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: steffe@msu.edu 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 289 4.9.5. Lubricated Squeezing Flow of Peanut Butter 291 Chapter 5. Viscoelasticity 294 5.1. Introduction 294 5.2. Transient Tests for Viscoelasticity 297 5.2.1. Mechanical Analogues 298 5.2.2. Step Strain (Stress Relaxation) 299 5.2.3. Creep and Recovery 304 5.2.4. Start-Up Flow (Stress Overshoot) 310 5.3. Oscillatory Testing 312 5.4. Typical Oscillatory Data 324 5.5. Deborah Number 332 5.6. Experimental Difficulties in Oscillatory Testing of Food 336 5.7. Viscometric and Linear Viscoelastic Functions 338 5.8. Example Problems 341 5.8.1. Generalized Maxwell Model of Stress Relaxation 341 5.8.2. Linearized Stress Relaxation 342 5.8.3. Analysis of Creep Compliance Data 343 5.8.4. Plotting Oscillatory Data 348 6. Appendices 350 6.1. Conversion Factors and SI Prefixes 350 6.2. Greek Alphabet 351 6.3. Mathematics: Roots, Powers, and Logarithms 352 6.4. Linear Regression Analysis of Two Variables 353 6.5. Hookean Properties 357 6.6. Steady Shear and Normal Stress Difference 358 6.7. Yield Stress of Fluid Foods 359 6.8. Newtonian Fluids 361 6.9. Dairy, Fish and Meat Products 366 6.10. Oils and Miscellaneous Products 367 6.11. Fruit and Vegetable Products 368 6.12. Polymer Melts 371 6.13. Cosmetic and Toiletry Products 372 6.14. Energy of Activation for Flow for Fluid Foods 374 6.15. Extensional Viscosities of Newtonian Fluids 375 6.16. Extensional Viscosities of Non-Newtonian Fluids 376 6.17. Fanning Friction Factors: Bingham Plastics 377 6.18. Fanning Friction Factors: Power Law Fluids 378 6.19. Creep (Burgers Model) of Salad Dressing 379 6.20. Oscillatory Data for Butter 380 6.21. Oscillatory Data Iota-Carrageenan Gel 381 6.22. Storage and Loss Moduli of Fluid Foods 382 ° ° vii Nomenclature 385 Bibliography 393 Index 412 viii Preface Growth and development of this work sprang from the need to provide educational material for food engineers and food scientists. The first edition was conceived as a textbook and the work continues to be used in graduate level courses at various universities. Its greatest appeal, however, was to individuals solving practical day-to-day prob- lems. Hence, the second edition, a significantly expanded and revised version of the original work, is aimed primarily at the rheological practitioner (particularly the industrial practitioner) seeking a broad understanding of the subject matter. The overall goal of the text is to present the information needed to answer three questions when facing problems in food rheology: 1. What properties should be measured? 2. What type and degree of deformation should be induced in the mea- surement? 3. How should experimental data be analyzed to generate practical information? Although the main focus of the book is food, scientists and engineers in other fields will find the work a convenient reference for standard rheological methods and typical data. Overall, the work presents the theory of rheological testing and provides the analytical tools needed to determine rheological properties from experimental data. Methods appropriate for common food industry applications are presented. All standard measurement techniques for fluid and semi-solid foods are included. Selected methods for solids are also presented. Results from numerous fields, particularly polymer rheology, are used to address the flow behavior of food. Mathematical relationships, derived from simple force balances, provide a funda- mental view of rheological testing. Only a background in basic calculus and elementary statistics (mainly regression analysis) is needed to understand thematerial. The text containsnumerous practicalexample problems, involving actual experimental data, to enhance comprehen- sion and the execution of concepts presented. This feature makes the work convenient for self-study. Specific explanations of key topics in rheology are presented in Chapter 1: basic concepts of stress and strain; elastic solids showing Hookean and non-Hookean behavior; viscometric functions including normal stress differences; modeling fluid behavior as a function of shear rate, temperature, and composition; yield stress phenomena, exten- sional flow; and viscoelastic behavior. Efficient methods of attacking problems and typical instruments used to measure fluid properties are discussed along with an examination of problems involving interfacial ix rheology, electrorheology, and on-line viscometry for control and mon- itoring of food processing operations. Common methods and empirical instruments utilized in the food industry are also introduced: Texture Profile Analysis, Compression-Extrusion Cell, Warner-Bratzler Shear Cell, Bostwick Consistometer, Adams Consistometer, Amylograph, Farinograph, Mixograph, Extensigraph, Alveograph, Kramer Shear Cell, Brookfield disks and T-bars, Cone Penetrometer, Hoeppler Vis- cometer, Zhan Viscometer, Brabender-FMC Consistometer. The basic equations of tube (or capillary) viscometry, such as the Rabinowitsch-Mooney equation, are derived and applied in Chapter 2. Laminar flow criteria and velocity profiles are evaluated along with data correction methods for many sources of error: kinetic energy losses, end effects, slip (wall effects), viscous heating, and hole pressure. Tech- niques for glass capillary and slit viscometers are considered in detail. A section on pipeline design calculations has been included to facilitate the construction of large scale tube viscometers and the design of fluid pumping systems. A general format, analogous to that used in Chapter 2, is continued in Chapter 3 to provide continuity in subject matter development. The main foci of the chapter center around the theoretical principles and experimental procedures related to three traditional types of rotational viscometers: concentric cylinder, cone and plate, and parallel plate. Mathematical analyses of data are discussed in detail. Errors due to end effects, viscous heating, slip, Taylor vortices, cavitation, and cone truncations are investigated. Numerous methods in mixer viscometry, techniques having significant potential to solve many food rheology problems, are also presented: slope and matching viscosity methods to evaluate average shear rate, determination of power law and Bingham plastic fluid properties. The vane method of yield stress evaluation, using both the slope and single point methods, along with a consider- ation of vane rotation during testing, is explored in detail. The experimental methods to determine extensional viscosity are explained in Chapter 4. Techniques presented involve uniaxial exten- sion between rotating clamps, biaxial extensional flow achieved by squeezing material between lubricated parallel plates, opposing jets, spinning, and tubeless siphon (Fano) flow. Related procedures, involving lubricated and nonlubricated squeezing, to determine shear flow behavior are also presented. Calculating extensional viscosity from flows through tapered convergences and flat entry dies is given a thorough examination. x Essential concepts in viscoelasticity and standard methods of investigating the phenomenon are investigated in Chapter 5. The full scope of viscoelastic material functions determined in transient and oscillatory testing are discussed. Mechanical analogues of rheological behavior are given as a means of analyzing creep and stress relaxation data. Theoretical aspects of oscillatory testing, typical data, and a discussion of the various modes of operating commercial instruments -strain, frequency, time, and temperature sweep modes- are presented. The Deborah number concept, and how it can be used to distinguish liquid from solid-like behavior, is introduced. Start-up flow (stress overshoot) and the relationship between steady shear and oscillatory properties are also discussed. Conversion factors, mathematical rela- tionships, linear regression analysis, and typical rheological data for food as well as cosmetics and polymers are provided in the Appendices. Nomenclature is conveniently summarized at the end of the text and a large bibliography is furnished to direct readers to additional infor- mation. J.F. Steffe June, 1996 xi Dedication To Susan, Justinn, and Dana. xiii [...]... Medicines, paints, spices in salad dressing Frosting, paints, printing inks Vats, small food containers, painting and coating Snack and pet foods, toothpaste, cereals, pasta, polymers Dough Sheeting Foods, cosmetics, toiletries Foods Paints, confectionery Food processing Food processing, blood flow Topical application of creams and lotions Brush painting, lipstick, nail polish Spray drying, spray painting,... understanding rheology is critical in optimizing product development efforts, processing methodology and final product quality To the extent possible, standard nomenclature (Dealy, 1994) has been used in the text One can think of food rheology as the material science of food There are numerous areas where rheological data are needed in the food industry: a Process engineering calculations involving a wide... and create difficult problems in process engineering design These problems are particularly prevalent in the plastic processing industries but also present in processing foods such as dough, particularly those containing large quantities of wheat protein Fig 1.10 illustrates several phenomena During mixing or agitation, a viscoelastic fluid may climb the impeller shaft in a phenomenon known as the Weissenberg... material is capable of sustaining; shear strength - the maximum shear stress a material is capable of sustaining; tensile strength - the maximum tensile stress a material is capable of sustaining; yield point - the first stress in a test where the increase in strain occurs without an increase in stress; 1.5.1 Time-Independent Material Functions 13 yield strength - the engineering stress at which a material... Familiar Materials and Processes ˙ γ (1/s) Situation Sedimentation of particles in a suspending liquid Leveling due to surface tension Draining under gravity 10-6 - 10-3 Extrusion 100 - 103 Calendering Pouring from a bottle Chewing and swallowing Dip coating Mixing and stirring Pipe flow Rubbing 101 101 101 101 101 100 102 Brushing 103 - 104 Spraying 103 - 105 High speed coating Lubrication 104 - 106... the increase in length This deformation may be thought of in terms of Cauchy strain (also called engineering strain): 1.3 Stress and Strain 5 εc = δL L − Lo L = = −1 Lo Lo Lo [1.1] or Hencky strain (also called true strain) which is determined by evaluating an integral from Lo to L : [1.2] L dL εh = ⌠ = ln(L/Lo ) ⌡Lo L L0 L Figure 1.2 Linear extension of a rectangular bar Cauchy and Hencky strains... that taking single point tests for determining the general behavior of non-Newtonian materials may cause serious problems Some quality control instruments designed for single point tests may produce confusing results Consider, for 26 Chapter 1 Introduction to Rheology example, the two Bingham plastic materials shown in Fig 1.16 The two curves intersect at 19.89 1/s and an instrument measuring the apparent... Herschel-Bulkley, have proven useful in developing mathematical models to solve food process engineering problems (Ofoli et al., 1987) involving wide shear rate ranges Additional rheological models have been summarized by Holdsworth (1993) The Casson equation has been adopted by the International Office of Cocoa and Chocolate for interpreting chocolate flow behavior The Casson and Bingham plastic models are similar... materials and increases with increasing shear rate in shear-thickening fluids (Fig 1.15) Apparent Viscosity, Pa s Time-Independent Fluids Bingham Herschel-Bulkley ( 0 < n < 1.0 ) Shear-Thickening Shear-Thinning Newtonian Shear Rate, 1/s Figure 1.15 Apparent viscosity of time-independent fluids A single point apparent viscosity value is sometimes used as a measure of mouthfeel of fluid foods: The human... (dynamic) mode Some rotational instruments function in the controlled stress mode facilitating the collection of creep data, the analysis of materials at very low shear rates, and the investigation of yield stresses This information is needed to understand the internal structure of materials The controlled rate mode is most useful in obtaining data required in process engineering calculations Mechanical . 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 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. been used in the text. One can think of food rheology as the material science of food. There are numerous areas where rheological data are needed in the food industry: a. Process engineering calculations