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- 1. RHEOLOGICAL METHODS INFOOD 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
- 2. Prof. James F. Steffe209 Farrall HallMichigan State UniversityEast Lansing, MI 48824-1323USAPhone: 517-353-4544FAX: 517-432-2892E-mail: steffe@msu.eduURL: 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 anymeans, 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
- 3. Table of ContentsPreface ixChapter 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 ............... 91Chapter 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
- 4. 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 ................... 250Chapter 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
- 5. 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 ................. 291Chapter 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 ................................................... 3486. 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
- 6. Nomenclature ......................................................................................... 385Bibliography ........................................................................................... 393Index ......................................................................................................... 412 viii
- 7. Preface Growth and development of this work sprang from the need toprovide educational material for food engineers and food scientists. Thefirst edition was conceived as a textbook and the work continues to beused in graduate level courses at various universities. Its greatestappeal, however, was to individuals solving practical day-to-day prob-lems. Hence, the second edition, a significantly expanded and revisedversion of the original work, is aimed primarily at the rheologicalpractitioner (particularly the industrial practitioner) seeking a broadunderstanding of the subject matter. The overall goal of the text is topresent the information needed to answer three questions when facingproblems 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 generatepractical information? Although the main focus of the book is food,scientists and engineers in other fields will find the work a convenientreference for standard rheological methods and typical data. Overall, the work presents the theory of rheological testing andprovides the analytical tools needed to determine rheological propertiesfrom experimental data. Methods appropriate for common food industryapplications are presented. All standard measurement techniques forfluid and semi-solid foods are included. Selected methods for solids arealso presented. Results from numerous fields, particularly polymerrheology, are used to address the flow behavior of food. Mathematicalrelationships, derived from simple force balances, provide a funda-mental view of rheological testing. Only a background in basic calculusand elementary statistics (mainly regression analysis) is needed tounderstand the material. The text contains numerous practical exampleproblems, involving actual experimental data, to enhance comprehen-sion and the execution of concepts presented. This feature makes thework convenient for self-study. Specific explanations of key topics in rheology are presented inChapter 1: basic concepts of stress and strain; elastic solids showingHookean and non-Hookean behavior; viscometric functions includingnormal stress differences; modeling fluid behavior as a function of shearrate, temperature, and composition; yield stress phenomena, exten-sional flow; and viscoelastic behavior. Efficient methods of attackingproblems and typical instruments used to measure fluid properties arediscussed along with an examination of problems involving interfacial ix
- 8. rheology, electrorheology, and on-line viscometry for control and mon-itoring of food processing operations. Common methods and empiricalinstruments utilized in the food industry are also introduced: TextureProfile Analysis, Compression-Extrusion Cell, Warner-Bratzler ShearCell, Bostwick Consistometer, Adams Consistometer, Amylograph,Farinograph, Mixograph, Extensigraph, Alveograph, Kramer ShearCell, 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 theRabinowitsch-Mooney equation, are derived and applied in Chapter 2.Laminar flow criteria and velocity profiles are evaluated along with datacorrection methods for many sources of error: kinetic energy losses, endeffects, 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 facilitatethe construction of large scale tube viscometers and the design of fluidpumping systems. A general format, analogous to that used in Chapter 2, is continuedin Chapter 3 to provide continuity in subject matter development. Themain foci of the chapter center around the theoretical principles andexperimental procedures related to three traditional types of rotationalviscometers: concentric cylinder, cone and plate, and parallel plate.Mathematical analyses of data are discussed in detail. Errors due toend effects, viscous heating, slip, Taylor vortices, cavitation, and conetruncations are investigated. Numerous methods in mixer viscometry,techniques having significant potential to solve many food rheologyproblems, are also presented: slope and matching viscosity methods toevaluate average shear rate, determination of power law and Binghamplastic 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 areexplained in Chapter 4. Techniques presented involve uniaxial exten-sion between rotating clamps, biaxial extensional flow achieved bysqueezing material between lubricated parallel plates, opposing jets,spinning, and tubeless siphon (Fano) flow. Related procedures,involving lubricated and nonlubricated squeezing, to determine shearflow behavior are also presented. Calculating extensional viscosity fromflows through tapered convergences and flat entry dies is given athorough examination. x
- 9. Essential concepts in viscoelasticity and standard methods ofinvestigating the phenomenon are investigated in Chapter 5. The fullscope of viscoelastic material functions determined in transient andoscillatory testing are discussed. Mechanical analogues of rheologicalbehavior are given as a means of analyzing creep and stress relaxationdata. Theoretical aspects of oscillatory testing, typical data, and adiscussion 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 distinguishliquid from solid-like behavior, is introduced. Start-up flow (stressovershoot) and the relationship between steady shear and oscillatoryproperties are also discussed. Conversion factors, mathematical rela-tionships, linear regression analysis, and typical rheological data forfood as well as cosmetics and polymers are provided in the Appendices.Nomenclature is conveniently summarized at the end of the text and alarge bibliography is furnished to direct readers to additional infor-mation. J.F. Steffe June, 1996 xi
- 10. DedicationTo Susan, Justinn, and Dana. xiii
- 11. Chapter 1. Introduction to Rheology1.1. Overview The first use of the word "rheology" is credited to Eugene C. Bingham(circa 1928) who also described the motto of the subject as π α ν τ α ρ ε ι("panta rhei," from the works of Heraclitus, a pre-Socratic Greek phi-losopher active about 500 B.C.) meaning "everything flows" (Reiner,1964). Rheology is now well established as the science of the deformationand flow of matter: It is the study of the manner in which materialsrespond to applied stress or strain. All materials have rheologicalproperties and the area is relevant in many fields of study: geology andmining (Cristescu, 1989), concrete technology (Tattersall and Banfill,1983), soil mechanics (Haghighi et al., 1987; Vyalov, 1986), plasticsprocessing (Dealy and Wissburn, 1990), polymers and composites(Neilsen and Landel, 1994; Yanovsky, 1993), tribology (study of lubri-cation, friction and wear), paint flow and pigment dispersion (Patton,1964), blood (Dintenfass, 1985), bioengineering (Skalak and Chien,1987), interfacial rheology (Edwards et al., 1991), structural materials(Callister, 1991), electrorheology (Block and Kelly, 1988), psychor-heology (Drake, 1987), cosmetics and toiletries (Laba, 1993b), andpressure sensitive adhesion (Saunders et al., 1992). The focus of thiswork is food where understanding rheology is critical in optimizingproduct development efforts, processing methodology and final productquality. 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. Thereare numerous areas where rheological data are needed in the foodindustry: a. Process engineering calculations involving a wide range of equip- ment such as pipelines, pumps, extruders, mixers, coaters, heat exchangers, homogenizers, calenders, and on-line viscometers; b. Determining ingredient functionality in product development; c. Intermediate or final product quality control; d. Shelf life testing; e. Evaluation of food texture by correlation to sensory data; f. Analysis of rheological equations of state or constitutive equations.Many of the unique rheological properties of various foods have beensummarized in books by Rao and Steffe (1992), and Weipert et al. (1993).
- 12. 2 Chapter 1. Introduction to Rheology Fundamental rheological properties are independent of the instru-ment on which they are measured so different instruments will yieldthe same results. This is an ideal concept and different instrumentsrarely yield identical results; however, the idea is one which distin-guishes true rheological material properties from subjective (empiricaland generally instrument dependent, though frequently useful)material characterizations. Examples of instruments giving subjectiveresults include the following (Bourne, 1982): Farinograph, Mixograph,Extensograph, Viscoamlyograph, and the Bostwick Consistometer.Empirical testing devices and methods, including Texture ProfileAnalysis, are considered in more detail in Sec. 1.13.1.2. Rheological Instruments for Fluids Common instruments, capable of measuring fundamental rheolog-ical properties of fluid and semi-solid foods, may be placed into twogeneral categories (Fig. 1.1): rotational type and tube type. Most arecommercially available, others (mixer and pipe viscometers) are easilyfabricated. Costs vary tremendously from the inexpensive glass capil-lary viscometer to a very expensive rotational instrument capable ofmeasuring dynamic properties and normal stress differences. Solidfoods are often tested in compression (between parallel plates), tension,or torsion. Instruments which measure rheological properties are calledrheometers. Viscometer is a more limiting term referring to devicesthat only measure viscosity. Rotational instruments may be operated in the steady shear (con-stant angular velocity) or oscillatory (dynamic) mode. Some rotationalinstruments function in the controlled stress mode facilitating thecollection of creep data, the analysis of materials at very low shear rates,and the investigation of yield stresses. This information is needed tounderstand the internal structure of materials. The controlled ratemode is most useful in obtaining data required in process engineeringcalculations. Mechanical differences between controlled rate and con-trolled stress instruments are discussed in Sec. 3.7.3. Rotational sys-tems are generally used to investigate time-dependent behavior becausetube systems only allow one pass of the material through the apparatus.A detailed discussion of oscillatory testing, the primary method ofdetermining the viscoelastic behavior of food, is provided in Chapter 5.
- 13. 1.2 Rheological Instruments for Fluids 3 Rotational Type Parallel Plate Cone and Plate Concentric Cylinder Mixer Tube Type Glass Capillary High Pressure Capillary PipeFigure 1.1. Common rheological instruments divided into two major categories: rotational and tube type. There are advantages and disadvantages associated with eachinstrument. Gravity operated glass capillaries, such as the Cannon-Fenske type shown in Fig. 1.1, are only suitable for Newtonian fluidsbecause the shear rate varies during discharge. Cone and plate systemsare limited to moderate shear rates but calculations (for small coneangles) are simple. Pipe and mixer viscometers can handle much largerparticles than cone and plate, or parallel plate, devices. Problemsassociated with slip and degradation in structurally sensitive materialsare minimized with mixer viscometers. High pressure capillariesoperate at high shear rates but generally involve a significant endpressure correction. Pipe viscometers can be constructed to withstandthe rigors of the production or pilot plant environment.
- 14. 4 Chapter 1. Introduction to Rheology All the instruments presented in Fig. 1.1 are "volume loaded" deviceswith container dimensions that are critical in the determination ofrheological properties. Another common type of instrument, known asa vibrational viscometer, uses the principle of "surface loading" wherethe surface of an immersed probe (usually a sphere or a rod) generatesa shear wave which dissipates in the surrounding medium. A largeenough container is used so shear forces do not reach the wall and reflectback to the probe. Measurements depend only on ability of the sur-rounding fluid to damp probe vibration. The damping characteristic ofa fluid is a function of the product of the fluid viscosity (of Newtonianfluids) and the density. Vibrational viscometers are popular as in-lineinstruments for process control systems (see Sec. 1.12). It is difficult touse these units to evaluate fundamental rheological properties of non-Newtonian fluids (Ferry, 1977), but the subjective results obtained oftencorrelate well with important food quality attributes. The coagulationtime and curd firmness of renneted milk, for example, have been suc-cessfully investigated using a vibrational viscometer (Sharma et al.,1989). Instruments used to evaluate the extensional viscosity of materialsare discussed in Chapter 4. Pulling or stretching a sample betweentoothed gears, sucking material into opposing jets, spinning, orexploiting the open siphon phenomenon can generate data for calcu-lating tensile extensional viscosity. Information to determine biaxialextensional viscosity is provided by compressing samples betweenlubricated parallel plates. Shear viscosity can also be evaluated fromunlubricated squeezing flow between parallel plates. A number ofmethods are available to calculate an average extensional viscosity fromdata describing flow through a convergence into a capillary die or slit.1.3. Stress and Strain Since rheology is the study of the deformation of matter, it is essentialto have a good understanding of stress and strain. Consider a rectan-gular bar that, due to a tensile force, is slightly elongated (Fig. 1.2). Theinitial length of the bar is Lo and the elongated length is L whereL = Lo + δL with δL representing the increase in length. This deformationmay be thought of in terms of Cauchy strain (also called engineeringstrain):
- 15. 1.3 Stress and Strain 5 δL L − Lo L [1.1] εc = = = −1 Lo Lo Loor Hencky strain (also called true strain) which is determined byevaluating an integral from Lo to L : [1.2] εh = ⌠ L dL = ln(L/Lo ) ⌡Lo L L0 L Figure 1.2. Linear extension of a rectangular bar. Cauchy and Hencky strains are both zero when the material isunstrained and approximately equal at small strains. The choice ofwhich strain measure to use is largely a matter of convenience and onecan be calculated from the other: εh = ln(1 + εc ) [1.3]εh is preferred for calculating strain resulting from a large deformation. Another type of deformation commonly found in rheology is simpleshear. The idea can be illustrated with a rectangular bar (Fig. 1.3) ofheight h . The lower surface is stationary and the upper plate is linearlydisplaced by an amount equal to δL . Each element is subject to the samelevel of deformation so the size of the element is not relevant. The angleof shear, γ, may be calculated as
- 16. 6 Chapter 1. Introduction to Rheology δL [1.4] tan(γ) = hWith small deformations, the angle of shear (in radians) is equal to theshear strain (also symbolized by γ), tan γ = γ. Lh Figure 1.3. Shear deformation of a rectangular bar. 22 x 2 21 23 11 x 1x3 33 Figure 1.4. Typical stresses on a material element. Stress, defined as a force per unit area and usually expressed inPascal (N/m2), may be tensile, compressive, or shear. Nine separatequantities are required to completely describe the state of stress in amaterial. A small element (Fig. 1.4) may be considered in terms of
- 17. 1.3 Stress and Strain 7Cartesian coordinates (x1, x2, x3). Stress is indicated as σij where the firstsubscript refers to the orientation of the face upon which the force actsand the second subscript refers to the direction of the force. Therefore,σ11 is a normal stress acting in the plane perpendicular to x1 in thedirection of x1 and σ23 is a shear stress acting in the plane perpendicularto x2 in the direction of x3. Normal stresses are considered positive whenthey act outward (acting to create a tensile stress) and negative whenthey act inward (acting to create a compressive stress). Stress components may be summarized as a stress tensor writtenin the form of a matrix: σ11 σ12 σ13 [1.5] σij = σ21 σ22 σ23 σ31 σ32 σ33A related tensor for strain can also be expressed in matrix form. Basiclaws of mechanics, considering the moment about the axis underequilibrium conditions, can be used to prove that the stress matrix issymmetrical: σij = σji [1.6]so σ12 = σ21 [1.7] σ31 = σ13 [1.8] σ32 = σ23 [1.9]meaning there are only six independent components in the stress tensorrepresented by Eq. [1.5]. Equations that show the relationship between stress and strain areeither called rheological equations of state or constitutive equations. Incomplex materials these equations may include other variables such astime, temperature, and pressure. A modulus is defined as the ratio ofstress to strain while a compliance is defined as the ratio of strain tostress. The word rheogram refers to a graph of a rheological relationship.
- 18. 8 Chapter 1. Introduction to Rheology1.4. Solid Behavior When force is applied to a solid material and the resulting stressversus strain curve is a straight line through the origin, the material isobeying Hooke’s law. The relationship may be stated for shear stressand shear strain as σ12 = Gγ [1.10]where G is the shear modulus. Hookean materials do not flow and arelinearly elastic. Stress remains constant until the strain is removedand the material returns to its original shape. Sometimes shaperecovery, though complete, is delayed by some atomistic process. Thistime-dependent, or delayed, elastic behavior is known as anelasticity.Hooke’s law can be used to describe the behavior of many solids (steel,egg shell, dry pasta, hard candy, etc.) when subjected to small strains,typically less than 0.01. Large strains often produce brittle fracture ornon-linear behavior. The behavior of a Hookean solid may be investigated by studyingthe uniaxial compression of a cylindrical sample (Fig. 1.5). If a materialis compressed so that it experiences a change in length and radius, thenthe normal stress and strain may be calculated: F F [1.11] σ22 = = A π(Ro )2 δh [1.12] εc = ho Ro Ro R hho Initial Shape Compressed Shape h Figure 1.5 Uniaxial compression of a cylindrical sample.
- 19. 1.4 Solid Behavior 9This information can be used to determine Young’s modulus (E ), alsocalled the modulus of elasticity, of the sample: σ22 [1.13] E= εcIf the deformations are large, Hencky strain (εh ) should be used tocalculate strain and the area term needed in the stress calculationshould be adjusted for the change in radius caused by compression: F [1.14] σ22 = π(Ro + δR)2A critical assumption in these calculations is that the sample remainscylindrical in shape. For this reason lubricated contact surfaces areoften recommended when testing materials such as food gels. Young’s modulus may also be determined by flexural testing ofbeams. In one such test, a cantilever beam of known length (a) isdeflected a distance (d) when a force (F) is applied to the free end of thebeam. This information can be used to calculate Young’s modulus formaterials having a rectangular or circular crossectional area (Fig. 1.6).Similar calculations can be performed in a three-point bending test (Fig.1.7) where deflection (d) is measured when a material is subjected to aforce (F) placed midway between two supports. Calculations are sightlydifferent depending on wether-or-not the test material has a rectangularor circular shape. Other simple beam tests, such as the double cantileveror four-point bending test, yield comparable results. Flexural testingmay have application to solid foods having a well defined geometry suchas dry pasta or hard candy. In addition to Young’s modulus, Poisson’s ratio (ν) can be definedfrom compression data (Fig. 1.5): lateral strain δR/Ro [1.15] ν= = axial strain δh/hoPoisson’s ratio may range from 0 to 0.5. Typically, ν varies from 0.0 forrigid like materials containing large amounts of air to near 0.5 for liquidlike materials. Values from 0.2 to 0.5 are common for biologicalmaterials with 0.5 representing an incompressible substance like potato
- 20. 10 Chapter 1. Introduction to Rheologyflesh. Tissues with a high level of cellular gas, such as apple flesh, wouldhave values closer to 0.2. Metals usually have Poisson ratios between0.25 and 0.35. a d F Rectangular Crossection Circular Crossection 3 3 3 E = 4Fa /(dbh ) E = 64Fa /(3d D4) h b D Figure 1.6 Deflection of a cantilever beam to determine Young’s modulus. a d F Rectangular Crossection Circular Crossection 3 3 3 4 E = Fa /(4dbh ) E = 4Fa /(3d D )Figure 1.7 Three-point beam bending test to determine Young’s modulus (b, h, and D are defined on Fig. 1.6).
- 21. 1.4 Solid Behavior 11 If a material is subjected to a uniform change in external pressure,it may experience a volumetric change. These quantities are used todefine the bulk modulus (K ): change in pressure change in pressure [1.16] K= = volumetric strain (change in volume/original volume)The bulk modulus of dough is approximately 106 Pa while the value forsteel is 1011 Pa. Another common property, bulk compressibility, isdefined as the reciprocal of bulk modulus. When two material constants describing the behavior of a Hookeansolid are known, the other two can be calculated using any of the fol-lowing theoretical relationships: 1 1 1 [1.17] = + E 3G 9K E = 3K(1 − 2ν) = 2G(1 + ν) [1.18] 3K − E E − 2G [1.19] ν= = 6K 2GNumerous experimental techniques, applicable to food materials, maybe used to determine Hookean material constants. Methods includetesting in tension, compression and torsion (Mohsenin, 1986; Pola-kowski and Ripling, 1966; Dally and Ripley, 1965). Hookean propertiesof typical materials are presented in the Appendix [6.5]. Linear-elastic and non-linear elastic materials (like rubber) bothreturn to their original shape when the strain is removed. Food maybe solid in nature but not Hookean. A comparison of curves for linearelastic (Hookean), elastoplastic and non-linear elastic materials (Fig.1.8) shows a number of similarities and differences. The elastoplasticmaterial has Hookean type behavior below the yield stress (σo ) but flowsto produce permanent deformation above that value. Margarine andbutter, at room temperature, may behave as elastoplastic substances.Investigation of this type of material, as a solid or a fluid, depends onthe shear stress being above or below σo (see Sec. 1.6 for a more detaileddiscussion of the yield stress concept and Appendix [6.7] for typical yieldstress values). Furthermore, to fully distinguish fluid from solid likebehavior, the characteristic time of the material and the characteristictime of the deformation process involved must be considered simulta-
- 22. 12 Chapter 1. Introduction to Rheologyneously. The Deborah number has been defined to address this issue.A detailed discussion of the concept, including an example involvingsilly putty (the "real solid-liquid") is presented in Sec. 5.5. Linear Elastic Elastoplastic Non-Linear Elastic o 12 12 12 Permanent Deformation Figure 1.8. Deformation curves for linear elastic (Hookean), elastoplastic and non-linear elastic materials. Food rheologists also find the failure behavior of solid food (partic-ularly, brittle materials and firm gels) to be very useful because thesedata sometimes correlate well with the conclusions of human sensorypanels (Hamann, 1983; Montejano et al., 1985; Kawanari et al., 1981).The following terminology (taken from American Society for Testing andMaterials, Standard E-6) is useful in describing the large deformationbehavior involved in the mechanical failure of food: elastic limit - the greatest stress which a material is capable of sus- taining without any permanent strain remaining upon complete release of stress; proportional limit - the greatest stress which a material is capable of sustaining without any deviation from Hooke’s Law; compressive strength - the maximum compressive stress a 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;
- 23. 1.5.1 Time-Independent Material Functions 13 yield strength - the engineering stress at which a material exhibits a specified limiting deviation from the proportionality of stress to strain.A typical characteristic of brittle solids is that they break when givena small deformation. Failure testing and fracture mechanics in struc-tural solids are well developed areas of material science (Callister, 1991)which offer much to the food rheologist. Evaluating the structuralfailure of solid foods in compression, torsion, and sandwich shear modeswere summarized by Hamann (1983). Jagged force-deformation rela-tionships of crunchy materials may offer alternative texture classifi-cation criteria for brittle or crunchy foods (Ulbricht et al., 1995; Pelegand Normand, 1995).1.5. Fluid Behavior in Steady Shear Flow1.5.1. Time-Independent Material FunctionsViscometric Functions. Fluids may be studied by subjecting themto continuous shearing at a constant rate. Ideally, this can be accom-plished using two parallel plates with a fluid in the gap between them(Fig. 1.9). The lower plate is fixed and the upper plate moves at aconstant velocity (u ) which can be thought of as an incremental changein position divided by a small time period, δL/δt . A force per unit areaon the plate is required for motion resulting in a shear stress (σ21) onthe upper plate which, conceptually, could also be considered to be alayer of fluid. The flow described above is steady simple shear and the shear rate(also called the strain rate) is defined as the rate of change of strain: ˙ dγ d δL = u [1.20] γ= = dt dt h hThis definition only applies to streamline (laminar) flow betweenparallel plates. It is directly applicable to sliding plate viscometerdescribed by Dealy and Giacomin (1988). The definition must beadjusted to account for curved lines such as those found in tube androtational viscometers; however, the idea of "maximum speed dividedby gap size" can be useful in estimating shear rates found in particularapplications like brush coating. This idea is explored in more detail inSec. 1.9.
- 24. 14 Chapter 1. Introduction to Rheology AREA FORCE u VELOCITY PROFILE h x 2 x 1 u=0 Figure 1.9. Velocity profile between parallel plates. Rheological testing to determine steady shear behavior is conductedunder laminar flow conditions. In turbulent flow, little information isgenerated that can be used to determine material properties. Also, tobe meaningful, data must be collected over the shear rate rangeappropriate for the problem in question which may vary widely inindustrial processes (Table 1.1): Sedimentation of particles may involvevery low shear rates, spray drying will involve high shear rates, andpipe flow of food will usually occur over a moderate shear rate range.Extrapolating experimental data over a broad range of shear rates isnot recommended because it may introduce significant errors whenevaluating rheological behavior. Material flow must be considered in three dimensions to completelydescribe the state of stress or strain. In steady, simple shear flow thecoordinate system may be oriented with the direction of flow so the stresstensor given by Eq. [1.5] reduces to σ11 σ12 0 [1.21] σij = σ21 σ22 0 0 0 σ33
- 25. 1.5.1 Time-Independent Material Functions 15Table 1.1. Shear Rates Typical of Familiar Materials and ProcessesSituation γ (1/s) ˙ ApplicationSedimentation of particles in 10-6 - 10-3 Medicines, paints, spices in a suspending liquid salad dressingLeveling due to surface ten- 10-2 - 10-1 Frosting, paints, printing inks sionDraining under gravity 10-1 - 101 Vats, small food containers, painting and coatingExtrusion 100 - 103 Snack and pet foods, tooth- paste, cereals, pasta, poly- mersCalendering 101 - 102 Dough SheetingPouring from a bottle 101 - 102 Foods, cosmetics, toiletriesChewing and swallowing 101 - 102 FoodsDip coating 101 - 102 Paints, confectioneryMixing and stirring 101 - 103 Food processingPipe flow 100 - 103 Food processing, blood flowRubbing 102 - 104 Topical application of creams and lotionsBrushing 103 - 104 Brush painting, lipstick, nail polishSpraying 103 - 105 Spray drying, spray painting, fuel atomizationHigh speed coating 104 - 106 PaperLubrication 103 - 107 Bearings, gasoline engines Simple shear flow is also called viscometric flow. It includes axialflow in a tube, rotational flow between concentric cylinders, rotationalflow between a cone and a plate, and torsional flow (also rotational)between parallel plates. In viscometric flow, three shear-rate-dependentmaterial functions, collectively called viscometric functions, are neededto completely establish the state of stress in a fluid. These may bedescribed as the viscosity function, η(γ), and the first and second normal ˙stress coefficients, Ψ1(γ) and Ψ2(γ) , defined mathematically as ˙ ˙ σ21 [1.22] η = f(γ) = ˙ ˙ γ σ11 − σ22 N1 [1.23] Ψ1 = f(γ) = ˙ = 2 (γ)2 ˙ (γ) ˙
- 26. 16 Chapter 1. Introduction to Rheology σ22 − σ33 N2 [1.24] Ψ2 = f(γ) = ˙ = 2 (γ)2 ˙ (γ) ˙The first (σ11 − σ22) and second (σ22 − σ33) normal stress differences areoften symbolically represented as N1 and N2 , respectively. N1 is alwayspositive and considered to be approximately 10 times greater than N2.Measurement of N2 is difficult; fortunately, the assumption that N2 = 0is usually satisfactory. The ratio of N1/σ12, known as the recoverable shear(or the recoverable elastic strain), has proven to be a useful parameterin modeling die swell phenomena in polymers (Tanner, 1988). Somedata on the N1 values of fluid foods have been published (see Appendix[6.6]). If a fluid is Newtonian, η(γ) is a constant (equal to the Newtonian ˙viscosity) and the first and second normal stress differences are zero.As γ approaches zero, elastic fluids tend to display Newtonian behavior. ˙Viscoelastic fluids simultaneously exhibit obvious fluid-like (viscous)and solid-like (elastic) behavior. Manifestations of this behavior, dueto a high elastic component, can be very strong and create difficultproblems in process engineering design. These problems are particu-larly prevalent in the plastic processing industries but also present inprocessing foods such as dough, particularly those containing largequantities of wheat protein. Fig. 1.10 illustrates several phenomena. During mixing or agitation,a viscoelastic fluid may climb the impeller shaft in a phenomenon knownas the Weissenberg effect (Fig. 1.10). This can be observed in homemixing of cake or chocolate brownie batter. When a Newtonian fluidemerges from a long, round tube into the air, the emerging jet willnormally contract; at low Reynolds numbers it may expand to a diameterwhich is 10 to 15% larger than the tube diameter. Normal stress dif-ferences present in a viscoelastic fluid, however, may cause jet expan-sions (called die swell) which are two or more times the diameter of thetube (Fig. 1.10). This behavior contributes to the challenge of designingextruder dies to produce the desired shape of many pet, snack, and cerealfoods. Melt fracture, a flow instability causing distorted extrudates, isalso a problem related to fluid viscoelasticity. In addition, highly elasticfluids may exhibit a tubeless siphon effect (Fig. 1.10).
- 27. 1.5.1 Time-Independent Material Functions 17 VISCOUS FLUID VISCOELASTIC FLUID WEISSENBERG EFFECT TUBELESS SIPHON JET SWELLFigure 1.10. Weissenberg effect (fluid climbing a rotating rod), tubeless siphon and jet swell of viscous (Newtonian) and viscoelastic fluids. The recoil phenomena (Fig. 1.11), where tensile forces in the fluidcause particles to move backward (snap back) when flow is stopped, mayalso be observed in viscoelastic fluids. Other important effects includedrag reduction, extrudate instability, and vortex inhibition. An excel-lent pictorial summary of the behavior of viscoelastic polymer solutionsin various flow situations has been prepared by Boger and Walters(1993). Normal stress data can be collected in steady shear flow using anumber of different techniques (Dealy, 1982): exit pressure differencesin capillary and slit flow, axial flow in an annulus, wall pressure inconcentric cylinder flow, axial thrust in cone and plate as well as parallelplate flow. In general, these methods have been developed for plasticmelts (and related polymeric materials) with the problems of the plasticindustries providing the main driving force for change.
- 28. 18 Chapter 1. Introduction to Rheology Cone and plate systems are most commonly used for obtainingprimary normal stress data and a number of commercial instrumentsare available to make these measurements. Obtaining accurate datafor food materials is complicated by various factors such as the presenceof a yield stress, time-dependent behavior and chemical reactionsoccurring during processing (e.g., hydration, protein denaturation, andstarch gelatinization). Rheogoniometer is a term sometimes used todescribe an instrument capable of measuring both normal and shearstresses. Detailed information on testing viscoelastic polymers can befound in numerous books: Bird et al. (1987), Barnes et al. (1989), Bogueand White (1970), Darby (1976), Macosko (1994), and Walters (1975). VISCOUS FLUID VISCOELASTIC FLUID START START STOP STOP RECOILFigure 1.11. Recoil phenomenon in viscous (Newtonian) and viscoelastic fluids. Viscometric functions have been very useful in understanding thebehavior of synthetic polymer solutions and melts (polyethylene, poly-propylene, polystyrene, etc.). From an industrial standpoint, the vis-cosity function is most important in studying fluid foods and much ofthe current work is applied to that area. To date, normal stress datafor foods have not been widely used in food process engineering. Thisis partly due to the fact that other factors often complicate the evaluationof the fluid dynamics present in various problems. In food extrusion,for example, flashing (vaporization) of water when the product exits the
- 29. 1.5.1 Time-Independent Material Functions 19die makes it difficult to predict the influence of normal stress differenceson extrudate expansion. Future research may create significantadvances in the use of normal stress data by the food industry.Mathematical Models for Inelastic Fluids. The elastic behavior ofmany fluid foods is small or can be neglected (materials such as doughare the exception) leaving the viscosity function as the main area ofinterest. This function involves shear stress and shear rate: the rela-tionship between the two is established from experimental data.Behavior is visualized as a plot of shear stress versus shear rate, andthe resulting curve is mathematically modeled using various functionalrelationships. The simplest type of substance to consider is the New-tonian fluid where shear stress is directly proportional to shear rate [forconvenience the subscript on σ21 will be dropped in the remainder of thetext when dealing exclusively with one dimensional flow]: σ = µγ ˙ [1.25]with µ being the constant of proportionality appropriate for a Newtonianfluid. Using units of N, m2, m, m/s for force, area, length and velocitygives viscosity as Pa s which is 1 poiseuille or 1000 centipoise (note: 1Pa s = 1000 cP = 1000 mPa s; 1 P = 100 cP). Dynamic viscosity andcoefficient of viscosity are synonyms for the term "viscosity" in referringto Newtonian fluids. The reciprocal of viscosity is called fluidity.Coefficient of viscosity and fluidity are infrequently used terms.Newtonian fluids may also be described in terms of their kinematicviscosity (ν) which is equal to the dynamic viscosity divided by density(µ/ρ). This is a common practice for non-food materials, particularlylubricating oils. Viscosity conversion factors are available in Appendix[6.1]. Newtonian fluids, by definition, have a straight line relationshipbetween the shear stress and the shear rate with a zero intercept. Allfluids which do not exhibit this behavior may be called non-Newtonian.Looking at typical Newtonian fluids on a rheogram (Fig. 1.12) revealsthat the slope of the line increases with increasing viscosity. Van Wazer et al. (1963) discussed the sensitivity of the eye in judgingviscosity of Newtonian liquids. It is difficult for the eye to distinguishdifferences in the range of 0.1 to 10 cP. Small differences in viscosityare clearly seen from approximately 100 to 10,000 cP: something at 800cP may look twice as thick as something at 400 cP. Above 100,000 cPit is difficult to make visual distinctions because the materials do not
- 30. 20 Chapter 1. Introduction to Rheology 3 2.5 40% fat cream, 6.9 cP Shear Stress, Pa 2 olive oil, 36.3 cP 1.5 castor oil, 231 cP 1 0.5 0 0 5 10 15 20 Shear Rate, 1/s Figure 1.12. Rheograms for typical Newtonian fluids.pour and appear, to the casual observer, as solids. As points of referencethe following represent typical Newtonian viscosities at room temper-ature: air, 0.01 cP; gasoline (petrol), 0.7 cP; water, 1 cP; mercury, 1.5cP; coffee cream or bicycle oil, 10 cP; vegetable oil, 100 cP; glycerol, 1000cP; glycerine, 1500 cP; honey, 10,000 cP; tar, 1,000,000 cP. Data formany Newtonian fluids at different temperatures are presented inAppendices [6.8], [6.9], and [6.10]. A general relationship to describe the behavior of non-Newtonianfluids is the Herschel-Bulkley model: σ = K(γ)n + σo ˙ [1.26]where K is the consistency coefficient, n is the flow behavior index, andσo is the yield stress. This model is appropriate for many fluid foods.Eq. [1.26] is very convenient because Newtonian, power law (shear-thinning when 0 < n < 1 or shear-thickening when 1 < n < ∞) and Bing-ham plastic behavior may be considered as special cases (Table 1.2, Fig.1.13). With the Newtonian and Bingham plastic models, K is commonlycalled the viscosity (µ) and plastic viscosity (µpl ), respectively. Shear-thinning and shear-thickening are also referred to as pseudoplastic anddilatent behavior, respectively; however, shear-thinning and
- 31. 1.5.1 Time-Independent Material Functions 21shear-thickening are the preferred terms. A typical example of ashear-thinning material is found in the flow behavior of a 1% aqueoussolution of carrageenan gum as demonstrated in Example Problem1.14.1. Shear-thickening is considered with a concentrated corn starchsolution in Example Problem 1.14.2.Table 1.2. Newtonian, Power Law and Bingham Plastic Fluids as Special Cases ofthe Herschel-Bulkley Model (Eq. [1.26]) Fluid K n σo Typical Examples Herschel-Bulkley >0 0<n<∞ >0 minced fish paste, raisin paste Newtonian >0 1 0 water, fruit juice, milk, honey, vegeta- ble oil shear-thinning >0 0<n<1 0 applesauce, banana (pseudoplastic) puree, orange juice concentrate shear-thickening >0 1<n<∞ 0 some types of (dilatent) honey, 40% raw corn starch solution Bingham plastic >0 1 >0 tooth paste, tomato paste An important characteristic of the Herschel-Bulkley and Binghamplastic materials is the presence of a yield stress (σo ) which representsa finite stress required to achieve flow. Below the yield stress a materialexhibits solid like characteristics: It stores energy at small strains anddoes not level out under the influence of gravity to form a flat surface.This characteristic is very important in process design and qualityassessment for materials such as butter, yogurt and cheese spread. Theyield stress is a practical, but idealized, concept that will receive addi-tional discussion in a later section (Sec. 1.6). Typical yield stress valuesmay be found in Appendix [6.7].
- 32. 22 Chapter 1. Introduction to Rheology Herschel-Bulkley Bingham Shear Stress, Pa Shear-Thinning Newtonian Shear-Thickening Shear Rate, 1/s Figure 1.13. Curves for typical time-independent fluids. Slope = Upper Region Slope = Shear Stress, Pa Middle Region Lower Region Shear Rate, 1/s Figure 1.14. Rheogram of idealized shear-thinning (pseudoplastic) behavior.
- 33. 1.5.1 Time-Independent Material Functions 23 Shear-thinning behavior is very common in fruit and vegetableproducts, polymer melts, as well as cosmetic and toiletry products(Appendices [6.11], [6.12], [6.13]). During flow, these materials mayexhibit three distinct regions (Fig. 1.14): a lower Newtonian regionwhere the apparent viscosity (ηo ), called the limiting viscosity at zeroshear rate, is constant with changing shear rates; a middle region wherethe apparent viscosity (η) is changing (decreasing for shear-thinningfluids) with shear rate and the power law equation is a suitable modelfor the phenomenon; and an upper Newtonian region where the slopeof the curve (η∞), called the limiting viscosity at infinite shear rate, isconstant with changing shear rates. The middle region is most oftenexamined when considering the performance of food processing equip-ment. The lower Newtonian region may be relevant in problemsinvolving low shear rates such as those related to the sedimentation offine particles in fluids. Values of ηo for some viscoelastic fluids are givenin Table 5.4. Numerous factors influence the selection of the rheological modelused to describe flow behavior of a particular fluid. Many models, inaddition to the power law, Bingham plastic and Herschel-Bulkleymodels, have been used to represent the flow behavior of non-Newtonianfluids. Some of them are summarized in Table 1.3. The Cross,Reiner-Philippoff, Van Wazer and Powell-Eyring models are useful inmodeling pseudoplastic behavior over low, middle and high shear rateranges. Some of the equations, such as the Modified Casson and theGeneralized Herschel-Bulkley, have proven useful in developingmathematical models to solve food process engineering problems (Ofoliet al., 1987) involving wide shear rate ranges. Additional rheologicalmodels have been summarized by Holdsworth (1993). The Casson equation has been adopted by the International Officeof Cocoa and Chocolate for interpreting chocolate flow behavior. TheCasson and Bingham plastic models are similar because they both havea yield stress. Each, however, will give different values of the fluidparameters depending on the data range used in the mathematicalanalysis. The most reliable value of a yield stress, when determinedfrom a mathematical intercept, is found using data taken at the lowestshear rates. This concept is demonstrated in Example Problem 1.14.3for milk chocolate.
- 34. 24 Chapter 1. Introduction to RheologyApparent Viscosity. Apparent viscosity has a precise definition. Itis, as noted in Eq. [1.22], shear stress divided by shear rate: ˙ σ [1.27] η = f(γ) = ˙ γWith Newtonian fluids, the apparent viscosity and the Newtonianviscosity (µ) are identical but η for a power law fluid is ˙ K(γ) ˙n [1.28] η = f(γ) = ˙ = K(γ)n − 1 ˙ γTable 1.3. Rheological Models to Describe the Behavior of Time-independent FluidsModel (Source) Equation*Casson (Casson, 1959) σ0.5 = (σo )0.5 + K1(γ)0.5 ˙Modified Casson (Mizrahi and Berk, ˙n σ0.5 = (σo )0.5 + K1(γ) 1 1972)Ellis (Ellis, 1927) n γ = K1σ + K2(σ) 1 ˙Generalized Herschel-Bulkley (Ofoli et n n σ 1 = (σo ) 1 + K1(γ) 2 ˙ n al., 1987)Vocadlo (Parzonka and Vocadlo, 1968) n1 σ = (σo ) ˙ 1/n1 + K1γPower Series (Whorlow, 1992) γ = K1σ + K2(σ)3 + K3(σ)5… ˙ σ = K1γ + K2(γ)3 + K3(γ)5… ˙ ˙ ˙Carreau (Carreau, 1968) (n − 1)/2 η = η∞ + (ηo − η∞) 1 + (K1γ)2 ˙ Cross (Cross, 1965) ηo − η∞ η = η∞ + 1 + K1(γ)n ˙Van Wazer (Van Wazer, 1963) ηo − η∞ η= n + η∞ 1 + K1γ + K2(γ) 1 ˙ ˙Powell-Eyring (Powell and Eyring, 1944) 1 σ = K1γ + sinh−1(K3γ) ˙ ˙ K2 Reiner-Philippoff (Philippoff, 1935) ηo − η∞ σ = η∞ + γ˙ 1 + ((σ)2/K1) * K1, K2, K3 and n1, n2 are arbitrary constants and power indices, respectively, determinedfrom experimental data.
- 35. 1.5.1 Time-Independent Material Functions 25Apparent viscosities for Bingham plastic and Herschel-Bulkley fluidsare determined in a like manner: K(γ) + σo ˙ σo [1.29] η = f(γ) = ˙ =K+ ˙ γ ˙ γ K(γ)n + σo ˙ σo [1.30] η = f(γ) = ˙ = K(γ)n − 1 + ˙ ˙ γ ˙ γη decreases with increasing shear rate in shear-thinning and Binghamplastic substances. In Herschel-Bulkley fluids, apparent viscosity willdecrease with higher shear rates when 0 < n < 1.0, but behave in theopposite manner when n > 1.0. Apparent viscosity is constant withNewtonian materials and increases with increasing shear rate inshear-thickening fluids (Fig. 1.15). Time-Independent Fluids Bingham Apparent Viscosity, Pa s 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 ameasure of mouthfeel of fluid foods: The human perception of thicknessis correlated to the apparent viscosity at approximately 60 s-1. Apparentviscosity can also be used to illustrate the axiom that taking single pointtests for determining the general behavior of non-Newtonian materialsmay cause serious problems. Some quality control instruments designedfor single point tests may produce confusing results. Consider, for
- 36. 26 Chapter 1. Introduction to Rheologyexample, the two Bingham plastic materials shown in Fig. 1.16. Thetwo curves intersect at 19.89 1/s and an instrument measuring theapparent viscosity at that shear rate, for each fluid, would give identicalresults: η = 1.65 Pa s. However, a simple examination of the materialwith the hands and eyes would show them to be quite different becausethe yield stress of one material is more than 4 times that of the othermaterial. Clearly, numerous data points (minimum of two for the powerlaw or Bingham plastic models) are required to evaluate the flowbehavior of non-Newtonian fluids. 100 Bingham Plastic Fluids 80 Shear Stress, Pa 60 40 20 Yield Stress = 25.7 Pa Yield Stress = 6.0 Pa Plastic Viscosity = .36 Pa s Plastic Viscosity = 1.35 Pa s 0 0 10 20 30 40 50 60 Shear Rate, 1/s Figure 1.16. Rheograms for two Bingham plastic fluids.Solution Viscosities. It is sometimes useful to determine the visco-sities of dilute synthetic or biopolymer solutions. When a polymer isdissolved in a solvent, there is a noticeable increase in the viscosity ofthe resulting solution. The viscosities of pure solvents and solutionscan be measured and various values calculated from the resulting data: ηsolution [1.31] relative viscosity = ηrel = ηsolvent specific viscosity = ηsp = ηrel − 1 [1.32]
- 37. 1.5.2 Time-Dependent Material Functions 27 ηsp [1.33] reduced viscosity = ηred = C ln ηrel [1.34] inherent viscosity = ηinh = C ηsp [1.35] intrinsic viscosity = ηint = C C →0where C is the mass concentration of the solution in units of g/dl org/100ml. Note that units of reduced, inherent, and intrinsic viscosityare reciprocal concentration (usually deciliters of solution per grams ofpolymer). The intrinsic viscosity has great practical value in molecularweight determinations of high polymers (Severs, 1962; Morton-Jones,1989; Grulke, 1994). This concept is based on the Mark-Houwinkrelation suggesting that the intrinsic viscosity of a dilute polymersolution is proportional to the average molecular weight of the soluteraised to a power in the range of 0.5 to 0.9. Values of the proportionalityconstant and the exponent are well known for many polymer-solventcombinations (Progelf and Throne, 1993; Rodriquez, 1982). Solutionviscosities are useful in understanding the behavior of some biopolymersincluding aqueous solutions of locust bean gum, guar gum, and car-boxymethylcellulose (Rao, 1986). The intrinsic viscosities of numerousprotein solutions have been summarized by Rha and Pradipasena(1986).1.5.2. Time-Dependent Material Functions Ideally, time-dependent materials are considered to be inelastic witha viscosity function which depends on time. The response of the sub-stance to stress is instantaneous and the time-dependent behavior isdue to changes in the structure of the material itself. In contrast, timeeffects found in viscoelastic materials arise because the response ofstress to applied strain is not instantaneous and not associated with astructural change in the material. Also, the time scale of thixotropy maybe quite different than the time scale associated with viscoelasticity:The most dramatic effects are usually observed in situations involvingshort process times. Note too, that real materials may be both time-dependent and viscoelastic!
- 38. 28 Chapter 1. Introduction to Rheology Time-Dependent Behavior Thixotropic Shear Stress, Pa Time-Independent Rheopectic Time at Constant Shear Rate, s Figure 1.17. Time-dependent behavior of fluids. Separate terminology has been developed to describe fluids withtime-dependent characteristics. Thixotropic and rheopectic materialsexhibit, respectively, decreasing and increasing shear stress (andapparent viscosity) over time at a fixed rate of shear (Fig. 1.17). In otherwords, thixotropy is time-dependent thinning and rheopexy is time-dependent thickening. Both phenomena may be irreversible, reversibleor partially reversible. There is general agreement that the term"thixotropy" refers to the time-dependent decrease in viscosity, due toshearing, and the subsequent recovery of viscosity when shearing isremoved (Mewis, 1979). Irreversible thixotropy, called rheomalaxis orrheodestruction, is common in food products and may be a factor inevaluating yield stress as well as the general flow behavior of a material.Anti-thixotropy and negative thixotropy are synonyms for rheopexy. Thixotropy in many fluid foods may be described in terms of thesol-gel transition phenomenon. This terminology could apply, forexample, to starch-thickened baby food or yogurt. After being man-ufactured, and placed in a container, these foods slowly develop a threedimensional network and may be described as gels. When subjected toshear (by standard rheological testing or mixing with a spoon), structureis broken down and the materials reach a minimum thickness where
- 39. 1.5.2 Time-Dependent Material Functions 29 Evidence of Thixotropy in Torque Decay Curves Shear Rate Rest Period 0 Stress Complete Recovery Partial Recovery No Recovery 0 time Figure 1.18. Thixotropic behavior observed in torque decay curves.they exist in the sol state. In foods that show reversibility, the networkis rebuilt and the gel state reobtained. Irreversible materials remainin the sol state. The range of thixotropic behavior is illustrated in Fig. 1.18. Sub-jected to a constant shear rate, the shear stress will decay over time.During the rest period the material may completely recover, partiallyrecover or not recover any of its original structure leading to a high,medium, or low torque response in the sample. Rotational viscometershave proven to be very useful in evaluating time-dependent fluidbehavior because (unlike tube viscometers) they easily allow materialsto be subjected to alternate periods of shear and rest. Step (or linear) changes in shear rate may also be carried outsequentially with the resulting shear stress observed between steps.Typical results are depicted in Fig. 1.19. Actual curve segments (suchas 1-2 and 3-4) depend on the relative contribution of structuralbreakdown and buildup in the substance. Plotting shear stress versusshear rate for the increasing and decreasing shear rate values can beused to generate hysteresis loops (a difference in the up and down curves)for the material. The area between the curves depends on the time-dependent nature of the substance: it is zero for a time-independent
- 40. 30 Chapter 1. Introduction to Rheologyfluid. This information may be valuable in comparing differentmaterials, but it is somewhat subjective because different step changeperiods may lead to different hysteresis loops. Similar information canbe generated by subjecting materials to step (or linear) changes in shearstress and observing the resulting changes in shear rates. Shear Rate Step Changes in Shear Rate Shear Stress 1 2 3 4 time Figure 1.19. Thixotropic behavior observed from step changes in shear rate. Torque decay data (like that given for a problem in mixer viscometrydescribed in Example Problem 3.8.22) may be used to model irreversiblethixotropy by adding a structural decay parameter to the Herschel-Bulkley model to account for breakdown (Tiu and Boger, 1974): σ = f(λ, γ) = λ(σo + K(γ)n ) ˙ ˙ [1.36]where λ, the structural parameter, is a function of time. λ = 1 before theonset of shearing and λ equals an equilibrium value (λe ) obtained aftercomplete breakdown from shearing. The decay of the structuralparameter with time may be assumed to obey a second order equation: dλ [1.37] = −k1(λ − λe )2 for λ > λe dt
- 41. 1.5.2 Time-Dependent Material Functions 31where k1 is a rate constant that is a function of shear rate. Then, theentire model is completely determined by five parameters: σo , K, n, k1(γ), ˙and λe . K, n and σo are determined under initial shearing conditionswhen λ = 1 and t = 0. In other words, they are determined from the initialshear stress in the material, observed at the beginning of a test, for eachshear rate considered. λ and λe are expressed in terms of the apparent viscosity (η = σ/γ) to ˙find k1. Equating the rheological model (Eq. [1.36]) to the definition ofapparent viscosity (which in this case is a function of both shear rateand the time-dependent apparent viscosity) yields an expression for λ: ηγ ˙ [1.38] λ= σo + K(γ) ˙nEq. [1.38] is valid for all values of λ including λe at ηe , the equilibriumvalue of the apparent viscosity. Differentiating λ with respect to time,at a constant shear rate, gives dλ dη γ ˙ [1.39] = n dt dt σo + K(γ) ˙Using the definition of dλ/dt , Eq. [1.37] and [1.39] may be combinedyielding dη γ ˙ [1.40] −k1(λ − λe )2 = ˙ n dt σo + K(γ) Considering the definition of λ given by Eq. [1.38], this may be rewrittenas ηγ ˙ ηe γ 2 dη ˙ γ ˙ [1.41] −k1 − = ˙ n σo + K(γ) dt σo + K(γ) ˙ n σ + K(γ)n ˙ o Simplification yields dη γ ˙ [1.42] = −k1 ˙ n (η − ηe ) 2 dt σo + K(γ) or
- 42. 32 Chapter 1. Introduction to Rheology dη [1.43] = −a1(η − ηe )2 dtwhere k1γ ˙ [1.44] a1 = σo + K(γ)n ˙Integrating Eq. [1.43] gives η [1.45] ⌠ (η − η )−2dη = ⌠ −a dt t ⌡ηo e ⌡0 1so 1 1 [1.46] = +a t η − ηe ηo − ηe 1where ηo is the initial value of the apparent viscosity calculated fromthe initial (t = 0 and λ = 1) shear stress and shear rate. Using Eq. [1.46], a plot 1/(η − ηe ) versus t , at a particular shear rate,is made to obtain a1. This is done at numerous shear rates and theresulting information is used to determine the relation between a1 andγ and, from that, the relation between k1 and γ. This is the final infor-˙ ˙mation required to completely specify the mathematical model given byEq. [1.36] and [1.37]. The above approach has been used to describe the behavior ofmayonnaise (Tiu and Boger, 1974), baby food (Ford and Steffe, 1986),and buttermilk (Butler and McNulty, 1995). More complex modelswhich include terms for the recovery of structure are also available(Cheng, 1973; Ferguson and Kemblowski, 1991). Numerous rheologicalmodels to describe time-dependent behavior have been summarized byHoldsworth (1993).1.5.3. Modeling Rheological Behavior of Fluids Modeling provides a means of representing a large quantity ofrheological data in terms of a simple mathematical expression. Rheo-grams, summarized in terms of the Herschel-Bulkley equation (Eq.[1.26]), represent one example of modeling. In this section we willexpand the idea to include temperature and concentration (or moisturecontent) effects into single empirical expressions. Many forms of theequations are possible and one master model, suitable for all situations,
- 43. 1.5.3 Modeling Rheological Behavior of Fluids 33does not exist. The equations covered here are acceptable for a largenumber of practical problems involving homogeneous materials whichdo not experience a phase change over the range of conditions underconsideration. The influence of temperature on the viscosity for Newtonian fluidscan be expressed in terms of an Arrhenius type equation involving theabsolute temperature (T ), the universal gas constant (R ), and the energyof activation for viscosity (Ea ): Ea [1.47] µ = f(T) = A exp RT Ea and A are determined from experimental data. Higher Ea valuesindicate a more rapid change in viscosity with temperature. The energyof activation for honey is evaluated in Example Problem 1.14.4. Considering an unknown viscosity (µ) at any temperature (T ) and areference viscosity (µr ) at a reference temperature (Tr ), the constant (A )may be eliminated from Eq. [1.47] and the resulting equation writtenin logarithmic form: µ Ea 1 1 [1.48] ln = − µr R T Tr In addition to modeling the viscosity of Newtonian fluids, an Arrheniusrelationship can be used to model the influence of temperature onapparent viscosity in power law fluids. Considering a constant shearrate, with the assumption that temperature has a negligible influenceon the flow behavior index, yields η Ea 1 1 [1.49] ln = − ηr R T Tr or η Ea 1 1 [1.50] = exp − ηr R T Tr Eq. [1.50] can be used to find η at any temperature (T ) from appropriatereference values (ηr , Tr ). Activation energies and reference viscositiesfor a number of fluid foods are summarized in Appendix [6.14].

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