Handbook of food engineering [Third edition] 9780429449734, 0429449739, 9780429831577, 0429831579, 9781466563131, 1466563133

Table of contents :
Content: Rheological properties of foods --
Advances in nanotechnology of food materials for food and non-food applications --
Reaction kinetics in food systems --
Phase and state transitions and transformations in food systems --
Transport and storage of food products --
Heating and cooling processes for foods --
Food freezing and frozen food storage --
Mass transfer in foods --
Evaporation and freeze concentration --
Membranes in food technology --
Food dehydration --
Thermal processing of canned foods --
Extrusion processes --
Food packaging --
Engineering considerations for cleaning and disinfection in the food industry.

Citation preview

Handbook of Food Engineering Third Edition

Handbook of Food Engineering Third Edition

Edited by

Dennis R. Heldman Daryl B. Lund Cristina M. Sabliov

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 ©  2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-4665-6312-4 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice:  Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at  http://www.taylorandfrancis.com  and the CRC Press Web site at  http://www.crcpress.com

Contents Preface...................................................................................................................................................... vii Editors........................................................................................................................................................ ix Contributors............................................................................................................................................... xi 1 Linear and Non-Linear Rheological Properties of Foods............................................................. 1 Ozlem C. Duvarci, Gamze Yazar, Hulya Dogan, and Jozef L. Kokini 2 Advances in Nanotechnology of Food Materials for Food and Non-Food Applications........ 153 Rohollah Sadeghi, Thanida Chuacharoen, Cristina M. Sabliov, Carmen I. Moraru, Mahsan Karimi, and Jozef L. Kokini 3 Reaction Kinetics in Food Systems.............................................................................................. 225 Ricardo Villota and James G. Hawkes 4 Phase and State Transitions and Transformations in Food Systems...................................... 485 Yrjö H. Roos 5 Transport and Storage of Food Products................................................................................... 551 M. A. Rao 6 Heating and Cooling Processes for Foods................................................................................... 597 R. Paul Singh and Gail Bornhorst 7 Food Freezing and Frozen Food Storage.................................................................................... 637 Dennis R. Heldman 8 Mass Transfer in Foods................................................................................................................ 683 Bengt Hallström, Vassilis Gekas, Ingegerd Sjöholm, and Anne Marie Romulus 9 Evaporation and Freeze Concentration...................................................................................... 705 Ken R. Morison and Richard W. Hartel 10 Membranes in Food Technology.................................................................................................. 765 Frank Lipnizki 11 Food Dehydration.......................................................................................................................... 799 Martin R. Okos, Osvaldo Campanella, Ganesan Narsimhan, Rakesh K. Singh, and A. C. Weitnauer 12 Thermal Processing of Canned Foods........................................................................................ 951 Arthur Teixeira 13 Extrusion Processes...................................................................................................................... 985 Leon Levine and Robert C. Miller

v

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Contents

14 Food Packaging........................................................................................................................... 1031 John M. Krochta 15 Engineering Considerations for Cleaning and Disinfection in the Food Industry............... 1125 Kylee R. Goode, David Phinney, Tony Hasting, and Peter Fryer Index......................................................................................................................................................1175

Preface The editors and authors of the Third Edition of the Handbook of Food Engineering hope the content will continue to provide students and food engineering professionals with the latest information needed to improve the efficiency of the food supply system. As the complexity of the system increases, the focus on processes used to convert raw food materials and ingredients into consumer food products becomes more important. As suggested in previous editions, there are three different audiences for the content of this Handbook: (1) practicing engineers in food, beverage, and related industries; (2) students preparing for careers as food engineers; and (3) other engineers and scientists seeking the latest information on unit operations and processes needed for process design and development. Hopefully, practicing engineers will use the content to improve processes throughout the food supply system. The Handbook should become the primary reference for students as a supplement to textbooks during the study of unit operations and process design and development. Other scientists and engineers should be able to locate the latest information and data needed when evaluating processes during development of new products or for quality assurance considerations. As with the previous editions, the Handbook of Food Engineering contains the latest information on the thermophysical properties of foods and kinetic constants needed to estimate changes in key components of foods during manufacturing and distribution. Illustrations are used to demonstrate the applications of the information to process design. Researchers should be able to use the information to pursue new directions in process development and design, and to identify future directions for research on physical properties of foods and kinetics of changes in food throughout the supply system. As in the second edition, the Appendix provides tables of data and illustrations for many food-related properties. The first four chapters of the Third Edition focus primarily on properties of foods and food ingredients with a new chapter on nanoscale applications in foods. Each of the eleven chapters that follow focus on one of the more traditional unit operations used throughout the food supply system. Major revisions and/ or updates have been incorporated into chapters on heating and cooling, membrane and extrusion processes, and cleaning operations. Most of the chapters have been revised and updated. The editors are dedicated to continuing the evolution of food engineering as an interface between traditional engineering disciplines and food science. As the focus on a more efficient food supply system continues to evolve, with a specific focus on reducing energy, fresh water, and waste, the need for more efficient unit operations and processes is very evident. These needs, driven by the consumer, have evolved as the demands for safe, high quality, nutritious, and health-enhancing foods continue to increase. As the applications of molecular biology, nanoscale sciences, and metabolomics evolve and begin to impact consumer food products, the design of processes used throughout the food supply system will improve. The importance of unit operation and process design from bench-scale to pilot plant scale-up to commercial scale becomes even more critical. Opportunities for optimization of processes that ensure food safety while achieving maximum food quality and throughput should become more evident when using the content of this Handbook. Finally, the incorporation of emerging technologies and unit operations into the processes used throughout the food supply chain can be accelerated using the background presented in this Handbook. The editors wish to acknowledge the contributions of all contributors to the Handbook of Food Engineering. Thirty-six authors have contributed to the 15 chapters in the Handbook. The content has been greatly enhanced, broadened, and expanded through the contributions of each author. Each author is a leading food engineer and scientist, and the co-editors are proud to include their contributions to the Third Edition of the Handbook of Food Engineering. Dennis R. Heldman Daryl B. Lund Cristina M. Sabliov vii

Editors Dennis R. Heldman was awarded a BS (1960) and MS (1962) degrees from The Ohio State University, and a PhD (1965) from Michigan State University. In 1966, he joined the faculty at Michigan State University, and began teaching and research in the area of food process engineering. He served as chair of the Agricultural Engineering Department at Michigan State University from 1975 to 1979. He joined the Campbell Soup Company in 1984, as the vice president of Process Research and Development. In 1986, he moved to the National Food Processors Association, as executive vice president of Scientific Affairs, CEO for The National Food Laboratory, and president of The Food Processors Institute. In 1991, he joined the Weinberg Consulting Group Inc, as a consultant on food regulatory issues. In 1992, he was appointed professor of Food Process Engineering at the University of Missouri and Leader for the Foods, Feeds and Products cluster in the Foods for the 21st Century program. Beginning in 1994, he served as unit leader for the Food Science and Engineering Unit, and in 1997, as director for the Office of Value-Added Agriculture Outreach. From 1998 to 2004, he was professor of Food Process Engineering at Rutgers, the State University of New Jersey, and director of the Cooperative Research & Development Program in the Center for Advanced Food Technology (CAFT). From 2004 to 2012, he was a consultant involved in applications of engineering concepts to food manufacturing for educational institutions, industry and government. In August 2012, he joined the faculty at The Ohio State University as Dale A. Sobering Endowed Professor of Food Engineering. In 2016, he assumed responsibilities as director of the Center for Advanced Processing and Packaging Studies (CAPPS). He has received numerous recognitions, including the FIEI Young Researchers’ Award from the American Society of Agricultural Engineers in 1974, elected fellow of the Institute of Food Technologists (IFT) in 1981, the DFISAASAE Food Engineering Award in 1981, elected fellow of the American Society of Agricultural Engineers in 1984, elected fellow in the International Academy of Food Science & Technology in 2007, recipient of the Life Achievement Award from the International Association for Engineering and Food, and the Frozen Food Foundation Freezing Research Award in 2011, the Carl R. Fellers Award from the IFT and Phi Tau Sigma in 2013, the Harold Macy Food Science and Technology Award from the Minnesota Section of IFT in 2017, and the Nicholas Appert Award from IFT in 2018. Daryl B. Lund earned a BS (1963) in mathematics and a PhD (1968) in food science with a minor in chemical engineering at the University of Wisconsin-Madison. During 21 years at the University of Wisconsin, he was a professor of food engineering in the food science department serving as chair of the department from 1984–1987. He has contributed over 150 scientific papers, edited 5 books, and co-authored one major textbook in the area of simultaneous heat and mass transfer in foods, kinetics of reactions in foods, and food processing. In 1988 he continued his administrative responsibilities by chairing the Department of Food Science at Rutgers University, and from December 1989 through July 1995 served as the executive dean of Agriculture and Natural Resources with responsibilities for teaching, research and extension at Rutgers University. In August 1995, he joined Cornell University as the Ronald P. Lynch Dean of Agriculture and Life Sciences. In January 2001, Dr. Lund became the executive director of the North Central Regional Association of State Agricultural Experiment Station Directors located at UW Madison. In this position he facilitated interstate collaboration on research and a greater integration between research and extension in the twelve-state region. He has received numerous awards in recognition of personal achievements:ASAE/DFISA Food Engineering Award, Irving Award from the American Distance Education Consortium (ADEC), International Congress on Engineering and Food (ICEF) Lifetime Achievement Award, and Institute of Food Technologists (IFT) International Award, Carl R. Fellers Award, and Nicolas Appert Award for ix

x

Editors

lifetime achievement. In 2016 he received the Lifetime Achievement Award from the International Union of Food Science and Technology (IUFoST). He is an elected fellow of the IFT, IUFoST, and charter inductee in the International Academy of Food Science and Technology (IAFoST). Cristina M. Sabliov is the Richard R. & Betty S. Fenton LSU Alumni Professor in the Biological and Agricultural Engineering (BAE) Department at Louisiana State University and LSU Agricultural Center. She received a BS in Food Technology, two MS degrees in Agricultural Engineering and Chemical Engineering, and PhDs in Food Science, and Biological and Agricultural Engineering. Currently, she serves as a graduate coordinator in the BAE Department. is leading an international renowned research program in the field of nanotechnology, specifically focused on polymeric nanoparticles designed for delivery of bioactive components for improved food quality and human health. She collaborates extensively across disciplines with colleagues from the US and abroad, to address complex research questions. To date, she has published 56 papers and has been cited more than 1,000 times. She has been recognized for the quality and impact of her work by LSU, as a recipient of Tiger Athletic Foundation Undergraduate Teaching Award, and Distinguished Faculty Award. The American Society of Agricultural and Biological Engineers awarded her the New Holland Young Researcher Award in 2011, and in 2016 She was inducted as a Fellow of the American Institute for Medical and Biological Engineering.

Contributors Gail Bornhorst Department of Biological and Agricultural Engineering University of California Davis, California Osvaldo Campanella Purdue University West Lafayette, Indiana Thanida Chuacharoen Department of Food Science and Technology Suan Sunandha Rajabhat University Bangkok, Thailand Ozlem C. Duvarci Purdue University West Lafayette, Indiana and Izmir Institute of Technology Izmir, Turkey Hulya Dogan Kansas State University Manhattan, Kansas Peter Fryer Centre for Formulation Engineering, School of Chemical Engineering University of Birmingham Birmingham, UK Vassilis Gekas Technical University of Crete Chania, Greece

Richard W. Hartel Department of Food Science University of Wisconsin Madison, Wisconsin Tony Hasting Tony Hasting Consulting Sharnbrook, Bedford, UK James G. Hawkes GLG, Inc. Naperville, Illinois Dennis R. Heldman The Ohio State University Columbus, Ohio Mahsan Karimi Department of Food Science and Technology Islamic Azad University Kermanshah, Iran Jozef L. Kokini Purdue University West Lafayette, Indiana John M. Krochta Department of Food Science and Technology Department of Biological and Agricultural Engineering University of California Davis, California Leon Levine Leon Levine & Associates, Inc. Albuquerque, New Mexico

Kylee R. Goode Centre for Formulation Engineering, School of Chemical Engineering University of Birmingham Birmingham, UK

Frank Lipnizki Department of Chemical Engineering Lund University Lund, Sweden

Bengt Hallström University of Lund Lund, Sweden

Robert C. Miller Consulting Engineer Auburn, New York xi

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Contributors

Carmen I. Moraru Department of Food Science Cornell University Ithaca, New York

Rohollah Sadeghi Department of Food Science Purdue University West Lafayette, Indiana

Ken R. Morison Department of Chemical and Process Engineering University of Canterbury Christchurch, New Zealand

R. Paul Singh Department of Biological and Agricultural Engineering University of California, Davis, California

Ganesan Narsimhan Purdue University West Lafayette, Indiana

Rakesh K. Singh University of Georgia Athens, Georgia

Martin R. Okos Purdue University West Lafayette, Indiana David Phinney The Ohio State University Columbus, Ohio M. A. Rao Cornell University Ithaca, New York Anne Marie Romulus Université Paul Sabatier Toulouse, France Yrjö H. Roos University of Cork Cork, Ireland Cristina M. Sabliov Department of Biological and Agricultural Engineering Louisiana State University Baton Rouge, Louisiana

Ingegerd Sjöholm University of Lund Lund, Sweden Arthur Teixeira University of Florida Gainesville, Florida Ricardo Villota GKF Foods Glenview, Illinois Angela C. Weitnauer Purdue University West Lafayette, Indiana Gamze Yazar Purdue University West Lafayette, Indiana

1 Linear and Non-Linear Rheological Properties of Foods Ozlem C. Duvarci, Gamze Yazar, Hulya Dogan, and Jozef L. Kokini CONTENTS 1.1 Introduction....................................................................................................................................... 2 1.2 Basic Concepts.................................................................................................................................. 3 1.2.1 Stress and Strain................................................................................................................... 3 1.2.2 Classification of Materials.................................................................................................... 4 1.2.3 Types of Deformation........................................................................................................... 4 1.2.3.1 Shear Flow............................................................................................................ 4 1.2.3.2 Extensional (Elongational) Flow.......................................................................... 7 1.2.3.3 Volumetric Flows.................................................................................................. 9 1.2.4 Response of Viscous and Viscoelastic Materials in Shear and Extension........................... 9 1.2.4.1 Stress Relaxation................................................................................................. 10 1.2.4.2 Creep....................................................................................................................11 1.2.4.3 Small Amplitude Oscillatory Measurements......................................................11 1.2.4.4 Interrelations between Steady Shear and Dynamic Properties...........................14 1.3 Methods of Measurement.................................................................................................................18 1.3.1 Shear Measurements........................................................................................................... 19 1.3.2 Small Amplitude Oscillatory Shear (SAOS) Measurements............................................. 24 1.3.3 Large Amplitude Oscillatory Shear (LAOS) Measurements............................................. 26 1.3.3.1 Time Dependency of Tomato Paste, Mayonnaise, and Soft and Hard Dough in SAOS and LAOS.................................................................................41 1.3.4 Extensional Measurements................................................................................................. 44 1.3.5 Stress Relaxation................................................................................................................ 53 1.3.6 Creep Recovery.................................................................................................................. 57 1.3.7 Transient Shear Stress Development...................................................................................61 1.3.8 Yield Stresses..................................................................................................................... 63 1.4 Constitutive Models........................................................................................................................ 64 1.4.1 Simulation of Steady Rheological Data............................................................................. 66 1.4.2 Linear Viscoelastic Models................................................................................................ 67 1.4.2.1 Maxwell Model................................................................................................... 71 1.4.2.2 Voigt Model........................................................................................................ 73 1.4.2.3 Multiple Element Models.................................................................................... 76 1.4.2.4 Mathematical Evolution of Nonlinear Constitutive Models............................... 78 1.4.3 Nonlinear Constitutive Models.......................................................................................... 81 1.4.3.1 Differential Constitutive Models........................................................................ 81 1.4.3.2 Integral Constitutive Models.............................................................................. 84 1.4.3.3 Simulation for Large Amplitude Oscillatory Flow............................................ 89 1.5 Molecular Information from Rheological Measurements.............................................................. 91 1.5.1 Dilute Solution Molecular Theories................................................................................... 91 1.5.2 Concentrated Solution Theories......................................................................................... 94 1.5.2.1 The Bird–Carreau Model................................................................................... 94 1.5.2.2 The Doi–Edwards Model.................................................................................... 98 1.5.3 Understanding Polymeric Properties from Rheological Properties..................................101 1

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1.5.3.1 Gel Point Determination....................................................................................101 1.5.3.2 Glass Transition Temperature and the Phase Behavior.................................... 105 1.5.3.3 Networking Properties...................................................................................... 109 1.6 Use of Rheological Properties in Practical Applications..............................................................111 1.6.1 Sensory Evaluations..........................................................................................................112 1.6.2 Molecular Conformations.................................................................................................115 1.6.3 Product and Process Characterization..............................................................................117 1.7 Numerical Simulation of Flows.....................................................................................................119 1.7.1 Numerical Simulation Techniques....................................................................................119 1.7.2 Selection of Constitutive Models.......................................................................................121 1.7.3 Finite Element Simulations...............................................................................................121 1.7.3.1 FEM Techniques for Viscoelastic Fluid Flows................................................. 122 1.7.3.2 FEM Simulations of Flow in an Extruder........................................................ 123 1.7.3.3 FEM Simulations of Flow in Model Mixers..................................................... 126 1.7.3.4 FEM Simulations of Mixing Efficiency........................................................... 128 1.7.4 Verification and Validation of Mathematical Simulations................................................133 1.8 Concluding Remarks..................................................................................................................... 136 References............................................................................................................................................... 138

1.1 Introduction Rheological properties are important to the design of flow processes, quality control, storage and processing stability measurements, predicting texture and learning about molecular and conformational changes in food materials (Dealy and Wang, 2013; Davis, 1973). The rheological characterization of foods provides important information for food scientists in ingredient selection strategies to design, improve and optimize their products, to select and optimize their manufacturing processes and to design packaging and storage strategies. Rheological studies become particularly useful when predictive relationships for rheological properties of foods can be developed, which start from the molecular architecture of the constituent species. Reliable and accurate steady rheological data are necessary to design continuous-flow processes, select and size pumps and other fluid-moving machinery and to evaluate heating rates during engineering operations, which include flow processes such as aseptic processing and concentration (Sheath, 1976; Holdsworth, 1971), and to estimate velocity, shear and residence-time distribution in food processing operations including extrusion and continuous mixing. Viscoelastic properties are also useful in processing and storage stability predictions. For example, during extrusion, viscoelastic properties of cereal flour doughs affect die swell and extrudate expansion. In batch mixing, elasticity is responsible for the rod climbing phenomenon, also known as the Weissenberg effect (Bird et al., 1987). To allow for elastic recovery of dough during cookie making, the dough is cut in the form of an ellipse which relaxes into a perfect circle. Creep and smal‑lamplitude oscillatory measurements are useful in terms of understanding the role of constituent ingredients on the stability of oil-in-water emulsions. Steady shear and creep measurements help identify the effect of ingredients that have stabilizing abilities, such as gums, proteins or other surface-active agents (Fischbach and Kokini, 1984). Dilute solution viscoelastic properties of biopolymeric materials, such as carbohydrates and protein, can be used to characterize their three-dimensional configuration in solution. Their configuration affects their functionality in many food products. It is possible to better predict and improve the flow behavior of food polymers through an understanding of how the molecular structure of foods affect their rheological properties (Liguori, 1985). Examples can be found in the improvement of the consistency and stability of emulsions by using polymers with enhanced surface activity and greater viscosity and elasticity. This chapter will review recent advances in basic rheological concepts, methods of measurement, molecular theories, linear and nonlinear constitutive models and numerical simulation of viscoelastic

3

Linear and Non-Linear Rheological Properties of Foods

flows. In this edition we have updated the new and emerging topics related to large Amplitude Oscillary Measurements (LAOS)

1.2 Basic Concepts 1.2.1 Stress and Strain Rheology is the science of the deformation and flow of matter. Rheological properties define the relationship between stress and strain/strain rate in different types of shear and extensional flows. The stress is defined as the force F acting on a unit area A. Since both force and area have directional as well as magnitude characteristics, stress is a second order tensor and typically has nine components. Strain is a measure of deformation, or relative displacement, and is determined by the displacement gradient. Since displacement and its relative change both have directional properties, strain is also a second order tensor with nine components. A rheological measurement is conducted on a given material by imposing a well-defined stress and measuring the resulting strain or strain rate, or by imposing a well-defined strain or strain rate and by measuring the stress developed. The relationship between these physical events leads to different kinds of rheological properties. When a force F is applied to a piece of material (Figure 1.1), the total stress acting on any infinitesimal element is composed of two fundamental classes of stress components (Darby, 1976; Chhabra, 2010; Campanella, 2011):

1. “normal stress” components, applied perpendicularly to the plane (τ11, τ22, τ33), 2. “shear stress” components, applied tangentially to the plane (τ12, τ13, τ21, τ23, τ31, τ32).

There are a total of nine stress components acting on an infinitesimal element (i.e. two shear and one normal stress components acting on each of the three planes). Individual stress components are referred to as τij, where i refers to the plane the stress acts on, and j indicates the direction of the stress component (Bird et al., 1987; van Vliet, 2014). The stress tensor can be written as a matrix of nine components as follows: é t11 ê t = ê t21 êë t31



t12 t22 t32

t13 ù ú t23 ú t33 úû

In general, the stress tensor in the deformation of an incompressible material is described by three shear stresses and two normal stress differences: Shear stresses:  τ12 (=τ21), τ13 (=τ31), τ23 (=τ32) Normal stress differences:  N1 = τ11 − τ22, N2 = τ22 − τ33 2 22 12 21 23

11

1 13

1-plane 3 FIGURE 1.1  Stress components on a cubical material element.

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Handbook of Food Engineering

1.2.2 Classification of Materials Rheological properties of materials are the result of their stress-strain behaviors. Ideal solid (elastic) and ideal fluid (viscous) behaviors represent two extreme responses of a material (Darby, 1976). An ideal solid material deforms instantaneously when a load is applied. It returns back to its original configuration instantaneously (complete recovery) upon removal of the load. Ideal elastic materials obey Hooke’s law, where the stress (τ) is directly proportional to the strain (γ). The proportionality constant (G) is called the modulus. t = Gg



An ideal fluid deforms at a constant rate under an applied stress, and the material does not regain its original configuration when the load is removed. The flow of a simple viscous material is described by Newton’s law, where the shear stress (τ) is directly proportional to the shear rate (g ). The proportionality constant (η) is called the Newtonian viscosity. t = hg



Most food materials exhibit characteristics of both elastic and viscous behavior, and are called viscoelastic. If viscoelastic properties are strain and strain rate independent, then these materials are referred to as linear viscoelastic materials. On the other hand, if they are strain and strain rate–dependent than they are referred to as nonlinear viscoelastic materials (Bird et al., 1987; Macosko, 1994; Ferry, 1980). A simple and classical approach to describe the response of a viscoelastic material is using mechanical analogs. Purely elastic behaviour is simulated by springs and purely viscous behaviour is simulated using dashpots. The Maxwell and Voigt models are the two simplest mechanical analogs of viscoelastic materials. They simulate a liquid (Maxwell) and a solid (Voigt) by combining a spring and a dashpot in series or in parallel, respectively. These mechanical analogs are the building blocks of constitutive models, as discussed in Section 1.4 in detail.

1.2.3 Types of Deformation 1.2.3.1 Shear Flow One of the most useful types of deformation for rheological measurements is simple shear. In simple shear, a material element is placed between two parallel plates (Figure 1.2) where the bottom plate is stationary and the upper plate is displaced in x-direction by ∆x by applying a force F tangentionally to the surface A. The velocity profile in simple shear is given by the following velocity components (Chhabra, 2010; Chhabra and Richardson, 2011): u x = g y, u y = 0 and uz = 0



The corresponding shear stress is given as: t=



F A

A F

∆y

vx

y x

FIGURE 1.2  Shear flow.

5

Linear and Non-Linear Rheological Properties of Foods If the relative displacement at any given point ∆y is ∆x, then the shear strain is given by g=



Dx Dy

If the material is a fluid, the force applied tangentionally to the surface will result in a constant velocity vx in x-direction. The deformation is described by the strain rate ( g ), which is the time rate of change of the shear strain: g =



d g d æ Dx ö dn x = ç ÷= dt dt è Dy ø dy

Shear strain defines the displacement gradient in simple shear. The displacement gradient is the relative displacement of two points divided by the initial distance between them. For any continuous medium the displacement gradient tensor is given as: é ¶u1 ê ¶x ê 1 ¶ui ê ¶u2 = ¶x j êê ¶x1 ê ¶u3 ê ¶x ë 1



¶u1 ù ¶x3 úú ¶u2 ú ¶x3 úú ¶u3 ú ¶x3 úû

¶u1 ¶x2 ¶u2 ¶x2 ¶u3 ¶x2

A non-zero displacement gradient may represent pure rotation, pure deformation or both (Darby, 1976). Thus, each displacement component has two parts:

¶ui 1 æ ¶u ¶u ö 1 æ ¶u ¶u ö = çç i + j ÷÷ + çç i - j ÷÷ ¶x j 2 ¶x j ¶xi ø 2 è ¶x j ¶xi ø è    Pure deformation

Pure rotation

Then the strain tensor (eij) can be defined as: æ ¶u ¶u ö eij = çç i + j ÷÷ è ¶x j ¶xi ø



Similarly, the rotation tensor (rij) can be defined as: æ ¶u ¶u rij = çç j - i ¶ x ¶ xj è i



ö ÷÷ ø

In simple shear, there is only one non-zero displacement gradient component that contributes to both strain and rotation tensors.



é ê0 ¶ui ê = ê0 ¶x j ê ê0 êë

¶ux ¶y 0 0

ù 0ú é0 ú du ê x 0ú = ê0 ú dy ê 0ú ë0 úû

1 0 0

0ù ú 0ú 0 ûú

The time derivative of strain tensor gives the rate of strain tensor (∆ij):

Dij =

æ ¶n ¶u ö ¶ ( eij ) = ¶¶t çç ¶¶xui + ¶x j ÷÷ = ¶¶xni + ¶x j ¶t i ø j i è j

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Handbook of Food Engineering Similarly, the time derivative of the rotation tensor gives the vorticity tensor (Ωij): Wij =



¶n ¶n ¶ rij ) = j - i ( ¶xi ¶x j ¶t

Simple shear flow, or viscometric flow, serves as the basis for many rheological measurement techniques (Bird et al., 1987). The stress tensor in simple shear flow is given as:

é0 ê t = ê t21 êë 0



t12 0 0

0ù ú 0ú 0 úû

There are three shear rate–dependent material functions used to describe material properties in simple shear flow: t12 g N t -t First normal stress coefficient; y1 ( g ) = 11 2 22 = 21 g g t22 - t33 N 2 = 2 Second normal stress coefficient; y 2 ( g ) = g 2 g Viscosity; m ( g ) =

Among the viscometric functions, the viscosity is the most important parameter for a food material. In the case of a Newtonian fluid, both the first and second normal stress coefficients are zero and the material is fully described by a constant viscosity over all shear rates studied. First normal stress data for a wide variety of food materials are available (Dickie and Kokini, 1982; Chang et al., 1990; Wang and Kokini, 1995a; Chesterton et al., 2011; Ng et al., 2011; Torres et al., 2013). Well-known practical examples demonstrating the presence of normal stresses are the Weissenberg or rod climbing effect and the die swell effect. Although the exact molecular origin of normal stresses is not well understood, they are considered to be the result of the elastic properties of viscoelastic fluids (Darby, 1976; Chhabra and Richardson, 2011), and are a measure of the elasticity of the fluids. Figure 1.3 shows the normal stress development for butter at

100.0 sec–1

1.0

(

11− 22)

(

11− 22)

∞

0.8

0.6

10.0 sec–1 1.0 sec–1

0.4

0.1 sec–1

0.2

0.0

0

20

40

60

80

100

120

Time (sec) FIGURE 1.3  Normal stress development for butter at 25°C. (Reproduced with permission from Kokini, J.L. and Dickie, A., Journal of Texture Studies, 12, 539–557, 1981.)

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Linear and Non-Linear Rheological Properties of Foods

25°C. Primary normal stress coefficients versus shear rate plots for various semi-solid food materials on log-log coordinates are shown in Figure 1.4 in the shear rate range 0.1 to 100 s−1. The Weissenberg effect and the die swell effect are known to occur due to the difference in the first normal force. Phan-Thien (2013) discusses the role of the first normal stresses in terms of how they affect the flow down an inclined plane. A Newtonian fluid seems to have a flat surface during the flow down an inclined channel, while a viscoelastic fluid shows a convex surface due to negative second normal force difference.

1.2.3.2 Extensional (Elongational) Flow Pure extensional flow does not involve shearing and is referred to as shear-free flow (Bird et al., 1987; Macosco, 1994). Extensional flows are generically defined by the following velocity field:

1 u x = - e (1 + b ) x 2



1 u y = - e (1 - b ) y 2



uz = +e z

where: 0≤b ≤ 1 e is the elongation rate (Bird et al., 1987; Chhabra and Richardson, 2011). There are three basic types of extensional flow: uniaxial, planar and biaxial, as shown in Figure 1.5. When a cubical material is stretched in one or two direction(s), it gets thinner in the other direction(s) as 105

Strick margarine

105

Apple butter

4

10

Strick butter Canned frosting

Mayonnaise 4

10

Mustard

1

(Pascal sec2)

Ketchup

103

103

102

102

101

101

100

100

10

1

10

2

0.1

1.0

10.0

100.0

10

1

10

2

0.1

1.0

10.0

100.0

–1

Shear rate (sec )

FIGURE 1.4  Steady primary normal stress coefficient ψ1 versus shear rate for semisolid foods at 25°C. (Reproduced with permission from Kokini, J.L. and Dickie, A., Journal of Texture Studies, 12, 539–557, 1981.)

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Handbook of Food Engineering Undeformed y 1

1 x 1 z

Deformed y

y

l =e

1l l

.

y

(t2–t1)

1l l

1l l

1

x 1l l z

l =e

.

x

(t1–t2)

(a)

x

(b)

z

z

1l l

l =e

.

(t2–t1)

(c)

FIGURE 1.5  Types of extensional flows (a) uniaxial, (b) biaxial, (c) planar. (Reproduced with permission from Bird, R.B., et al., Dynamics of Polymeric Liquids, John Wiley and Sons Inc., New York, 1987.)

the volume of the material remains constant. During uniaxial extension, the material is stretched in one direction which results in a corresponding size reduction in the other two directions. In biaxial stretching, a flat sheet of material is stretched in two directions with a corresponding decrease in the third direction. In planar extension, the material is stretched in one direction with a corresponding decrease in thickness while the height remains unchanged. The velocity distribution in Cartesian coordinates and the resulting normal stress differences and viscosities for these three extensional flows are given in Table 1.1 (Bird et al., 1987). The concept of extensional flow measurements goes back to 1906 with measurements conducted by Trouton. Trouton established a mathematical relationship between extensional viscosity and shear TABLE 1.1 Velocity Distribution and Material Functions in Extensional Flow

Velocity distribution

Uniaxial (b = 0, e >0)

Biaxial (b = 0, e 0)

1 u x = - e x 2

u x = +e x

u x = -e x

u y = -2e x

uy = 0

uz = +e z

uz = +e z

σ11 − σ 22 and σ33 − σ 22

σ11 − σ 22

1 u y = - e y 2 uz = +e z Normal stress differences Viscosity

s11 - s22 and s11 - s33 hE =

s11 - s22 s11 - s33 = e e

hB =

s11 - s22 s33 - s22 = e e

hP =

s11 - s22 e

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Linear and Non-Linear Rheological Properties of Foods

viscosity. The dimensionless ratio known as the Trouton number (N T ) is used to compare the relative magnitude of extensional (ηE, ηB or ηP) and shear (η) viscosities:

NT =

Extensional viscosity Shear viscosity

The Trouton ratio for a Newtonian fluid is 3, 6 and 4 in uniaxial, biaxial and planar extensions, respectively (Dealy, 1984). h h h h= E = B = P 3 6 4

1.2.3.3 Volumetric Flows When an isotropic material is subjected to identical normal forces (e.g. hydrostatic pressure) in all directions, it deforms uniformly in all axes resulting in a uniform change (decrease or increase) in dimensions of a cubical element (Figure 1.6). In response to the applied isotropic stress, the specimen changes its volume without any change in its shape. This uniform deformation is called volumetric strain. An isotropic decrease in volume is called a compression, and an isotropic increase in volume is referred to as dilation (Darby, 1976). In this case, all shear stress components will be zero and the normal stresses will be constant and equal:

é1 ê sij = s ê0 êë0

0 1 0

0ù ú 0ú 1 úû

The bulk elastic properties of a material determine how much it will compress under a given amount of isotropic stress (pressure). The modulus relating hydrostatic pressure and volumetric strain is called the bulk modulus (K), which is a measure of the resistance of the material to the change in volume (Ferry, 1980). It is defined as the ratio of normal stress to the relative volume change: K=



s DV V

1.2.4 Response of Viscous and Viscoelastic Materials in Shear and Extension Viscoelastic properties can be measured by experiments which examine the relationship between stress and strain and strain rate in time-dependent experiments. These experiments consist of (1) stress relaxation, (2) creep and (3) small amplitude oscillatory measurements. Stress relaxation (or creep) consists of instantaneously applying a constant strain (or stress) to the test sample and measuring the change in stress (or strain) as a function of time. Dynamic testing consists of applying an oscillatory stress (or strain) to the test sample and determining its strain (or stress) response as a function of frequency. All linear viscoelastic rheological measurements are related and it is possible to calculate one from the other (Ferry, 1980; Macosko, 1994; Banks et al., 2011; Münstedt and Schwarzl, 2014).

V

V′ ∆V=V–V ′

FIGURE 1.6  Volumetric strain.

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Handbook of Food Engineering

1.2.4.1 Stress Relaxation In a stress relaxation test, a constant strain (γ0) is applied to the material at time t0 and the change in the stress over time, τ(t), is measured (Darby, 1976; Macosko, 1994). Ideal viscous, ideal elastic and typical viscoelastic materials show different responses to the applied step strain, as shown in Figure 1.7. When a constant stress is applied at t0, an ideal (Newtonian) fluid responds with an instantaneous infinite stress. An ideal (Hooke) solid responds with instantaneous constant stress at t0 and stress remains constant for t > t0. Viscoelastic materials respond with an initial stress growth which is followed by a decay in time. Upon removal of strain, viscoelastic fluids equilibrate to zero stress (complete relaxation) while viscoelastic solids store some of the stress and equilibrate to a finite stress value (partial recovery) (Darby, 1976). The relaxation modulus, G(t), is an important rheological property measured during stress relaxation. It is the ratio of the measured stress to the applied initial strain at constant deformation. The relaxation modulus has units of stress (Pascals in SI): G (t ) =



t g0

A logarithmic plot of G(t) versus time is useful in observing the relaxation behavior of different classes of materials, as shown in Figure 1.8. In glassy polymers, there is a little stress relaxation over many decades of logarithmic time scale. Crosslinked rubber shows a short time relaxation followed by a constant modulus, caused by the network structure. Concentrated solutions show a similar qualitative response but only at very small strain levels caused by entanglements. High molecular weight concentrated polymeric liquids show a nearly constant equilibrium modulus followed by a sharp fall after long periods of time caused by disentanglement. Molecular weight has a significant impact on relaxation time, the smaller the molecular weight the shorter the relaxation time. Moreover, a narrower Input

Responses Ideal fluid

t0

t

t0

Ideal solid

t

t0

Viscoelastic

t

t0

Solid Fluid t

FIGURE 1.7  Response of ideal fluid, ideal solid, and viscoelastic materials to imposed step strain. (From Darby, R., Viscoelastic Fluids: An Introduction to Their Properties and Behavior, Dekker Inc., New York, 1976.)

Crosslinking

Glass G0 log G

Mw

Rubber (concentrated suspension) Polymeric liquid

Dilute solution

log t FIGURE 1.8  Typical relaxation modulus data for various materials. (Reproduced with permission from Macosko, C.W., Rheology: Principles, Measurements and Applications, VCH Publishers, Inc., New York, 1994.)

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Linear and Non-Linear Rheological Properties of Foods

molecular weight distribution results in a much sharper drop in relaxation modulus. Uncrosslinked polymers, dilute solutions and suspensions show complete relaxation after short time periods. In these materials, G(t) falls rapidly and eventually vanishes (Ferry, 1980; Macosko, 1994; Gallegos and Martinez Boza, 2010).

1.2.4.2 Creep In a creep test, a constant stress (τ0) is applied at time t0 and removed at time t1, and the corresponding strain γ(t) is measured as a function of time. As in the case with stress relaxation, various materials respond in different ways, as shown by typical creep data given in Figure 1.9. A Newtonian fluid responds with a constant rate of strain from t0 to t1; the strain attained at t1 remains constant for times t > t1 (no strain recovery). An ideal (Hooke) solid responds with a constant strain from t0 to t1, which is recovered completely at t1. A viscoelastic material responds with a nonlinear strain. Strain level approaches a constant rate for a viscoelastic fluid and a constant magnitude for a viscoelastic solid. When the imposed stress is removed at t1, the solid recovers completely at a finite rate, but the recovery is incomplete for the fluid (Darby, 1976). The rheological property of interest is the ratio of strain to stress, as a function of time is referred to as the creep compliance, J(t). J (t ) =



g (t ) t0

The compliance has units of Pa−1 and describes how compliant a material is. The greater the compliance, the easier it is to deform the material. By monitoring how the strain changes as a function of time, the magnitude of elastic and viscous components can be evaluated using available viscoelastic models. Creep testing also provides means to determine the zero shear viscosity of fluids, such as polymer melts and concentrated polymer solutions, at extremely low shear rates. Creep data are usually expressed as logarithmic plots of creep compliance versus time (Figure 1.10). Glassy materials show a low compliance due to the absence of any configurational rearrangements. Highly crystalline or concentrated polymers exhibit creep compliance increasing slowly with time. More liquid-like materials, such as low molecular weight or dilute polymers, show higher creep compliance and faster increase in J(t) with time (Ferry, 1980).

1.2.4.3 Small Amplitude Oscillatory Measurements In small amplitude oscillatory flow experiments, a sinusoidal oscillating stress or strain with a frequency (ω) is applied to the material and the oscillating strain or stress response is measured along with the phase difference between the oscillating stress and strain. The input strain (γ) varies with time according to the relationship g = g 0 sin wt

Input

Responses Ideal fluid

Ideal solid

Viscoelastic

Fluid t0

t1

t

t0

t1

t

t0

t1

t

t0

t1

t

solid

FIGURE 1.9  Response of ideal fluid, ideal solid, and viscoelastic materials to imposed instantaneous step stress. (From Darby, R., Viscoelastic Fluids: An Introduction to Their Properties and Behavior, Dekker Inc., New York, 1976 and Findley, W.N., et al., Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover Publications Inc., New York, 2013.)

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Handbook of Food Engineering

Steady-state

Dilute solution log J (t )

Polymeric liquid Mw Crosslinking

Rubber (concentrated suspension) Glass

log t FIGURE 1.10  Typical creep modulus data for various materials. (From Ferry, J., Viscoelastic Properties of Polymers, John Wiley and Sons, New York, 1980.)

and the rate of strain is given by g = g 0w cos wt



where γ0 is the amplitude of strain. The corresponding stress (τ) can be represented as t = t0 sin ( wt + d )

where: τ0 δ δ = 0 δ = 90° 0   1.0: Concave upward, sensation grows more and more rapidly as the stimulus increases b  c*) were dependent upon the degree of branching, i.e. the higher the branching, the higher the gradients, whereas there was no significant difference in the dilute region (c