to the total barrel circumference. Considering that the drag flow occurs as a result of <br>adherence to the barrel surface, it seems that the drag flow correction factor FDTW <br>
+overestimates the drag flow considerably. If the barrel circumference is reduced by <br>a factor f, one would expect a proportional reduction in the drag flow rate. Thus, the <br>drag flow correction factor should be approximately:<br>
+ (10.85)<br>
+The difference between the two drag flow correction factors is shown in Fig. 10.44.<br>
+<b> Figure 10.44</b> <br>
+Comparison of two drag flow correction <br>
+factors<br>
+<hr>
+<A name=745></a> <br>
+10.5Non-Intermeshing Twin Screw Extruders  <b>733</b><br>
+In the normal range of f, Eq. 10.83 overestimates the drag flow correction factor <br>by about 10 to 20%. Another drawback of Eq. 10.82 is the fact that the slots in the <br>barrel plate are considered infinitely thin. This corresponds to assuming an essen-<br>tially zero apex area. Furthermore, the pressure flow correction assumes that pres-<br>sure leakage because of the open barrel occurs in the crosshatched area shown in <br>Fig. 10.45.<br>
+<b> Figure 10.45</b> <br>
+Assumed area <br>
+of pressure-induced <br>
+leakage<br>
+However, a top view (see Fig. 10.46) shows that in a matched screw configuration, <br>the screw flights will prevent such leakage from taking place.<br>
+<b> Figure 10.46</b> <br>
+Top view of a matched screw geometry<br>
+If the slot width is considered infinitely small, pressure leakage can only take place <br>in the down-channel direction. This would indicate that the pressure flow does not <br>require a correction factor when the screw flights are matched. However, if the <br>screw configuration is staggered, as shown in Fig. 10.47, the pressure flow does <br>require a correction term.<br>
+<b> Figure 10.47</b> <br>
+Staggered screw configuration<br>
+In actual NOCT extruders, the apex area has a non-zero value. The effect of the apex <br>area on the pumping performance is quite important, as discussed by Nichols and <br>Yao [18]. In actual experiments on a two-inch NOCT extruder with dimethylsiloxane, <br>
+<hr>
+<A name=746></a><b>734</b>  <b> 10Twin Screw Extruders</b><br>
+Nichols [19] found less than satisfactory agreement between actual output values <br>and output predictions based on Eq. 10.82. The actual apex area AT, shown in <br>
+Fig. 10.40, is:<br>
+ (10.86)<br>
+Nichols [24] developed a modified output model in cooperation with Lindt. The <br>model is based on a flat three-plate model with the thickness of the center plate <br> <br>having a finite thickness equal to the apex width Wa; see Fig. 10.48.<br>
+<b>Figure 10.48</b>Modified flat plate model for NOCT extruder<br>
+This model yields the following output relationship for Newtonian fluids:<br>
+ (10.87)<br>
+where:<br>
+ (10.88)<br>
+and:<br>
+ (10.89)<br>
+where Fd and Fp are the shape factors for drag flow and pressure flow as given by <br>
+Eqs. 7.218 and 7.219, respectively.<br>When Wa = 0, Eqs. 10.88 and 10.89 become the same as Eqs. 10.83 and 10.84, <br>
+respectively. In addition to the correction factors established from the three-plate <br>model, another correction is introduced to account for leakage in the apex area. The <br>apex area is approximated by a triangle as shown in Fig. 10.49.<br>
+<hr>
+<A name=747></a> <br>
+10.5Non-Intermeshing Twin Screw Extruders  <b>735</b><br>
+<b> Figure 10.49</b> <br>
+Apex approximation<br>
+The pressure flow rate through a triangle is expressed as (see reference 32 of <br>Chapter 1):<br>
+ (10.90)<br>
+where Wa is the apex width and Ha the height of the triangle shown in Fig. 10.49, <br>
+and Mo is a shape factor for a triangle.<br>The pressure gradient over the flight is taken as:<br>
+ (10.91)<br>
+where S is the pitch and b the axial flight width.<br>With Eqs. 10.87 through 10.91, an additional pressure flow correction term FPCRT2 <br>
+can now be formulated:<br>
+ (10.92)<br>
+where Mo is shown graphically in reference 32 of Chapter 1.<br>The two pressure flow correction factors are now combined to give:<br>
+ (10.93)<br>
+which yields the final result obtained by Nichols [24]:<br>
+ (10.94)<br>
+Comparison of predictions from Eq. 10.94 to experimental results [19] yielded <br> <br>considerably improved agreement as compared to predictions from Eq. 10.82. A <br>drawback of Eq. 10.94 is the fact that the derivation of the pressure flow correction <br>term FPCRT2 for the apex area is not consistent with the flat three-plate model. Fur-<br>
+thermore, the approximated apex area is smaller than the actual apex area as shown <br>in Fig. 10.49. The graphical results of the shape factor for triangles shown in Chap-<br>ter 1 [32] do not extend beyond a height-to-base ratio of unity. In reality, the Ha/Wa <br>
+<hr>
+<A name=748></a><b>736</b>  <b> 10Twin Screw Extruders</b><br>
+ratio will usually be larger than unity; thus, the graphical results in reference [32] <br>cannot be used. Finally, the actually pressure gradient over the screw flight also <br>contains a drag-induced component, which is not taken into account in Eqs. 10.92 <br>through 10.94.<br>Another approach used to predict the output-pressure characteristics of non-<br>intermeshing extruders is to abandon the three-plate model and follow more closely <br>the analysis used for single screw extruders. The circumference of the barrel is <br>interrupted for a fraction 1-f; see Fig. 10.50.<br>
+<b> Figure 10.50</b> <br>
+Effective barrel circumference<br>
+Since drag flow occurs as a result of polymer melt adhering to the barrel surface, the <br>reduced barrel circumference will affect the drag flow rate. The drag flow per revolu-<br>tion is found by moving the barrel with respect to the screw over a distance of D. <br>With a full barrel circumference, this results in a volume per revolution equal to:<br>
+ (10.95)<br>
+This results in the familiar drag flow rate equation:<br>
+ (10.96)<br>
+However, when the barrel circumference is reduced by 1­f, the volume dragged for-<br>ward per revolution will be reduced correspondingly:<br>
+ (10.97)<br>
+The actual drag flow rate becomes:<br>
+ (10.98)<br>
+<hr>
+<A name=749></a> <br>
+10.5Non-Intermeshing Twin Screw Extruders  <b>737</b><br>
+The pressure flow in the screw channel outside of the apex area will be the same as <br>it is in a single screw extruder. However, in the apex area there will be an additional <br>pressure-induced leakage flow. If the apex area is approximated by an isosceles tri-<br>angle of width Wa and height Ha (see Fig. 10.51), an expression can be derived for <br>
+leakage in this region.<br>
+<b> Figure 10.51</b> <br>
+Apex area approximation<br>
+An expression for the pressure flow through an isosceles triangle was derived by <br>Bird et al. [25] by following the variational principle due to von Helmholtz. Other <br>expressions have been proposed by Kozicki et al. [26, 27], who used a simple geo-<br>metric parameter method to predict the pressure drop-flow rate relationship in flow <br>channels of arbitrary cross-section. Following Bird's approach, the output-pressure <br>relationship for an isosceles triangle can be written as:<br>
+ (10.99)<br>
+where:<br>
+ (10.99a)<br>
+For values of Wa less Ha (m &lt; 0.5), Eq. 10.99 can be approximated reasonably well <br>
+by:<br>
+ (10.100)<br>
+where:<br>
+ (10.101)<br>
+ (10.102)<br>
+The difference between Eq. 10.99 and Eq. 10.100 over most of the range is only <br>about 2 or 3%; see also Fig. 10.52.<br>
+<hr>
+<A name=750></a><b>738</b>  <b> 10Twin Screw Extruders</b><br>
+<b>Figure 10.52</b>Output versus m according to Eqs . 10 .99 and 10 .100<br>
+Thus, the pressure-induced leakage flow through the apex area can be written as:<br>
+ (10.103)<br>
+where ga is the axial pressure gradient.<br>In addition to the pressure-induced leakage flow through the apex area, there is the <br>usual leakage flow through the flight clearance; refer to Section 7.4.1.<br>
+ (10.104)<br>
+Thus, the output per screw for a non-intermeshing twin screw extruder can be <br> <br>written  as:<br>
+ (10.105)<br>
+This relationship is valid for a matched screw geometry. The apex leakage term  12 <br>
+is determined by the apex width cubed; see Eq. 10.103. Thus, increases in the apex <br>angle a will cause very strong increases in the apex leakage flow. Figures 10.53 <br>
+and 10.54 compare output predictions made with Eq. 10.105 to the experimental <br>results obtained by Nichols with dimethyl-siloxane polymeric fluids [19].<br>It can be seen that reasonable agreement is obtained between predictions and actual <br>data.<br>
+<hr>
+<A name=751></a> <br>
+10.5Non-Intermeshing Twin Screw Extruders  <b>739</b><br>
+<b>Figure 10.53</b>Comparison of output predictions using Eq . 10 .105 to experimental data <br>
+by Nichols [19]; viscosity 12 Pa·s<br>
+<b>Figure 10.54</b>Comparison of output predictions using Eq . 10 .105 to experimental data <br>
+by Nichols [19]; viscosity 58 Pa·s<br>
+The analysis of the conveying characteristics becomes considerably more compli-<br>cated when the screws are placed in a staggered configuration. In this case, there <br>will be considerably more leakage in the apex region as illustrated in Fig. 10.55.<br>With the matched screw configuration, the tangential pressure profiles in the two <br>screws are symmetrical. Thus, there will be little interscrew material transfer. In the <br>staggered screw configuration, however, the tangential pressure profiles are non-<br>symmetrical because the flights are 180° offset. When the flight of the left screw is <br>
+<hr>
+<A name=752></a><b>740</b>  <b> 10Twin Screw Extruders</b><br>
+approaching the apex area, the pressure on the left side (PL) of the apex area will be <br>
+larger than the pressure on the right side (PR). As a result, material will flow through <br>
+the apex area from left to right, as shown in Fig. 10.55.<br>
+<b> Figure 10.55</b> <br>
+Interscrew material transfer with staggered screw <br>
+configuration<br>
+When the flight of the right screw approaches the apex area, PR will be larger than <br>
+PL, causing a flow from right to left. Therefore, in the staggered configuration there <br>
+will be a significant amount of interscrew material transfer. This material transfer <br>will change direction every half turn of the screw. In the staggered screw geometry, <br>the pressure-induced leakage flow through the apex region will be larger because of <br>the larger area. In the staggered configuration, the drag-induced leakage flow through <br>the apex region also has to be taken into account because of the non-symmetrical <br>tangential pressure profiles. The apex leakage flow can be written as:<br>
+ (10.106)<br>
+where  ld is the drag-induced leakage flow and  lp is the pressure-induced leakage <br>
+flow.<br>The drag-induced leakage flow can be determined from the drag-induced pressure <br>gradient gd2. If the effective channel height is Heff and the effective channel width <br>
+Weff, then the drag-induced leakage flow can be written as:<br>
+ (10.107)<br>
+In order to determine the drag-induced leakage flow, the drag-induced pressure gra-<br>dient needs to be known. Consider the simplified situation shown in Fig. 10.56.<br>
+<b> Figure 10.56</b> <br>
+Simplified geometry to determine the <br>
+drag-induced pressure gradient<br>
+<hr>
+<A name=753></a> <br>
+10.5Non-Intermeshing Twin Screw Extruders  <b>741</b><br>
+The drag-induced pressure gradient in the clearance gd2 can be related to the drag-<br>
+induced pressure gradient in the channel gd1 by stating that the net cross-channel <br>
+flow equals the flow through the clearance:<br>
+ (10.108)<br>
+where:<br>
+ (10.109)<br>
+ (10.110)<br>
+In the case of pure drag flow, the following relationship must be satisfied:<br>
+ (10.111)<br>
+With these equations, the following expression for the drag-induced pressure gra-<br>dient in the clearance is obtained:<br>
+ (10.112)<br>
+Considering that the actual clearance value ranges from the flight clearance to the <br>channel depth, the pressure gradient over the screw flight can be approximated by:<br>
+ (10.113)<br>
+The resulting leakage flow can be approximated by:<br>
+ (10.114)<br>
+The total drag-induced leakage flow in the apex region now becomes:<br>
+ (10.115)<br>
+The pressure-induced leakage flow in the apex region can be approximated by:<br>
+ (10.116)<br>
+<hr>
+<A name=754></a><b>742</b>  <b> 10Twin Screw Extruders</b><br>
+The output per screw for the staggered screw geometry can be written as:<br>
+ (10.117)<br>
+Figure 10.57 compares output predictions made with Eq. 10.117 to the experimen-<br>tal results obtained by Nichols [19].<br>
+<b> Figure 10.57</b> <br>
+Output prediction using Eq . 10 .117 to <br>
+experimental results by Nichols [19], <br>
+ <br>viscosity 58 Pa·s<br>
+The agreement between predictions and experimental results is reasonable. Both <br>the drag flow rate and the pressure-generating capacity of the staggered screws are <br>lower than the same screws in a matched configuration. However, the mixing capa-<br>bility of the staggered screws will be significantly better as a result of the interscrew <br>material transfer. The mixing process in non-intermeshing twin screw extruders <br>was studied by Howland and Erwin [28] for screws in a matched configuration. They <br>found that the mixing efficiency of the twin screw extruder was markedly better <br>than the single screw extruder. Howland and Erwin did not report on the mixing <br>efficiency of the twin screw extruder with the staggered screw configuration. How-<br>ever, it can be expected that this will be considerably better than the mixing effi-<br>ciency of screws in a matched configuration.<br>A different type of NOCT extruder is the Farrel Continuous Mixer (FCM). It is a <br>short twin screw mixer that runs at high speed, up to 1200 rpm for the smallest unit <br>(2 FCM). A schematic picture of an FCM is shown in Fig. 10.58.<br>
+<b> Figure 10.58</b> <br>
+Farrel continuous mixer (FCM)<br>
+<hr>
+<A name=755></a><IMG src="Polymer Extrusion by Chris Rauwendaal-755_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-755_12.jpg"><br>
+ <br>
+10.6Coaxial Twin Screw Extruders  <b>743</b><br>
+The length of the screws is about 5 D, where the first 2 D is conventional screw <br>geometry and the last 3 D a sigma type geometry, similar to the Banbury mixer. The <br>actual mixing takes place in the sigma screw section. Since the twin screw mixer <br>does not generate much pressure, material is dumped into a discharge extruder or a <br>gear pump for pressure generation. A typical residence time in the FCM is about <br>three to five seconds. The mixing action, therefore, is very intensive because energy <br>is dissipated in the material at a very fast rate. This type of mixing is particularly <br>useful in applications where dispersive mixing is required. The intensive mixing <br>action is attended with high shear stresses, causing effective breakdown of gels and <br>agglomerates in the polymer.<br>
+<b></b><br>
+ <b>10.6Coaxial Twin Screw Extruders</b><br>
+An unusual type of twin screw extruder is the coaxial twin screw (CTS) extruder. <br>The CTS extruder is basically a single screw machine where the main screw is hol-<br>low towards the end of the screw. In the hollow portion of the main screw an inner <br>screw is placed to aid in the conveying process in the extruder. The inner screw is <br>generally stationary with a cantilever support against a disk at the end of the barrel. <br>Thus, the I.D. of the main screw forms a rotating barrel for the stationary inner <br>screw. If the flights of the inner screw have an opposite pitch to the flights of the <br>main screw, the material transport in the inner screw will be forward. If the flights <br>of the inner screw have the same pitch as the flights of the main screw, the material <br>transport in the inner screw will be backward.<br>There are two versions of CTS extruders commercially available. In one version, <br>molten material from the outer screw is transferred to the inner screw where it is <br>pumped towards the die. In this case, the inner screw is used for forward pumping; <br>see Fig. 10.59.<br>
+<b>Figure 10.59</b>Inner screw for forward melt conveying<br>
+<hr>
+<A name=756></a><IMG src="Polymer Extrusion by Chris Rauwendaal-756_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-756_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-756_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-756_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-756_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-756_6.jpg"><br>
+<b>744</b>  <b> 10Twin Screw Extruders</b><br>
+If the channel depth of the outer screw is reduced to zero, all the polymer melt is <br>pumped toward the die by the inner screw. This inner melt removal (IMR) screw was <br>invented by Kovacs [20] of Midland-Ross Corporation. It is essentially a very compli-<br>cated version of a barrier screw and does not offer any obvious advantages over <br>conventional barrier screws.<br>Another version of a CTS extruder is the solids draining screw (SDS) originally <br>developed by Klein and Tadmor [21] of Scientific Process &amp; Research, Inc. In this <br>design, unmolten polymer drains into the inner screw; see Fig. 10.60.<br>
+<b>Figure 10.60</b>Solids draining screw (SDS)<br>
+Material is transported upstream in the inner screw and plasticated. At the end of <br>the inner screw, the polymer, which is now molten, is pumped back into the channel <br>of the main screw. A modified version of the SDS screw for use in a molding machine <br>was patented on September 22, 1981 [22]. Another modification of the SDS screw <br>involves using a barrier type main screw to improve the solids draining process. <br>This barrier SDS screw was patented on June 14, 1983 [23]. The SDS screw is <br>claimed to give higher output and lower energy consumption. However, from a func-<br>tional analysis it is difficult to see why recirculation of a fraction of the polymer flow <br>would increase output or reduce power consumption.<br>The mechanical design of CTS extruders is considerably more complex than conven-<br>tional single screw extruders. Since material has to leak through holes in the main <br>screw, there is a chance of plugging and of stagnant areas. Also, maintenance and <br>operating procedures of CTS extruders will be considerably more complex than single <br>screw extruders.<br>
+<hr>
+<A name=757></a> <br>
+10.7Devolatilization in Twin Screw Extruders  <b>745</b><br>
+<b></b><br>
+ <b>10.7Devolatilization in Twin Screw Extruders</b><br>
+Twin screw extruders are finding increasing use in specialty operations such as <br>reactive processing of polymers and devolatilization. Twin screw extruders are used <br>as continuous chemical reactors for polymerization and polymer modifications, e.g., <br>grafting of side groups.<br>Both co-rotating (e.g., [30­32]) and counter-rotating (e.g., [33­35]) twin screw <br>extruders are used for this purpose, intermeshing as well as non-intermeshing [36]. <br>In the extrusion of reacting materials, another degree of difficulty is added to the <br>description of the process because the material properties will change as the reac-<br>tion progresses along the machine. The theory of extrusion of reacting materials is <br>still in a stage of development. However, one aspect of specialty polymer processing <br>operations, namely continuous devolatilization in twin screw extruders, has reached <br>a point where a reasonably accurate description of the process is possible.<br>Todd [37] proposed an equation to describe devolatilization in co-rotating twin screw <br>extruders based on the penetration theory discussed in Section 5.4 and Section 7.6. <br>The equation contains the Peclet number (see Eq. 7.371), which represents the effect <br>of longitudinal backmixing. The Peclet number must be measured or estimated to <br>predict the devolatilizing performance of an extruder. Todd selected a Peclet number <br>of 40 to correlate predictions to experimental results. A similar approach was followed <br>by Werner [38]. A visualization study was made by Han and Han [39], particularly to <br>study foam devolatilization. They found substantial entrainment of the bubbles in a <br>circulatory flow region in a partially filled screw devolatilizer. Collins, Denson, and <br>Astarita [40] published an experimental and theoretical study of devolatilization in a <br>co-rotating twin screw extruder. The experimentally determined mass transfer coeffi-<br>cients were about one-third those predicted by the mathematical model. They con-<br>cluded, therefore, that the effective surface area for mass transfer is substantially <br>less than the sum of the areas of the screws and barrel.<br>Secor [41] presented an intermeshing model for devolatilization in co-rotating twin <br>screw extruders that incorporates the major characteristics of fluid motion. These <br>characteristics were experimentally observed in a twin screw extruder with a trans-<br>parent barrel. The observed flow pattern consisted of alternating rotation in the <br> <br>tangential direction with the screw and axial forward motion at the entrance to the <br>intermeshing region; see Fig. 10.61.<br>These fluid flow patterns are fully consistent with the flow patterns described in <br>Section 10.3. The model is based on the following assumptions:<br>1.  All the fluid is exposed for a series of intervals, each of duration , in which <br>
+devolatilization occurs.<br>
+2.  Between exposures, perfect mixing occurs during the forward axial motion in a <br>
+time that is very small compared with .<br>
+<hr>
+<A name=758></a><b>746</b>  <b> 10Twin Screw Extruders</b><br>
+3.  The diffusion coefficient is constant.<br>4.  The fluid layers in the screw channel are effectively of infinite depth.<br>5.  The volumetric flow rate of the fluid is constant.<br>6.  No flow of fluid relative to the underlying flight faces takes place during devolatili-<br>
+zation.<br>
+7.  Nucleation and bubble growth are negligible.<br>
+b<br>
+­<br>
+<b> Figure 10.61</b> <br>
+Predominant fluid flow pattern<br>
+A material balance on the volatile component can be written as:<br>
+ (10.118)<br>
+where<br>C0  =  concentration of the volatile component in the feed to the extruder in gr/cm3<br>C1  =   average concentration of the volatile component in the liquid at the end of the <br>
+first exposure in interval in gr/cm3<br>
+  =  volumetric liquid flow rate in cm3/s<br>
+1  =  average rate of evaporation in the first exposure interval in gr/s<br>If the surface concentration of the volatile component is maintained at zero, the rate <br>of evaporation is:<br>
+ (10.119)<br>
+This equation corresponds to Eq. 7.434 for devolatilization in single screw extru-<br>ders. In Eq. 10.119, the variables are:<br>Ae  =  area for evaporation in cm2/s<br>D  =  diffusion coefficient in cm2/s<br>  =  exposure time in s<br>The effective area for evaporation is given by:<br>
+ (10.120)<br>
+<hr>
+<A name=759></a> <br>
+10.7Devolatilization in Twin Screw Extruders  <b>747</b><br>
+where A is the total leading face area of a 360° section of a single screw and f is the <br>ratio of channel area outside the intermeshing region to the channel area inside the <br>intermeshing region. Thus, the exposure time  can be written as:<br>
+ (10.121)<br>
+where N is the rotational speed of the screws in rev/s. With Eqs. 10.120 and 10.121, <br>the rate of evaporation becomes:<br>
+ (10.122)<br>
+Substituting Eq. 10.122 into Eq. 10.118 gives:<br>
+ (10.123)<br>
+For the n-th exposure, Eq. 10.123 becomes:<br>
+ (10.124)<br>
+For a sequence of n exposures:<br>
+ (10.125)<br>
+Each exposure is followed by a short axial movement, which has been determined to <br>be equal to the mean flight thickness  . If the total length of the screws is L, the <br>number of exposures is given by:<br>
+ (10.126)<br>
+Thus, Eq. 10.125 can be written as:<br>
+ (10.127)<br>
+Equation 10.127 is graphically represented in Fig. 10.62.<br>
+<hr>
+<A name=760></a><b>748</b>  <b> 10Twin Screw Extruders</b><br>
+<b> Figure 10.62</b> <br>
+Graphical representation of <br>
+Eq . 10 .127<br>
+The concentration ratio decreases with increasing diffusivity, screw speed, area, <br>and number of exposures. The concentration ratio increases with increasing flow <br>rate. When the following inequality is fulfilled:<br>
+ (10.128)<br>
+Equation 10.127 reduces to:<br>
+ (10.129)<br>
+with an error of less than 1%.<br>Experiments were performed on a 20 cm twin screw extruder with a screw length of <br>27 cm. The liquid was a polybutene and the volatile halocarbon. Figure 10.63 shows <br>the correlation between experimental results and theoretical predictions.<br>The agreement between theory and experiments is quite good. Even though bubble <br>formation was observed during some of the experiments, the mass transfer rate was <br>not significantly affected. Reasons for this include axial short-circuiting of fluid <br>along the bottom of the figure-eight bore and incomplete mixing at points of transfer <br>between the screws. Another reason is the likelihood of the bubble actually being <br>retained by the liquid phase. This is supported by visual observations by Han and <br>Han [39].<br>Evaporation from the liquid on the barrel was assumed to be of minor importance <br>compared with evaporation from the liquid in the screw channels. This assumption <br>is supported by experimental work by Biesenberger and Lee [42] who found the <br>
+<hr>
+<A name=761></a> <br>
+10.8Commercial Twin Screw Extruders  <b>749</b><br>
+contribution of devolatilization through the barrel film to be essentially negligible. <br>This is an important point because if the contribution through the melt film is <br> <br>negligible in a twin screw extruder, it should also be of minor importance in a single <br>screw extruder. One note of caution is that Secor's experiments were performed on <br>a machine with a rather unusual geometry. The screw channel had a very small <br>width to depth ratio (W/H), the actual value being W/H = 0.384, and the screw <br>length was very short (1.36 D). In this case, the area of the melt pool is much larger <br>than the area of the melt film. Thus, in this geometry, the contribution of the melt <br>film has to be very small from simple geometric arguments. In most commercial <br>extruders, however, the W/H ratio is at least an order of magnitude higher and the <br>screw length is also at least an order of magnitude higher. Thus, the Secor model <br>may not give equally good results with twin screw extruders of more standard geo-<br>metry.<br>
+<b> Figure  10.63</b> <br>
+Correlation of experimental <br>
+results [41] and theoretical <br>
+predictions<br>
+<b></b><br>
+ <b>10.8Commercial Twin Screw Extruders</b><br>
+Since twin screw extruders have become an important segment of the field of ex- <br>trusion it is reasonable to discuss important aspects of commercial twin screw <br>machinery. A good reference on capabilities of twin screw extruders is the book <br>edited by David Todd [43]. In this book, some of the main suppliers of twin screw <br>extruders describe their machinery and capabilities; the companies are Leistritz, <br>Werner &amp; Pfleiderer, Berstorff, APV (Baker Perkins), Japan Steel Works, Welding <br>Engineers, Farrel, and Buss. The book describes both co- and counter-rotating twin <br>screw extruders and internal mixers. Another good reference on twin screw extru-<br>ders is the book by James White [44]. This book gives an excellent historical over-<br>
+<hr>
+<A name=762></a><b>750</b>  <b> 10Twin Screw Extruders</b><br>
+view of twin screw extruders with broad coverage of early patents. White's book also <br>covers theoretical analysis in detail; there is less emphasis on practical applications.<br>Roberto Colombo and Lavorazione Materie Plastiche (LMP) of Turin, Italy, developed <br>and patented the first commercial co-rotating twin screw extruder in 1938. LMP <br>sold their first twin screw extruders to I.G. Farbenindustrie in Germany in 1939. <br>Meskat and Erdmenger were involved in a separate development at I.G. Farben-<br>industrie in the 1940s. These workers developed and patented the fully intermesh-<br>ing self-cleaning twin screw extruders. After I.G. Farbenindustrie was broken up <br>after World War II, Meskat and Erdmenger worked for Bayer AG. In 1953 Bayer AG <br>worked with Werner &amp; Pfleiderer to develop a new generation of intermeshing self-<br>wiping twin screw extruders; these machines were manufactured and sold by W&amp;P.<br>In 1957 W&amp;P introduced the first high speed, co-rotating, twin screw compounding <br>extruder and has been a major player in the twin screw extruder industry ever <br>since. W&amp;P offers a wide range of sizes from as small as 25 mm to as large as 380 <br>mm. Data on the various W&amp;P extruders is shown in Table 10.3.<br>
+<b>Table 10.3</b>Technical Data for ZSK Type Twin Screw Compounders by W&amp;P<br>
+Model<br>
+Torque Motor Max. Screw Flight Max. Screw<br>
+Machine dimen-<br>
+Total<br>
+per<br>
+power screw diam. depth L/D shaft<br>
+sions,<br>
+weight<br>
+screw at Nmax speed [mm] [mm] ratio height<br>
+length, width, <br>
+[kg]<br>
+[N·m]<br>
+[kW]<br>
+[rpm]<br>
+[mm]<br>
+height<br>
+L×W×H (mm)<br>
+25 WLE 82<br>
+21.5<br>
+1200<br>
+25<br>
+4.2<br>
+42<br>
+1100<br>
+1800×780×1400<br>
+900<br>
+40 MC<br>
+425 110 1200<br>
+40<br>
+7.1<br>
+48<br>
+1100<br>
+3900×720×1400<br>
+2400<br>
+50 MC<br>
+815 215 1200<br>
+50<br>
+8.9<br>
+48<br>
+1100<br>
+4600×840×1400<br>
+2750<br>
+58 MC 1250 325 1200<br>
+58<br>
+10.3<br>
+48<br>
+1100<br>
+5200×1000×1500 5600<br>
+70 MC 2275 600 1200<br>
+70<br>
+12.5<br>
+48<br>
+1100<br>
+5800×1100×1550 8000<br>
+92 MC 5000 1100 1000<br>
+92<br>
+16.3<br>
+48<br>
+1100<br>
+7300×1150×1650<br>
+14000<br>
+133 MC 15100 3300 1000<br>
+133<br>
+23.5<br>
+48<br>
+1100<br>
+9900×1700×2000<br>
+28500<br>
+170 SC<br>
+25000 1925 350<br>
+170<br>
+25.5<br>
+48<br>
+1100 12500×2200×1600 32000<br>
+177 MC 35000 2700 350<br>
+177<br>
+31.5<br>
+42<br>
+1100 13000×2300×1600 37000<br>
+240 SC<br>
+70000 4150 270<br>
+236<br>
+33<br>
+42<br>
+1300 16000×3300×1960 60000<br>
+250 MC 87500 5200 270<br>
+248<br>
+44<br>
+42<br>
+1300 16000×3300×1960 60000<br>
+300 SC<br>
+130000 7720 270<br>
+298<br>
+39.8<br>
+42<br>
+1500 20000×4000×2250 105000<br>
+320 MC 170000 10100 270<br>
+315<br>
+55.8<br>
+42<br>
+1500 20000×4000×2250 105000<br>
+380 MC 356000 17200 220<br>
+380<br>
+67.4<br>
+42<br>
+2000 29000×4800×4200 210000<br>
+WLE = World-Lab-Extruder<br>
+MC = Mega Compounder<br>
+SC = Super Compounder<br>
+<hr>
+<A name=763></a> <br>
+10.8Commercial Twin Screw Extruders  <b>751</b><br>
+<b>Table 10.4</b>Historical Development of Extruder Generations<br>
+1957<br>
+Introduction of first commercial ZSK extruder (1st generation)<br>
+1963<br>
+Change to variable design (2nd generation)<br>
+1975<br>
+Change from triple- to double-flighted screw geometry (3rd generation) 60% increase in <br>
+ <br>volume<br>
+1978<br>
+40% increase in torque (4th generation)<br>
+1983<br>
+30% volume increase with 15% increase in torque (5th generation, Super Compounder)<br>
+1995<br>
+30% increase in torque (6th generation, Mega Compounder)<br>
+From 1957 to 1995 both the volume and torque capability have more than doubled <br>in ZSK extruders. The original W&amp;P twin screw extruders (ZSK53, 83, 120, 160) <br>used a three-lobe (triple-flighted) screw geometry. As discussed in Section 10.3.2 <br>triple-flighted screws have relatively low open area and thus low conveying capabil-<br>ity because of their low O.D./I.D. ratio, around 1.2:1. The first two-lobe machines <br>(ZSK57, 90, 130, 170) had greater free volume but used the same gearbox as the <br>three-lobe extruders. As a result, no additional power transmission could be incor-<br>porated.<br>In the late 1970s, W&amp;P redesigned the gearbox for more power, and the shafts were <br>changed to a six-key design for better torque transmission. The fifth generation <br>(Super Compounder) was designed with a 24-spline shaft to allow greater power <br>transmission and greater free volume. The sixth generation (Mega Compounder) <br>increased torque capability another 30%; this extruder can run at screw speeds up <br>to 1200 rpm. Table 10.5 summarizes the developments of the W&amp;P twin screw <br>extruders.<br>
+<b>Table 10.5</b>Comparison of Six Generations of ZSK Extruders by W&amp;P<br>
+Generation number<br>
+Number of flights<br>
+O.D./I.D. ratio<br>
+Torque/centerline<br>
+distance cubed<br>
+1<br>
+3<br>
+3.7­3.9<br>
+2<br>
+3<br>
+1.22<br>
+4.7­5.5<br>
+3<br>
+3<br>
+1.44<br>
+4.7­5.5<br>
+4<br>
+2 or 3<br>
+1.22 or 1.44<br>
+7.2­8.0<br>
+5<br>
+2<br>
+1.55<br>
+8.7<br>
+6<br>
+2<br>
+1.55<br>
+11.3<br>
+The ratio of torque to centerline distance cubed is expressed in Nm/cm3. Older ZSK <br>extruders have O.D./I.D. ratios that vary slightly with machine size. The O.D./I.D. <br>ratio of the Super and Mega Compounders does not vary with machine size. This <br>geometric consistency simplifies scale-up.<br>
+<hr>
+<A name=764></a><b>752</b>  <b> 10Twin Screw Extruders</b><br>
+Distributive mixing is achieved in kneading disks and slotted mixing sections placed <br>along the screw. W&amp;P has developed a mixing tip [45] that has a protruding helical <br>flight as shown in Fig. 10.64.<br>The mixing tip shown in Fig. 10.64 was designed to improve thermal homogeneity. In <br>tests, it was found that the mixing tip reduced the melt temperature variation from <br>60°C with a standard tip to 20°C with a mixing tip. Obviously, the benefits of a mix-<br>ing tip are not limited to co-rotating twin screw extruders. Mixing tips can be used <br>beneficially in counter-rotating twin screw extruders and single screw extruders.<br>
+<b> Figure 10.64</b> <br>
+Mixing tip (MSSP) by Werner &amp; Pfleiderer<br>
+Most suppliers of co-rotating twin screw extruders, such as W&amp;P, supply only one <br>type of extruder. However, there are a few manufacturers that supply both co- and <br>counter-rotating twin screw extruders; examples are Leistritz and Japan Steel Works. <br>In fact, both these companies manufacture machines that can change from a co-<br>rotating mode of operation to a counter-rotating mode. Obviously, the screws have to <br>be changed when a switch is made from co- to counter-rotational operation and vice <br>versa. Co-rotating extruders require screws with equal pitch (both right handed or <br>left handed), while counter-rotating extruders require screws with opposite pitch <br>(one right handed and the other left handed). Such machines offer a great deal of <br>flexibility and versatility.<br>The main specifications of TEX- extruders by Japan Steel Works are shown in Table <br>10.6.<br>
+<b>Table 10.6</b>Main Specifications of TEX- Extruders by JSW<br>
+Model<br>
+Diameter [mm]<br>
+Torque [kg·m]<br>
+Power [kW]<br>
+Speed [rpm]<br>
+TEX30<br>
+32.0<br>
+41<br>
+15<br>
+358<br>
+TEX44<br>
+47.0<br>
+129<br>
+45<br>
+339<br>
+TEX54<br>
+58.0<br>
+243<br>
+75<br>
+300<br>
+TEX65<br>
+69.0<br>
+410<br>
+132<br>
+314<br>
+TEX77<br>
+82.5<br>
+700<br>
+220<br>
+306<br>
+TEX90<br>
+96.5<br>
+1121<br>
+355<br>
+309<br>
+TEX105<br>
+113.0<br>
+1799<br>
+560<br>
+303<br>
+TEX120<br>
+129.5<br>
+2708<br>
+850<br>
+306<br>
+TEX140<br>
+152.0<br>
+4379<br>
+1350<br>
+300<br>
+TEX160<br>
+174.0<br>
+6569<br>
+2000<br>
+297<br>
+TEX180<br>
+196.0<br>
+9389<br>
+2875<br>
+298<br>
+<hr>
+<A name=765></a> <br>
+10.8Commercial Twin Screw Extruders  <b>753</b><br>
+The smaller extruders can run at higher screw speeds with correspondingly higher <br>power rating. The TEX extruders are available with shallow and deep-flighted <br>screws. Screws and barrel are completely segmented. Typical L/D values of the TEX <br>extruders range from 30:1 to 60:1. Studies at JSW [43] and elsewhere have shown <br>that counter-rotating extruders are more efficient than co-rotating extruders with <br>respect to feeding, melting, devolatilization, and dispersive mixing. These are impor-<br>tant functions in compounding extruders. These results are surprising in light of <br>the fact that the majority of twin screw extruders used in compounding are co-rotat-<br>ing twin screw extruders.<br>The reason that co-rotating twin screw extruders are so widely used in compound-<br>ing is probably related to the fact that in the past more developmental work has <br>been directed to co-rotating than counter-rotating TSEs. When a twin screw extruder <br>is considered for a particular compounding task or for reactive extrusion, it should <br>be remembered that a counter-rotating TSE may be more appropriate than a co-<br>rotating TSE. Also, with new developments of mixing devices for single screw <br>extruders the use of single screw extruders in compounding is likely to increase.<br>There are many manufacturers (over 50) of twin screw compounders. These manu-<br>facturers offer a variety of ancillary equipment such as loss-in-weight feeders, twin <br>screw side feeders, vent stuffers, screw segment disassembly units, screen changers, <br>etc. There are many different screw and barrel materials available with special wear <br>and/or corrosion-resistant characteristics. This is important because TSEs are often <br>used with abrasive and corrosive materials.<br>
+<b>10.8.1Screw Design Issues for Co-Rotating Twin Screw Extruders</b><br>
+The main difference in screw design for twin and single screw extruders is that in <br>TSEs the root diameter of the screws is constant in most cases, while in SSEs the root <br>diameter usually varies considerably along the length of the screw. In intermeshing <br>TSEs the root diameter has to be constant if self-cleaning action is to be achieved--<br>this is desired in most cases. Obviously, in non-intermeshing extruders the root <br>diameter can vary (and often does vary) along the length.<br>Another important difference between SSEs and TSEs is that SSEs are normally <br>flood fed while high speed TSEs are starve fed. In starve fed extruders the regions of <br>partial fill contribute little to melting, mixing, and temperature rise; the partially <br>filled regions are pressureless. Most of the melting, mixing, and temperature <br>increase occurs in fully filled regions of the extruder. Starve-feeding adds an impor-<br>tant control element to the process because it allows for a variation of the effective <br>length of the extruder. Generally, the effective length can be increased by raising <br>the feed rate and/or reducing the screw speed. Increasing the effective length usu-<br>ally reduces the specific energy consumption in the extruder; this results in less <br>mixing and lower stock temperatures.<br>
+<hr>
+<A name=766></a><b>754</b>  <b> 10Twin Screw Extruders</b><br>
+In high speed TSEs, the fully filled length is relatively short, typically 20 to 40% of <br>the length of the extruder. The fully filled regions occur where restrictive elements <br>are placed along the screw and at the end of the screw where the diehead pressure <br>has to be developed. Restrictive elements are often placed just upstream of a vent <br>port to create a melt seal. In a starve fed extruder, a melt seal is necessary to be able <br>to draw a vacuum at the vent port. This is illustrated in Fig. 10.65.<br>
+<b> Figure 10.65</b> <br>
+A left-handed screw element used before <br>
+vent port to create a melt seal<br>
+The length of the filled section depends on the pressure generating capability of <br>the screw section upstream of the restrictive element and on the level of pressure <br>that is required to override the restrictive element. A small pitch restrictive element <br>will require more pressure than a large pitch element and, therefore, will create a <br>stronger seal. A longer restrictive element will also create a stronger seal. Similarly, <br>a neutral kneading block will not create as strong a seal as a reversing kneading <br>block.<br>The mixing action of conveying elements is limited; the same is true of single screw <br>extruders. As a result, the mixing action has to be achieved by placing mixing ele-<br>ments along the screw. In the past, the only mixing elements were kneading disks. <br>Kneading elements are basically screw elements with a 90° helix angle. This means <br>the flight runs in the axial direction. The benefit of kneading disks is that they are <br>completely self-wiping just like conveying elements. For distributive mixing, narrow <br>disk kneading blocks are used while for dispersive mixing wide disk kneading <br>blocks are used.<br>The reason that wide kneading disks achieve good dispersive mixing is that a sub-<br>stantial amount of material is forced into the high-stress region of the kneading <br>disk. When the disk is wide, there is little chance for material to bypass the high-<br>stress region; see Fig. 10.66.<br>
+<b> Figure 10.66</b> <br>
+Mixing with wide (left) and narrow (right) <br>
+kneading disks<br>
+<hr>
+<A name=767></a> <br>
+10.8Commercial Twin Screw Extruders  <b>755</b><br>
+With wide kneading disks, a large amount of material is drawn into the tip clearance <br>and the material is exposed to strong shear and elongational stresses in the process. <br>With narrow kneading disks, most material will bypass the high stress region of the <br>tip clearance. As a result, the dispersive mixing action is limited; however, the dis-<br>tributive mixing capability is improved because of the frequent splitting of the flow.<br>The mixing action of kneading blocks depends not only on the width of the knead-<br>ing disks, but it also depends on the stagger angle. In a bilobal screw geometry, a <br>stagger angle of 90° will create a neutral kneading block. If the stagger angle is <br>between zero and 90° the kneading block becomes a forwarding section. If the <br>stagger angle is between zero and ­90° the kneading block becomes reversing; see <br>Fig. 10.67. The mixing action in neutral and reversing kneading blocks is better <br>than in forwarding kneading blocks.<br>
+<b> Figure 10.67</b> <br>
+Example of forwarding (left) and neutral <br>
+Forward 30  stagge<br>
+<b>o</b><br>
+r<br>
+Neutral 90  stagger<br>
+<b>o</b><br>
+kneading block (right)<br>
+Even though narrow kneading disks provide reasonable distributive mixing, certain <br>applications required better distributive mixing. Therefore, single screw extrusion <br>technology was applied, using a variety of screw elements with slotted flights; see <br>Section 8.7. Some TSE manufacturers resisted this trend, because slotted mixing <br>elements are not completely self-wiping. As a result, slotted mixers compromise the <br>self-cleaning capability of TSEs. In practice, however, the benefits of slotted mixers <br>have been greater than the partial loss of self-cleaning action.<br>Slotted mixers are usually flighted elements with axial or angled slots machined <br>into the flights. The helix angle of the flights can range from zero to 90°. With a zero <br>degree helix angle the flight becomes a circumferential ring and the mixer looks <br>like a gear type or torpedo mixing element; see Fig. 10.68.<br>
+<b> Figure 10.68</b> <br>
+Example of gear type mixing element<br>
+<hr>
+<A name=768></a><IMG src="Polymer Extrusion by Chris Rauwendaal-768_1.jpg"><br>
+<b>756</b>  <b> 10Twin Screw Extruders</b><br>
+To improve both the distributive and dispersive mixing, tapered slots can be used <br>according to the CRD mixing technology as discussed in Chapters 7 and 8. Tapered <br>slots create elongational flow as the polymer passes through the slots. This improves <br>dispersive mixing as well as distributive mixing. A set of CRD mixing elements for a <br>twin screw extruder is shown in Fig. 10.69.<br>
+<b> Figure 10.69</b> <br>
+Example of CRD mixing elements for twin screw <br>
+extruder<br>
+<b>10.8.2Scale-Up in Co-Rotating Twin Screw Extruders</b><br>
+For scaling extruders similar in design the throughput can be scaled as:<br>
+ (10.130)<br>
+The power consumption in kilowatts (kW) can be determined from:<br>
+ (10.131)<br>
+where the screw speed N is expressed in revolutions per minute and the gearbox <br>rating in Nm.<br>The specific mechanical energy consumption (SMEC) is obtained by dividing the <br>power consumption by the mass flow rate (throughput):<br>
+ (10.132)<br>
+When the throughput is expressed in kg/hr, the units of SMEC become kWhr/kg. <br>In scale-up it is often desirable to keep the specific mechanical energy consumption <br>constant.<br>If the process is heat transfer limited, the following relationship can be used to <br>determine the throughput of the larger extruder:<br>
+ (10.133)<br>
+<hr>
+<A name=769></a> <br>
+10.8Commercial Twin Screw Extruders  <b>757</b><br>
+The average shear rate is proportional to the screw speed; it can be determined <br>from:<br>
+ (10.134)<br>
+The factor Km is determined by the screw geometry: Km equals the screw circum-<br>
+ference divided by the average channel depth. The Km value is about 0.4 for third- <br>
+and fourth-generation ZSK extruders with screw speed N expressed in rev/min. The <br>value of the average channel depth is about 85% of the channel depth of the screw. <br>For a ZSK57 the value K57 = 0.367; for a ZSK130 the value K130 = 0.427. The K <br>values for Super Compounders and Mega Compounders are lower because of the <br>larger channel depth of these machines.<br>If the shear rates are to be the same in the small and large extruder, then the screw <br>speed of the target extruder can be adjusted to achieve equal shear rates. The screw <br>speed of the large extruder D1 can be determined from:<br>
+ (10.135)<br>
+The effective volume is the product of the internal volume ratio and the screw speed <br>ratio of the two extruders:<br>
+ (10.136)<br>
+The target TSE must run at the same degree of fill to have the mixing elements per-<br>form the same as on the model extruder. The output rate of the target extruder is <br>determined by making the rate ratio the same as the effective volume ratio.<br>
+ (10.137)<br>
+We will take a ZSK57 that runs at 150 kg/hr, 300 rpm, and 90% torque as an exam-<br>ple. We will determine what rate and torque can be expected on a ZSK130 at equal <br>shear rate and SMEC. With K57 = 0.367 and K130 = 0.427 the screw speed of the <br>ZSK130 will be 300 * 0.367/0.427 = 258 rpm. The effective volume of the ZSK57 is <br>V57 = 1.67 l/m * 0.057 m = 0.0953 l. The effective volume of the ZSK130 is V130 = <br>
+7.808 l/m * 0.130 m = 1.015 l. Thus, the effective volume ratio becomes: Ve1/Ve2 = <br>
+(1.015/0.095) * (258/300) = 9.17. The rate on the ZSK130 will be:<br>
+ (10.138)<br>
+<hr>
+<A name=770></a><b>758</b>  <b> 10Twin Screw Extruders</b><br>
+If the gearbox rating of the ZSK57 is 1000 [Nm], the power consumption will be:<br>
+ (10.139)<br>
+With a throughput rate of 150 [kg/hr] the specific mechanical energy consumption <br>is SMEC = 28.27/150 = 0.1885 [kWhr/kg]. The power consumption of the ZSK130 at <br>constant SMEC will be kW130 = 0.1885 * 1376 = 259 [kW]. If the ZSK130 has a gear-<br>
+box rating of 13,000 [Nm], the torque on the ZSK130 will be:<br>
+ (10.140)<br>
+This means that there is enough extra torque to operate the ZSK130 at a screw <br>speed of 258 [rpm] and a throughput of 1376 [kg/hr].<br>
+<b></b><br>
+ <b>10.9Overview of Twin Screw Extruders</b><br>
+The following three tables give a brief overview of important aspects of twin screw <br>extruders. Table 10.7 compares intermeshing to non-intermeshing twin screw extru-<br>ders.<br>
+<b>Table 10.7</b>Comparison of Intermeshing to Non-Intermeshing Twin Screw Extruders<br>
+Intermeshing TSE<br>
+Non-intermeshing TSE<br>
+Self-wiping action possible<br>
+No self-wiping action possible<br>
+Good melting characteristics<br>
+Fair melting capability<br>
+Good distributive mixing<br>
+Good distributive mixing<br>
+Good dispersive mixing<br>
+Poor dispersive mixing<br>
+Good degassing<br>
+Good degassing<br>
+L/D up to about 60:1<br>
+L/D up to over 100:1<br>
+Large market share in polymer industry<br>
+Small market share in polymer industry<br>
+Table 10.8 compares co- to counter-rotating twin screw extruders.<br>Table 10.9 gives an overview of some of the most important performance charac-<br>teristics of twin screw extruders and single screw extruders. The performance char-<br>acteristics compared are feeding capability, dispersive mixing capability, distribu-<br>tive mixing capability, the ability of the machine to run at high screw speeds, the <br>self-cleaning action of the extruder, the pressure generating capability, and the <br>degassing capability. This table provides a useful overall view of the performance <br>capabilities of the most important commercial extruders.<br>
+<hr>
+<A name=771></a> <br>
+10.9Overview of Twin Screw Extruders  <b>759</b><br>
+<b>Table 10.8</b>Comparison of Co- to Counter-Rotating Twin Screw Extruders<br>
+Co-rotating TSE<br>
+Counter-rotating TSE<br>
+Screws have equal pitch<br>
+Screws have opposite pitch<br>
+Effective self-wiping action<br>
+Less effective self-wiping action<br>
+Limited number of screw geometries allow fully  Wider range of screw geometries possible<br>
+self-wiping action<br>Sliding action in intermeshing region, most <br>
+Milling type of intermeshing, material likely to be <br>
+material will bypass intermeshing region<br>
+drawn into intermeshing region<br>
+Fair pumping capability, limited pressure <br>
+Good pumping capability, good pressure <br>
+ <br>development  capability<br>
+ <br>development  capability<br>
+Good melting characteristics<br>
+Excellent melting capability<br>
+Good distributive mixing with effective <br>
+Good distributive mixing with effective distributive <br>
+ <br>distributive mixing elements<br>
+mixing element<br>
+Good dispersive mixing with effective dispersive  Inherently better dispersive mixing capability<br>
+mixing elements<br>Can run at very high screw speeds, up to <br>
+Can run at moderately high screw speeds, up to <br>
+1400 rpm<br>
+about 500 rpm<br>
+Good degassing<br>
+Excellent degassing<br>
+Large market share in compounding application,  Small market share in compounding, very large mar-<br>
+very small market share in profile extrusion<br>
+ket share in profile extrusion (low speed extruders)<br>
+Large market share in polymer industry<br>
+Large market share in polymer industry<br>
+<b>Table 10.9</b>Comparison of Various Single and Twin Screw Extruders<br>
+Extruder<br>
+Feeding<br>
+Disp.<br>
+Distr.<br>
+Screw<br>
+Self-<br>
+Pressure<br>
+De- <br>
+mixing<br>
+mixing<br>
+speed<br>
+cleaning<br>
+build-up<br>
+gassing<br>
+SSE<br>
+0<br>
++<br>
++<br>
++<br>
+­<br>
++<br>
+0<br>
+Pin barrel<br>
+0<br>
++<br>
++<br>
++<br>
+0<br>
+0<br>
+0<br>
+Kneader<br>
++<br>
++<br>
+++<br>
+++<br>
+++<br>
+­<br>
++<br>
+PGE<br>
+0<br>
+++<br>
+++<br>
+­<br>
+++<br>
+­<br>
+­<br>
+KCK<br>
++<br>
+++<br>
+++<br>
+0<br>
+0<br>
+0<br>
++<br>
+CICO<br>
++<br>
+­<br>
+0<br>
+­<br>
++<br>
++<br>
+0<br>
+CSCO<br>
++<br>
++<br>
+++<br>
+++<br>
+++<br>
+0<br>
++<br>
+CICT<br>
+++<br>
+­<br>
+0<br>
+­<br>
++<br>
+++<br>
++<br>
+HSCT<br>
+++<br>
+++<br>
++<br>
++<br>
++<br>
++<br>
+++<br>
+NOCT<br>
++<br>
++<br>
++<br>
+­<br>
+­<br>
+­<br>
+BIM<br>
++<br>
++<br>
++<br>
+0<br>
+­<br>
+­<br>
+­<br>
+CIM<br>
++<br>
++<br>
++<br>
+++<br>
+­<br>
+­<br>
+0<br>
+SSE is single screw extruder<br>
+PGE is planetary gear extruder<br>
+KCK is Kishihiro Continuous Kneader<br>
+CICO is closely intermeshing co-rotating twin screw extruder (low speed)<br>
+CSCO is closely self-wiping co-rotating twin screw extruder (high speed)<br>
+CICT is closely intermeshing counter-rotating twin screw extruder (low speed)<br>
+HSCT is high speed counter-rotating twin screw extruder<br>
+NOCT is non-intermeshing counter-rotating twin screw extruder<br>
+BIM is batch internal mixer<br>
+CIM is continuous internal mixer<br>
+<hr>
+<A name=772></a><b>760</b>  <b> 10Twin Screw Extruders</b><br>
+Table 10.9 compares important characteristics of different types of extruders; the <br>ranking ranges from--(poor or low) to ++ (very good or very high). The comparison <br>in Table 10.9 is a global comparison. Obviously, differences in screw and barrel <br>geometry can change characteristics significantly. For instance, the mixing charac-<br>teristics of CSCO extruders are poor when simple conveying screws are used; the <br>same is true for single screw extruders. However, CSCO extruders can achieve very <br>good distributive mixing and good dispersive mixing when appropriate mixing ele-<br>ments are used along the length of the screws.<br>From an overall point of view the CICT and HSCT twin screw extruders have attrac-<br>tive characteristics. They have good feeding, mixing, and degassing capabilities; <br>they can handle a high level of fillers and have good self-cleaning action. From a <br>performance point of view the kneader is quite competitive. However, from a price <br>point of view these machines are substantially more expensive than single screw <br>extruders. As a result, single screw extruders will likely maintain their dominance <br>in the polymer processing industry.<br>
+References<br>1.  K. Eise, S. Jakopin, H. Herrmann, U. Burkhardt, and H. Werner, Adv. Plast. Technol., <br>
+April, 18­39 (1981)<br>
+2.  K. Burkhardt, H. Herrmann, S. Jakopin, Plast. Compd., Nov./Dec., 73­78 (1978)<br>3.  H. Herrmann and U. Burkhardt, Kunststoffe, 11, 753­758 (1978)<br>4.  C.D. Denson, B.K. Hwang, Jr., Polym. Eng. Sci., 20, 965­971 (1980)<br>5.  C.E. Wyman, Polym. Eng. Sci., 15, 606­611 (1975)<br>6.  J. Maheshri and C.E. Wyman, Ind. Eng. Chem. Fundam., 18, 226­233 (1979)<br>7.  J.C. Maheshri, Ph.D. thesis, Univ. of New Hampshire, Durham, NH (1977)<br>8.  M. L. Booy, Polym. Eng. Sci., 18, 973­984 (1978)<br>9.  H. Werner, Ph.D. thesis, Univ. of Munich, Germany (1976)<br>10.  G. Schenkel, "Kunststoff-Extrudertechnik," Carl Hanser Verlag, Munich (1963)<br>11.  Z. Doboczky, Plastverarbeiter, 16, 57­67 (1965)<br>12.  Z. Dodoczky, Plastverarbeiter, 16, 395­400 (1965)<br>13.  P. Klenk, Plastverarbeiter, 22, 33­38 (1971)<br>14.  P. Klenk, Plastverarbeiter, 22, 105­109 (1971)<br>15.  L.P.B.M. Janssen, "Twin Screw Extrusion," Elsevier, Amsterdam (1978)<br>16.  L.P.B.M. Janssen, L.P.H.R.M. Mulders, and J.M. Smith, Plast. Polym., June, 93­98 <br>
+(1974)<br>
+17.  A. Kaplan and Z. Tadmor, Polym. Eng. Sci., 14, 58­66 (1974)<br>18.  R.J. Nichols and J. Yao, SPE ANTEC, San Francisco, 416­422 (1982)<br>19.  R.J. Nichols, SPE ANTEC, Chicago, 130­133 (1983)<br>
+<hr>
+<A name=773></a> References <br>
+<b>761</b><br>
+20.  L. Kovacs, U.S. Patent 3,689,182<br>21.  I. Klein and Z. Tadmor, U.S. Patent 3,924,842<br>22.  R. Klein and I. Klein, U.S. Patent 4,290,702<br>23.  R. Klein, Edison, and I. Klein, U.S. Patent 4,387,997<br>24.  R.J. Nichols, SPE ANTEC, New Orleans (1984)<br>25.  R.B. Bird, R.C. Armstrong, and O. Hassager, "Dynamics of Polymeric Liquids," Volume 1, <br>
+Fluid Mechanics, Wiley, NY (1977)<br>
+26.  W. Kozicki, C.H. Chou, and C. Tiu, Chem. Eng. Sci., 21, 665 (1966)<br>27.  W. Kozicki, C.J. Hsu, and C. Tiu, Chem. Eng. Sci., 22, 487 (1967)<br>28.  C. Howland and L. Erwin, SPE ANTEC, Chicago, 113­116 (1983)<br>29.  M.L. Booy, Polym. Eng. Sci., 20, 1220­1228 (1980)<br>30.  W.A. Mack and R. Herber, Chem. Eng. Prog., 72, Jan., 64­70 (1976)<br>31.  L. Wielgolinski and J. Nangeroni, Adv. Polym. Technol., 3, 99­105 (1984)<br>32.  M. Eyrich, 3rd Int. Congress on Reactive Processing of Polymers, Strasbourg, France, <br>
+Sept., 165­180 (1984)<br>
+33.  L.P.B.M. Janssen, B.J. Schaart, and J.M. Smith, Polymer Extrusion II Conference, Lon-<br>
+don, England, May, 15.1­15.7 (1982)<br>
+34.  N.P. Stuber and M. Tirrell, 3rd Int. Congress on Reactive Processing of Polymers, Stras-<br>
+bourg, France, Sept., 193­201 (1983)<br>
+35.  J.A. Speur and L.P.B.M. Janssen, 3rd Int. Congress on Reactive Processing of Polymers, <br>
+Strasbourg, France, Sept., 363­372 (1984)<br>
+36.  R.J. Nichols and R.K. Senn, Paper presented at 53rd Annual Meeting of the Society of <br>
+Rheology, Louisville, KY, Oct. 15 (1981)<br>
+37.  D. B. Todd, SPE ANTEC, 472­475 (1974)<br>38.  H.W. Werner, Kunststoffe, 71, 18­26 (1981)<br>39.  H.P. Han and C.D. Han, SPE ANTEC, Washington, DC (1985)<br>40.  G.P. Collins, C.D. Denson, and G. Astarita, AIChE J., Aug., 1288­1296 (1985)<br>41.  R.M. Secor, 3rd Int. Congress on Reactive Processing of Polymers, Strasbourg, France, <br>
+Sept., 153­164 (1984)<br>
+42.  J.A. Biesenberger and S.T. Lee, SPE ANTEC, Washington, DC (1985)<br>43.  D.B. Todd (Ed.), "Plastics Compounding, Equipment and Processing," Carl Hanser <br>
+ <br>Verlag, Munich (1998)<br>
+44.  J.L. White, "Twin Screw Extrusion, Technology and Principles," Carl Hanser Verlag, <br>
+Munich (1990)<br>
+45.  A. Grimminger et al., U.S. Patent 4,863,364 (1989)<br>
+<hr>
+<A name=774></a><b> 11 Troubleshooting </b><br>
+<b>Extruders</b><br>
+Troubleshooting is often the most critical element of extrusion engineering because <br>of the huge financial impact that extrusion problems can have. As a result, this topic <br>may be the most important in the whole book. For that reason it was decided to <br>devote a separate publication to this subject. The book "Troubleshooting the Ex -<br>trusion Process" [159] is an expanded text of this chapter, with many detailed case <br>studies. Therefore, for more details on troubleshooting the reader is referred to that <br>book.<br>
+<b></b><br>
+ <b>11.1 Requirements for Efficient </b><br>
+<b>Troubleshooting</b><br>
+Before dealing with specific extrusion problems, there are some issues that should <br>be addressed first. When an extruder develops a problem, it is very important to be <br>able to diagnose the extruder quickly and accurately in order to minimize downtime <br>or off-quality product. Important requirements for efficient troubleshooting are good <br>instrumentation and good understanding of the extrusion process. Instrumentation <br>is very important in process control, but it is absolutely essential in troubleshooting. <br>Without good instrumentation, troubleshooting is a guessing game at best, no mat-<br>ter how well one understands the entire process. Thus, lack of instrumentation can <br>prove to be very costly if it delays solving a certain problem for even a limited length <br>of time.<br>Important prerequisites to an efficient problem-solving process are:<br>
+ <br>
+ Good instrumentation<br>
+ <br>
+ Good understanding of the extrusion process<br>
+ <br>
+ Collect and analyze historical data<br>
+ <br>
+ Team building<br>
+ <br>
+ Good information on the condition of the equipment<br>
+ <br>
+ Good information on the feedstock<br>
+<hr>
+<A name=775></a><b>764</b>  <b> 11Troubleshooting Extruders</b><br>
+<b>11.1.1Instrumentation</b><br>
+The extrusion process is largely a black box process. In other words, it is not pos-<br>sible to visually observe what goes on inside the extruder. We can see material going <br>into the extruder and material coming out of the extrusion die. However, what <br> <br>happens between the feed opening and the die exit cannot be seen on normal ex -<br>truders, because the process is obscured by the extruder barrel. That means that <br>we are largely dependent on instrumentation to determine what happens inside <br>the extruder. We can think of instrumentation as our "<i>window to the process.</i>"<br>It is not sufficient to have ample instrumentation on the extruder; it is also impor-<br>tant to make sure that the sensors and readouts are working correctly. For instance, <br>if a temperature zone along the extruder is showing an excessively low or high tem-<br>perature, it should be verified that the temperature reading is correct. The measur-<br>ing instruments have to be correctly calibrated, and it should be ascertained that <br>the instrument is capable of measuring the variation in the parameter it is supposed <br>to monitor. In SPC, specific procedures have been developed to determine the cap-<br>ability of the measuring instrument [78].<br>
+<b>11.1.2Understanding of the Extrusion Process</b><br>
+In order to solve extrusion problems efficiently, one has to have a good understand-<br>ing of the extrusion process. For people new to extrusion, it is recommended to take <br>classes that cover material characteristics of plastics, typical features of extrusion <br>machinery, instrumentation and operating control, the inner workings of the extrud-<br>ers, as well as screw and die design. Classes are available from a variety of sources. <br>Some colleges have classes on extrusion. Many organizations provide continuing <br>education short courses on extrusion. Further, there are a number of training pro-<br>grams available [79] such as video training programs, interactive computer-based <br>training, and web-based training.<br>In many extrusion operations, the primary mode of training is on-the-job training. <br>However, it should be realized that on-the-job training is often the least effective and <br>most expensive method of training. Extruders are expensive machines that have to <br>be operated correctly to produce good parts. If extruders are not operated correctly, <br>out-of-spec parts may be produced or the extruder may be damaged. It is also impor-<br>tant to realize that extruders are potentially dangerous devices. Serious accidents <br>can occur when extruders are not operating properly. Therefore, it is imperative that <br>people operating extrusion equipment receive comprehensive safety training.<br>
+<hr>
+<A name=776></a> <br>
+11.1Requirements for Efficient Troubleshooting  <b>765</b><br>
+<b>11.1.3Collect and Analyze Historical Data (Timeline)</b><br>
+To understand why a process is not behaving correctly, we have to compare the cur-<br>rent process conditions to previous conditions when the problem did not exist; <br>this is also referred to as constructing a timeline. This means collecting not only <br>process information from the extruder, such as temperatures, pressures, motor load, <br>line speeds, barrel dimensions, screw dimensions, etc., but also collecting informa-<br>tion on the material and any other variables that can affect the process. Changes in <br>the process can occur not only because of machine parameters, but also because of <br>material changes. For instance, a change in the stabilizer level in the plastic can <br>cause degradation problems without any changes in the machine conditions and <br>settings; see Section 11.1.6.<br>The importance of a timeline is based on the fact that the process was running well <br>for a certain period of time. In order for the process to become unstable, there has to <br>be an identifiable change or changes that precipitated the process upset. The task is <br>to identify these changes and correct them to get the process back in control. The <br>timeline creation process starts during a period of process stability and ends some <br>time after the process upsets were noticed. All events, even those remotely con-<br>nected to the process, are listed on the timeline. Once the timeline is finished it <br>becomes a helpful tool in identifying the event(s) that precipitated the problem.<br>It should be noted that not all changes have an immediate effect on the process. In <br>some cases, there can be a considerable incubation time before the effects of a <br>change become noticeable. This, of course, complicates the troubleshooting process; <br>it is important to keep this in mind and not jump to conclusions. The author experi-<br>enced a case where a disastrous wear problem was related to an event that took <br>place four months earlier. The wear remained insignificant until about four months <br>after a new feed housing was installed. However, when the rapid wear started, the <br>screw was destroyed within a time period of only 48 hours.<br>Figure 11.1 shows an example of a timeline leading up to a gel problem.<br>In constructing the timeline make sure to list all events that can potentially affect <br>the process. Obvious events are things like power outage, new or refurbished ex -<br>truder screw, resin lot change, etc. Some events are less obvious but may still affect <br>the process such as construction in the area, changes in materials handling, main-<br>tenance on plant water system, operator training, power surges, etc.<br>
+<hr>
+<A name=777></a><b>766</b>  <b> 11Troubleshooting Extruders</b><br>
+November 1999<br>
+New extruder screw installed 11-28-99<br>
+Resin lot 110199<br>
+December 1999<br>
+Resin lot 011200<br>
+January 2000<br>
+Replaced brushed on DC motor, 01-22-00<br>
+Replaced die heater and thermocouples, 01-27-00<br>
+Replaced oil in gearbox, 02-15-00<br>
+February 2000<br>
+Power outage, 02-26-00<br>
+New extruder operator John Haynes, 03-07-00<br>
+Replaced temperature sensor in water line, 03-16-00<br>
+March 2000<br>
+Resin lot 032300 <br>
+Changed barrel temperature profile, 04-12-00<br>
+April 2000<br>
+Installed refurbished extrusion die, 04-23-00<br>
+Replaced desiccant in dryer, 05-02-00<br>
+Replaced gearpump, 05-09-00<br>
+May 2000<br>
+Resin lot 052400<br>
+Miked screw and barrel, 05-29-00 (in spec)<br>
+Gel problem, 6-05-2000<br>
+June 2000<br>
+<b>Figure 11.1</b>Example of timeline leading up to a gel problem<br>
+<b>11.1.4Team Building</b><br>
+If the scope of a problem is small, a single individual can go through the problem <br>solving process and there is no need to organize a team. In many cases, however, <br>problems involve different departments and functions and require a wide range of <br>skills to come to a solution. In such cases, problem solving requires a team effort. <br>Extrusion problems often require input from materials QC, purchasing, main-<br>tenance, engineering, and possibly other departments.<br>
+<b>11.1.5Condition of the Equipment</b><br>
+When a problem develops on an extruder, it is important to have good information <br>on the condition of the equipment. Extruders should be well maintained, and good <br>maintenance records should be available so that the condition of the various com-<br>ponents of the machine can be assessed. Maintenance recommendations from the <br>extruder manufacturer should be followed to ensure good performance.<br>Extruder screws and barrels will wear over time. The wear rate depends on many <br>factors. Extruder screws can last for several years or only several weeks. It is impor-<br>
+<hr>
+<A name=778></a> <br>
+11.1Requirements for Efficient Troubleshooting  <b>767</b><br>
+tant to measure the I.D. of the barrel and the O.D. of the screw on a regular basis <br>(at least once a year) so that the life of the screw and barrel can be predicted. This <br>allows screws and barrel to be replaced at predetermined intervals without unpleas-<br>ant surprises.<br>
+<b>11.1.6Information on the Feedstock</b><br>
+The performance of an extruder is determined as much by the characteristics of the <br>feedstock as it is by the machine. Feedstock properties that affect the extrusion pro-<br>cess include bulk properties, melt flow properties, and thermal properties. Impor-<br>tant bulk flow properties are the bulk density, compressibility, particle size, particle <br>shape, external and internal coefficient of friction, and agglomeration tendency. <br>Important melt flow properties are the shear and elongational viscosity as a func-<br>tion of strain rate and temperature. The commonly used melt indexer provides only <br>limited information on the melt viscosity. Important thermal properties include the <br>specific heat, the glass transition temperature, the crystalline melting point, the <br>latent heat of fusion, the thermal conductivity, the density, the degradation tempe-<br>rature, and the induction time as a function of temperature.<br>A change in the material can cause a problem in extrusion when it affects one or <br>more polymer properties that determine the extrusion behavior of the material. If a <br>material problem is suspected, one should first examine the quality control (QC) <br>records on incoming material to see if a change in feedstock properties was deter-<br>mined. Unfortunately, often the only QC test on incoming material is a melt index <br>(MI) test. This test is only able to detect a very limited number of material-related <br>extrusion problems. Thus, in many cases, material testing may have to be more <br>extensive than the regular QC testing.<br>There are a number of problems associated with making measurements on the criti-<br>cal properties of the feedstock. The total number of properties that need to be meas-<br>ured is about ten, with some of the measurements being rather time-consuming. <br>Thus, it may take considerable time to fully characterize the extrusion properties of <br>a material; this does not help when a quick solution is required. Another problem is <br>the fact that some important properties are difficult to measure and require a high <br>degree of accuracy and reproducibility. The most notable property in this respect is <br>the external coefficient of friction. A further problem can be that instruments to <br>measure all the pertinent properties may not be readily available. Not all companies <br>can afford to maintain a fully instrumented laboratory to completely analyze the <br>extrusion characteristics of a certain compound. Finally, even after a material is <br>fully characterized and no significant changes in properties have been found, there <br>is no guarantee that the extrusion problem is not material related because the mate-<br>rial sample used for testing may not have been a representative sample. Since most <br>
+<hr>
+<A name=779></a><b>768</b>  <b> 11Troubleshooting Extruders</b><br>
+tests are done on samples of about 0.01 kg or less and most extruders run at a <br>throughput of several hundred to several thousand kg/hr, there is a considerable <br>chance that the test sample is not representative of the entire feedstock.<br>A practical test for a material-related extrusion problem is to extrude some material <br>from an old batch to see if the problem will disappear. If this is indeed the case, then <br>this provides a very strong indication that the problem is material related. For this <br>reason, it is helpful to retain some material of older batches, which will also provide <br>a reference for more detailed measurements.<br>If the problem is material related, there are two possible solutions. The easiest solu-<br>tion from an extrusion point of view is to change the material back to the way it was <br>before the problem developed. However, this may not always be possible for other <br>reasons. Thus, if the change in the material is permanent, then the extrusion pro-<br>cess will have to be adjusted to accommodate the material change. At this point, the <br>nature of the problem may change from an upset to a development problem. The <br>chance of solving the problem will depend on the nature and the magnitude of the <br>change in the material.<br>
+<b></b><br>
+ <b>11.2Tools for Troubleshooting</b><br>
+There are a number of tools that will help during the troubleshooting process. A dis-<br>cussion of a number of the important tools follows.<br>
+<b>11.2.1Temperature Measurement Devices</b><br>
+A useful tool is a pyrometer with a surface contact probe and a melt probe (needle <br>probe). The contact probe can be used to check for heater burnout, barrel tempera-<br>tures, die temperatures, and temperature distribution and variation. The melt probe <br>can be used to check melt thermocouple accuracy and to measure the actual melt <br>temperature as it exits the die. The melt temperature at the die exit can be higher <br>than the melt probe temperature at the end of the extruder barrel.<br>Another useful troubleshooting tool is the infrared thermometer. Figure 11.2 shows <br>an example of a handheld infrared thermometer.<br>The non-contacting IR thermometer allows temperature measurement in spots that <br>are difficult to reach with a contacting thermometer. Also, the IR thermometer <br>allows measurement of polymer melt temperature without damaging the extruded <br>product. It allows determination of the melt temperature variation across the melt <br>stream coming out of a sheet die. Large melt temperature variations will generally <br>create problems downstream.<br>
+<hr>
+<A name=780></a><IMG src="Polymer Extrusion by Chris Rauwendaal-780_1.jpg"><br>
+ <br>
+11.2Tools for Troubleshooting  <b>769</b><br>
+<b>Figure 11.2</b>Example of a handheld infrared thermometer<br>
+<b>11.2.2Data Acquisition Systems (DAS)</b><br>
+Data acquisition systems are extremely useful in extrusion because problems often <br>occur when the operator is not watching the instrument panel of the extruder. Even <br>if the operator is watching the instrument panel, he can only observe a limited num-<br>ber of variables at one time. A DAS that captures and saves important process data <br>is indispensable in troubleshooting. When a problem occurs at 2:30 AM, it is very <br>difficult for a process engineer coming in at 7:00 AM to reconstruct the events at <br>2:30 AM if important process data was not recorded at that particular time.<br>A simple DAS is a chart recorder that can track important variables like screw <br>speed, diehead pressure, melt temperature, motor amperage, etc. More useful is a <br>computer-based DAS; these come in two forms: portable data collectors/machine <br>analyzers and fixed-station data acquisition system.<br>
+<b>11.2.2.1Portable Data Collectors/Machine Analyzers<br></b>Portable data collectors, PDCs, are similar to check sheets in that they can be easily <br>moved around. In injection molding, these devices are often referred to as portable <br>machine analyzers, PMAs. They have some important advantages:<br>
+ <br>
+ They can record data in computer readable form.<br>
+ <br>
+ They can take data directly from electronic sensors and gauges. This makes PMAs <br>
+fast and minimizes errors.<br>
+<hr>
+<A name=781></a><b>770</b>  <b> 11Troubleshooting Extruders</b><br>
+ <br>
+ Data can be analyzed internally to yield information on mean, range, maximum <br>
+value, minimum value, standard deviation, etc.<br>
+ <br>
+ They are available with limit checking; the PMA can give an alarm when data just <br>
+taken is out of specification.<br>
+PMAs can collect variables as well as attributes data. The use of PMAs has increased <br>considerably as the prices have come down to levels that are affordable even for <br>small operations [162]. Several PMAs are presently available for less than $10,000.<br>Most PMAs offer the user some flexibility in assigning inputs to the data acquisition <br>channels; this even extends to auxiliary equipment, e.g., dryers, and external sig-<br>nals, such as plant ambient temperature and relative humidity. Important inputs <br>are:<br>
+ <br>
+ Melt pressure(s)<br>
+ <br>
+ Melt temperature(s)<br>
+ <br>
+ Screw speed<br>
+ <br>
+ Motor load<br>
+ <br>
+ Temperature feed housing<br>
+ <br>
+ Barrel temperatures<br>
+ <br>
+ Die temperatures<br>
+ <br>
+ Line speed<br>
+ <br>
+ Extrudate dimensions<br>
+ <br>
+ Heating power at various temperature zones<br>
+ <br>
+ Cooling rate at various temperature zones<br>Other parameters to be monitored may depend on the specifics of the operation. For <br>instance, in vented extrusion it is often important to monitor the vacuum level at the <br>vent port. For in-depth process analysis a system capable of handling 32 channels or <br>more should be used.<br>
+<b>11.2.2.2Fixed Station Data Acquisition Systems<br></b>As the name implies, fixed station data acquisition systems are fixed to one location, <br>either because of size or because the wiring makes it very difficult to move the unit. <br>A fixed station DAS can have a wide range of capabilities. A simple DAS may record <br>data from only one extruder, i.e., a dedicated DAS. A number of machine suppliers <br>now offer extruders with integrated data acquisition and SPC capability.<br>A more sophisticated DAS may be able to record data from various sources, analyze <br>the data, present control charts, show trend plots for different variables, etc. These <br>systems are often referred to as plant-wide monitoring systems. Central computer-<br>based systems are now available that allow the user to view the operation of plant <br>equipment and change the operating parameters on any selected equipment. A fixed <br>
+<hr>
+<A name=782></a> <br>
+11.2Tools for Troubleshooting  <b>771</b><br>
+station DAS can take many different forms depending on the application. A good <br>DAS can be a very valuable tool in improving process control as well as in problem <br>solving. For the application of DAS to extrusion operations, the following capabili-<br>ties are useful:<br>
+ <br>
+ <i>Monitoring of many variables.</i> A typical extrusion process requires about 40 to 80 <br>
+process parameters to be monitored.<br>
+ <br>
+ <i>Data on slowly changing variables</i> should be taken at least once a second. For <br>
+ <br>rapidly changing variables, such as melt pressure and hydraulic pressure, data <br>should be collected at higher frequency, typically 100 points/s. Some high-end <br>systems sample up to 100,000 points/s.<br>
+ <br>
+ <i>Trending</i>--the capability to display the variation of one or more process para-<br>
+meters over a particular time period. It is useful if the scales in these displays are <br>adjustable. This capability is extremely helpful in troubleshooting and problem <br>solving.<br>
+ <br>
+ <i>Determination of statistical parameters</i> on process and product parameters, such as <br>
+mean, standard deviation, control limits, etc.<br>
+ <br>
+ <i>Alarms for out-of-spec data</i> and/or indications of assignable causes of variation.<br>
+ <br>
+ <i>Recipes</i>--the capability to store important process parameters for different prod-<br>
+ucts. This allows previous process conditions to be reproduced quickly and relia-<br>bly.<br>
+ <br>
+ <i>Production summaries</i>--the capability to follow the amount of material being pro-<br>
+duced and present summaries per shift, per day, per week, etc. This is a useful <br>management tool for analyzing productivity on different production lines.<br>
+The following features have to be considered:<br>
+ <br>
+ <i>User friendliness.</i> The system should be easy to use, intuitive, and should not <br>
+require very long training for operators to become accustomed to it.<br>
+ <br>
+ <i>Accessibility.</i> Is access from a remote computer possible? With remote access, inte-<br>
+gration into a plant-wide information control system is possible.<br>
+ <br>
+ <i>Connectivity.</i> Can the DAS software work together with other software packages?<br>
+ <br>
+ <i>Upgradability.</i> Can new upgraded versions of software and/or hardware be readily <br>
+implemented?<br>
+ <br>
+ <i>Cost.</i> An analysis should be made to determine whether the cost savings in terms <br>
+of improved production efficiency and quality outweigh the cost of a data acquisi-<br>tion system.<br>
+The different methods of data collection each have their advantages and disadvan-<br>tages. Check sheets are slow, prone to human errors and transcription errors; how-<br>ever, they are also inexpensive, flexible, efficient for small amounts of data, and easy <br>to use. Fixed station data acquisition systems are fast, not prone to human errors <br>and transcription errors, can handle large amounts of data; however, they tend to be <br>
+<hr>
+<A name=783></a><b>772</b>  <b> 11Troubleshooting Extruders</b><br>
+expensive and more difficult to use. Thus, check sheets and fixed station data acqui-<br>sition systems have complementary areas of applications. Portable data collectors <br>fall between check sheets and fixed station data acquisition systems. Table  11.1 <br>shows a comparison of different data recording methods.<br>For plant-wide monitoring systems, the extrusion process can be integrated with <br>upstream and downstream operations. Bar coding can be used to achieve parts trace-<br>ability and integrated with automatic product tracking and warehousing. With these <br>systems, accurate data can be collected for job costing, such as cycle times, yields, <br>efficiencies, scrap, labor allocation, and downtime--essential information for effi-<br>cient plant management. Functions that can be included in plant-wide monitoring <br>systems are preventive maintenance, production scheduling, inventory, production <br>reporting, order entry, job histories, real-time alarms, quality control, SPC, etc.<br>Some plant-wide monitoring systems allow changes to be made by remote control <br>from a central terminal, in such parameters as screw speed or barrel temperature. <br>Obviously, the results from these changes have to be monitored carefully, because <br>they can make the process not only better, but also worse.<br>
+<b>11.2.3Light Microscopy</b><br>
+This technique allows the observation of a polymeric sample in order to obtain im -<br>portant information about its structure, processing or manufacturing, and failure or <br>damage causes. Therefore, it is a very common technique for quality control, trou-<br>bleshooting, and failure analysis.<br>In the case of a high-magnification microscope, the possible magnification ranges <br>are 50, 100, 200, and 500X. The illumination can be transmitted, reflected, or polar-<br>ized light. In a low-magnification microscope, the magnification ranges vary from <br>7 to 80X. There is the possibility of a three-dimensional view. Figure 11.3 shows a <br>microscope.<br>For sample preparation, a microtome with a range from 0 to 340 microns with 0.5 <br>micron resolution and a polishing machine are required.<br>
+<b>Table 11.1</b>Comparison of Data Recording Methods<br>
+Check sheets<br>
+Portable data collectors<br>
+Fixed station DAS<br>
+Speed<br>
+slow<br>
+fair<br>
+fast<br>
+Human errors<br>
+substantial<br>
+small<br>
+very small<br>
+Transcription errors<br>
+substantial<br>
+no<br>
+no<br>
+Efficient for:<br>
+small amount of data<br>
+large amount of data<br>
+huge amount of data<br>
+Cost<br>
+minor<br>
+fair<br>
+substantial<br>
+Flexibility<br>
+very good<br>
+fair<br>
+good<br>
+User skill required<br>
+low<br>
+medium<br>
+high<br>
+<hr>
+<A name=784></a><IMG src="Polymer Extrusion by Chris Rauwendaal-784_1.jpg"><br>
+ <br>
+11.2Tools for Troubleshooting  <b>773</b><br>
+<b> Figure 11.3</b> <br>
+Light microscope<br>
+The application field of this technique includes:<br>
+ <br>
+ Analysis of defects in plastic products<br>
+ <br>
+ Dispersion of a filler or a pigment in a polymeric matrix<br>
+ <br>
+ Residual stresses under polarized light<br>
+ <br>
+ Dimensional studies<br>
+ <br>
+ Differences due to orientation<br>
+ <br>
+ Crystallization analysis<br>
+ <br>
+ Polymer reinforcement analysis<br>
+ <br>
+ Coextruded film analysis using colorimetry<br>Colorimetry involves the treatment of a sample to impart color to selected materials <br>in the sample. For instance, in multi-layer film it can be difficult to distinguish <br>the different layers when natural polymers are used (no colorants added). Certain <br>chemi cals color certain polymers but do not color others. For example, iodine will <br>color nylon-6 red, EVOH brown, and typical tie layers gray, but will not change the <br>color of PE or PET.<br>
+<b>11.2.4Thermochromic Materials</b><br>
+Thermochromic substances can be very useful in temperature-related problems. <br>These materials change color irreversibly at a particular temperature, and they are <br>used in a number of applications. For instance, thermochromic paints are used on <br>heat shrinkable sleeves so that the installer who is heating the sleeve with a torch <br>can see whether the product has reached proper installation temperature.<br>
+<hr>
+<A name=785></a><b>774</b>  <b> 11Troubleshooting Extruders</b><br>
+It is difficult to measure the actual polymer melt temperatures inside an extruder. <br>The reason is that an immersion probe cannot be used because the rotating screw <br>will shear it off. Flush-mounted temperature sensors do not give a good indication of <br>the actual stock temperature. Thermochromic materials can be added to the feed-<br>stock to determine whether the stock temperatures in the extruder exceed the color <br>transition temperature of the material.<br>Mennig [160] published a paper quite some time ago on a preliminary investigation <br>into the use of temperature-indicating materials in extrusion. The thermochromic <br>materials that were used in the study were not clearly identified and suffered from <br>the fact that the color transition temperature was time dependent. Obviously, this <br>limits the usefulness of these materials for accurate temperature indication.<br>Rauwendaal [152] conducted a study using thermochromic powders used commer-<br>cially in thermochromic paints to study melt temperature in extruders. The study <br>involved a small laboratory twin screw extruder processing a highly filled HDPE. It <br>was found that the material degraded inside the extruder even though the barrel <br>temperatures and exit melt temperature were quite reasonable. The barrel tempera-<br>tures were set a 200°C and the exit melt temperature was measured at about 215 to <br>220°C.<br>These are normal temperatures for HDPE and should not cause degradation under <br>normal circumstances. However, it was suspected that the melt temperatures inside <br>the extruder were much higher than the barrel temperature. Therefore, a thermo-<br>chromic powder with a color transition temperature of 250°C was added to the <br>feed. With this material, a very clear discoloration was noticed in the material exit-<br>ing the die. Next, a thermochromic powder was added with a color transition tem-<br>perature of 300°C. This material unexpectedly resulted in very clear discoloration <br>just as in the first material, indicating that stock temperatures in the extruder <br>exceeded 300°C.<br>At this point, the cause of degradation was quite clear. However, the location of high <br>stock temperatures needed to be determined as well. Therefore, the screws were <br>pulled and the color change along the screw was visually observed. It was found that <br>the color change took place in a screw section with high-restriction kneading blocks. <br>Clearly, the shearing action in this part of the screw caused excessively high melt <br>temperatures, resulting in degradation of the polymer. Based on these findings, the <br>screw geometry was changed to a less severe mixing geometry and the problem <br>disappeared.<br>In this study the thermochromic powders were found to be very useful because they <br>provided information very difficult to obtain by any other means. The drawback of <br>this method is that thermochromic powders are not readily available. It would be <br>helpful if a little troubleshooting kit were available with several thermochromic <br>powders with different color transition temperatures.<br>
+<hr>
+<A name=786></a> <br>
+11.2Tools for Troubleshooting  <b>775</b><br>
+<b>11.2.5Thermal Analysis</b><br>
+In thermal characterization, a controlled amount of heat is applied to a sample and <br>its effect measured and recorded. In isothermal operations, the effect is recorded as <br>a function of time at constant temperature. In a programmed temperature operation, <br>the temperature is changed in a predetermined fashion, e.g., at a certain rate, and <br>the effect is recorded as a function of temperature.<br>The main thermal analysis techniques are differential thermal analysis (DTA), dif-<br>ferential scanning calorimetry (DSC), thermo-gravimetric analysis (TGA), thermo-<br>mechanical analysis (TMA), and dynamic mechanical analysis (DMA). These meth-<br>ods are discussed in more detail in Section 6.3.8.<br>
+<b>11.2.6Miscellaneous Tools</b><br>
+A tape measure can be used to measure distances from about 20 cm up to several <br>meters. Dial calipers are useful for closely measuring extruded products, dimen-<br>sions of the extruder screws, extrusion tooling, etc. A stopwatch is an indispensable <br>tool for measuring screw speed, line speed, blender calibration, etc. A scale can be <br>used to measure the output of an extruder. A small voltmeter is very useful in mak-<br>ing sure that voltage and resistance levels of various components are at their <br>required values. A millivolt source can be used for verifying thermocouple inputs <br>and to check controller response and line continuity.<br>A digital still camera is a valuable tool in troubleshooting to accurately document <br>the configuration of the extrusion line, details of the screw geometry, die geometry, <br>wear regions on the screw, contamination in the product, unusual build-up on the <br>screen pack, etc. A digital camera is an excellent tool in the preparation of clear <br>operating procedures and other process documentation. The digital photos can be <br>easily transferred to a PC, and the photos can be used in reports, manuals, presen-<br>tations, etc.<br>A digital video camera is also a very useful troubleshooting tool because of its abil-<br>ity to capture motion. This allows determination of the frequency of pulsing of the <br>extrudate coming out of the die. Another benefit of the video camera is that it can <br>record sound as well as images. The soundtrack can be analyzed separately. For <br>instance, the sound of a high-frequency vibration can be captured on video and <br>downloaded to a PC, and the frequency of the vibration can be determined from the <br>soundtrack. Digital still cameras are better for high-quality capture of a static event. <br>However, a video camera is an excellent tool for capturing dynamic events. In trou-<br>bleshooting, the dynamic events are often the most important.<br>
+<hr>
+<A name=787></a><b>776</b>  <b> 11Troubleshooting Extruders</b><br>
+<b></b><br>
+ <b>11.3Systematic Troubleshooting</b><br>
+<b>11.3.1Upsets versus Development Problems</b><br>
+In this chapter, the problems that will be primarily focused on are upsets. These are <br>problems that occur in an existing extrusion line for some unknown reason. If the <br>extrusion line has been running fine for a considerable period of time, then it is <br>clear that there must be a solution to the problem. Thus, the objective of trouble-<br>shooting is to find the cause of the upset and to eliminate it. On the other hand, <br>when one deals with a development problem, there may not be a solution. In a devel-<br>opment problem, one tries to establish a condition that has not been achieved before. <br>If the desired condition is physically impossible, then clearly there is no solution to <br>the problem. From a functional analysis of the process, one should be able to deter-<br>mine the bounds of the conditions that can be realized in practice.<br>
+<b>11.3.2Machine-Related Problems</b><br>
+In machine-related problems, mechanical changes in the extruder cause a change in <br>extrusion behavior. These changes can affect the drive system, the heating and cool-<br>ing system, the feed system, the forming system, or the actual geometry of screw <br>and barrel. The main components of the drive are the motor, the reducer, and the <br>thrust bearing assembly. Drive problems manifest themselves either in variations <br>in rotational speed and/or the inability to generate the required torque. Problems <br>in the reducer and thrust bearings are often associated with clear audible signals of <br>mechanical failure. If the problem is suspected to be the drive, make sure that the <br>load conditions do not exceed the drive capability.<br>
+<b>11.3.2.1Drive System<br></b>Older motor drive systems generally consist of a DC brush motor, a power conver-<br>sion unit (PCU), and operator controls. A frequent problem with the motor itself is <br>worn brushes; these should be replaced at regular intervals as recommended by the <br>manufacturer. In troubleshooting an extruder drive, one should follow the proce-<br>dure recommended by the manufacturer of the drive. A typical troubleshooting <br>guide for a DC motor is shown in Table 11.2.<br>
+<hr>
+<A name=788></a> <br>
+11.3Systematic Troubleshooting  <b>777</b><br>
+<b>Table 11.2</b>Troubleshooting Guide for DC Motor<br>
+Problem<br>
+Possible cause<br>
+Action<br>
+Motor will not start<br>
+Low armature voltage<br>
+Make sure motor is connected to proper <br>
+voltage<br>
+Weak field<br>
+Check for resistance in the shunt field <br>
+circuit<br>
+Open circuit in armature or field<br>
+Check for open circuit<br>
+Short circuit in armature or field<br>
+Check for short circuit<br>
+Motor runs too slow<br>
+Low armature voltage<br>
+Check for resistance in armature <br>
+circuit<br>
+Overload<br>
+Reduce load or use larger motor<br>
+Brushes ahead of neutral<br>
+Determine proper neutral <br>
+position for brush location<br>
+Motor runs too fast<br>
+High armature voltage<br>
+Reduce armature voltage<br>
+Weak field<br>
+Check for resistance in shunt <br>
+field circuit<br>
+Brushes behind neutral<br>
+Determine proper neutral <br>
+position for brush location<br>
+Brushes sparking<br>
+Brushes worn<br>
+Replace<br>
+Brushes not seated properly<br>
+Reseat brushes<br>
+Incorrect brush pressure<br>
+Measure brush pressure and <br>
+correct<br>
+Brushes stuck in holder<br>
+Free brushes, make sure brushes <br>
+are of proper size<br>
+Commutator dirty<br>
+Clear commutator<br>
+Commutator rough or eccentric<br>
+Resurface commutator<br>
+Brushes off neutral<br>
+Determine proper neutral <br>
+position for brush location<br>
+Short circuit in commutator<br>
+Check for shorted commutator, <br>
+check for metallic particles <br>
+between commutator segments<br>
+Overload<br>
+Reduce load or use larger motor<br>
+Excessive vibration<br>
+Check driven machine for <br>
+balance<br>
+Brush chatter<br>
+Incorrect brush pressure<br>
+Measure and correct<br>
+High mica<br>
+Undercut mica<br>
+Incorrect brush size<br>
+Replace with proper size<br>
+Bearings hot<br>
+Belt too tight<br>
+Reduce belt tension<br>
+Misaligned<br>
+Check alignment and correct<br>
+Bent shaft<br>
+Straighten shaft<br>
+Bearing damage<br>
+Inspect and replace<br>
+<hr>
+<A name=789></a><b>778</b>  <b> 11Troubleshooting Extruders</b><br>
+<b>11.3.2.2The Feed System<br></b>The most important component of the feed system in a flood-fed extruder is the feed <br>hopper with possible stirrer and/or discharge screw. A mechanical malfunction of <br>this system can be determined by visual inspection. If the feed hopper is equipped <br>with a discharge screw (crammer feed), constancy of the drive should be checked. <br>For proper functioning of a crammer feeder, the drive of the crammer feeder should <br>have a torque feedback control to ensure constant feeding and to avoid overfeeding.<br>Many extruders have square feed hoppers with rapid compression in the converging <br>region. Extruder manufacturers often choose this geometry because of the ease of <br>manufacture. However, this hopper geometry does not promote steady flow. When <br>flow instabilities occur in a feed hopper, the extruder operator will often hit the <br> <br>hopper with a heavy object to get the flow going again. As a result, hoppers that <br>cause flow problems often shows signs of abuse such as mars, dents, scrapes, etc. <br>Such mars are a strong indication of poor feed hopper design.<br>
+<b>11.3.2.3Different Feeding Systems<br></b>The feeding system of the extruder is of utmost importance in achieving a stable and <br>consistent extrusion process. In flood-feeding, the design of the feed hopper deter-<br>mines to a large extent how stable the bulk material can flow through the hopper. <br>This subject is covered in Section 7.2.1. In starve-fed extruders, the stability of the <br>process is to a large extent determined by the quality of the feeders. Feeders are <br>either volumetric or gravimetric. Feeders are often integrated into an overall control <br>system for the extrusion line or even a complete plant.<br>Volumetric feeders are basically speed-controlled and deliver a constant mass flow <br>rate as long as the bulk density of the feed material is constant. When variations <br>occur in bulk density, volumetric feeders are less suitable. Gravimetric feeders con-<br>trol the mass flow rate generally by weighing the total weight of the feeder and the <br>material in the feeder. The discharge rate is controlled such that there is a linear <br>reduction in weight with time. When the gravimetric feeder is recharged with mate-<br>rial, it generally switches to volumetric mode for a brief time to handle the disturb-<br>ance in weight during this time. The book by Wilson [161] contains a large amount <br>of detail on feeding systems.<br>
+<b>11.3.2.4Heating and Cooling System<br></b>The heating and cooling system is used to achieve a certain degree of control of the <br>polymer melt temperature. However, stock temperature deviations do not necessarily <br>indicate a heating or cooling problem because heat is added directly to (or removed <br>from) the barrel and only indirectly to (from) the polymer. The local barrel tempera-<br>ture as measured with a temperature sensor determines the amount of barrel heat-<br>ing or cooling. The temperature that is controlled is actually a barrel temperature.<br>
+<hr>
+<A name=790></a> <br>
+11.3Systematic Troubleshooting  <b>779</b><br>
+The stock temperature is generally controlled by changing the setpoint of the tem-<br>perature zones along the extruder. However, due to the slow response of the melt <br>temperature to changes in heat input, only very gradual stock temperature changes <br>can be effectively controlled by setpoint changes. Rapid stock temperature fluctua-<br>tions, a cycle time of less than about five minutes, can usually not be reduced with <br>a melt temperature control system. Such fluctuations are indicative of conveying <br>instabilities in the extrusion process and can only be effectively reduced by elimi-<br>nating the cause of the conveying instability.<br>The heating system can be checked by changing the setting to a much higher tem-<br>perature, for instance 50°C above the regular setting. The heater should turn on a <br>full 100% and the measured barrel temperature should start rising in about one to <br>two minutes. If the heater does not turn full on, the barrel temperature measure-<br>ment is in error or there is a problem in the electronic circuit of the temperature <br>controller. If the heater turns full on but the temperature does not start to rise within <br>two to four minutes, either the barrel temperature measurement is incorrect or <br>there is poor contact between heater and barrel.<br>The cooling system can be checked by changing the setting to a much lower tem-<br>perature, for instance 50°C below the regular setpoint. The cooling should turn full <br>on and the measured barrel temperature should start to drop in about one to two <br>minutes. If the cooling does not turn full on, the barrel temperature measurement is <br>in error or there is a problem in the circuit of the temperature controller. If the cool-<br>ing turns full on but the temperature does not start to drop within two to four min-<br>utes, either the barrel temperature is incorrect or the cooling device is inoperable. <br>This checkout procedure is summarized in Table 11.3.<br>
+<b>Table 11.3</b>Heating and Cooling System Check<br>
+Heating system: Increase setpoint of temperature zone by 50°C:<br>Heater turns on full blast and the barrel <br>
+Heating system normal<br>
+temperature rises in about 2 minutes<br>Heater turns on full blast but the barrel <br>
+Poor contact of heater to barrel, insufficient <br>
+temperature does not change<br>
+heating capacity, temperature sensor failure<br>
+Heater output does not change<br>
+Heater failure, controller bad<br>
+Cooling system: Reduce setpoint of temperature zone by 50°C:<br>Cooling on full blast and the barrel <br>
+Cooling system normal<br>
+temperature drops in about 2 minutes<br>Cooling on full blast but the barrel <br>
+Temperature sensor failure, insufficient <br>
+temperature does not change<br>
+cooling capacity, cooling system not <br>
+functioning at all<br>
+Cooling output does not change<br>
+Cooling system bad, controller bad<br>
+<hr>
+<A name=791></a><b>780</b>  <b> 11Troubleshooting Extruders</b><br>
+If a substantial amount of cooling is required to maintain the desired stock tempe-<br>rature, this is generally a strong indication of excessive internal heat generation by <br>frictional and viscous dissipation. The internal heat generation can be reduced by <br>lowering screw speed or by changing the screw design. The main screw design vari-<br>able that affects the viscous heating is the channel depth. Increasing the channel <br>depth will reduce the shear rates and viscous heating.<br>Mechanical changes in the forming system relate to the extrusion die and down-<br>stream equipment. These elements can be subjected to simple visual inspection, <br>and mechanical changes can thus be easily determined. Changes to the geometry of <br>screw and barrel are often caused by wear. Since wear is a very important phenom-<br>enon in extrusion, it will be discussed in detail in the following section.<br>
+<b>11.3.2.5Wear Problems<br></b>Wear occurs in all machinery with moving parts. Unfortunately, extruders are no <br>exception. Wear problems in extrusion can take many shapes and forms. A good <br>general text on wear in polymer processing is the book by Mennig [80].<br>Wear in extruders generally causes an increase in the clearance between screw <br>flight and barrel. Wear often occurs towards the end of the compression section. <br>This type of wear is more likely to occur when the screw has a high compression <br>ratio. Wear in the compression section of this type of screw reduces the melting <br>capacity and will lead to temperature non-uniformities and pressure fluctuations. <br>Wear in the metering section of the screw will reduce the pumping capacity; how-<br>ever, the reduction in pumping capacity is generally quite small as long as the wear <br>does not exceed two to three times the design clearance. An increased flight clear-<br>ance will also reduce the effectiveness of the heat transfer from the barrel to the <br>polymer melt and vice versa; this may contribute to temperature non-uniformities in <br>the polymer melt.<br>Wear can only be detected by disassembling the extruder and by inspection of the <br>screw and barrel. If the wear is serious enough to affect the extruder performance, <br>it will often be noticeable with the naked eye. However, it is recommended to meas-<br>ure the I.D. of the barrel and the O.D. of the screw over the length of the machine. If <br>this is done regularly, then it is easy to determine how fast wear is progressing with <br>time. By extrapolating to the maximum allowable wear, a determination can be <br>made at what point in time the screw and/or barrel should be replaced or rebuilt. If <br>replacement resulting from wear is necessary after several years of operation, the <br>easiest solution is to simply replace the worn parts. However, if replacement result-<br>ing from wear becomes necessary within a short period of time, for instance several <br>months, then simple replacement will not provide an acceptable solution. In short-<br>term wear problems, the cost of downtime and replacement parts can easily become <br>unacceptable, and the solution has to be found in reducing the actual wear rate <br>instead of simply replacing the worn parts. To reduce the wear rate, one has to <br>
+<hr>
+<A name=792></a> <br>
+11.3Systematic Troubleshooting  <b>781</b><br>
+understand the wear mechanism(s) in order to determine the most effective way to <br>reduce wear.<br>
+<b>11.3.2.5.1Wear Mechanisms</b><br>
+Five mechanisms of wear can be distinguished:<br>1.  Adhesive wear<br>2.  Abrasive wear<br>3.  Laminar wear<br>4.  Surface-fatigue wear<br>5.  Corrosive wear<br>When wear occurs, often more than one mechanism is at work. Adhesive wear <br>occurs with metal-to-metal contact under high stresses. Since the actual contact <br>area is much smaller than the apparent contact area, local welds can form at points <br>of contact. This phenomenon is often referred to as cold welding. The sliding motion <br>causes a rupture in the weld region, and small fragments of the weld region are car-<br>ried away with one member of the sliding system. Usually fragments of the softer <br>material transfer to the harder material. Adhesive junctions are only formed between <br>clean surfaces. The attrition rate depends on the shear strength of the adhesive <br>junctions. Adhesive wear is generally more severe with sliding contact of similar <br>metals. Adhesive wear between similar metals is often referred to as galling. In slid-<br>ing motion between dissimilar metals, the adhesive junction will contain a spectrum <br>of compositions. Adhesive wear can be significantly reduced when the spectrum of <br>compositions in a junction contains brittle intermetallic compounds that fracture <br>easily. Lubricants are often used to reduce the chance of adhesive wear. When oxide <br>layers form at the interface, this will also reduce adhesive wear because oxides will <br>not bond.<br>Abrasive wear occurs by a micro-cutting process. In two-body adhesive wear, the <br>asperities of the harder member penetrate the softer one and remove material as a <br>result of the sliding motion. In three-body abrasive wear, hard particles are embed-<br>ded in the material of at least one member of the wear system. The hardness ratio <br>has been found to be the most important material characteristic in abrasive wear, <br>although the influence of fracture toughness seems to play a role [1]. Krushchov [2] <br>found that the wear resistance of pure metals and annealed steels increases propor-<br>tionally with hardness. Strain hardening or precipitation hardening does not result <br>in improved abrasive wear resistance because the micro-cutting process already <br>yields maximum local strain hardening.<br>Laminar wear occurs when the shear strength in the heterogeneous interfacial layer <br>is higher than the shear strength of the homogeneous portion of the interfacial <br>layer. Laminar wear takes place only at the thin outer layers of the interface. Lami-<br>nar wear is sustained only if the outer layer of the heterogeneous interface continu-<br>
+<hr>
+<A name=793></a><b>782</b>  <b> 11Troubleshooting Extruders</b><br>
+ously regenerates as an oxidic or other reactive layer. The formation of wear-reduc-<br>ing reactive layers can be controlled by additives in the lubricant. Laminar wear, to <br>a certain extent, is a mild form of corrosive wear. A mild corrosive or oxidative <br>action affects only a thin layer of the newly generated metallic surface. When the <br>new surface layer is formed, the reaction will stop.<br>In surface fatigue wear, there is a separation of microscopic and macroscopic mate-<br>rial particles from the surface, which is caused by fatigue crazing, cracking, and <br>breakup under specific mechanical, thermal, and chemical loads in rolling contact <br>between two surfaces. Fatigue cracking is initiated by alternating thermal or me -<br>chanical loads. This type of wear can occur even with direct metallic contact. Surface <br>fatigue wear is characterized by considerable induction times and relatively large <br>depth of penetration. A familiar example is the pitting of roller bearings and gears.<br>In corrosive wear, a chemical reaction attacks at least one of the sliding surfaces. <br>Corrosive wear in extrusion occurs usually in combination with one or more of the <br>other wear mechanisms. The combined chemical and mechanical attack of the slid-<br>ing surface can cause wear rates far in excess of what would be expected based on <br>their individual contribution. In extrusion, the most important wear mechanisms <br>are adhesive, abrasive, and corrosive wear.<br>
+<b>11.3.2.5.2Test Methods for Wear</b><br>
+There are basically two ways to test the wear characteristics in the extrusion pro-<br>cess. One method is to run the actual machine under normal operating conditions <br>and to measure the progress of wear at regular intervals. This approach is time con-<br>suming and expensive, but it does yield accurate and representative results. How-<br>ever, it does not allow a simple analysis of the parameters that influence the wear <br>process.<br>An interesting technique to do wear studies relatively fast on actual extruders was <br>developed at the IKT in Stuttgart, Germany, by Fritz and coworkers [31]. A reference <br>surface of the machine is made radioactive by proton and neutron bombardment to <br>a depth of 30 to 80 m. The impulse rate from the measuring isotope reduces line-<br>arly with activation depth. This allows accurate measurement of wear over short <br>time periods. It was found that the wear process could be accurately characterized <br>in about one to three hours. In the particular study mentioned [31], wear was meas-<br>ured in the feed section of a screw; the extruder was equipped with a grooved barrel <br>section. The abrasive filler was titanium dioxide, which was added to the virgin poly-<br>mer as a masterbatch. When the virgin polymer was in pellet form, considerable <br>wear occurred, while no wear was measured when the virgin polymer was in pow-<br>der form.<br>Another method involves testing on model systems. In such a test, a test specimen is <br>subjected to certain load conditions to simulate actual service conditions. Such wear <br>
+<hr>
+<A name=794></a><IMG src="Polymer Extrusion by Chris Rauwendaal-794_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-794_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-794_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-794_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-794_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-794_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-794_7.jpg"><br>
+ <br>
+11.3Systematic Troubleshooting  <b>783</b><br>
+testers allow the tribological relevant loads to be pre-selected; tribological parame-<br>ters such as temperature, coefficient of friction, etc., can be measured and recorded <br>continuously. This method allows a relatively quick and inexpensive determination <br>of wear characteristics. However, information obtained from a wear tester can only <br>be transferred to practice if the wear conditions in the model system are essentially <br>the same as those in the real system, i.e., the extruder. Many mistakes have been <br>made in transferring information from a short wear test to actual extruders, simply <br>because of the differences in the tribological conditions of the wear process. It is <br>often not realized that the tribological parameters, friction and wear, are not mate-<br>rial properties, but properties of a complex system. Thus, the transfer from a model <br>system to an actual extruder has to be made cautiously. Measurements on the actual <br>extruder are required to ensure that results from the model system are also valid for <br>the real system. A good review and analysis of test methods for wear in the polymer <br>processing industry was given by Mennig and Volz [3]. Considering that there are <br>many different types of wear in polymer processing, there is, unfortunately, no uni-<br>versal wear tester. Mennig and Volz distinguish four types of testing: metal-liquid <br>wear, metal-solid polymer wear, corrosive wear, and metal-to-metal wear.<br>As early as 1944, a test device was proposed by Mehdorn [4] to measure metal- <br>liquid wear. This test was to simulate wear conditions in a press used for injection of <br>thermosets. The test geometry is shown in Fig. 11.4.<br>
+In<br>
+Out<br>
+Spacer<br>
+Sample<br>
+<b> Figure 11.4</b> <br>
+Wear test device proposed by Mehdorn<br>
+A molten or liquid mass of polymer is forced onto a test specimen, from which it is <br>deflected; the material exits through a small clearance of 0.4 mm. This test device <br>gives relatively quick results. Disadvantages are the complex geometry of the clear-<br>ance, non-uniform flow conditions at the specimen, and that increased wear changes <br>the resistance to flow and thus the wear conditions. Another method was developed <br>by Bauer, Eichler, and John [5] in 1967. Figure 11.5 shows the geometry of their test <br>apparatus.<br>
+<hr>
+<A name=795></a><IMG src="Polymer Extrusion by Chris Rauwendaal-795_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-795_10.jpg"><br>
+<b>784</b>  <b> 11Troubleshooting Extruders</b><br>
+Sample<br>
+<b> Figure  11.5</b> <br>
+Wear test proposed by Bauer et al .<br>
+The specimen is a diamond-shaped obstruction in the center of a flow channel. A <br>commercial wear test apparatus based on this geometry is the Tribotest from Bra-<br>bender OHG in Duisburg, Germany. This test is often referred to as the "Siemens-<br>Method" wear test. Eichler and Frank [6] modified this test to make it more suitable <br>for injection molding. Entirely different test geometry was developed at the DKI <br>(Deutsches Kunststoff Institut, Darmstadt). This test utilized a flat plate geometry as <br>shown in Fig. 11.6.<br>
+<b> Figure 11.6</b> <br>
+Sample<br>
+DKI flat plate wear test apparatus<br>
+The rectangular test gap has a length of 12 mm, a width of 10 mm, and a height that <br>is adjustable from 0.1 to 1.0 mm. This geometry has been used for studies with ther-<br>moplastics [7] as well as with thermosets [8]. A modification of the flat plate wear <br>tester is the BASF wear tester. This test simulates the wear process in a molding <br>machine. Another test apparatus developed at the DKI is the ring method, shown in <br>Fig. 11.7.<br>This test simulates conditions occurring in the annular space between the tip of <br>the screw flight and the extruder barrel [7, 9]. Plumb and Glaeser [10] developed a <br>test method for filled elastomers based on a capillary rheometer. The specimen in <br>this test is a cone-shaped torpedo in the flow channel. The flow conditions change in <br>
+<hr>
+<A name=796></a><IMG src="Polymer Extrusion by Chris Rauwendaal-796_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_23.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_24.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_25.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_26.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_27.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_28.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_29.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_30.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_31.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_32.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_33.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_34.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_35.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_36.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_37.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_38.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_39.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_40.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_41.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_42.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_43.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_44.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_45.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_46.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_47.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_48.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_49.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_50.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_51.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_52.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_53.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_54.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_55.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-796_56.jpg"><br>
+ <br>
+11.3Systematic Troubleshooting  <b>785</b><br>
+axial direction and with it, the local wear rate. A test developed at Georgia Marble <br>Company, a supplier of calcium carbonate, uses an aluminum breaker plate at the <br>end of a screw extruder to evaluate abrasive wear of mineral fillers. The amount of <br>wear is determined by measuring the weight loss of the breaker plate over a certain <br>run time [25].<br>
+Force<br>
+Melt out<br>
+Inner ring Outer ring<br>
+Melt in<br>
+<b>Figure 11.7</b>DKI ring wear test apparatus<br>
+Metal-to-solid polymer wear occurs in the solids conveying zone of the extruder. The <br>introduction of the grooved barrel extruder has significantly increased the interest <br>and concern about wear in this portion of the extruder. Grooved barrel sections <br> <br>substantially increase the shear and normal stresses between the polymer solid bed <br>and the metal surfaces. As a result, grooved barrel sections are much more suscep-<br>tible to wear than smooth barrels. The first systematic study of wear in the solids <br>conveying section of extruders was made by Fritz [11]. He used a diamond-shaped <br>specimen that protruded into the screw channel.<br>A model system was developed at the DKI in the form of a disk wear tester. This <br>universal disk-tribometer was discussed by Volz [12]. The concept of the disk-tribo-<br>meter is partially based on a modified friction tester developed at Enka Glanzstoff <br>[13]. The geometry of the disk-tribometer is schematically shown in Fig. 11.8.<br>
+Sample<br>
+Disk can be heated<br>
+Polymer<br>
+and cooled<br>
+<b> Figure 11.8</b> <br>
+Universal disk tribometer<br>
+<hr>
+<A name=797></a><IMG src="Polymer Extrusion by Chris Rauwendaal-797_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-797_13.jpg"><br>
+<b>786</b>  <b> 11Troubleshooting Extruders</b><br>
+The disk can be heated and cooled. The annular groove at the bottom surface of <br>the disk contains the metal specimen. The disk-tribometer allows measurement of <br>frictional force, normal force, and wear at contact pressures of up to 150 MPa. A test <br>for corrosive wear was first proposed by Calloway, Morrison, and Williams [14]. <br>They used a vacuum pyrolysis at normal process temperature.<br>A metal specimen is suspended in the extracted volatiles for 24 hours at room tem-<br>perature. A similar apparatus was used by Mahler [7] and Braun and Maelhammar <br>[15] in order to test for corrosion at elevated temperatures and pressures. A disad-<br>vantage of vacuum pyrolysis is the fact that the corrosion conditions are different <br>from those existing in the actual extrusion process. This drawback is reduced in the <br>test procedure used by Knappe and Mahler [9].<br>Other tests have been described by Moslé et al. [16] and Maelhammar [17]. The lat-<br>ter test is a combination of metal-liquid wear and corrosive wear. The apparatus <br>was modified by Volz [18] for thermosets. The volatiles are extracted from the poly-<br>mer melt, which has been prepared in an injection molding machine and has been <br>sheared through a test gap. Through electrochemical measurements, Volz could <br>prove that significant differences exist in the corrosive action of volatiles separated <br>from injection molded samples of thermosetting polymers.<br>Tests for metal-to-metal wear can utilize the standard test methods, provided the <br>proper intermediate material can be introduced between the metallic surfaces. Bros-<br>zeit utilized the cylinder-disk apparatus to study metal-to-metal wear [19]. Saltz-<br>man, et al. [20­22] used the Alpha LFW-1 test apparatus; see Fig. 11.9.<br>
+FN=136 kp<br>
+Swing arm sample holder<br>
+Stationary sample<br>
+Rotating sample<br>
+Liquid<br>
+<b> Figure 11.9</b> <br>
+Alpha LFW-1 wear test apparatus<br>
+A stationary block is forced against a rotating ring by a dead-weight load. The bot-<br>tom part of the ring is immersed in a water-oil emulsion. The presence of the water-<br>oil emulsion is a drawback in this test because the actual wear behavior in an ex -<br>truder with a polymer melt as the intermediate material between screw and barrel <br>is bound to be substantially different from the wear behavior in the LFW-1 test appa-<br>
+<hr>
+<A name=798></a> <br>
+11.3Systematic Troubleshooting  <b>787</b><br>
+ratus. No data have been published on metal-to-metal wear with a polymer melt as <br>the intermediate material.<br>
+<b>11.3.2.5.3Causes of Wear</b><br>
+In polymer-metal wear, the main causes of wear are abrasive wear and corrosive <br>wear. Abrasive wear is generally due to abrasive components in the polymer matrix. <br>Whether these components are classified as fillers, reinforcements, or additives, <br>they can cause significant wear when the filler is hard and available in significant <br>amounts. Factors affecting the wear are particle hardness, particle size, particle <br>shape, and loading [23]. One indication of the abrasive wear ability of a filler is its <br>ranking on the Mohs scale. This scale ranks a material from 1 to 10 according to its <br>ability to scratch another material or to be scratched by another material. A very soft <br>material, such as talc, is ranked at the bottom of the scale, rank 1, and a very hard <br>material, such as diamond, is ranked at the top of the scale, rank 10. The Mohs scale <br>ranking of several fillers is given in Table 11.4.<br>
+<b>Table 11.4</b>Mohs Scale Ranking of Various Fillers<br>
+Calcined Kaolin<br>
+7<br>
+Silica<br>
+6.5<br>
+Glass<br>
+6<br>
+Perlite<br>
+5.5<br>
+Wollastonite<br>
+5.5<br>
+Mica<br>
+3<br>
+Calcium Carbonate<br>
+3<br>
+Kaolin<br>
+2<br>
+Alumina trihydrate<br>
+1<br>
+Talc<br>
+1<br>
+It should be realized, however, that the filler hardness only partially determines the <br>wear characteristics of a filled compound. The particle size is another important <br>parameter. Generally, the severity of wear reduces with reducing particle size. In <br>glass fiber reinforced compounds, the wear reduces when the fiber length reduces <br>[12]. This is believed to be due to greater mobility and reduced kinetic energy of the <br>smaller glass fibers. The particle shape has a very strong influence on the wear <br>characteristics. Experiments done by Mahler [26] with glass fiber reinforced nylon <br>and glass bead reinforced nylon showed the wearing intensity of the fiber reinforced <br>compound to be 14 times higher than the compound reinforced with glass spheres. <br>The ability to wear is larger when the particle has sharp corners and a large aspect <br>ratio. The particle shape that best minimizes wear is a spherical shape. Unfortu-<br>nately, the spherical shape is often undesirable with respect to mechanical proper-<br>ties, electrical properties, etc.<br>
+<hr>
+<A name=799></a><b>788</b>  <b> 11Troubleshooting Extruders</b><br>
+In a study on glass-reinforced polymers, Mahler [7] found that with some polymers <br>the wear intensity could be much higher than with others because of corrosive wear <br>in addition to abrasive wear. He found that the wear intensity of glass fiber re inforced <br>nylon-6,6 against 9S20K steel was about 13 times higher than reinforced SAN and <br>PC. A similar finding was made by Olmsted [27] in injection molding of glass fiber <br>reinforced nylon-6, where severe wear occurred on both the screw and the barrel. <br>The wear was found to be primarily corrosive-type wear, caused by a silane wetting <br>agent on the fiber. The decomposition temperature of the wetting agent was lower <br>than the process temperature and, hence, degradation occurred with resulting cor-<br>rosive attack of screw and barrel. However, similar work by Mahler [7] with nylon-<br>6,6 filled with glass fibers coated with an aminosilane coupling agent did not reveal <br>any corrosive wear resulting from the coupling agent.<br>Surface treatment of fillers can reduce wear. A comparison between coated and <br>un coated calcium carbonate showed the wear intensity of a rigid PVC compound <br>with the coated filler to be about 3 to 9 times lower than the same compound with <br>the uncoated filler [25]. The wear intensity was measured by using the aluminum <br>breaker plate test discussed earlier. When incorporating abrasive fillers, such as <br>glass fibers, it is good practice to add the filler at a location where the polymer is <br>already molten. This practice allows the melt to coat the filler and reduce the wear <br>intensity. When abrasive fillers are added with solid polymer particles, the abrasive <br>action is much more severe and rapid wear will occur in the solids conveying sec-<br>tion of the extruder. This is why glass fiber is generally added in a downstream bar-<br>rel opening or in a downstream extruder in the case of a tandem extrusion set-up.<br>Corrosive wear also occurs in non-filled polymers; well-known examples are fluoro-<br>polymers and chlorine containing polymers, such as HPFA and PVC. Fluoropolymers <br>have a tendency to form hydrofluoric acid at high temperatures in combination with <br>air and moisture. PVC tends to generate hydrochloric acids at elevated temperatures. <br>The corrosion problem is generally more severe with rigid PVC than with flexible <br>PVC. With polymers like these, the metal parts in contact with the polymer should be <br>made out of a corrosion-resistant metal, such as Hastelloy, 17­4 PH, 15­5 PH, etc.<br>Corrosion can also occur with hygroscopic polymers, such as ABS, PA, PET, PMMA, <br>etc., when moisture is released under high temperature and pressure, forming high-<br>pressure steam. Braun and Maelhammer [15] found that PA 6,6 splits into various <br>corrosive components. Calloway et al. [14] found that the corrosive attack of HIPS is <br>dependent on the chlorine and sulfur content of the carbon blacks. Moslé et al. [16] <br>found that degradation products of ABS can cause corrosive attack in extruders.<br>Sometimes abrasive components in the compound are foreign objects resulting from <br>contamination or human error. Hard foreign objects, such as wrenches, bolts, knives, <br>etc., can cause severe wear in a very short period of time. Magnetic traps are avail-<br>able to catch metallic objects. However, there is not a simple method to successfully <br>remove all foreign objects from the feed stock. Good housekeeping procedures and <br>
+<hr>
+<A name=800></a> <br>
+11.3Systematic Troubleshooting  <b>789</b><br>
+conscientious personnel will go a long way in reducing the chance of foreign objects <br>ending up in an extruder. One word of caution is in order for barrier-type screws. <br>If small particles are present in the polymer feedstock that do not melt at operating <br>temperatures, they will get trapped at the end of the barrier section when the par-<br>ticle size is larger than the barrier clearance. This can result in very rapid wear, as <br>the author has been able to verify personally.<br>Another important cause of wear in extrusion equipment is metal-to-metal wear. <br>Unfortunately, relatively little is known about this type of wear. A number of circum-<br>stances can cause metal-to-metal contact between screw and barrel. In addition, <br>metal-to-metal contact will occur at start-up. It can also occur as a result of misalign-<br>ment, or a warped barrel, or a warped screw. Metal-to-metal contact can occur at the <br>feed opening of an extruder as described by Luelsdorf [28], particularly when the <br>feed opening is offset and when it forms a sharply tapered angle with the circum-<br>ference of the screw.<br>The author has also experienced cases where metal-to-metal wear occurred in the <br>feed throat of an extruder as a result of improper feed opening geometry. Radio-<br>graphic analysis of wear particles indicate [28] that temperatures in the contact <br>zone between screw and barrel may exceed 800°C. This is confirmed by personal <br>observations of a screw subjected to severe metal-to-metal wear with a very notice-<br>able purple discoloration of the metal in the wear region. Discoloration indicates <br>exposure to temperatures over 500°C.<br>Metal-to-metal wear can also occur in intermeshing twin screw extruders. Counter-<br>rotating twin screw extruders are more susceptible to metal-to-metal wear than <br> <br>co-rotating twin screw extruders. Counter-rotating twin screw extruders, therefore, <br>generally operate at rather low screw speeds. Unfortunately, co-rotating extruders <br>also can experience serious metal-to-metal wear problems.<br>Lai Fook and Worth [29] proposed modified flight geometries to increase the center-<br>ing force on the screw in order to reduce metal-to-metal contact. The two flight <br>geometries they proposed based on theoretical calculations are shown in Fig. 11.10.<br>Actual measurements of the tangential pressure profile differed from predicted val-<br>ues by a factor of about 10. This indicates that the analysis employed was not entirely <br>realistic; in particular, neglecting side leakage results in large errors in the pre-<br>dicted values. No data was presented on the actual centering force acting on the <br>screw. Winter [30] analyzed the non-isothermal flow of a power law fluid in the <br>flight clearance; solutions were obtained by employing a numerical procedure. Very <br>high temperature increases were found to occur in the polymer melt in the clear-<br>ance; these temperature changes affect the velocity profile. Since the mass flow rate <br>cannot change, Winter adjusted the pressure gradient along the gap to satisfy the <br>continuity equation. This led to a prediction of large negative pressure gradients at <br>the leading edge of the flight and large positive pressure gradients at the trailing <br>edge of the flight, as shown qualitatively in Fig. 11.11.<br>
+<hr>
+<A name=801></a><b>790</b>  <b> 11Troubleshooting Extruders</b><br>
+<b> Figure 11.10</b> <br>
+Flight geometry to reduce the chance <br>
+of metal-to-metal wear<br>
+<b> Figure 11.11</b> <br>
+Pressure profile in the flight clearance as <br>
+ <br>predicted by Winter [30]<br>
+Extreme values of the actual pressure occur at about 1/3 from the leading edge, and <br>the calculated values range from 5 to 20 MPa.<br>The predicted pressure profile is obviously a direct result of the assumptions made <br>in the calculations. Winter assumed isothermal conditions at the barrel wall and <br>adiabatic conditions at the flight tip. With stock temperature increases in the order <br>of 100°C and more, it is unlikely that the isothermal boundary condition is valid <br>for the barrel. For the same reason, it is unlikely that the adiabatic boundary condi-<br>tion is valid for the flight tip, particularly since the rest of the screw will be at much <br>lower temperature. Unfortunately, it is difficult to measure actual temperature and <br>pressure profiles. Thus, the predicted temperature and pressure profiles have not <br>been compared to experimental results.<br>Winter postulated that the pressure minimum in the middle of the clearance could <br>cause the screw to be pushed against the barrel by pressure on the other side of the <br>screw. This would only be true if the clearance pressure profile changed along the <br>helical length of the screw flight and if the pressure profiles in the screw channel <br>itself do not play a role of significance. Most likely, the actual situation will be con-<br>siderably more complex. Following Lai Fook and Worth, Winter recommended be -<br>
+<hr>
+<A name=802></a> <br>
+11.3Systematic Troubleshooting  <b>791</b><br>
+veling the flight tip to create a tapered gap between flight and barrel. Another re- <br>commendation was to alter the thermal boundary conditions, for instance by heating <br>the barrel above the temperature of the screw flight. It is interesting to note that the <br>chance of metal-to-metal contact is reduced when the flight clearance and helix <br>angle are increased and when the flight width is decreased, according to Winter's <br>analysis. If this is true, then these measures will have a dual benefit because they <br>will also reduce the power consumption. Unfortunately, actual experience contra-<br>dicts Winter's recommendations.<br>A disadvantage of the stepped or beveled flight geometry is that when the extruder <br>is running empty or partially empty, the apparent contact area between screw and <br>barrel will be considerably reduced. As a result, the stresses at the actual contact <br>area will be considerably increased and wear, particularly adhesive wear, is more <br>likely to occur. Nonetheless, Volz [12] reports that flight lands with hydrodynamic <br>slide bearing function are used by a number of extruder manufacturers. However, <br>it does not appear that this flight geometry is widely used in the industry.<br>Metal-to-metal contact between screw and barrel will occur when the extruder is <br>empty because of gravitational sag of the screw. This type of wear will reach a maxi-<br>mum at the very end of the screw and reduce in the upstream direction. In practice, <br>however, the location of maximum wear generally occurs towards the end of the <br>compression section of the screw. This indicates that the screw is supported by the <br>polymer melt at the end of the screw and is subjected to a substantial lateral force in <br>the compression section of the screw. It is unlikely that this lateral force is caused <br>by temperature-induced pressure gradients in the flight clearance because the flow <br>process in the flight clearance will not change significantly from the start of melting <br>to the very end of the screw. Therefore, it seems more likely that the lateral force is <br>caused by the conveying process in the screw channel.<br>In the melting zone of the extruder, there is a continuous deformation of the solid <br>bed. In the compression section of the screw, the solid bed is compressed between <br>the root of the screw and the barrel. Very large pressure can be built up in this <br>screw section, particularly when the compression ratio is high. These rapid pres-<br>sure changes along the screw can easily cause an imbalance of the lateral forces <br>acting on the screw. Thus, it seems that rapid pressure changes along the screw <br>channel are the most likely cause of metal-to-metal contact between screw and bar-<br>rel. Considering the frequent occurrence of this wear problem, it is surprising how <br>little attention this problem has received as judged from the open technical litera-<br>ture.<br>
+<b>11.3.2.5.4Solutions to Wear Problems</b><br>
+The key to find the best solution to a wear problem is to identify the cause of wear <br>and the wear mechanism. For example, if the screw O.D. is worn but not the flight <br>flanks and the root diameter, then the problem is probably caused by contact be -<br>
+<hr>
+<A name=803></a><b>792</b>  <b> 11Troubleshooting Extruders</b><br>
+tween screw and barrel. In this case, one could put hardfacing on the tip of the <br>flight. However, this would not eliminate the cause of the problem, but it may reduce <br>the magnitude of the problem. The actual problem can be eliminated by a change in <br>screw design or by another method that will alter the pressure profile along the <br>screw, e.g., by starve feeding the extruder.<br>Corrosive wear can usually be identified by a pitted, worn surface. The best solution <br>to corrosive wear is to eliminate the corrosive component from the compound. How-<br>ever, this is often not possible for other reasons. In this case, one has to use corro-<br>sion-resistant materials of construction, such as stainless steel, Inconel, Hastelloy, <br>etc. In order to select the best material of construction, one should know the chemi-<br>cal species that are causing the corrosive attack. Various metal handbooks contain <br>information about the chemical resistance of many metals against a number of <br>chemical species. Figure 11.12 shows a flow chart that allows systematic trouble-<br>shooting of wear problems.<br>
+Signs of wear problems:<br>* metal particles in extruded product<br>* metal particles on screen pack<br>
+<b>Wear problem</b><br>
+* unusual noises from extruder<br>* high motor load<br>* high temperatures<br>
+Remove corrosive substance<br>
+Yes<br>
+Replace worn parts<br>
+Use corrosion resistant metals<br>
+Long term?<br>
+Yes<br>
+No<br>
+Wear resistant surfaces<br>
+No<br>
+Yes<br>
+Corrosive wear?<br>
+Abrasive wear?<br>
+Change sequence of addition<br>Check pressure gradients<br>
+Yes<br>
+Soft screw surface<br>
+Improper cleaning method,<br>
+Yes<br>
+Yes<br>
+Misalignment<br>
+Wear located at<br>
+e.g. hard metal brushes<br>
+In the feed section?<br>
+Metal particles in feed<br>
+tip of screw flight?<br>
+Improper purging compound<br>
+Insufficient clearance<br>
+No<br>
+No<br>
+No<br>
+Yes<br>
+Compression ratio too high<br>
+Compression length too short<br>
+Yes<br>
+Compression section?<br>
+Abrasive fillers in<br>
+Wear at screw root<br>
+Incorrect screw/barrel material<br>
+the compound?<br>
+and flight flanks?<br>
+Warped screw or barrel<br>
+No<br>
+Yes<br>
+Yes<br>
+No<br>
+Metering section?<br>
+Misalignment<br>
+Screw run dry too long<br>
+Use less abrasive filler<br>
+Incorrect screw/barrel material<br>
+Coat filler<br>
+Yes<br>
+Add filler downstream<br>
+Wear at shank?<br>
+Incorrect dimensions<br>
+Apply wear resistant coating<br>
+Debris in shank region<br>
+to root and flight flanks<br>
+No<br>
+Incorrect barrel liner material<br>
+Yes<br>
+Wear at barrel but<br>
+No wear resistant barrel liner<br>
+not at screw?<br>
+Nitrided barrel worn through<br>
+very thin hardened layer<br>
+<b>Figure 11.12</b>Flow chart for troubleshooting wear problems<br>
+A large number of materials are available for the screw and barrel. Most extruder <br>barrels in the U.S. have a liner, which is centrifugally cast into the barrel. The barrel <br>liner is made of a wear-resistant material, often boron-stabilized white irons with a <br>Rockwell C hardness of about 65 containing iron chromium boron carbides. Bimetal-<br>lic barrels provide better wear resistance than nitrided barrels as reported, for <br>
+<hr>
+<A name=804></a> <br>
+11.3Systematic Troubleshooting  <b>793</b><br>
+instance, by O'Brien [32] and Thursfield [33]. The liner material can be formulated <br>to give good abrasion resistance, good corrosion resistance, or a combined abrasion <br>and corrosion resistance. It should be kept in mind that the correct choice of screw <br>material depends to some extent on the liner material, particularly if metal-to-metal <br>contact takes place. Recommended screw materials for several commercial barrel <br>liners are shown in Table 11.5. The recommendations are based on metal-to-metal <br>wear tests on the Alpha LFW-1 wear test machine. As discussed earlier, this test <br>does not simulate actual conditions in an extruder too well; however, results from <br>more realistic tests are not available in the open literature.<br>
+<b>Table 11.5</b>Recommended Screw Flight Materials, Courtesy of [22]<br>
+Barrel liner material<br>Xaloy 101<br>
+Xaloy 306<br>
+Xaloy 800<br>
+Screw material<br>Colmonoy 56<br>
+Colmonoy 6<br>
+Colmonoy 6<br>
+Colmonoy 6<br>
+Colmonoy 56<br>
+Colmonoy 56<br>
+Haynes 711<br>
+Colmonoy 63<br>
+Xaloy 008<br>
+Colmonoy 5<br>
+Stellite 1<br>
+Nye-Carb<br>
+Colmonoy 63<br>
+Nye-Carb<br>
+Xaloy 830<br>
+Stellite 1<br>
+Stellite 6<br>
+Ferro-Tic<br>
+Ferro-Tic (Iron)<br>
+Stellite 6H (severe wear)<br>
+Stellite 6H (severe wear)<br>
+HC-250<br>Colmonoy 84<br>Triballoy T-700<br>
+The most common screw material is 4140 steel. Advantages of 4140 steel are low <br>price, good machinability, and the ability to be used with hardfacing and chrome <br>plating. A disadvantage of the 4140 steel is its relatively poor wear resistance, as <br>discussed, for instance, by Hoffmann [34]. As a result, 4140 screws are often flame-<br>hardened, plated, or hardfaced when used in more demanding applications. Chrome <br>plating is often used on extruder screws. This produces a hard corrosion-resistant <br>layer, up to 70 Rc hardness. The layer is usually quite thin, about 25 to 75 m, and <br>
+does not form a hermetic seal to most corrosive substances. Thus, the chrome plat-<br>ing does not substantially improve the corrosion resistance of the screw due to its <br>porosity. Chrome plating is quite resistant to abrasive wear, but because of its lim-<br>ited thickness, it does not provide much protection.<br>Nickel plating is also used quite frequently on extruder screws. It is applied with a <br>thickness similar to chrome plating. The surface hardness of nickel plating tends to <br>be somewhat lower than chrome plating. However, the thickness of the nickel coating <br>is generally more uniform and the coating is much less porous than chrome plating. <br>Therefore, nickel plating usually offers better protection than chrome plating.<br>
+<hr>
+<A name=805></a><b>794</b>  <b> 11Troubleshooting Extruders</b><br>
+A very large number of proprietary plating materials and plating processes are <br>available today. Without exception, the claims made by the supplier of the proprie-<br>tary plating are quite impressive. Unfortunately, these claims are rarely based on <br>long-term extrusion tests. Thus, one has to be quite cautious in selecting a proprie-<br>tary plating.<br>An interesting proprietary plating is the Poly-Ond plating. It is basically an electro-<br>less nickel plating impregnated with a fluoropolymer. This yields a moderately hard, <br>corrosion-resistant layer with a low coefficient of friction. The low coefficient of <br> <br>friction makes it attractive to use in injection molds and on extruder screws. Luker <br>from Killion Extruders reported [35] on tests with Poly-Ond plated extruder screws. <br>He reported output increases from 5 to 36% for a number of different polymers. <br>Another Teflon-impregnated nickel plating is Nedox plating as discussed by Levy <br>[74]. A Teflon-impregnated chrome plating was discussed by Tropler [75].<br>Metal coatings such as chrome and nickel plating are generally more effective in <br>reducing wear than hardening of the base material. Various methods are used for <br>hardening, such as flame hardening, nitriding, carburizing, and induction harden-<br>ing. All of these techniques are basically case-hardening processes with limited <br>depth of penetration and limited improvement in wear characteristics. Further, heat-<br>treated steels have reduced hardness and wear resistance at elevated temperatures.<br>
+<b>Hardfacing Materials<br></b>Another technique used for improving the wear resistance of screws and barrels is <br>hardfacing or hardsurfacing. Hardfacing materials are generally nickel or cobalt <br>based containing various metal carbides, such as chromium carbide, tungsten car-<br>bide, etc. The most common alloys are Stellite and Colmonoy, but many other mate-<br>rials are available today. Hardfacing materials are applied by welding, spraying, or <br>casting with a thickness ranging from 1 to 3 mm. There are several steps involved <br>in the hardfacing process as shown in Fig. 11.13. On a worn screw the flights are <br>ground to a uniform undersize, the hardfacing material is welded onto the flights, <br>the hardfacing is ground to the original O.D., and the sides of the flights are <br>machined to provide a smooth transition from the flight flank to the hardfacing <br>material.<br>A critical step in the application of the hardfacing is the preheating of the screw <br>before application and, more importantly, the gradual cooling of the screw after the <br>hardfacing is applied. If the screw cools too rapidly, thermal stresses will develop in <br>the hardfacing and cracks will form. Considering that hardfacing materials are hard <br>and brittle it is quite easy for small cracks to form. In fact, for many hardfacing <br>materials small hairline cracks cannot be completely avoided. In general, the higher <br>the carbon content of the hardfacing material, the greater its tendency to crack and <br>the better its wear characteristics.<br>
+<hr>
+<A name=806></a> <br>
+11.3Systematic Troubleshooting  <b>795</b><br>
+<b>Figure 11.13</b>Steps involved in applying hardfacing to a worn screw<br>
+Screws using the high carbon, highly wear resistant hard facing materials generally <br>will show some degree of small hairline cracks. If such hardfacing does not show <br>any hairline cracks, the hardfacing material may be suspect. It is possible that <br>softer, less wear resistant hard facing is actually used. It is also possible that the <br>hardfacing material has been diluted too much with the base screw material; this <br>will result in a softer hardfacing material that will wear rapidly. For this reason it is <br>a good idea to measure the hardness of the hardfacing after the screw is manu-<br>factured. If the hardness is substantially below the normal value, it is likely that the <br>hardfacing is overdiluted or that the wrong hardfacing was applied.<br>Cracks that are cause for concern are those that traverse completely through the <br>hardfacing material and then turn in a circumferential direction. When such cracks <br>occur, parts of the hardfacing may come loose. This may happen when the hard-<br>facing is not properly applied. It may also occur when the screw is not handled <br>properly before installation or if the screw is exposed to excessive stresses during <br>the extrusion process.<br>Most hardfacing materials do not plate well, i.e., a wavy line occurs at the inter-<br>section of the welded material. This problem can be eliminated by the use of an <br>inlay, as shown in Fig. 11.14. Here, two basic hardfacing geometries are shown: one <br>that has the hardfacing applied to the full width of the flight and the other that has <br>an inlay with hardfacing. Inlays can only be applied to new extruder screws.<br>
+<b> Figure 11.14</b> <br>
+Full width and inlay hardfacing<br>
+<hr>
+<A name=807></a><b>796</b>  <b> 11Troubleshooting Extruders</b><br>
+Applying hardfacing to small screws (diameter less than 40 mm) is more difficult <br>than to large screws. For this reason, smaller screws are often manufactured with-<br>out hardfacing. For small screws, it makes economic sense to make the screw out of <br>a wear-resistant tool steel such as D2 or CPM because the weight of the screw is <br>small and the material cost relatively low. For large screws, the material cost is a <br>more important issue because of the large weight of the screw. As a result, for large <br>screws it is more important to use a low-cost steel.<br>Hardfacing can offer substantial improvements in wear resistance over heat treated <br>steels. Lucius reports increases in service life by a factor of 2 to 25 by using hard-<br>facing on a screw used in extrusion of glass fiber reinforced nylon-6,6 [36]. Two <br>common hardfacing materials are Stellite (trademark Cabot Corp.) and Colmonoy <br>(trademark Wall Colmonoy Corp.). Colmonoy is a nickel-based alloy containing chro-<br>mium, iron, boron, and silicon. Stellite is a cobalt-based alloy containing chromium <br>and tungsten. Other hardfacing materials are Haynes 711, HC-250, Triballoy T-700, <br>Ferro-Tic HT6 and M6, Nye-Carb, and Xaloy 008, 830, 101, 306, and 800. Properties <br>of various hardfacing materials were given in Table 3.4. A larger list of hardfacing <br>materials is shown in Table 11.6.<br>
+<b>Table 11.6</b>Properties of Hardfacing Materials<br>
+Product<br>
+Base<br>
+Hardness<br>
+Cracking<br>
+%C<br>
+%Cr<br>
+%W<br>
+%B<br>
+Cost/lb<br>
+material<br>
+Rc<br>
+tendency<br>
+[$]<br>
+Stellite 1<br>
+Cobalt<br>
+48­54<br>
+High<br>
+2.5<br>
+30.0<br>
+12<br>
+-<br>
+25­40<br>
+Stellite 6<br>
+Cobalt<br>
+37­42<br>
+Medium<br>
+1.1<br>
+28.0<br>
+4<br>
+-<br>
+25­40<br>
+Stellite 12<br>
+Cobalt<br>
+41­47<br>
+Medium<br>
+1.4<br>
+29.0<br>
+8<br>
+-<br>
+25­40<br>
+Colmonoy 5<br>
+Nickel<br>
+45­50<br>
+Medium<br>
+0.65<br>
+11.5<br>
+-<br>
+2.5<br>
+15­25<br>
+Colmonoy 56<br>
+Nickel<br>
+50­55<br>
+High<br>
+0.70<br>
+12.5<br>
+-<br>
+2.7<br>
+15­25<br>
+Colmonoy 6<br>
+Nickel<br>
+56­61<br>
+Very high<br>
+0.75<br>
+13.5<br>
+-<br>
+3.0<br>
+15­25<br>
+Colmonoy 83<br>
+Nickel<br>
+50­55<br>
+High<br>
+2.0<br>
+20.0<br>
+34<br>
+1.0<br>
+40­50<br>
+N-45<br>
+Nickel<br>
+30­40<br>
+Medium<br>
+0.3<br>
+11.0<br>
+-<br>
+2.2<br>
+15­25<br>
+N-50<br>
+Nickel<br>
+40­45<br>
+Medium<br>
+0.4<br>
+12.0<br>
+-<br>
+2.4<br>
+15­25<br>
+N-56<br>
+Nickel<br>
+45­50<br>
+High<br>
+0.6<br>
+13.5<br>
+-<br>
+2.8<br>
+15­25<br>
+Metal-to-metal wear tests of these materials were described by McCandles and <br>Maddy [22]. Welding techniques for applying these materials are tungsten inert gas <br>(TIG), transferred arc plasma, and oxyacetylene.<br>Molybdenum-based hard alloys have also been used on extruder screws. These <br>materials are relatively soft, about 40 Rockwell C, but have good lubricity. In some <br>instances, wear resistance with the molybdenum hardfacing improved about 500% <br>over the more common, but harder, hardfacing compounds. Molybdenum-based <br>hardfacing alloys are used primarily with nitrided barrels, not only in single screw <br>
+<hr>
+<A name=808></a> <br>
+11.3Systematic Troubleshooting  <b>797</b><br>
+extruders but also in twin screw extruders. The use of molybdenum hardfacing in <br>bimetallic barrels often results in rapid wear of the screw. Since most extruders in <br>the U.S. have bimetallic barrels, molybdenum-based hardfacing for extruder screws <br>has not found widespread use.<br>In some cases, improvements in wear resistance can be obtained by diffusion coat-<br>ing hardfacing alloys. Panzera and Saltzman [37] tested treated and untreated hard-<br>facing alloys against carburized SAE 4620 rings using the Alpha Model LFW-1 wear <br>tester. Three case-hardening processes were selected: aluminum diffusion coating, <br>boron diffusion coating, and ion nitriding. It was found that cobalt-based hardfacing <br>alloys exhibited significant improvement in wear resistance against carburized <br>SAE 4620 steel; however, nickel-based hardfacing alloys were unaffected by ion <br>nitriding. Aluminized cobalt-based alloys showed improved wear resistance after a <br>porous outer layer was ground off. Boriding reduces the wear resistance of both <br>nickel and cobalt-based facing alloys. Removing the porous outer layer improved the <br>wear properties of the borided nickel-based alloy.<br>The effect of work hardening of hardfacing alloys was also studied by Panzera and <br>Saltzman. The work hardening was done by shot peening. The cobalt-based alloy <br>hardened to a depth of about 250 m; the nickel-based alloy did not work harden. It <br>was found that work hardening the cobalt-based alloy did not improve its wear <br>resistance against carborized SAE4620 steel as measured on the LFW-1 wear tests. <br>Another process that has been used to surface-harden extruder screws is chemical <br>vapor deposition (CVD). The process and some of its applications have been de -<br>scribed by Bonetti [73]. This process has been used to apply a thin layer, approxi-<br>mately 4 micron, of a very hard titanium nitride coating to the screw surface. Hard-<br>ness values of about 110 Rockwell C can be obtained. Obviously, with such extreme <br>hardness of the screw, be very careful that the barrel material is com patible with <br>the screw material. Rebuilding of extruder screws is covered in Section 8.9.<br>
+<b>11.3.2.6Screw Binding<br></b>There is a special problem that can occur in extruders where the screw suddenly <br>stops rotating and gets stuck in the extruder barrel and/or feed housing. This prob-<br>lem does not occur very often but when it occurs it wreaks havoc with the machine. <br>The most frequent cause of this problem is differential thermal expansion between <br>the screw and barrel. Corrosion-resistant screws commonly used in the extrusion <br>of fluoropolymers are particularly susceptible to screw binding. There are certain <br>characteristics of the corrosion-resistant materials that make these screws suscep-<br>tible to locking up in the extruder. This usually results in considerable damage to <br>the machine and substantial downtime. We will analyze the mechanism of screw <br>binding and give recommendations on how to prevent this problem [147].<br>
+<hr>
+<A name=809></a><b>798</b>  <b> 11Troubleshooting Extruders</b><br>
+<b>11.3.2.6.1Basic Facts</b><br>
+In the extrusion of fluoropolymers, the extruder screw and die generally have to be <br>made out of a highly corrosion-resistant material. Common materials used for this <br>purpose are Hastelloy, Monel, Inconel, and Duranickel. These materials not only <br>have corrosion resistance that is much better than the typical 4140 steel used for <br>most extruder screws, but there are other properties that are quite different from <br>4140 steel as well. Because of these other properties these corrosion-resistant <br>screws are much more susceptible to getting stuck in an extruder barrel than screws <br>made of 4140 steel. Since screw binding usually results in considerable damage <br>to the extruder with associated downtime and cost, not to mention aggravation, it <br>is important for processors to be aware of the pitfalls of using screws made out of <br>highly corrosion-resistant materials.<br>
+<b>11.3.2.6.2The Mechanics of Screw Binding</b><br>
+When a screw is installed in an extruder, the typical radial clearance between the <br>screw and the barrel is 0.001 D, where D is the diameter of the extruder. This is the <br>clearance at room temperature. When the machine is in operation the actual clear-<br>ance between the screw and the barrel can be quite different. There are two main <br>reasons for the change in clearance under actual processing conditions. One reason <br>is temperature; the other is compressive load on the screw. When the processing <br>temperature is much greater than room temperature, the clearance can change <br>when a) the coefficient of thermal expansion (CTE) of the screw and barrel is differ-<br>ent and b) the temperature of the screw is different from the barrel.<br>
+<b>11.3.2.6.3Changes in Clearance Because of Temperature Differences</b><br>
+When the screw and barrel increase in temperature, both the screw and barrel will <br>increase in diameter due to thermal expansion. If the CTE of the screw is greater <br>than the barrel, the clearance between the screw and barrel will reduce with increas-<br>ing temperature. Values of the CTE for several materials are shown in Table 11.7 <br>together with data on thermal conductivity and elastic modulus.<br>For a 25.40 mm barrel running at 333.3°C above room temperature, the increase in <br>I.D. is 9.652E­2 mm when the CTE is 11.34/°C. For a 25.3492 mm Monel screw <br>with a CTE of 13.86E­6/°C running at 333.3°C above room temperature, the <br>increase in screw diameter will be 0.1168 mm. Thus, the difference between the <br>thermal expansion of the screw and barrel diameter is about 0.02 mm or 0.01 mm <br>based on the radius. If the radial clearance is 0.0254 mm, the clearance will reduce <br>to 0.01524 mm due to the differential thermal expansion. Thus, the clearance is <br>reduced but still greater than zero provided both the screw and barrel are at the <br>same temperature.<br>In an operating extruder, however, it is not very likely that the screw and barrel are <br>at the same temperature. The largest difference between the screw and barrel tem-<br>
+<hr>
+<A name=810></a> <br>
+11.3Systematic Troubleshooting  <b>799</b><br>
+perature is likely to occur in the feed throat. The feed throat of most extruders is <br>water-cooled and therefore close to room temperature in many cases. The screw <br> <br>temperature in the feed section, however, can be (and in many cases will be) much <br>higher because the high temperatures in the compression and metering section will <br>raise the feed section temperature due to thermal conduction.<br>If the feed throat is maintained at room temperature and the screw temperature <br>in the feed section is 166.7°C higher, then the screw diameter will increase <br>from 25.3492 mm to 25.4076 mm if the CTE = 13.86E­6/°C. This corresponds to <br>0.0023 mm per mm of screw diameter. The screw diameter now is larger than the <br>diameter of the feed throat and the screw will bind! The question can be asked, <br>is this temperature difference not equally likely to occur in a screw made of 4140 <br>steel? The answer is no and the reason has to do with the thermal conductivity of <br>these materials. The thermal conductivity of corrosion-resistant metals tends to be <br>considerably lower than that of steel, by a factor of three to five: see Table 11.7.<br>
+<b>Table 11.7</b>Thermal Properties and Elastic Modulus for Several Screw Materials<br>
+Material<br>
+Coefficient of Thermal<br>
+Thermal<br>
+Elastic<br>
+Expansion, [/°C]<br>
+conductivity J/ms[°K]<br>
+modulus [MPa]<br>
+Hastelloy C276<br>
+11.16E­6<br>
+11.25<br>
+2.00E5<br>
+Inconel 718<br>
+12.96E­6<br>
+11.42<br>
+2.00E5<br>
+Inconel 625<br>
+12.78E­6<br>
+9.86<br>
+2.07E5<br>
+Monel 400<br>
+13.86E­6<br>
+21.80<br>
+1.79E5<br>
+Monel 500<br>
+13.68E­6<br>
+17.47<br>
+1.79E5<br>
+4140 steel<br>
+11.34E­6<br>
+42.56<br>
+2.00E5<br>
+4340 steel<br>
+11.34E­6<br>
+42.21<br>
+2.00E5<br>
+17-4 stainless<br>
+10.44E­6<br>
+17.82<br>
+2.00E5<br>
+316 stainless<br>
+18.54E­6<br>
+16.09<br>
+2.00E5<br>
+304 stainless<br>
+18.72E­6<br>
+16.26<br>
+2.00E5<br>
+The lower thermal conductivity of corrosion-resistant materials will reduce the <br>amount of heat that can be transferred from the screw shank to the reducer. As a <br>result, the shank and feed section of the screw will be at higher temperature as com-<br>pared to a high-conductivity screw. Figure 11.15 shows the thermal expansion <br>graphed against the temperature difference between the screw and barrel.<br>The graph shows two typical values of the coefficient of thermal expansion. From <br>Fig. 11.15 it is clear that it takes only a temperature difference of about 167 to <br>222°C to have the screw lock up in the barrel, considering that the coefficient of <br>thermal expansion is in the range of 10E­6 to 17E­6/°C. It is clear that if a polymer <br>is processed at 371°C, it is quite possible that the screw temperature will be more <br>than 167°C above the barrel temperature.<br>
+<hr>
+<A name=811></a><b>800</b>  <b> 11Troubleshooting Extruders</b><br>
+<b> Figure  11.15</b> <br>
+Thermal expansion vs . temperature <br>
+difference<br>
+When viscous heating is significant, the screw temperature will tend to be higher <br>than the barrel temperature, at least with a neutral screw. Janssen et al. [148] found <br>that in extruders without screw cooling the screw temperature gives a much better <br>indication of the mean polymer temperature than does the barrel temperature. <br>Finite element analysis of non-isothermal, non-Newtonian flow in extruders [84] <br>also found that screw temperatures tend to be higher than barrel temperatures <br>when viscous heating is significant. As a result, the screw temperature in the meter-<br>ing section of the screw may be significantly higher than the barrel temperature. <br>Therefore, the temperature difference between screw and barrel in the feed section <br>may be greater than what might be assumed based on the measured barrel tempera-<br>tures.<br>Few publications are available that provide data on screw and plastic temperatures <br>along an extruder. The publication by Marshall et al. [149] provides some interest-<br>ing experimental data. They confirm that the screw temperature in the metering <br>section is higher than the barrel temperature. Further, they measured screw tem-<br>peratures in the feed section and found temperatures in the range of 115 to 127°C <br>(240 to 260°F) with barrel temperatures of 190°C (375°F). When barrel tempera-<br>tures are around 371°C (700°F), it can be expected that the feed section of the <br>screw will be in the range of 204 to 260°C (400 to 500°F) if not higher.<br>
+<b>11.3.2.6.4Analysis of Temperature Distribution in Extruder Screws</b><br>
+In order to confirm whether the mechanism proposed in the previous section is cor-<br>rect, predictions of the temperature distribution in extruder screw processing of <br>FEP were made using finite element analysis. The program used is FEHT [150], <br>developed at the University of Wisconsin-Madison.<br>
+<hr>
+<A name=812></a> <br>
+11.3Systematic Troubleshooting  <b>801</b><br>
+Figure 11.16 shows the thermal boundary conditions that were used in the analysis; <br>697 nodes were used with 1280 triangular elements.<br>
+<b>Figure 11.16</b>Schematic of thermal boundary conditions for FEA<br>
+The feed throat temperature is set at 15.6°C, the barrel temperatures are set at 288, <br>371, 371, and 371°C, the screw shank is set at 93°C, the screw tip is at 371°C, and <br>the heat flux at the screw centerline is zero in the radial direction. The program <br>does not take into account viscous dissipation or convection; the heat transfer is <br>purely by conduction. The thermal conductivity of the FEP is taken as 0.246 J/ms°K.<br>Figure 11.17 shows the predicted temperature distribution in a screw that is made <br>of 4140 steel. Figure 11.18 shows the temperature distribution in a Monel screw.<br>
+<b>Figure 11.17</b>Predicted temperature distribution in a 4140 steel screw<br>
+<b>Figure 11.18</b>Predicted temperature distribution in Monel screw<br>
+Comparing Figs. 11.17 and 11.18 it is clear that with the Monel screw higher tem-<br>peratures occur in the feed section of the screw. This must be due to the thermal <br>conductivity since this is the only difference between the two cases. These predic-<br>tions confirm that a lower thermal conductivity of the screw material can lead to <br>higher temperatures in the feed section of the screw. In the case of the Monel screw <br>shown in Fig. 11.18, the screw temperatures in the feed section range from about <br>150 to 260°C (300 to 500°F).<br>
+<hr>
+<A name=813></a><b>802</b>  <b> 11Troubleshooting Extruders</b><br>
+<b>11.3.2.6.5Change in Clearance Because of Compressive Load</b><br>
+When an extruder screw develops pressure in the plastic melt to force it through a <br>die, the pressure at the end of the screw will cause a compressive thrust load on the <br>screw. As a result, the length of the screw will reduce, while at the same time the <br>diameter of the screw will increase. The relative increase in the screw diameter can <br>be expressed as:<br>
+ (11.1)<br>
+Values of the elastic modulus for several materials are shown in Table 11.7. If the <br>pressure is 34.5 MPa (5000 psi) and the modulus 2.07E5 MPa (30E6 psi), the D/D = <br>8.3E­5. Thus, for a 25.4 mm (1 in) screw, the increase in diameter will be 0.0021 mm <br>(0.000083 in). This means that the increase in diameter due to compressive load is <br>quite small compared to the effect of differential thermal expansion. As a result, the <br>effect of radial expansion due to compressive load is likely to be only a minor factor <br>in the chance of the screw locking up in the barrel.<br>
+<b>11.3.2.6.6Conclusions and Recommendations</b><br>
+The analysis above confirms that corrosion-resistant screws do indeed have a greater <br>chance of locking up in an extruder than screws made of regular 4140 steel. The <br>main culprit appears to be the low thermal conductivity of highly corrosion-resist-<br>ant metals, causing a large temperature difference between the feed throat and <br>the feed section of the screw. The higher screw temperature will cause the screw to <br>expand much more than the feed throat and the barrel, causing the screw to bind. <br>Finite element analysis results confirm that a lower thermal conductivity of the <br>screw leads to higher temperatures in the feed section of the screw.<br>The reason that screw binding problems often occur with highly corrosion-resistant <br>materials is that these screws are typically used for fluoropolymers that are pro-<br>cessed at high temperatures, around 370°C or 700°F. In this case, there are several <br>factors that make screw binding more likely. One, at the high process tempe ratures <br>there will be a higher temperature difference between the screw and barrel in the <br>feed section. Two, the highly corrosion-resistant material of the screw has a much <br>lower thermal conductivity than 4140 steel and, therefore, will tend to have an even <br>higher temperature difference between screw and barrel in the feed section. Three, <br>the highly corrosion-resistant material of the screw will have a higher coefficient of <br>thermal expansion than 4140 steel and thus expand more.<br>There are a number of measures that can be taken to reduce the chance of screw <br>binding. They are:<br>
+ <br>
+ Screw cooling of the feed section<br>
+ <br>
+ Increased temperatures on the feed throat<br>
+ <br>
+ Reduced temperatures in the transition section<br>
+<hr>
+<A name=814></a> <br>
+11.3Systematic Troubleshooting  <b>803</b><br>
+ <br>
+ Reduced temperatures in the metering section<br>
+ <br>
+ Increased clearance in the feed throat region<br>In most cases, the best way to avoid binding problems is to reduce the screw dia-<br>meter in the feed section by at least 0.002 mm per mm (0.002 per in) of screw dia-<br>meter. Because most plastics are fed in pellet form, increasing the flight clearance <br>in the early part of the feed section is most likely not going to have an effect on the <br>performance of the extruder. On the other hand, an increased flight clearance in the <br>feed section will substantially reduce the chance of the screw locking up in the <br>extruder barrel or feed throat.<br>
+<b>11.3.3Polymer Degradation</b><br>
+Polymer degradation is a frequent problem in extrusion. Degradation usually mani-<br>fests itself as discoloration, loss of volatile components (smoking), or loss of mechan-<br>ical properties. According to the mode of initiation, the following types of degrada-<br>tion can be distinguished:<br>1. Thermal<br>2. Chemical<br>3. Mechanical<br>4. Radiation<br>5. Biological<br>Degradation processes are generally quite complex; often more than one type of <br>degradation is operational, e.g., thermo-oxidative degradation, thermo-mechanical <br>degradation, etc. This situation is quite similar to wear in extruders, where usually <br>more than one wear mechanism is operational at any one time.<br>
+<b>11.3.3.1Types of Degradation<br></b>In extrusion, the first three types of degradation are the most important: thermal, <br>mechanical, and chemical degradation.<br>
+<b>11.3.3.1.1Thermal Degradation</b><br>
+Thermal degradation occurs when a polymer is exposed to an elevated temperature <br>in an inert atmosphere under exclusion of other compounds. The resistance against <br>such degradation depends on the nature and the inherent thermal stability of the <br>polymer backbone. There are three main types of thermal degradation: depolymeri-<br>zation, random chain scission, and unzipping of substituent groups.<br>Depolymerization or unzipping is a reduction in length of the main chain by sequen-<br>tial elimination of monomer units. Polymers that degrade by this mechanism are <br>polymethylmethacrylate, polyformaldehyde, polystyrene, etc. Polystyrene unzips to <br>
+<hr>
+<A name=815></a><b>804</b>  <b> 11Troubleshooting Extruders</b><br>
+some extent during degradation, although only about 40% is converted to monomer. <br>Random scission occurs in many polyolefins because of their simple carbon chain <br>backbone. Unzipping of substituent groups is an important thermal degradation <br>mechanism since it is the primary breakdown process for polyvinylchloride.<br>It is often difficult to distinguish between thermal and thermo-chemical degradation <br>because polymers are rarely chemically pure. Impurities and additives can react <br>with the polymeric matrix at sufficiently high temperatures.<br>
+<b>11.3.3.1.2Mechanical Degradation</b><br>
+Mechanical degradation refers to molecular scission induced by the application of <br>mechanical stresses. The stresses can be shear stresses, elongational stresses, or a <br>combination of the two. Mechanical degradation of polymers can occur in the solid <br>state, in the molten state, and in solution. An extensive review of the field of mechan-<br>ically induced reactions in polymers was published by Casale and Porter [38]. In an <br>extruder, mechanical stresses are applied mostly to the molten polymer.<br>Various theoretical approaches have been developed to describe mechanical de -<br>gradation. One of the earlier studies was made by Frenkel [39] and Kauzmann and <br>Eyring [40]. They proposed that linear macromolecules are extended in a shear field <br>in the direction of motion. The strain of the molecules is primarily concentrated at <br>the middle of the chain. No degradation is expected when the degree of polymeri-<br>zation is below a certain critical value. Bueche [41] predicts that entanglements <br>produce preferential tension in the mid-section of macromolecules. Thus, chain <br> <br>scission is more likely to occur in the center of the chain. He also predicts that main <br>chain rupture increases dramatically with increasing molecular weight.<br>These theoretical considerations suggest that mechanical degradation in polymer <br>melts or solutions is a non-random process, producing new low molecular weight <br>species with molecular weights of one-half, one-fourth, one-eighth, etc., the original <br>molecular weight. Mechanical degradation in polymer melts is essentially always <br>combined with thermal degradation, and possibly chemical degradation, because of <br>the elevated temperature of the melt. When a polymer melt is exposed to intense <br>mechanical deformation, local temperatures can rise substantially above the bulk <br>temperature if the rate of deformation is non-uniform.<br>Thus, bulk temperature measurements may not properly reflect actual stock tem-<br>peratures. This is the case in screw extruders where very high local temperatures <br>can occur. The same holds true for high intensity internal mixers. In such devices, <br>pure mechanical degradation is unlikely to occur. Therefore, degradation processes <br>in polymer melts involving mechanical stresses tend to be rather complex.<br>Some workers have reported that degradation at processing conditions is almost <br>exclusively thermal [43, 44], while others conclude that degradation is mainly <br>mechanical [45, 46]. Most workers, however, deduce that, though the nature of deg-<br>
+<hr>
+<A name=816></a> <br>
+11.3Systematic Troubleshooting  <b>805</b><br>
+radation is basically thermal, there is a distinct reduction in the temperature needed <br>for reaction due to the mechanical energy stored within the polymer chains as a <br>result of the mechanical deformation. This corresponds to a shear-induced change <br>in the potential energy function for thermal bond rupture as proposed by Arisawa <br>and Porter [42]. What this means in practice is that the polymer induction time <br>determined under quiescent conditions will be longer than the actual induction <br>time if the polymer is exposed to a mechanical deformation.<br>Because of the aforementioned complications in mechanical degradation in polymer <br>melts, mechanical degradation can be more easily studied in polymer solutions. <br>Casale and Porter [38] have reviewed most work in this area up to 1978. Work in <br>this area published between 1978 and 1984 is summarized in a later publication <br>[77]. More recent work by Odell, Keller, and Miles [47] describes an elegant tech-<br>nique to continuously monitor the molecular weight distribution (MWD) of a poly-<br>mer solution undergoing mechanical deformation. They use a cross-slot device to <br>apply a pure elongational flow field to dilute solutions of narrow MWD atactic poly-<br>styrene. By measuring birefringence, information was obtained on the MWD of <br>this polymer. They observed repeated breakage of the stretched molecules at their <br>centers, as shown by the MWD before and after mechanical deformation of the poly-<br>mer; see Fig. 11.19.<br>
+<b> Figure 11.19</b> <br>
+MWD after mechanical deformation [47]<br>
+<b>11.3.3.1.3Chemical Degradation</b><br>
+Chemical degradation refers to processes induced under the influence of chemicals <br>in contact with a polymer. These chemicals can be acids, bases, solvents, reactive <br>gases, etc. In many cases, a significant conversion is only observed at elevated tem-<br>peratures because of high activation energy for these processes.<br>Two important types of chemical degradation are solvolysis and oxidation. Solvoly-<br>sis reactions concern the breaking of C­X bonds, where X represents a non-carbon <br>atom. Hydrolysis is an important type of solvolysis; schematically the reaction can <br>be described as follows:<br>
+<hr>
+<A name=817></a><b>806</b>  <b> 11Troubleshooting Extruders</b><br>
+This type of degradation occurs in polyesters, polyethers, polyamides, polyure-<br>thanes, and polydialkylsiloxanes. Polymers that tend to absorb water are more likely <br>to undergo hydrolysis. Thus, in the extrusion of polyester and polyamide, it is very <br>important that the polymer be properly dried before extrusion. The stability of poly-<br>mers against solvolytic agents is important in many applications. Some important <br>polymers that have poor stability against acids and bases at room temperature are <br>PVC, PMMA, PA, PC, PETP, PU, PAN, and POM. Polyolefins and fluoropolymers tend <br>to have good stability against these solvolytic agents.<br>Oxidative degradation is a very common type of degradation in polymers. In extru-<br>sion, oxidation occurs at elevated temperatures; thus, the degradation becomes a <br>thermo-oxidative degradation.<br>Polymer degradation starts with the initiation of free radicals. Free radicals have a <br>high affinity for reacting with oxygen to form unstable peroxy radicals. The new <br>peroxy radicals will abstract neighboring labile hydrogens, producing unstable <br>hydroperoxides and more free radicals that will start the same process again. This <br>results in an autocatalytic process, i.e., one that self-propagates once the process <br>has started. Under continuous initiation, the reaction rate is accelerated, resulting <br>in an exponential increase conversion with reaction time. The process will stop <br>when a reacting chemical species is depleted or when the propagation is inhibited <br>by reaction products.<br>The oxidative degradation in polymers is generally combated with the addition of <br>antioxidants. The purpose of the antioxidant is the interception of radicals or pre-<br>vention of radical initiation during the various phases of a polymer's life: polymeri-<br>zation, processing, storage, and end use. According to their functionality, antioxi-<br>dants can be classified as primary or secondary antioxidants. Primary antioxidants <br>or chain terminators interrupt chain reactions by tying up free radicals. They are <br>also referred to as free-radical scavengers. Secondary antioxidants, or preventive <br>antioxidants, destroy hydroperoxides. They are also referred to as peroxide decom-<br>posers. Primary antioxidants consist primarily of hindered phenols and aromatic <br>amines. These materials tie up polymeric peroxy radicals through hydrogen dona-<br>tion, forming polymeric hydroperoxide groups and relatively stable antioxidant <br> <br>species. Secondary antioxidants consist of various phosphorous or sulfur containing <br>compounds, particularly phosphites and thioesters. These materials reduce hydro-<br>peroxides to inert products, thus preventing the proliferation of alkoxy and hydroxy <br>radicals. Selecting an effective antioxidant package is a key factor to the success of <br>a plastic product. Some of the factors that should be considered in the selection of <br>an antioxidant are toxicity, volatility, color, extractability, odor, compatibility, supply, <br>cost, and performance.<br>
+<hr>
+<A name=818></a> <br>
+11.3Systematic Troubleshooting  <b>807</b><br>
+<b>11.3.3.2Degradation in Extrusion<br></b>Degradation during the extrusion process will often be a combination of thermal, <br>mechanical, and chemical degradation. Factors that are important in determining <br>the rate of degradation are:<br>1.  Residence time and residence time distribution (RTD)<br>2.  Stock temperature and distribution of stock temperatures<br>3.  Deformation rate and deformation rate distribution<br>4.  Presence of solvolytic agents, oxygen, or other degradation promoting agents<br>5.  Presence of antioxidants and other stabilizers<br>The first three factors are strongly influenced by the machine geometry and by the <br>operating conditions. The presence of solvolytic agents or oxygen can be influenced <br>by operating conditions, e.g., oxygen can be eliminated from the extruder by putting <br>the feed hopper under a nitrogen blanket. The presence of antioxidants and other <br>stabilizers is part of the material selection process. Proper selection of a stabilizer <br>package is very important; however, the details to determine the right stabilizer <br>package are outside the scope of this book.<br>
+<b>11.3.3.2.1Residence Time Distribution</b><br>
+Knowledge of the residence time distribution (RTD) of an extruder provides valuable <br>information about the details of the conveying process in the machine. The RTD is <br>directly determined by the velocity profiles in the machine. Thus, if the velocity <br>profiles are known, the RTD can be calculated. Various workers have made theore-<br>tical calculations of the RTD in single screw extruders [48­50]. Obviously, theore-<br>tical calculations of the RTD require knowledge of the velocity profiles in the <br>machine. Thus, the predicted RTD is only as accurate as the velocity profiles that <br>form the basis of the calculations. In single screw extruders, the velocity profiles <br>can be determined reasonably well, although usually a substantial number of sim-<br>plifying assumptions are made. In other screw extruders, e.g., twin screw extrud-<br>ers, calculation of velocity profiles is rather complex and thus prediction of the RTD <br>more difficult.<br>Experimental determination of the RTD of an extruder yields information about the <br>conveying process in the extruder. This information is useful in a number of areas, <br>not just to analyze the chance of degradation in the machines. The RTD can be used <br>to analyze the mixing process in an extruder. When an extruder is used as a con-<br>tinuous chemical reactor, the RTD provides important information for process <br>design and process analysis. The RTD also provides a good selection criterion, e.g., <br>an extruder used in profile extrusion should have a narrow RTD and short residence <br>time. Experimental studies of RTD in single screw extruders have been reported by <br>Wolf and White [51], Bigg and Middleman [49], Schott and Saleh [55], Rauwendaal <br>[52], Golba [53], and Kemblowski and Sek [54]. Experimental studies of RTD in twin <br>
+<hr>
+<A name=819></a><b>808</b>  <b> 11Troubleshooting Extruders</b><br>
+screw extruders have been reported by Todd [56], Janssen et al. [57, 58], Rauwendaal <br>[52], Walk [59], and Nichols et al. [60].<br>The RTD is determined by measuring the output response of a change in input. This <br>is referred to as the stimulus response method as discussed by Levenspiel [61] and <br>Himmelblau and Bischoff [62].<br>The system is disturbed by a stimulus and the response of the system to the stimu-<br>lus is measured. Two common stimulus response techniques are the step input <br>response and the pulse input response; see Fig. 11.20.<br>Other stimuli that can be used are a random input and a sinusoidal input. The <br>response of a step input is an S-shaped curve; see Fig. 11.20 top. The response of a <br>pulse input is a bell-shaped curve; see Fig. 11.20 bottom. The ideal pulse input is of <br>infinitely short duration; such an input is called a delta function or impulse. The <br>normalized response to a delta function is called the C curve. Thus, the total area <br>under the curve equals unity.<br>
+<b> Figure 11.20</b> <br>
+Response to step change and pulse <br>
+input<br>
+The definition of RTD functions is due to Danckwerts [63]. The internal RTD func-<br>tion g(t)dt is defined as the fraction of fluid volume in the system with a residence <br>time between t and t + dt.<br>The external RTD function f(t)dt is defined as the fraction of exiting flow rate with a <br>residence time between t and t + dt. The cumulative internal RTD function G(t) is <br>defined as:<br>
+ (11.2)<br>
+G(t) represents the fraction of fluid volume in the system with a residence time <br>between 0 and t. The cumulative external RTD function is defined as:<br>
+ (11.3)<br>
+where to is the minimum residence time.<br>
+<hr>
+<A name=820></a> <br>
+11.3Systematic Troubleshooting  <b>809</b><br>
+F(t) represents the fraction of exiting flow rate with a residence time equal to or <br>shorter than t. For very long times t, both functions G and F become equal to unity:<br>
+ (11.4)<br>
+The mean residence time   is given by the following expression:<br>
+ (11.5)<br>
+The mean residence time is determined by the volume of the machine V, the degree <br>of fill X of the machine, and the volumetric flow rate  :<br>
+ (11.6)<br>
+The relationship between the internal RTD function and the external RTD function <br>is given by:<br>
+ (11.7)<br>
+In the flow of a Newtonian fluid through a pipe, the RTD can be calculated rather <br>easily by using the expression for the velocity profile given earlier. The external <br>RTD function is:<br>
+ (11.8)<br>
+where the minimum residence time is:<br>
+ (11.9)<br>
+Figure 11.21 shows a typical cumulative RTD curve for a single screw extruder as <br>determined experimentally [52].<br>The curve is for a 25-mm extruder running at 20 rpm with an output of 2.3 kg/hr. <br>The mean residence time in this example is 5.9 minutes. This type of information is <br>useful because one can easily tell how large a fraction of the material spends how <br>long a time in the machine. For instance, in Fig. 11.21, more than 1% of the material <br>is exposed to a residence time of three times the mean time, i.e., 17.7 minutes! If <br>the induction time of the material at process temperature is less than 17.7 minutes, <br>one can expect more than 1% of the material to be degraded.<br>
+<hr>
+<A name=821></a><b>810</b>  <b> 11Troubleshooting Extruders</b><br>
+<b> Figure 11.21</b> <br>
+RTD for a single screw <br>
+extruder<br>
+Figure 11.22 shows several cumulative RTD curves for an intermeshing counter-<br>rotating twin screw extruder [52].<br>
+<b> Figure 11.22</b> <br>
+RTD for an intermeshing <br>
+counter-rotating twin screw <br>
+extruder<br>
+It can be seen that the shape of the curve changes substantially when the process-<br>ing conditions are changed. The narrowest RTD is obtained by running the extruder <br>at low speed and high output. Figure 11.23 shows several cumulative RTD curves for <br>an intermeshing co-rotating twin screw extruder [52].<br>It is clear that the curves indicate considerable deviations from positive conveying <br>characteristics for the co-rotating twin screw extruder.<br>A major advantage of these normalized RTD curves is that conveying characteristics <br>of different extruders can be directly compared. From comparison of Figs. 11.21­11.23 <br>it is clear that the conveying characteristics of the single screw extruder are quite <br>positive compared to the two twin screw extruders. This is partially due to the plug <br>flow of the solid bed in the single screw extruder. The solid bed in a twin screw <br>
+<hr>
+<A name=822></a> <br>
+11.3Systematic Troubleshooting  <b>811</b><br>
+extruder is not continuous and generally does not extend over a long length of the <br>machine. It should be realized that the RTD is strongly dependent on the screw <br>design and the operating conditions. This point was discussed in some detail by <br>Kemblowski and Sek [54] with regard to single screw extruders and by the author <br>[52] with regard to twin screw extruders.<br>
+<b> Figure 11.23</b> <br>
+RTD for an intermeshing <br>
+co-rotating twin screw <br>
+extruder [52]<br>
+<b>11.3.3.2.2Temperature Distribution Simple Calculations</b><br>
+Obviously, the residence time and its distribution only partially determine the <br>chance of degradation in an extruder. The other factors that play an important role <br>are the actual stock temperatures and the strain rates to which the polymer is <br>exposed. The actual stock temperatures and strain rates are closely related. In the <br>extruder, there are two major areas of concern: the screw channel and the flight <br>clearance. Janssen, Noomen, and Smith [65] studied temperature distribution of the <br>polymer melt in the screw channel. Temperature distribution of the polymer right <br>after the end of the screw was measured, for instance, by Anders, Brunner, and Pan-<br>haus [66]. The temperature variations in the screw channel at the end of the screw <br>were reported to be less than 5 to 10°C and relatively close to the barrel tempera-<br>ture. More recently, Noriega et  al. [145] measured melt temperature distribution <br>with a thermocomb and found temperature variations as high as 20 to 30°C.<br>The situation in the screw clearance is substantially different from the screw chan-<br>nel. The strain rates in the screw channel are relatively low, but the melt tempera-<br>ture variations can be high [84]. In the screw clearance, however, the strain rates <br>are very high, and the stock temperature increase can also be very high. This can be <br>verified by the following simple analysis. The shear rate in the clearance is approxi-<br>mately the Couette shear rate:<br>
+ (11.10)<br>
+<hr>
+<A name=823></a><b>812</b>  <b> 11Troubleshooting Extruders</b><br>
+The corresponding viscous heat generation per unit volume for a power law fluid is:<br>
+ (11.11)<br>
+If it is assumed that there is no exchange of heat between the polymer melt and the <br>screw and between the polymer melt and the barrel, the average adiabatic tempera-<br>ture rise can be determined from:<br>
+ (11.12)<br>
+where  R is the average residence time of the polymer melt in the flight clearance. <br>
+The average residence time in the flight clearance is approximately:<br>
+ (11.13)<br>
+Combining Eqs. 11.11, 11.12, and 11.13, the average adiabatic temperature rise in <br>the clearance can be written as:<br>
+ (11.14)<br>
+The average temperature rise is directly proportional to the consistency index m <br>and the tangential flight width w/sin. The temperature rise is strongly dependent <br>on the radial clearance , the power law index of the polymer melt n, and the screw <br>speed N. Figure 11.24 shows the effect of flight clearance  and the power law <br>index n for a 114-mm (4.5-in) extruder running at 100 rpm; the specific heat is <br>2250 J/kg°C, the melt density is 900 kg/m3, and the consistency index is 104 Pasn.<br>It is clear that the adiabatic temperature rise in the flight clearance can be very <br>high. However, the actual temperature rise will be less than the adiabatic tempe-<br>rature rise because there will be transfer of heat to the screw and to the barrel. In <br>reality, the thermal boundary conditions at the barrel and the flight land will be <br>somewhere between adiabatic and isothermal.<br>The temperature rise in the clearance can be substantially reduced by simply reduc-<br>ing the flight width and increasing the flight helix angle. These same measures will <br>also substantially reduce the power consumption of the extruder. Thus, proper <br>design of the screw flight is of great importance when it comes to reducing power <br>consumption and reducing the chance of degradation in the extruder.<br>Another reason that the flight clearance is so important in degradation processes <br>occurring in the extruder is the fact that, in addition to high stock temperatures, the <br>polymer melt is exposed to very high strain rates, both elongational and shear. As <br>
+<hr>
+<A name=824></a> <br>
+11.3Systematic Troubleshooting  <b>813</b><br>
+discussed earlier, this causes a flow-induced change in the potential energy function <br>for thermal bond rupture. Thus, the degradation will be more severe than it would <br>be based on just the effect of temperature. Obviously, another important point is to <br>eliminate dead spots in the screw design and in the die design. Hang-up of material <br>can be very detrimental and should be avoided if at all possible. For instance, fluted <br>mixing sections with a 90° helix angle should not be used with polymers that have <br>a tendency to degrade because such mixing sections have stagnating regions.<br>
+<b>Figure 11.24</b>Adiabatic temperature rise versus flight clearance<br>
+The values of the temperature increase in the flight clearance calculated with Eq. <br>11.14 are surprisingly high, particularly considering the very short residence time <br>of the polymer in the flight clearance, which is usually in the order of 0.1 s. Ob -<br>viously, in reality the temperature rise will not be as high as the adiabatic tempera-<br>ture rise because there will be exchange of heat with the screw and with the barrel.<br>The lowest temperature rise will occur in the extreme case that both screw flight <br>surface and barrel surface can be maintained at constant temperature, i.e., iso-<br>thermal boundary conditions. This situation was analyzed by Meijer, Ingen Housz, <br>and Gorissen [67] with the primary purpose to determine the thermal development <br>length. They assumed that the clearance flow is dominated by drag flow in the <br> <br>tangential direction. The thermal development length for the Newtonian case was <br>found to be approximately 0.36 Npe, where Npe is the Peclet number. Thus, the length <br>
+required for thermal development can be written as:<br>
+ (11.15)<br>
+where  is the thermal diffusivity.<br>
+<hr>
+<A name=825></a><b>814</b>  <b> 11Troubleshooting Extruders</b><br>
+The thermal development length is directly proportional to the barrel velocity vb <br>
+(and thus the screw speed) and to the radial clearance squared. With thermal diffu-<br>sivity values of about 10­7 m2/s, the thermal entrance length will be the same order <br>of magnitude as the tangential flight width when the clearance has the normal <br>design value ( 0.001 D). Thus, the temperature profile at the exit of the flight <br>clearance will be very close to the fully developed temperature profile. The fully <br>developed temperature profile for the isothermal case can be written as [67]:<br>
+ (11.16)<br>
+The maximum temperature Tmax that can develop in the isothermal case is:<br>
+ (11.17)<br>
+The first term to the right of the equal sign represents the viscous temperature rise. <br>If the viscosity in the flight clearance is written as a power law fluid the viscous <br>temperature rise can be written as:<br>
+ (11.18)<br>
+The viscous temperature rise with isothermal conditions is plotted against the flight <br>clearance in Fig. 11.25. The isothermal viscous temperature rise data were calcu-<br>lated for a 114 mm extruder running at 100 rpm with a polymer melt with a consist-<br>ency index of m = 104 Pasn and a thermal conductivity k = 0.25 J/ms°C.<br>It is interesting to see that the isothermal viscous temperature rise increases with <br>clearance while the adiabatic temperature rise decreases with clearance. For both <br>thermal boundary conditions, the temperature rise increases strongly with the <br>power law index of the polymer melt. This indicates that highly shear thinning poly-<br>mers (low power law index) will have much lower melt temperature rise in the flight <br>clearance than weakly shear thinning polymers.<br>In reality, true isothermal conditions may not be achieved because the high heat <br>fluxes at the screw and barrel interface required to maintain isothermal conditions <br>may not be physically possible. Thus, the actual maximum stock temperature in the <br>clearance will be somewhere between the adiabatic and the isothermal case. It <br>should be noted that the expressions for the melt temperature rise are valid only for <br>purely viscous fluids with a temperature independent viscosity in pure drag flow. <br>Obviously, for a temperature dependent fluid the melt temperature rise will be <br>reduced relative to the temperature independent fluid. Also, the pressure gradient <br>in the flight clearance will affect the velocities and temperatures. With pressure <br>
+<hr>
+<A name=826></a> <br>
+11.3Systematic Troubleshooting  <b>815</b><br>
+normally reducing from the pushing to the trailing flight flank the pressure gra-<br>dient in the flight clearance will usually be negative. This will reduce the melt tem-<br>perature rise in the flight clearance relative to the pure drag flow case.<br>
+<b>Figure 11.25</b>Isothermal viscous temperature rise vs . flight clearance<br>
+The melt temperature rise values in the flight clearance for both adiabatic and iso-<br>thermal conditions are shown in Fig. 11.26.<br>
+<b>Figure 11.26</b>Adiabatic and isothermal melt temperature rise vs . flight clearance<br>
+<hr>
+<A name=827></a><b>816</b>  <b> 11Troubleshooting Extruders</b><br>
+At small values of the flight clearance the adiabatic temperature rise is much greater <br>than the isothermal temperature rise. However, at some clearance value the adiaba-<br>tic and isothermal curves intersect. At clearance values higher than the intersection <br>the isothermal temperature rise is actually greater than the adiabatic temperature <br>rise. For low values of the power law index the crossover clearance (where Tadiabatic = <br>
+Tisothermal) is about 0.001 D, which is a typical flight clearance in single screw extrud-<br>
+ers. At higher values of the power law index the crossover clearance becomes larger.<br>The isothermal melt temperature rise values for large values of the flight clearance <br>( &gt; 0.001 D) are probably unrealistic due to the fact that the thermal development <br>length will be greater than the width of the flight; see Eq. 11.15. In this case, fully <br>developed temperatures cannot be reached in the flight clearance, and Eqs. 11.17 <br>and 11.18 will not yield accurate values for the melt temperature rise. Figure 11.26 <br>shows that melt temperature rise values from about 25°C to over 100°C can be <br>expected in a typical flight clearance ( = 0.001 D) of a 114 mm extruder running at <br>100 rpm.<br>
+<b>11.3.3.2.3Temperature Distribution Numerical Calculations</b><br>
+Winter [30] has performed numerical calculations of the developing temperature <br>profile in the flight clearance for power law fluids. He assumed isothermal condi-<br>tions at the barrel wall and adiabatic conditions at the screw flight surface. These <br>assumptions are considerably more realistic than the purely adiabatic case or the <br>purely isothermal case, although a better boundary condition would probably be a <br>prescribed maximum heat flux. Winter calculates a typical maximum temperature <br>increase of about 150°C. This value is closer to the maximum temperature rise in <br>the adiabatic case than the maximum temperature rise in the isothermal case. <br>These analyses indicate that the temperature rise in the flight clearance can be <br>quite significant and can play a very important role in degradation in extruders.<br>Rauwendaal [84] developed a finite element method (FEM) program to determine <br>temperature profiles in the melt conveying zone of extruders. This FEM program <br>allows the calculation of three-dimensional velocities and temperatures at any point <br>in the screw channel. The program is based on a 2½-D analysis, which means that <br>the velocities are assumed to change little in the down-channel direction.<br>The temperature field is shown in Chapter 12, Fig. 12.7. The barrel surface is set at <br>175°C and the screw surface is taken as adiabatic (zero heat flux). The melt tem-<br>peratures at any point in the channel are considerably higher than the barrel tem-<br>perature. The highest temperatures occur at about two-thirds of the channel height; <br>this is where the cross-channel velocities are the lowest. The highest temperatures <br>are about 31°C above the barrel temperature. This agrees well with experimental <br>results published earlier [83].<br>
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+ <br>
+11.3Systematic Troubleshooting  <b>817</b><br>
+The isotherms in the bottom half of the screw channel clearly show the effect of <br>cross-channel circulation on the temperature field. In fact, the lower temperatures at <br>the flight flanks and the bottom of the screw channel are a direct result of the re-<br>circulation in the channel. The re-circulation causes the low-temperature melt close <br>to the barrel surface to move close to the screw surface, keeping the melt tempera-<br>tures low in this region.<br>It is interesting to note that the melt temperatures in the flight clearance are lower <br>than in the screw channel. Since the shear rate in the flight clearance is higher than <br>it is anywhere else, one might expect the highest temperatures to occur there as <br>well. The key to the low temperatures in the clearance is the fact that the clearance <br>is quite thin. The high level of viscous heat generated in the clearance is efficiently <br>conducted away to the barrel because of its close proximity. This effect was con-<br>firmed by results of numerical analysis by Pittman [86] and an analytical solution of <br>non-isothermal drag flow by Rauwendaal and Ingen Housz [87]. Further confirma-<br>tion was provided by experimental work on leakage flow [83].<br>When the flight clearance is increased under the same conditions, the major change <br>occurs in the temperature field as shown in Figure 11.27. The temperatures in the <br>lower portion of the channel increase significantly with increasing clearance. This <br>is due to a thicker, relatively stagnant layer of polymer melt at the barrel surface. <br>This insulating, stagnant layer inhibits heat transfer between the material in the <br>screw channel and the barrel. These results agree well with experimental work [83].<br>
+<b> Figure 11.27</b> <br>
+Melt temperature distribution with <br>
+increased flight clearance<br>
+A larger flight clearance not only increases the maximum temperature in the chan-<br>nel, but the size of the high-temperature region also expands considerably. This is <br>due to a weaker re-circulating flow with a large clearance. With a large clearance, <br>the flow rate through the clearance is large, causing a corresponding reduction in <br>the re-circulating flow. A large flight clearance reduces melt temperature control. <br>
+<hr>
+<A name=829></a><b>818</b>  <b> 11Troubleshooting Extruders</b><br>
+The high melt temperatures close to the screw surface can lead to degradation. This <br>is because the residence times are longest at the screw surface. The combination of <br>high temperatures and long residence times at the screw surface with large flight <br>clearance make degradation more likely.<br>
+<b>11.3.3.2.4Conclusions from Finite Element Analysis</b><br>
+FEA can be very helpful in analyzing details of flow and heat transfer inside extrud-<br>ers that are very difficult to determine on operating extruders. FEA can predict <br>three-dimensional velocity profiles in screw extruders. In addition, the pressure and <br>temperature fields can be determined.<br>When the viscous heat generation is important, high melt temperature regions form <br>in the center of the channel. This is due to the thermal convection caused by the <br>recirculating flow pattern in the screw channel. Thus, the melt temperature non-<br>uniformities that form are inherent to the flow in single screw extruders.<br>When the flight clearance increases, the melt temperatures inside the screw chan-<br>nel can rise considerably. Also, the high-temperature region expands towards the <br>root of the screw. This can have serious consequences, because the residence times <br>at the screw root are quite long and this can easily lead to degradation of the plastic.<br>The results indicate that it is very important to make sure that the flight clearance <br>between screw and barrel is within reasonable limits. In practice, the radial flight <br>clearance should be no more than 0.003 D, with D being the diameter of the screw. <br>Also, the inherently non-uniform melt temperatures make distributive mixing sec-<br>tions almost indispensable.<br>The thermal development length increases with the screw diameter as shown in <br>Fig. 11.28. For small diameters (D  60 mm), the thermal development time is about <br>10 to 20 s. For larger diameters (D &gt; 100 mm), the thermal development time is <br>greater than 30 s. Considering that a typical residence time in the melt conveying <br>zone of an extruder is about 15 to 20 s, it is clear that in large extruders fully devel-<br>oped temperature conditions will not always be achieved in the extruder. Fully <br>developed conditions may occur at low screw speed; however, at high screw speed it <br>is unlikely that fully developed temperatures can be reached within the length of <br>the extruder.<br>The increases in melt temperature and the temperature non-uniformities in large <br>extruders are greater than in small extruders. This is due to the fact the surface-to-<br>volume ratio becomes less favorable with increasing screw diameter. As a result, in <br>large extruders it is more difficult to keep the melt temperatures low and uniform. <br>Of course, this is well known in practice, and this is the reason that large extruders <br>generally run at lower screw speed than small extruders. Also, it is more important <br>to incorporate good mixing along the extruder screw when the screw diameter is <br>large and when the extruder is operated at high screw speed.<br>
+<hr>
+<A name=830></a> <br>
+11.3Systematic Troubleshooting  <b>819</b><br>
+<b> Figure 11.28</b> <br>
+Evolution of bulk average <br>
+ <br>temperature over time<br>
+<b>11.3.3.2.5Reducing Degradation</b><br>
+The following changes to the process can reduce the chance of degradation in the <br>extruder:<br>1.  Reduce residence time and achieve a narrow RTD<br>2.  Reduce stock temperature and avoid high peak temperatures<br>3.  Eliminate degradation-promoting substances<br>The residence time can generally be reduced by designing the screw for maximum <br>throughput. Low stock temperatures and reduced peak temperatures can be obtained <br>by designing the screw for minimum specific energy consumption. Further, stagnat-<br>ing regions should be avoided if at all possible. Thus, the design of both the screw <br>and die has to be made as streamlined as possible.<br>In some special cases, it may be beneficial to design an extruder screw for low out-<br>put and small inventory. An example of such a case can be a medical extrusion of <br>a very small catheter where production occurs at a screw speed of 2 rpm. At such <br>a low screw speed, the average residence time in the extruder will range from 30 to <br>60 minutes, and degradation is likely to occur at such long residence times. In such <br>a case, it is beneficial to design an extruder screw that has low output per revolution <br>and a small total channel volume. This can be achieved by reducing the flight helix <br>angle, reducing the channel depth, increasing the flight width, and increasing the <br>number of flights. With such a low output, the screw speed may increase from 2 to <br>10 rpm and the average residence reduce from 60 minutes to less than 10 minutes.<br>High stock temperatures are likely to be a problem in extrusion operations where <br>the extruder is run at high screw speed and where the polymer melt viscosity is <br>high. The main screw design variable that affects viscous heating is the channel <br>depth. Increasing the channel depth will reduce the shear rate and thus the viscous <br>heating. There are limits to how deep the screw can be cut. One limit is the physical <br>
+<hr>
+<A name=831></a><b>820</b>  <b> 11Troubleshooting Extruders</b><br>
+strength of the screw. Another limit is the solids conveying and melting capacity of <br>the screw. It is not possible to increase the depth of the screw channel to the point <br>where the melt conveying rate exceeds the melting rate. This is a case where a bar-<br>rier type extruder screw can be useful.<br>One of the most demanding situations for controlling melt temperature is in foam <br>extrusion. In the secondary extruder of a tandem foam extrusion line, the cooling <br>screw is usually made with very deep channels to minimize viscous heating. Also, <br>multiple (usually six) thin flights with a large helix angle are used to increase the <br>heat transfer capability of the screw.<br>If degradation occurs by a thermo-oxidative mechanism, air should be excluded <br>from the extruder. This can be done by putting a nitrogen blanket on the feed hop-<br>per, vent port, or at the die, depending on where the air is introduced. If degradation <br>occurs by hydrolysis, moisture has to be excluded from the process. If degradation <br>occurs by a chemical reaction with the metal surfaces of screw and barrel, a non-<br>reacting material of construction has to be selected for the screw and barrel.<br>
+<b>11.3.4Extrusion Instabilities</b><br>
+Variations in extruder performance is perhaps the most frequent problem encoun-<br>tered in extrusion. One possible reason for the frequent occurrence of instabilities is <br>the fact that they can have a large number of causes, some of which are:<br>
+ <br>
+ Bulk flow problems in the feed hopper<br>
+ <br>
+ Solids conveying problems in the extruder<br>
+ <br>
+ Insufficient melting capacity<br>
+ <br>
+ Solid bed breakup<br>
+ <br>
+ Melt temperature non-uniformities in the die<br>
+ <br>
+ Barrel temperature fluctuations<br>
+ <br>
+ Screw temperature fluctuations<br>
+ <br>
+ Variations in the take-up device<br>
+ <br>
+ Melt fracture/shark skin<br>
+ <br>
+ Variations in screw speed<br>
+ <br>
+ Barrel wear/screw wear<br>
+ <br>
+ Insufficient mixing capacity<br>
+ <br>
+ Very low diehead pressure<br>
+ <br>
+ Insufficient pressure-generating capacity<br>Proper instrumentation is vitally important to be able to diagnose a problem quickly <br>and accurately as discussed in Chapter 1. A prerequisite for stable extrusion is a <br>good extruder drive, good temperature control system, good take-up device, and <br>
+<hr>
+<A name=832></a> <br>
+11.3Systematic Troubleshooting  <b>821</b><br>
+most importantly, a good screw design. Probably more instabilities result from <br>improper screw design than from any other cause. However, a change in screw <br>design is often only considered as the very last option.<br>The extruder drive should be able to hold the screw speed constant to about 0.1% or <br>better; the same holds true for the take-up device. However, this is not always the <br>case on actual extrusion lines. The extruder should be equipped with some type of <br>proportioning temperature control, preferably a PID-type control or better. On-off <br>temperature control is inappropriate for most extrusion operations.<br>
+<b>11.3.4.1Frequency of Instability<br></b>Various workers [68, 69] have classified extrusion instabilities based on the time <br>frame in which they occur. The frequency of the instability is often an indication of <br>the cause of the problem. Most of the earlier works distinguished only three or four <br>types of instabilities based on the frequency. However, it is probably more appro-<br>priate to distinguish at least five types of instabilities:<br>1.  High-frequency instabilities occurring faster than the frequency of screw rota-<br>
+tion<br>
+2.  Screw frequency instabilities occurring at the same rate as the frequency of <br>
+screw rotation<br>
+3.  Low-frequency instabilities, about 5 to 10 times slower than the frequency of <br>
+screw rotation<br>
+4.  Very slow fluctuations occurring at a frequency of at least several minutes<br>5.  Random fluctuations<br>
+<b>11.3.4.1.1High-Frequency Instabilities</b><br>
+High-frequency instabilities are often associated with die flow instabilities, such as <br>melt fracture, shark skin, or draw resonance. They can also be caused by drive prob-<br>lems, melt temperature non-uniformities, or vibration.<br>
+<b>Shark Skin<br></b>Shark skin manifests itself as a regular ridged surface distortion, with the ridges <br>running perpendicular to the extrusion direction. A less severe form of shark skin is <br>the occurrence of matness of the surface, where the glossy surface cannot be main-<br>tained. Shark skin is generally thought to be formed in the die land or at the exit. It <br>is dependent primarily on the temperature and the linear extrusion speed. Factors <br>such as shear rates, die dimensions, approach angle, surface roughness, L/D ratio, <br>and material of construction seem to have little or no effect on shark skin.<br>The mechanism of shark skin is postulated to be caused by the rapid acceleration of <br>the surface layers of the extrudate when the polymer leaves the die. If the stretching <br>rate is too high, the surface layer of the polymer can actually fail and form the char-<br>
+<hr>
+<A name=833></a><b>822</b>  <b> 11Troubleshooting Extruders</b><br>
+acteristic ridges of the shark skin surface [96]. High-viscosity polymers with narrow <br>molecular weight distribution (MWD) seem to be most susceptible to shark skin <br>instability [97, 98].<br>The shark skin problem can generally be reduced by reducing the extrusion velocity <br>and increasing the die temperature, particularly at the land section. There is some <br>evidence that running at very low temperatures can also reduce the problem [99]. <br>Selection of a polymer with a broad MWD will also be beneficial in reducing shark <br>skin. Using an external lubricant can also reduce the problem. This can be done by <br>using an additive to the polymer or by coextruding a thin, low-viscosity outer layer.<br>
+<b>Melt Fracture<br></b>Melt fracture is a severe distortion of the extrudate, which can take many different <br>forms: spiraling, bambooing, regular ripple, random fracture, etc.; see Chapter 7, <br>Fig. 7.108. It is not a surface defect like shark skin, but is associated with the entire <br>body of the molten extrudate. However, many workers do not distinguish between <br>shark skin and melt fracture, but lump all these flow instabilities together under <br>the term melt fracture. There is a large amount of literature on the subject of melt <br>fracture (e.g., [100­112]); however, there is no clear agreement as to the exact cause <br>and mechanism of melt fracture. It is quite possible that the mechanism is not the <br>same for different polymers and/or different flow channel geometries [113]. Linear <br>polymers tend to develop an instability of the shear flow in the die land, while <br>branched polymers tend to develop instabilities in the converging region of the die <br>flow channel.<br>However, there is relatively uniform agreement that melt fracture is triggered when <br>a critical wall shear stress is exceeded in the die. This critical stress is in the order <br>of 0.1 to 0.4 MPa (15 to 60 psi). A number of mechanisms have been proposed to <br>explain melt fracture. Some of the more popular ones are:<br>
+ <br>
+ Critical elastic deformation in the entry zone<br>
+ <br>
+ Critical elastic strain<br>
+ <br>
+ Slip-stock flow in the die<br>The effect of the entry zone has been demonstrated by many workers. In general, the <br>smaller the entry angle, the higher the deformation rate at which instability occurs. <br>Gleissle [114] has proposed a critical elastic strain as measured by recoverable <br>strain. Based on measurements with 11 fluids, he proposed the existence of a criti-<br>cal value of the ratio of first normal stress difference to the shear stress, the average <br>value being 4.63 for 11 widely different fluids with a standard deviation of about 5%. <br>Much larger differences were found in the critical shear stress; the average being <br>3.7E5 Pa with a standard deviation of about 55%. In 1961, Benbow, Charley, and <br>Lamb [111, 115] introduced the slip-stick mechanism to explain flow instability and <br>extrudate distortion. Above a certain critical stress, the polymer melt is believed to <br>
+<hr>
+<A name=834></a> <br>
+11.3Systematic Troubleshooting  <b>823</b><br>
+experience intermittent slipping due to lack of adhesion between the melt and the <br>die wall in order to relieve excessive deformation energy absorbed because of flow <br>through a die. A large number of workers have observed slippage by various tech-<br>niques.<br>More recent work by Utracki and Gendron on pressure oscillations in extrusion of <br>polyethylenes [116] led them to conclude that the pressure oscillation does not seem <br>to be related to elasticity or slip. They conclude that the parameter responsible for <br>pressure oscillations is the critical strain (Hencky) value c of the melt. For LLDPE, <br>
+c &lt; 3, for HDPE, c &lt; 2, while for LDPE, c &gt; 3.5. The instability seems to be based on <br>
+the inability of the polymer melt to sustain levels of strain larger than the critical <br>strain.<br>Streamlining the flow channel geometry has been found to reduce the tendency for <br>melt fracture in branched polymers. Increased temperatures, particularly at the wall <br>of the die land, enable higher extrusion rates before melt fracture appears. The crit-<br>ical wall shear stress appears to be relatively independent of the die length, radius, <br>and temperature. The critical stress seems to vary inversely with molecular weight, <br>but seems to be independent of MWD. Certain polymers exhibit a super-extrusion <br>region, above the melt fracture range, where the extrudate is not distorted [117]. <br>This process is particularly advantageous with polymers that melt fracture at rela-<br>tively low rates, such as FEP. In super-extrusion, the polymer melt is believed to slip <br>relatively uniformly along the die wall. The occurrence of slip in extruder dies has <br>been studied by a number of investigators, e.g., [118, 119]. However, it is still not <br>clear whether the slip is actual loss of contact of polymer melt and metal wall or <br>whether it is failure of a thin polymeric layer very close to the metal surface.<br>The melt fracture problem can be reduced by streamlining the die, increasing the <br>temperature at the die land, running at lower rates, reducing the MW or the poly-<br>mer melt viscosity, increasing the cross-sectional area of the exit flow channel, or by <br>using an external lubricant. In some instances, the melt fracture problem can be <br>solved by going to super-extrusion; this process is used particularly often in the <br>wire coating industry where high line speeds are quite important for economic pro-<br>duction.<br>
+<b>Draw Resonance<br></b>Draw resonance occurs in processes where the extrudate is exposed to a free sur-<br>face stretching flow, such as blown film extrusion, fiber spinning, and blow molding. <br>It manifests itself in a regular cyclic variation of the dimensions of the extrudate. An <br>extensive review [113] and an analysis [120] of draw resonance were done by Petrie <br>and Denn. Draw resonance occurs above a certain critical draw ratio while the poly-<br>mer is still in the molten state when it is taken up and rapidly quenched after take-<br>up.<br>
+<hr>
+<A name=835></a><b>824</b>  <b> 11Troubleshooting Extruders</b><br>
+Draw resonance will occur when the resistance to extensional deformation decreases <br>as the stress level increases. The total amount of mass between die and take-up may <br>vary with time because the take-up velocity is constant but the extrudate dimen-<br>sions are not necessarily constant. If the extrudate dimensions reduce just before <br>the take-up, the extrudate dimensions above it have to increase. As the larger extru-<br>date section is taken up, a thin extrudate section can form above it; this can con-<br>tinue for a long time. Thus, a cyclic variation of the extrudate dimensions can occur. <br>Draw resonance does not occur when the extrudate is solidified at the point of take-<br>up, because the extrudate dimensions at the take-up are then fixed [121, 122]. Iso-<br>thermal draw resonance is found to be independent of the flow rate. The critical <br>draw ratio for almost Newtonian fluids such as nylon, polyester, polysiloxane, etc. is <br>approximately 20. The critical draw ratio for strongly non-Newtonian fluids such <br>as polyethylene, polypropylene, polystyrene, etc. can be as low as 3 [123]. The ampli-<br>tude of the dimensional variation increases with draw ratio and drawdown length.<br>Various workers have performed theoretical studies of the draw resonance problem <br>by linear stability analysis. Pearson and Shah [124, 125] studied inelastic fluids and <br>predicted a critical draw ratio of 20.2 for Newtonian fluids. Fisher and Denn [126] <br>confirmed the critical draw ratio for Newtonian fluids. Using a linearized stability <br>analysis for fluids that follows a White-Metzner equation, they found that the criti-<br>cal draw ratio depends on the power law index n and a viscoelastic dimensionless <br>number. The dimensionless number is a function of the die take-up distance, the <br>tensile modulus, and the velocity at the die. Through their analysis, Fisher and Denn <br>were able to determine stable and unstable operating regions. In some instances, <br>draw resonance instability can be eliminated by increasing the draw ratio, although <br>under most operating conditions, draw resonance is eliminated by reducing the <br>draw ratio.<br>White and Ide [127­130] demonstrated experimentally and theoretically that poly-<br>mers whose elongational viscosity increases with time or strain do not exhibit draw <br>resonance, but undergo cohesive failure at high draw ratios. A polymer that behaves <br>in such a fashion is LDPE. Polymers whose elongational viscosity decreases with <br>time or strain do exhibit draw resonance at low draw ratios and fail in a ductile fash-<br>ion at high draw ratios. Examples of polymers that behave in such a fashion are <br>HDPE and PP. Lenk [131] proposed a unified concept of melt flow instability. His <br>main conclusions are that all flow instabilities originate at the die entrance and that <br>melt fracture and draw resonance are not distinct and separate flow phenomena; <br>both are caused by elastic effects that have their origin at the die entrance. Lenk's <br>analysis, however, is purely qualitative and does not offer much help in the engi-<br>neering design of extrusion equipment or in determining how to optimize process <br>conditions to minimize instabilities.<br>
+<hr>
+<A name=836></a> <br>
+11.3Systematic Troubleshooting  <b>825</b><br>
+<b>11.3.4.1.2Screw Frequency Instabilities</b><br>
+Screw frequency instabilities occur to a small extent in essentially every extrusion <br>operation. This can be caused by variation in the intake of polymer from the feed <br>hopper to the feed housing when the flow is interrupted every time the flight passes <br>by the feed opening. This can cause a cyclic pressure change that can be detected if <br>the extruder has accurate pressure readout. One way to reduce the unsteady intake <br>of polymer from the feed hopper is to use a double-flighted screw geometry in the <br>feed section. Generally, a better solution is to change the shape of the feed opening. <br>According to Wheeler [132], the screw frequency instability is more likely to occur <br>at very low diehead pressures.<br>Screw frequency variations can also be caused by the pressure difference across the <br>screw flight. This pressure difference is responsible for the re-circulating flow in the <br>cross-channel direction. When pressure is measured along the screw or at the very <br>end of the screw, the pressure pattern will have a sawtooth shape; see Fig. 11.29.<br>
+<b> Figure 11.29</b> <br>
+Pressure variation over time ("screw beat")<br>
+In most cases, the major cause of screw frequency instabilities will be the pressure <br>difference between the leading and trailing edge of the flight in the pump section. <br>This pressure fluctuation is often called "screw beat." This pressure difference is <br>inherent to the conveying process. It occurs even if no pressure is developed in the <br>pump section because this pressure difference is a drag-induced pressure differ-<br>ence. If the flight clearance can be neglected the pressure difference P across the <br>screw flight is:<br>
+ (11.19)<br>
+where  is the viscosity, D the screw diameter, N the rotational speed,  the screw <br>flight helix angle, p the number of parallel flights, and H the channel depth.<br>This expression is valid for a Newtonian fluid. The pressure difference increases <br>with viscosity, diameter, screw speed, and helix angle; the pressure difference <br>reduces with channel depth. When the helix angle increases from 17.5° to 25.0°, <br>the pressure difference will double. Thus, large helix angle screws will exhibit more <br>screw frequency pressure fluctuations than small helix angle screws. The screw <br>
+<hr>
+<A name=837></a><b>826</b>  <b> 11Troubleshooting Extruders</b><br>
+frequency pressure fluctuation can be reduced by placing a multi-flighted screw <br> <br>section at the end of the screw, such as a Saxton [95] or CRD [88­94] mixing section <br>as shown in Fig. 11.30. A multi-flighted mixing section at the end of the screw will <br>reduce the amplitude of the pressure fluctuation but increase its frequency.<br>
+<b> Figure 11.30</b> <br>
+Pressure fluctuation with single- and <br>
+multi-flighted screws<br>
+It should be noted that these pressure fluctuations will be most severe at the very <br>end of the screw, but they will reduce with increasing distance from the screw <br>because the polymer melt is slightly compressible. Thus, the actual pressure fluc-<br>tuation will be very much dependent on the location of the pressure transducer. <br>When a breaker plate is used, the pressure fluctuation will reduce significantly <br>because the breaker plate will largely break up the flight-induced pressure fluctua-<br>tion as shown in Fig. 11.29.<br>Obviously, the screw frequency pressure fluctuation will be problematic when the <br>value of P is large relative to the actual diehead pressure. This will occur when the <br>diehead pressure is low, as pointed out by Wheeler [68], when the polymer melt <br>viscosity is high, the screw diameter large, the screw speed high, the helix angle or <br>pitch large, or when the channel depth is shallow.<br>
+<b>11.3.4.1.3Low-Frequency Instabilities</b><br>
+Low-frequency instabilities have been associated with solid bed breakup [69, 70]. <br>Fenner et al. have attempted to theoretically predict solid bed breakup [69]. They <br>proposed that solid bed breakup is due to acceleration of the solid bed in the plasti-<br>cating zone of the extruder and claimed that no solid bed acceleration occurs in the <br>absence of a melt film between the solid bed and the screw. In practice, formation of <br>a melt film can be avoided by cooling the screw [71]. Earlier, Maddock [72] found <br>that screw cooling helped in reducing surging. The most likely reason that screw <br>cooling reduces surging is that it reduces the throughput rate by a substantial <br>amount, about 20% in the experiments of Fenner and Edmondson [71]. Therefore, <br>the melting process will be completed over a shorter axial distance, reducing the <br>stresses acting on the solid bed. Solid bed breakup is also more likely to occur on <br>screws with a high compression ratio. Fenner et al. [69, 71] found solid bed breakup <br>with high compression ratio screws (3:1 and 4:1), but did not find solid bed breakup <br>
+<hr>
+<A name=838></a> <br>
+11.3Systematic Troubleshooting  <b>827</b><br>
+with a low compression ratio screw (2.25:1). A low compression ratio screw would <br>seem a better solution than a high compression ratio screw with screw cooling. <br>Another method by which formation of a melt film on the screw surface can be <br>avoided is by using barrier screw geometry.<br>Fluctuations occurring over about 10 to 30 s can be caused by temperature fluctua-<br>tions along the extruder barrel. The temperature fluctuations may not be noticeable <br>from the temperature readouts. This can be due to the slow response of many tem-<br>perature sensors and because the sensors are often located a considerable distance <br>from the polymer/metal interface. However, if the actual temperature at the inter-<br>face fluctuates, there will be a corresponding fluctuation in the flow rate. In time-<br>proportioning temperature control systems, power is added or removed at relatively <br>short intervals, typically about 15 to 20 s. These bursts of heating or cooling energy <br>will cause short-term changes in the polymer/metal interface temperature with <br> <br>corresponding variations in throughput rate. The variation in throughput can be as <br>much as 5 to 10%. Therefore, from a stability point of view, the true proportioning <br>temperature control is significantly better than the time-proportioning temperature <br>control.<br>Throughput variations caused by wall temperature changes have been described <br>by Gitschner and Lutterbeck [76]. They were able to show a very clear correlation <br>between the on-and-off cycling of barrel cooling and the diehead pressure fluctua-<br>tions. They also found that the pressure fluctuations correlated very closely with the <br>resulting throughput fluctuations. It should be clear that these temperature-induced <br>throughput fluctuations could be considerably more severe in the case of on-off tem-<br>perature control.<br>
+<b>11.3.4.1.4Very Slow Fluctuations</b><br>
+Very slow fluctuations are often associated with poor temperature control, changes <br>in ambient conditions (room temperature, relative humidity), plant voltage varia-<br>tion, and similar causes. A steady, slow reduction in output is often caused by build-<br>up of contaminants on the screen pack.<br>
+<b>11.3.4.1.5Random Fluctuations</b><br>
+Random fluctuations are often associated with irregular feeding. Maddock [72] dis-<br>cussed a case where the extruder performance was very sensitive to the level of fill <br>in the feed hopper. Random fluctuations can also result from a combination of cyclic <br>fluctuations. Figure 11.31 shows the pattern of a regular sinusoidal variation.<br>Figure 11.32 shows a combination of three sinusoidal variations with different fre-<br>quency and amplitude. This variation appears to be random; however, it is made up <br>of three different components with each component being a very regular sinusoidal <br>variation.<br>
+<hr>
+<A name=839></a><b>828</b>  <b> 11Troubleshooting Extruders</b><br>
+<b> Figure 11.31</b> <br>
+Regular sinusoidal variation<br>
+<b> Figure 11.32</b> <br>
+Combination of three sinusoidal variations<br>
+This situation frequently occurs in extrusion because in many cases the process is <br>affected by variation from different sources. The variation from different sources <br>will typically have different frequencies and amplitudes. Obviously, the more <br>sources of variation act on the process, the more complicated and random the infor-<br>mation from the process will tend to be.<br>The pattern of variation as shown in Fig. 11.32 is often detected by measuring melt <br>pressure. Obviously, the response time of the measurement has to be short enough <br>to capture high-frequency variations occurring in the process. Fast Fourier Trans-<br>form (FFT) analysis can be used to analyze a complex signal and decompose it into <br>the base frequencies. As a result, FFT is a powerful tool in troubleshooting complex <br>extrusion problems. Becker et al. [133, 134] used FFT of melt pressure signals to <br>analyze extrusion instabilities. Reinhard et  al. [144] describe the application of <br>spectral analysis to surging problems in extrusion.<br>
+<b>11.3.4.2Functional Instabilities<br></b>Another method of classifying instabilities is by the functional zone in which the <br>instability originates. Thus, the following instabilities can be distinguished:<br>
+ <br>
+ Solids conveying instabilities<br>
+ <br>
+ Plasticating instabilities<br>
+ <br>
+ Melt conveying instabilities<br>
+ <br>
+ Devolatilization instabilities<br>
+ <br>
+ Mixing instabilities<br>
+ <br>
+ Die forming instabilities<br>We will discuss each of these in detail.<br>
+<hr>
+<A name=840></a> <br>
+11.3Systematic Troubleshooting  <b>829</b><br>
+<b>11.3.4.2.1Solids Conveying Instabilities</b><br>
+Solids conveying instabilities have three major causes: flow problems in the feed <br>hopper, internal deformation of the solid bed in the screw channel, and insufficient <br>friction against the barrel surface. Flow problems in the feed hopper can be detected <br>by observing the flow from the feed hopper when it is disconnected from the <br>extruder. Solids conveying problems in the extruder itself are difficult to diagnose. <br>One method that can be used is to Teflon-coat the screw. Even though the coating <br>may not last very long, it will substantially reduce the retarding force acting on <br>the solid bed and thus improve solids conveying. If the coating eliminates the insta-<br>bility, this is a strong indication of a solids conveying problem. A more permanent <br>solution can be provided by a grooved barrel section or a nickel-plated screw impreg-<br>nated with a fluorocarbon polymer.<br>The stability of solids conveying is strongly related to the uniformity of the feed-<br>stock. The best stability is achieved with uniform pellet size and pellet shape. Large <br>variations in particle size and shape invariably lead to variations in extruder perform-<br>ance. This is often observed when regrind is added to the virgin feedstock. Since <br>the regrind usually has non-uniform particle size and shape, increasing amounts of <br>regrind will reduce the stability of the extrusion process. In many cases, this reduc-<br>tion in stability will put an upper limit on the amount of regrind that can be added <br>to the extruder.<br>
+<b>11.3.4.2.2Plasticating Instabilities</b><br>
+Plasticating problems are likely to occur on screws with a high compression ratio <br>and a short compression section length. They are also likely to occur when the over-<br>all length of the extruder is short. A short extruder will run into melting-related <br>instabilities at lower output than a longer extruder. This is one of the reasons behind <br>the trend in the extrusion industry to go to longer extruders. In the 1950s and <br>1960s, most single screw extruders were about 20 D long. In the 1970s and 1980s, <br>most extruders were about 25 D long. In the 1990s and 2000s, most single screw <br>extruders are about 30 D long.<br>Insufficient melting capacity can be diagnosed by preheating the feedstock. If pre-<br>heating reduces or eliminates the instability, then the problem is most likely insuf-<br>ficient melting capacity. Melting can be improved either by changing the processing <br>conditions or by changing the screw geometry. Processing conditions that can <br>improve melting:<br>
+ <br>
+ Preheating of the feedstock<br>
+ <br>
+ Increasing the barrel temperature when running at low screw speed<br>
+ <br>
+ Reducing the barrel temperature when running at high screw speed<br>
+ <br>
+ Reducing the screw speed<br>
+ <br>
+ Increasing the barrel pressure<br>
+<hr>
+<A name=841></a><b>830</b>  <b> 11Troubleshooting Extruders</b><br>
+Screw design changes that can improve the melting capability are:<br>
+ <br>
+ Increased flight helix angle in transition section<br>
+ <br>
+ Reduced flight clearance in transition section<br>
+ <br>
+ Increased length of transition section<br>
+ <br>
+ Multi-flighted design in transition section<br>
+ <br>
+ Use of fluted mixing section at the end of the transition section<br>
+<b>11.3.4.2.3Melt Conveying Instabilities</b><br>
+Most melt conveying or pumping problems are caused by improper design of the <br>metering section. The most common problem is excessive channel depth; the second <br>most common problem is insufficient length of the metering section. If the channel <br>depth is too large, the metering end of the screw can be cooled to reduce the effec-<br>tive depth. Conversely, if the channel depth is too small the metering end of the <br>screw can be cooled to increase the effective depth of the channel and increase the <br>melt conveying capability. However, if the metering depth is incorrect it is better to <br>switch to a screw design with proper dimensions of the metering section.<br>In some cases, the melt conveying capability cannot be improved sufficiently by a <br>simple change in screw design. An example is a two-stage extruder screw that has to <br>operate at high discharge pressure. If the second stage of the screw is not long <br>enough, it may not be possible for the screw to generate the required pressure. Such <br>a condition will result in vent flow; this is molten polymer flowing out of the vent <br>port. One possible solution to such a problem is to place a gear pump between the <br>extruder and the die. In this set-up the gear pump can generate the required die-<br>head pressure, and the screw only has to generate enough pressure to feed the gear <br>pump. Other possible solutions for vent flow problems are:<br>
+ <br>
+ Reduce diehead pressure<br>
+ <br>
+ Increase the length of the extruder<br>
+ <br>
+ Use internal screw heating in the second stage of the screw<br>
+ <br>
+ Reduce the barrel temperature in the second stage<br>
+<b>11.3.4.2.4Devolatilization Instabilities</b><br>
+Devolatilization instabilities can be caused by plugging of the vent port, variation in <br>the vacuum level, or by variations of the volatile level in the feedstock. The effi-<br>ciency of devolatilization can be improved by:<br>
+ <br>
+ Increasing barrel temperatures in the first stage<br>
+ <br>
+ Increasing the vacuum level at the vent port<br>
+ <br>
+ Preheating the feed stock<br>
+ <br>
+ Use of a stripping agent<br>
+<hr>
+<A name=842></a> <br>
+11.3Systematic Troubleshooting  <b>831</b><br>
+Screw design can affect the devolatilization efficiency a great deal. A multi-flighted <br>extraction section can improve degassing. Further, it is important that the polymer <br>is fully melted in the first stage of the screw. To ensure complete melting it is bene-<br>ficial to place a fluted mixing section at the end of the first stage of the screw.<br>
+<b>11.3.4.2.5Mixing-Related Instabilities</b><br>
+Extrusion instabilities are often related to insufficient mixing capacity of the screw. <br>Mixing can be improved a small amount by increasing the diehead pressure. How-<br>ever, this is a relatively ineffective method to improve the mixing capacity of the <br>extruder: it also increases the chance of degradation. The mixing capacity of a screw <br>can be improved significantly by adding one or more mixing sections to the design <br>of the screw.<br>The polymer melt has to be well mixed when it leaves the extruder screw and enters <br>into the die. For this reason it is beneficial to have an efficient distributive mixing <br>device at the very end of the screw. Mixing was discussed in detail in Section 7.7 <br>and mixing devices and mixing screws in Section 8.7.<br>
+<b>Solving Mixing Problems<br></b>In addition to have efficient mixing devices along the screw, there are several other <br>issues that are important in achieving good mixing. The method of feeding plays an <br>important role in the mixing action in the extruder.<br>
+<b>Flood Feeding versus Starve Feeding<br></b>Most single screw extruders are flood fed; however, flood feeding is often detrimen-<br>tal to achieving good mixing in the extruder. With flood feeding, high pressures are <br>generated in the solids conveying and plasticating zones of the extruder. These high <br>pressures tend to agglomerate ingredients that later need to be dispersed and dis-<br>tributed [135­137]. Obviously, this can be highly counter-productive.<br>In starve feeding the material is metered into the extruder with a feeder. As a result, <br>there is no accumulation of material at the feed opening. The first several turns of <br>the screw are partially filled with material without any pressure development in <br>this part of the extruder. The screw channel does not become completely filled until <br>some distance from the feed opening; at this point, the pressure will start building <br>up in the extruder. In effect, starve feeding reduces the effective length of the <br>extruder.<br>One of the benefits of starve feeding is that the pressures along the extruder are <br>lower than in flood feeding. Therefore, there is less chance of agglomeration, result-<br>ing in improved mixing action in the extruder. Recently, a number of workers have <br>analyzed the effect of starve feeding on the mixing capability of extruders and injec-<br>tion molding machines [138­142]. Without exception, all these investigators found <br>
+<hr>
+<A name=843></a><b>832</b>  <b> 11Troubleshooting Extruders</b><br>
+major improvements in mixing quality in starve feeding compared to flood feeding. <br>Starve feeding has become the standard mode of operation for twin screw extruders <br>used in compounding. However, the benefits of starve feeding are not limited to twin <br>screw extruders. Mixing in single screw extruders can be improved significantly by <br>using starve feeding.<br>
+<b>Initial Scale of Segregation<br></b>The mixing action that is required in an extruder is determined both by the initial <br>and final scale of segregation. The required final scale of segregation is usually <br>around 1 micron. If the initial scale of segregation is the size of a pellet (about 3000 <br>micron), the mixing action will have to produce a 3000-fold reduction in the scale <br>of segregation. In simple shear mixing, this will require a total strain rate of 3000 <br>units [143]. If the average shear rate is 50 s­1 the polymer melt will have to be <br>exposed to this shear rate for 60 s to produce a total shear strain of 3000. Consider-<br>ing that a typical residence time in the melt conveying zone is 15 to 20 s, it is clear <br>that the mixing action will be insufficient to reduce the scale of segregation down to <br>the micron level.<br>This situation can be improved if we start the mixing process with ingredient in <br>powder form rather than in pellet form. If the size of the powder particles is 100 <br>micron and the powder is well mixed before extrusion, the initial scale of segrega-<br>tion will be 100 micron. In this case, the mixing action will have to produce only a <br>100-fold reduction in the scale of segregation to achieve a final scale of one micron. <br>At a shear rate of 50 s­1 this will take a shear time of 2 s. Clearly, there is a very good <br>chance that a single screw extruder will be able to accomplish this mixing task <br>without much trouble. The difference in the scale of segregation is illustrated in <br>Fig. 11.33.<br>
+<b> Figure 11.33</b> <br>
+A coarse scale of segregation (left) and a fine scale <br>
+of segregation (right)<br>
+One of the most difficult mixing tasks is to mix a low percentage of a pelletized color <br>concentrate (CC) with natural pellets. Assume that the pellet size is 3 mm (3000 <br>micron) and that 1% CC is added to the natural pellets. This means that for every CC <br>pellet there will be 100 natural pellets. The initial scale of segregation in this case <br>will be about 30 mm (30,000 micron). Since not all CC pellets will have the same <br>distance from one another, the actual scale of segregation can be as high as 100 mm <br>(100,000 micron). If we want to achieve a final scale of segregation of 1 micron, the <br>mixing action will have to achieve a 100,000-fold reduction in scale. If the average <br>
+<hr>
+<A name=844></a> <br>
+11.3Systematic Troubleshooting  <b>833</b><br>
+shear rate is 50 s­1 the polymer melt will have to be exposed to this shear rate for <br>200 s (more than 3 minutes) to produce a total shear strain of 100,000. It is very <br>unlikely that this can be accomplished in a simple conveying screw in a single screw <br>extruder. As a result, a very efficient mixing screw will have to be used to accom-<br>plish this mixing task successfully.<br>
+<b>11.3.4.3Solving Extrusion Instabilities<br></b>There are many different causes of extrusion instabilities. Even though the mecha-<br>nism of the instability is not always clear, the following measures often reduce <br>extrusion instabilities:<br>
+ <br>
+ Reduce the screw speed<br>
+ <br>
+ Reduce the screw temperature<br>
+ <br>
+ Reduce the barrel temperature at the delivery end<br>
+ <br>
+ Reduce the channel depth in the metering section<br>
+ <br>
+ Increase the length of the compression section<br>
+ <br>
+ Increase the rear barrel temperatures<br>
+ <br>
+ Increase the diehead pressure<br>The first approach to the problem is generally adjustment of the temperature profile <br>or other process conditions. If temperature adjustment does not solve the problem, <br>one should check the hardware: thermocouples, controllers, screw, barrel, drive, etc. <br>If the problem is not associated with the hardware, it must be a functional problem <br>and one should determine what functional zone is causing the problem. The trouble-<br>shooting flow chart in Fig. 11.34 can help in systematically exploring the possible <br>causes of the instability.<br>If the problem cannot be solved by changing the processing conditions, which is, of <br>course, the first choice, then one can generally solve the problem either by making <br>a material change or by making a change in screw or barrel design. In most cases, <br>material changes are not possible. In that case, the problem usually has to be solved <br>by a new screw design. Another option is to add a gear pump at the end of the <br>extruder. Figure 11.35 illustrates some of the important interactions that take place <br>during the extrusion process.<br>Because of the complicated interactions that occur during the extrusion process, it <br>is often difficult to predict the effect of a change in process conditions or in screw <br>design. For instance, increasing the barrel temperature will generally increase the <br>polymer melt temperature at the discharge end of the extruder. However, it is also <br>possible for the melt temperature to reduce with increasing barrel temperatures. <br>This is possible because, when the barrel temperature is increased, the local melt <br>viscosity can reduce, which will reduce the local viscous heating. This effect can <br>result in lower melt temperature at the discharge end of the extruder.<br>
+<hr>
+<A name=845></a><b>834</b>  <b> 11Troubleshooting Extruders</b><br>
+<b>Cyclic variations</b><br>
+Melt fracture/shark skin<br>Draw resonance<br>
+Yes<br>
+Melt temperature variation<br>Screw speed variation<br>
+Cycle time &lt; 1 screw revolution<br>
+Ta ke-up speed variation<br>Vibration<br>
+No<br>
+Yes<br>
+Fluctuation in feed rate<br>
+Cycle time = 1 screw revolution<br>
+&quot;Screw beat&quot;<br>
+No<br>
+Yes<br>
+Cycle time = 5-20 screw revolutions<br>
+Melting instabilities<br>
+No<br>
+Yes<br>
+Cycle time =2-15 minutes<br>
+Te mperature fluctuations<br>
+No<br>
+Ambient variations<br>
+Yes<br>
+Cycle time = hours-days<br>
+Shift changes<br>Day/night differences<br>
+No<br>
+Yes<br>
+Irregular feeding<br>
+Random fluctuations<br>
+Combination of cyclic variations<br>Contamination of feed stock<br>Feed stock variation<br>Plant voltage variation<br>Random changes in conditions<br>
+<b>Figure 11.34</b>Troubleshooting flow chart for fluctuation in extruder performance<br>
+Time<br>
+Degradation<br>
+Shear working<br>
+Te mperature<br>
+Pressure<br>
+<b> Figure 11.35</b> <br>
+Interactions during the extrusion <br>
+Visco-elastic properties<br>
+ <br>process<br>
+<b>11.3.5Air Entrapment</b><br>
+Air entrapment is a rather common problem in extrusion. It is caused by air being <br>dragged in with particulate material from the feed hopper. Under normal conditions, <br>the compression of the solid particulate material in the feed section will force the air <br>out of the solid bed. However, under some circumstances the air cannot escape back <br>to the feed hopper and travels with the polymer until it exits from the die. As the air <br>pockets exit from the extruder, the sudden exposure to a much lower ambient pres-<br>sure may cause the compressed air bubbles to burst in an explosive manner. How-<br>ever, even without the bursting of the air bubbles, the extrudate is generally ren-<br>dered unacceptable because of the air inclusions.<br>
+<hr>
+<A name=846></a> <br>
+11.3Systematic Troubleshooting  <b>835</b><br>
+There are a number of possible solutions to air entrapment. The first approach <br>should be to change the temperature in the solids conveying zone to achieve a more <br>positive compacting of the solid bed. Often, a temperature increase of the first barrel <br>section reduces the air entrapment; however, in some cases, a lower temperature <br>causes an improvement. In any case, the temperatures in the solids conveying zone <br>are important parameters in the air entrapment process. It should be realized that <br>both the barrel and screw temperatures are important. Thus, if a screw temperature <br>adjustment capability is available, it should definitely be used to reduce the air <br>entrapment problem.<br>The next step is an increase in the diehead pressure to alter the pressure profile <br>along the extruder and to achieve a more rapid compacting of the solid bed. The <br>diehead pressure can be increased by simply adding screens in front of the breaker <br>plate. Another possible solution is to starve feed the extruder; however, this may <br>reduce extruder output and requires additional hardware, i.e., an accurate feeding <br>device.<br>The aforementioned recommended solutions can be implemented rather easily. <br>However, if these measures do not solve the problem, more drastic steps have to be <br>taken. One possibility that needs to be explored is a change in particle size or shape. <br>If this is a reasonable option, it will most likely solve the problem. A rather safe solu-<br>tion is to utilize a vacuum feed hopper system; however, these systems are rather <br>complex and expensive. Another possible solution is to use a grooved barrel section. <br>Pressure development in a grooved barrel section is much more rapid than in a <br>smooth barrel. Thus, a grooved barrel section causes a rapid compacting of the solid <br>bed and, therefore, less chance of air entrapment. Instead of grooving the barrel, <br>one can opt for reducing the friction on the screw, which would have a similar effect. <br>A coating that might be used for this purpose is described by Luker [35]. Air entrap-<br>ment is also often successfully eliminated by vented extrusion using a multi-stage <br>extruder screw. Increasing the compression ratio of the screw is also likely to reduce <br>air entrapment.<br>It should be noted that bubbles in the extrudate are not only a sign of air entrap-<br>ment, but it may also be an indication of moisture, surface agents, volatile species <br>in the polymer itself, or degradation as shown in the fishbone diagram shown in <br>Fig. 11.36.<br>
+Air entrapment<br>
+Shrink voids<br>
+Degradation<br>
+Volatiles<br>
+Voids in<br>
+product<br>
+Vent flow<br>
+Contamination<br>
+Plugged vent port<br>
+<b> Figure 11.36</b> <br>
+Particle size<br>
+Vacuum too low<br>
+Fishbone chart for voids in <br>
+and shape<br>
+Inefficient venting<br>
+extruded product<br>
+<hr>
+<A name=847></a><b>836</b>  <b> 11Troubleshooting Extruders</b><br>
+Thus, before concentrating on solving an apparent air entrapment problem, one <br>should make sure that the problem is indeed caused by air entrapment. In some <br>cases, the pellets contain small air bubbles within the pellet itself. In this case, one <br>of the few possible solutions is vented extrusion; most of the other recommended <br>solutions will not work in this situation. The fishbone diagram shown above helps in <br>systematic troubleshooting. The diagram can be used to put together a troubleshoot-<br>ing flow chart for this kind of problem. Figure 11.37 shows an example of a trouble-<br>shooting flow chart for voids in the extruded product.<br>
+<b>Voids in extruded product</b><br>
+Reduce cooling rate by: <br>
+Yes<br>
+Cooling too fast?<br>
+ * Lower melt temperature <br> * Increase distance die/water trough <br>
+Check moisture level if necessary <br>
+ * Increase water trough temperature <br>
+No<br>
+Dry compound before extrusion <br>
+ * Use multiple, short water troughs <br>
+Yes<br>
+Remove volatile component from compound <br>
+ * Reduce line speed<br>
+Volatiles?<br>
+Lower temperatures in extrusion<br>
+ * Heat extrudate at die exit<br>
+No<br>
+Change barrel temperatures <br>
+Reduce stock temperatures <br>
+Yes<br>
+Air entrapment?<br>
+Increase barrel pressure <br>
+Reduce residence times <br>
+Use larger particle size feed stock <br>
+Reduce holdup in extruder <br>
+No<br>
+Screw with higher compression ratio <br>
+Reduce holdup in die <br>
+Screw with shorter feed section <br>
+Add stabilizer to compound <br>
+Yes<br>
+Degradation?<br>
+Grooved feed extruder <br>
+Remove degradation promoting substances <br>
+Vented extruder <br>
+Use nitrogen blanket at feed port<br>
+Vacuum feed hopper system<br>
+No<br>
+Yes<br>
+Increase particle size<br>
+Small particle size?<br>
+No<br>
+Eliminate vent flow <br>
+Yes<br>
+Remove vent port buildup <br>
+Inefficient venting?<br>
+Increase vacuum at vent port <br>Improve screw geometry<br>
+No<br>
+Yes<br>
+Eliminate contamination<br>
+Contamination?<br>
+<b>Figure 11.37</b>Troubleshooting flow chart for voids in the extruded product<br>
+<b>11.3.6Gels, Gel Content, and Gelation</b><br>
+The term "gel" has different meanings in the polymer extrusion industry. The term <br>gel is used as "gel content" in crosslinked polymers. Determination of gel content is <br>described in ASTM D2765, standard test methods for determination of gel content <br>and swell ratio of crosslinked polyethylene plastics [163]. The gel content (insoluble <br>fraction) produced in ethylene plastics by crosslinking is determined by extracting <br>with solvents such as decahydronaphthalene or xylenes.<br>The term "gelation" is used in extruded products made of rigid polyvinylchloride <br>(PVC). The term gelation is the fusion of the primary particles of the PVC [164]. In <br>
+<hr>
+<A name=848></a><IMG src="Polymer Extrusion by Chris Rauwendaal-848_1.jpg"><br>
+ <br>
+11.3Systematic Troubleshooting  <b>837</b><br>
+the plastics industry the term gelation is only used for PVC plastics. Insufficient <br>gelation leads to premature failure of PVC products [164, 165]. Two techniques are <br>widely used to determine the level of gelation in PVC. One is the acetone immersion <br>test described in ASTM D2152; the other is the dichloromethane test described in <br>ISO 9852. Gramann and Cruz describe the use of testing by differential scanning <br>calorimetry to determine the extent of gelation in rigid PVC [165].<br>Low levels of gelation are generally associated with stock temperatures that are too <br>low in the PVC extrusion process. Proper levels of gelation typically require stock <br>temperatures above 190°C. However, at temperatures above 200°C, rigid PVC <br>(RPVC) degrades rapidly. As a result, the process window for RPVC is quite narrow, <br>and close control of stock temperatures is essential in making a high-quality RPVC <br>product. Also, proper melt temperature measurement in RPVC extrusion is critical. <br>If melt temperature is not properly measured, the melt temperature cannot be con-<br>trolled and improper gelation levels can result in the extruded product.<br>Gels are generally defined as small, more or less round defects in extruded pro-<br>ducts, especially film or thin walled tubing. Some people define gels as any particle <br>in an extruder plastic product that has visual properties different from the rest of <br>the product. This includes discolored specks, contamination, crosslinked polymer <br>droplets, etc.<br>We will define gels as small spherical droplets or specks with a distinct boundary that <br>can be observed by simple visual inspection. The material making up the gel particle <br>is basically the same as the polymer of the surrounding film. Therefore, a gel particle <br>is different from contamination. In many cases the gel particle has no discoloration; <br>see for instance Fig. 11.38. Small droplets with strong discoloration are generally <br>referred to as discolored specks; this topic is discussed in Section 11.3.7.4.<br>Section 11.3.7.4 also shows expressions that can be used to relate the frequency of <br>discolored specks in the raw material to the frequency of discolored specks in the <br>extruded product. These expressions can also be used to relate the frequency of gels <br>in the raw material to the frequency of gels in the extruded product.<br>A photograph of a gel defect in blown film is shown in Fig. 11.38.<br>
+<b> Figure 11.38</b> <br>
+Transparent gel particle in HDPE film, <br>
+ <br>courtesy of Dr . Cantor<br>
+<hr>
+<A name=849></a><IMG src="Polymer Extrusion by Chris Rauwendaal-849_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-849_2.jpg"><br>
+<b>838</b>  <b> 11Troubleshooting Extruders</b><br>
+The gels shown in Figs. 11.38 and 11.39 were created in a blown film extrusion <br> <br>process. The polymer is an HDPE (ExxonMobil Paxon AA45-004) with a melt index <br>of 0.35 gr/10 min. The gel in Fig. 11.38 is about 0.5 mm in diameter; it has no sig-<br>nificant discoloration.<br>Figure 11.39 shows two pictures of gel defects with discoloration.<br>
+<br>
+<b>Figure 11.39</b>Photographs of gel defects in polyethylene film, courtesy of Dr . Cantor<br>
+The gel defects are approximately 1 mm in size and show brownish discoloration, <br>indicative of oxidative degradation.<br>There used to be an ASTM standard for test method for counting gels in film; this <br>was ASTM D3351-93. However, this standard was withdrawn and has not been <br>replaced. Gels can range in size from 100 micron or smaller to as large as 1500 <br>micron or even larger. Sizes smaller than 100 micron are difficult to detect; see Fig. <br>11.40. The pellets in this figure have a length of 3.5 mm and height of 1.5 mm.<br>
+250 micron<br>
+31 micron<br>
+125 micron<br>
+16 micron<br>
+8 micron<br>
+62 micron<br>
+8 micron<br>
+<b>Figure 11.40</b>Pellets with defects ranging from 250 to 8 micron<br>
+<hr>
+<A name=850></a> <br>
+11.3Systematic Troubleshooting  <b>839</b><br>
+Gels are usually crosslinked polymer particles or highly entangled particles that <br>behave as crosslinked particles. Gels can be generated in polymerization, pelletiz-<br>ing, resin transfer, transport to processor, conveying through transfer lines, con-<br>tamination, extrusion, in regrind, etc. It is important to understand that gels are not <br>only formed inside the extruder at the processor. Gels generated in the extrusion <br>process are referred to as E-gels.<br>Gels created in polymerization and downstream operations such as pelletizing are <br>referred to as P-gels. Resin producers are well aware of this fact. Specialized equip-<br>ment is available that allows resin producers to detect defects in the pellets. An <br>example of a company that manufactures high-speed inspection systems is Optical <br>Control Systems GmbH (OCS); see [166]. This equipment allows analysis of millions <br>of pellets. As a result, resin producers can determine how many gels are produced in <br>their resin manufacturing processes.<br>
+<b>11.3.6.1Measuring Gels<br></b>When there is a problem with gels in an extruded product, it is important to deter-<br>mine whether the gels are in the incoming raw material or created in the extrusion <br>process. Clearly, this is an issue where the resin supplier may be in disagreement <br>with the processor! To determine whether the incoming material contains gels, <br>the material must be tested for gels. One method to test for P-gels is to press a thin <br>plaque using the polymer pellets as supplied by the resin producer and visually <br>examine the plaque for gels. The plaque has to be prepared in such a way as to <br>minimize the exposure to high temperatures to make sure that gels are not created <br>in the sample preparation process. The number of gels per unit area can be counted <br>using an overhead projector and polarized film to project an image on a screen. The <br>number of gels per unit area is a measure of the amount of gels in the material. <br>Obviously, the conditions used to press the plaque and the thickness of the plaque <br>have to be standardized for the measurements to be meaningful.<br>For extruded film there is an existing standard for manual gel counting that does <br>not appear to be used much today [155]. There are automatic gel counting methods <br>that are used by a number of companies; these methods are based on laser or CCD <br>camera technology. However, these methods are not standardized and each company <br>tends to use their own procedures. A proposed ASTM procedure has been submitted <br>for these automatic gel detection methods. However, many issues remain to be re -<br>solved such as a standardized method to report the results, what size gels to count, <br>how to report the distribution of gel sizes, etc. Various end-use applications have <br>different requirements with regard to gels. As a result, it is difficult to develop one <br>standardized test method that satisfies all requirements.<br>In fiber extrusion, in particular PP, a screen build-up test is used in some cases. <br>This test reflects not just the gel level in the polymer because other materials can <br>be trapped in the screen as well. Also, not all gel particles may be captured in the <br>
+<hr>
+<A name=851></a><b>840</b>  <b> 11Troubleshooting Extruders</b><br>
+screen. For best gel capture capability a 3-D fiber filter with a specified rating should <br>be used. The build-up can be quantified by monitoring the increase in pressure drop <br>over time. The time to reach a specific pressure drop is a measure of the percentage <br>of gels greater than a certain size removed in the filtration process. A regular wire <br>mesh screen pack is not suitable for this test because wire mesh screens have very <br>limited gel capture capability.<br>Gels can be characterized using hot-stage microscopy (HSM). This method allows <br>slow heating of a film sample with a gel on a microscope hot stage to a temperature <br>above the melting point of the polymer. Transmitted light is passed through cross-<br>polarized filters. By analyzing the melting point of the film and the gel, different <br>types of gels can be identified. Birefringence effects allow further identification.<br>Gels caused by contamination will have a different chemical composition from the <br>polymer used to make the product. As a result, chemical analysis can determine <br>whether a gel particle is contamination or if it has the same chemical composition <br>as the polymer. Gels can be analyzed by micro-infrared analysis. The infrared spec-<br>trum can determine whether the gel is from a foreign material or if it similar to the <br>polymer.<br>Crosslinked gels can have crystallinity; therefore, they can be birefringent under <br>polarized light. A hot-stage polarizing microscope is useful for this analysis. To <br>determine whether the gel is crosslinked the gel can be heated above the melting <br>point of the polymer. At this elevated temperature the gel can be stressed by care-<br>fully applying a force to the gel. A crosslinked gel will appear birefringent under <br>polarized light. If the size and shape of the gel remains after cooling, the gel is <br>crosslinked. If the gel was not crosslinked but highly entangled, the gel would dis-<br>appear after the stress was applied and subsequent cooling.<br>Fines are created when off-spec product is shredded into regrind. If the fines are not <br>removed they will tend to show up as gels in the extruded product. The thermo-<br>mechanical action of the shredder creates fines that do not melt like regular pellets. <br>For that reason it is important to remove fines from the regrind before it is reintro-<br>duced to the extruder. Special equipment is available to remove fines from pellets; <br>these machines are called dedusters.<br>
+<b>11.3.6.2Gels Created in the Extrusion Process<br></b>To avoid E-gels, it is important to avoid dead spots in the extruder. This can be <br>accomplished by making sure that both the screw and the die have a streamlined <br>design. Mixing sections with stagnating regions, such as the Maddock mixing sec-<br>tion, should be avoided. It is also important that the screw, barrel, and die surfaces <br>are smooth without grooves, scratches, or gouges that might collect melted plastic <br>and cause degradation. Another method of reducing gel formation in the extruder <br>is to start up the extruder with a highly stabilized version of the plastic, or even a <br>
+<hr>
+<A name=852></a> <br>
+11.3Systematic Troubleshooting  <b>841</b><br>
+different plastic, to coat the critical surfaces with a degradation-resistant layer of <br>plastic. This can reduce the chance of degradation and gel formation.<br>It is also important to check the resin feed tubes, blenders, feeders, hoppers, and <br>other bulk handling hardware components for fines, streamers, or contamination <br>from another plastic. To avoid fines, streamers, and contamination, the bulk hand-<br>ling equipment should be completely blown down and cleaned when a material <br>change is made.<br>
+<b>11.3.6.3Removing Gels Produced in Polymerization<br></b>Howard [168] discusses gels created in polymerization. He mentions that the most <br>common defect found by resin suppliers is the crosslinked gel. These result from <br>dislodged pieces of reactor or separator "plaque." These crosslinked gels tend to be <br>evenly distributed in the polymer and are carried through processing extruders in a <br>predictable manner.<br>Some resin producers supply their customers with spec sheets that show the maxi-<br>mum level of gels in the resin. An example is shown in Fig. 11.41 [167].<br>
+Certificate 7654321                The Dow Chemical Company                                Page       1 <br> <br>Date: 10/20/2008                Certificate of Analysis              Shipped: 10/20/2008 <br>
+ <br>Valued <br>Dow Customer:<br>
+XYZ Corp. <br>1234 MAIN STREET <br>CITY                   ST 000000-0000    UNITED STATES <br>
+ <br>Cust P.O.:    1212121212                                                               Dlvy Note: 12345678 10 <br>                                                                                                         Order No.: 8754321<br>
+Material:<br>
+NORDEL* MG 47085 Hydrocarbon Rubber <br>25 KG Bag <br>40 Bags on a Pallet<br>
+       Spec: 000000 <br>
+Cust Mtl: <br> <br>Batch:<br>
+ABC0000000  ABC0000000 <br>
+ <br>Dlvy Qty:<br>
+BG       10      172 <br>
+Vehicle:         X0000 <br> <br>Ship from:  THE DOW CHEMICAL COMPANY     LA PORTE,  TX UNITED STATES <br> <br>                                         Results                                     Limits <br>Feature                     Units      For Batch No.        Minimum    Maximum         Method <br>--------------------------------------------------------------------------------------------------------------------- <br>Mooney Viscosity     unit                  85                       80              90                Calculated <br>ML1+4 @ 125oC (Polymer) <br> <br>Mooney Viscosity     unit               105                        97             113                Calculated <br>ML1+4 @ 125oC (Standard Compound) <br> <br>Carbon Black        P/100R              24                        22              26                Mass Balance <br> <br>Ethylene                  % wt             69.5                    68.0            71.0               ASTM D3900 <br> <br>ENB<br>
+           % wt               4.5                      4.0            5.0               ASTM D6047 <br>
+ <br>Gels<br>
+         ppm (v)        &lt; 12.0                      ---            12.0              Dow Method <br>
+ <br> <br> <br> <br>
+&lt; 12.0<br>
+Jane Doe <br>
+<b> Figure  11.41</b> <br>
+Quality Systems Specialist <br> <br>
+Certificate of resin producer for <br>
+For inquiries please contact Customer Service or local sales. <br>English: 800-232-2436      French: 800-565-1255<br>
+maximum gel level<br>
+The processor should know the level of gels in the incoming raw material to know <br>the gel level and to determine whether this level is relatively constant or subject to <br>substantial variation.<br>
+<hr>
+<A name=853></a><b>842</b>  <b> 11Troubleshooting Extruders</b><br>
+Unfortunately, it is quite difficult to remove P-gels in a regular extrusion process. <br>Adding dispersive mixing elements to the extruder screw usually does not achieve <br>sufficient mixing to eliminate P-gels. Also, the screen pack typically used before the <br>breaker plate does not have enough gel capture capability to significantly reduce <br>the gel level. New dispersive mixers capable of generating elongational flow such as <br>the CRD mixer (see Section 3.4.2.5) can disperse gels [88­94] and provide a valu-<br>able tool in reducing gel problems.<br>One of the best tools available to remove gels is a depth filtration medium, such as <br>sintered metal or a random 3-D fiber. These depth filters have been in use for dec-<br>ades and have proven themselves in high-quality film and fiber applications where <br>gels have to be kept to the lowest possible level. Several companies sell large area <br>depth filtration devices that are well suited for gel removal. Unfortunately, such <br> <br>filters are expensive, maintenance intensive, and require replacement on a regular <br>basis. However, they are effective in removing gels from the plastic melt. Figure <br>11.42 shows a flow chart for troubleshooting gel problems.<br>
+<b>Gel problem</b><br>
+Measure P-gels in incoming resin <br>
+No<br>
+Can resin supplier bring P-gels <br>
+Use SPC to analyze data. <br>
+Problem solved<br>
+under control?<br>
+Are P-gels in statistical control?<br>
+Yes<br>
+Yes<br>
+Yes<br>
+No<br>
+Can resin supplier <br>
+Change resin or change <br>
+Yes<br>
+reduce P-gel level?<br>
+Is average P-gel level too high?<br>
+resin supplier<br>
+No<br>
+No<br>
+Yes<br>
+Go to flow chart for <br>
+Change resin <br>
+Are stock temperatures too high?<br>
+high melt temperature<br>
+Change resin supplier <br>
+Use elongational dispersive <br>
+No<br>
+mixing device(s) <br>
+Use 3D random metal fiber <br>
+Yes<br>
+Reduce residence times by: <br>
+filter<br>
+Are the residence times too long?<br>
+Increasing throughput <br>
+Reducing screw volume <br>
+No<br>
+Reducing adaptor/die volume <br>
+Eliminate dead spots, dents, <br>
+Improve stabilizer package <br>
+No<br>
+Does the material have sufficient <br>
+scratches, etc.<br>
+Change material <br>
+thermal stability to be extruded? <br>
+Change to other process<br>
+(check induction time)<br>
+Yes<br>
+Apply low friction coating <br>
+Yes<br>
+Low friction surface treatment <br>
+Eliminate contamination<br>
+Contamination in the material?<br>
+Use self-wiping extruder <br>
+Change compound, for instance <br>
+No<br>
+Yes<br>
+add fluoroelastomer in combination <br>
+Material building up on screw or <br>
+with antioxidant additives <br>
+other surfaces (plate-out)?<br>
+Change compounding procedure<br>
+<b>Figure 11.42</b>Flow chart to troubleshoot gel problems<br>
+<hr>
+<A name=854></a> <br>
+11.3Systematic Troubleshooting  <b>843</b><br>
+<b>11.3.7Die Flow Problems</b><br>
+Die flow problems typically result in appearance problems. These can be related to <br>melt fracture, die lip build-up, gels, v- or w-patterns, specks, color variation, lines, <br>and change in optical properties (e.g., transparency, mattness, gloss, haze).<br>
+<b>11.3.7.1Melt Fracture<br></b>Melt fracture was discussed in Section 7.5.3.2. It manifests itself as extrudate sur-<br>face roughness, shark skin, orange peel, and other distortions. Melt fracture can be <br>reduced or eliminated by:<br>
+ <br>
+ Streamlining the die flow channel<br>
+ <br>
+ Reducing the shear stress in the land region (operating below the critical shear <br>
+stress for melt fracture)<br>
+ <br>
+ Use a processing aid (e.g., a fluoroelastomer for polyethylene)<br>
+ <br>
+ Use "super-extrusion" (operating above the critical shear stress for melt fracture)<br>
+ <br>
+ Ultrasonic vibration<br>Streamlining the die flow channel is always a good idea but it will increase the cost <br>of a die. For a high-volume product, it generally makes sense to design and manu-<br>facture a fully streamlined die. For a small-volume product, it may not make eco-<br>nomic sense to design and manufacture a fully streamlined die.<br>Reducing the shear stress in the land region can be done by:<br>
+ <br>
+ Increasing the die land temperature<br>
+ <br>
+ Opening up the die land region (increase die gap)<br>
+ <br>
+ Reduce the extrusion rate<br>
+ <br>
+ Use process aid (e.g., external lubricant, viscosity depressant)<br>
+ <br>
+ Increase the melt temperature<br>
+ <br>
+ Reducing the polymer melt viscosity<br>
+ <br>
+ Use a more shear thinning plastic<br>Several polymer processing aids (PPA) are available to eliminate or reduce melt frac-<br>ture. An effective method to eliminate melt fracture in high molecular weight poly-<br>olefins is to add a small amount of fluoroelastomer [153], about 500 to 1000 ppm <br>(parts per million). When a fluoroelastomer PPA is added to a polyolefin it usually <br>takes a certain amount of time for a critical coating of fluoropolymer to form on the <br>die. This conditioning time can vary from 5 minutes to more than 1 hour [154].<br>Silicon-based polymers such as polydimethyl siloxanene (PDMS) have been used as <br>polymer processing aids for many years. Dow Corning has developed ultra-high <br>molecular weight PDMS additives that work as process aids in polyethylene and <br>polypropylene. Because these materials solidify with the polymer, they reportedly <br>
+<hr>
+<A name=855></a><b>844</b>  <b> 11Troubleshooting Extruders</b><br>
+do not affect printability or paint adhesion. HMW PDMS has been used to reduce <br>surface roughness of extruded LLDPE tape [158].<br>Super-extrusion is a technique whereby the shear stress in the die land is above <br>the critical shear rate for melt fracture; see Section 7.5.3.2. This is possible with <br>polymers that exhibit a second stable region above the melt fracture region. Linear <br>polymers such as HDPE, FEP, and PFA exhibit super-extrusion behavior. The melt <br>fracture behavior can be determined on a capillary rheometer by running a polymer <br>melt at different shear rates and observing the corresponding condition of the extru-<br>date. A typical flow curve for a linear polymer is shown in Fig. 11.43.<br>
+<b> Figure 11.43</b> <br>
+Flow curve of a linear polymer showing melt <br>
+fracture and super-extrusion regions<br>
+Melt fracture can be avoided by keeping the shear stress in the die below the critical <br>level for melt fracture. This requires operating at a shear rate below the lower criti-<br>cal shear rate for melt fracture. Melt fracture can also be avoided by running under <br>conditions where the shear stress in the die is above the critical shear stress level <br>for melt fracture. In this case, one has to operate at a shear rate above the upper <br>critical shear rate for melt fracture.<br>Ultrasonic vibration of the die is done by mounting external transducers on the die, <br>which deliver ultrasonic energy in the kHz range. Little quantitative information is <br>available on this technique; however, it is known that it is successfully practiced in <br>the extrusion industry. The principle behind ultrasonic vibration is related to the <br>shear thinning characteristics of polymers. As discussed in Chapter 6, the melt vis-<br>cosity of polymers reduces by orders of magnitude when the rate of deformation in <br>increased. This applies not only to steady deformation but also to cyclic deformation. <br>Therefore, when a polymer melt is exposed to a high-frequency vibration, its vis-<br>cosity will reduce by a large amount depending on the degree of shear thinning.<br>With ultrasonic die vibration the polymer melt layer at the die wall is most exposed <br>to the high-frequency deformation. This causes a large drop in melt viscosity at the <br>die wall with several beneficial effects, such as:<br>
+ <br>
+ Reduced diehead pressure<br>
+ <br>
+ Reduced extrudate swell<br>
+<hr>
+<A name=856></a> <br>
+11.3Systematic Troubleshooting  <b>845</b><br>
+ <br>
+ Reduced melt fracture<br>
+ <br>
+ Reduced die lip build-up (die drool)<br>
+<b>11.3.7.2Die Lip Build-Up (Die Drool)<br></b>Die lip build-up is a common problem in the extrusion industry; it is a condition <br>where material accumulates right at the die exit as illustrated in Fig. 11.44.<br>
+Die drool<br>
+<b> Figure  11.44</b> <br>
+Illustration of die lip build-up<br>
+Material build-up right at the die exit can cause lines in the extruded product. This <br>problem is often referred to as "die drool." It typically results from incompatible <br>components in the compound, even though it can also happen in non-compounded <br>plastics. Die drool can be caused by gas or moisture in the molten plastic, degrada-<br>tion, or poor dispersion of fillers or additives. Die drool can be reduced either by <br>changing the material, the process, or the die design.<br>To reduce die drool by changing the material:<br>
+ <br>
+ Remove the incompatible component<br>
+ <br>
+ Add a fluoroelastomer<br>
+ <br>
+ Add a compatibilizer<br>
+ <br>
+ Change the compounding procedure<br>To reduce die drool by changing the process:<br>
+ <br>
+ Adjust the die temperature (usually lower)<br>
+ <br>
+ Blow air at the die exit<br>
+ <br>
+ Use scraper at the die exit<br>
+ <br>
+ Use ultrasonic vibration<br>To reduce die drool by changing the die design:<br>
+ <br>
+ Use a low-friction coating in the die<br>
+ <br>
+ Use another die material, for instance ceramic<br>
+ <br>
+ Use a longer land length<br>
+ <br>
+ Use small taper in the land region of the die<br>
+<hr>
+<A name=857></a><b>846</b>  <b> 11Troubleshooting Extruders</b><br>
+It is also important to inspect the condition of the die internal surfaces for scratches, <br>bad plating, a pitted surface, or general poor surface quality. Any of these conditions <br>have to be corrected to minimize die lip build-up. It is interesting to note that several <br>of the measures that reduce die drool also reduce melt fracture. Processing aids that <br>reduce melt fracture often reduce die drool as well [155­157].<br>
+<b>11.3.7.3V- or W-Patterns<br></b>These patterns are often related to line tension, more specifically uneven line ten-<br>sion. In advanced cast film lines (BOPP or BOPET), there are tension-control systems <br>that allow tension adjustment in specific regions of the film. In these lines, a gear <br>pump is often necessary to minimize output variation. The diehead pressure in such <br>extrusion operations can be very high (up to 600 bar).<br>These patterns can also be related to uneven flow out of the die. Check the die design <br>and make sure the melt temperature is uniform in the extrusion direction as well as <br>across the die. In some cases, a static mixer can be used; it can be placed even inside <br>the die just before the manifold region of the die.<br>
+<b>11.3.7.4Specks and Discoloration<br></b>Discolored specks are a common problem in extrusion and molding. This problem is <br>similar to another common defect, which is the problem of gels. Like gels, discolored <br>specks are formed not only in extrusion and molding at the processor but also in <br>polymerization at the resin producer. As a result, in order to get a handle on the <br>problem, we need to know how many specks are in the incoming raw material. We <br>will distinguish between a speck formed in polymerization and located inside the <br>pellet (P-speck) and a speck formed in extrusion (E-speck).<br>
+<b>11.3.7.4.1Specks Formed in Polymerization</b><br>
+One of the challenges in testing pellets produced at the resin supplier is that pellets <br>are quite small. A typical plastic pellet may have a diameter and length of about <br>3 mm with a volume of about 20 mm3. If we assume a density of 1 gr/cm3, the mass <br>of a typical pellet will be about 20 milligram. This means 1 kg of resin will have <br>about 50,000 pellets. If we run an extruder at 300 kg/hr, approximately 1,500,000 <br>pellets will pass through the extruder every hour.<br>Today commercial instruments are available that allow analysis of millions of pel-<br>lets. Optical Control Systems (OCS) GmbH produces systems for defect detection. <br>OCS makes instruments for defect detection in pellets as well as in extruded sheet <br>and film. Pellet defect detection systems by OCS are used by many resin producers. <br>Therefore, most resin companies know how many defects occur in the pellets they <br>produce. Figure 11.45 shows an example of pellets with discolored specks.<br>
+<hr>
+<A name=858></a><IMG src="Polymer Extrusion by Chris Rauwendaal-858_1.jpg"><br>
+ <br>
+11.3Systematic Troubleshooting  <b>847</b><br>
+<b>Figure 11.45</b>Pellet with black and discolored specks (courtesy OCS)<br>
+If we have one pellet with a discolored speck for every N pellets, we can determine <br>the average incidence of specks in the extruded product. If the cross-sectional area <br>of the extruded product is Ae and the volume of the pellet or powder particle is Vp, <br>
+the average length over which a P-speck will occur in the extruded product is:<br>
+Lps = NVp/Ae <br>
+(11.20)<br>
+As an example, we will consider the extrusion of tubing with an inside diameter of <br>10.0 mm with a wall thickness of 1.0 mm. The cross-sectional area of this product is:<br>
+A<br>
+2<br>
+2<br>
+e = 0.25(Do  ­ Di ) = 0.25 · 3.14 (144 ­ 100) = 34.45 mm2 <br>
+(11.21)<br>
+The volume of the pellet is 20 mm3. We will consider a plastic raw material that con-<br>tains one pellet with a discolored P-speck for every 10,000 pellets; this corresponds <br>to 100 ppm. With this input data, we find the average length over which a P-speck <br>will occur is 5806 mm or 5.8 m. If we generate specks in the extrusion process, the <br>average length over which a speck occurs in the extruded product will be less than <br>5.8 meter.<br>If we cannot tolerate one speck every 5.8 meter there is only one option: reducing <br>the number of P-specks in the raw material. Since resin producers can scan millions <br>of pellets for specks, it is also possible to remove pellets with specks from the pellet <br>stream. In other words, the processor can specify the ppm level of pellets with <br>specks that can be tolerated by the processor.<br>Clearly, sorting the pellets carries a certain cost to the resin producer. The resin <br>producer will pass this cost on to the customer. Alternatively, the customer can do <br>this sorting in-house. In this case, the customer will have to purchase the pellet <br>scanning and sorting (S&amp;S) system. If in-house scanning and sorting is less expen-<br>sive than the cost of pre-sorted pellets, it makes sense for the processor to perform <br>this task in-house.<br>
+<hr>
+<A name=859></a><IMG src="Polymer Extrusion by Chris Rauwendaal-859_1.jpg"><br>
+<b>848</b>  <b> 11Troubleshooting Extruders</b><br>
+Pellets can be scanned and sorted not only for discolored specks but also for irre-<br>gular pellets, for instance, pellets with tails. Figure 11.46 shows a picture of pellets <br>with tails; such pellets can cause problems in extrusion and are preferably elimi-<br>nated from the feed stream.<br>
+<b>Figure 11.46</b>Pellets with tails (courtesy OCS)<br>
+11.3.7.4.1.1Specks Formed in Extrusion<br>Discoloration of the plastic inside the extruder can be caused by degradation, con-<br>tamination, and several other causes. Figure 11.47 shows a listing of possible causes <br>of specks generated in the extrusion process.<br>
+incorrect process conditions<br>
+poor design screenpack-breakerplate assembly<br>
+rough screw, barrel, die surface<br>
+poor startup/shutdown<br>
+worn screw/barrel<br>
+Discolored <br>
+poor changeover<br>
+E-specks<br>
+degradation<br>
+poor die design<br>
+contamination <br>
+poor adaptor design<br>
+e.g. bag fibers <br>
+       airborne particles <br>
+poor screw design<br>
+poor screen pack design<br>
+       fines <br>
+       etc.<br>
+<br>
+<b>Figure 11.47</b>Fishbone chart for discolored E-specks<br>
+<hr>
+<A name=860></a><IMG src="Polymer Extrusion by Chris Rauwendaal-860_1.jpg"><br>
+ <br>
+11.3Systematic Troubleshooting  <b>849</b><br>
+The causes listed in Fig. 11.47 by no means present a complete listing. Figure 11.47 <br>shows some of the more common causes of discolored specks; however, other causes <br>can certainly play a role. A detailed discussion of all the causes listed in Fig. 11.47 <br>is beyond the scope of this book, but we will discuss one cause in detail: screw wear.<br>
+<b>11.3.7.4.2Screw Wear</b><br>
+Screw wear is a fact of life. The question is not whether or not screw wear occurs; <br>while running an extruder, wear takes place at all times. Therefore, the pertinent <br>question is how fast the wear reduces the outside diameter (O.D.) of the screw. More <br>importantly, we need to know when the wear has progressed to the point where it <br>starts causing unacceptable problems. At that point, the worn screw will need to be <br>replaced with a new or refurbished screw.<br>In order to determine when the screw needs to be replaced, it is necessary to meas-<br>ure the O.D. of the screw over the entire length. This has to be done regularly! Spe-<br>cial tools are available to measure the O.D. of an extruder screw; for example, see <br>Fig. 11.48. In a typical extrusion operation, the screw and barrel should be meas-<br>ured at least once a year. Sometimes a company believes that they need not monitor <br>screw and barrel wear. As a result, this company has no idea how badly the wear is <br>affecting their process and product. When product quality starts deteriorating <br>because of wear the appropriate corrective action is not obvious.<br>A simple problem that could have been solved quickly and less expensively can <br>potentially become very expensive if the corrective action is not obvious. A problem <br>that could have been solved by installing a spare screw may cost $500,000 if the <br>solution to the problem is not taken. The spare screw may cost $15,000; but not <br>being able to solve the problem in a timely fashion will cost 10 to 100 times more!<br>
+<b> Figure 11.48</b> <br>
+Special micrometer to measure <br>
+the outside diameter<br>
+<hr>
+<A name=861></a><b>850</b>  <b> 11Troubleshooting Extruders</b><br>
+By careful process monitoring we can often detect the effects of wear. The process <br>parameters that are affected by wear are melt temperature, output, and motor load. <br>As wear progresses, the quality of the extruded product tends to deteriorate. This <br>may manifest itself as discoloration, streaks, discolored specks, holes, etc. It is very <br>important to monitor the specific energy consumption (SEC) and specific extruder <br>throughput (SET) because changes in these parameters often correlate with wear. <br>SEC is the ratio of motor power divided by the throughput. It is the mechanical <br>power consumed per unit mass of plastic. The SEC tends to correlate with the melt <br>temperature. The SEC is normally expressed in kWh/kg. A typical value of the SEC <br>is 0.25 kWh/hr for extrusion of polyolefins.<br>SET is the output divided by screw speed; it is often expressed in kg/h/rpm. For <br>instance, for a 75-mm extruder the SET may be 2.0 kg/h/rpm; this corresponds to <br>0.033 kg/rev. The SET is the amount of resin extruded per screw revolution.<br>Both SEC and SET are normalized performance parameters that allow comparison of <br>data achieved at different process conditions, i.e., screw speed. As wear progresses, <br>the SEC values tend to increase and the SET values tend to reduce.<br>Why should we be concerned about screw and barrel wear? As wear progresses the <br>gap between the screw and barrel increases. This will reduce the pumping capabil-<br>ity of the screw. However, the more critical issue may be the fact that the thickness <br>of the stagnant melt layer on the barrel increases. This will increase degradation <br>inside the extruder and reduce extrudate quality.<br>Screw wear will create a thicker insulating melt layer at the barrel surface. This will <br>inhibit heat transfer between the barrel and the melt in the screw channel. As a <br>result, the control of melt temperature will be diminished and excessively high melt <br>temperatures are more likely. This further increases the chance of degradation.<br>There are additional adverse effects of screw and barrel wear. Increases in clearance <br>will reduce the ability of the screw flights to wipe the barrel surface. This will <br>increase the average residence and broaden the residence time distribution. As a <br>result, changeover from one resin to another will take longer. Also, purging will take <br>longer, as well as startup and shutdown. This means that more scrap will be pro-<br>duced, and the downtime will increase with a corresponding reduction in uptime. <br>This can have a significant negative effect on the production cost.<br>
+<b>Conclusions<br></b>Discolored specks are ubiquitous in the plastic extrusion industry. Controlling <br>specks starts with quantifying P-specks in the incoming raw material. Once we can <br>control P-specks to an acceptable level, we can address E-specks. Important factors <br>that affect E-specks have been identified with one factor, screw wear, discussed in <br>detail.<br>
+<hr>
+<A name=862></a> <br>
+11.3Systematic Troubleshooting  <b>851</b><br>
+Specks are a common problem in extruded products, particularly in thin or transpar-<br>ent products. The specks can be black, brown, yellow, or almost any other color dif-<br>ferent from the matrix material. Specks are usually caused by contamination, degra-<br>dation, or wear. Degradation can manifest itself as discoloration, specks, pinholes, <br>loss of volatiles (smoking), or loss of physical properties in the extruded product.<br>To find the cause of specks and discoloration:<br>
+ <br>
+ Check for contamination<br>
+ <br>
+ Check for high temperatures<br>
+ <br>
+ Check for stagnation<br>
+ <br>
+ Check thermal stability of the plastic (perhaps improve stabilizer package)<br>
+ <br>
+ Check for foreign particles (wear)<br>Degradation can generally be reduced by:<br>
+ <br>
+ Reducing stock temperatures in the extruder<br>
+ <br>
+ Reducing the residence time in the extruder<br>
+ <br>
+ Eliminating the presence of degradation-promoting substances, e.g., oxygen<br>
+ <br>
+ Adding a thermal stabilizer or improve the stabilizer package<br>Poor color uniformity can be caused by mixing problems in the extruder, variation <br>in the color additive or color concentrate, variation in the addition of the color addi-<br>tive or concentrate, and compatibility problems between the virgin and masterbatch. <br>Mixing of a small amount of color concentrate (CC) in a natural polymer is actually <br>quite difficult as discussed in Section 7.7.4. Mixing can be improved by improving <br>the mixing capability of the extruder screw. Another way to improve mixing is to <br>reduce the initial scale of segregation of the mixture. This can be done by:<br>
+ <br>
+ Reducing the particle sizes of the CC and virgin material<br>
+ <br>
+ Using liquid colorants<br>Liquid colorants can create problems in extrusion such as solids conveying prob-<br>lems. This can be avoided by using a porous carrier resin. Several resin suppliers <br>(e.g., DSM, Akzo, Montell) now have porous carriers for use with liquid additives; <br>these can be colorants or other additives, such as antioxidants, peroxides, silanes, <br>etc. Mixing can further be improved by starve feeding the extruder as discussed in <br>Section 11.3.4.2.5.<br>
+<b>11.3.7.5Lines in Extruded Product<br></b>Lines can be caused by the die, breaker plate, extruder screw, die lip build-up, and <br>downstream equipment such as calibrators, cooling baths, catapullers, etc. If the <br>line is visible right at the die exit, it must be formed at the die exit (e.g., by die <br>drool), in the die (poor internal surface conditions in the die or build-up in the die), <br>or upstream (breaker plate, screen pack, screw). A single line is often formed in a <br>
+<hr>
+<A name=863></a><b>852</b>  <b> 11Troubleshooting Extruders</b><br>
+crosshead die. With in-line dies, the number of lines often will match the number <br>of spider supports in the die. The breaker plate can cause a large number of lines in <br>the product.<br>In blown film extrusion, the number of lines in the film frequently corresponds to <br>the number of ports in the die. As a result, these lines are often referred to as port <br>lines. These port lines are obviously related to the die design but they are also very <br>dependent on the melt flow characteristics of the polymer. High molecular weight <br>polymers have very long relaxation times and are more likely to exhibit port lines or <br>other types of die lines.<br>Lines can also originate downstream of the die, for instance in the calibrator. Lines <br>can result from an object touching the extrudate or local cold or hot sections. These <br>problems can generally be diagnosed easily by observing at what point in the pro-<br>duction line the surface line originates.<br>
+<b>11.3.7.5.1Weld Lines</b><br>
+Lines in the extruded product can result from weld lines. These form when the poly-<br>mer melt is split and recombined in the die or even before the die. Weld lines are <br>also called knit lines; they can form in tubing and pipe dies where a mandrel is held <br>in place by spider supports. The polymer melt is split at the start of the spider leg <br>and flows together again behind (downstream) the spider support. Because of the <br>limited mobility of long polymer molecules, it takes a certain amount of time for the <br>molecules to re-entangle. This re-entanglement process is also called a "healing" <br>process. Longer molecules take longer to re-entangle. As a result, high molecular <br>weight (high viscosity) polymers are more susceptible to weld lines than low mole-<br>cular weight (low viscosity) polymers.<br>The severity of the weld line problem will be determined by:<br>1.  The length of time from the point where the melt streams recombined to the exit <br>
+of the die (residence time)<br>
+2.  The healing time of the polymer melt<br>If the residence time is longer than the healing time, the weld line will disappear <br>inside the die and not cause a problem in the extruded product. However, if the resi-<br>dence time is shorter than the healing time, the weld line will not disappear inside <br>the die and the weld line will cause a problem in the extruded product. The weld <br>line problem can be reduced or eliminated by increasing the residence time in the <br>die or reducing the healing time of the polymer melt.<br>The residence time in the die can be increased by reducing the flow rate (extruder <br>throughput) or by changing the die geometry. The flow splitter has to be located as <br>far away from the die exit as possible. Some die geometries reduce weld line prob-<br>lems. For instance, spiral mandrel dies for pipe, tubing, and blown film spread out <br>
+<hr>
+<A name=864></a> <br>
+11.3Systematic Troubleshooting  <b>853</b><br>
+the weld line as the melt flows through the spiral mandrel section; this largely elim-<br>inates problems with weld lines. Tubing and pipe dies with a rotating mandrel and/<br>or die can also effectively spread out the weld lines and eliminate weld line prob-<br>lems. Some dies incorporate relaxation zones to enhance the healing process. Relax-<br>ation zones are basically local regions in the die flow channel where the cross-sec-<br>tional area of the channel is increased.<br>The healing time depends on the molecular characteristics of the polymer and the <br>melt temperature. Reducing the molecular weight of the polymer will speed up the <br>re-entanglement process. Also, higher melt temperatures will increase the mobility <br>of the polymer molecules and reduce the healing time. The molecular architecture <br>also plays an important role. The molecules of linear polymers tend to align more <br>readily and, as a result, entangle more slowly when separate flow streams meet. <br>This is a particular problem in liquid crystalline polymers (LCPs), which have rod-<br>like molecules. As a result, LCPs are highly susceptible to weld lines. Another <br>method to promote re-entanglement is to subject the material in the die to a high-<br>frequency vibration. Some processors use ultrasonic vibration of the die by mount-<br>ing external transducers that deliver ultrasonic energy in the kilohertz range. <br>Because polymers are not only shear thinning but also frequency thinning, the <br>effective viscosity of the polymer melt is reduced by high-frequency vibration. Other <br>benefits of high-frequency vibration are reduced extrudate swell, reduced extrudate <br>distortion at the die exit, reduced melt fracture, and reduced die lip build-up (die <br>drool).<br>
+<b>11.3.7.6Optical Properties<br></b>Optical properties such as transparency, matness, gloss, and haze are strongly deter-<br>mined by the cooling conditions of the extruded product. In crystalline polymers, <br>the crystal growth is very much temperature and stress dependent. As a result, the <br>morphology of the extruded product will depend on how rapidly the polymer melt is <br>cooled as it leaves the die. Slow cooling generally promotes crystal growth. Rapid <br>cooling reduces crystallization; in fact, in some semi-crystalline polymers the crys-<br>tallinity can be suppressed completely with rapid cooling.<br>In sheet extrusion where the molten sheet of polymer is forced through a set of rolls, <br>the surface conditions are strongly determined by the surface texture of the rolls. If <br>the rolls are polished, they will impart a polished surface onto the polymer sheet--<br>this is why these rolls are often called polishing rolls. Obviously, a variety of differ-<br>ent textures can be machined into the rolls and, consequently, a number of different <br>textures can be imparted to the extruded sheet.<br>
+<hr>
+<A name=865></a><b>854</b>  <b> 11Troubleshooting Extruders</b><br>
+References<br>1.  E. Wagner, Ph.D. thesis, Technische Hochschule Darmstadt, Germany (1978)<br>2.  M.M. Kruschchov, Ind. Lab (USSR), 20, 372­376 (1962)<br>3.  G. Mennig and P. Volz, Kunststoffe, 70, 385­390 (1980)<br>4.  W. Mehdorn, Kunststoffe, 34, 133­136 (1944)<br>5.  W. Bauer, K. Eichler, and W. John, Kunststoffe, 57, 53­55 (1967)<br>6.  K. Eichler and W. Frank, Ind. Anzeiger, 95, 2033­2035 (1973)<br>7.  W.D. Mahler, Ph.D. thesis, Technische Hochschule Darmstadt, Germany (1975)<br>8.  P. Volz, W.D. Mahler, and G. Menning, Kunststoffe, 66, 428­434 (1976)<br>9.  W. Knappe and W.D. Mahler, Kunststoff-Rundschau, 19, 45­51 (1972)<br>10.  R.S. Plumb and W.A. Glaeser, Wear, 46, 219­229 (1978)<br>11.  H.G. Fritz, Kunststoffe, 65, 176­182 and 258­274 (1975)<br>12.  P. Volz, Kunststoffe, 69, 758­771 (1979)<br>13.  P. Volz, Kunststoffe, 67, 279­283 (1976)<br>14.  G.P. Calloway, E.D. Morrison, and R.F. Williams, Jr., SPE ANTEC, 354­360 (1972)<br>15.  D. Braun and G. Maelhammer, Angew. Makromol. Chem., 69, 157­167 (1978)<br>16.  H.G. Moslé, H.F. Schmidt, and J. Schroeder, Kunststoffe, 67, 220­223 (1977)<br>17.  G. Maelhammar, Ph.D. thesis, Technische Hochschule Darmstadt, Germany (1978)<br>18.  P. Volz, Kunststoffe, 69, 259­262 (1979)<br>19.  E. Broszeit, Ph.D. thesis, Technische Hochschule Darmstadt, Germany (1972)<br>20.  G.A. Saltzman and J.H. Olson, SPE ANTEC, 173­175 (1974)<br>21.  V. Murer and G.A. Saltzman, Kunststoffe, 66, 219­220 (1976)<br>22.  W.W. McCandles and W.D. Maddy, Plast. Technol., Feb., 89­93 (1981)<br>23.  S.H. Collins, Plast. Compd., May/June, 113­124 (1982)<br>24.  S.H. Collins, Plast. Compd., July/Aug., 16­28 (1982)<br>25.  E.L. Moon and R.A. Hunter, "An Abrasion Study of Surface Treated Calcium Carbonate <br>
+Fillers in Rigid PVC," Technical Bulletin, Georgia Marble Corp., Atlanta, GA<br>
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+Vista, Florida (1983)<br>
+<hr>
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+(1979)<br>
+59.  C.J. Walk, SPE ANTEC, San Francisco, 423­426 (1982)<br>60.  R.J. Nichols, J.C. Golba, and P.K. Shete, paper presented at the AIChE Annual Meeting, <br>
+Paper No. 59F (1983)<br>
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+(1978)<br>
+<hr>
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+Van Nostrand Reinhold, New York (1970)<br>
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+published in Plast. Eng., June, 41­43 (1998)<br>
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+the Netherlands (1988)<br>
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+277­283 (1998)<br>
+89.  C. Rauwendaal, "New Dispersive Mixers for Single and Twin Screw Extruders," Annual <br>
+Meeting of the Polymer Processing Society, Yokohama, Japan, June 8­12 (1998)<br>
+90.  C.J. Rauwendaal, "A New Dispersive Mixer," Plast. Addit. Compd., Sept. (1999)<br>91.  C.J. Rauwendaal, "Eliminate Gels and other Mixing Problems," Film Conference 99, <br>
+Somerset, NJ, 141­148, Dec. 7­9 (1999)<br>
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+pounding Conference, Cleveland, OH, 219­228, Nov. 14­15 (2000)<br>
+93.  U.S. Patent 5,932,159, C.J. Rauwendaal (1999)<br>94.  U.S. Patent 6,136,246, C.J. Rauwendaal, P. Gramann, B. Davis, T. Osswald (2000)<br>95.  U.S. Patent 3,006,029 (1961)<br>96.  F.N. Cogswell, J. Non-Newtonian Fluid Mech., 2, 373­47 (1977)<br>
+<hr>
+<A name=868></a> References <br>
+<b>857</b><br>
+97.  M.T. Dennison, Trans. J. Plast. Inst., 35, 803­808 (1967)<br>98.  I.J. Benbow and E.R. Howells, Trans. J. Plast. Inst., 30, 240­254 (1960)<br>99.  British Patent 32559/72<br>100. I.P. Tordella, in "Rheology," Vol. 4, Chapter 3, F.R. Eirich (Ed.), Academic Press, New <br>
+York (1969)<br>
+101.  J.L. White, Appl. Polym. Symp., No. 20, 155 (1973)<br>102.  J.M. Lupton and J.W. Regester, Polym. Eng. Sci., 5, 235 (1965)<br>103.  I.L. Blyler and A.C. Hart, Polym. Eng. Sci., 10, 193 (1970)<br>104.  S.M. Barnett, Polym. Eng. Sci., 7, 168 (1967)<br>105. E. Boudreaux and J.A. Cuculo, J. Macromol. Sci.--Rev. Macromol. Chem., C16, 39­77 <br>
+(1977­1978)<br>
+106.  C.D. Han and R.R. Lamonte, Polym. Eng. Sci., 11, 385 (1971)<br>107.  J.L. den Otter, Rheol. Acta, 10, 200­207 (1971)<br>108.  T.W. Huseby, Trans. Soc. Rheol., 10, 181­190 (1966)<br>109.  A.P. Metzger and C.W. Hamilton, SPE Trans., 4, 107­112 (1964)<br>110.  G.V. Vinogradov et al., Polym. Eng. Sci., 12, 323­334 (1972)<br>111.  J.J. Benbow and P. Lamb, SPE Trans., 3, 7­17 (1963)<br>112.  W. Phillippoff and F.H. Gaskins, Trans. Soc. Rheol., I, 263­284 (1958)<br>113.  C.J.S. Petrie and M.M. Denn, AIChE J., 22, 109­236 (1976)<br>114.  W. Gleissle, Rheol. Acta, 21, 484 (1982)<br>115.  J.J. Benbow, R.V. Charley, and P. Lamb, Nature, 192, 223 (1961)<br>116.  L.A. Utracki and R. Gendron, J. Rheol., 5, 28, 601­623 (1984)<br>117. U.S. Patent 2,991,508 by R.T. Fields and C.F.W. Wolf to E.I. DuPont de Nemours and <br>
+Company, issued July 11 (1961)<br>
+118.  A.M. Kraynik and W.R. Schowalter, J. Rheol., 25, 95­114 (1981)<br>119.  R.A. Worth, J. Parnaby, and H.A.A. Helmy, Polym. Eng. Sci., 17, 257 (1977)<br>120.  C.J.S. Petrie and M.M. Denn, AIChE J., 22, 236­246 (1976)<br>121.  J.C. Miller, SPE Trans., 3, 134 (1963)<br>122.  S. Kase, J. Appl Polym. Sci., 18, 3279 (1974)<br>123.  G.F. Cruz-Saenz, G.J. Donnelly, and C.B. Weinberger, AIChE J., 22, 441 (1976)<br>124.  J.R.A. Pearson and Y.T. Shah, Trans. Soc. Rheol., 16, 519 (1972)<br>125.  J.R.A. Pearson and Y.T. Shah, Ind. Eng. Chem. Fundam., 13, 134 (1979)<br>126.  R.J. Fischer and M.M. Denn, Chem. Eng. Sci., 30, 1129 (1975)<br>127.  Y. Ide and J. L. White, J. Appl. Polym. Sci., 20, 2511­2531 (1976)<br>128.  Y. Ide and J.L. White, J. Non-Newtonian Fluid Mech., 2, 281­298 (1977)<br>129.  Y. Ide and J.L. White, J. Appl. Polym. Sci., 22, 1061­1079 (1978)<br>130.  J.L. White and Y. Ide, J. Appl. Polym. Sci., 22, 3058­3074 (1978)<br>131.  R.S. Lenk, J. Appl. Polym. Sci., 22, 1781­1785 (1970)<br>
+<hr>
+<A name=869></a><b>858</b>  <b> 11Troubleshooting Extruders</b><br>
+132.  N.C. Wheeler, Annual Convention Wire Association, Baltimore, MD, Oct. 22­25 (1962)<br>133.  J. Becker, P. Bengtsson, C. Klason, J. Kubat, and P. Saha, Int. Polym. Process., VI, no. 4, <br>
+318­325 (1991)<br>
+134. J. Becker, C. Klason, J. Kubat, and P. Saha, Int. Polym. Process., VI, no. 4, 326­331 <br>
+(1991)<br>
+135.  G.M. Gale, 49th SPE ANTEC, 95 (1991)<br>136.  G.M. Gale, Adv. Polym. Technol., 16, no. 4, 251 (1997)<br>137.  M.H. Mack, Plast. Eng., 47, no. 12, 39 (1991)<br>138.  I.R. Thompson, G. Donoian, and J.C. Christiano, 57th SPE ANTEC, 145 (1999)<br>139.  P. Elemans and J.M. van Wunnik, 58th SPE ANTEC, 265­267 (2000)<br>140.  P. Elemans, 58th SPE ANTEC, 2582­2586 (2000)<br>141. K. Luker, paper presented at the Continuous Compounding Conference, Beachwood, <br>
+Ohio, Nov. 14­15 (2000)<br>
+142.  K. Luker, paper presented at the Wood Composite Conference, Dec. 5­6 (2000)<br>143. C.J. Rauwendaal, "Polymer Mixing, A Self-Study Guide," Carl Hanser Publishers, <br>
+Munich (1998)<br>
+144. R.H. Reinhard, A.R. Crixell, C.L. Loney, and P.W. Tidwell, 32nd SPE ANTEC, 40­42 <br>
+(1974)<br>
+145. T. Osswald, P. Gramann, B. Davis, M. del Pilar Noriega, and O.A. Estrada, 57th SPE <br>
+ANTEC, New York, 167­176 (1999), also the annual meeting of the Polymer Processing <br>Society in Den Bosch, the Netherlands (1999)<br>
+146. P. Limbach, in "Extrusion Solutions," publication by the SPE Extrusion Division, p. 6 <br>
+(1999)<br>
+147.  C.J. Rauwendaal and P. Gramann, 56th SPE ANTEC, Atlanta, GA, 102­106 (1998)<br>148.  L.B.P.M. Janssen, G.H. Noomen, and J.M. Smith, Plast. Polym., Aug., 135­140 (1975)<br>149.  D.I. Marshall, I. Klein, and R.H. Uhl, SPE J., 20, no. 4, April, 329 (1964)<br>150. S.A. Klein, W.A. Beckman, and G.E. Myers, FEHT: Finite Element Analysis, Vol. 6.98, <br>
+University of Wisconsin-Madison (1997)<br>
+151.  T. Butler, Private communication (2001)<br>152.  C.J. Rauwendaal, Unpublished study (1978)<br>153.  S.E. Amos, G.M. Giacoletto, J.H. Horns, C. Lavallée, and S.S. Woods, in "Plastics Addi-<br>
+tives Handbook," H. Zweifel (Ed.), 5th ed., Carl Hanser Publishers, Munich (2000)<br>
+154.  T.J. Blong, K. Focquet, C. Lavallée, SPE ANTEC Tech. Papers, 3011 (1997)<br>155.  C.M. Chan, "Viscosity and the Formation of Die Drool at the Polymer-Metal Interfaces," <br>
+Int. Polym. Process., 10 (1995)<br>
+156.  D.E. Priester, G.R. Chapman, "Reducing Die Build Up in Extrusion Applications," Tech-<br>
+nical Information, DuPont Dow Elastomers<br>
+157.  S.E. Amos, SPE ANTEC Tech. Papers, pp.133­143 (1997)<br>158. D.E. Hauenstein, D.J. Cimbalik, and P.G. Pape, SPE ANTEC Tech. Papers, 3002­3010 <br>
+(1997)<br>
+<hr>
+<A name=870></a> References <br>
+<b>859</b><br>
+159. M. del Pilar Noriega and C.J. Rauwendaal, "Troubleshooting the Extrusion Process," <br>
+Carl Hanser Publishers, Munich (2001)<br>
+160.  G. Mennig, 36th SPE ANTEC, Washington, DC, 529­531 (1978)<br>161. D.H. Wilson, "Feeding Technology for Plastics Processing," Carl Hanser Publishers, <br>
+Munich (1998)<br>
+162.  J. Ogando, Plast. Technol., Feb., 54­63 (1995)<br>163.  ASTM D2765, Standard Test Methods for Determination of Gel Content and Swell Ratio <br>
+of Crosslinked Polyethylene Plastics<br>
+164. H. Overeijnder, "Extrusion of PVC Pipe," Course "Extrusie" at Twente University of <br>
+Technology, Feb. 4­6 (2013)<br>
+165.  P. Gramann and J. Cruz, "Determining the Quality of Failed PVC Pipe," SPE Plast. Res. <br>
+Online, 10.1002/spepro.000056<br>
+166. "Pellet, Powder, Quality Control," brochure by Optical Control Systems, GmbH (www.<br>
+ocsgmbh.com)<br>
+167. "Fewer Gels Means More Peace of Mind," publication by Dow Chemical, (http://msds <br>
+search.dow.com/PublishedLiteratureDOWCOM/dh_08cb/0901b803808cbf1e.<br>pdf?filepath=elastomers/pdfs/noreg/265-12101.pdf&amp;fromPage=GetDoc)<br>
+168. K.A. Howard, "Gel Defects in LDPE Film--Resolving Polymer and Process Effects," <br>
+TAPPI 2001 PLACE Conference (2001)<br>
+<hr>
+<A name=871></a><b> 12 Modeling and </b><br>
+<b>Simulation of the </b><br>
+<b>Extrusion Process<br></b><i>Paul J. Gramann, Bruce A. Davis, and Tim A. Osswald</i><br>
+<b></b><br>
+ <b>12.1Introduction</b><br>
+In most polymer processes, the quality of the final part is greatly dependent on the <br>melting, flow and mixing of the polymer. The optimization of the equipment and <br>manufacturing process, as done today, is time consuming and expensive. It is often <br>necessary to build complex flow visualization equipment, i.e., model extruders with <br>transparent barrels, to qualify the flow during processes. Quantifying flow and heat <br>transfer is an even more intimidating task. Furthermore, reproducing the proper-<br>ties of a particular blend from batch to batch can be extremely difficult. Obviously, <br>these barriers make numerical simulation a viable alternative when optimizing and <br>analyzing the extrusion process.<br>Traditionally, when simulating polymer processes, the main concern of the engineer <br>has been to accurately represent the material behavior using complex models. <br>Although many problems still exist regarding polymer material models and will <br>continue to be a field of research, today one can easily deal with the shear thinning <br>behavior, temperature dependence and to some degree the viscoelasticity of poly-<br>mers. In fact, to date a large number of processes have been realistically simulated <br>in polymer processing ranging from mold filling with fiber orientation, shrinkage <br>and warpage, to extrusion with viscoelastic effects. However, only a few fully three-<br>dimensional models of realistic processes have been solved. Simulating a fully <br>three-dimensional process involves intensive labor, trying to accurately represent <br>the geometry of the device and also requires large amounts of computation time and <br>data storage. Obviously, computational demands have been reduced by the enor-<br>mous increase in computational power available to the engineer at the desktop. <br>However, the labor intensity and requirements of computer performance are multi-<br>plied by the added complexity of moving boundaries. Two types of moving bounda-<br>ries are very common in polymer processing: <i>moving free boundaries</i> and <i>moving </i><br>
+<i>solid boundaries</i>.* Moving free boundary problems are encountered in such areas <br>
+as mold filling, extrudate swell, coating problems, inside the extruder at the screw, <br>to name a few. Solid moving boundaries are those where the actual cavity that con-<br>
+* The name "moving solid boundary problems" for this category of problems was introduced by Prof . C .L . <br>
+Tucker III in his keynote talk "Mathematical Modeling: ON to Maturity!," PPS 9 Manchester, 1993<br>
+<hr>
+<A name=872></a><b>862</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+tains the polymer changes in shape during the process, i.e., the rotating mixing <br>heads in an internal batch mixer, or the screw and mixing elements in a single <br>screw extruder.* The most complex process involving solid moving boundaries is <br>the intermeshing twin screw extruder. Here, the domain of interest, i.e., the poly-<br>mer, is constantly changing shape as the screws, mixing heads, and kneading blocks <br>rotate. The self-wiping arrangement of these systems adds to the complexity of the <br>problem, since it involves very small gaps, which introduce numerical complica-<br>tions. One should regard the complete simulation of the single or twin screw extru-<br>sion processes including melting, melt conveying, mixing, and die flow to the pre-<br>diction of final morphology, including coalescence and a partially filled system, as <br>one of the grand challenge problems in polymer processing.<br>Before attempting to solve such problems, their complex geometry and processing <br>conditions can be simplified to a form in which they can be modeled with two-<br>dimensional simulations. Simplifications may include laying-flat curved surfaces, <br>neglecting thinner dimensions, assuming planar problems, etc., at the cost of pos-<br>sibly losing important features that dominate the flow and heat transfer in the pro-<br>cess. The advent of more powerful computers and efficient numerical techniques are <br>now beginning to make it possible to simulate three-dimensional problems of com-<br>plex geometry with non-linear material behavior.<br>This chapter gives a general overview of the state-of-the art techniques used for the <br>modeling and simulation of the extrusion process. Recent developments of computa-<br>tional and numerical technologies are presented along with a discussion of the <br>direction this growing field is taking. A brief background on numerical techniques <br>and basic modeling in polymer processing is presented. A discussion on two-dimen-<br>sional models that are used to simulate three-dimensional flows is followed by recent <br>advancements in full three-dimensional models.<br>
+<b></b><br>
+ <b>12.2Background</b><br>
+<b>12.2.1Analytical Techniques</b><br>
+Before describing numerical methods, the technique of using analytical or pseudo-<br>analytical solutions will be discussed. Strictly speaking, a problem has an analytical <br>solution if a mathematical equation can fully describe the phenomena examined. <br>This is usually reserved for simple geometries, with simple conditions and proper-<br>ties. However, by applying specific assumptions and limiting the scope of the prob-<br>lem to be solved, analytical techniques can be applied to more realistic situations. In <br>
+* An assumption that the device is completely filled is taken, which may not always be realistic .<br>
+<hr>
+<A name=873></a> 12.2Background  <b>863</b><br>
+1922, Rowell and Finlayson [1] solved the down channel velocity profile for a screw <br>pump.<br>The major design variables of the screw, including throughput, pressure, and power <br>consumption are described by Tadmor et al. [2]. The value of being able to predict <br>the output, melt temperature, and pressure development along the screw is fairly <br>obvious, but computer simulation can show even more. One example is the rate of <br>solids melting and the corresponding solid bed width. In the feed section of the <br>extruder, the solid bed occupies 100% of the channel width. As the solids melt, the <br>solid bed will occupy less of the channel so that ideally no solids remain at the end <br>of the extruder. Figure 12.1(a) shows a plot of the solid bed width as predicted by <br>simulation* for a given design and operating condition [3]. The x-axis represents <br>the fraction of the channel occupied by the solid bed and the y-axis the relative posi-<br>tion along the screw. The solid bed width begins with a value of 1.0 in the feed sec-<br>tion and gradually reduces to 0.0 at around 60% along the screw length, which <br>allows room for mixing sections to be used.<br>
+1.000<br>
+0.500<br>
+0.000<br>
+Dimensionless Solid Bed Width<br>
+0.0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Dimensionless Screw Length<br>
+(a)<br>
+1.000<br>
+0.500<br>
+0.000<br>
+Dimensionless Solid Bed Width<br>
+0.0<br>
+0.2<br>
+0.4<br>
+0.6<br>
+0.8<br>
+1.0<br>
+Dimensionless Screw Length<br>
+(b)<br>
+<b>Figure 12.1</b>Predicted solid bed profile down the channel of a single screw extruder [4] <br>
+(a) Solid bed melts completely before metering section (b) solid bed is not completely melted <br>
+when it reaches the metering section<br>
+* Flow 2000(TM) Suite of CAE Tools for Extrusion, Compuplast International<br>
+<hr>
+<A name=874></a><b>864</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Modifying the screw design, material or operating conditions can result in a com-<br>pletely different solid bed profile as shown in Fig. 12.1(b) [4]. In this case, the simu-<br>lation predicts that the solid bed width reduces much slower and that melting is not <br>completed until the very end of the screw. This is clearly a less desirable condition <br>than the one shown in Fig. 12.1(a) because there is the potential for some material <br>not to melt and mix properly before leaving the extruder. An extreme case of this <br>can result if the reduction in the solid bed is slower than the compression rate of <br>the screw. In this case, the volume of the channel is reducing faster than the volume <br>of the solid bed. This forces the solid bed to expand, accelerate, or break-up. Prema-<br>ture solid bed break-up can lead to extrudate surging and a poorly mixed material.<br>The method of applying analytical techniques for the simulation of the extrusion <br>process is currently the most common to design an extrusion system.* When the <br>fine details of flow and how it influences the quality of the extrusion process are of <br>interest, more detailed numerical techniques are typically preferred.<br>
+<b>12.2.2Numerical Methods</b><br>
+In order to predict and model complex polymer flows, a basic understanding of the <br>mathematics that govern the flow is necessary. Regardless of the complexity of the <br>flow, it must satisfy certain physical laws. These laws can be expressed in mathe-<br>matical terms as the conservation of mass, the conservation of momentum, and the <br>conservation of energy. In addition to these three conservation equations, there may <br>also be one or more constitutive equations, which describe material properties, i.e., <br>shear thinning behavior. Since these equations may also be coupled together, i.e., <br>temperature dependent viscosity, the solution becomes even more complex. The <br>goal of the modeler is to take a physical problem, apply these mathematical equa-<br>tions, and solve them to predict the flow phenomena. Although analytical solutions <br>to the conservation equations for some simple two-dimensional shapes are availa-<br>ble, when more complex two-dimensional problems need to be solved or a three-<br>dimensional analysis is required, numerical methods are to be used.<br>Beyond using analytical solutions, there are three basic classes of numerical tech-<br>niques that are commonly used to solve complex fluid flow problems: the finite dif-<br>ference method (FDM), the finite element method (FEM), and the boundary element <br>method (BEM) [5]. Each of these methods has its advantages and disadvantages and, <br>therefore, one may be preferred for a certain type of process or material. Each tech-<br>nique has been adapted in some form for specific problems encountered in polymer <br>processing. Although it is not the purpose of this chapter to provide a detailed deri-<br>
+*  Examples of commercial programs of this type are: Flow 2000(TM) from Compuplast Intl ., Extrud 2000 from <br>
+SPR, Inc ., REX from the University of Paterborn<br>
+<hr>
+<A name=875></a> 12.2Background  <b>865</b><br>
+vation of the three numerical methods mentioned, it is necessary to provide a gen-<br>eral description of each.<br>
+<b>12.2.2.1Finite Difference Method<br></b>The finite difference method started gaining prevalence in the 1930s for use in hand <br>calculations and is the simplest to use and understand. Figure 12.2(a) shows the <br>grid constructed to represent the geometry of a two-dimensional domain.<br>Once the grid is created, the governing differential equations are rewritten in a dis-<br>cretized form and then applied at each nodal point. The resulting system of alge-<br>braic equations can then be solved by standard Gaussian elimination or by more <br>elaborate numerical algorithms. Because of the simplicity of the method, it can be <br>implemented in a wide variety of problems. Since the method discretizes the gov-<br>erning equations at the start of the analysis, it relatively easy to model non-linear <br>problems. The finite difference method is straight forward to program and can have <br>quick computation times. While the simple nature of the finite difference method <br>allows for easy programming, this simplicity also yields certain limitations and <br>other disadvantages. The first consideration when implementing the FDM is that it <br>is best suited for cases that have relatively simple geometries. Even though more <br>complex geometries have been modeled with special differential equations or coor-<br>dinate transformations, limitations still exist, and the other methods presented in <br>this chapter often prove to be more efficient. Because discretization of the actual <br>governing equations occurs at the start of the analysis, more error is introduced <br>early in its derivation, which is then carried through the computation. Therefore, <br>FDM can have problems with obtaining a convergent solution for non-linear prob-<br>lems. Since the FDM is a domain method, it typically takes considerable effort to <br>discretize the geometry of interest, which can severely limit its application to realis-<br>tic devices. In addition, FDM is not well suited to model problems with moving solid <br>boundaries.<br>
+a.<br>
+b.<br>
+c.<br>
+a<br>
+b<br>
+c<br>
+<b>Figure 12.2</b>Mesh representation for common numerical methods (a) FDM (b) FEM (c) BEM<br>
+<hr>
+<A name=876></a><b>866</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+<b>12.2.2.2Finite Element Method<br></b>In contrast to FDM, the finite element method (FEM) is a relatively new technique <br>used for solving fluid flow problems. Popularized in the 1960s along with the advent <br>of digital computers, FEM has become the basis for most commercial structural <br>dynamic and fluid flow simulation programs. Like FDM, FEM is a domain method in <br>which the entire geometry to be modeled must be discretized into nodes and ele-<br>ments. The mesh shown in Fig. 12.2(b) represents the discretization required for <br>FEM to model a two-dimensional geometry. Although several different methods are <br>available to obtain the final equations, the Galerkin method [5] of weighted residu-<br>als is normally preferred in fluid flow problems. Once the mesh has been created, <br>the governing differential equations are then expressed in integral form and numer-<br>ically integrated to obtain an algebraic system of equations. Because of the nature of <br>the finite element method, it is capable of modeling much more complex geometries <br>than FDM. It can also provide quite accurate solutions to the field variables, such as <br>fluid velocities or pressures, for a wide variety of problems that include non-linear <br>flows. However, higher order derivative solutions, such as velocity gradients, tend to <br>be less accurate. Without complex adaptive meshing techniques, FEM is also diffi-<br>cult to use for problems with moving solid boundaries. Since the governing equa-<br>tions are approximated with the Galerkin method, they have a certain amount of <br>intrinsic error even before numerical errors are accounted for, which is carried <br>throughout the computation. This can cause the FEM to become unstable in highly <br>non-linear situations. Although this can be partially alleviated by special upwinding <br>techniques [6], it nonetheless increases the amount of computation effort. In addi-<br>tion, because the solution is computed only at the nodes and the velocity field must <br>be interpolated, the tracking of particles in the flow field in not easily accomplished <br>with FEM.<br>To overcome the limitations that exist with FEM when dealing with moving bound-<br>aries, Avalosse et al. [7] used a special FEM method referred to as the Mesh Super-<br>position Technique (MST). With this technique, a finite element mesh is created for <br>each part of the system. For example, when analyzing a batch mixer, a fully three-<br>dimensional mesh is generated for the bowl without the mixer. Another three-<br>dimensional mesh of the mixing apparatus alone without the bowl is also created. <br>Both meshes are then combined to create an overlapping of both regions. The over-<br>lapping region is accounted for by using a penalty formation that imposes the proper <br>velocity. For example, with extrusion, the nodes from the mesh of the barrel that are <br>within the domain of the screw are given the rotational speed of the screw, every-<br>where else the nodes are handled in the normal way. With this technique, Avalosse <br>et al. [7] were able to take into account non-isothermal, non-Newtonian effects when <br>simulating solid moving boundaries.<br>The finite element has proved to be ideal when simulating the mold filling, fiber <br>orientation, shrinkage, and warpage of thin plastic parts. With more difficulty, it has <br>
+<hr>
+<A name=877></a> 12.2Background  <b>867</b><br>
+been used to simulate the fluid flow in batch mixers, dies, and single and twin screw <br>extruders.<br>
+<b>12.2.2.3Boundary Element Method<br></b>In contrast to both the finite difference and finite element methods, the boundary <br>element method (BEM) is a technique that only requires the boundary or surfaces of <br>the geometry be discretized. As shown in Fig. 12.2(c), a two-dimensional geometry <br>only requires a discretization of the curve that makes up the boundary of the part. <br>In essence, the order of analysis being made is reduced by one. For example, the <br>two-dimensional geometry shown in Fig. 12.2(c) is meshed (discretized) with one-<br>dimensional elements. Similarly, a three-dimensional geometry is meshed with two-<br>dimensional elements because only the surface of the geometry is defined. It is <br>important to note that a two-dimensional BEM analysis is still two-dimensional and <br>a three-dimensional BEM analysis is three-dimensional because the velocity and <br>velocity gradients and other attributes can be calculated at any place in the domain <br>(where the fluid is) of the geometry.<br>BEM gained prevalence around the same time as FEM, but because of the relatively <br>complex mathematics involved with BEM, it has been relatively slow to gain the <br>same level of acceptance that FEM did in the engineering community, and has pri-<br>marily been used by mathematicians. The formulation of the boundary element <br>method begins with a different form of the governing equations, which are expressed <br>in terms of domain integrals. These integrals are manipulated by Green-Gauss <br>transformations until they are reduced to boundary integrals [8­11]. The integrals <br>are then numerically evaluated to yield an algebraic system of equations. Up to the <br>point of evaluating the integrals, no approximations have been made in the govern-<br>ing equations. Thus, the boundary element method, unlike the FDM or FEM, does <br>not introduce any error to the solution until the boundary is discretized--the bound-<br>ary element solution is <i>exact</i> until the geometry is meshed.* Another advantage <br>
+of BEM is the fact that the accuracy of higher order derivatives is excellent. This <br>becomes extremely important when calculating heat transfer effects or tracking <br> <br>particles. Here, the boundary element method is well suited to track particles in the <br>flow of material since the solution at any location in the fluid can be obtained quite <br>easily and very accurately.<br>The reduction of the dimensionality is a key advantage to modeling polymer flows <br>in complex geometries with this technique. Because only the boundary or surface of <br>the domain needs to be defined, which is relatively simple with the many commer-<br>cially available solid modelers, the amount of work for the engineer is dramatically <br>reduced. Moreover, this method can handle moving solid boundaries with relative <br>
+* An exact solution with regards to the material model used, which up to this point has been limited to <br>
+Newtonian flows .<br>
+<hr>
+<A name=878></a><b>868</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+ease because the mesh moves right along with the moving boundary. For example, <br>when simulating the extrusion process, the mesh describes the geometry of the <br>screw and when it rotates the nodes and elements that make up the screw simply <br>rotate with it.<br>Although BEM is quite an elegant technique for linear problems, it loses many of <br>its advantages when non-linear problems are investigated. The standard BEM is <br>only able to handle non-linearities by using domain meshing, thus eliminating the <br>boundary-only discretization. Recently, Nardini and Brebbia [12] have developed <br>a  variation of BEM, known as the Dual Reciprocity Method (DRM), which has the <br>ability to solve non-linearities with a boundary-only formulation. The application of <br>DRM to polymer processing fluid flow applications is still an open problem [9, 13] <br>and will be examined later in this chapter.<br>
+<b>12.2.3Remeshing Techniques in Moving Boundary Problems</b><br>
+The major difficulty, which arises when simulating a mixing process, is the tran-<br>sient free surface or solid moving boundaries, respectively. The material constantly <br>changes shape as it flows, making it necessary to redefine the geometry of the <br>domain of interest after each successive time step. Redefining the finite element <br>mesh or finite difference grid is the most tedious part of simulations when dealing <br>with moving boundary problems and many times makes it unreasonable to simu-<br>late.<br>Wang, Hieber and Wang [14] implemented a mesh editing procedure or dynamic <br>mesh generator into an injection molding simulation.* After carefully choosing a <br>time step and advancing the flow fronts, the user is required to fill the gap between <br>the old and the updated melt fronts with new triangular finite elements. Obviously, <br>this procedure not only requires extensive user interaction but also makes the mesh <br>sizing dependent on the size of the chosen time step.<br>The compression mold filling simulation of Lee, Folger and Tucker [15] used a finite <br>element mesh to represent the initial charge. The same mesh was used after each <br>time step to fit the shape of the charge. This was accomplished with a finite element <br>calculation, which used the displacements of the nodes on the free flow fronts as <br>boundary conditions. This procedure is analogous to drawing the original mesh on a <br>sheet of rubber and then stretching it to conform to the shape of the charge at any <br>time step. The mesh stretching technique kept elements from becoming so distorted <br>as to cause large numerical errors. This technique required a minimum amount of <br>computation and ran automatically once the problem was set. However, it does not <br>
+* A review of simulation processes for injection molding can be found in a chapter by Davis, B .A . and Rios, <br>
+A .C . in the Injection Molding Handbook edited by Osswald, T .A ., Turng, T ., and Gramann, P .J ., Carl Hanser <br>
+(2001)<br>
+<hr>
+<A name=879></a> 12.2Background  <b>869</b><br>
+handle problems that have multiple charges, mold inserts, or problems where the <br>initial shape of the charge differs greatly from the mold shape or final shape of the <br>charge. However, the technique applies very well to the blow molding and thermo-<br>forming process. Here, the initial finite element mesh that represents the parison or <br>sheet is stretched to fit the shape of the material as it is formed into its final shape. <br>Kouba and Vlachopoulos [16] and deLorenzi and Nied [17] used such a technique <br>to model membrane stretching during blow molding and thermoforming. Although <br>the processes are basically three-dimensional, they can be represented with two-<br>dimensional plate elements oriented in three-dimensional space.<br>Brown [18] developed another mesh generation scheme that has been extensively <br>used, which begins by covering the entire mold surface with elements. The initial <br>charge is described by specifying the location of its boundary. The technique in -<br>cludes in its finite element calculations only those elements that form part of the <br>charge. The elements are either "full," "empty," or "partially filled." The elements <br>that are partially filled are temporarily distorted in order to make the element <br>boundaries coincide with the flow fronts. Although this method is promising, it <br>requires some user interaction. Problems are encountered when two element sides <br>lie on the free flow front. Due to the nature of the shape functions for each element, <br>the corner node that lies on the flow front will never move as it will always have zero <br>velocity. Crochet et al. [19] developed a similar technique to simulate the injection <br>mold filling process of complex non-planar parts. Their mesh fitting technique was <br>extended to simulate flows with solid moving boundaries such as the ones encoun-<br>tered in internal batch mixing and extrusion processes.<br>Tadmor, Broyer and Gutfinger [20, 21] used a spatial finite difference formulation to <br>solve two-dimensional flow problems in complex geometrical configurations. Using <br>a Hele-Shaw [22] formulation to simulate the flow, their method is applicable to <br>flows in narrow gaps of variable thickness, such as injection molding of thin parts <br>and flows inside certain extrusion dies. This technique is known as the Flow Analy-<br>sis Network (FAN), and works well for Newtonian and non-Newtonian fluids. The <br>method uses an Eulerian grid of cells that covers the flow cavity. A fill factor is asso-<br>ciated with each cell, a number that varies between zero and one. A fill factor of zero <br>denotes an empty cell, and a fill factor of one denotes a cell that is full of material. <br>The fluid is assumed to be concentrated at the center of each cell. A local mass bal-<br>ance is made around each cell, which results in a set of linear algebraic equations <br>with pressures at the center of the cells as the unknown parameters. The pressure <br>field that results from solving the set of equations is used to calculate the flow distri-<br>bution between the cells, which in turn is used to advance the flow inside the cavity <br>by updating the cell fill factors. A major disadvantage of this technique is that rela-<br>tively fine meshes are required, especially if curved boundaries are present in the <br>geometry. This disadvantage can be overcome by using finite difference operators, <br>however, this makes the simulation awkward and difficult to use.<br>
+<hr>
+<A name=880></a><b>870</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Osswald and Tucker [23] and Wang et al. [24] modified the flow analysis network <br>to model the non-isothermal flow of non-Newtonian fluids inside thin three-dimen-<br>sional cavities using finite elements. The technique, which is commonly known as <br>the control volume approach (CVA) requires that the three-dimensional molding <br>surface be divided in flat three- or four-noded finite elements. Cells or control vol-<br>umes are generated by connecting element centroids with element mid-sides. When <br>applying the mass balance to each cell, the resulting equations are the same as <br>those that result from applying the Galerkin method to the governing equation for <br>pressure. This allows the use of standard finite element assembling techniques <br>when generating the set of linear algebraic equations.<br>
+<b>12.2.4Rheology</b><br>
+Most polymer processes are dominated by the shear strain rate.* Consequently, the <br>viscosity used to characterize the fluid is based on shear deformation measurement <br>devices. The rheological models that are used for these types of flows are usually <br>termed <i>Generalized Newtonian Fluids </i>(<i>GNF</i>)<i>.</i> In a GNF model, the stress in a fluid is <br>
+dependent on the second invariant of the stain rate tensor, which is approximated <br>by the shear rate in most shear dominated flows. The temperature dependence of <br>GNF fluids is generally included in the coefficients of the viscosity model. Various <br>models are currently being used to represent the temperature and strain rate de -<br>pendence of the viscosity.<br>The power law model proposed by Ostwald [25] and de Waele [26] is a simple model <br>that accurately represents the shear thinning region in the viscosity curve, but <br>neglects the Newtonian plateau at small and large strain rates. The major disadvan-<br>tage of this model is that the viscosity approaches infinity at low stain rates and zero <br>at high strain rates. The infinite viscosity leads to erroneous results in problems <br>where there is a region of zero shear rate such as in the center of a tube. However, <br>this problem can be overcome by using a truncated model where a constant vis-<br>cosity is assumed in the strain rate region of Newtonian behavior. A model that fits <br>the complete range of strain rate was developed by Bird and Carreau [27]. The Bird-<br>Carreau model accurately models the Newtonian plateau observed at low and some-<br>times high strain rates, and the shear thinning region in-between.<br>The tendency of polymer molecules to "curl-up" while they are being stretched in <br>shear flows results in normal stresses in the fluid that greatly affect the flow field in <br>certain cases. Additionally, most polymer melts exhibit an elastic as well as a vis-<br>cous response to strain. This puts them under the category of viscoelastic materials. <br>There are no precise models accurately representing this behavior in polymers. <br>
+* A resource for viscosity vs . shear rate and other multipoint and single point data is the CAMPUS® materials <br>
+databank, which can be found on-line at www .campusplastics .com<br>
+<hr>
+<A name=881></a> <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>871</b><br>
+However, various combinations of elastic and viscous elements have been used to <br>approximate the material behavior of polymer melts.* Some models are combina-<br>tions of springs and dashpots to represent the elastic and viscous responses, respec-<br>tively. The most common ones being the Maxwell model for a polymer melt and the <br>Kelvin or Voight model for a solid. One model that represents shear thinning be -<br>havior, normal stresses in shear flow and elastic behavior of certain polymer melts <br>is the K-BKZ model [28­29].<br>Elongational or "shear-free" flows are the least studied types of flows that occur in <br>polymer processing. A major reason for this is that they are not as common as shear <br>flows that dominate extrusion and injection molding. However, in certain polymer <br>processes, such as fiber spinning, blow molding, thermoforming, foaming, and com-<br>pression molding, under specific processing conditions, the major mode of deforma-<br>tion is elongational. Moreover, the elongational viscosity that is needed for simula-<br>tion is difficult to measure and thus requires expensive equipment.<br>
+<b></b><br>
+ <b>12.3Simulating 3-D Flows with 2-D Models</b><br>
+The flow and heat transfer in polymer processes is essentially three-dimensional. <br>Historically, complex systems such as flow inside extruders or dies have been sim-<br>plified from three-dimensions to one-dimensional or two-dimensional channel flow <br>systems. Sometimes, the geometry of a system is simple enough that it can be sim-<br>plified to a planar type flow problem. Other times, the thickness of the cavity or die <br>is thin enough that the lubrication approximation can be used to model the process. <br>Though there is some degree of loss of accuracy caused by neglecting or approxi-<br>mating one of the dimensions, these types of simulations offer good insight into the <br>process and have been used for many years to help design plastic parts and to ana-<br>lyze and optimize the polymer processing operation. The simplifications taken with <br>this type of simulation will be discussed in more detail in this section.<br>
+<b>12.3.1Simulating Flows in Internal Batch Mixers with 2-D Models</b><br>
+A high intensity mixer commonly used in the plastic and rubber industry is the <br>Banbury type mixer. The figure eight shaped chamber with the spiral lobed rotors <br>creates a complex, transient flow. This flow is ideal for mixing different polymers <br>into a homogenous blend, but makes analysis extremely difficult. In the Banbury <br>
+* For a more detailed presentation concerning rheological models the reader is referred to Bird, R .B ., <br>
+Armstrong, R .C ., and Hassager, O ., Dynamics of Polymeric Liquids, Vol . 1, Wiley, New York (1987)<br>
+<hr>
+<A name=882></a><b>872</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+mixer, flow exists in the axial direction of the two rotors, however, the majority of <br>the mixing occurs when the polymer is exchanged between the two lobes. Thus, it is <br>common to use a 2-dimensional model to analyze the mixing that occurs in these <br>types of processes. Yang et al. [30] found that in spite of neglecting the axial flow, <br>the 2-dimensional model predicts the flow field characteristics of the chamber quite <br>well.<br>Using a finite element fluid dynamics analysis package, FIDAP [31], Cheng and <br>Manas-Zloczower [32] simulated the isothermal flow patterns in the Banbury type <br>mixer. To represent the dynamic motion that is present with the moving rotors, they <br>selected eighteen different geometries to represent the mixing cycle. To describe the <br>rheological behavior of the fluid, the power law model was used. To characterize the <br>flow and assess the efficiency of dispersive mixing, Cheng and Manas-Zloczower <br>[33] used the flow number , defined as<br>
+ (12.1)<br>
+A value of 0.5 for  signifies simple shear, while values of 0.0 and 1.0 represent pure <br>rotation and pure elongation, respectively. During mixing, Elmendorp [34] experi-<br>mentally observed that in the dispersion of liquids with high viscosity ratios, elon-<br>gational flows are more effective than shear flows. Thus, using the simulation, areas <br>of high  indicate efficient dispersion of agglomerates into the liquid.<br>During experiments, tracking tracer ink through a process is a common method to <br>use when attempting to understand the occurring mixing phenomena. Using tra-<br>ditional methods of simulation, the finite difference and finite element methods, <br>moving boundaries of mixers make the tracking of "ink" lines very cumbersome. <br>This is due to the difficulty encountered when reorganizing the finite difference grid <br>or finite element mesh to rapidly fit different mixer geometries after every consecu-<br>tive time step when the computational domain changes shape. Furthermore, when <br>computing the internal velocities an interpolation function must be used, making it <br>difficult to track particles accurately through the process. A method that overcomes <br>these difficulties and is well suited to analyze these types of processes is the bound-<br>ary element method. As the domain changes shape, i.e., rotors rotate, the boundary <br>elements follow along because they describe the shape of the device, thus eliminat-<br>ing the need for remeshing. Moreover, when capturing the velocity at key points in <br>the domain, interpolation methods are not needed, allowing for more accurate track-<br>ing of particles through the process. Several authors, Gramann et al. [8, 11, 36], <br>Stradins [37], and Davis et al. [38] have used this method to analyze the flow, heat <br>transfer, and mixing that occur in internal batch mixers during processing.<br>
+<hr>
+<A name=883></a> <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>873</b><br>
+As they simulated the flow in a Banbury type mixer, Gramann and Osswald [11] <br>were able to track particles throughout the processes. Figure 12.3 shows the defor-<br>mation that can take place as the rotors turn.<br>Here, one half of one chamber has a different color to help visualize the mixing that <br>takes place. As the rotors turn, the mixing effect is easily seen. From Fig.12.3 it can <br>be seen that though there is a lot of deformation, exchange between chambers is <br>quite low. A small portion of material is squeezed from one side to the other. The <br>exchange of fluid between mixer halves is necessary to optimize mixing. Using this <br>type of simulation, the geometry of the rotors and chamber wall can be easily modi-<br>fied allowing the engineer to find the optimum geometry.<br>
+0o<br>
+90o<br>
+o<br>
+180o<br>
+270<br>
+<b> Figure 12.3</b> <br>
+Simulated fluid deformation in a <br>
+360o<br>
+450o<br>
+Banbury-type mixer<br>
+When mixing one or more polymeric materials into a homogeneous blend, the major <br>task of the mixing equipment is to break up the individual agglomerates or droplets <br>and distribute them throughout. In the compounding industry, a great deal of diffi-<br>culty is encountered when trying to reproduce blends from batch to batch. Using <br>simulation, the stress and velocity fields during blending can be predicted. With <br>this information, an engineer has the ability to characterize the mixing by calculat-<br>ing how much deformation is produced in the mixer. When simulating the blending <br>of multiple fluids, two important properties must be considered: the viscosity of <br>each fluid and the effect of surface tension. Figure 12.4(a) depicts the initial state of <br>a rotor-cylinder mixer with a circular outline of particles within the matrix and a <br>drop or sub-domain of different viscosity, shown in black. Figure 12.4(b) shows the <br>deformation of a drop that has half the viscosity of the surrounding fluid. As the <br>viscosity ratio of the drop to the surrounding fluid increases, Figs. 12.4(c­d), the <br>deformation becomes significantly less.<br>
+<hr>
+<A name=884></a><b>874</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+µ2<br>
+a<br>
+µ1<br>
+b<br>
+<b> Figure 12.4</b> <br>
+Deformation of a droplet with a varying viscosity ratio <br>
+(continuous phase/droplet = 1/2) (a) initial state <br>
+c<br>
+d<br>
+(b) 1/2 = 2 (c) 1/2 = 1/2 (d) 1/2 = 1/5<br>
+When including surface tension effects, forces that tend to keep the drop spherical <br>resist deformation even further. The basic parameter that must be considered when <br>breaking-up droplets with surface tension effects is the Capillary (or Weber) num-<br>ber defined as<br>
+ (12.2)<br>
+where   is the magnitude of the deviatoric stress, R the radius of the dispersed <br>phase and s the surface tension. The interfacial stress is the ratio of the surface <br>
+tension to the radius of curvature of the droplet. In order to break-up droplets the <br>value of Ca must reach the critical Ca number, which is about 1 for shear flow and <br>about 0.4 for elongational flow, when the drop viscosity equals the matrix viscosity. <br>Hence, this critical Capillary number, Cacrit, is more difficult to achieve as the drop-<br>
+let becomes smaller during the mixing process. Using the boundary element <br>method, Biswas and Osswald [39] simulated droplet deformation, and Stone and <br>Leal [40] the break-up of an initially stretched droplet during relaxation in a quies-<br>cent matrix.<br>As mentioned earlier in this chapter, the boundary element technique is not well <br>suited for solving non-linear problems. An extension of this technique that has <br>shown great promise in taking into account non-linearities, while still requiring <br>only a boundary or surface mesh, is the Dual Reciprocity boundary element Method <br>(DRM) [12, 13]. The DRM is essentially a collocation method in which the non-linear <br>terms are collected into an extra term in the governing equation and then mathe-<br>matically manipulated to a boundary-only formulation. Mätzig et al. [13, 38] used <br>this method to model the convective and viscous dissipation that occurs in a single <br>rotor mixer. Their analysis used a temperature dependent viscosity model and an <br>iterative approach between the BEM flow and DRM heat transfer solutions to con-<br>verge on the viscosity. The resulting streamlines, found by particle tracking points, <br>are shown in Fig. 12.5.<br>
+<hr>
+<A name=885></a> <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>875</b><br>
+Outer Streamline<br>
+Outer Streamline<br>
+T<br>
+R<br>
+R<br>
+W<br>
+<b> Figure 12.5</b> <br>
+Inner Streamline<br>
+Inner Streamline<br>
+Calculated streamlines in a mixer using BEM<br>
+The boundary conditions for this mixer were a zero heat flux for the rotor and a con-<br>stant temperature of 180°C on the barrel. Polyethylene was used for the simulation <br>where the material properties were: thermal conductivity 63000 g cm/s3°C, ther-<br>mal diffusivity 0.0021 cm2/s, density 0.95 g/cm3, specific heat 0.55 cm2/s2°C and a <br>viscosity of 53,000 g/s cm at 185°C and 13,000 g/s cm at 250°C. The temperature, <br>shown in Figs. 12.6(a­b), of the outer streamline was computed for two different <br>rotor speeds, 0.125 rev/s and 1.0 rev/s. In each graph, one curve shows the com-<br>bined effect of viscous heating and thermal conductivity and the other curve has the <br>additional effect of convection. The high strain rates present during the mixing pro-<br>cess generate heat by viscous dissipation, greatly influencing the temperature of the <br>polymer inside the cavity. The importance of the energy generated by viscous dissi-<br>pation and its transport by convection become more significant as the rotor speed <br>is increased. When the rotor speed is 0.125 rev/s the viscous dissipation causes a <br>7°C rise in temperature, Fig 12.6(a).<br>As expected, the temperature profile caused by viscous heating (neglecting convec-<br>tion) is symmetric on both sides of the rotor. In the wide gap region of the mixer, <br>signified as "W," the temperature of the fluid is approximately that of the barrel wall <br>temperature. This is caused by the low viscous heating in this area and direct heat <br>conduction to the barrel wall. The high temperature areas of the graph correspond <br>to the two recirculation areas of the mixer, "R." Although, the viscous heating is <br>significant in the area of the rotor tip, "T," the heat conduction to the barrel wall low-<br>ers the temperature of the fluid due to the close proximity of the material to the wall <br>and the narrow gap. When the energy transport by convection is included, the tem-<br>perature profile is slightly shifted with more variability in temperature in the recir-<br>culation area. However, because of the low rotational speed, the energy transported <br>by convection is relatively small.<br>
+<hr>
+<A name=886></a><b>876</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+187<br>
+Viscous heating effects<br>
+186<br>
+Viscous heating and <br>
+convective transport effects<br>
+185<br>
+184<br>
+183<br>
+Temperature (C)<br>
+182<br>
+181<br>
+180 0<br>
+20<br>
+40<br>
+60<br>
+80<br>
+Outer Streamline <br>
+(a)<br>
+270<br>
+260<br>
+Viscous heating effects<br>
+Viscous heating and <br>
+250<br>
+convective transport effects<br>
+240<br>
+230<br>
+220<br>
+<b> Figure 12.6</b> <br>
+Temperature (C)<br>
+Calculated temperature of a <br>
+210<br>
+particle traveling on the outer <br>
+200<br>
+streamline (Fig . 12 .5) caused by <br>
+190<br>
+viscous heating with and with-<br>
+out convective transport; <br>
+180 0<br>
+20<br>
+40<br>
+60<br>
+80<br>
+(a) rotor speed of 0 .125 rev/s <br>
+Outer Streamline<br>
+(b)<br>
+(b) rotor speed of 1 .0 rev/s<br>
+Increasing the speed of the rotor to 1.0 rev/s the temperature increase caused by <br>viscous dissipation can be quite high. In the analysis, a low Nahme-Griffith number <br>was assumed, hence, a constant viscosity throughout the domain. This assumption <br>does not hold for high temperature variation, however, the results shown here give a <br>good qualitative value when studying the effects of viscous dissipation and convec-<br>tive energy transport. When compared to the 0.125 rev/s rotor speed results, the <br>temperatures increase maintaining a similar profile, Fig 12.6(b). However, when the <br>effects of energy transport by convection are included, the temperature profiles sig-<br>nificantly decrease with the highest variation in the recirculation regions. The <br>Graetz number, ratio of convection transport to conduction, for the system analyzed <br>was on the order of 10,000. Most numerical solutions will have great difficulty at <br>this high of a Graetz number and become unstable, and as a result, up-winding tech-<br>niques are required. However, for the BEM analysis no special upwinding tech-<br>niques were needed.<br>
+<hr>
+<A name=887></a> <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>877</b><br>
+<b>12.3.2Simulating Flows in Extrusion with 2-D Models</b><br>
+At some stage of manufacture, virtually all polymers go through some type of ex -<br>truder one or more times. This may include the extrusion that occurs during the <br>production of pellets, or the final extrusion to produce the finished product. There is <br>a growing need to be able to accurately model and predict the phenomena occurring <br>within the extruder so that it is possible to optimize both the extruder and the prop-<br>erties of the final product without the use of timely and expensive experiments. <br>Extrusion, the most widely used process in the polymer processing industry--almost <br>all injection molding machines have an extruder (plasticating unit) attached to it, <br>and its flow phenomena have been well studied experimentally [41­49] and due to <br>its geometric complexity to a lesser degree numerically [11, 50­56]. Experimental <br>data is of extreme importance when studying the extruder, however, the time con-<br>sumption and expense of these experiments creates the need for numerical simula-<br>tion. Moreover, experimental set-ups are sometimes difficult to control and measure, <br>and many times introduce unexpected variables, such as leakage. Despite contro-<br>versies* involving the assumptions of the analytical models, referred to as classical <br>theory, these models have been used successfully since Rowell and Finlayson [1] <br>modeled the extruder with a moving barrel and a stationary unwrapped rectangular <br>channel to represent the screw. The barrel is represented as a flat plate moving over <br>the rectangular channel; this system is called the flat plate system (FPS). Obviously, <br>the FPS neglects the curvature of the channel.<br>It is interesting to note that, in the FPS, moving the screw relative to a stationary <br>barrel introduces a much larger error than moving the barrel relative to a stationary <br>screw. This error is assessed relative to an analysis using cylindrical coordinates <br>(CCS). The CCS analysis shows no difference between a rotating screw and a rotat-<br>ing barrel as discussed in Chapter 7, Section 7.4.3.4.<br>A simulation program to model the cross flow and down channel flow phenomena in <br>single screw extruders was developed by Rauwendaal, Muller, and Anderson [56]. <br>This model accounts for the non-isothermal, non-Newtonian flow effects in a single <br>screw extruder using a two dimensional finite element formulation. Although this <br>model is, in essence 2-D and represents an idealization to the extruder flow, it does <br>incorporate down-channel flow, leakage, viscous heating, and non-Newtonian effects <br>in the analysis. This is done by assuming that the changes in flow in down channel <br>direction are small relative to the changes in cross channel and normal direction. <br>This is frequently called a 2.5-D analysis because the mesh is 2-D but the calculated <br>
+* The authors Campbell, G .A ., Sweeney, P .A . and Fenton [49] claimed that significant differences exist between <br>
+flow and throughput if the barrel is rotated in lieu of the screw . The paper by Rauwendaal, C ., Oss wald, <br>
+T .A ., Tellez, G ., and Gramann, P .J . [50] used analytical, experimental and numerical techniques to show that <br>
+rotating the barrel is a correct method to analyze the fluid flow in extrusion .<br>
+<hr>
+<A name=888></a><IMG src="Polymer Extrusion by Chris Rauwendaal-888_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_15.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_16.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_17.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_18.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_19.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_20.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_21.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_22.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_23.jpg"><br>
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+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1756.jpg"><br>
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+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1778.jpg"><br>
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+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1785.jpg"><br>
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+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1798.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1799.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1800.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1801.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1802.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1803.jpg"><br>
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+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1806.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1807.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1808.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1809.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1810.jpg"><br>
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+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1813.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1814.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1815.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1816.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1817.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-888_1818.jpg"><br>
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+<b>878</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+velocities, stresses, and temperatures are 3-D. An example of this simulation pro-<br>gram is shown in Fig. 12.7.<br>Figure 12.7 presents the temperature distribution in a single-flighted 38 mm-dia-<br>meter screw extruder. For clarity, the graphic display is exaggerated in the channel <br>height direction. Due to cross flow the highest melt temperature occurs in the mid-<br>region of the channel, while the outside region of the channel is at a lower tempera-<br>ture. The dramatic difference in temperature in this relatively small area can be the <br>source of instabilities or other problems. The use of mixing sections can help allevi-<br>ate this and create a more homogeneous temperature in the extrudate.<br>
+<b> Figure 12.7</b> <br>
+Predicted melt tempera-<br>
+ture in the screw channel <br>
+using FEM<br>
+A screw extruder can have one or more screws. The most common multi-screw ex -<br>truder is the twin screw extruder, which has two screws. These types of extruders <br>can have the screws rotate in the same direction, called co-rotating, or in an opposite <br>direction, called counter rotating twin screw extruders. The flow that is created in <br>these types of extruders is of great interest since they have the ability to produce <br>very homogeneous blends. However, the complexity of the movement of fluid that is <br>generated by two moving screws that are fully intermeshing is a daunting task, both <br>experimentally and with simulation. Obviously, the method of using a stationary <br>screw with a rotating barrel, as used in the analysis of the single screw, is not appli-<br>cable. Here, a moving solid boundaries simulation must be utilized if particles are to <br>be tracked.<br>Rios [57] used a two-dimensional boundary element simulation program to simulate <br>the crosssectional flow in several different co and counter rotating twin screw geo-<br>metries. To create the geometries needed for the self-cleaning twin screw cross sec-<br>tion, equations by Booy [58] were used. All geometries were created using Auto-<br>CADTM or ProEngineerTM. To quantify the mixing, numerous mass-less particles <br>were placed throughout the domain of the mixer and the velocity and velocity gradi-<br>ents of each were computed. With this information, particles were tracked through-<br>out the process giving a visual representation on how the material mixes, strain <br>rates were computed to predict a stress level, and the flow number, Eq. 12.1, was <br>calculated to determine the type (rotation, shear or elongation) of flow. Though a <br>
+<hr>
+<A name=889></a> <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>879</b><br>
+two-dimensional analysis was used to analyze a highly three dimensional flow, a <br>great deal of insight was gained on how these extruders work and comparisons from <br>one system to another was easily made. Figure 12.8 shows the simulated velocity <br>vectors at points throughout the domain of a cross section of a self-cleaning, double-<br>flighted co-rotating twin screw extruder in two different positions.<br>
+<b> Figure 12.8</b> <br>
+Velocity vectors in a double-flighted co-rotating twin screw extruder <br>
+at different rotor positions<br>
+The nodes and elements that were used for the simulation are shown. Because the <br>boundary element method was used, nodes and elements are needed only at the <br>boundaries of the barrel chamber and screws. As the screws rotate, the nodes and <br>elements move as well, essentially eliminating the need for remeshing. The velocity <br>vectors give an indication of stagnant and recirculation zones and are used to <br>advance the particle to the next time step. Figure 12.9 shows the movement of a <br>line, which is made up of many points, through the mixer as the screws turns.<br>
+<b> Figure 12.9</b> <br>
+Deformation of a tracer line in a double-flighted <br>
+twin screw<br>
+The influence of the intermeshing region can be seen as the deformation of the line <br>increases. Portions of the line that were separated enough so that individual parti-<br>cles can be seen indicate that a great deal of deformation has occurred. More insight <br>on the flow field was obtained by calculating the flow number, Eq. 12.1, of the par-<br>ticles as they flow through the system. Rios [57] found that the double-flighted co-<br>rotating twin screw produces a flow that is primarily shear with some elongation.<br>
+<hr>
+<A name=890></a><b>880</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Recall from Section 12.3.1 that a flow number of 0.5 indicates shear flow while a <br>flow number of 1.0 indicates pure elongational flow. For mixing purposes, a flow <br>number near 1.0 is preferred. Figures 12.10(a­c) illustrate that the co-rotating, <br> <br>double-flighted twin screw produces a flow that is mainly shear to elongation.<br>Rios [57] made a similar analysis on a single- and triple-flighted, co-rotating twin <br>screw extruder. In the analysis, an average strain rate and volumetric strain rate <br>were calculated for each system and compared, Figs. 12.10(a­c).<br>
+60<br>
+.·x V<br>
+×  ol = 4426<br>
+Vol = 4426<br>
+.· = 26.1<br>
+50<br>
+= 26.1<br>
+ = 0.53<br>
+= 0.53<br>
+40<br>
+oints<br>
+30<br>
+20<br>
+ercentage of P<br>P<br>
+10<br>
+(a)<br>
+0<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Flow Number<br>
+60<br>
+.·x V<br>
+×  ol = 7300<br>
+V .ol = 7300<br>
+50<br>
+· = 45.9<br>
+= 45.9<br>
+ = 0.50<br>
+= 0.50<br>
+40<br>
+oints<br>
+30<br>
+20<br>
+ercentage of P<br>P<br>
+10<br>
+(b)<br>
+0<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+Flow Number<br>
+60<br>
+.·x V<br>
+×  ol = 5347<br>
+Vol = 5347<br>
+50<br>
+. = 55.6<br>
+· = 55.6<br>
+= 0.53<br>
+ = 0.53<br>
+40<br>
+oints<br>
+30<br>
+20<br>
+ercentage of P<br>P<br>
+10<br>
+0<br>
+0.0<br>
+0.1<br>
+0.2<br>
+0.3<br>
+0.4<br>
+0.5<br>
+0.6<br>
+0.7<br>
+0.8<br>
+0.9<br>
+1.0<br>
+(c)<br>
+Flow Number<br>
+<b>Figure 12.10</b>Flow number and strain rate for the twin screw extruder . (a) single-flighted <br>
+(b) double-flighted (c) triple-flighted<br>
+<hr>
+<A name=891></a> <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>881</b><br>
+Figures 12.10(a­c) shows that the flow number distribution for the single- and <br> <br>double-flighted screws are nearly identical, whereas, the triple-flighted screw pro-<br>duces mainly shear flow due to the small gaps that are present with this geometry. <br>Interestingly, the geometry that created the highest volumetric strain rate was the <br>double-flighted screw, which is the most commonly used screw geometry for twin <br>screw extruders.<br>
+<b>12.3.3Simulating Flows in Extrusion Dies with 2-D Models</b><br>
+Several types of extrusion dies are used in the polymer industry, including: tubing, <br>film-blowing, wire-coating, profile, and sheeting. Die design is a difficult task that is <br>often performed by trial-and-error. Various aspects of the flow through extrusion <br>dies affect the quality of the final product. In addition, flow in dies has several pro-<br>cessing variables that contribute to the quality of the final product. The difficulty in <br>achieving acceptable die designs makes simulation and numerical optimization a <br>viable and useful tool, preferably used before ever cutting metal. Through simula-<br>tion, the engineer can also gain a better understanding and control of the process-<br>ing parameters that affect product quality. In the past few years, there has been an <br>increase in work done regarding die optimization using computer simulation. Several <br>researchers have used simulation to analyze flow through extrusion dies [59­65].<br>The process of wire coating is used extensively throughout the wire and cable indus-<br>try. Wire coating dies are discussed in Chapter 9, Section 9.3 and shown in Fig. 9.19. <br>During this continuous process, one or more layers of polymer are coated onto the <br>wire or cable in a single step. In the coextrusion process, two melt streams are sup-<br>plied by two different extruders. Hen and Mitsoulis [61] analyzed this process using <br>both the lubrication approximation theory and the finite element method. In their <br>analysis, they modified an initial die design to meet criteria needed for better ope-<br>ration. The criterion was to eliminate all recirculation regions throughout the die, <br>have smooth stresses along the die wall--especially at the converging point of the <br>two polymer melts, and have the interface of the two polymers melts lie parallel to <br>the wire. If recirculation areas appear in the die during the process, the polymer can <br>degrade in these regions. Smooth stresses and a parallel interface are needed for a <br>uniform coating. Using lubrication approximation theory and simulating HDPE <br>and PS as the two polymer streams, they analyzed the effect of the flow rate ratio of <br>these two polymers. To take into account the double layer flow, a double-node tech-<br>nique was used at the interface.<br>Simulating an isothermal flow and investigating several different combinations of <br>flow rates, Hen and Mitsoulis found that by having PS as the outer layer and HDPE <br>as the inner layer, a recirculation area is formed that cannot be eliminated by chang-<br>ing the flow rate ratio alone. To find the affect that the viscosity ratio (outer layer <br>viscosity/inner layer viscosity) has they used the finite element method to model <br>
+<hr>
+<A name=892></a><b>882</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+flow through a die. Simulating a Newtonian flow, they found that with a viscosity <br>ratio of 0.2, a large recirculation region exists; see Fig. 12.11(a).<br>Whereas, with a viscosity ratio of 2.0, smooth streamlines appear, see Fig. 12.11(b). <br>They found that the recirculation regions disappear when the viscosity ratio is <br>greater than 1.5, however, when having a viscosity ratio greater than 1.0, interfacial <br>instabilities occur [66]. Thus, to eliminate the recirculation areas and at the same <br>time have no interfacial instabilities (viscosity ratio greater than one), the effect of <br>constraining the flow area in the die was examined. In the analysis, the final thick-<br>ness ratio (outer layer thickness/inner layer thickness) of the two layers was varied <br>(1/15, 1/3, 1/1) to see which one would give a uniform interface and also have the <br>smoothest streamline pattern. From the simulation, the reduced flow area that gave <br>a 1/3 thickness ratio and reduced the die-taper provided the best results--a level <br>interface, no recirculation areas, and a smooth shear stress transition at the die <br>wall, see Fig 12.11(c). Next, using this die configuration with the favorable viscosity <br>and thickness ratios, they simulated the flow with nonisothermal effects. Because of <br>the high speeds that are present with extrusion, viscous dissipation and energy <br>transport by convection must be considered. To handle the high convective flows, <br>their simulation used a streamline-upwind Petrov-Galerkin technique along with <br>higher order elements. From the analysis, they found that the streamlines for the <br>non-isothermal simulation were, in essence, identical to the isothermal case--an <br> <br>isothermal simulation predicted quite well the phenomena that occur.<br>
+<b></b><br>
+<b> Figure 12.11</b> <br>
+Calculated streamlines in a coextrusion wiring-coating <br>
+die . (a) viscosity ratio of 0 .2 (b) viscosity ratio of 2 .0 <br>
+(c) thickness ratio of 0 .3333<br>
+<hr>
+<A name=893></a><IMG src="Polymer Extrusion by Chris Rauwendaal-893_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-893_2.jpg"><br>
+ <br>
+12.3Simulating 3-D Flows with 2-D Models  <b>883</b><br>
+Utilizing extrusion to make intricate profiles is a major advantage of this process, <br>but one that requires a great deal of die design knowledge. An incorrect die can <br>produce a profile that is dramatically different than expected. Figure 12.12(a) dem-<br>onstrates an example of such a case [67].<br>
+(a)<br>
+(b)<br>
+<br>
+<b>Figure 12.12</b>Cable tray profile made from PVC . (a) Result of a poorly designed die (b) Result of <br>
+a properly designed die [67]<br>
+Figure 12.12(a) shows two arms of a cable tray designed to be smooth and flat that <br>have taken a wavy shape, making it completely useless. Using a two-dimensional <br>simulation program* one can analyze why this problem occurred. Figure 12.13(a) <br>clearly shows a non-uniform velocity profile indicating a poor distribution of mate-<br>rial.<br>Specifically, a large portion of the material is predicted to flow out of the large trian-<br>gular sections at well over 200 mm/s while the average velocity (and average line <br>speed) will be closer to 60 mm/s. This discrepancy in the velocity field for the cross <br>section results in the waviness shown. Once the cause of the problem is determined, <br>the software can be used as a tool to analyze design changes before they are imple-<br>mented in the actual die. The calculated velocity in a modified die that places a <br>restriction in the triangular sections of the die is shown in Fig. 12.13(b). The velo-<br>cities are uniformly balanced across the die, which produces a part that does not <br>have a wavy shape, Fig. 12.12(b) [67].<br>While an experienced die designer may intuitively know where flow balancing is <br>required within a profile, it is difficult to quantify this. This is compounded if a <br>requirement comes along for a profile to be produced from a new material with which <br>the designer has little experience. The ability to simulate the flow distribution based <br>on the geometry and the flow characteristics of the polymer allows the designer to <br>more accurately determine the flow channel details for successful extrusion.<br>
+* The simulation results for this die were performed with Flow 2000 by Compuplast, Inc .<br>
+<hr>
+<A name=894></a><IMG src="Polymer Extrusion by Chris Rauwendaal-894_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-894_2.jpg"><br>
+<b>884</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+<b>a) </b><br>
+<b>b) </b><br>
+<b>Figure 12.13</b>Computed velocity in a cable tray profile die . (a) Original die (Fig . 12 .12[a]) <br>
+(b) Corrected die (Fig . 12 .12[b])<br>
+During the extrusion process, viscoelastic behavior of polymers is readily exhibited <br>as swelling at the end of the die. This behavior is the result of the "memory" pheno-<br>mena associated with polymeric materials, normal stresses and sudden changes in <br>boundary conditions--as the material separates from the die. For short dies with <br>a large contraction, the material will tend to "remember" its original configuration <br>and swell at the die exit. Whereas in longer dies with small contractions the mate-<br>rial will "forget" its original state and swelling is less prominent. The number of <br>rheological models that correctly predict this phenomena is small [68]. A commonly <br>used model is the K-BKZ model proposed independently by Kaye [28] and Bernstein <br>et al. [29]. Kiriakides et al. [64] used special integration procedures to apply the <br>K-BKZ model when simulating flows in common dies. They found good agreement <br>between experimental and numerical results, including small vortexes that form <br>and change in size and intensity at different flow rates.<br>
+<hr>
+<A name=895></a> <br>
+12.4Three-Dimensional Simulation  <b>885</b><br>
+<b></b><br>
+ <b>12.4Three-Dimensional Simulation</b><br>
+The complex three-dimensional geometries common in polymer processing equip-<br>ment, molds, and dies make it difficult to analyze their fields using two-dimensional <br>models. Although experiments often give good insight into a problem, they are <br>costly and the results are difficult to analyze for a quantitative evaluation of the pro-<br>cess. For example, measuring temperature fields is often impossible.<br>These shortcomings can be ameliorated using numerical techniques. The advan-<br>tages offered by numerical simulation open up vast resources, many of which have <br>not been fully explored. As shown earlier in this chapter, many complex flows can be <br>analyzed with two-dimensional models that give the engineer a broad understand-<br>ing of the occurring phenomena. However, when accurate detail of the flow field <br>is sought during the evaluating processes, which have "strong" three-dimensional <br>flows, a full three-dimensional simulation must be used. Adding one more dimen-<br>sion to the problem dramatically increases the complexity of the model as well as <br>the computational time. However, with the advent of more advanced computers and <br>the development of more efficient numerical techniques, along with the industrial <br>increase in demand for higher quality, three-dimensional simulation is becoming a <br>reality.<br>This section presents various examples and applications of three-dimensional simu-<br>lation using finite element and boundary element techniques. These are presented <br>to offer the reader an idea of what is possible with the current state-of-the-art of <br>simulation programs. As examples, the authors have chosen cases that are of great <br>interest to the academic, as well as, the industrial research community. These are <br>internal batch mixers, extrusion dies and extrusion mixing sections.<br>
+<b>12.4.1 Simulating Flows in the Banbury Mixer with Three-Dimensional </b><br>
+<b>Models</b><br>
+As discussed earlier, a common type of batch mixer used in the rubber and plastics <br>industry is the Banbury mixer. The previous sections included a review of the exper-<br>imental and two-dimensional studies that have been completed to analyze the flow <br>and mixing behavior in this type of mixer. However, the flow in these mixers is <br>three-dimensional with an important axial flow component that contributes to mix-<br>ing. Using six sequential geometries to represent the entire mixing process, Yang <br>and Manas-Zloczower [30] used a three-dimensional finite element simulation, <br>FIDAPTM [31], to simulate the flow patterns inside the mixer. The polymer used in <br>their simulation had a power law index, n, of 0.22 and a consistency index, m, of <br>9.87×104 N  s0.22/m2. The calculated velocity contours in the axial, z, direction <br>located at Z = 9 cm are shown in Fig. 12.14(a).<br>
+<hr>
+<A name=896></a><b>886</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+<b> Figure 12.14</b> <br>
+Calculated velocity in the axial direction <br>
+(a) and the flow number (b) in the Banbury <br>
+mixer using FEM<br>
+The fluid in the upper region of the mixer flows backward while the fluid in the <br>lower portion flows forward. From the simulation it was also determined that the <br>minimum pressure regions were located behind the rotor tips while the local maxi-<br>mum pressure regions were located in front of the rotor. Figure 12.14(b) shows the <br>predicted flow number distribution at one position of the rotors. Analyzing the flow <br>number, along with the strain rate, through time one can use this information to <br>help design the mixer for optimal mixing conditions. Figure 12.14 shows that this <br>mixer is shear dominated with isolated areas of elongation flow. Figures 12.15(a­b) <br>show the simulated velocity profiles of both the 2-D and 3-D models, respectively. In <br>the bridge region of the mixer there are differences in velocity vectors for the two <br>models, whereas, away from this region the velocity vectors are quite similar.<br>
+<b> Figure 12.15</b> <br>
+Calculated in-plane velocity of the Banbury mixer using <br>
+FEM (a) 2-dimensional analysis (b) 3-dimensional analysis<br>
+<hr>
+<A name=897></a> <br>
+12.4Three-Dimensional Simulation  <b>887</b><br>
+The comparison between the two models shows that a two-dimensional model will <br>give the engineer a broad view of the flow patterns that occur, however, when a more <br>precise velocity field is needed to accurately qualify the mixing effect, a three-<br>dimensional model must be used.<br>
+<b>12.4.2Simulating Flows in Extrusion Dies with 3-Dimensional Models</b><br>
+Due to the flexibility and low cost of the overall extrusion process, complex profiles <br>are commonly extruded. Such shapes lead to full three-dimensional flows inside the <br>die, which can be studied in detail*only through simulating with a full three-dimen-<br>sional simulation program. Various researchers have performed experimental and <br>numerical investigations on flow through extrusion dies [69­78].<br>The coat-hanger die, used in the fabrication of large polymer sheets, is a commonly <br>used die in the polymer industry. The geometry of the die is designed so that the <br>extruded sheet maintains a constant thickness and uniform temperature. The opti-<br>mal geometry that transports the polymer from the circular cross section at the exit <br>of the extruder into one that produces a uniform thickness sheet has been exten-<br>sively researched [2, 79­87] Using a three-dimensional finite element flow pro-<br>gram, Dooley [62] optimized the geometry of a sheeting die. To obtain an optimal <br>geometry, one that has a uniform velocity at the exit, the finite element method was <br>used. Numerous rheological models were used to simulate the flow in the die: New-<br>tonian, Power Law, Carreau, Polynomial, and the Cross-model. Figures 12.16(a­c) <br>show the simulated pressure contours in the die.<br>
+<b></b><br>
+<b> Figure 12.16</b> <br>
+Calculated pressure profile for three different <br>
+coat hanger sheeting dies<br>
+* A complex die can often be simplified into simple elements that can be modeled using analytical models . <br>
+For further reading on this topic, the reader is referred to [83] .<br>
+<hr>
+<A name=898></a><b>888</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+The pressure profile throughout sheeting die "A" is not parallel to the die exit, while <br>it is in sheeting die "C." This creates a more uniform velocity at the exit of the die. To <br>evaluate each die, results were compared to experimental data. Figures 12.17(a­c) <br>show the comparison between experimental and numerical data as the percent devi-<br>ation from uniform flow versus normalized die width for the three die geometries. <br>From Fig. 12.17 it can be seen that the numerical results correctly predict the trends <br>in experimental data. Geometry "C" creates the most uniform flow and is clearly the <br>best design both experimentally and numerically.<br>
+w<br>
+40<br>
+w<br>
+40<br>
+Numerical<br>
+Numerical<br>
+30<br>
+30<br>
+m Flo<br>
+Experimental<br>
+m Flo<br>
+Experimental<br>
+or<br>
+or<br>
+20<br>
+20<br>
+10<br>
+10<br>
+0<br>
+0<br>
+-10<br>
+-10<br>
+viation from Unif<br>
+viation from Unif<br>
+-20<br>
+-20<br>
+-30<br>
+-30<br>
+ercent De<br>
+ercent De<br>
+P -40<br>
+P -40<br>
+0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+100<br>
+0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+100<br>
+Percentage of Die Width<br>
+Percentage of Die Width<br>
+(a)<br>
+( )<br>
+<b></b><br>
+(b)<br>
+w<br>
+40<br>
+Numerical<br>
+30<br>
+m Flo<br>
+Experimental<br>
+or<br>
+20<br>
+10<br>
+0<br>
+-10<br>
+viation from Unif<br>
+-20<br>
+-30<br>
+ercent De<br>P -40<br>
+<b> Figure 12.17</b> <br>
+0<br>
+10<br>
+20<br>
+30<br>
+40<br>
+50<br>
+60<br>
+70<br>
+80<br>
+90<br>
+100<br>
+Comparison of experimental data with numeri-<br>
+Percentage of Die Width<br>
+cal simulation for the three coat hanger sheet-<br>
+(c)<br>
+ing dies shown in Fig . 12 .16<br>
+Multi-layer polymer sheets or films are produced using a coextrusion process in <br>which two or more polymers are extruded and joined together. This allows the pro-<br>cessor to combine desirable properties of multiple polymers into one structure. For <br>example, in packaging it is common to have a layer comprised of recycled polymer <br>sandwiched between virgin and barrier layers. The coextrusion process is typically <br>configured in either of two possible die designs. One option is to use a different <br>manifold for each layer. The layers are then joined together just before the exit of <br>the die. Here, the design of each die is specific for the polymer passing through it. <br>Techniques for die design include non-Newtonian and non-isothermal effects [88], <br>as well as die body deflection [71].<br>Another configuration used for coextrusion is a single die for all layers with a feed-<br>block that combines and distributes the various layers so that they all maintain a <br>uniform thickness across the exit of the feedblock. The design of the die is made to <br>produce a product with the desired width and thickness while maintaining thick-<br>
+<hr>
+<A name=899></a> <br>
+12.4Three-Dimensional Simulation  <b>889</b><br>
+ness uniformity in all layers. With this type of die, the existence of an interface <br>between the layers in the feedblock and the die make simulation quite difficult. The <br>objective when designing a single manifold die for coextrusion is similar to design-<br>ing a single layer--produce a uniform flow rate across the exit of the die. The goal <br>is to create a sheet, film, or coating that is uniform in thickness. However, with the <br>coextrusion process there is the additional requirement that each layer should be <br>uniform in thickness as well.<br>Gifford [74] used the finite element method to examine the effect of the flow rate <br>ratio (flow rate of one layer/flow rate of the second layer) and viscosity ratio (mate-<br>rial viscosity of one layer/material viscosity of the second layer) on the die exit flow <br>distribution. Decreasing the viscosity ratio (A/B) results in the interface between <br>
+the two materials moving upward toward the less viscous material. Figure 12.18(a) <br>shows the interface of the two polymers at several different viscosity ratios.<br>
+(a)<br>
+<b> Figure 12.18</b> <br>
+Calculated effect of viscosity ratio <br>
+(a) and flow rate (b) on the inter-<br>
+face of two coextruded polymers <br>
+(b)<br>
+using FEM<br>
+The smallest viscosity ratio simulated, (A /B = 0.1), demonstrates that the smaller <br>
+viscosity material is starting to flow around the higher viscosity material producing <br>a nonuniform layer thickness--a phenomenon called <i>viscous encapsulation.</i> The <br>
+effect of the flow rate ratio (QA/QB) on the interface between the two polymers is <br>
+<hr>
+<A name=900></a><IMG src="Polymer Extrusion by Chris Rauwendaal-900_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-900_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-900_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-900_4.jpg"><br>
+<b>890</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+shown in Fig. 12.18(b). As this ratio decreases, the interface rises upward toward <br>the lower viscosity material. However, the interface between the two becomes <br>smoother and yields a more stable interface.<br>Layer nonuniformity in coextruded products is also common when the viscosities of <br>the polymers are nearly the same, implying that another factor besides viscous <br>encapsulation is affecting the flow during coextrusion. To study this phenomenon, <br>Dooley et al. [76] investigated the effect of polymer viscoelasticity on layer thickness <br>uniformity of multi-layer coextruded structures. This was done experimentally by <br>coextruding multi-layer structures through die channels of different cross-sectional <br>shapes and observing the location of the interface. Here, the experiments were con-<br>ducted with identical materials in each layer that were pigmented to allow obser-<br>vation of the layer interface. Setting the experiment like this eliminated any effects <br>of viscous encapsulation and demonstrated the effect of viscoelasticity on coextru-<br>sion. The materials studied in the experiment were polystyrene, polyethylene and <br>polycarbonate. Based on the storage moduli, the polystyrene is the most elastic, fol-<br>lowed by the polyethylene resin and then the polycarbonate resin.<br>In the experiment, the shapes (square, teardrop, circular, and rectangular) that <br>were used are commonly found in the design of feed blocks, dies, and transfer lines. <br>Figure 12.19(a) shows the initial cross section of the material as it flows through a <br>square channel, which measures 0.95×0.95 cm.<br>
+(a)<br>
+(b)<br>
+(c)<br>
+(d)<br>
+<b>Figure 12.19</b>Experimental cross section of a two-layer coextruded structure; (a) initial <br>
+condition; (b) polystyrene; (c) polyethylene; (d) polycarbonate 50 cm downstream [76]<br>
+Notice that the white material occupies approximately 20% of the area while the <br>black occupies 80%. It should be reiterated that both materials are identical except <br>for the color pigment--the change in viscosity caused by the color pigment is insig-<br>nificant.* Figures 12.19(b­d) show a cross section of the square die 50 cm down-<br>stream for the polystyrene, polyethylene, and polycarbonate, respectively. In both <br>the polystyrene and polyethylene samples, a thin black substrate layer is shown <br>moving up along the channel walls while the interface in the center is moving up <br>towards the top of the channel. Interestingly, the black substrate material, which <br>flowed along the walls, turned after it reached the corner and flowed back towards <br>
+* The black and white pigmented material was switched for each layer to ensure that identical results were <br>
+observed .<br>
+<hr>
+<A name=901></a><IMG src="Polymer Extrusion by Chris Rauwendaal-901_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-901_2.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>891</b><br>
+the center at 45°--viscous encapsulation does not explain this layer rearrange-<br>ment. Dooley et al. [76] explained this movement of layers in the polystyrene and <br>polyethylene samples as the existence of secondary flows produced by differences in <br>normal forces created when flowing through a non-radial symmetric channel. The <br>amount of relative layer movement in the three samples corresponds to their storage <br>modulus. Here, the polystyrene has the greatest layer movement and the highest <br>storage modulus, while the polycarbonate has the smallest layer movement and the <br>lowest storage modulus.<br>To help understand and quantitatively evaluate the secondary movement shown <br>above, Debbaut et  al. [75, 77] augmented this experimental work with a three-<br>dimensional flow simulation* that incorporated viscoelastic effects. The finite ele-<br>ment method, using a 4-mode Giesekus model as the viscoelastic constitutive <br>equation,** was used for the simulation. The polymer used for the experiment and <br>simulation was a low-density polyethylene. Figures 12.20 and 12.21 show the exper-<br>imental observations and the numerical predictions of the deformations of the inter-<br>face for the rectangular straight channel [78], and for the teardrop channel [75], <br>respectively.<br>
+(a)<br>
+(b)<br>
+(a)<br>
+(b)<br>
+z = 5.08 cm<br>
+z = 2.54 cm<br>
+z = 17.8 cm<br>
+z = 10.2 cm<br>
+z = 30.5 cm<br>
+z = 27.9 cm<br>
+z = 43.2 cm<br>
+z = 40.6 cm<br>
+z = 50.8 cm<br>
+z = 53.3 cm<br>
+<b>Figure 12.20</b>Flow of polyethylene <br>
+<b>Figure 12.21</b>Flow of polyethylene trough a <br>
+trough a square channel for both <br>
+teardrop channel for both experimental (a) and <br>
+ <br>experimental (a) and  numerical (b) [78]<br>
+numerical (b) [75]<br>
+*  Polyflow simulation program, 16 Place de l'Université, B-1348 Louvain-la-Neuve, Belgium<br>
+** The application of the White-Metzner, Phan-Thien Tanner and Giesekus models was done by Dietsche, L ., <br>
+and Dooley, J ., SPE ANTEC, 53, 188 (1995)<br>
+<hr>
+<A name=902></a><b>892</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Figures 12.20 and 12.21 both show excellent agreement between the experiments <br>and the predictions with some slight differences, which could be attributed to the <br>selection of a particular fluid model used with its material parameters. A simulation <br>analysis of this type is indispensable when investigating abnormalities in die flows <br>and allows for the quantification of the processing conditions and material proper-<br>ties on the development of secondary motions.<br>
+<b>12.4.3Simulating Flows in Extrusion with 3-Dimensional Models</b><br>
+The helical geometry of the screw creates important three-dimensional flow effects <br>that influence the overall performance of the extrusion process. The complexity and <br>three-dimensionality of the screw kept researchers from simulating the flow and <br>heat transfer in the actual geometry, leaving experimental work as the only means <br>to analyze the actual process. To circumvent the time consuming disadvantages of <br>experiments and to take advantage of simulation, recently more research has been <br>performed on the extrusion process using finite element and boundary element <br>techniques. This work is presented in this section.<br>
+<b>12.4.3.1Regular Conveying Screw<br></b>To study the flow in the metering section of a single screw extruder, Spalding et al. <br>[51] used the finite element flow simulation program FIDAP [31] and verified the <br>results with experiments. The resin used was a low-density polyethylene with a melt <br>flow index of 2.0, solid density of 0.922 g/cm3, a melt density of 0.74 g/cm3, thermal <br>conductivity of 0.182 W/(m °C) and a heat capacity of 1260 J/(kg °C). A single-<br>flighted, square-pitched screw had a flight width of 7.94 mm and a flight clearance <br>of 0.07 mm was simulated. For the numerical calculations, a non-isothermal flow of <br>a non-Newtonian liquid was modeled. The effect of the flight land and the curved <br>flight radii were also included in the geometric representation. Spalding et al. [51] <br>simulated the process with both a rotating barrel and stationary screw and a station-<br>ary barrel and rotating screw. The mesh used to represent the extruder channel <br>contained 51,714 elements and 50,197 nodes. The processing  conditions used were <br>a flow rate of 41 kg/hr, rotational speed of 60 rpm, a barrel temperature of 205°C, a <br>screw temperature of 200°C and an inlet temperature of 205°C. The average pres-<br>sure gradient computed was around ­1.0 MPa/turn, which is in excellent agree-<br>ment with the experimentally measured pressure gradient of ­0.96 MPa/turn. The <br>computed shear rate and temperature fields at the specified conditions are shown in <br>Fig. 12.22.<br>
+<hr>
+<A name=903></a> <br>
+12.4Three-Dimensional Simulation  <b>893</b><br>
+<b>Figure 12.22</b>Calculated shear rate and temperature contour of a cross-section in the metering <br>
+section of a single screw extruder using fully 3D FEM [51]<br>
+At these processing conditions, with a LDPE resin, the temperature rise due to vis-<br>cous dissipation is minimal. Figure 12.22 shows a slight temperature rise near the <br>pushing flight due to heat conduction through the hot barrel and convection back to <br>the advancing pushing flight.* When analyzing the results for both a rotating barrel <br>and a stationary screw and a stationary barrel with a rotating screw, Spalding et al. <br>[51] found the pressure and temperature fields to be identical. Results from their <br>research are extremely valuable since they show a viable mean to actually look into <br>the extruder and to gain insight into the temperature, pressure and velocity field <br>development.<br>
+<b>12.4.3.2Energy Transfer Mixer<br></b>A mixing section that has been used to improve thermal mixing and to lower the <br>extrudate temperature is the wave-type or Energy Transfer (ET) screws. These mix-<br>ing sections typically have two or more channels that create cross-channel mixing <br>by having their channel depths vary periodically out of phase with one another. The <br>flights between the channels are strategically undercut to permit flow of material <br>between the channels. Thus, as the depth of one channel decreases it forces the <br>material to the other channel where the depth is increasing. To study the thermal <br>mixing ability of these type of screws, Somers et al. [89] used the 3-dimensional <br>simulation program FIDAP [31] to simulate particle trajectory and heat transfer <br>effects, including conduction, convection, and viscous dissipation.<br>
+*  Spalding, M .A ., personal communication, April, 1994<br>
+<hr>
+<A name=904></a><IMG src="Polymer Extrusion by Chris Rauwendaal-904_1.jpg"><br>
+<b>894</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+The non-Newtonian, non-isothermal analysis included the simulation of the ET mix-<br>ing section as well as a conventional metering section for comparison reasons. The <br>mesh for the ET section consisted of 155,520 8-noded brick elements, while the <br>mesh for the conventional system had 186,192 8-noded brick elements. To analyze <br>the thermal mixing, Somers et al. [89] simulated one channel (A) to be fed a fluid <br>at a temperature of 230°C, while the other channel (B) was fed a fluid at 190°C <br>(Fig. 12.23). Likewise, the inlet fluid for the conventional system was specified as <br>230°C, and 190°C for the pushing and trailing sides, respectively. Figure 12.23 <br>shows the calculated temperature contours for the ET at cross-sectional planes taken <br>down the screw.<br>
+<b> Figure  12.23</b> <br>
+Calculated temperature contour in the Energy <br>
+Transfer section [89]<br>
+Recall that Channel A in Fig. 12.23 will become deeper and Channel B shallower <br>and the undercut connecting the channels will permit fluid to transfer between <br>them. As one moves down Channel A, the temperature decreases as cooler material <br>from Channel B transfers in and heat is conducted out through the barrel wall and <br>screw surfaces. Inversely, the fluid in Channel B will increase in temperature due to <br>the combination of viscous dissipation and conduction of heat in through the barrel <br>wall and screw root. At an axial distance of 4.2 diameters, a relatively large thermal <br>gradient is created in Channel A due to viscous dissipation, while the thermal gradi-<br>ents in Channel B are minimal. In the conventional screw, the cross-flow mixes the <br>two fluids of different temperature through convection. After 1 diameter, the tem-<br>perature difference between the channel halves was significantly reduced. Results <br>for the total bulk temperature for both systems show that the temperature gradients <br>imposed at the inlet were eliminated after about 1.5 diameters for the ET and 1 <br>diameter for the conventional section; see Fig. 12.24. The bulk temperature for the <br>ET after several diameters is predicted to be several degrees higher than the conven-<br>tional screw. This is due to the high level of viscous dissipation generated as mate-<br>rial passes from one channel to the other.<br>
+<hr>
+<A name=905></a><IMG src="Polymer Extrusion by Chris Rauwendaal-905_1.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>895</b><br>
+230<br>
+230<br>
+Channel  A<br>
+225<br>
+225<br>
+Channel  A<br>
+220<br>
+220<br>
+215<br>
+215<br>
+o<br>
+o<br>
+C 210<br>
+C 210<br>
+Channel  B<br>
+Channel  B<br>
+205<br>
+205<br>
+200<br>
+200<br>
+ET Sectio n<br>
+Con ventional Syste m<br>
+195<br>
+195<br>
+190<br>
+190<br>
+0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+0<br>
+1<br>
+2<br>
+3<br>
+4<br>
+5<br>
+L/D<br>
+L/D<br>
+(b)<br>
+(a)<br>
+<b>Figure 12.24</b>Calculated mean temperature of regions in the (a) Energy Transfer section and the <br>
+(b) conventional system using fully 3-D FEM [89]<br>
+Though the temperatures generated are predicted to be higher with the ET mixing <br>section, the distributive mixing intuitively is greater than that found in the conven-<br>tional screw. To analyze this mixing, Plumely et al. [90] calculated the trajectory of <br>two particles in both systems--one for each channel of the ET, shown in Fig. 12.25. <br>Recall that the particle path in the conventional screw has a repetitive helical <br>motion. In the ET section the particle cross over from one channel to the other and <br>increases the distributive mixing effect.<br>
+<b> Figure 12.25</b> <br>
+Particle tracking using FEM <br>
+(a) Energy Transfer section <br>
+(b) conventional system [90]<br>
+<b>12.4.3.3Twin Screw Extruder<br></b>Twin screw extruders (TSE) have been used for the processing of viscous materials <br>for several decades. The TSE has additional mixing capabilities that are not found in <br>the typical single screw extruder. The flow in this device is quite complex and highly <br>3-D due to the movement of two screws, thus making simulation a daunting task. <br>However, the increase in mixing ability over the single screw has led many research-<br>ers to study this device both numerically and experimentally.<br>
+<hr>
+<A name=906></a><b>896</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Using the Mesh Superposition Technique (MST), described earlier, Avalosse et al. <br>[91] studied the non-isothermal flow of polypropylene in co-rotating and counter-<br>rotating twin screw extruders. Recall that this technique simplifies the meshing of <br>the geometric entities and avoids remeshing at each time step. To validate the simu-<br>lation, results were compared to experimental measurements made on a Japan Steel <br>Works TEX30, a 30 mm twin screw extruder, where the screws were made to rotate <br>at 200 rpm with a flow rate of 15 kg/hour. Here, the screw was designed so that it <br>could operate as both co- and counter rotating. The numerical measurements were <br>taken along the line of A­A' and B­B', shown in Fig. 12.26.<br>
+<b> Figure 12.26</b> <br>
+Mesh of twin screw with places of measure-<br>
+ment shown<br>
+Figure 12.27(a) displays the pressure values along A­A. Except for the higher angle <br>values, there is very good agreement with the measured pressure. Avalosse et al. <br>[91] explained that the discrepancy occurs because at higher angle values the gap <br>size decreases, thus reducing the mesh density in this area. The predicted tempera-<br>tures also compare very well to the measured values, as shown in Fig. 12.27(b).<br>
+<b>Figure 12.27</b>Experimental and numerical pressure (a) and temperature (b) profile in a twin <br>
+screw extruder along plane A-A shown in Fig . 12 .26<br>
+In the next step of their analysis, Avalosse et al. [91] used simulation to compare the <br>co-rotating and counter-rotating twin screw to one another. Figure 12.28(a) shows <br>the calculated pressure and Fig. 12.28(b) shows the temperature along B­B' (the <br>symmetric of line AA' at the bottom of the X=0 plane) for both systems. There is <br>an obvious difference between the two systems with the counter-rotating system <br>
+<hr>
+<A name=907></a><IMG src="Polymer Extrusion by Chris Rauwendaal-907_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-907_2.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>897</b><br>
+producing a smoother pressure profile. The co-rotating system appears to generate <br>significantly more shear heating with a 6°C higher temperature is some regions. <br>Moreover, it was found that the co-rotating system required 20% more torque to run <br>than the counter-rotating system.<br>
+1.4e6<br>
+230<br>
+Contra Rot<br>
+1.2e6<br>
+228<br>
+Co-Rot<br>
+226<br>
+1.0e6<br>
+224<br>
+8.0e5<br>
+222<br>
+6.0e5<br>
+220<br>
+4.0e5<br>
+218<br>216<br>
+2.0e5<br>
+Contra Rot<br>
+214<br>
+Co-Rot<br>
+0.0e0<br>
+212<br>
+-2.0e5<br>
+210<br>
+0<br>
+10 20 30 40 50 60 70 80 90 100<br>
+0<br>
+10 20 30 40 50 60 70 80 90 100<br>
+(a)<br>
+(b)<br>
+<b>Figure 12.28</b>Numerically calculated pressure (a) and temperature (b) profile in a twin screw <br>
+extruder along plane B-B shown in Fig . 12 .26<br>
+Avalosse et al. [91] used the simulation program to further study the co-rotating <br>twin screw system with and without kneading blocks, Fig. 12.29.<br>
+a<br>
+b<br>
+<b>Figure 12.29</b>Partial finite element mesh used to analyze the twin screw extruder; <br>
+(a) conventional system (b) kneading block mixing section<br>
+The calculated velocity vectors of a conventional co-rotating system with no mixing <br>sections and the kneading block region are shown in Fig. 12.30.<br>
+<hr>
+<A name=908></a><IMG src="Polymer Extrusion by Chris Rauwendaal-908_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-908_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-908_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-908_4.jpg"><br>
+<b>898</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+a<br>
+b<br>
+<b>Figure 12.30</b>Calculated velocity vectors in the twin screw extruder using FEM; (a) conventional <br>
+system (b) kneading block mixing section<br>
+Figures 12.31 to 12.33 show the pressure profile, strain rate and flow number of the <br>kneading block region along a plane in the domain of the mixer at one time step, <br>respectively.<br>
+<b>Figure 12.31</b>Calculated pressure in the <br>
+<b>Figure 12.32</b>Calculated shear rate in the <br>
+kneading block region of a twin screw extruder  kneading block region of a twin screw extruder <br>
+using MST-FEM<br>
+using MST-FEM<br>
+<hr>
+<A name=909></a><IMG src="Polymer Extrusion by Chris Rauwendaal-909_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-909_2.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>899</b><br>
+<b> Figure 12.33</b> <br>
+Calculated flow number in the kneading block <br>
+region of a twin screw extruder using MST-FEM<br>
+Simulation programs like this create abundant information and one needs to know <br>how to interpret these data to make it useful, i.e., predict mixing. One method is to <br>track particles through a system to determine where they go, if there are any stag-<br>nant regions and how long they stay in the system. Figure 12.34 shows the tracking <br>of particles that originated on a plane at the inlet to the mixer. After only a short <br>period, the particles become distributed throughout the twin screw. Using some of <br>the techniques described earlier, the distributive and dispersive mixing capability <br>of this system can be quantitatively analyzed.<br>
+<b>Figure 12.34</b>Calculated particle tracking in a co-rotating twin screw extruder using MST-FEM<br>
+<hr>
+<A name=910></a><IMG src="Polymer Extrusion by Chris Rauwendaal-910_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-910_2.jpg"><br>
+<b>900</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+The addition of fibers to a polymer melt in order to increase the mechanical proper-<br>ties of a part is common practice. The most common fiber used for plastics is glass; <br>however, wood fiber is making inroads in many important applications, along with <br>carbon, aramid, and boron for advanced engineered materials. The amount of fiber, <br>how it is oriented in the part, and the length of the fiber influence greatly the effec-<br>tiveness of increasing, or possibly decreasing, the properties of the part. The extru-<br>sion process is commonly used as the device to mix fibers into a resin to create a <br>composite blend. However, during this process, fibers become oriented and fiber <br>length is reduced. This is caused by the flow field and high stresses in the extruder. <br>To investigate the degradation of fiber length, Krawinkel et al. [92] used the bound-<br>ary element method to calculate the evolution of stresses in the kneading block and <br>double-flighted screw of the twin screw extruder. Figure 12.35 shows the calculated <br>particle tracking in a double-flighted twin screw while Fig. 12.36 shows the velocity <br>and stress gradient at a plane in the kneading block region.<br>
+<b> Figure  12.35</b>Calculated particle <br>
+ <br>trajectories inside a double-flighted <br>
+ <br>co-rotating twin screw extruder<br>
+(a)<br>
+<b> Figure 12.36</b> <br>
+Calculated velocity (a) and stress <br>
+(b) contour at a plane along the <br>
+ <br>kneading block region in a twin screw <br>
+(b)<br>
+extruder using BEM [92]<br>
+<hr>
+<A name=911></a> <br>
+12.4Three-Dimensional Simulation  <b>901</b><br>
+As expected, the highest stresses occur at the nip region of the mixer. Knowing the <br>stress history of the mixer, Krawinkel et al. used the following relationship [93] to <br>solve for the fiber length for a given critical stress.<br>
+ (12.3)<br>
+Developing or synthesizing new polymeric materials is becoming increasingly ex -<br>pensive and difficult. However, it is possible to develop new engineering materials <br>by mixing two or more polymers or by modifying existing ones by adding various <br>ingredients. These polymer blends can be made to provide a wide range of proper-<br>ties. The morphology of these blends plays a critical role in the development of <br>these properties, and the final morphology is a direct result of how the polymer <br>blend was mixed. The ability to qualitatively and quantitatively predict mixing <br>through simulation has led to a better understanding on how materials are mixed or <br>de-mixed and has led to the development of a new generation of mixers.<br>
+<b>12.4.3.4Rhomboidal Mixers and Fluted Mixers (Leroy/Maddock)<br></b>A device that has become a standard in single screw, and to a lesser degree, in twin <br>screw extrusion, is the rhomboidal-pineapple mixing section. This device must be <br>well designed for the number of rhomboids in the axial and circumferential direc-<br>tions, length, pitch, and channel depth to create the optimal mixing environment. <br>Using the boundary element method along with experiments Gramann et al. [94] <br>and Rios et al. [95] studied the rhomboid mixing section. Gramann et al. [94] showed <br>with both simulation and a flow visualization experiment that multiple stagnant <br>regions may occur on the sides of the rhomboid depending on its shape. In the <br>numerical analysis, particles were tracked through the mixer generating the stream-<br>lines shown in Fig. 12.37.<br>Figure 12.37 reveals a large region of stagnant fluid on the top of the rhomboid, <br>which is detrimental to mixing and potentially damaging for polymeric materials <br>that easily degrade. To verify these results, Gramann et al. [94] built an experimen-<br>tal set-up consisting of a 25.5 mm clear acrylic cylinder and five scaled-up rhom-<br>boids. A Newtonian fluid, Dow Corning 200, 10,000 centiStokes polydimethylsil-<br>oxane, was used as the medium fluid. A vertical ink-line was placed in front of the <br>5 rhomboids in the visual experiment; see Fig. 12.38(a). Figures 12.38(b­d) show <br>the deformation of the ink-line after rotation and clearly show a stagnant region on <br>the upper surface of the rhomboid.<br>
+<hr>
+<A name=912></a><IMG src="Polymer Extrusion by Chris Rauwendaal-912_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_4.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_5.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_6.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_7.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_8.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_9.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_10.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_11.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_12.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_13.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_14.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-912_15.jpg"><br>
+<b>902</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Vertical ink line<br>
+Stagnation<br>
+<b>Figure 12.37</b>Predicted <br>
+<b>Figure 12.38</b>Deformation of an ink line in a  visual <br>
+streamlines in a rhomboid mixing <br>
+experiment of a rhomboid mixing section <br>
+section using BEM<br>
+Rios et al. [95] investigated several different configurations of the rhomboid mixing <br>section by changing the two helix angles that define the mixing section. The differ-<br>ent sections analyzed are shown in Fig. 12.39, where the notation of each indicates <br>the pitch in both directions.<br>
+<b> Figure 12.39</b> <br>
+Geometries of rhomboidal mixing sections <br>
+analyzed by [95]<br>
+The geometries were compared according to mixing efficiency, pressure, and energy <br>consumption. The experiments for this study were performed on three of the mixing <br>sections shown in Fig. 12.39, using a high density polyethylene on a highly instru-<br>mented Extrudex ED-N-45­25D single screw extruder equipped with a 45 mm <br>three-zone screw. The performance of each mixer was evaluated by means of a char-<br>acteristic curve and a subjective comparison of micrographs taken from the ex -<br>
+<hr>
+<A name=913></a><IMG src="Polymer Extrusion by Chris Rauwendaal-913_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-913_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-913_3.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>903</b><br>
+trudate; see Fig. 12.40, of material that originally contained a yellow master batch <br>pigment.<br>
+a<br>
+b<br>
+c<br>
+<b>Figure 12.40</b>Extrudate micrographs 50X (a) rhomboid 1D3D (b) rhomboid 1D6D and <br>
+(c) pineapple 1 .6D<br>
+The numerically calculated characteristic curves of the nine rhomboids are shown <br>in Fig. 12.41. Here, data on the positive side of the chart (right) represent a mixing <br>device that produces pressure and the negative side (left) consumes pressure. Here, <br>it can be seen that the pineapple 1.6D is the highest pressure consumer while <br>the rhomboid 1D4D has the highest pumping capability. The pineapple 1.6D has the <br>most restrictions or rhomboids, through which the material must flow and experi-<br>ences a bigger pressure loss. The neutral effect (zero flow rate at a zero pressure <br>difference) for the two pineapple mixers is caused by the two counter helixes--one <br>helix pumps material forward while the other pumps backwards.<br>
+0.030<br>
+Rhomb 1D2D<br>Rhomb 1D3D<br>
+0.025<br>
+Rhomb 1D4D<br>Rhomb 2D3D<br>Rhomb 2D4D<br>
+0.020<br>
+Rhomb 1D6D<br>
+* m<br>
+0.015<br>
+0.010<br>
+0.005<br>
+0.00<br>
+-30<br>
+-20<br>
+-10<br>
+0<br>
+10<br>
+20<br>
+30<br>
+Dp*<br>
+<b>Figure 12.41(a)</b>Numerical calculated characteristic curves of rhomboidal mixing sections<br>
+<hr>
+<A name=914></a><b>904</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+0.030<br>
+0.025<br>
+Pineapple 2D<br>
+Rhomb -1D6D<br>
+0.020<br>
+Pineapple 1.6D<br>
+*<br>
+Rhomb 1D6D<br>
+m<br>
+0.015<br>
+0.010<br>
+0.005<br>
+0.00<br>
+-30<br>
+-20<br>
+-10<br>
+0<br>
+10<br>
+20<br>
+30<br>
+Dp*<br>
+<b>Figure 12.41(b)</b>Numerical calculated characteristic curves of rhomboidal mixing sections<br>
+The residence time distribution was calculated numerically by tracking particles <br>(stimulus) and measuring the time that each particle spends in the mixing device. <br>The cumulative residence time distribution (CRTD) is computed by integrating the <br>residence time distribution and is shown in Fig. 12.42 for five of the rhomboids. <br>Recall that the mixer that gives steepest CRTD curve has the worst distributive mix-<br>ing since all the stimulus exits at nearly the same time. Figure 12.42 shows that the <br>rhomboid 1D3D has the steepest CRTD while the two pineapple mixers give a much <br>better response. Comparing these results to the micrographs shown in Fig. 12.40, <br>and performing a qualitative analysis, the pineapple 1.6D exhibits the thinnest <br>stria tions and the rhomboid 1D3D developed the thickest, indicating that the pine-<br>apple mixer is a better distributive mixer.<br>
+1.0<br>
+0.9<br>
+0.8<br>
+0.7<br>
+TD 0.6<br>CR<br>
+Rhomb 1D3D<br>
+0.5<br>
+Pineapple 2D<br>
+0.4<br>
+Rhomb 2D4D<br>
+0.3<br>
+Pineapple 1.6D<br>
+0.2<br>
+Rhomb -1D6D<br>
+0.1<br>
+0<br>
+0<br>
+1<br>
+2<br>
+3<br>
+t/ t<br>
+<b>Figure 12.42</b>Calculated residence time distribution of rhomboidal mixing sections using BEM<br>
+<hr>
+<A name=915></a><IMG src="Polymer Extrusion by Chris Rauwendaal-915_1.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>905</b><br>
+The counter part to the distributive rhomboid mixing section is the Maddock or <br>LeRoy dispersive mixing section. This device is commonly used to ensure that any <br>unmelted polymer particles entering are subjected to high shear and are melted <br>before leaving the extruder. This mixer consists of several sets of semicircular <br>grooves that run parallel or helically to the axis of the screw. One groove acts as an <br>inlet while the other acts as an outlet with the two grooves connected by an under-<br>cut flight that produces high shear as the material passes through. Fluted mixers <br>were discussed in detail in Section 8.7.1. The LeRoy/Maddock mixer has been ana-<br>lyzed numerical by several investigators using FDM [96], FEM [97], and BEM [8]. <br>Figure 12.43 presents the results of tracking particles through the inlet and outlet <br>of one repeating set of grooves for this mixer.<br>
+Outlet<br>
+Point A<br>
+Point B<br>
+<b> Figure 12.43</b> <br>
+Point C<br>
+Calculated streamlines in one repeating <br>
+ <br>section of a LeRoy-Maddock dispersive mixing <br>
+Inlet<br>
+section<br>
+After the material enters from the bottom, it goes through a cross-flow before going <br>over the undercut flight. Because all material entering at one instant does not pass <br>over the flight at the same time, this mixer does a good job at distributive mixing as <br>well as at dispersive mixing. Looking at the strain rate and flow number of three <br>specific particles, shown in Fig.12.44, as they travel through the mixer one can get <br>a feel for how this mixer creates a dispersive mixing environment. Figure 12.44(a) <br>shows the history of the strain rates for these three particles. From Fig. 12.44 it is <br>obvious when the particle travels over the undercut flight by the rapid increase in <br>strain rate. For example, particle C goes over the flight quickly after entering the <br>mixer while particles A and B go through a cross-flow before going over. The flow <br>number, Fig. 12.44(b), for the particles indicate that the mixer is mainly a shear-<br>type mixer, i.e., flow number averages around 0.5. However, a short elongational <br>flow generated as the particle enters and exits the narrow region, which should be <br>expected due to the funneling effect that occurs.<br>
+<hr>
+<A name=916></a><b>906</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+40<br>
+1<br>
+point - a<br>
+point - a<br>
+point - b<br>
+30<br>
+0.8<br>
+point - b<br>
+point - c<br>
+point - c<br>
+20<br>
+0.6<br>
+Flow Number<br>
+Strainrate (1/s) 10<br>
+0.4<br>
+0 0<br>
+1<br>
+2<br>
+0.2 0<br>
+1<br>
+2<br>
+0.5<br>
+1.5<br>
+2.5<br>
+0.5<br>
+1.5<br>
+2.5<br>
+time<br>
+time<br>
+(a)<br>
+(b)<br>
+<b>Figure 12.44</b>Calculated strain rates (a) and flow number (b) of three particles as they travel <br>
+through a LeRoy-Maddock dispersive mixing section<br>
+<b>12.4.3.5Turbo-Screw<br></b>During the production of foamed polymeric products, keeping the amount of heat <br>generation and the thermal gradients to a minimum are extremely important. The <br>quality of the foam product is mainly dictated by cell size and uniformity. If the tem-<br>perature becomes too high, the cell structure of the foam will break down leaving <br>large cells with random sizes throughout the extrudate. The Turbo-Screw has been <br>used to create efficient mixing and heat transfer for foam extrusion [98]. Figure <br>12.45 shows a perspective view of this multi-flighted mixing section.<br>This screw uses deep screw channels to minimize heat generation while utilizing <br>numerous openings through the flight that allow the polymer melt to flow from one <br>channel to the next. This mixing section has been found to increase output from <br>45% to 70% over conventional foamed extrusion operations. To analyze the effective-<br>ness of this mixing section, Fogarty et al. [99] used the boundary element method <br>to calculate the streamlines in two different screw geometries where Screw B was <br>made to have larger holes in the screw flight than Screw C, see Fig. 12.45. Figures <br>12.46(a­b) show the particle tracking for Screw B from a side and front view, while <br>Figs. 12.47(a­b) show the results for Screw C.<br>
+<b>Figure 12.45</b>A perspective view of the Turbo-cool screw and schematic of two different <br>
+openings put into flight wall<br>
+<hr>
+<A name=917></a><IMG src="Polymer Extrusion by Chris Rauwendaal-917_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-917_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-917_3.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-917_4.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>907</b><br>
+(a)<br>
+(b)<br>
+(a)<br>
+(b)<br>
+<b>Figure 12.46</b>Particle tracking in Screw B of the Turbo Screw . Note: Screw moves from left to <br>
+right<br>
+(a)<br>
+(a)<br>
+(b)<br>
+(b)<br>
+<b>Figure 12.47</b>Particle tracking in Screw C of the Turbo Screw . Note: Screw moves from left to <br>
+right<br>
+The particles follow a spiral pattern with a significant number of particles flowing <br>through the flight hole in Screw B, while no particles flow through the hole in Screw <br>C, thus making it less of a distributive mixer than Screw B. Here, the material close <br>to the leading flight is at a relatively low temperature. Some of this material will end <br>up flowing through one of the holes where it will combine with the hotter material <br>on the trailing flight and barrel resulting in a cooler, more thermally consistent <br>melt.<br>
+<hr>
+<A name=918></a><IMG src="Polymer Extrusion by Chris Rauwendaal-918_1.jpg"><br>
+<b>908</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+<b>12.4.3.6CRD Mixer<br></b>Most extrusion dispersive mixers are ineffective because shear is the main mode of <br>deformation and the plastic melt is exposed to a high stress region only once. A <br>mixer developed to overcome these shortcomings, is the Chris Rauwendaal Disper-<br>sive (CRD) mixer [100­104] shown in Fig. 12.48, see also Section 8.7.1.1.<br>
+<b> Figure 12.48</b> <br>
+CRD dispersive/distributive mixing <br>
+ <br>section<br>
+These mixers create elongational flow by incorporating a curved flight flank with a <br>larger than normal flight clearance and tapered slots machined into the flights. The <br>flow in the CRD mixer is a combination of shear and elongational flow with the latter <br>dominating in the wedge shaped regions of the mixer. The slots also serve to increase <br>the distributive as well as dispersive mixing. If the material is not randomized in its <br>passage through a mixer, only the outer shells of fluid will be dispersed leaving the <br>inner shells undispersed [105]. Therefore, it is critical to incorporate both distribu-<br>tive and dispersive mixing within one mixer. The initial design of the CRD mixer <br>was developed using the concept of the passage distribution function [106], while <br>the final geometry was developed using a detailed three-dimensional boundary ele-<br>ment flow analysis [107]. Here, simulation was used to give a complete description <br>of the flow so that stresses, the number of passes over the mixing flights, the num-<br>ber of passes through the tapered slots and residence time could be quantified for a <br>large number of particles. The simulated tracking of particles, as they travel over <br>the curved flight flank of the CRD and the tapered slot in the flight, are shown in <br>Fig. 12.49.<br>In Fig. 12.49, the flow number of the particles is displayed with a color contour and <br>the strain rate is shown for two particles in a separate graph. As expected, as the <br>material goes through the wedge shaped area, it experiences an extensional flow <br>and high strain rate--both required for effective dispersive mixing. The distributive <br>mixing of this device can be seen by simulating particles that are initially grouped <br>in the same area and observing how they spread apart as they travel through the <br>mixer. The amount of pressure that the material must overcome influences the <br>degree of mixing that will occur [107]. This effect, along with the distributive mix-<br>ing in the CRD, is shown in Fig. 12.50, for a high and a low pressure, respectively.<br>
+<hr>
+<A name=919></a><IMG src="Polymer Extrusion by Chris Rauwendaal-919_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-919_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-919_3.jpg"><br>
+ <br>
+12.4Three-Dimensional Simulation  <b>909</b><br>
+Flow Number<br>0.0<br>
+0.25<br>
+0.5<br>
+0.75<br>
+1.0<br>
+rotation<br>
+shear<br>
+elongation<br>
+CRD Flight<br>
+<b> Figure 12.49(a)</b> <br>
+Flow of two tracer points over the CRD flight <br>
+showing the history of the Flow Number and <br>
+Dimensionless Strain Rate<br>
+top particle<br>
+bottom particle<br>
+strain rate<br>
+Flow Number<br>0.0<br>
+0.25<br>
+0.5<br>
+0.75<br>
+1.0<br>
+rotation<br>
+shear<br>
+elongation<br>
+CRD Flight<br>
+<b> Figure 12.49(b)</b> <br>
+Flow of two tracer points through the CRD <br>
+flight slot showing the history of the Flow <br>
+Dimensionless Strain Rate<br>
+Number and strain rate<br>
+<b> Figure 12.50(a)</b> <br>
+Computed particle tracking in the CRD <br>
+mixer with a high p simulated across the <br>
+Initial position of<br>tracking particles<br>
+mixing section<br>
+<b> Figure 12.50(b)</b> <br>
+Computed particle tracking in the CRD <br>
+mixer with a low p simulated across the <br>
+Initial position of<br>tracking particles<br>
+mixing section<br>
+<hr>
+<A name=920></a><IMG src="Polymer Extrusion by Chris Rauwendaal-920_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-920_2.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-920_3.jpg"><br>
+<b>910</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+<b>12.4.4Static Mixers</b><br>
+When adding compounds to a melt stream to produce a specific color or enhance <br>properties, it is often desirable to produce the required mixing in the absence of <br>moving parts. In these cases, a static mixer is necessary to produce a homogenized <br>uniform melt. Fluids entering a static mixer are typically divided by baffles and mix-<br>ing occurs by the repeated splitting and recombination of flow streams. The repeated <br>dividing flows improve uniformity in composition, concentration, viscosity, and tem-<br>perature. Figure 12.51 shows a common static mixer that is configured with cross-<br>ing fingers, which split and divide that material as it passes through.<br>This mixer has been studied using both FEM and BEM to evaluate its mixing cap-<br>ability. Figure 12.52 shows simulated particle tracking through one repeating sec-<br>tion of the mixer using BEM.<br>
+<b>Figure 12.51</b>SMX static mixer <br>
+<b>Figure 12.52</b>Computed particle tracking in <br>
+ <br>
+one repeating unit of the SMX static mixer <br>
+using BEM<br>
+Figure 12.53 shows the predicted pressure profile from low to high pressure (inlet <br>to exit).<br>
+<b> Figure 12.53</b> <br>
+Computed pressure distribution of several repeating units <br>
+of the SMX static mixer using BEM<br>
+In this analysis, the history of each particle can be monitored for such quantities as <br>velocity and velocity gradients as it travels through the mixer. With this informa-<br>tion, the distributive and dispersive mixing of the device can be evaluated qualita-<br>tively and quantitatively. For instance, the distributive mixing can be calculated <br>using the residence time distribution by monitoring the time it takes for the parti-<br>
+<hr>
+<A name=921></a> <br>
+12.4Three-Dimensional Simulation  <b>911</b><br>
+cles to go through the system. Stagnant areas, which are a common problem with <br>these mixers, can be found by viewing the particle paths or the streamlines of the <br>mixer. The dispersive mixing, or ability to break-up liquid agglomerates or solid <br>particulates, can be calculated by considering the stress and flow number (type of <br>flow) history of the particles as they pass through the system. A more complete <br>analysis can be made by taking into account coalescence [108].<br>The distributive mixing of a number of the current generation of static mixers are <br>known to be quite good, however, their dispersive mixing is such that they often are <br>unable to sufficiently break-up the dispersed phase. To alleviate some of the short-<br>comings found in many static mixers Gramann et al. [109] used the boundary ele-<br>ment method to design a new type of static mixer that creates elongational flows for <br>highly effective dispersive mixing, while creating distributive mixing by repeatedly <br>splitting and folding the material. The Dispersive/Distributive Static Mixer (DDSM) <br>[110] creates a dispersive environment by utilizing baffles that define a converging <br>pathway for the mixing materials. Distributive mixing is accomplished by providing <br>a series of baffles along the length of the mixing tube with each baffle angled rela-<br>tive to previous baffles. Figure 12.54 shows two units of the mixer that are repeated <br>in series in the direction of flow.<br>
+<b>  Figure  12.54</b> <br>
+Two repeating units of the Dispersive/Distributive Static Mixer<br>
+The calculated streamlines are shown in Fig. 12.55 with corresponding values of the <br>flow number Fig. 12.56.<br>From Fig. 12.56 it is obvious that the mixer is creating a high degree of elongational <br>flow. The arrangement of the baffles is such that the fluid element experiences a <br>continual elongational flow, ensuring enough time for break-up. Further, all mate-<br>rial passing through the mixer is assured to encounter this high stress region a <br>number of times corresponding to the number of mixer elements put in series.<br>
+<hr>
+<A name=922></a><IMG src="Polymer Extrusion by Chris Rauwendaal-922_1.jpg"><br>
+<IMG src="Polymer Extrusion by Chris Rauwendaal-922_2.jpg"><br>
+<b>912</b>  <b> 12Modeling and Simulation of the Extrusion Process</b><br>
+Rotation<br>
+Shear<br>
+Elongation<br>
+0.0<br>
+0.5<br>
+1.0<br>
+<b> <br>Figure  12.55</b>Computed particle <br>
+<b> <br>Figure  12.56</b> <br>
+tracking in one module of the Dispersive/ <br>
+Computed flow number history of several  particles <br>
+ <br>Distributive Static Mixer using BEM <br>
+as they travel through one unit of the Dispersive/<br>
+Distributive Static Mixer<br>
+<b></b><br>
+ <b>12.5Conclusions</b><br>
+While the extrusion process has been utilized for commercial products since the <br>mid-1800s, modeling and computer simulation is merely in its infancy. However, the <br>tremendous advantages offered by simulation make it well suited to further advance <br>the extrusion process through the next century. Simulation and CAE of the extru-<br>sion process and equipment permits a systematic design rather than <i>ad hoc</i> designs <br>
+based solely on experience. While traditional experimental design of the extrusion <br>process is focused on producing an acceptable product, oftentimes there is no ex -<br>plicit understanding of how process variables interrelate with one another and <br>affect the final extrudate. Simulation provides insight into how process variables <br>influence the flow field and contribute to the quality of the final product. Addition-<br>ally, simulation provides feedback on variables that cannot easily be measured, e.g., <br>temperature in the channel, during the actual extrusion process. Furthermore, <br>simu lation permits the engineer to quantify the effects of modifying a single vari-<br>able on the final extrudate quality.<br>Simulation also offers significant advantages in the design of extruder screws and <br>mixing devices. Computer simulation allows an engineer the possibility to "try out" <br>new designs on the computer rather than expend time and costs associated with <br>developing and testing a prototype. These virtual designs permit numerous trials to <br>fine-tune the design in a cost effective manner prior to production.<br>
+<hr>
+<A name=923></a> References <br>
+<b>913</b><br>
+With the increasing requirements to decrease costs, improve product quality, and <br>increase productivity, computer aided engineering (CAE) provides an important tool <br>to improve quality and throughput and, thus, profitability. It should be remembered, <br>however, that modeling and computer simulation cannot capture all problems accu-<br>rately; experimental verification continues to be critically important. Therefore, it <br>is the combination of CAE and experimental work, which will provide the surest <br>avenue to process improvements and innovations.<br>
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+<hr>
+<A name=927></a> References <br>
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+5,932,159 (1999)<br>
+104. Rauwendaal, C.J., Gramann, P.J., Davis, B.A., Osswald, T.A., "Screw Extruder with <br>
+Improved Dispersive Mixing Elements," US Patent 6,136,246 (2000)<br>
+105.  Kwon, T.H., Joo, J.W., Kim, S.J., <i>Polym. Eng. Sci.,</i> Vol. 34, 3 (1994)<br>106.  Tadmor, Z., Manas-Zloczower, I., <i>Adv. Polym. Tech.,</i> Vol. 3 (1983)<br>107.  BEMflow: A boundary element fluid dynamics simulation program, The Madison Group: <br>
+Polymer Processing Research Corporation<br>
+108. Janssen, J.M.H., Ph.D. Thesis, Eindhoven University of Technology, The Netherlands <br>
+(1993)<br>
+109. Gramann, P.J., Davis, B.A., Osswald, T.A., Rauwendaal, C.J., "A New Dispersive and <br>
+Distributive Static Mixer for the Compounding of Highly Viscous Materials," SPEAN-<br>TEC (1999)<br>
+110. Davis, B.A., Gramann, P.J., Osswald, T.A., New Dispersive Static Mixer, US Patent <br>
+5,971,603 (1999)<br>
+<hr>
+<A name=928></a><b>Conversion </b><br>
+<b>Constants</b><br>
+<b>Length<br></b>1 kilometer<br>
+km<br>
+= 1E3 m (meter)<br>
+1 centimeter<br>
+cm<br>
+= 1E-2 m<br>
+1 millimeter<br>
+mm<br>
+= 1E-3 m<br>
+1 micron<br>
+µm<br>
+= 1E-6 m<br>
+1 nanometer<br>
+nm<br>
+= 1E-9 m<br>
+1 Ångstrom<br>
+Å<br>
+= 1E-10 m<br>
+1 inch<br>
+in<br>
+= 2.54E-2 m<br>
+1 milliinch<br>
+mil<br>
+= 2.54E-5 m<br>
+1 foot<br>
+ft<br>
+= 0.3048 m<br>
+1 mile<br>
+mile<br>
+= 1609 m<br>
+<b>Volume<br></b>1 cubic decimeter<br>
+dm3<br>
+= 1E-3 m3 (cubic meter)<br>
+1 cubic centimeter<br>
+cc<br>
+= 1E-6 m3<br>
+1 liter<br>
+1<br>
+= 1E-3 m3<br>
+1 deciliter<br>
+dl<br>
+= 1E-4 m3<br>
+1 milliliter<br>
+ml<br>
+= 1E-6 m3<br>
+1 cubic inch<br>
+in3<br>
+= 1.639E-5 m3<br>
+1 cubic foot<br>
+ft3<br>
+= 2.832E-2 m3<br>
+1 barrel<br>
+barrel<br>
+= 0.159 m3<br>
+1 gallon (US)<br>
+gal US<br>
+= 3.785E-3 m3<br>
+1 gallon (UK)<br>
+gal UK<br>
+= 4.546E-3 m3<br>
+<hr>
+<A name=929></a><b>920</b>  <b> Conversion  Constants</b><br>
+<b>Mass<br></b>1 ton<br>
+t<br>
+= 1E3 kg (kilogram)<br>
+1 gram<br>
+gr<br>
+= 1E-3 kg<br>
+1 ounce<br>
+oz<br>
+= 2.83E-2 kg<br>
+1 pound<br>
+lbm<br>
+= 0.4536 kg<br>
+I ton (US)<br>
+to<br>
+= 907 kg<br>
+1 ton (UK)<br>
+ton<br>
+= 1016 kg<br>
+<b>Density<br></b>1 gram per cc<br>
+gr/cc<br>
+= 1E3 kg/m3<br>
+1 pound per cu ft<br>
+lb/ft3<br>
+= 16.02 kg/m3<br>
+1 pound per cu in<br>
+lb/in3<br>
+= 2.77E4 kg/m3<br>
+1 gram per cc<br>
+gr/cc<br>
+= 3.61E-2 lbs/inch3<br>
+<b>Force<br></b>1 dyne<br>
+dyn<br>
+= 1E-5 N (Newton)<br>
+1 kilogram-force<br>
+kgf<br>
+= 9.81 N<br>
+1 ton-force<br>
+tf<br>
+= 9810 N<br>
+1 pound-force<br>
+lbf<br>
+= 4.448 N<br>
+<hr>
+<A name=930></a> <br>
+Conversion Constants  <b>921</b><br>
+<b>Stress<br></b>1 megapascal<br>
+MPa<br>
+= 1E6 Pa (Pascal)<br>
+1 dyne per cm2<br>
+dyn/cm2<br>
+= 0.1 Pa<br>
+1 Newton per m2<br>
+N/m2<br>
+= 1 Pa<br>
+1 Joule per m3<br>
+J/m3<br>
+= 1 Pa<br>
+1 atmosphere<br>
+atm<br>
+= 1.013E5 Pa<br>
+1 mm mercury (torr.)<br>
+mm Hg<br>
+= 133.3 Pa<br>
+1 mm water<br>
+mm H2O<br>
+= 9.81 Pa<br>
+1 bar<br>
+bar<br>
+= 1E5 Pa<br>
+1 pound per inch2<br>
+psi<br>
+= 6890 Pa<br>
+1 kgf per cm2<br>
+= 9.81E4 Pa<br>
+1 megapascal<br>
+MPa<br>
+= 145 psi<br>
+<b>Viscosity<br></b>1 poise<br>
+poise<br>
+= 0.1 Pas<br>
+1 poise<br>
+poise<br>
+= 1 dyns/cm2<br>
+1 pound-sec per in2<br>
+psisec<br>
+= 6897 Pas<br>
+I poise<br>
+poise<br>
+= 14.5E-6 psisec<br>
+<hr>
+<A name=931></a><b>922</b>  <b> Conversion  Constants</b><br>
+<b>Energy/Work<br></b>1 erg<br>
+dyncm<br>
+= 1E-7 J (Joule)<br>
+1 Newton-meter<br>
+Nm<br>
+= 1 J<br>
+1 watt-second<br>
+Ws<br>
+= 1 J<br>
+1 kgf-meter<br>
+kgfm<br>
+= 9.81 J<br>
+1 foot-pound<br>
+ftlbf<br>
+= 1.356 J<br>
+1 horsepower-hour<br>
+hph<br>
+= 2.685E6 J<br>
+1 kilowatt-hour<br>
+kWh<br>
+= 3.60E6 J<br>
+1 calorie<br>
+cal<br>
+= 4.19 J<br>
+1 Brit. thermal unit<br>
+Btu<br>
+= 1055 J<br>
+1 inch-pound<br>
+inlbf<br>
+= 0.113 J<br>
+<b>Power<br></b>1 kilowatt<br>
+kW<br>
+= 1000 W (Watt)<br>
+1 horsepower<br>
+hp<br>
+= 745.7 W<br>
+1 foot-pound per sec<br>
+ftlbf/s<br>
+= 1.356 W<br>
+<b>Specific Energy<br></b>1 calorie per gram<br>
+cal/gr<br>
+= 4190 J/kg<br>
+1 Btu per pound<br>
+Btu/lb<br>
+= 2326 J/kg<br>
+1 hph per pound<br>
+hph/lb<br>
+= 5.92E6 J/kg<br>
+1 hph per pound<br>
+hph/lb<br>
+= 1.644 kWh/kg<br>
+1 kWh per kg<br>
+kWh/kg<br>
+= 3.60E6 J/kg<br>
+1 kWh per kg<br>
+kWh/kg<br>
+= 0.608 hph/lb<br>
+<hr>
+<A name=932></a> <br>
+Conversion Constants  <b>923</b><br>
+<b>Thermal Conductivity<br></b>1 cal/cms°C<br>
+= 419 J/ms°K<br>
+1 kcal/mh°C<br>
+= 1.163 J/ms°K<br>
+1 Btu/fth°F<br>
+= 1.73 J/ms°K<br>
+I Btu/in/ft2h-°F<br>
+= 0.144 J/ms°K<br>
+1 Btu/fts°F<br>
+= 6230 J/ms°K<br>
+1 W/m°K<br>
+= 1 J/ms°K<br>
+<b>Temperature</b><br>
+Unit<br>
+Degrees Kelvin<br>
+Degrees Celcius<br>
+Degrees Fahrenheit<br>
+Symbol<br>
+°K<br>
+°C<br>
+°F<br>
+Boiling point water<br>
+373.15<br>
+100.00<br>
+212.00<br>
+Melting point ice<br>
+273.15<br>
+0.00<br>
+32.00<br>
+Absolute zero<br>
+0.00<br>
+­273.15<br>
+-459.67<br>
+1 degree Fahrenheit  °F  = 1/1.8°C<br>
+1 degree Celcius <br>
+°C  = 1.8°F<br>
+1 degree Kelvin <br>
+°K  = 1°C<br>
+<hr>
+<A name=933></a><hr>
+<A name=934></a><b>Index</b><br>
+Numeric<br>
+Banbury type mixer871, 885<br>
+ <br>
+­ modeling871<br>
+4-mode Giesekus model891<br>
+ <br>
+­ simulating873<br>
+barrel temperature measurement100<br>
+A<br>
+barrier flight extruder screws569<br>
+ <br>
+­ patent history569<br>
+abrasive wear787<br>
+Barr screw575<br>
+absorptivity170<br>
+basic guidelines in profile design654<br>
+AC drive systems49<br>
+batch extruders2<br>
+ <br>
+­ adjustable frequency drive49<br>
+BEM simulation611<br>
+ <br>
+­ adjustable transmission ratio drive49<br>
+Bessel functions725<br>
+actual energy requirement34<br>
+best straight line (BSL)93<br>
+actual stock temperatures, and degradation <br>
+bimetallic barrels67<br>
+811<br>
+Bingham model213<br>
+adiabatic temperature401<br>
+Biot number167, 424<br>
+adjustable frequency drives49, 51<br>
+blister ring587, 618<br>
+adjustable groove depth299<br>
+blown film extrusion676<br>
+adjustable grooved feed extruder297<br>
+BMK mixer<i>see</i>SMV mixer<br>
+adjustable transmission ratio drive49<br>
+Boltzmann constant169<br>
+air entrapment<br>
+boundary element method (BEM)610, 653, 872<br>
+ <br>
+­ solutions to834<br>
+breakdown force470<br>
+analytical solution862<br>
+breaker plate72<br>
+analytical techniques862<br>
+Brinkman number164, 638<br>
+angle of repose193<br>
+Brownian motion469<br>
+apex area730<br>
+brushless DC drives55<br>
+Archimedean solids transport269<br>
+BT mixer585<br>
+arching268<br>
+buckling515<br>
+arching or bridging in hopper flow260<br>
+bulk density191, 193<br>
+area draw ratio (ADR)672<br>
+bypass vented extruder559<br>
+AutoCADTM878<br>
+autogenous process78<br>
+automatic gel detection methods839<br>
+C<br>
+automatic reset123, 124<br>
+calibrator692<br>
+autoscreen74<br>
+ <br>
+­ external calibrators693<br>
+average flow velocity in the thin section660<br>
+ <br>
+­ internal calibrator693<br>
+Avogadro's number158<br>
+ <br>
+­ slide calibrator692<br>
+axial mixing (backmixing)483, 484<br>
+ <br>
+­ Technoform process694<br>
+ <br>
+­ vacuum calibrator692<br>
+B<br>
+CAMPUS, Computer Aided Material Preselection249<br>
+cantilever515<br>
+backmixing (axial mixing)483, 484<br>
+capacitance measurement112<br>
+ <br>
+­ prevention488<br>
+Capillary (or Weber) number874<br>
+"back relieving" the land658<br>
+capillary rheometer223<br>
+balance by channel height659<br>
+ <br>
+­ melt index tester223<br>
+<hr>
+<A name=935></a><b>926</b>  <b> Index</b><br>
+capillary transducers89<br>
+conventional multi-flighted screw568<br>
+Carreau model213<br>
+conveying mechanism33<br>
+cascade devolatilization562<br>
+conveying process, self-wiping extruders713<br>
+Cauchy's equation150<br>
+co-rotating extruders, screw geometry708<br>
+cavity transfer mixing (CTM)620<br>
+corrosive wear788<br>
+chain drives50<br>
+Couette flow169<i>see</i>drag flow<br>
+change-over23<br>
+Couette shear rate405, 520, 594<br>
+ <br>
+­ time23<br>
+Coulomb friction260<br>
+channel depth537<br>
+crammer feeders68, 258<br>
+characterization of the mixture442<br>
+CRD (Chris Rauwendaal Dispersive mixing) barrier <br>
+chemical degradation805<br>
+screw581<br>
+Chris Rauwendaal Dispersive (CRD)908<br>
+CRD fluted mixer586, 603, 613, 614<br>
+Chung5<br>
+critical screw speed400<br>
+classification of polymer extruders14<br>
+critical tensile deformation rate range675<br>
+classification of twin screw extruders25<br>
+critical Weber (Capillary) number475<br>
+clearance70, 350<br>
+cross-channel flow361<br>
+closely intermeshing co-rotating (CICO) twin screw <br>
+crosshead die669<br>
+extruder701<br>
+crosslinking1<br>
+closely intermeshing counter-rotating extruder <br>
+cross-sectional mixing483<br>
+(CICT)702, 720<br>
+CT mixer621<br>
+closely self-wiping co-rotating extruders (CSCO)705<br>
+cumulative residence time distribution (CRTD)904<br>
+coalescence479<br>
+curved plate model277<br>
+coat hanger die664, 666<br>
+cylindrical coordinate system (CCS)413<br>
+coaxial twin screw (CTS) extruder743<br>
+coefficient of cohesion193<br>
+D<br>
+coefficient of friction195, 290<br>
+ <br>
+­ of the bulk material194<br>
+data acquisition systems (DAS)769<br>
+coefficient of variation (COV)464<br>
+DC brush motor776<br>
+coextrusion686<br>
+dead spots675<br>
+color concentrates (CC)614<br>
+degradation245<br>
+color measurement116<br>
+ <br>
+­ polymer624, 626<br>
+commercial twin screw machinery749<br>
+ <br>
+­ reducing819<br>
+common screw materials70<br>
+del Pilar Noriega5<br>
+ <br>
+­ european equivalents70<br>
+design parameters for spiral mandrel die679<br>
+comparison of various temperature sensors98<br>
+determinant, det211<br>
+comparison, type of thrust bearings65<br>
+developing melt temperatures390<br>
+compatibilizers480<br>
+devolatilization175, 180, 435, 554<br>
+compression characteristics of bulk materials194<br>
+ <br>
+­ continous745<br>
+compression mold filling simulation868<br>
+ <br>
+­ efficiency830<br>
+compression ratio (Xc)518, 536<br>
+ <br>
+­ particulate polymer180<br>
+computer aided engineering3<br>
+ <br>
+­ polymer melts181<br>
+computer simulation610<br>
+devolatilizing extruder screws553<br>
+ <br>
+­ melting theories332<br>
+diameter draw ratio (DDR)672<br>
+conservation of energy864<br>
+diaphragms92<br>
+conservation of mass864<br>
+die72<br>
+conservation of momentum864<br>
+die assembly72<br>
+"constant torque" characteristic60<br>
+die drool<i>see</i>die lip build-up<br>
+constructing a timeline765<br>
+die flow instabilities431<br>
+contiguous solids melting (CSM)304, 333<br>
+die flow problems843<br>
+continuous extruders2, 13, 28<br>
+die forming zone419<br>
+ <br>
+­ disk extruders28<br>
+die geometry, effect on flow distribution680<br>
+controllers, factory-tuned and non-adjustable <br>
+diehead pressure87, 419<br>
+ <br>parameters139<br>
+dielectric heating171<br>
+controller transfer curve (power input curve)122<br>
+die lip build-up845<br>
+control volume approach (CVA)870<br>
+dies, film and sheet663<br>
+conventional extruders23<br>
+die swell<i>see</i>extrudate swell<br>
+<hr>
+<A name=936></a> Index  <b>927</b><br>
+die temperature adjustment665<br>
+elastic melt extruder35<br>
+differences in screw design753<br>
+elastomers1<br>
+differential thermal analysis (DTA)247<br>
+electric friction clutch drive50<br>
+dilatant fluid205<br>
+electric heating75<br>
+dimensional analysis161<br>
+electric resistance heaters75<br>
+dimensionless numbers161, 163<br>
+electronic data acquisition systems (DAS)625<br>
+dimensionless throughput364<br>
+electronic integrator123<br>
+dimensionless viscosity423<br>
+Ellis model213<br>
+discharge screw (crammer feed)778<br>
+elongation203<br>
+discolored specks846<br>
+elongational deformation203<br>
+ <br>
+­ in the raw material837<br>
+elongational flows<br>
+discontinuous extruders2, 13<br>
+ <br>
+­ shear-free flows871<br>
+disk extruder<i>see</i>screwless extruder<br>
+elongational viscosity205<br>
+ <br>
+­ diskpack extruder31<br>
+empirical identifications technique142<br>
+ <br>
+­ spiral disk extruder31<br>
+ <br>
+­ deterministic functions142<br>
+ <br>
+­ stepped disk extruder28<br>
+ <br>
+­ stochastic (random) identification functions142<br>
+dispersed solids melting (DSM)304, 333, 617<br>
+energy balance equation151, 242<br>
+Dispersive/Distributive Static Mixer (DDSM)911<br>
+energy consumption23<br>
+ <br>
+­ CRD mixing technology463<br>
+energy efficiency for pressure generation34, 43<br>
+dispersive mixers604, 618<br>
+Energy Transfer (ET) screws622, 893<br>
+dispersive mixing469<br>
+entropy155, 157<br>
+ <br>
+­ elements584<br>
+equation of continuity149<br>
+ <br>
+­ liquid-liquid systems471<br>
+equation of motion<i>see</i>momentum balance <br>
+distributive mixers623<br>
+ <br>equation<br>
+distributive mixing441, 619<br>
+equilibrium melt temperature<i>see</i>fully developed <br>
+double wave mixing screw601, 622<br>
+melt temperature<br>
+down channel velocity863<br>
+equivalent spherical diameter (esd)200<br>
+drag flow202<br>
+Esde22<br>
+ <br>
+­ of power law fluids374<br>
+E-speck846<br>
+ <br>
+­ rate635<br>
+Euler equation150<br>
+ <br>
+­ term43<br>
+extended mandrel674<br>
+drag induced solids conveying270<br>
+extensional flow mixer (EFM)602<br>
+draw ratio balance (DRB)673<br>
+extensive mixing441<br>
+draw resonance434, 823<br>
+external coefficient of friction194<br>
+Dray and Lawrence screw578<br>
+extrudate swell<i>see</i>die swell<br>
+Dray fluted mixing section587<br>
+extrudate thickness, measure110<br>
+drop rupture467<br>
+extruder1<br>
+drum extruder30<br>
+ <br>
+­ barrel64<br>
+DSM process<i>see</i>dispersed solids melting theory<br>
+ <br>
+­ cooling77<br>
+dual output temperature controllers128<br>
+extruder drive49<br>
+dual reciprocity boundary element method <br>
+ <br>
+­ AC motor drive systems49<br>
+(DRM)874<br>
+ <br>
+­ DC motor drive systems49<br>
+dual sensor temperature control125<br>
+ <br>
+­ hydraulic drives49<br>
+Dulmage mixing section619<br>
+extruder screw13, 70<br>
+dynamic analysis231<br>
+ <br>
+­ functions626<br>
+dynamic process model141<br>
+extruder screw for rubber<br>
+dynamic testing of pressure tranducers94, 95<br>
+ <br>
+­ EVK screw (Wener &amp; Pfeiderer)20<br>
+ <br>
+­ Pirelli rubber extruder screw19<br>
+E<br>
+ <br>
+­ Plastiscew19<br>
+ <br>
+­ QSM Extruder20<br>
+effective angle of friction, e199<br>
+ <br>
+­ Transfermix20<br>
+effective yield locus (EYL)198<br>
+extruders without gear reducer23<br>
+Egan mixing section587<br>
+extrusion instabilities820<br>
+E-gels839<br>
+ <br>
+­ solutions to833<br>
+Ehring model213<br>
+elastic encapsulation691<br>
+<hr>
+<A name=937></a><b>928</b>  <b> Index</b><br>
+F<br>
+G<br>
+Farrel continuous mixer (FCM)742<br>
+Gaussian elimination865<br>
+feedback ratio436<br>
+gear pump extruder27<br>
+feed block dies686<br>
+gear reducer23<br>
+feed-controlled extrusion7<br>
+gel836, 837, 839<br>
+feed hopper68<br>
+ <br>
+­ content836<br>
+feed port design65<br>
+ <br>
+­ crosslinked840<br>
+feedstock properties767<br>
+ <br>
+­ defects with discoloration838<br>
+feed throat64<br>
+gelation836<br>
+F-H relationship<i>see</i>Flory-Huggins relationship<br>
+general characteristics of the standard extruder549<br>
+fibrillation482<br>
+Generalized Newtonian Fluids (GNF)870<br>
+Fick's law175<br>
+geometrical encapsulation691<br>
+filtration systems74<br>
+geometrical relationships257<br>
+finite element analysis (FEM)653, 800<br>
+geometry, extruder screw255<i>see</i>single stage<br>
+finite element fluid dynamics analysis package, <br>
+ <br>
+­ feed section of screw13<br>
+ <br>FIDAP872, 892<br>
+ <br>
+­ metering section (pump section)13<br>
+first law of thermodynamics151, 153<br>
+ <br>
+­ transition section (compression section)13<br>
+fishtail die663<br>
+Gibbs free energy159<br>
+five-layer blown film coextrusion dies687<br>
+gloss853<br>
+flash devolatilization182<br>
+gloss, quantitative measurement115<br>
+flat plate model275<br>
+GMP, good manufacturing practices625<br>
+flat plate system (FPS)208, 877<br>
+G-process676<br>
+flat slanted flight flanks<br>
+Graetz number166, 638<br>
+ <br>
+­ trapezoidal flight geometry279<br>
+gravity flow261, 266<br>
+flat three-plate model734<br>
+grooved barrel sections66, 284, 288<br>
+flex lip adjustment664<br>
+grooved feed extruder296<br>
+flight flank347<br>
+grooves, number682<br>
+flight geometries514, 539<br>
+grooves, wear286<br>
+flood feeding7, 302, 831<br>
+Flory-Huggins (F-H) relationship177<br>
+H<br>
+Flow Analysis Network (FAN)869<br>
+flow behavior of polymer melt653<br>
+hardfacing or hardsurfacing794<br>
+flow distribution682<br>
+hardfacing techniques644<br>
+flow function270<br>
+ <br>
+­ laser hardfacing646<br>
+flow number610<br>
+ <br>
+­ metal inert gas (MIG) welding646<br>
+ <br>
+­ distribution881<br>
+ <br>
+­ oxyacetylene welding644<br>
+flow patterns699<br>
+ <br>
+­ plasma transfer arc (PTA) welding645<br>
+flow rate43, 265, 415<br>
+ <br>
+­ tungsten inert gas (TIG) welding645<br>
+fluoropolymers673<br>
+haze853<br>
+flux vector (FV)52<br>
+ <br>
+­ and luminous transmittance of transparent plastics, <br>
+form factor54<br>
+measurement115<br>
+Fortran program363, 493<br>
+HDPE39<br>
+Fourier number168, 310<br>
+ <br>
+­ melt extruded, properties39<br>
+Fourier's law of heat transfer152, 160, 176<br>
+ <br>
+­ solid-state extruded, properties39<br>
+four-motor CMG torque drive22<br>
+heating extruders75<br>
+four-screw extruder26<br>
+ <br>
+­ electric heating75<br>
+Frequon173<br>
+ <br>
+­ fluid heating75<br>
+frictional heat67<br>
+ <br>
+­ steam heating75<br>
+ <br>
+­ generation282, 290<br>
+heat transfer160<br>
+frictional properties, polymer196<br>
+ <br>
+­ conductive heat transfer160<br>
+friction, attributions195<br>
+ <br>
+­ convective heat transfer161<br>
+fully developed melt temperature<i>see</i>equilibrium <br>
+helix angle532, 538, 578, 594, 683<br>
+melt temperature<br>
+Henry's law177<br>
+functional instabilities828<br>
+Henry's law constant177<br>
+funnel flow260<br>
+high-frequency instabilities821<br>
+funneling268<br>
+high-pressure extrusion669<br>
+<hr>
+<A name=938></a> Index  <b>929</b><br>
+high speed co-rotating twin screw extruders, compared <br>
+L<br>
+to low speed twin screw extruders698<br>
+high-speed extrusion22<br>
+laminar mixing (laminar flow)457<br>
+high-speed single screw extruders (HS-SSE)22, 23<br>
+land length657, 673<br>
+high stress region, HSR602<br>
+leakage722<br>
+high-viscosity polymers23<br>
+leakage flow350<br>
+Hi-mixer463<br>
+LeRoy/Maddock mixer585<br>
+history of polymer extrusion6<br>
+light microscope772<br>
+homopolymers245<br>
+linear low-density polyethylene, LLDPE546<br>
+horseshoe die667<br>
+linear variable differential transformer (LVDT)110<br>
+hot-stage microscopy (HSM), to characterize <br>
+lines in extruded product851<br>
+gels840<br>
+liquid crystalline polymers (LCPs)853<br>
+hydraulic drive<br>
+lobal mixing603<br>
+ <br>
+­ hydrostatic drive55<br>
+low-frequency instabilities826<br>
+hygroscopic resins564<br>
+hysteresis93, 118<br>
+M<br>
+ <br>
+­ curve118<br>
+machinery, wear780<br>
+Maddock mixing section585<br>
+I<br>
+Maddock or LeRoy dispersive mixing section905<br>
+improper screw geometry509<br>
+magneto-strictive ultrasound transducer112<br>
+increase in the screw diameter802<br>
+Maillefer extruder screw572, 573<br>
+induction time245<br>
+mass balance equation149<br>
+infrared (IR) heating171<br>
+mass flow (hopper flow)260<br>
+infrared (IR) melt temperature measurement106<br>
+material consumed in the change-over24<br>
+infrared (IR) sensors98<br>
+mathematical modeling610<br>
+infrared thermometer768<br>
+maximum melting efficiency571<br>
+Ingen Housz barrier screw581<br>
+maximum normal stress512<br>
+initial scale of segregation832<br>
+maximum shear stress513<br>
+inner melt removal (IMR)744<br>
+maximum stress511<br>
+instrumentation85, 86<br>
+Maxwell model871<br>
+insufficient melting capacity829<br>
+"MC3" and "MC4" screw576<br>
+insufficient mixing capacity of the screw831<br>
+mechanical adjustable speed (MAS)50<br>
+integrator123<br>
+mechanical adjustment of the die flow channel655<br>
+interface distortion690<br>
+mechanical degradation804<br>
+intermeshing co-rotating extruders701<br>
+mechanical specific energy consumption23<br>
+intermeshing twin screw extruder (TSE)304, 862<br>
+medical devices, molecular degradation630<br>
+internal coefficient of friction194<br>
+medical extrusion624<br>
+ion-nitriding67<br>
+ <br>
+­ automation625<br>
+iron-constantan thermocouple (TC)98<br>
+ <br>
+­ good manufacturing practices (GMP)625<br>
+ISG mixer461<br>
+ <br>
+­ requirements625<br>
+ITX5<br>
+medical tubing672<br>
+melt conveying of isothermal fluids342<br>
+melt conveying or pumping problems830<br>
+J<br>
+melt conveying theory of Newtonian fluids440<br>
+Janssen model481<br>
+melt fracture432, 822, 843, 844<br>
+melt index (MI)223<br>
+melting mechanism, for diskpack<br>
+K<br>
+ <br>
+­ dissipative melt mixing (DMM)34<br>
+K-BKZ model884<br>
+ <br>
+­ drag melt removal (DMR)34<br>
+Kelvin model871<br>
+melting models326<br>
+Kenics mixer462<br>
+melting point245<br>
+Kennaway19<br>
+melting rate for a non-zero clearance534<br>
+Kim screw579<br>
+melting theory for temperature-dependent fluids <br>
+kinematic viscosity167<br>
+319<br>
+Kirchhoff's law170<br>
+melting zone (plasticizing zone)103<br>
+<hr>
+<A name=939></a><b>930</b>  <b> Index</b><br>
+melt pressure86, 87<br>
+nuclear radiation sensors113<br>
+melt pressure control systems141<br>
+numerical techniques864<br>
+melt temperature22<br>
+ <br>
+­ boundary element method (BEM)867<br>
+ <br>
+­ distribution627<br>
+ <br>
+­ finite difference method (FDM)865<br>
+ <br>
+­ parameters407<br>
+ <br>
+­ finite element method (FEM)866<br>
+ <br>
+­ variations628<br>
+Nusselt number166<br>
+membrane stretching model during blow molding and <br>
+thermoforming869<br>
+O<br>
+mesh editing procedure<br>
+ <br>
+­ dynamic mesh generator868<br>
+objective of an extrusion die654<br>
+mesh generation scheme869<br>
+offset121, 124<br>
+mesh superposition technique (MST)896<br>
+one-dimensional flow analysis356<br>
+metallic filter medium73<br>
+one-dimensional power law analysis525<br>
+metallocenes546<br>
+on-off controller133<br>
+metal-to-metal wear789<br>
+on-off control of temperature117, 118<br>
+mettalic filter medium<br>
+on-off differential118<br>
+ <br>
+­ relative performance comparison73<br>
+optical properties853<br>
+microwave heating173<br>
+optimize the channel depth and helix angle521<br>
+minimum set of instrumentation85<br>
+optimizing output519<br>
+mixing441<br>
+optimum channel depth519<br>
+ <br>
+­ element584<br>
+optimum channel depth H*542<br>
+ <br>
+­ of immiscible fluids in a twin screw extruder482<br>
+optimum helix angle *541<br>
+ <br>
+­ zone in the extruder442<br>
+optimum screw geometry for melt conveying519<br>
+modeling, linear control system130<br>
+oxidative degradation806<br>
+molecular weight (MW)624<br>
+ <br>
+­ reductions624<br>
+P<br>
+momentum150<br>
+ <br>
+­ convection of momentum150<br>
+"pancake" coextrusion die687<br>
+momentum balance equation150<br>
+parallel flights532<br>
+moving boundaries861<br>
+parallel plate analysis with moving barrel411<br>
+ <br>
+­ moving free boundaries861<br>
+particles<br>
+ <br>
+­ moving solid boundaries861<br>
+ <br>
+­ broken solids201<br>
+multi-flighted screw geometry568<br>
+ <br>
+­ granules201<br>
+multiflux mixer460<br>
+ <br>
+­ pellets201<br>
+multi-manifold external-combining dies688<br>
+ <br>
+­ powders201<br>
+multi-manifold internal combining dies686<br>
+ <br>
+­ semi-free-flowing granules201<br>
+multi-manifold sheet die688<br>
+particle size200<br>
+multi-screw extruder878<br>
+Peclet number NPe166, 389, 439<br>
+multi-vent devolatilization561<br>
+PET powder564<br>
+Petrov-Galerkin technique882<br>
+N<br>
+P-gels839<br>
+piezoelectric pressure transducers90, 93<br>
+Nahme number321, 324<br>
+piezoelectric ultrasonic transducer112<br>
+ <br>
+­ also Griffith number167<br>
+piezo-resistive pressure transducers91<br>
+Newtonian flow rate equation528<br>
+piezo-resistive transducer88<br>
+Newtonian fluids205, 210, 343, 367<br>
+pineapple mixer620<br>
+Newton-Raphson method542<br>
+pin mixing section619<br>
+nitriding67<br>
+pipe dies668<br>
+ <br>
+­ gas nitriding67<br>
+piping260<br>
+ <br>
+­ liquid bath nitriding67<br>
+Planck constant169<br>
+noise level23<br>
+Planck's law of radiation169<br>
+non-intermeshing twin screw extruders730<br>
+planetary roller extruder25<br>
+ <br>
+­ are counter-rotating (NOCT extruders)730<br>
+plasticating548, 551, 562<br>
+non-Newtonian and non-isothermal, polymer melt316<br>
+plasticating extrusion2<br>
+non-Newtonian fluid205, 210<br>
+plasticating instabilities829<br>
+non-return valve (NRV)615<br>
+plasticating zone<i>see</i>melting zone<br>
+<hr>
+<A name=940></a> Index  <b>931</b><br>
+plug flow203<br>
+proportional band119, 120, 123<br>
+ <br>
+­ zone261<br>
+proportional control123<br>
+plugging536<br>
+proportional controller119, 123<br>
+pneumatic pressure transducers90<br>
+proportional plus integral (PI) controllers124<br>
+Poiseuille equation187<br>
+proportional plus integral plus derivative (PID) control <br>
+Poiseuille flow169<br>
+system124<br>
+Poisson ratio244<br>
+ <br>
+­ PID controller136<br>
+polymer degradation803<br>
+P-speck846<br>
+polymer density 236<br>
+Pulsar mixer622<br>
+polymer processing aids (PPA)843<br>
+Pulsar mixing section622<br>
+polymers1, 2, 219<br>
+pumping efficiency in diskpack extruder42<br>
+ <br>
+­ elastomers1<br>
+pump ratio (Xp)555<br>
+ <br>
+­ thermoplastics1<br>
+pushrod transducer88<br>
+ <br>
+­ thermosets1<br>
+pyrometer768<br>
+polypropylene23<br>
+polystyrene23<br>
+R<br>
+polytetrafluoroethylene (PTFE)<br>
+ <br>
+­ extrusion of37<br>
+Rabinowitsch and Weissenberg, fluid relationship213<br>
+portable data collectors (PDCs)769<br>
+Rabinowitsch correction222<br>
+portable machine analyzers (PMAs)769<br>
+Rabinowitsch equation213<br>
+positive displacement699<br>
+radial clearance543, 544<br>
+positive displacement devices27<i>see</i>gear pump <br>
+radiation, absorption113<br>
+extruder<br>
+radio frequency heating<i>see</i>dielectric heating<br>
+Potente5<br>
+ram extruder<br>
+power consumption (z)43, 353, 391, 540<br>
+ <br>
+­ multi-ram extruder41<br>
+ <br>
+­ in the melting zone330<br>
+random fluctuations in instability827<br>
+ <br>
+­ measurement108<br>
+rapid flow260<br>
+power conversion unit (PCU)776<br>
+rate of solids melting863<br>
+power factor108<br>
+Rayleigh disturbances472<br>
+power law equation209, 212, 216<br>
+reaction-curve method138<br>
+power law fluid356, 588<br>
+rear valving559<br>
+power law index209, 529<br>
+rearward devolatilization560<br>
+power law model870, 872<br>
+rebuilding extruder barrels646<br>
+power law of Ostwald and de Waele<i>see</i>power law <br>
+rebuilding extrusion equipment642<br>
+equation<br>
+reducer59<br>
+Prandtl number167<br>
+reducing gel formation in the extruder840<br>
+pressure differential (P)512<br>
+reduction in noise level23<br>
+pressure distribution262<br>
+residence time23, 24, 449<br>
+pressure drop465, 596, 601<br>
+ <br>
+­ distribution functions (RTD)455, 466, 485, 807<br>
+pressure flow202<br>
+resistance temperature (RT) curve97<br>
+pressure flow rate through a triangle735<br>
+resistive temperature sensors<br>
+pressure flow term43<br>
+ <br>
+­ conductive-type temperature (RTD)97<br>
+pressure transducers88, 89, 90, 91, 92<br>
+ <br>
+­ semiconductor type97<br>
+ <br>
+­ selection93<br>
+resonance frequency516<br>
+problem-solving process763<br>
+Reynolds number163<br>
+process model137, 138<br>
+rheological models887<br>
+process monitoring, of wear850<br>
+rheological properties191<br>
+process reaction curve<br>
+rheometer220<br>
+ <br>
+­ response curve128<br>
+ <br>
+­ capillary rheometer220<br>
+process transfer curve122<br>
+ <br>
+­ cone and plate rheometer227<br>
+ProEngineerTM878<br>
+ <br>
+­ slit die rheometer228<br>
+profile dies684<br>
+rheopectic219<br>
+profile extrusion684<br>
+rhomboid mixing section902<br>
+profiling691<br>
+roller die21<br>
+profitability of an extrusion operation23<br>
+rotating drum devolatilizer (RDD)564<br>
+properties of the bulk material191<br>
+rotational speed, measure109<br>
+<hr>
+<A name=941></a><b>932</b>  <b> Index</b><br>
+RPVC extrusion837<br>
+single screw residence time distribution (RTD)487<br>
+rubber elasticity, theory157<br>
+single stage13<br>
+rubber extruders17<br>
+sinh law213<br>
+ <br>
+­ cold feed18<br>
+six-step variable voltage inverter52<br>
+ <br>
+­ hot feed18<br>
+slenderness ratio513<br>
+slide-plate screen changers73<br>
+S<br>
+slot flank angle610<br>
+slot geometry609<br>
+sag at the end of the screw514<br>
+slotted mixing sections620<br>
+Saxton mixer619, 620<br>
+slow fluctuations in instability827<br>
+Saxton mixing section609<br>
+slow frictional flow260<br>
+scale-up635, 641<br>
+SMV mixer<i>see</i>BMK mixer<br>
+ <br>
+­ factors639<br>
+solids conveying6, 192, 537, 562<br>
+ <br>
+­ method635<br>
+ <br>
+­ angle 272, 277<br>
+scanning and sorting (S&amp;S) system847<br>
+ <br>
+­ instabilities829<br>
+screen<br>
+ <br>
+­ rate286, 637<br>
+ <br>
+­ autoscreen system74<br>
+ <br>
+­ zone258<br>
+screens72<i>see</i>metallic filter medium<br>
+solids draining screw (SDS)744<br>
+screw beat825<br>
+solid-state extrusion38<br>
+screw binding797<br>
+ <br>
+­ direct solid-state38<br>
+ <br>
+­ problems802<br>
+ <br>
+­ hydrostatic38<br>
+screw extruders<br>
+spatial finite difference formulation869<br>
+ <br>
+­ multi screw13<br>
+specifications on pressure transducers93<br>
+ <br>
+­ single screw13<br>
+specific energy consumption (SEC)23, 301, 542, 850<br>
+screw frequency instabilities825<br>
+specific enthalpy241<br>
+screw geometry540<br>
+specific extruder throughput (SET)850<br>
+screwless extruders28<br>
+specific heat239<br>
+ <br>
+­ drum extruder30<br>
+spherulitic crystal morphology238<br>
+screw, outside diameter (O .D .)849<br>
+spiral disk extruder31<br>
+screw shank,  rear vacuum seal round69<br>
+spiral mandrel die671, 676, 678<br>
+screw wear849<br>
+square feed hoppers68<br>
+second law of thermodynamics156<br>
+square pitch extruder screw509<br>
+Secor's model for devolatilization in co-rotating twin <br>
+standardize evaluation and testing of temperature <br>
+screw extruders745<br>
+sensors105<br>
+self-tuning temperature controllers140<br>
+standard or conventional extruder screw23, 549<br>
+self-wiping co-rotating twin screw extruders705<br>
+Staromix mixer621<br>
+setpoint119, 121, 133<br>
+starve feeding303, 831<br>
+shark skin431, 821<br>
+state-of-the art techniques862<br>
+shear202<br>
+static coefficient of friction194<br>
+ <br>
+­ rate203<br>
+static mixers, geometry460<br>
+ <br>
+­ strain204<br>
+static mixer, simulation910<br>
+ <br>
+­ ­rate870<br>
+static mixing459<br>
+ <br>
+­ strength197<br>
+stationary screw, barrel rotates, assumption411<br>
+ <br>
+­ stress43<br>
+statistical process control (SPC)625<br>
+ <br>
+­ viscosity204<br>
+statistical rated life of a thrust bearing, formula62<br>
+sheet extrusion853<br>
+Stefan-Boltzmann law169<br>
+shish-kebab crystal morphology238<br>
+stock temperature measurement102<br>
+short extruder829<br>
+ <br>
+­ flush-mounted temperature sensor104<br>
+short residence times24<br>
+ <br>
+­ straight immersion temperature sensor104<br>
+simple conveying screws629<br>
+Stokes drag force479<br>
+simple proportional controller123<br>
+strain distribution functions (SDF)455<br>
+simulating a mixing process868<br>
+strain gauge transducer88<br>
+simulating polymer processes861<br>
+strain-induced crystallization159<br>
+simulation873, 912<br>
+strain rates, and degradation811<br>
+single screw extruders (SSE)22<br>
+Strata-blend mixer623<br>
+single screw extrusion22<br>
+super-extrusion843, 844<br>
+<hr>
+<A name=942></a> Index  <b>933</b><br>
+T<br>
+thrust bearing assembly61<br>
+time constant of the closed-loop system132<br>
+Tadmor melting model326<br>
+timeline, leading up to problem765<br>
+taper angle674<br>
+time-proportioning power controller126<br>
+T-die663<br>
+tipstreaming477<br>
+Technoform process694<br>
+torque510<br>
+temperature86<br>
+torque (T)207<br>
+temperature controller126<br>
+torsional braid analysis (TBA)248<br>
+ <br>
+­ analog126<br>
+total flight width (pw)544<br>
+ <br>
+­ digital126<br>
+total process control140<br>
+temperature control system116<br>
+total strain453<br>
+ <br>
+­ closed loop116<br>
+tracking tracer ink872<br>
+ <br>
+­ feedback116<br>
+transfer function for a reverse-acting controller119<br>
+temperature measurement96<br>
+transparency853<br>
+ <br>
+­ radiation pyrometers97<br>
+troubleshooting763<br>
+ <br>
+­ resistive temperature sensors97<br>
+ <br>
+­ feeding system778<br>
+ <br>
+­ thermocouple temperature sensors97<br>
+ <br>
+­ heating and cooling system778<br>
+temperature profiles367, 420<br>
+ <br>
+­ machine-related problems776<br>
+test methods for P-gels839<br>
+ <br>
+­ tools768<br>
+test methods for wear782<br>
+Troubleshooting the Extrusion Process, book763<br>
+The Madison Group610<br>
+true proportional power controller<br>
+thermal analysis techniques775<br>
+ <br>
+­ current proportioning controller126<br>
+thermal barrier65<br>
+ <br>
+­ phase-angle-fired proportional controller127<br>
+thermal characteristics of the system128<br>
+true total extrusion process control141<br>
+thermal characterization247<br>
+tubing671<br>
+thermal conductivity234, 235<br>
+ <br>
+­ dies668<br>
+thermal degradation803<br>
+Turbo-ScrewTM620, 906<br>
+thermal diffusivity167, 242<br>
+Twente Mixing Ring (TMR)621<br>
+thermal homogenization467<br>
+twin ram extruder41<br>
+thermal optical analysis (TOA)248<br>
+twin screw devolatilization systems561<br>
+thermal properties191, 233<br>
+twin screw extruder (TSE)22, 24, 895<br>
+thermal stability23<br>
+ <br>
+­ advantages over single screw extruders697<br>
+ <br>
+­ of a polymer246<br>
+ <br>
+­ characteristics699<br>
+thermal time distribution467<br>
+two-dimensional BEM analysis610<br>
+thermistors97<br>
+two-dimensional flow analysis361<br>
+thermochromic materials774<br>
+two-dimensional power law analysis525<br>
+ <br>
+­ for temperature related problems773<br>
+two-stage devolatilizing extruder screw554<br>
+thermochromic powders774<br>
+types of mixing motion<br>
+thermocouple (TC) temperature sensors97­99<br>
+ <br>
+­ convective motion441<br>
+thermodynamics153<br>
+ <br>
+­ molecular diffusion441<br>
+thermoelectric effect97<br>
+ <br>
+­ turbulent motion441<br>
+thermoelectric transducers97<br>
+thermogravimetric analyzer (TGA)247<br>
+thermomechanical analysis (TMA)248<br>
+U<br>
+thermoplastics1<br>
+ultrahigh molecular weight polyethylene (UHMWPE)<br>
+thermosets1<br>
+ <br>
+­ extrusion of37<br>
+thixotropic219<br>
+ultrasonic transducers112<br>
+three-dimensional BEM package610<br>
+ultrasonic transit time ("Laufzeit")113<br>
+three-dimensional finite element simulation, <br>
+ultrasonic vibration843, 844<br>
+ <br>FIDAPTM885<br>
+ultrasound transmission time (UTT)105<br>
+three-dimensional models861<br>
+understanding of the extrusion process764<br>
+three-dimensional simulation885<br>
+uniformity690<br>
+three-phase full wave rectifier54<br>
+Union Carbide (UC) mixing section585<br>
+three-stage extruder screw562<br>
+throttle ratio, ratio between pressure flow to drag <br>
+flow43, 353<br>
+<hr>
+<A name=943></a><b>934</b>  <b> Index</b><br>
+V<br>
+W<br>
+vacuum feed hopper69<br>
+wall draw ratio (WDR)672<br>
+variable decreasing pitch (VDP)19<br>
+wear<br>
+variable depth mixers622<br>
+ <br>
+­ abrasive787<br>
+variable pitch extruder screw550<br>
+ <br>
+­ causes of wear787<br>
+variable reducing pitch (VRP) screw551<br>
+ <br>
+­ corrosive788<br>
+velocity patterns699<br>
+ <br>
+­ five mechanisms of wear781<br>
+velocity profiles379, 420, 715<br>
+ <br>
+­ grooves286<br>
+ <br>
+­ for a Newtonian fluid42<br>
+ <br>
+­ machinery780<br>
+vented extruders16<br>
+ <br>
+­ metal-to-metal789<br>
+ <br>
+­ with axial adjustment capability of the screw559<br>
+ <br>
+­ monitoring850<br>
+vernier screw mechanism50<br>
+ <br>
+­ screw849<br>
+viscoelastic fluids653<br>
+ <br>
+­ test methods782<br>
+viscometric flow210<br>
+weighted average total strain (WATS)455<br>
+viscosity, and pressure214<br>
+weld lines in product852<br>
+viscosity, and temperature214<br>
+whirling516<br>
+viscous dissipation390<br>
+Wien's law169<br>
+viscous encapsulation691, 889, 891<br>
+Williams-Landel-Ferry (WLF) equation216<br>
+viscous flow<i>see</i>rapid flow<br>
+wire coating881<br>
+viscous heat generation168<br>
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