RHEOLOGICAL METHODS IN FOOD PROCESS ENGINEERING Second Edition James F. Steffe, Ph.D., P.E. Professor of Food Process Engineering Dept. of Food Science and Human Nutrition Dept. of Agricultural Engineering Michigan State University Freeman Press 2807 Still Valley Dr. East Lansing, MI 48823 USA Prof. James F. Steffe 209 Farrall Hall Michigan State University East Lansing, MI 48824-1323 USA Phone: 517-353-4544 FAX: 517-432-2892 E-mail: [email protected] URL: www.egr.msu.edu/~steffe/ Copyright 1992, 1996 by James F. Steffe. All rights reserved. No part of this work may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or other- wise, without the prior written permission of the author. Printed on acid-free paper in the United States of America Second Printing Library of Congress Catalog Card Number: 96-83538 International Standard Book Number: 0-9632036-1-4 Freeman Press 2807 Still Valley Dr. East Lansing, MI 48823 USA Table of Contents Preface ix Chapter 1. Introduction to Rheology 1 1.1. Overview ...................................................................................... 1 1.2. Rheological Instruments for Fluids .......................................... 2 1.3. Stress and Strain .......................................................................... 4 1.4. Solid Behavior ............................................................................. 8 1.5. Fluid Behavior in Steady Shear Flow ....................................... 13 1.5.1. Time-Independent Material Functions ............................ 13 1.5.2. Time-Dependent Material Functions ............................... 27 1.5.3. Modeling Rheological Behavior of Fluids ....................... 32 1.6. Yield Stress Phenomena ............................................................. 35 1.7. Extensional Flow ......................................................................... 39 1.8. Viscoelastic Material Functions ................................................ 47 1.9. Attacking Problems in Rheological Testing ............................ 49 1.10. Interfacial Rheology ................................................................. 53 1.11. Electrorheology ......................................................................... 55 1.12. Viscometers for Process Control and Monitoring ................ 57 1.13. Empirical Measurement Methods for Foods ........................ 63 1.14. Example Problems .................................................................... 77 1.14.1. Carrageenan Gum Solution ............................................. 77 1.14.2. Concentrated Corn Starch Solution ................................ 79 1.14.3. Milk Chocolate .................................................................. 81 1.14.4. Falling Ball Viscometer for Honey ................................. 82 1.14.5. Orange Juice Concentrate ................................................ 86 1.14.6. Influence of the Yield Stress in Coating Food ............... 91 Chapter 2. Tube Viscometry 94 2.1. Introduction ................................................................................. 94 2.2. Rabinowitsch-Mooney Equation .............................................. 97 2.3. Laminar Flow Velocity Profiles ................................................ 103 2.4. Laminar Flow Criteria ................................................................ 107 2.5. Data Corrections ......................................................................... 110 2.6. Yield Stress Evaluation .............................................................. 121 2.7. Jet Expansion ............................................................................... 121 2.8. Slit Viscometry ............................................................................ 122 2.9. Glass Capillary (U-Tube) Viscometers .................................... 125 2.10. Pipeline Design Calculations .................................................. 128 2.11. Velocity Profiles In Turbulent Flow ....................................... 138 2.12. Example Problems .................................................................... 141 2.12.1. Conservation of Momentum Equations ........................ 141 2.12.2. Capillary Viscometry - Soy Dough ................................ 143 2.12.3. Tube Viscometry - 1.5% CMC ......................................... 146 2.12.4. Casson Model: Flow Rate Equation ............................... 149 2.12.5. Slit Viscometry - Corn Syrup .......................................... 150 2.12.6. Friction Losses in Pumping ............................................. 152 2.12.7. Turbulent Flow - Newtonian Fluid ................................ 155 2.12.8. Turbulent Flow - Power Law Fluid ................................ 156 v Chapter 3. Rotational Viscometry 158 3.1. Introduction ................................................................................. 158 3.2. Concentric Cylinder Viscometry .............................................. 158 3.2.1. Derivation of the Basic Equation ...................................... 158 3.2.2. Shear Rate Calculations ...................................................... 163 3.2.3. Finite Bob in an Infinite Cup ............................................. 168 3.3. Cone and Plate Viscometry ....................................................... 169 3.4. Parallel Plate Viscometry (Torsional Flow) ............................ 172 3.5. Corrections: Concentric Cylinder ............................................. 174 3.6. Corrections: Cone and Plate, and Parallel Plate ..................... 182 3.7. Mixer Viscometry ....................................................................... 185 3.7.1. Mixer Viscometry: Power Law Fluids ............................. 190 3.7.2. Mixer Viscometry: Bingham Plastic Fluids ..................... 199 3.7.3. Yield Stress Calculation: Vane Method ........................... 200 3.7.4. Investigating Rheomalaxis ................................................ 208 3.7.5. Defining An Effective Viscosity ........................................ 210 3.8. Example Problems ...................................................................... 210 3.8.1. Bob Speed for a Bingham Plastic Fluid ............................ 210 3.8.2. Simple Shear in Power Law Fluids .................................. 212 3.8.3. Newtonian Fluid in a Concentric Cylinder ..................... 213 3.8.4. Representative (Average) Shear Rate .............................. 214 3.8.5. Concentric Cylinder Viscometer: Power Law Fluid ...... 216 3.8.6. Concentric Cylinder Data - Tomato Ketchup ................. 218 3.8.7. Infinite Cup - Single Point Test ......................................... 221 3.8.8. Infinite Cup Approximation - Power Law Fluid ........... 221 3.8.9. Infinite Cup - Salad Dressing ............................................ 223 3.8.10. Infinite Cup - Yield Stress Materials .............................. 225 3.8.11. Cone and Plate - Speed and Torque Range ................... 226 3.8.12. Cone and Plate - Salad Dressing ..................................... 227 3.8.13. Parallel Plate - Methylcellulose Solution ....................... 229 3.8.14. End Effect Calculation for a Cylindrical Bob ................ 231 3.8.15. Bob Angle for a Mooney-Couette Viscometer .............. 233 3.8.16. Viscous Heating ................................................................ 235 3.8.17. Cavitation in Concentric Cylinder Systems .................. 236 3.8.18. Mixer Viscometry .............................................................. 237 3.8.19. Vane Method - Sizing the Viscometer ........................... 243 3.8.20. Vane Method to Find Yield Stresses .............................. 244 3.8.21. Vane Rotation in Yield Stress Calculation .................... 247 3.8.22. Rheomalaxis from Mixer Viscometer Data ................... 250 Chapter 4. Extensional Flow 255 4.1. Introduction ................................................................................. 255 4.2. Uniaxial Extension ...................................................................... 255 4.3. Biaxial Extension ......................................................................... 258 4.4. Flow Through a Converging Die .............................................. 263 4.4.1. Cogswell’s Equations ......................................................... 264 4.4.2. Gibson’s Equations ............................................................. 268 4.4.3. Empirical Method ............................................................... 271 4.5. Opposing Jets .............................................................................. 272 4.6. Spinning ....................................................................................... 274 4.7. Tubeless Siphon (Fano Flow) .................................................... 276 vi 4.8. Steady Shear Properties from Squeezing Flow Data ............. 276 4.8.1. Lubricated Squeezing Flow ............................................... 277 4.8.2. Nonlubricated Squeezing Flow ........................................ 279 4.9. Example Problems ...................................................................... 283 4.9.1. Biaxial Extension of Processed Cheese Spread ............... 283 4.9.2. Biaxial Extension of Butter ................................................ 286 4.9.3. 45° Converging Die, Cogswell’s Method ........................ 287 4.9.4. 45° Converging Die, Gibson’s Method ...........................