Reological%20Methods%20in%20Food%20Processing.pdf

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RHEOLOGICAL MET HODS IN

FOOD PROCESS ENGINEERING

Second Edition

James F. Steffe, Ph.D., P.E.

Professor of Food Process Engin eering Dept. of Food Science and H uman Nu tri t ion

Dept. of Agricul tura l Engineering Michigan State Universi ty

Free m an Press2807 Still Valley Dr.

East Lansing, MI 48823USA


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Prof. James F. Steffe209 Farrall HallMichigan State University East Lansing, MI 48824-1323USA

Phone: 517-353-4544FAX: 517-432-2892E-mail: [email protected]: 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

Freem an Press2807 Still Valley Dr.

East Lansing, MI 48823USA


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Table of Contents

Preface ix

Chapter 1. Introduction to Rheology 11.1. Overview ...................................................................................... 11.2. Rheological Instruments for Fluids .......................................... 21.3. Stress and Strain .......................................................................... 41.4. Solid Behavior ............................................................................. 81.5. Fluid Behavior in Steady Shear Flow ....................................... 13

1.5.1. Time-Independent Material Functions ............................ 131.5.2. Time-Dependent Material Functions ............................... 271.5.3. Modeling Rheological Behavior of Fluids ....................... 32

1.6. Yield Stress Phenomena ............................................................. 351.7. Extensional Flow ......................................................................... 391.8. Viscoelastic Material Functions ................................................ 471.9. Attacking Problems in Rheological Testing ............................ 491.10. Interfacial Rheology ................................................................. 531.11. Electrorheology ......................................................................... 551.12. Viscometers for Process Control and Monitoring ................ 571.13. Empirical Measurement Methods for Foods ........................ 631.14. Example Problems .................................................................... 77

1.14.1. Carrageenan Gum Solution ............................................. 771.14.2. Concentrated Corn Starch Solution ................................ 791.14.3. Milk Chocolate .................................................................. 811.14.4. Falling Ball Viscometer for Honey ................................. 821.14.5. Orange Juice Concentrate ................................................ 861.14.6. Influence of the Yield Stress in Coating Food ............... 91

Chapter 2. Tube Viscometry 942.1. Introduction ................................................................................. 94

2.2. Rabinowitsch-Mooney Equation .............................................. 972.3. Laminar Flow Velocity Profiles ................................................ 1032.4. Laminar Flow Criteria ................................................................ 1072.5. Data Corrections ......................................................................... 1102.6. Yield Stress Evaluation .............................................................. 1212.7. Jet Expansion ............................................................................... 1212.8. Slit Viscometry ............................................................................ 1222.9. Glass Capillary (U-Tube) Viscometers .................................... 1252.10. Pipeline Design Calculations .................................................. 1282.11. Velocity Profiles In Turbulent Flow ....................................... 1382.12. Example Problems .................................................................... 141

2.12.1. Conservation of Momentum Equations ........................ 1412.12.2. Capillary Viscometry - Soy Dough ................................ 1432.12.3. Tube Viscometry - 1.5% CMC ......................................... 1462.12.4. Casson Model: Flow Rate Equation ............................... 1492.12.5. Slit Viscometry - Corn Syrup .......................................... 1502.12.6. Friction Losses in Pumping ............................................. 1522.12.7. Turbulent Flow - Newtonian Fluid ................................ 1552.12.8. Turbulent Flow - Power Law Fluid ................................ 156

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Chapter 3. Rotational Viscometry 1583.1. Introduction ................................................................................. 158

3.2. Concentric Cylinder Viscometry .............................................. 1583.2.1. Derivation of the Basic Equation ...................................... 1583.2.2. Shear Rate Calculations ...................................................... 1633.2.3. Finite Bob in an Infinite Cup ............................................. 168

3.3. Cone and Plate Viscometry ....................................................... 1693.4. Parallel Plate Viscometry (Torsional Flow) ............................ 1723.5. Corrections: Concentric Cylinder ............................................. 1743.6. Corrections: Cone and Plate, and Parallel Plate ..................... 1823.7. Mixer Viscometry ....................................................................... 185

3.7.1. Mixer Viscometry: Power Law Fluids ............................. 1903.7.2. Mixer Viscometry: Bingham Plastic Fluids ..................... 1993.7.3. Yield Stress Calculation: Vane Method ........................... 2003.7.4. Investigating Rheomalaxis ................................................ 2083.7.5. Defining An Effective Viscosity ........................................ 210

3.8. Example Problems ...................................................................... 210

3.8.1. Bob Speed for a Bingham Plastic Fluid ............................ 2103.8.2. Simple Shear in Power Law Fluids .................................. 2123.8.3. Newtonian Fluid in a Concentric Cylinder ..................... 2133.8.4. Representative (Average) Shear Rate .............................. 2143.8.5. Concentric Cylinder Viscometer: Power Law Fluid ...... 2163.8.6. Concentric Cylinder Data - Tomato Ketchup ................. 2183.8.7. Infinite Cup - Single Point Test ......................................... 2213.8.8. Infinite Cup Approximation - Power Law Fluid ........... 2213.8.9. Infinite Cup - Salad Dressing ............................................ 2233.8.10. Infinite Cup - Yield Stress Materials .............................. 2253.8.11. Cone and Plate - Speed and Torque Range ................... 2263.8.12. Cone and Plate - Salad Dressing ..................................... 2273.8.13. Parallel Plate - Methylcellulose Solution ....................... 2293.8.14. End Effect Calculation for a Cylindrical Bob ................ 2313.8.15. Bob Angle for a Mooney-Couette Viscometer .............. 233

3.8.16. Viscous Heating ................................................................ 2353.8.17. Cavitation in Concentric Cylinder Systems .................. 2363.8.18. Mixer Viscometry .............................................................. 2373.8.19. Vane Method - Sizing the Viscometer ........................... 2433.8.20. Vane Method to Find Yield Stresses .............................. 2443.8.21. Vane Rotation in Yield Stress Calculation .................... 2473.8.22. Rheomalaxis from Mixer Viscometer Data ................... 250

Chapter 4. Extensional Flow 2554.1. Introduction ................................................................................. 2554.2. Uniaxial Extension ...................................................................... 2554.3. Biaxial Extension ......................................................................... 2584.4. Flow Through a Converging Die .............................................. 263

4.4.1. Cogswell’s Equations ......................................................... 2644.4.2. Gibson’s Equations ............................................................. 2684.4.3. Empirical Method ............................................................... 271

4.5. Opposing Jets .............................................................................. 2724.6. Spinning ....................................................................................... 2744.7. Tubeless Siphon (Fano Flow) .................................................... 276

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4.8. Steady Shear Properties from Squeezing Flow Data ............. 2764.8.1. Lubricated Squeezing Flow ............................................... 277

4.8.2. Nonlubricated Squeezing Flow ........................................ 2794.9. Example Problems ...................................................................... 2834.9.1. Biaxial Extension of Processed Cheese Spread ............... 2834.9.2. Biaxial Extension of Butter ................................................ 2864.9.3. 45 Converging Die, Cogswell’s Method ........................ 2874.9.4. 45 Converging Die, Gibson’s Method ............................ 2894.9.5. Lubricated Squeezing Flow of Peanut Butter ................. 291

Chapter 5. Viscoelasticity 2945.1. Introduction ................................................................................. 2945.2. Transient Tests for Viscoelasticity ............................................ 297

5.2.1. Mechanical Analogues ....................................................... 2985.2.2. Step Strain (Stress Relaxation) .......................................... 2995.2.3. Creep and Recovery ........................................................... 3045.2.4. Start-Up Flow (Stress Overshoot) ..................................... 310

5.3. Oscillatory Testing ...................................................................... 3125.4. Typical Oscillatory Data ............................................................ 3245.5. Deborah Number ........................................................................ 3325.6. Experimental Difficulties in Oscillatory Testing of Food ..... 3365.7. Viscometric and Linear Viscoelastic Functions ...................... 3385.8. Example Problems ...................................................................... 341

5.8.1. Generalized Maxwell Model of Stress Relaxation ........ 3415.8.2. Linearized Stress Relaxation ............................................. 3425.8.3. Analysis of Creep Compliance Data ................................ 3435.8.4. Plotting Oscillatory Data ................................................... 348

6. Appendices 3506.1. Conversion Factors and SI Prefixes .......................................... 3506.2. Greek Alphabet ........................................................................... 3516.3. Mathematics: Roots, Powers, and Logarithms ....................... 352

6.4. Linear Regression Analysis of Two Variables ........................ 3536.5. Hookean Properties .................................................................... 3576.6. Steady Shear and Normal Stress Difference ........................... 3586.7. Yield Stress of Fluid Foods ........................................................ 3596.8. Newtonian Fluids ....................................................................... 3616.9. Dairy, Fish and Meat Products ................................................. 3666.10. Oils and Miscellaneous Products ........................................... 3676.11. Fruit and Vegetable Products ................................................. 3686.12. Polymer Melts ........................................................................... 3716.13. Cosmetic and Toiletry Products ............................................. 3726.14. Energy of Activation for Flow for Fluid Foods .................... 3746.15. Extensional Viscosities of Newtonian Fluids .............. .......... 3756.16. Extensional Viscosities of Non-Newtonian Fluids .............. 3766.17. Fanning Friction Factors: Bingham Plastics .......................... 3776.18. Fanning Friction Factors: Power Law Fluids ........................ 3786.19. Creep (Burgers Model) of Salad Dressing ............................. 3796.20. Oscillatory Data for Butter ...................................................... 3806.21. Oscillatory Data Iota-Carrageenan Gel ................................. 3816.22. Storage and Loss Moduli of Fluid Foods .............................. 382

°

°

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Nomenclature ......................................................................................... 385

Bibliography ........................................................................................... 393

Index ......................................................................................................... 412

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Preface

G rowth and development of th is work sprang from the need to

provide educat ional ma teria l for food engineers an d food scientist s. The

first edition was conceived as a textbook and the work continues to be

used in gra duate level courses a t var ious universi t ies. I t s greatest

appeal, however, was to individuals solving practical day-to-day prob-

lems. Hence, th e second edition, a significan tly expa nded an d revised

version of the or ig inal work, is a imed pr imar i ly a t the rheologica l

practit ioner (par ticularly th e industria l practit ioner) seeking a broad

understa nding of the subject m at ter . The overall goal of the text is to

present the information needed to answer three questions when facing

problems in food rheology: 1. Wha t properties should be mea sured? 2.

What type and degree of deformation should be induced in the mea-surement? 3. How should experimental data be analyzed to generate

practical informa tion? Although the ma in focus of the book is food,

scientists a nd engineers in other fields will find the w ork a convenient

reference for st an dar d rheological methods and typical da ta .

O verall, the work presents the theory of rheological test ing and

provides the an a lytica l tools needed to determ ine rheological properties

from experiment a l dat a . Methods appropriat e for common food indust ry

applicat ions ar e presented. All stand ar d measurement techniques for

fluid an d semi-solid foods a re included. Selected methods for solids ar e

also presented. Results from numerous fields, part icularly polymer

rheology, are used to ad dress the flow beha vior of food. Mat hemat ical

relat ionships, derived fr om simple force bala nces, provide a funda-ment a l view of rheological testing . Only a ba ckground in basic ca lculus

and elementary sta t ist ics (mainly regression analysis) is needed to

understa nd them at erial. The text conta ins numerous practical exam ple

problems, involving a ctual experimental da ta , to enha nce comprehen-

sion and th e execution of concepts presented. This feat ure ma kes the

work convenient for self-stu dy.

S pecific explanations of key topics in rheology are presented in

Chapter 1: basic concepts of stress and strain; elast ic solids showing

Hookean and non-Hookean behavior; viscometric functions including

norma l stress differences; modeling fluid beha vior as a fun ction of shear

rate, temperature, and composit ion; yield stress phenomena, exten-

siona l flow; a nd viscoelastic beha vior. Efficient methods of at ta ckingproblems and typical instruments used to measure fluid properties are

discussed along with an examination of problems involving interfacial

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rheology, electrorheology, a nd on-line viscometr y for contr ol and mon-

itoring of food processing opera tions. Comm on meth ods and empiricalinstruments utilized in the food industry are also introduced: Texture

Profile Analysis, Compression-Extrusion Cell, Warner-Bratzler Shear

Cell, Bostwick Consistometer , Adams Consistometer , Amylograph,

Far inograph, Mixograph, Extensigraph, Alveograph, Kra mer Shear

Cell, Brookfield disks and T-bars, Cone Penetrometer, Hoeppler Vis-

cometer, Zhan Viscometer, Brabender-FMC Consistometer.

T he basic equa tions of tube (or capillary) viscometry, such as the

Rabinowitsch-Mooney equation, are derived and applied in Chapter 2.

La mina r flow criteria a nd velocity profiles a re evalua ted along w ith dat a

correction methods for many sources of error: kinetic energy losses, end

effects, slip (wa ll effects), viscous heat ing, a nd hole pressure. Tech-

niques for glass capillary and slit viscometers are considered in detail.A section on pipeline design calculat ions ha s been included to facilitat e

the const ruction of large scale tu be viscometers a nd t he design of fluid

pumping syst ems.

A general format, analogous to that used in Chapter 2, is continued

in Ch a pter 3 to provide cont inuity in subject ma tt er development. The

ma in foci of the cha pter center ar ound the t heoretical principles and

experiment a l procedures relat ed to th ree tr a ditiona l types of rota tiona l

viscometers: concentric cylinder, cone and plate, and parallel plate.

Mathemat ica l a nalyses of da ta are discussed in deta i l . Errors due to

end effects, viscous heating, slip, Taylor vortices, cavitation, and cone

trunca tions are investiga ted. Numerous methods in mixer viscometry,

techniques having significant potential to solve many food rheologyproblems, are also presented: slope and matching viscosity methods to

evaluat e avera ge shear ra te, determinat ion of power law an d Bingha m

plastic fluid properties. The van e method of yield stress evaluat ion,

using both the slope a nd single point m ethods, along with a consider-

ation of vane rotation during testing, is explored in detail.

T he experimental methods to determine extensiona l viscosity ar e

expla ined in Ch a pter 4. Techniq ues presented involve uniaxia l exten-

sion between rotat ing clamps, biaxial extensional flow achieved by

squeezing m at erial betw een lubricat ed para llel plates, opposing jets,

spinning, and tubeless siphon (Fano) f low. Rela t ed procedures,

involving lubricat ed a nd nonlubricated sq ueezing, to determine shear

flow beha vior a re a lso presented. C a lculat ing extensional viscosity fromflows through tapered convergences and f la t entry dies is g iven a

thorough examina tion.

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E ssent ia l concepts in viscoelast icity and standa rd methods of

investigating t he phenomenon a re investigat ed in C ha pter 5. The fullscope of viscoelastic material functions determined in transient and

oscillatory testing ar e discussed. Mecha nical a na logues of rh eological

behavior a re given as a means of ana lyzing creep an d stress relaxat ion

data . Theoret ica l aspects o f osci lla tory test ing , typica l da ta , a nd a

discussion of the various modes of operating commercial instruments

-stra in, frequency, t ime, an d t empera ture sw eep modes- ar e presented.

The Debora h num ber concept, a nd h ow it can be used to dist inguish

liquid from solid-like beha vior, is int roduced. St ar t-up flow (stress

overshoot) and t he relat ionship betw een stea dy shea r a nd oscillatory

properties ar e also discussed. Conversion factors, mat hemat ical rela-

tionships, linear regression analysis, and typical rheological data for

food a s w ell as cosmetics an d polymers a re provided in the Appendices.

Nomenclature is conveniently summarized at the end of the text and a

large bibliography is furnished to direct readers to addit ional infor-

mat ion.

J .F. S teffe

J une, 1996

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Dedicat ion

To Susan, J ustinn, a nd Da na .

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Chapter 1. Introduct ion to Rheology

1.1. Overview

The first use of the word "r heology" is credited to Eu gene C. Bingh a m

(circa 1928) wh o also described th e mott o of t he subject a s

("pa nta rhei," from the w orks of H eraclitus, a pre-Socra tic Gr eek phi-

losopher active about 500 B.C.) meaning "everything flows" (Reiner,

1964). Rh eology is now well esta blished a s the science of th e deforma tion

and f low of mat ter : I t is the study of the manner in w hich ma ter ia ls

respond t o applied st ress or st ra in . All mat er ia ls have r heologica l

propert ies and th e area is relevan t in ma ny fields of study: geology an d

mining (Cristescu, 1989), concrete technology (Tattersall and Banfill,

1983), soil mechanics (Haghighi et al., 1987; Vyalov, 1986), plastics

processing (Dealy an d Wissburn, 1990), polymers an d composites

(Neilsen a nd L a ndel, 1994; Yan ovsky, 1993), tr ibology (study of lubri-

cation, fr ict ion and wea r), paint flow an d pigment dispersion (P at ton,

1964), blood (Dintenfa ss, 1985), bioengineering (Ska lak an d Chien,

1987), interfacial rheology (Edwards et al., 1991), structural materials

(Ca llister, 1991), electr orheology (Block a nd Kelly, 1988), psychor-

heology (Dra ke, 1987), cosmetics a nd toiletries (La ba, 1993b), an d

pressure sensitive adhesion (Saunders et al., 1992). The focus of this

work is food where understanding rheology is crit ical in optimizing

product development efforts , processing met hodology a nd fina l product

qua lity . To the extent possible, st an dar d nomenclature (Dealy, 1994)has been used in the text.

One can th ink of food rheology as t he ma ter ia l science of food. There

are numerous areas where rheologica l da ta are needed in the food

industry :

a. Process engineering calculations involving a wide range of equip-

ment such as pipelines, pumps, extruders, mixers, coaters, heat

excha ngers, homogenizers, calenders, a nd on-line viscometers ;

b. Determining ingredient functionality in product development;

c. Intermediate or final product quality control;

d. Sh elf life test ing;

e. Evaluation of food texture by correlation to sensory data;

f. Ana lysis of rheological equa tions of sta te or constitu tive equa tions.

Many of the unique rheological properties of various foods have been

sum ma rized in books by Ra o a nd S teffe (1992), an d Weipert et a l. (1993).

παντα ρει


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2 Chapter 1. Introduction to Rheology

Funda menta l rheological properties a re independent of th e instru-

ment on which they a re measur ed so different inst ruments w ill yield

the same results. This is an ideal concept and different instruments

rarely yield identical results; however, the idea is one which dist in-

guishes tr ue rheological ma teria l propert ies from subjective (empirical

and genera l ly inst rument dependent , though f requent ly useful)

ma terial cha ra cterizations. Exam ples of instrum ents giving subjective

results in clude th e followin g (B ourne, 1982): Fa rinogra ph, Mixogra ph,

Extensograph, Viscoamlyograph, and the Bostwick Consistometer .

Empir ica l test ing devices and methods, including Texture P rofi le

Ana lysis, a re considered in more deta il in S ec. 1.13.

1.2. Rheological Instruments for Fluids

Common instruments, capa ble of measuring fundam ental r heolog-

ical properties of fluid and semi-solid foods, may be placed into two

genera l cat egories (Fig. 1.1): rotat iona l type and t ube type. Most ar e

commer cially ava ilable, others (mixer a nd pipe viscometers ) a re easily

fabricated. Costs var y tremendously from th e inexpensive glass capil-

lary viscometer to a very expensive rotat ional instrument capable of

measur ing dyna mic proper t ies and normal st ress differences. Solid

foods a re often test ed in compression (betw een para llel plat es), tension,

or torsion. I nst rum ents which mea sure rheologica l propert ies a re called

rheometers. Viscometer is a more limiting t erm referring to devicestha t only measur e viscosity.

Rotat iona l instruments ma y be opera ted in the steady sh ear (con-

stant angular velocity) or oscillatory (dynamic) mode. Some rotational

instruments funct ion in the controlled st ress mode faci l i t a t ing the

collection of creep da ta , the ana lysis of mat eria ls at very low shear ra tes,

an d the investigat ion of yield stresses. This informa tion is needed to

understand the internal st ructure of ma ter ia ls . The controlled ra te

mode is most useful in obtaining data required in process engineering

calculat ions. Mechan ical differences betw een cont rolled rat e a nd con-

trolled stress instr uments a re discussed in Sec. 3.7.3. Rotat iona l sys-

tems are genera lly used to investiga te tim e-dependent beha vior because

tube systems only a llow one pass of the ma teria l th rough the appar a tus .

A detailed discussion of oscillatory testing, the primary method of

determ ining th e viscoelas tic beha vior of food, is provided in Cha pter 5.


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1.2 Rheological Instruments for Fluids 3

Figure 1.1. Common rheological instruments divided into two major categories:rotational and tube type.

There a re advan tages and d isadvan tages a ssocia ted w i th each

instrument . Gr avity opera ted glass capil lar ies, such a s the Ca nnon-

Fenske type shown in Fig. 1.1, a re only suitable for Newtonian fluids

because the shear ra te varies during discha rge. Cone and plate systems

are limited to moderate shear rates but calculations (for small cone

a ngles) a re simple. P ipe a nd mixer viscometers can ha ndle much larg er

par t icles tha n cone and pla t e , or para l le l pla te , devices. P roblems

associat ed with slip and degrada tion in structura lly sensit ive ma terials

are m inimized w ith mixer viscometers. High pressure capil lar ies

operate a t h igh shear ra tes but genera l ly involve a signif icant end

pressure correction. Pipe viscometers can be constructed to withstand

the rigors of the production or pilot pla nt environment .

Rotational Type

Parallel Plate Cone and Plate

Concentric Cylinder

Mixer

Tube Type

PipeGlass Capillary High Pressure Capillary


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All the inst rum ents presented in Fig. 1.1 a re "v olume loaded" devices

with conta iner dimensions tha t are cr i t ica l in the determinat ion of

rheological properties. Another comm on type of inst rum ent, known a s

a vibrat iona l viscometer, uses the principle of " surface load ing" w here

the sur face of a n imm ersed probe (usua lly a sphere or a rod) genera tes

a shear wa ve which dissipates in the surrounding medium. A large

enough cont a iner is used so shea r forces do not reach the wa ll a nd reflect

back to the probe. Measurements depend only on ability of the sur-

rounding fluid to da mp probe vibra tion. The dam ping cha ra cteristic of

a fluid is a function of th e product of the fluid viscosity (of Newt onian

fluids) an d the density. Vibra tional viscometers a re popular as in-line

inst rum ents for process cont rol system s (see Sec. 1.12). It is difficult touse these units to evaluate fundamental rheological properties of non-

Newt onian fluids (Ferry, 1977), but th e subjective results obta ined often

correlat e well with importa nt food qua lity at tr ibutes. The coagula tion

time and curd firmness of renneted milk, for example, have been suc-

cessfully investigated using a vibrational viscometer (Sharma et al. ,

1989).

Inst ruments used to evalua te the extensiona l viscosity of ma terials

are discussed in Cha pter 4. P ulling or st retching a sample between

toothed gears, sucking mat er ia l in to opposing je ts , spinning, or

exploit ing t he open siphon phenomenon can genera te da ta for calcu-

lat ing tensile extensiona l viscosity. Informat ion to determine biaxial

extensiona l viscosity is provided by compressing sa mples between

lubricat ed par allel plat es. Shea r viscosity can a lso be evalua ted from

unlubr ica ted squeezing f low between para l lel pla tes. A number of

meth ods a re a va ilable to calculat e a n avera ge extensional viscosity from

data describing flow through a convergence into a capillary die or slit.

1.3. Stress and Strain

Sin cer heology is the stu dy of th e deforma tion ofm a tt er, it is essentia l

to have a good understa nding of stress an d stra in. Consider a recta n-

gula r ba r t ha t, due to a t ensile force, is slight ly elonga ted (Fig. 1.2). The

ini t ia l len g th of t h e b a r is a n d t h e elon g a t ed len g t h i s w h er e

with representing the increase in length. This deforma tion

ma y be thought of in terms of Cauchy st ra in (also called engineering

strain):

Lo L

L = Lo + δ L δ L


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1.3 Stress and Strain 5

[1.1]

or Hencky st ra in (a lso ca l led t rue st ra in) which is determined by

eva lu a t in g a n int eg r a l f rom t o :

[1.2]

Figure 1.2. Linear extension of a rectangular bar.

Cauchy and Hencky s t r a ins a re both zero when the ma ter ia l i s

unstra ined and approximat ely equal a t small st ra ins. The choice of

wh ich str a in measu re to use is lar gely a ma tt er of convenience a nd one

can be calculated from the other:

[1.3]

is preferred for ca lculat ing stra in resulting from a la rge deforma tion.

Another t ype of deforma tion commonly found in r heology is simple

shear. The idea can be illustrat ed with a recta ngular bar (Fig. 1.3) of

height . The lower surfa ce is sta tiona ry and the upper plat e is linear lydisplaced by an a mount equal to . E ach element is subject to the sam e

level of deform a tion so th e size of th e element is not releva nt . The an gle

of shear, , ma y be calculat ed as

εc =

δ L Lo

= L − Lo

Lo=

L

Lo− 1

Lo L

εh = ⌠ ⌡ L

o

L dL

L = ln( L/ L

o)

LL0

εh = ln(1 + ε

c)

εh

h δ L

γ


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[1.4]

With sma ll deforma tions, the an gle of shear (in radia ns) is equa l to th e

s hea r s t ra i n (a l so s ym boli zed b y ), .

Figure 1.3. Shear deformation of a rectangular bar.

Figure 1.4. Typical stresses on a material element.

Stress, defined as a force per unit area and usually expressed inP a sca l (N/m 2), may be tensile, compressive, or shear . Nine separ at e

qua ntit ies a re required to completely describe the sta te of stress in a

ma terial. A sma ll element (Fig. 1.4) ma y be considered in t erms of

tan(γ) = δ L

h

γ tan γ = γ

h

L

11

21

23

22

33

1

2

3

x

x

x


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1.3 Stress and Strain 7

Car tes ian coord ina tes ( ). S t ress i s indica ted a s where the f ir s t

subscript refers t o the orienta tion of th e face upon w hich the force a cts

a nd t he second subs cript refers to th e direction of the force. Therefore,

i s a norma l s t ress ac t ing in the p lane perpendicu la r to in the

direct ion of and is a shear st ress act ing in the plane perpendicular

to in the direction of . Normal stresses ar e considered posit ive when

they act outwa rd (acting to creat e a t ensile stress) an d negat ive when

they act inward (acting to create a compressive stress).

Stress components may be summarized as a stress tensor writ ten

in the form of a ma trix:

[1.5]

A related t ensor for stra in can a lso be expressed in ma trix form. Ba sic

laws of mechanics, consider ing t he moment about t he a xis under

equilibrium condit ions, can be used to prove tha t t he str ess ma trix is

symmetrical:

[1.6]

so

[1.7]

[1.8]

[1.9]

mea ning th ere are only six independent components in th e stress tensor

represented by Eq. [1.5].

Eq uat ions that sh ow the relat ionship between stress and str ain ar e

either ca lled rheological equat ions of sta te or const itut ive equa tions. In

complex materials these equations may include other variables such as

time, tempera ture, a nd pressure. A modulus is defined as t he rat io of

stress to stra in wh ile a compliance is defined a s the ra t io of stra in to

str ess. The word rheogram refers to a gra ph of a rheological rela tionship.

x1, x2, x3 σij

σ11 x1 x1 σ23

x2 x3

σij =

σ11 σ12 σ13σ21 σ22 σ23σ31 σ32 σ33

σij = σ ji

σ12 = σ21

σ31 = σ13

σ32 = σ23


18/427


1.4. Solid Behavior

When force is applied to a solid ma terial a nd th e result ing stress

versus stra in curve is a st ra ight line thr ough the origin, the mat erial is

obeying Hooke’s law . The relat ionship may be sta ted for sh ear st ress

and shear st ra in as

[1.10]

where is the shear modulus. Hookean ma ter ia ls do not f low a nd are

l inear ly e last ic . St ress remains constant unt i l the st ra in is removed

and t he ma ter ia l r e turns to i t s o r ig ina l shape . Somet imes sha pe

recovery, th ough complete, is delayed by s ome at omistic process. This

time-dependent, or delayed, elast ic behavior is known as an elasticity.

Hooke’s la w ca n be used to describe the beha vior of ma ny s olids (steel,egg shell, dry past a, h ar d candy, etc.) when subjected to small st ra ins,

typically less tha n 0.01. La rge stra ins often produce brit tle fractur e or

non-linear beha vior.

The behavior of a Hookean solid may be investigated by st udying

the unia xial compression of a cylindrical sa mple (Fig. 1.5). If a ma teria l

is compressed so th a t it experiences a cha nge in lengt h an d ra dius, then

the norma l stress and st ra in may be calculated:

[1.11]

[1.12]

Figure 1.5 Uniaxial compression of a cylindrical sample.

σ12 = Gγ

G

σ22 =F

A =

F

π( Ro)2

εc = δhho

R

hh o

R R

h

Initial Shape Compressed Shape

oo


19/427

1.4 Solid Behavior 9

This informa tion can be used to determin e Young’s modulus ( ), also

called the modulus of elast icity, of the sa mple:

[1.13]

I f the deformat ions a re large , Hencky st ra in ( ) should be used to

ca lcula te st ra in and the area term needed in the st ress ca lcula t ion

should be adjusted for the chang e in ra dius caused by compression:

[1.14]

A crit ical a ssumption in these calculations is tha t t he sam ple remains

cylindrical in sha pe. For this reason lubricated contact surfa ces ar e

often recommended when testing materials such as food gels.

Young’s modulus may a lso be determined by f lexura l test ing of

beams. In one such test , a cant i lever beam of known length (a) is

deflected a d ist a nce (d) w hen a force (F) is a pplied to the free end of th e

beam . This informa tion ca n be used to ca lculat e Young’s modulus for

materials having a rectangular or circular crossectional area (Fig. 1.6).

Similar calculations can be performed in a three-point bending test (Fig.

1.7) w here deflection (d) is mea sured w hen a ma teria l is subjected t o a

force (F) placed midw a y betw een tw o supports. Ca lculat ions a re sight ly

different depending on wet her-or-not th e test mat eria l ha s a recta ngula r

or circular sha pe. Other simple beam tests , such a s the doubleca ntilever

or four-point bending t est , yield compar able results. Flexura l test ing

ma y ha ve applica tion to solid foods havin g a w ell defined geometr y such

as dry pasta or hard candy.

In add it ion t o Young’s m odulus, P oisson’s r at io ( ) can be defined

from compression data (Fig. 1.5):

[1.15]

P oisson’s ra tio ma y ran ge from 0 to 0.5. Typically, va ries from 0.0 for

rigid like ma teria ls conta ining lar ge a mounts of a ir to nea r 0.5 for liquid

like mat er ia ls . Values from 0.2 to 0.5 ar e common for biologica l

ma terials w ith 0.5r epresenting an incompressible substan ce likepotato

E

E = σ22εc

εh

σ22 =F

π( Ro + δ R )2

ν

ν = lateral strain

axial strain =

δ R / Roδh /ho

ν


20/427


flesh. Tissu es w ith a hi gh level of cellula r ga s, such as apple flesh, would

ha ve values closer to 0.2. Metals usua lly have P oisson rat ios between

0.25 a nd 0.35.

Figure 1.6 Deflection of a cantilever beam to determine Young’s modulus.

Figure 1.7 Three-point beam bending test to determine Young’s modulus (b, h,and D are defined on Fig. 1.6).

E = 4Fa /(dbh ) E = 64Fa /(3d D )Rectangular Crossection Circular Crossection

3 3 43

h

b D

d

F

a

E = Fa /(4dbh ) E = 4Fa /(3d D )

Rectangular Crossection Circular Crossection433 3

F

d

a


21/427

1.4 Solid Behavior 11

If a material is subjected to a uniform change in external pressure,

it ma y experience a volumetric cha nge. These quant it ies are used to

define th e bulk modulus ( ):

[1.16]

The bulk modulus of dough is approximat ely P a while the value for

steel is P a . Another common proper ty , bulk compressibi li ty , is

defined as the r eciproca l of bulk modulus.

When tw o ma teria l consta nt s describing the beha vior of a Hookean

solid a re known, the other t wo can be calculated using a ny of the fol-

lowing theoretical relationships:[1.17]

[1.18]

[1.19]

Numerous experimental techniques, applicable to food materials, may

be used to determine Hookean ma terial consta nts. Methods include

testing in tension, compression and torsion (Mohsenin, 1986; Pola-

kowski a nd Ripling, 1966; Da lly a nd R ipley, 1965). H ookean propert ies

of typical ma teria ls a re presented in t he Appendix [6.5].

Linear-elastic a nd non-linear elast ic ma terials (like rubber) both

return to their original shape when the stra in is removed. Food may

be solid in nat ure but not H ookean . A compa rison of curves for linear

elastic (Hookean), elast oplast ic an d n on-linear elast ic ma terials (Fig.

1.8) shows a nu mber of simila rities a nd differences. The ela stoplast ic

ma teria l ha s H ookean type beha vior below the yield str ess ( ) but flows

to produce permanent deforma t ion above tha t va lue. Margar ine and

butter, at room temperature, may behave as elastoplastic substances.

Investigat ion of this type of ma terial, a s a solid or a fluid, depends on

the shear str ess being a bove or below (see Sec. 1.6 for a more deta iled

discus sion of the yield st ress concept an d Appendix [6.7]for ty pical yieldstress va lues). Furt hermore, to fully dist inguish fluid from solid like

behavior, the chara cterist ic t ime of the mat erial and t he cha ra cterist ic

time of the deformation process involved must be considered simulta-

K

K = change in pressure

volumetric strain =

change in pressure

(change in volume/original volume)

106

1011

1

E =

1

3G +

1

9K

E = 3K (1 − 2 ν) = 2G(1 + ν)

ν = 3K − E

6K =

E − 2G2G

σo

σo


22/427


neously. The Debora h number ha s been defined to addr ess this issue.

A detailed discussion of the concept, including an example involving

silly putt y (the " rea l solid-liquid" ) is presented in Sec. 5.5.

Figure 1.8. Deformation curves for linear elastic (Hookean), elastoplastic andnon-linear elastic materials.

Food rheologists a lso find th e failur e behavior of solid food (pa rt ic-

ularly, brit t le mat erials an d firm gels) to be very useful because these

dat a sometimes correlat e well with the conclusions of human sensory

pan els (Ha ma nn, 1983; Mont eja no et al., 1985; Ka wa na ri et a l., 1981).

The following ter min ology (ta ken from America n Society for Testin g and

Mat erials, St an dar d E -6) is useful in describing th e large deforma tionbehav ior involved in the mecha nical fa ilure of food:

elast ic limit - the great est stress w hich a ma terial is capa ble of sus-

taining without any permanent strain remaining upon complete

release of stress;

proportional limit - the great est stress wh ich a ma terial is capable of

susta ining without a ny deviation from Hooke’s La w;

compressive strength - the ma ximum compressive stress a ma terial

is capable of susta ining;

shear str ength - the maximum shear st ress a ma terial is capa ble of

susta ining;

tensile strength - the ma ximum tensile stress a ma terial is capable

of susta ining;

yield point - the first stress in a test where the increase in strain

occurs without an increase in stress;

o

Linear Elastic Elastoplastic Non-Linear Elastic

Permanent

Deformation

12 12 12


23/427

1.5.1 Time-Independent Material Functions 13

yield strength - the engineering str ess at w hich a ma terial exhibits

a specified limiting d eviat ion from th e proport ionalit y of str ess to

st ra in .

A typical chara cterist ic of brit t le solids is tha t t hey break w hen given

a sma ll deforma tion. Fa ilure testing an d fracture mecha nics in struc-

tur a l solids ar e well developed a rea s of mat erial science (Ca llister, 1991)

which of fer m uch t o the food rheologist . Eva luat ing the st r uctura l

failur e of solid foods in compression, torsion, a nd sa ndw ich shear modes

were summa rized by H am an n (1983). J agged force-deforma tion rela-

tionships of crunchy materials may offer alternative texture classifi-

cat ion crit eria for britt le or crunchy foods (U lbricht et a l., 1995; P eleg

a nd Norma nd, 1995).

1.5. Fluid Behavior in Steady Shear Flow

1.5.1. Time-Independent Material Functions

Viscometric Functions. Fluids may be studied by subjecting them

to continuous shearing at a consta nt ra te. Ideally, this can be accom-

plished using two para llel plates with a fluid in the ga p between them

(Fig. 1.9). The lower plat e is fixed an d th e upper plat e moves at a

const a nt velocity ( ) w hich can be th ought of as an increment a l cha nge

in posit ion divided by a small t ime per iod, . A force per unit area

on the plat e is required for motion result ing in a shear str ess ( ) on

the upper plate which, conceptually, could also be considered to be a

lay er of fluid.

The flow described above is steady simple shear and the shear rate

(also called the st ra in ra te) is defined as the ra te of cha nge of stra in:

[1.20]

This defini t ion only applies to st reamline (laminar) f low between

para l le l pla t es. I t is direct ly applicable to sl iding pla te viscometer

descr ibed by D ealy a nd Gia comin (1988). The def init ion must be

adjusted to a ccount for curved lines such a s t hose found in tube a nd

rotat iona l viscometers; however, the idea of " ma ximum speed dividedby gap size" can be useful in estimating shea r ra tes found in part icular

a pplica tions like brush coa ting . This idea is explored in more deta il in

Sec. 1.9.

u

δ L/δt σ21

γ̇ =d γ dt

=d

dt

δ Lh

=

u

h


24/427


Figure 1.9. Velocity profile between parallel plates.

Rheological testing t o determ ine steady sh ear beha vior is conducted

under lamina r flow condit ions. In t urbulent flow, lit t le informa tion is

genera ted tha t can be used to determine ma terial properties. Also, to

be mean ing fu l, da t a must be col lected over the shear r a te r ange

appropria te for the problem in quest ion which may vary widely in

indust ria l processes (Ta ble 1.1): Sediment a tion of pa rt icles ma y involve

very low shear rates, spray drying will involve high shear rates, and

pipe flow of food will usually occur over a moderate shear rate range.

Extrapolating experimental data over a broad range of shear rates is

not recommended because it may introduce significant errors when

evaluating rheological behavior.

Ma teria l flow m ust be considered in thr ee dimensions to completely

describe the sta te of stress or stra in. In stea dy, simple shear flow the

coordinate system ma y beoriented wit h the direction of flow so the stress

tensor given by Eq . [1.5] reduces to

[1.21]

hVELOCITY PROFILE

FORCE

AREA

u

u = 01

2

x

x

σij =

σ11

σ12

0

σ21 σ22 00 0 σ33


25/427


Table 1.1. Shear Rates Typical of Familiar Materials and Processes

Situation (1/s) Application

Sedimentation of particles in 10-6 - 10-3 Medicines, paints, spices in

a suspending liquid salad dressing

Leveling due to surface ten- 10-2 - 10-1 Frosting, paints, printing inks

sion

Draining under gravity 10-1 - 101 Vats, small food containers,

painting and coating

Extrusion 100 - 103 Snack and pet foods, tooth-

paste, cereals, pasta, poly-

mers

Calendering 10

1

- 10

2

Dough SheetingPouring from a bottle 101 - 102 Foods, cosmetics, toiletries

Chewing and swallowing 101 - 102 Foods

Dip coating 101 - 102 Paints, confectionery

Mixing and stirring 101 - 103 Food processing

Pipe flow 100 - 103 Food processing, blood flow

Rubbing 102 - 104 Topical application of creams

and lotions

Brushing 103 - 104 Brush painting, lipstick, nail

polish

Spraying 103 - 105 Spray drying, spray painting,

fuel atomization

High speed coating 104 - 106 Paper

Lubrication 103 - 107 Bearings, gasoline engines

Simple shear flow is a lso called viscometric flow. It includes a xial

flow in a tube, rotational flow between concentric cylinders, rotational

flow between a cone and a plate, and torsional flow (also rotat ional)

between para llel plat es.I n viscometric flow, three shear-ra te-dependent

ma teria l functions, collectively called viscometr ic functions, a re needed

to completely establish t he sta te of st ress in a f luid . These may be

described as the viscosity function, , an d the first an d second normal

s t r ess coef fi ci en t s , a n d , d ef in ed m a t h em a t i ca l ly a s

[1.22]

[1.23]

γ̇

η(γ̇)Ψ1(γ̇) Ψ2(γ̇)

η = f (γ̇) = σ21

γ̇

Ψ1 = f (γ̇) = σ11 − σ22

(γ̇)2 =

N 1

(γ̇)2


26/427


[1.24]

Th e f ir st ( ) a n d s econ d ( ) n or m a l s t res s d if fer en ces a r e

of ten symbol ica l ly represented a s and , r espect ive ly . i s a lways

posit ive an d considered to be approximat ely 10 t imes great er than .

Measurement of is di ff icult ; for tunat ely , the assumption that

is usually sa t isfactory . The rat ioof , known as the recoverableshear

(or the recovera ble elas tic stra in), ha s proven to be a useful par a meter

in modeling die swell phenomena in polymers (Ta nner, 1988). Some

da ta on the va lues of fluid foods have been published (see Appendix

[6.6]).

I f a f lu id is Newton ian , i s a const an t (equa l t o the Newton ian

viscosity) and the first an d second normal s tress differences ar e zero.

As a pproaches zero, elast ic fluids tend to display Newt onian behavior.

Viscoelastic fluids simultaneously exhibit obvious fluid-like (viscous)

an d solid-like (elast ic) behavior. Man ifesta t ions of this behavior, due

to a high elast ic component, can be very strong and create difficult

problems in process engineering design. These problems ar e pa rt icu-

larly prevalent in th e plastic processing industries but also present in

processing foods such as dough, part icularly those containing large

qua ntit ies of wheat protein.

Fig. 1.10 illustr a tes severa l phenomena. Dur ing mixing or a gita tion,

a viscoelast ic fluid may climb th e impeller sha ft in a phenomenon known

a s th e Weissenberg effect (Fig. 1.10). This ca n be observed in home

mixing of cake or chocolat e brownie bat ter . When a Newtonian fluid

emerges from a long, round tube into t he a ir , the emerging jet wil l

norma lly contr a ct; a t low Reynolds number s it may expand to a dia meter

which is 10 to 15%larger tha n the tube diam eter. Normal stress dif-

ferences present in a viscoelast ic fluid, however, ma y cause jet expa n-

sions (called die swell) wh ich ar e tw o or more times the dia meter of the

tube (Fig. 1.10). This behavior contributes to the challenge of designing

extruder dies to produce the desired sha pe of ma ny pet, sna ck, a nd cerealfoods. Melt fracture, a flow inst ability causing distorted extruda tes, is

a lso a problem relat ed to fluid viscoelast icity. In add ition, highly elast ic

fluids may exhibit a tubeless siphon effect (Fig. 1.10).

Ψ2 = f (γ̇) = σ22 − σ33

(γ̇)2 =

N 2

(γ̇)2

σ11 − σ22 σ22 − σ33 N 1 N 2 N 1

N 2

N 2 N 2 = 0

N 1/σ12

N 1

η(γ̇)

γ̇


27/427


Figure 1.10. Weissenberg effect (fluid climbing a rotating rod), tubeless siphonand jet swell of viscous (Newtonian) and viscoelastic fluids.

The r ecoil phenomena (Fig. 1.11), w here t ensile forces in t he fluid

cause part icles to move ba ckwa rd (sna p ba ck)w hen flow is stopped, ma y

a lso be observed in viscoelast ic fluids. Other importa nt effects include

dra g reduction, extruda te insta bility , a nd vortex inhibit ion. An excel-

lent pictoria l summ a ry of the beha vior of viscoelast ic polymer solutions

in var ious f low si tuat ions has been prepared by B oger and Walters

(1993).

Normal stress data can be collected in steady shear flow using a

number of different techniques (Dea ly, 1982): exit pressur e differences

in capil lary a nd sl i t f low, axia l f low in a n a nnulus, wall pressure in

concentric cylinder flow, a xial thr ust in cone a nd plate as w ell a s para llel

plate flow. In general, t hese methods ha ve been developed for plastic

melts (a nd relat ed polymeric ma teria ls) wit h the problems of th e plast ic

industries providing the main driving force for change.

VISCOUS FLUID VISCOELASTIC FLUID

WEISSENBERG

EFFECT

TUBELESS

SIPHON

JET SWELL


28/427


Cone and pla te systems are most commonly used for obta ining

primary normal stress data and a number of commercial instruments

are ava i lable to make these measurements. Obta ining accurate da ta

for food ma teria ls is complica ted by va rious factors such as t he presence

of a y ield st ress, t ime-dependent behavior and chemical react ions

occurring dur ing processing (e.g., hydr a tion, protein denat ura tion, and

sta rch gelat inization). Rheogoniometer is a term sometimes used to

describe an instrument capable of measuring both normal and shear

stresses. Deta iled informa tion on testing viscoelastic polymers can be

found in num erous books: B ird et a l. (1987), Ba rn es et al. (1989), Bogue

and White (1970), Darby (1976), Macosko (1994), and Walters (1975).

Figure 1.11. Recoil phenomenon in viscous (Newtonian) and viscoelastic fluids.

Viscometric functions h ave been very useful in understa nding t he

behav ior of synt hetic polymer s olutions a nd m elts (polyethylene, poly-

propylene, polystyrene, etc.). Fr om a n indust rial sta ndpoint , t he vis-

cosity function is most importa nt in stud ying fluid foods a nd much of

the current w ork is applied to tha t a rea. To dat e, normal stress da ta

for foods ha ve not been widely used in food process engineering. This

is par t ly due tot he fact tha t other factors often complicat e the evaluat ion

of the fluid dyna mics present in va rious problems. In food extrusion,

for example, fla shing (va poriza tion) of w a ter wh en the product exits the

START

STOP

RECOIL

VISCOUS FLUID VISCOELASTIC FLUID

START

STOP


29/427


die makes it difficult to predict the influence of norma l stress differences

on ext ruda te expansion . Fu ture resea rch may crea te s ign if ican t

adva nces in the use of normal str ess data by the food industry.

MathematicalModels for Inelastic Fluids. The ela st ic beha vior of

ma ny fluid foods is sma ll or can be neglected (ma teria ls such as dough

are the exception) leaving the viscosity function as the main area of

interest . This function involves shear st ress and shea r ra te: the rela-

t ionsh ip between the two is es t ab l ished f rom exper imenta l da t a .

B ehavior is visualized as a plot of shear st ress versus shear r at e, and

the resulting curve is ma them a tically modeled using va rious functiona l

relat ionships. The simplest type of substance to consider is t he New-

tonia n fluid where shear st ress is directly proport ional to shear ra te [for

convenience th e subscript on will be dropped in th e rema inder of th e

text w hen dealing exclusively with one dimensiona l flow]:

[1.25]

with being the consta nt of proportionality appropriate for a Newtonian

fluid. Using units of N, m2, m, m /s for force, ar ea, lengt h a nd velocity

gives viscosity a s P a s w hich is 1 poiseuille or 1000 centipoise (note: 1

P a s = 1000 cP = 1000 mP a s ; 1 P = 100 cP ). Dyna mic viscosity and

coefficient of viscosity a re synonym s for th e term " viscosity" in referring

to Newtonian f luids. The reciprocal o f viscosity is ca l led f luidity .

Coeff icient of viscosity and f luidity are infrequent ly used terms.

Newtonian fluids may also be described in terms of their kinematic

viscosity ( ) wh ich is equa l to th e dyna mic viscosity divided by densit y

( ). This is a common pra ctice for non-food ma terials, par t icular ly

lubricat ing oils. Viscosity conversion factors a re ava ilable in Appendix

[6.1].

Newtonian fluids, by definit ion, have a straight line relat ionship

between the shear stress an d the shear r at e with a zero intercept. All

fluids wh ich do not exhibit this beha vior may be called non-Newt onian.

Looking at typical Newtonian fluids on a rheogram (Fig. 1.12) reveals

that the slope of the line increases with increasing viscosity.

Van Wa zer et a l. (1963) discus sed the sensit ivit y of the eye in judg ing

viscosity of Newtonian liquids. It is difficult for the eye to distinguish

differences in the ra nge of 0.1 to 10 cP . Sm a ll differences in viscositya re clea rly seen fr om approximat ely 100 to 10,000 cP : something a t 800

cP ma y look tw ice a s th ick as somethin g at 400 cP . Above 100,000 cP

it is difficult t o make visual dist inctions because the m at erials do not

σ21

σ = µγ̇

µ

νµ/ρ


30/427


Figure 1.12. Rheograms for typical Newtonian fluids.

pour a nd a ppear, to the casu a l observer, a s solids. As points of reference

the following represent typical Newtonian viscosities at room temper-

at ure: a ir , 0.01 cP ; ga soline (petrol), 0.7 cP ; w at er, 1 cP ; mercury, 1.5

cP ; coffee cream or bicycle oil, 10 cP ; vegeta ble oil, 100 cP ; glycerol, 1000

cP; glycerine, 1500 cP ; honey, 10,000 cP; ta r , 1,000,000 cP . Da ta for

many Newtonian f luids a t di f ferent temperatures are presented in

Appendices [6.8], [6.9], a nd [6.10].

A general relat ionship to describe the behavior of non-Newtonian

fluids is th e Hers chel-B ulkley model:

[1.26]

where is the consistency coefficient , is the flow beha vior index, an d

is the yield stress. This model is a ppropriat e for ma ny fluid foods.

Eq . [1.26] is very convenient because Newtonian, power law (shear-

t hinning w hen or shea r-t hickening w hen ) a nd B ing-

ha m plas tic behavior ma y be considered a s special ca ses (Ta ble 1.2, Fig.

1.13). With the Newtonian and Bingha m plast ic models, is commonly

called the viscosity ( ) an d plastic viscosity ( ), respectively. Shear -

thin ning an d shear -thickening are also referred to a s pseudoplast ic a nd

d il a t en t b eh a v ior , r espect i vely; h ow e ver , sh ea r -t h in n in g a n d

0 5 10 15 200

0.5

1

1.5

2

2.5

3

Shear Rate, 1/s

40% fat cream, 6.9 cP

olive oil, 36.3 cP

castor oil, 231 cP

0 5 10 15 200

0.5

1

1.5

2

2.5

3

Shear Rate, 1/s

S h e a r S t r e s s , P a

40% fat cream, 6.9 cP

olive oil, 36.3 cP

castor oil, 231 cP

σ = K (γ̇)n + σo

K n

σo

0 < n


31/427


shear-thickening are the preferred t erms. A typica l example of a

shear-thinning ma terial is found in t he flow behavior of a 1%aqueous

solut ion of carrageenan gum as demonstra ted in Example Problem

1.14.1. Sh ear -th ickening is considered with a concent ra ted corn sta rch

solution in Exa mple P roblem 1.14.2.

Table 1.2. Newtonian, Power Law and Bingham Plastic Fluids as Special Cases of the Herschel-Bulkley Model (Eq. [1.26])

Fluid K n 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, tomatopaste

An importa nt cha ra cterist ic of the H erschel-B ulkley a nd B ingham

plastic materia ls is the presence of a yield stress ( ) which represents

a finite str ess requir ed to achieve flow. B elow th e yield str ess a ma teria l

exhibits solid like cha ra cterist ics: I t st ores energy at sm all str ains a nd

does not level out under the influence of gravity to form a flat surface.

This chara cter ist ic is very importa nt in process design and quali ty

a ssessment for mat erials such as butter , yogurt an d cheese sprea d. The

yield stress is a practical, but idealized, concept that will receive addi-

tiona l discus sion in a la ter section (Sec. 1.6). Typica l yield str ess valu es

ma y be found in Appendix [6.7].

σo

∞

∞

σo


32/427


Figure 1.13. Curves for typical time-independent fluids.

Figure 1.14. Rheogram of idealized shear-thinning (pseudoplastic) behavior.

Newtonian

Shear-Thickening

Shear-Thinning

Herschel-BulkleyBingham

S h e a r S t r e s s ,

P a

Shear Rate, 1/s

S h e a r S t r e s s ,

P a

Shear Rate, 1/s

Lower Region

Upper Region

Middle RegionSlope =

Slope =


33/427


Shear-thinning behavior is very common in fruit and vegetable

products, polymer melts , as well as cosmet ic and to i le t ry products

(Appendices [6.11], [6.12], [6.13]). Dur ing flow, t hese m a teria ls m a y

exhibit three dist inct regions (Fig. 1.14): a lower Newtonian region

where the appar ent viscosity ( ), called the limiting viscosity a t zero

shea r ra te, is const a nt wit h chan ging shear ra tes; a middle region wh ere

the apparent viscosity ( ) is changing (decreasing for shear-thinning

fluids) with shear r at e and the power la w equa tion is a suita ble model

for t he phenomenon; a nd a n upper Newt onian region where t he slope

of the curve ( ), called the limiting viscosity a t infinite shear ra te, is

consta nt w ith changing shear ra tes. The middle region is most often

examin ed when considerin g th e performa nce of food processing equ ip-ment . The lower Newtonian region ma y be relevant in problems

involving low shear rates such as those related to the sedimentation of

fine par ticles in fluids. Values of for some viscoelast ic fluids a re given

in Table 5.4.

Numerous factors influence the selection of the rheological model

used to describe flow behavior of a pa rt icular fluid. Many models, in

addit ion to the power law, Bingham plast ic a nd Herschel-Bulkley

models, ha ve been used to represent th e flow beha vior of non-Newt onian

f lu ids . Some o f them a re summar ized in Table 1.3. The C ross ,

Reiner-Philippoff, Van Wazer and Powell-Eyring models are useful in

modeling pseudoplastic behavior over low, middle and high shear rateranges. Some of the equations, such as the Modified Casson and the

General ized Herschel-B ulkley , have proven useful in developing

ma th ema tica l models to solve food process engineering problems (Ofoli

et al. , 1987) involving wide shear ra te ra nges. Additional rheological

models have been summ a rized by H oldsworth (1993).

The Ca sson equat ion ha s been adopted by the Int ernat iona l Office

of Cocoa a nd C hocola te for inter preting chocolate flow behavior. The

Ca sson an d B ingham pla stic models are similar because they both h ave

a yield stress. Ea ch, however, will give different values of the fluid

parameters depending on the data range used in the mathemat ica l

an alysis. The most reliable value of a yield str ess, when determinedfrom a ma thema tical intercept, is found using data ta ken at the lowest

shea r ra tes. This concept is demonstr a ted in Exam ple P roblem 1.14.3

for milk chocolate.

ηo

η

η∞

ηo


34/427


Apparent Viscosity. Apparent viscosity has a precise definition. It

is, as noted in Eq. [1.22], shear stress divided by shear rate:

[1.27]

With Newtonian f luids, the apparent viscosity and the Newtonian

viscosity ( ) ar e identical but for a power law fluid is

[1.28]

T abl e 1.3. Rh eol ogi cal M od el s t o D escr i be t he B eh avi or of T im e -independent Flu i ds

Model (Source) Equation*

Casson (Casson, 1959)

Modified Casson (Mizrahi and Berk,

1972)

Ellis (Ellis, 1927)

Generalized Herschel-Bulkley (Ofoli et

al., 1987)

Vocadlo (Parzonka and Vocadlo, 1968)

Power Series (Whorlow, 1992)

Carreau (Carreau, 1968)

Cross (Cross, 1965)

Van Wazer (Van Wazer, 1963)

Powell-Eyring (Powell and Eyring, 1944)

Reiner-Philippoff (Philippoff, 1935)

*

and arearbitraryconstantsand power indices, respectively, determinedfrom experimental data.

η = f (γ̇) = σγ̇

µ η

η = f (γ̇) =K (γ̇)n

γ̇ = K (γ̇)n − 1

σ0.5 = (σo)0.5 + K 1(γ̇)

0.5

σ0.5 = (σo)0.5 + K 1(γ̇)

n1

γ̇ = K 1σ + K 2(σ)n

1

σn1 = (σ

o)

n1 + K 1(γ̇)n2

σ = (σo)1/n1 + K 1γ̇

n1

γ̇ = K 1σ + K 2(σ)3 + K 3(σ)

5…

σ =K

1

˙

γ +K

2

˙

(γ)

3

+K

3

˙

(γ)

5

…

η = η∞ + (ηo − η∞)1 + (K 1γ̇)

2

(n − 1)/2

η = η∞ + ηo − η∞1 + K 1(γ̇)

n

η = ηo − η∞

1 + K 1γ̇ + K 2(γ̇)n1

+ η∞

σ = K 1γ̇ +

1

K 2

sinh

−1(K 3γ̇)

σ = η∞ +

ηo − η∞1 + ((σ)2/K 1)

γ̇

K 1, K 2, K 3 n1,n2


35/427


Appar ent viscosit ies for B ingham plastic a nd Herschel-B ulkley fluids

ar e determined in a like mann er:

[1.29]

[1.30]

decreases with increasing shear ra te in shear -thinning an d Bingha m

plastic substances. In H erschel-B ulkley fluids, apparent viscosity w ill

d ecr ea s e w i th h ig her s hea r r a t es w h en , b ut b eh a ve in t h e

oppos it e m a n n er w h en App a r en t v iscos it y i s con s ta n t w i t h

Newtonian ma ter ia ls and increases w i th increas ing shear r a te in

shear-thickening fluids (Fig. 1.15).

Figure 1.15. Apparent viscosity of time-independent fluids.

A single point apparent viscosity va lue is somet imes used as a

mea sur e of mout hfeel of fluid foods: The hum a n perception of th ickness

is correla ted to the appa rent viscosity a t approximat ely 60 s-1. Apparent

viscosity can also be used to illustr a te the axiom tha t ta king single pointtests for determining the general behavior of non-Newtonian materials

ma y cause serious problems. S omeq uality control instrument s designed

for single point t ests ma y produce confusing results. Consider, for

η = f (γ̇) =K (γ̇) + σo

γ̇ = K +

σoγ̇

η = f (γ̇) =K (γ̇)n + σo

γ̇ = K (γ̇)n − 1 +

σoγ̇

η

0


36/427


exam ple, the two Bingha m plastic mat erials shown in Fig. 1.16. The

tw o curves intersect at 19.89 1/s a nd a n instr ument mea suring t he

a pparent viscosity at th a t shea r ra te, for each fluid, would give identica l

results : = 1.65 P a s . However , a simple examinat ion of the mat er ia l

wit h the ha nds a nd eyes would show them to be quit e different because

the yield str ess of one ma terial is more tha n 4 t imes tha t of the other

ma teria l. C learly , num erous da ta points (minim um of tw o for the power

law or Bingham plast ic models) are required to evaluate the f low

behav ior of non-Newt onian fluids .

Figure 1.16. Rheograms for two Bingham plastic fluids.

Solution Viscosities. It is sometimes useful to determ ine the visco-

sities of dilute synthetic or biopolymer solutions. When a polymer is

dissolved in a solvent, there is a noticeable increase in the viscosity of

the r esult ing solution. The viscosit ies of pure solvents a nd solutions

can be meas ured and va rious values calculat ed from the result ing dat a:

[1.31]

[1.32]

η

0 10 20 30 40 50 600

20

40

60

80

100

Shear Rate, 1/s

S h e a r S t r e s s , P a

Yield Stress = 25.7 Pa

Plastic Viscosity = .36 Pa s

Yield Stress = 6.0 Pa

Plastic Viscosity = 1.35 Pa s

Bingham Plastic Fluids

relative viscosity = ηrel = ηsolutionηsolvent

specific viscosity = ηsp

= ηrel −1


37/427

1.5.2 Time-Dependent Material Functions 27

[1.33]

[1.34]

[1.35]

where is the mass concentra tion of the solution in units of g/dl or

g/100ml. Note tha t unit s of reduced, inherent, a nd int rinsic viscosity

a re reciprocal concentra tion (usually deciliters of solution per gra ms of

polymer). The intrins ic viscosity ha s great pr a ctical va lue in molecular

weight determin a tions of high polymers (Severs, 1962; Morton-J ones,

1989; G rulke, 1994). This concept is ba sed on th e Ma rk-Houwink

rela t ion suggest ing tha t the intr insic viscosity of a di lute polymer

solution is proportiona l to t he a verage molecular weight of t he solute

ra ised to a power in th e ra nge of 0.5 to 0.9. Valu es of the proportiona lity

consta nt a nd the exponent a re well known for ma ny polymer-solvent

combina tions (Pr ogelf an d Throne, 1993; Rodriquez, 1982). Solution

viscosities a re useful in unders ta nding the beha vior of some biopolymers

including aqueous solutions of locust bean gum, guar gum, and car-

boxymet hylcellulose (Ra o, 1986). The intr insi c viscosit ies of numer ous

protein solut ions ha ve been summarized by Rha and P radipasena(1986).

1.5.2. Time-Dependent Material Functions

Ideally, t ime-dependent materia ls ar e considered to be inelast icw ith

a viscosity function wh ich depends on time. The response of the sub-

sta nce to stress is inst an ta neous a nd t he t ime-dependent beha vior is

due to cha nges in the structure of the mat erial itself. In contra st , t ime

effects found in viscoelastic materials arise because the response of

stress to applied stra in is not insta nta neous an d not a ssociat ed with a

str uctura l cha nge in the mat eria l. Also, th e tim e scale of thixotropy may

be quite different than the t ime scale associated with viscoelasticity:

The most dramatic effects are usually observed in situations involving

short process t imes. Note too, tha t real ma terials ma y be both t ime-

dependent and viscoelastic!

reduced viscosity = ηred

= ηspC

inherent viscosity = ηinh

= lnηrel

C

intrinsic viscosity = ηint =

ηspC

C → 0

C


38/427


Figure 1.17. Time-dependent behavior of fluids.

Separate terminology has been developed to describe fluids with

time-dependent characteristics. Thixotropic and rheopectic materials

exhibit , respect ively , decreasing and increasing shear st ress (and

a pparent viscosity) over time at a fixed ra te of shear (Fig. 1.17). In other

words, thixotropy is t ime-dependent thinning and rheopexy is t ime-

dependent thickening. B oth phenomena ma y be irreversible, reversibleor par t ia l ly reversible . There is genera l agreement tha t t he term

"thixotropy" refers to the time-dependent decrease in viscosity, due to

shearing, and the subsequent recovery of viscosity when shearing is

removed (Mewis, 1979). Ir reversible th ixotr opy, ca lled rheomala xis or

rheodestruction, is common in food products and may be a factor in

evalua ting yield str ess as well a s the genera l flow behav ior of a ma teria l.

Ant i-th ixotr opy a nd nega tive thixotr opy ar e synonyms for rheopexy.

Thixotropy in many fluid foods may be described in terms of the

sol-gel tra nsit ion phenomenon. This t erminology could a pply, for

exam ple, to sta rch-thickened baby food or yogurt . After being man-

ufa ctured, a nd placed in a conta iner, these foods slowly develop a t hreedimensiona l netw ork and ma y be described as gels. When subjected to

shea r (by sta nda rd rheological testin g or mixing wit h a spoon), str ucture

is broken down and the materials reach a minimum thickness where

Thixotropic

Time-Independent

Rheopectic

S h

e a

r S

t r e s

s ,

P a

Time at Constant Shear Rate, s

Time-Dependent Behavior


39/427


Figure 1.18. Thixotropic behavior observed in torque decay curves.

they exist in the sol sta te. In foods tha t show reversibility, the netw ork

is rebuilt a nd the gel stat e reobta ined. Irreversible ma terials rema in

in the sol sta te.

The ra nge of thixotropic beha vior is illustrat ed in Fig. 1.18. Sub-

jected to a constant shear r at e, the shear st ress will decay over t ime.During t he rest period th e ma terial m ay completely recover, part ially

recover or not recover an y of its original s tructure leading to a high,

medium, or low torque response in th e sa mple. Rota tiona l viscometers

have proven t o be very useful in evaluat ing t ime-dependent f luid

behav ior because (unlike tube viscometers ) they ea sily allow ma teria ls

to be subjected to alternate periods of shear and rest.

Step (or l inear) changes in shear ra te may a lso be carr ied out

sequentially with the result ing shear stress observed between steps.

Typical results a re depicted in Fig. 1.19. Actua l curve segment s (such

as 1-2 and 3-4) depend on the rela t ive contr ibut ion of st ructura l

breakdown and buildup in the substance. P lott ing shear st ress versus

shear rate for the increasing and decreasing shear rate values can beused to genera te hyst eresis loops (a differencein the up a nd down curves)

for the ma terial. The area between the curves depends on the t ime-

dependent na ture of t he substa nce: it is zero for a t ime-independent

0

S t r e s s

0

time

RestPeriod

S h e a r R a t e

Complete RecoveryPartial Recovery

No Recovery

Evidence of Thixotropy in Torque Decay Curves


40/427


f lu id. This in forma t ion may be va luable in compar ing d if ferent

ma terials, but it is somewha t subjective because different step change

periods may lead to different hyster esis loops. Sim ilar informa tion can

be genera ted by subjecting ma teria ls to step (or linea r) chan ges in shea r

stress a nd observing the result ing changes in shear ra tes.

Figure 1.19. Thixotropic behavior observed from step changes in shear rate.

Torqu e deca y da ta (like th a t given for a problem in mixer viscometr y

described in E xam ple P roblem 3.8.22)m a y be used to model irr eversible

thixotropy by adding a st ructura l decay param eter to the H erschel-

B ulkley model to account for brea kdown (Tiu a nd B oger, 1974):

[1.36]

where , the st ructura l param eter , is a funct ion of t ime. before the

onset of shearing and equals an equilibrium value ( ) obta ined aft er

complete breakdown from shear ing. The decay of the st ructura l

para meter with t ime ma y be assumed to obey a second order equa tion:

[1.37]

time

S h e a r S t r e s s

S h e a r R a t e

1

23 4

Step Changes in Shear Rate

σ = f (λ, γ̇) = λ(σo + K (γ̇)n)

λ λ = 1λ λe

d λdt

= −k 1(λ − λe)2 for λ > λ

e


41/427


where is a ra t e constant tha t is a funct ion of shear ra t e . Then, the

en t ir e m od el i s com plet el y d et er m in ed b y f ive p a r a m et er s : ,

a n d . a n d a r e d et er m in ed u nd er in it ia l s hea r in g con dit ion s

when and . In other words , they a re determined f rom the in it i a l

shea r str ess in the ma teria l, observed at t he beginning of a t est, for each

shear r at e considered.

and a re expressed in t e rms of the apparen t v iscos i t y ( ) t o

find . Eq ua ting th e rheological model (Eq . [1.36]) to th e definition of

appar ent viscosity (which in this case is a function of both shear ra te

a nd the time-dependent appar ent viscosity) yields an expression for :

[1.38]

Eq. [1.38] is va l id for a l l va lues of including a t , the equil ibr ium

value of the appar ent viscosity. Differentia t ing with respect to t ime,

at a constant shear rate, gives

[1.39]

Using the definit ion of , Eq . [1.37] and [1.39] may be combined

yielding[1.40]

Considering the definition of given by Eq. [1.38], th is may be rew ritt en

a s

[1.41]

Simplification yields

[1.42]

or

k 1

σo, K , n , k 1(γ̇)

λe K , n σoλ = 1 t = 0

λ λe η = σ/γ̇ k 1

λ

λ = ηγ̇ σo + K (γ̇)

n

λ λe ηeλ

d λdt

=d ηdt

γ̇ σo + K (γ̇)

n

d λ/dt

−k 1(λ − λe)2 =

d ηdt

γ̇ σo + K (γ̇)

n

λ

−k 1

ηγ̇ σo + K (γ̇)

n

−

ηeγ̇

σo + K (γ̇)n

2

=d ηdt

γ̇ σo + K (γ̇)

n

d ηdt = −k 1

γ̇ σo + K (γ̇)

n

(η − ηe)

2


42/427


[1.43]

where

[1.44]

Int egra ting E q. [1.43] gives

[1.45]

so

[1.46]

where is the init ia l va lue of the apparent viscosity ca lcula ted from

t h e in it ia l ( a n d ) s hea r s t res s a n d s hea r r a t e.

U s ing E q . [1. 46], a plot ver sus , a t a pa r t i cu la r sh ea r r a t e ,

i s made to obt a in . This is done a t numerous shear r a tes and the

result ing informa tion is used to determine the relat ion between an d

and, from tha t , the rela t ion between and . This is the f ina l infor-

ma tion required to completely specify the ma th ema tica l model given by

Eq. [1.36] and [1.37].

The above approach has been used to descr ibe the behavior ofma yonna ise (Tiu a nd B oger, 1974), ba by food (Ford a nd St effe, 1986),

an d butt ermilk (B utler a nd McNulty, 1995). More complex models

which include terms for the recovery of structure are also available

(Cheng, 1973; Ferguson and Kemblowski, 1991). Numerous rheological

models to describe tim e-dependent beha vior have been summa ri

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