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The Authors: Dr.-Ing. Natti S. Rao, 327 Route 216, Ghent, NY 12075, USA Dr. Gunter Schumacher, Am Bollerweg 6, 75045 Walzbachtal-Johlingen, Germany Distributed in the USA and in Canada by Hanser Gardner Publications, Inc. 6915 Valley Avenue, Cincinnati, Ohio 45244-3029, USA Fax: (513) 527-8801 Phone: (513) 527-8977 or 1-800-950-8977 Internet: http://www.hansergardner.com Distributed in all other countries by Carl Hanser Verlag Postfach 86 04 20, 81631 Munchen, Germany Fax: +49 (89) 98 48 09 Internet: http://www.hanser.de The use of general descriptive names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Library of Congress Cataloging-in-Publication Data Rao, Natti S. Design formulas for plastics engineers.-- 2nd ed. / Natti S. Rao, Gunter Schumacher. p. cm. Includes bibliographical references and index. ISBN 1-56990-370-0 (pbk.) 1. Plastics. I. Schumacher, Gunter. II. Title. TPl 140.R36 2004 668.4-dc22 2004017192 Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der DeutschenNationalbibliografie; detaillierte bibliografische Daten sind im Internetiiber abrufbar. ISBN 3-446-22674-5 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or by any information storage and retrieval system, without permission in wirting from the publisher. © Carl Hanser Verlag, Munich 2004 Production Management: Oswald Immel Typeset by Manuela Treindl, Laaber, Germany Coverconcept: Marc Muller-Bremer, Rebranding, Munchen, Germany Coverdesign: MCP • Susanne Kraus GbR, Holzkirchen, Germany Printed and bound by Druckhaus "Thomas Miintzer", Bad Langensalza, Germany
Preface Today, designing of machines and dies is done to a large extent with the help of computer programs. However, the predictions of theses programs do not always agree with the practical results, so that there is a need to improve the underlying mathematical models. Therefore, knowledge of the formulas, on which the models are based and the limits of their applicability is necessary if one wants to develop a new program or improve one already in use. Often the plastics engineer has to deal with different fields of engineering. The search for the appropriate equations in the various fields concerned can be time-consuming. A collection of formulas from the relevant fields and their applications, as given in this book, make it easier to write one's own program or to make changes in an existing program to obtain a better fit with the experiments. It is often the case that different equations are given in the literature on plastics technology for one and the same target quantity. The practicing engineer is sometimes at a loss judging the validity of the equations he encounters in the literature. During his long years of activity as an R&D engineer in the polymer field at the BASF AG and other companies, Natti Rao tested many formulas published while solving practical problems. This book presents a summary of the important formulas and their applications, which Natti Rao, in cooperation with the well-known resin and machine manufacturers, successfully applied to solve design and processing problems. The formulas are classified according to the fields, rheology, thermodynamics, heat transfer, and part design. Each chapter covers the relevant relations with worked-out examples. A separate chapter is devoted to the practical equations for designing extrusion and injection molding equipment with detailed examples in metric units. In addition, this work contains new, straightforward, practical relationships that have been developed and tested in recent years in solving design problems in the area of extrusion and injection molding. The topic of polymer machine design has been dealt with in several books. However, in these books the know-how was presented in a way that the vast majority of plastics engineers cannot easily apply it to the problems in their day-to-day work. By means of thoroughly worked-out, practical examples this book elucidates the computational background of designing polymer machinery in a manner which every engineer can understand and easily apply in daily practice. We wish to express our thanks to our colleagues at the University of Massachusetts at Lowell, USA, for fruitful discussions. Our thanks are also due to Faculty Innovation Center of the University of Austin, Texas, USA, for help in preparing the manuscript. Austin, USA Karlsruhe, Germany Natti S. Rao, Ph. D. Gunter Schumacher, Ph. D.
Page 1: Natti S. Rao Gunter Schumacher D e
Page 5 and 6: Figure 1.1 Deformation of a Hookean
Page 7 and 8: Figure 1.4 Shear flow The shear or
Page 9 and 10: where c and m are empirical constan
Page 11 and 12: Apparent viscosity 7\Q True viscosi
Page 13 and 14: Pa Shear stress T Figure 1.11 Deter
Page 15 and 16: Shift Factor for Crystalline Polyme
Page 17 and 18: The power law exponent is obtained
Page 19 and 20: 1.3.7.5 Klein's Viscosity Formula [
Page 21 and 22: where Mw = molecular weight JC —
Page 23 and 24: Steady state shear compliance J°t
Page 25 and 26: Ig 6\ Ig G" lga; Figure 1.22 Storag
Page 27 and 28: Characterization of the Transient S
Page 29 and 30: Time/ Figure 1.28 Tensile creep at
Page 31 and 32: 1.4.2.2 Nonlinear Viscoelastic Beha
Page 33 and 34: Figure 1.37 Maxwell fluid [1 ] The
Page 35 and 36: Die swel B1 Figure 1.40 Dependence
Page 37 and 38: 2 Thermodynamic Properties of Polym
Page 39 and 40: where u = internal energy T = tempe
Page 41 and 42: Enthalpy /7-/720 kWh kg Kl kg polyn
Page 43 and 44: Table 2.1 Approximate Values for th
Page 45 and 46: Analogous to Ohm's law in electric
Page 47 and 48: Figure 3.3 One-dimensional heat tra
Page 49 and 50: Figure 3.5 Heat flow in a multilaye
Page 51 and 52: The combination of convection and c
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The equation for an infinite cylind
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The foregoing equations apply to ca
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Figure 3.11 Midplane temperature fo
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Figure 3.12 Temperature distributio
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For drag flow the velocity gradient
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Lewis number: ratio of thermal diff
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The Reynolds number ReL, based on t
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At thermal equilibrium according to
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The rate of heat generation in a pl
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The mass of the fluid permeating th
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4 Designing Plastics Parts The defo
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The parabolic failure criterion is
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Figure 4.3 Secant modulus [4] Stres
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where P = load (N) Lybyd = dimensio
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5 Formulas for Designing Extrusion
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for values of the ratio n (R0 + R1)
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melt density pm = 0.7 g/cm 3 Soluti
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and finally the pressure drop Ap fr
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Pressure drop Ap from Equation 5.2
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Proportionality factor K from Equat
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n = 3.043 RTh= 1.274 mm Shear rate
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The relation of a slit is H = heigh
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Shear stress (N/m 2 ) Figure 5.5 Ef
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Figure 5.7 Surface distortion on a
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Residence time t (s) LDPE m = 40 kg
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Table 5.1 Dimensions of Square Scre
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Pressure drop in screen Mesh size F
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In practice, this efficiency is als
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5.2.2 Melt Conveying Starting from
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md = 46.42 kg/h Equation 5.31 and E
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Solution Power Zc in the screw chan
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Indices: m: melt f: melt film b: ba
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Example with symbols and units a) S
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X- Q Figure 5.24 Velocity and tempe
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(5.54) As seen from the equations a
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Figure 5.26 Three-zone screw [8] Th
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The profiles of stock temperature a
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where HF = feed depth H = metering
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Screw speed (rpm) Screw diameter D
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5.2.6 Mechanical Design of Extrusio
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Next Page Screw Vibration When the
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5.3.1 Pressure Drop in Runner As th
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Examples of calculating pressure dr
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Example with symbols and units The
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Flow length L mm Spiral height H Fi
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Solution The conversion factors for
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Figure 5.43 shows a sample plot of
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With the values for the properties
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Cooling time mm Distance between mo
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Table 5.2 Results of Optimization o
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Per cent unmelted Axial distance al
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A* [%] LDPE Axial length (screw dia
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A* [%] A* [%] LDPE LDPE Axial lengt
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Flow length (mm) Flow length (mm) F
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Flow length L (mm) Flow length L (m
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[26] VDI Warmeatlas, VDI Verlag, Du
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A Final Word The aim of this book i
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166 Index terms Links Index terms L
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168 Index terms Links Index terms L
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viii Contents 1.4 Viscoelastic Beha
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x Contents 5.2 Extrusion Screws ...
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