Transport Properties of Foods
FOOD SCIENCE AND TECHNOLOGY A Series of Monographs, Textbooks, and Reference Books
EDITORIAL BOARD
Senior Editors Owen R. Fennema University of Wisconsin-Madison Marcus Karel Rutgers University (emeritus) Gary W. Sanderson Universal Foods Corporation (retired) Pieter Walstra Wageningen Agricultural University John R. Whitaker University of California-Davis Additives P. Michael Davidson University of Tennessee-Knoxville Dairy science James L. Steele University of Wisconsin-Madison Flavor chemistry and sensory analysis John Thorngate University of Idaho-Moscow Food engineering Daryl B. Lund Cornell University Health and disease Seppo Salminen University of Turku, Finland Nutrition and nutraceuticals Mark Dreher Mead Johnson Nutritionals Processing and preservation Gustavo V. Barbosa-Canovas Washington State University-Pullman Safety and toxicology Sanford Miller University of Texas-Austin
1. Flavor Research: Principles and Techniques, R. Teranishi, I. Hornstein, P. Issenberg, and E. L. Wick 2. Principles of Enzymology for the Food Sciences, John R. Whitaker 3. Low-Temperature Preservation of Foods and Living Matter, Owen R. Fennema, William D. Powrie, and Elmer H. Marth 4. Principles of Food Science Part I: Food Chemistry, edited by Owen R. Fennema Part II: Physical Methods of Food Preservation, Marcus Karel, Owen R. Fennema, and Daryl B. Lund 5. Food Emulsions, edited by Stig E. Friberg 6. Nutritional and Safety Aspects of Food Processing, edited by Steven R. Tannenbaum 7. Flavor Research: Recent Advances, edited by R. Teranishi, Robert A. Flath, and Hiroshi Sugisawa 8. Computer-Aided Techniques in Food Technology, edited by Israel Saguy 9. Handbook of Tropical Foods, edited by Harvey T. Chan 10. Antimicrobials in Foods, edited by Alfred Larry Branen and P. Michael Davidson
11. Food Constituents and Food Residues: Their Chromatographic Determination, edited by James F. Lawrence 12. Aspartame: Physiology and Biochemistry, edited by Lewis D. Stegink and L. J. Filer, Jr. 13. Handbook of Vitamins: Nutritional, Biochemical, and Clinical Aspects, edited by Lawrence J. Machlin 14. Starch Conversion Technology, edited by G. M. A, van Beynum and J. A. Roe Is 15. Food Chemistry: Second Edition, Revised and Expanded, edited by Owen R. Fennema 16. Sensory Evaluation of Food: Statistical Methods and Procedures, Michael O'Mahony 17. Alternative Sweeteners, edited by Lyn O'Brien Nabors and Robert C. Gelardi 18. Citrus Fruits and Their Products: Analysis and Technology, S. V. Ting and Russell L. Rouseff 19. Engineering Properties of Foods, edited by M, A. Rao and S. S. H. Rizvi 20. Umami: A Basic Taste, edited by Yojiro Kawamura and Morley R. Kare 21. Food Biotechnology, edited by Dietrich Knorr 22. Food Texture: Instrumental and Sensory Measurement, edited by Howard R. Moskowitz 23. Seafoods and Fish Oils in Human Health and Disease, John E. Kinsella 24. Postharvest Physiology of Vegetables, edited by J. Weichmann 25. Handbook of Dietary Fiber: An Applied Approach, Mark L. Dreher 26. Food Toxicology, Parts A and B, Jose M. Concon 27. Modern Carbohydrate Chemistry, Roger W. Binkley 28. Trace Minerals in Foods, edited by Kenneth T. Smith 29. Protein Quality and the Effects of Processing, edited by R. Dixon Phillips and John W. Finley 30. Adulteration of Fruit Juice Beverages, edited by Steven Nagy, John A. Attaway, and Martha E. Rhodes 31. Foodborne Bacterial Pathogens, edited by Michael P. Doyle 32. Legumes: Chemistry, Technology, and Human Nutrition, edited by Ruth H. Matthews 33. Industrialization of Indigenous Fermented Foods, edited by Keith H. Steinkraus 34. International Food Regulation Handbook: Policy • Science • Law, edited by Roger D. Middlekauffand Philippe Shubik 35. Food Additives, edited by A. Larry Branen, P. Michael Davidson, and Seppo Salminen 36. Safety of Irradiated Foods, J. F. Diehl 37. Omega-3 Fatty Acids in Health and Disease, edited by Robert S. Lees and Marcus Karel 38. Food Emulsions: Second Edition, Revised and Expanded, edited by Kare Larsson and Stig E. Friberg 39. Seafood: Effects of Technology on Nutrition, George M. Pigott and Barbee W. Tucker 40. Handbook of Vitamins: Second Edition, Revised and Expanded, edited by Lawrence J. Machlin
41. Handbook of Cereal Science and Technology, Klaus J. Lorenz and Karel Kulp 42. Food Processing Operations and Scale-Up, Kenneth J. Valentas, Leon Levine, and J. Peter Clark 43. Fish Quality Control by Computer Vision, edited by L. F. Pau and R. Olafsson 44. Volatile Compounds in Foods and Beverages, edited by Henk Maarse 45. Instrumental Methods for Quality Assurance in Foods, edited by Daniel Y. C. Fung and Richard F. Matthews 46. Listeria, Listeriosis, and Food Safety, Elliot T. Ryserand Elmer H. Marth 47. Acesulfame-K, edited by D. G. Mayerand F. H. Kemper 48. Alternative Sweeteners: Second Edition, Revised and Expanded, edited by Lyn O'Brien Nabors and Robert C. Gelardi 49. Food Extrusion Science and Technology, edited by Jozef L. Kokini, ChiTang Ho, and Mukund V. Karwe 50. Surimi Technology, edited by Tyre C, Lanierand Chong M. Lee 51. Handbook of Food Engineering, edited by Dennis R. Heldman and Daryl B. Lund 52. Food Analysis by HPLC, edited by Leo M, L. Nollet 53. Fatty Acids in Foods and Their Health Implications, edited by Ching Kuang Chow 54. Clostridium botulinum: Ecology and Control in Foods, edited by Andreas H. W. Hauschild and Karen L. Dodds 55. Cereals in Breadmaking: A Molecular Colloidal Approach, Ann-Charlotte Eliasson and Kare Larsson 56. Low-Calorie Foods Handbook, edited by Aaron M. Altschul 57. Antimicrobials in Foods: Second Edition, Revised and Expanded, edited by P. Michael Davidson and Alfred Larry Branen 58. Lactic Acid Bacteria, edited by Seppo Salminen and Atte von Wright 59. Rice Science and Technology, edited by Wayne E, Marshall and James I. Wadsworth 60. Food Biosensor Analysis, edited by Gabriele Wagner and George G. Guilbault 61. Principles of Enzymology for the Food Sciences: Second Edition, John R. Whitaker 62. Carbohydrate Polyesters as Fat Substitutes, edited by Casimir C. Akoh and Barry G. Swanson 63. Engineering Properties of Foods: Second Edition, Revised and Expanded, edited by M. A. Rao and S. S. H. Rizvi 64. Handbook of Brewing, edited by William A. Hardwick 65. Analyzing Food for Nutrition Labeling and Hazardous Contaminants, edited by Ike J. Jeon and William G. Ikins 66. Ingredient Interactions: Effects on Food Quality, edited by Anilkumar G. Gaonkar 67. Food Polysaccharides and Their Applications, edited by Alistair M. Stephen 68. Safety of Irradiated Foods: Second Edition, Revised and Expanded, J. F. Diehl 69. Nutrition Labeling Handbook, edited by Ralph Shapiro
70. Handbook of Fruit Science and Technology: Production, Composition, Storage, and Processing, edited by D. K. Salunkhe and S. S. Kadam 71. Food Antioxidants: Technological, Toxicological, and Health Perspectives, edited by D. L. Madhavi, S. S. Deshpande, and D. K. Salunkhe 72. Freezing Effects on Food Quality, edited by Lester E. Jeremiah 73. Handbook of Indigenous Fermented Foods: Second Edition, Revised and Expanded, edited by Keith H. Steinkraus 74. Carbohydrates in Food, edited by Ann-Charlotte Eliasson 75. Baked Goods Freshness: Technology, Evaluation, and Inhibition of Staling, edited by Ronald E. Hebeda and Henry F, Zobel 76. Food Chemistry: Third Edition, edited by Owen R. Fennema 77. Handbook of Food Analysis: Volumes 1 and 2, edited by Leo M. L. Nollet 78. Computerized Control Systems in the Food Industry, edited by Gauri S. Mittal 79. Techniques for Analyzing Food Aroma, edited by Ray Marsili 80. Food Proteins and Their Applications, edited by Srinivasan Damodaran and Alain Paraf 81. Food Emulsions: Third Edition, Revised and Expanded, edited by Stig E. Friberg and Kare Larsson 82. Nonthermal Preservation of Foods, Gustavo V. Barbosa-Canovas, Usha R. Pothakamury, Enrique Palou, and Barry G. Swanson 83. Milk and Dairy Product Technology, Edgar Spreer 84. Applied Dairy Microbiology, edited by Elmer H. Marth and James L. Steele 85. Lactic Acid Bacteria: Microbiology and Functional Aspects: Second Edition, Revised and Expanded, edited by Seppo Salminen and Atte von Wright 86. Handbook of Vegetable Science and Technology: Production, Composition, Storage, and Processing, edited by D. K. Salunkhe and S. S. Kadam 87. Polysaccharide Association Structures in Food, edited by Reginald H. Walter 88. Food Lipids: Chemistry, Nutrition, and Biotechnology, edited by Casimir C. Akoh and David B. Min 89. Spice Science and Technology, Ken/7 Hirasa and Mitsuo Takemasa 90. Dairy Technology: Principles of Milk Properties and Processes, P. Walstra, T. J. Geurts, A. Noomen, A. Jellema, and M. A. J. S. van Boekel 91. Coloring of Food, Drugs, and Cosmetics, Gisbert Otterstatter 92. Listeria, Listeriosis, and Food Safety: Second Edition, Revised and Expanded, edited by Elliot T. Ryser and Elmer H. Marth 93. Complex Carbohydrates in Foods, edited by Susan Sungsoo Cho, Leon Prosky, and Mark Dreher 94. Handbook of Food Preservation, edited by M. Shafiur Rahman 95. International Food Safety Handbook: Science, International Regulation, and Control, edited by Kees van der Heijden, /Waged Younes, Lawrence Fishbein, and Sanford Miller 96. Fatty Acids in Foods and Their Health Implications: Second Edition, Revised and Expanded, edited by Ching Kuang Chow
97. Seafood Enzymes: Utilization and Influence on Postharvest Seafood Quality, edited by Norman F. Haard and Benjamin K. Simpson 98. Safe Handling of Foods, edited by Jeffrey M. Farber and Ewen C. D. Todd 99. Handbook of Cereal Science and Technology: Second Edition, Revised and Expanded, edited by Karel Kulp and Joseph G. Ponte, Jr. 100. Food Analysis by HPLC: Second Edition, Revised and Expanded, edited by Leo M. L. Nollet 101. Surimi and Surimi Seafood, edited by Jae W. Park 102. Drug Residues in Foods: Pharmacology, Food Safety, and Analysis, Nickos A. Botsoglou and Dimitrios J. Fletouris 103. Seafood and Freshwater Toxins: Pharmacology, Physiology, and Detection, edited by Luis M. Botana 104. Handbook of Nutrition and Diet, Babasaheb B. Desai 105. Nondestructive Food Evaluation: Techniques to Analyze Properties and Quality, edited by Sundaram Gunasekaran 106. Green Tea: Health Benefits and Applications, Yukihiko Hara 107. Food Processing Operations Modeling: Design and Analysis, edited by Joseph Irudayaraj 108. Wine Microbiology: Science and Technology, Claudio Delfini and Joseph V. Formica 109. Handbook of Microwave Technology for Food Applications, edited by Ashim K. Datta and Ramaswamy C. Anantheswaran 110. Applied Dairy Microbiology: Second Edition, Revised and Expanded, edited by Elmer H. Marth and James L. Steele 111. Transport Properties of Foods, George D. Saravacos and Zacharias B. Maroulis 112. Alternative Sweeteners: Third Edition, Revised and Expanded, edited by Lyn O'Brien Nabors
Additional Volumes in Preparation Food Additives: Second Edition, Revised and Expanded, edited by Alfred Larry Branen, P. Michael Davidson, Seppo Salminen, and John H. Thorngate Control of Foodborne Microorganisms, edited by Vijay K. Juneja and John N. Sofos Handbook of Dietary Fiber, edited by Susan Sungsoo Cho and Mark L. Ore her
Flavor, Fragrance, and Odor Analysis, edited by Ray Marsili
Transport Properties of Foods George D. Sanauacos Rutgers University New Brunswick, New Jersey and National Technical University of Athens
Athens, Greece
Zacharias B. Maroulis
National Technical University of Athens Athens, Greece
MARCEL DEKKER, INC.
NEW YORK • BASEL
ISBN: 0-8247-0613-7
This book is printed on acid-free paper.
Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above.
Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA
To our wives Katie G. Saravacos andRena Z. Maroulis for their encouragement and support
Preface The basic transport properties of momentum (flow), heat and mass are an important part of the engineering properties of foods, which are essential in the design, operation, and control of food processes and processing equipment. They are also useful in the quantitative analysis and evaluation of food quality and food safety during processing, packaging, storage and distribution of foods. The engineering properties are receiving increasing attention recently due to the need for more efficient processes and equipment for high quality and convenient food products, under strict environmental and economic constraints. The fundamentals of transport properties were developed in chemical engineering for simple gases and liquids, based on molecular dynamics and thermodynamics. However, the complex structure of solid, semi-solid, and fluid foods prevents the direct use of molecular dynamics for the prediction of the transport properties of foods. Thus, experimental measurements and empirical correlations are essential for the estimation of these important food properties. The need for reliable experimental data on physical properties of foods, especially on transport properties, was realized by the development of national and international research programs, like the European cooperative projects COST 90 and COST 90 bis, which dealt with such properties as viscosity, thermal conductivity, and mass diffusivity of foods. One outcome of these projects was the importance of context (relevancy) of the measurement and sample conditions. This explains the wide variation of the food transport properties, particularly mass diffusivity. Statistical analysis of compiled literature data may yield general conclusions and certain empirical "constants", which characterize the transport property (thermal conductivity or moisture diffusivity) of a given food or food class. All transport properties are structure-sensitive at the three levels, i.e. molecular, microstructural, and macrostructural. Correlation of food macrostructure to transport properties is relatively easy by means of measurements of density, porosity, and shrinkage. Correlation to molecular and microstructural (cellular) structure, although more fundamental, is difficult and requires further theoretical and applied work before wider application in food systems. The material of this book is arranged in a logical order: The introduction, Chapter 1, summarizes the contents of the book, emphasizing the need for a unified approach to the transport properties based on certain general principles. Chapter 2 introduces the fundamental transport properties as applied to simple gases and liquids. The three levels of food structure, molecular, micro- and macrostruc-
vi
Preface
ture, as related to transport properties are reviewed in Chapter 3. A unified treatment of the rheological properties of fluid foods is presented in Chapter 4. The theory, measurement and experimental data of moisture diffusivity are discussed in Chapter 5, while a statistical treatment of the literature data on moisture diffusivity is presented in Chapter 6. The diffusion of solutes in food systems is discussed in Chapter 7, with special reference to flavor retention and food packaging films and coatings. Thermal conductivity and thermal diffusivity are discussed in Chapter 8. Finally, heat and mass transfer coefficients are treated together in Chapter 9. We wish to acknowledge the contributions and help of several persons to our efforts over the years to prepare and utilize the material used in this book: Our colleagues, associates, and graduate students D. Marinos-Kouris, A. Drouzas, C. Kiranoudis, M. Krokida, N. Panagiotou, N. Zogzas of the National Technical University, Athens; M. Solberg, M. Karel, J. Kokini, K. Hayakawa, V. Karathanos, S. Marousis, K. Shah, and N. Papantonis of CAFT and Rutgers University; M. Bourne and A. Rao of Cornell University, Geneva, NY; A. Kostaropoulos of the Agricultural University of Athens; and V. Gekas of the Technical University of Crete. We also appreciate the discussions with the members of the European groups of cooperative projects COST 90 and COST 90 bis, especially R. Jowitt and W. Spiess. Special thanks are due to Dr. Magda Krokida for her substantial contributions in compilation and statistical analysis of the extensive literature data on transport properties of foods, and her continued help in preparing the illustrations and typing the manuscript. Finally, we wish to thank the staff of the publisher Marcel Dekker, Inc., especially Maria Allegra and Theresa Dominick, for their help and encouragement. We hope that this book will help the efforts to develop and establish food engineering as a basic discipline in the wide area of food science and technology. We welcome any comments and criticism from the readers. We regret any errors in the text that may have escaped our attention.
GeorgeD. Saravacos Zacharias B. Maroulis
Contents Preface
1. Introduction
1
I. RHEOLOGICAL PROPERTIES II. THERMAL TRANSPORT PROPERTIES III. MASS TRANSPORT PROPERTIES
3 3 4
2. Transport Properties of Gases and Liquids
7
I. INTRODUCTION II. ANALOGIES OF TRANSPORT PROCESSES III. MOLECULAR BASIS OF TRANSPORT PROCESSES A. Ideal Gases B. Thermodynamic Quantities C. Real Gases IV. PREDICTION OF TRANSPORT PROPERTIES OF FLUIDS A. Real Gases B. Liquids C. Comparison of Liquid/Gas Transport Properties D. Gas Mixtures V. TABLES AND DATA BANKS OF TRANSPORT PROPERTIES
7 8 9 9 10 12 14 15 16 18 19 19
3. Food Structure and Transport Properties I. INTRODUCTION II. MOLECULAR STRUCTURE A. Molecular Dynamics and Molecular Simulations
29 29 29 29 vii
viii
Contents
B. Food Materials Science C. Phase Transitions D. Colloid and Surface Chemistry III. FOOD MICROSTRUCTURE AND TRANSPORT PROPERTIES A. Examination of Food Microstructure B. Food Cells and Tissues C. Microstructure and Food Processing D. Microstructure and Mass Transfer IV. FOOD MACROSTRUCTURE AND TRANSPORT PROPERTIES A. Definitions B. Food Macrostructure and Transport Properties C. Determination of Food Macrostructure D. Macrostructure of Model Foods E. Macrostructure of Fruit and Vegetable Materials
30 30 31 32 32 32 34 34 36 36 40 45 46 50
4. Rheological Properties of Fluid Foods
63
I. INTRODUCTION II. RHEOLOGICAL MODELS OF FLUID FOODS A. Structure and Fluid Viscosity B. Non-Newtonian Fluids C. Effect of Temperature and Concentration D. Dynamic Viscosity III. VISCOMETRIC MEASUREMENTS A. Viscometers B. Measurements on Fluid Foods IV. RHEOLOGICAL DATA OF FLUID FOODS A. Edible Oils B. Aqueous Newtonian Foods C. Plant Biopolymer Solutions and Suspensions D. Cloudy Juices and Pulps E. Emulsions and Complex Suspensions V. REGRESSION OF RHEOLOGICAL DATA OF FOODS A. Edible Oils B. Fruit and Vegetable Products C. Chocolate
63 66 66 68 71 73 74 74 78 79 79 80 85 89 90 92 92 94 100
Contents
5. Transport of Water in Food Materials
ix
105
I. INTRODUCTION II. DIFFUSION OF WATER IN SOLIDS A. Diffusion of Water in Polymers III. DETERMINATION OF MASS DIFFUSIVITY IN SOLIDS A. Sorption Kinetics B. Permeability Methods C. Distribution of Diffusant D. Drying Methods E. Simplified Methods F. Simulation Method G. Numerical Methods H. Regular Regime Method I. Shrinkage Effect IV. MOISTURE DIFFUSIVITY IN MODEL FOOD MATERIALS A. Effect of Measurement Method B. Effect of Gelatinization and Extrusion C. Effect of Sugars D. Effect of Proteins and Lipids E. Effect of Inert Particles F. Effect of Pressure G. Effect of Porosity H. Effect of Temperature I. Drying Mechanisms V. WATER TRANSPORT IN FOODS A. Mechanisms of Water Transport B. Effective Moisture Diffusivity C. Water Transport in Cellular Foods D. Water Transport in Osmotic Dehydration E. Effect of Physical Structure F. Effect of Physical/Chemical Treatments G. Characteristic Moisture Diffusivities of Foods
105 106 107 109 110 114 118 120 123 124 124 12 5 126 127 127 13 0 133 13 5 137 138 140 141 143 144 144 145 146 147 150 152 155
6. Moisture Diffusivity Compilation of Literature Data for Food Materials
163
I. INTRODUCTION II. DATA COMPILATION
163 164
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Contents
III. MOISTURE DIFFUSIVITY OF FOODS AS A FUNCTION OF MOISTURE CONTENT AND TEMPERATURE
197
7. Diffusivity and Permeability of Small Solutes in Food Systems
237
I. INTRODUCTION A. Diffusivity of Small Solutes B. Measurement of Diffusivity II. DIFFUSIVITY IN FLUID FOODS A. Dilute Solutions B. Concentrated Solutions III. DIFFUSION IN POLYMERS A. Diffusivity of Small Solutes in Polymers B. Glass Transition C. Clustering of Solutes in Polymers D. Prediction of Diffusivity
237 237 239 241 241 242 243 244 246 247 248
IV. DIFFUSION OF SOLUTES IN FOODS A. Diffusivity of Salts
251 251
B. Diffusivity of Organic Components
252
C. Volatile Flavor Retention D. Flavor Encapsulation V. PERMEABILITY IN FOOD SYSTEMS A. Permeability B. Food Packaging Films C. Food Coatings D. Permeability/Diffusivity Relation
254 258 259 259 261 262 263
8. Thermal Conductivity and Diffusivity of Foods I. INTRODUCTION II. MEASUREMENT OF THERMAL CONDUCTIVITY AND DIFFUSIVITY A. Thermal Conductivity B. Thermal Diffusivity III. THERMAL CONDUCTIVITY AND DIFFUSIVITY DATA OF FOODS
269 269 270 270 273
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A. Unfrozen Foods B. Frozen Foods C. Analogy of Heat and Mass Diffusivity D. Empirical Rules IV. MODELING OF THERMAL TRANSPORT PROPERTIES A. Composition Models B. Structural Models V. COMPILATION OF THERMAL CONDUCTIVITY DATA OF FOODS VI. THERMAL CONDUCTIVITY OF FOODS AS A FUNCTION OF MOISTURE CONTENT AND TEMPERATURE
9. Heat and Mass Transfer Coefficients in Food Systems I. INTRODUCTION II. HEAT TRANSFER COEFFICIENTS A. Definitions B. Determination of Heat Transfer Coefficients C. General Correlations of the Heat Transfer Coefficient D. Simplified Equations for Air and Water III. MASS TRANSFER COEFFICIENTS A. Definitions B. Determination of Mass Transfer Coefficients C. Empirical Correlations D. Theories of Mass Transfer IV. HEAT TRANSFER COEFFICIENTS IN FOOD SYSTEMS A. Heat Transfer in Fluid Foods B. Heat Transfer in Canned Foods C. Evaporation of Fluid Foods D. Improvement of Heat/Mass Transfer V. HEAT TRANSFER COEFFICIENTS IN FOOD PROCESSING: COMPILATION OF LITERATURE DATA VI. MASS TRANSFER COEFFICIENTS IN FOOD PROCESSING: COMPILATION OF LITERATURE DATA
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275 276 276 279 280 280 283 289 326
359 359 360 360 361 362 364 364 364 365 366 367 369 369 371 372 373 374 391
Appendix: Notation
403
Index
407
1 Introduction
The transport properties of momentum (flow), heat and mass of unit operations are an important part of the physical and engineering properties of foods, which are necessary for the quantitative analysis, design, and control of food processes and food quality. The transport of momentum (rheological properties) and heat (thermal conductivity) have received more attention in the past (Rao, 1999; Rahman, 1995). However, mass transport is getting more attention recently, due to its importance to several traditional and new food processing operations (Saravacos, 1995). The transport properties of gases and liquids have been studied extensively and they are a basic element in the design of chemical processes and processing equipment (Reid et al., 1987). The theoretical analysis and applications of transport phenomena have been advanced by a unified treatment of the three basic transport processes (Brodkey and Hershey, 1988). The adoption of transport phenomena in food systems is expected to advance the emerging field of food engineering (Gekas, 1992). However, foods are complex heterogeneous and sensitive materials, mostly solids or semisolids, and application of the principles of transport phenomena requires sustained experimental and theoretical efforts. Application of modern computer aided design (CAD) to food processing has been limited by the lack of reliable transport data for the various food processes and food materials. Mathematical modeling and simulations have made considerable progress, but the accuracy of the available scattered data is not adequate for quantitative applications. Of particular importance is the need for mass transport properties (Saravacos and Kostaropoulos, 1995; 1996). While analysis and computation of the transport properties of gases and liquids is based on molecular dynamics, experimental measurements are necessary for the food materials and food processing systems.
2
Chapter 1
Theoretical analysis and experimental techniques of mass diffusion in polymeric materials, developed in polymer science (Vieth, 1991) are finding important applications in food materials science and in food process engineering. Molecular dynamics and molecular simulation techniques, developed for the prediction of mass diffusion in polymer science (Theodorou, 1996), could conceivably be utilized in food systems, although the complexity of foods would make such an effort very difficult. The transport properties are directly related to the microstructure of food materials, but limited studies and applications have been reported in the literature (Aguilera and Stanley, 1999; Aguilera, 2000). Food microstructure plays a particularly important role in mass transfer at the cellular level, for example in fruits and vegetables during osmotic dehydration. Food macrostructure has been used widely to analyze and model transport mechanisms, particularly mass diffusivity and thermal conductivity. Simple measurements of density, porosity and shrinkage can provide quantitative information on the heat and mass transport properties in important food processing operations, such as dehydration and frying. A thorough analysis of the transport properties should involve the momentum, heat and mass transport mechanisms at the molecular, microstructural, and macrostructural levels. Such a unified analysis might reveal any analogies among the three transport processes, which would be very helpful in prediction and empirical correlations of the properties, like the analogies for gases and liquids. Reliable data on transport properties of foods are essential because of the various non-standardized methods used, and the variability of composition and structure of food materials. An international effort to obtain standardized data of rheological properties (viscosity), heat conductivity, and mass diffusivity was made in the European collaborative research projects COST 90 and COST 90bis (Jowitt et al., 1983; 1987). The viscosity and thermal conductivity of foods were investigated in a U.S. Department of Agriculture (USDA) cooperative research project (Okos, 1987). Accurate and useful data were obtained for viscosity and thermal conductivity, but only limited mass diffusivity data were obtained, demonstrating that mass transport is a much more complicated process. An important conclusion of these projects is relevancy, i.e. each property refers to a given set of experimental conditions and sample material. A comprehensive treatment of the transport properties of foods should be based on the transport at the molecular, microstructural and macrostructural levels, and should consider the available literature data in a generalized form of statistical analysis.
Introduction
3
I. RHEOLOGICAL PROPERTIES
Food rheology has been primarily concerned with food texture and food quality. However, rheological data of fluid foods are essential in the analysis and design of important food processing operations, like pumping, heating and cooling, evaporation, and thermal processing (both in cans and aseptic processing). Most fluid foods are non-Newtonian fluids, and empirical Theological data are necessary (Rao, 1999). Statistical (regression) analysis of published rheological data can provide useful correlations for groups and typical fluid foods (see Chapter 4). The effect of temperature on the viscosity of fluid foods appears to be related to the molecular and microstrure of the material: High energies of activation for flow (about 50 kJ/mol) are observed in concentrated aqueous sugar solutions and fruit juices, while very low values (near 10 kJ/mol) characterize the highly non-Newtonian (and viscous) fruit purees and pulps (Saravacos, 1970).
II. THERMAL TRANSPORT PROPERTIES
Thermal conductivity represents the basic thermal transport property, and it shows a wider variation than thermal diffusivity, which can be estimated accurately from the thermal conductivity. The thermal conductivity of fluid foods is a weak function of their composition, and simple empirical models can be used for its estimation. Structural models are needed to model the thermal conductivity of solid foods, which varies widely, due to differences in micro- and macrostructure of the heterogeneous materials. Heat and mass transfer analogies in porous foods may be related to the known analogies of gas systems. Application of structural models of thermal conductivity to model foods has demonstrated the importance of porosity of granular or porous materials (Maroulis et al., 1990). Regression analysis of published data of thermal conductivity of various foods, as a function of moisture and temperature, can provide useful empirical parameters characteristic of each material. Such parameters are the thermal conductivity and the energy of activation of dry and infinitely wet materials (see Chapter 8).
Chapter 1 III. MASS TRANSPORT PROPERTIES
The diffusion model, developed for mass transport in fluid systems (Cussler, 1997), has been applied widely to mass transfer in food materials, assuming that the driving force is a concentration gradient. Since mass transfer in heterogeneous systems may involve other mechanisms than molecular diffusion, the estimated mass transport property is an effective (or apparent) diffusivity. Most of the published data on mass transport in food systems refer to moisture (water) diffusivity (Marinos-Kouris and Maroulis, 1995), since the transport of water is of fundamental importance to many food processes, like dehydration, and to food quality changes during storage. Mass transport in foods is strongly affected by the molecular, micro- and macrostructure of food materials. The crucial role of porosity in moisture transfer has been demonstrated by measurements on model foods of various structures, and on typical food materials (Marousis et al., 1991; Saravacos, 1995). The effect of temperature on moisture diffusivity may provide an indication whether mass transfer is controlled by air or liquid/solid phase of the food material. Low energies of activation for diffusion (about 10 kJ/mol) are obtained in porous materials, while high values (near 50 kJ/mol) are observed in nonporous products. The wide range of moisture diffusivities reported in the literature is caused primarily by the large differences in mass diffusivity among the vapor, liquid, and solid phases present in heterogeneous food materials. The diffusivity in the solid phase is also affected strongly by the physical state, i.e. glassy, rubbery or crystalline. Application of polymer science to food systems containing biopolymers can improve the understanding of the underlying transport mechanisms (see Chapters 5 and 7). Statistical (regression) analysis of published literature data on moisture diffusivity, using an empirical model as a function of moisture content and temperature, can provide useful parameters, such as diffusivity and activation energy in the dry and infinitely wet phases (see Chapter 6). Cellular models for mass transfer can provide an insight into the process of osmotic dehydration, where water and solutes are transported simultaneously. However, the diffusion model is often used, because of its simplicity, for the estimation of mass diffusivity of water and solutes during the osmotic process. Mass transport of important food solutes, such as nutrients and flavor/aroma components, is usually treated as a diffusion process, and effective mass diffusivities are used in various food processes and food quality changes, like aroma retention (see Chapter 7).
Introduction
REFERENCES
Aguilera, J.M. 2000. Microstructure and Food Product Engineering. Food Technol 54(ll):56-65. Aguilera, J.M., Stanley, D.W. 1999. Microstructural Principles of Food Processing, Engineering. 2nd ed. Gaithersburg, MD: Aspen Publ. Brodkey, R.S., Hershey, H.C. 1988. Transport Phenomena. A Unified Approach. New York: McGraw-Hill. Cussler, E.L. 1997. Diffusion Mass Transfer in Fluid Systems. Cambridge, UK: Cambridge University Press. Gekas, V. 1992. Transport Phenomena of Foods and Biological Materials. New York: CRC Press. Jowitt, R., Escher, F., Hallstrom, H., Meffert, H.F.Th., Spiess, W.E.L., Vos, G., eds. 1983. Physical Properties of Foods. London: Applied Science Publ. Jowitt, R., Escher, F., Kent, M., McKenna, B., Roques, M., eds. 1987. Physical Properties of Foods 2. London: Elsevier Applied Science. Marinos-Kouris, D., Maroulis, Z.B. 1995. Transport Properties in the Air-Drying of Solids. In: Handbook of Industrial Drying, 2nd ed. Vol.1, Mujumdar, A.S. ed. New York: Marcel Dekker. Maroulis, Z.B., Drouzas, A.E., Saravacos, G.S. 1990. Modeling of Thermal Conductivity of Granular Starches. J Food Eng 11:255-271. Marousis, S.N., Karathanos, V.T., Saravacos, G.S. 1991. Effect of Physical Structure of Starch Materials on Water Diffusivity. J Food Proc Preserv 15:183195. Okos, M.R., ed. 1987. Physical and Chemical Properties of Foods. ASAE Publication No. Q0986, St. Joseph, MI. Rahman, S. 1995. Food Properties Handbook. Boca Raton, FL: CRC Press. Rao, M.A. 1999. Rheology of Fluid and Semisolid Foods. Gaithersburg, MD: Aspen Publ. Reid, R.C., Prausnitz, J.M., Poling, B.E. 1987. The Properties of Gases and Liquids. 4th ed. New York: McGraw- Hill. Saravacos, G.D. 1970. Effect of Temperature on the Viscosity of Fruit Juices and Purees. J Food Sci 35:122-125. Saravacos, G.D. 1995. Mass Transfer Properties of Foods. In: Engineering Properties of Foods. 2nd ed. Rao, M.A., Rizvi, S.S.H. eds. New York: Marcel Dekker, pp. 169-221. Saravacos, G.D., Kostaropoulos, A.E. 1995. Transport Properties in Processing of Fruits and Vegetables. Food Technol 49(9):99-105. Saravacos, G.D., Kostaropoulos, A.E. 1996. Engineering Properties in Food Processing Simulation. Computers Chem Engng 20:S461-S466. Theodorou, D.N. 1996. Molecular Simulations of Sorption and Diffusion in Amorphous Polymers. In: Diffusion in Polymers. Neogi, P. ed. New York: Marcel Dekker, pp. 67-142.
Chapter 1
Vieth, W.R. 1991. Diffusion In and Through Polymers. Munich, Germany: Hanser Publ.
Transport Properties of Gases and Liquids
I. INTRODUCTION
The physical processes and unit operations of process engineering are based on the transport phenomena of momentum, heat, and mass (Bird et al., 1960; Geankoplis, 1993). The transport phenomena, originally developed in chemical engineering, can be applied to the processes and unit operations of food engineering (Gekas, 1992). The analogy of momentum, heat and mass transport facilitates a unified mathematical treatment of the three fundamental transport processes (Brodkey and Hershey, 1988). The transport properties of simple gases and liquids have been investigated more extensively than the corresponding properties of solids and semisolids. Molecular dynamics and thermodynamics have been used to predict, correlate and evaluate the transport properties of simple gases and liquids (Reid et al., 1987). Empirical prediction methods, based on theoretical principles, have been used to predict the transport properties of dense gases and liquids, compiling tables and data banks, which are utilized in process design and processing operations. This chapter presents a review of the molecular and empirical prediction of transport properties of gases and liquids, with examples of simple fluids of importance to food systems, like air and water. The theoretical treatment of simple fluids is useful in analyzing and evaluating the transport properties of complex food materials.
Chapter 2 II. ANALOGIES OF TRANSPORT PROCESSES
The transport processes of momentum (fluid flow), heat and mass can be expressed mathematically by analogous constitutive equations of the general form (one-dimensional transport):
(2-1)
where fx is the transport rate, 8 is the transport coefficient or property, and dy/dx is the transport gradient. The negative sign of Eq. (2-1) denotes that the transport is in the ^-direction, i.e. the transport gradient (dy/dx) is negative (Brodkey and Hershey, 1988). Equation (2-1) is a generalized expression of the empirical laws of Newton, Fourier and Pick for the transport of momentum, heat and mass, respectively:
r» = -T] (du/dx)
(2-2)
(q/A\ = -A (dT/dx)
(2-3)
(j/A)x = -D (dC/dx)
(2-4)
The basic transport properties, defined by the last equations, are: 77: viscosity, Pa s or kg/m s X: thermal conductivity, W/m K D: mass diffusivity, m2/s In the transport equations, T = F/A is the shear stress in the y-direction for flow in the ^-direction (Pa), u is the velocity in the ^-direction (m/s), T is the temperature (K), and C is the concentration (kg/m3). The following transport properties are also used: v = Tj/p:
momentum diffusivity or kinematic viscosity (m2/s)
a = ^1 p cp : thermal diffusivity (m2/s) where p (kg/m3) is the density and cp (J/kg K) is the heat capacity at constant pressure of the fluid. Thus, all three transport properties, v, a, D are expressed in the same units, m2/s.
Transport Properties of Gases and Liquids
9
It should be noted that, although the three transport processes are expressed mathematically by the same generalized transport equation (2-1), the mechanisms of transport of momentum, heat and mass may be quite different.
III. MOLECULAR BASIS OF TRANSPORT PROCESSES Molecular dynamics, which is concerned with intermolecular forces and molecular movement, can be utilized in the prediction of transport properties of simple fluids. The mechanism of the three transport processes are different (Brodkey and Hershey, 1988). Thus, momentum transport is caused by the relative motion of fluid layers parallel to the direction of flow; heat conduction is caused by collision of the molecules, without substantial movement of the species; and mass diffusion is caused by movement of molecules in mixtures. A. Ideal Gases
The ideal gases are considered to consist of rigid spherical molecules obeying the laws of mechanics, and their transport properties can be predicted by the kinetic theory of gases. Although very few gases approach ideality (Ar, Xe, Kr), the concept of ideal gas is useful in understanding the transport properties. The mean free path /lm of a gas, defined as the average distance of molecular movement before collision with another molecule or surface, is given by the equation (Brodkey and Hershey, 1988): X =5/0= "' '
T n*> 'l
(2-5)
where u is the mean velocity (m/s), 9 is the collision frequency (1/s), d is the molecular diameter (m), P is the pressure (Pa), and kB is the Boltzmann constant kB = 1.38xlO' 23 J/moleculeK. From Eq. (2-5) it follows that, ^mP = constant
(2-6)
This means that the mean free path of the molecular motion is inversely proportional to the pressure.
10
Chapter 2
The transport properties of ideal gases, viscosity 77, thermal conductivity /I, and mass diffusivity D are given by the following equations: (2-7)
(2-8)
D = - u Am
(2-9)
Since v = r/jp and a = A//? cp , Eqs. (2-7) to (2-9) yield the following analogies: ), = cvi] = pcvD
(2-10)
v = ay = D
(2-11)
where y = cp /cy , and cp, cv are the heat capacities at constant pressure and constant volume, respectively.
B. Thermodynamic Quantities The transport processes of momentum, heat, and mass take place in systems that are removed from thermodynamic equilibrium. Thermodynamics cannot predict transport properties, but some thermodynamic quantities are used in molecular and empirical predictions (Prausnitz et al., 1999). Pressure-volume-temperature (PVT) data are needed in calculations and correlations of transport properties of fluids. PVT data are usually obtained from the cubic equations of state, such as the Redlich-Kwong equation of state (Smith and van Ness, 1987):
V-b
Transport Properties of Gases and Liquids
11
Equation (2-12) yields the known cubic equation of van der Waals and it reduces
to the ideal gas law (PV=RT), if the empirical constants a and b are taken equal to zero. The cubic equations of state can be transformed to polynomials of third degree in respect to volume V. The empirical constants of the Redlich-Kwong equation are related to the critical properties P0 Tc of the fluid: (2-13) ^008664^
(2 _ M)
The critical properties of a fluid, Pc, Vc, and Tc, are characteristic for each fluid, and they are used to calculate the reduced properties: pr = p/pc,
v, = V/Vc,
Tr = T/Tc
(2-15)
The residual or excess properties of a fluid are the differences between the real and the ideal properties. Thus, the residual volume VKS is defined as: (2-16) RT
therefore V = z——
(2-17)
where z = PV/RTis the compressibility factor, which expresses the non-ideality of the fluid. The theorem of corresponding states indicates that all real gases, when compared at some reduced pressure and temperature, have the same compressibility factor z and all deviate from the ideal gas behavior to almost the same degree. The molecular structure of real nonspherical gases is characterized by the acentric factor co, which is related to the molecular shape. For ideal gases ca = 0, and for real gases, the acentric factor is related to the reduced pressure Pr by the equation: a> = 1.0-log(P,)rr.,7
For many fluids the normal boiling point is approximately equal to 0.7TC.
(2-18)
12
Chapter 2
Table 2.1 Critical Constants and Acentric Factors of Selected Fluids Vc , crrrVmol
r c ,K
P c ,bar
Oxygen
73.4
154.6
50.4
Nitrogen
89.8 93.9 130.4 167.1
126.2
33.9 73.8 50.4 61.4
Fluid
Carbon Dioxide Ethylene Ethanol Water
55.1
304.1 282.4
531.9 647.3
221.2
zc
0.288 0.290 0.274 0.280 0.240 0.235
CO
0.025 0.039 0.239 0.089 0.644 0.344
ource: Data from Reid et al., 1987.
Table 2.1 shows the critical constants V0 Tc, P0 zc and the acentric factor a> for selected fluids of interest to food systems.
C. Real Gases
The intermolecular forces of fluids constitute the basis of all transport processes at the molecular level. Their origin and determination is treated in the specialized literature (Maitland et al., 1981). In real gases (monoatomic, polyatomic, nonpolar, and polar) the Chapman-Enskog theory of nonuniform gases is usually applied. In the prediction of the transport properties, the following empirical parameters are usually employed (Reid et al., 1987): collision diameter a, potential energy s, and collision integral Q. The potential energy u(r) or the Lennard-Jones 12-6 potential of two spherical nonpolar molecules is given by the equation:
u(r)=4s
(2-19)
where cr is the characteristic collision diameter, similar to molecular diameter d of the kinetic theory, £ is the minimum value of u(r), and r is the intermolecular distance. The parameters cr and s are determined from empirical thermodynamic equations as functions of the critical pressure Pc, the critical temperature Tc and the acentric factor co of the fluid:
=1CT10(2.3551-0.00876;)
(2-20)
Transport Properties of Gases and Liquids
13
Table 2.2 Intermolecular Constants of Selected Components Gas Oxygen Nitrogen
Air Carbon Dioxide Ethylene
Ethanol Water
a, nm 0.35 0.38 0.37 0.39 0.42 0.45 0.26
£/kB,K 106.7 71.4 78.6 195.2 224.7 362.6 809.1
Source: Data from Reid et al., 1987. Inm = 10A = 1 0 m
s/kBTc =0.17915 + 0.169o>
(2-21)
Table 2.2 shows the empirical constants crand e of gases important to food systems. The collision integral Q of real fluids is a measure of the active cross section of collision, depending on temperature, which is related to the intermolecular potential by a complex integral (Assael et al., 1996). The collision cross section for a
molecule is the area, perpendicular to the direction of movement, within which the center of a second molecule should be located in order to collide. The collision integral is related to the transport properties of viscosity Qn and mass diffusivity £2D. The two integrals are different, and the collision integral for viscosity Qn is usually estimated more accurately than QQ, since viscosity 77 is determined more accurately than mass diffusivity D. The following empirical equation is used for estimation of fin (Brodkey and Hershey, 1988):
where T" = TkB/ sis the reduced temperature. The collision integral for diffusion f2D of nonpolar fluids is estimated as a complex function of the reduced temperature T*. For polar fluids, the collision integral for diffusion is estimated from the equation:
.
/r
where 5* is the dipole dimensionless number, defined by the equation:
(2-23)
14
Chapter 2
where (DPM) is the dipole moment (given in thermodynamic tables) and V0, Tb are the molar volume (m3/mol) at the normal boiling point (K). Water, a typical polar molecule, has a value DPM= 1.8 debye.
IV. PREDICTION OF TRANSPORT PROPERTIES OF FLUIDS
The transport properties of fluids can be considered that they consist of three contributions, as shown in the following equation (Assael et al., 1996; Millet et al., 1996): X(p,T)=X.(T) + AX(p,T) + A* (p,T)
(2-25)
where X(p, T) is the transport property (TJ, /I, D), X0(T) is the transport property of
the dilute fluid (gas), AX(p, T) is the excess contribution of the real fluid, and AXc(p,T) is the critical contribution. The terms X0(T)+ AX(p,T) represent the basic part of the transport property, while the critical contribution AXc(p, T) becomes of importance near the critical temperature. The transport properties are affected mainly by the temperature T and the density or concentration p, while pressure may have an effect in some special (e.g. critical) conditions. The excess contributions are important in predictions of viscosity (At] = 77rjq) and thermal conductivity (AA = /L-Ag), and they can be estimated when some data are available in the literature. The critical contribution is more important for A than for 77. The accuracy of prediction is higher for viscosity (1-3%) than for thermal conductivity (about 10%). The mass diffusivity is predicted with lower accuracy (10-50%), especially at high concentrations of the diffusant (Brodkey and Hershey, 1988).
Transport Properties of Gases and Liquids
15
A. Real Gases
The transport properties of real gases, viscosity 77 (Pa s), thermal conductivity /I (W/m K), and mass diffusivity D (m2/s) can be predicted by the ChapmanEnskog equations, based on the intermolecular parameters (Brodkey and Hershey, 1988): = 2.669x10"
(Mr)"
= 8.3224x10'
D = 1.883x10-
(2-26)
(2-27)
(2-28)
where M is the molecular weight, T is the temperature (K), P is the pressure (Pa), a is the collision diameter (m) and /?7i QD are the collision integrals.
Equations (2-26) to (2-28) show that 77 and /I increase with the square root of temperature, while D is a function of the cubic power of temperature. Equation (2-28) indicates that, at constant temperature, PD=constant, i.e. the mass diffusivity is inversely proportional to the pressure. Pressure has a negligible effect on /I of gases up to 10 bar, but it is important at higher pressures, especially near the critical condition. The viscosity 77 and the thermal conductivity /I of real gases are correlated by empirical equations, which facilitate the interconversion of the two transport properties. For monoatomic gases the Eucken factor is used: AM
= 2.5
(2-29)
where M is the molecular weight (kg/kmol) and cv is the heat capacity at constant volume (kJ/kmol K). For polyatomic gases, the Eucken factor is given by the equation: AM , 2.25 , 2.25 —— = 1 + ——— = 1 + cvIR c IR-\
(2-30)
16
Chapter 2
The heat capacities cv and cp are related by the equation:
cp=cy+R
(2-31)
where R = 8.314 kJ/kmol K is the gas constant.
B. Liquids
In liquids, the intermolecular forces are stronger than in gases, due to the close proximity of the molecules. Prediction of the transport properties by molecular parameters is difficult, and empirical relationships are normally used. Experimental measurements of the transport properties of liquids are necessary to validate the empirical prediction equations. Measurement techniques are discussed in the treatment of transport properties of food materials (see Chapters 5 and 7).
/. Viscosity The Eyring theory of rate processes yields an empirical expression for viscosity, which is similar to the Arrhenius equation: (2-32) where A and B are empirical constants for the particular liquid. The effect of temperature is sharper at higher viscosities, i.e. the sensitivity of viscosity to temperature variations depends primarily on the value of viscosity. The viscosity-temperature relationship for liquids is expressed by the empirical Lewis-Squires equation and diagram (Reid et al, 1987; Syncott, 1996): -0,66,
-0,66,
233
when 77, r/0 are the viscosities (Pa s) at temperatures T, T0, respectively.
Transport Properties of Gases and Liquids
17
2. Thermal Conductivity The thermal conductivity of liquids can be estimated from empirical equations as a function of the temperature T, like the following expression: /i = A + BT + CT2+DT3
(2-34)
where the constants A, B, C, D are given in tables (Reid et al., 1987). In most liquids, except water and some alcohols, /I is a negative function of temperature. Pressure has a negligible effect on A up to 50-80 bar, but it becomes important near the critical point, where the gas behaves like a liquid.
3. Mass Diffusivity The diffusivity of a species A in a liquid medium B can be estimated from the Wilke-Chang equation (Brodkey and Hershey, 1988): (2-35)
where rj is the viscosity of the liquid (Pa s), T is the temperature (K), MB is the molecular weight of B, VA is the molar volume of A at the boiling point (m3/kmol), and
is an interaction parameter, e.g. 2.26 for water and 1.5 for ethanol. In general, the mass diffusivity of a particle in a liquid medium is given by the Stokes - Einstein equation: D=
If B T
(2-36)
where rp is the particle radius (m), TJB is the viscosity of the liquid medium (Pa s), T is the temperature (K), and &s=1.38xlO"23 J/molecule K is the Boltzmann constant. From both Eqs. (2-35) and (2-36) it follows that = constant
(2-37)
i.e. the mass diffusivity is inversely proportional to the viscosity of the solution.
18
Chapter 2
The Eyring theory of rate processes predicts for mass diffusivity an Arrhenius-type relationship, analogous to viscosity: D = Ae\p(-ED/RT)
(2-38)
where A is a constant and ED is the energy of activation for diffusion (kJ/kmol). Experimental measurements of mass diffusivity in liquids are less accurate than for viscosity, especially at high concentrations of the diffusant. Mass diffusivity in aqueous systems is important in food and biological materials (Cussler, 1997) and is treated in Chapter 7 of this book. The mass diffusivity in electrolytes is affected strongly by the ionic species of the solution. In dilute solutions of a single salt the Nernst-Haskel equation is used to estimate the diffusivity D°B as a function of the temperature, the valences of anions and cations, the ionic conductances, and the Faraday constant (Reid et al., 1987). For concentrated solutions, the empirical Gordon equation is used,
which corrects the D°AB for viscosity, molality and ionic activity of the solute. The mass diffusivity of a typical electrolyte, sodium chloride, goes through a minimum at normality 0.2 N (DAB=l .2\\0'9 m2/s at 18.5 °C), and it increases at lower and higher concentrations.
C. Comparison of Liquid/Gas Transport Properties
Both viscosity 77 and thermal conductivity /I of liquids are much higher than the corresponding properties of gases. These differences reflect the stronger intermolecular forces of the dense liquid state. However, the mass diffusivity in the liquid state is much lower than in dilute gases, due to the difficulty of mass transport in dense molecular systems. Selected values of transport properties of fluids of importance to food systems are given in Tables 2.3-2.5. Typical values for air and water at 25 °C are the following: •
air:
77 = 0.017 mPas,
A = 0.025 W/m K
•
water:
r;=0.90mPas,
A = 0.62W/mK
DAB= 1.7x1 0"5m2/s • oxygen/water: DAB- 1.7x1 0"9m2/s • oxygen/air:
For comparison of transport properties in the liquid (L), and gas (G) state, the following approximate ratios are useful:
tljjj0 = A t / A G = 10 to 1000, DJDG si/10000
(2-39)
Transport Properties of Gases and Liquids
19
D. Gas Mixtures
The viscosity of steam/air mixtures, which is important in retort processing of packaged foods, is given by the empirical equation (Kisaalita et al., 1986):
^=vayJ°y*+t?,y,Tby'
(2-40)
where rja, rjs are the air and steam viscosities respectively, o=0.039, 6=0.0163, ya, ys are the mass fractions of air and steam, respectively, and Tis the temperature in °C. The calculated values of varied from 0.0144 Pa s (100 °C, yg= 0.1) to 0.0204 (140 °C, y, = 0.5).
V. TABLES AND DATA BANKS OF TRANSPORT PROPERTIES The transport properties are usually included in tables and data banks of physical properties of materials. Data on transport properties of common fluids are found in the well-known Perry's Chemical Engineers' Handbook (1984), in Reid et al. (1987) and in Cussler (1997). The extensive compilation of thermophysical properties of matter by Touloukian (1971) includes transport property data. Extensive compilation of physical properties, including transport properties of fluids, are available in computer databases and data banks, such as the Dechema databank of the German Chemical Society (Eckermann, 1983), and the PPDS data bank of the National Engineering Laboratory, U.K. (PPDS, 1996). The Design Institute for Physical Property Data (DIPPR) of the American Institute of Chemical Engineers (AIChE) recently published tables of transport properties for binary mixtures (DIPPR, 1997) and a data bank of the same properties (DIPMIX, 1997). Compilations of data and plots of physical properties of pure compounds are available for Windows (DIPPR, 1998). Packages of extensive data banks of physical properties, requiring mainframe computers, are used in process equipment design and in processing operations. Software of selected physical properties for desktop computers and Windows is available (CEP, 1999). Most of the data on transport properties of fluids published in tables and data banks are related to the chemical, petroleum, and petrochemical industries. Limited information is available on food and biological products, which are, for the most part, solid or semisolid materials.
Chapter 2
20
Tables 2.3 and 2.4 give some selected values of transport properties of simple fluids of importance to food processing and food engineering. They are useful in determining and evaluating the transport properties of food materials, which are treated in subsequent chapters of this book. More data for viscosity and thermal conductivity of air and water (liquid and vapor) as a function of temperature are presented in Figures 2.1 and 2.2.
Table 2.3 Viscosity 77 and Thermal Conductivity A of Selected Gases and Liquids (25°C) Material
r|, mPa s
A., W/m K
Air
0.017
0.025
Oxygen
0.018
0.020
Nitrogen
0.018
0.026
Carbon dioxide
0.015
0.016
Ethylene
0.012
0.020
Ethanol Water vapor
0.025
0.015
0.010
0.020
Water
0.90
0.62
Ethanol
1.04
0.15
Gases
Liquids
Source: Data from Perry, 1984 and Reid et al., 1987.
Table 2.4 Mass Diffusivity (DAB) Diffusant (A)
in Air (B)
in Water (B)
(atm pressure, 25°C)
(dilute solutions, 25°C)
DAB, xlO' 5 m 2 /s
DAB, xlO' 9 m 2 /s
Oxygen
1.7
1.7
Nitrogen
1.8
1.9
Carbon dioxide
1.9
2.0
Ethanol
1.3 2.1 2.0 -
1.3
Ethylene
Water Salt (NaCl)
Source: Data from Perry, 1984 and Reid et al., 1987.
1.9 1.1 1.2
21
Transport Properties of Gases and Liquids
B
B.
E
0.01
0.001 200
300
400
Temperature (°C)
Figure 2.1 Viscosity of air and water versus temperature. (Adapted from Pakowskietal., 1991.)
Chapter 2
22
•o
e o
0.01
200
300
400
Temperature (°C)
Figure 2.2 Thermal conductivity of air and water versus temperature. (Adapted from Pakowski et al., 1991.)
Transport Properties of Gases and Liquids
23
A third degree polynomial was fitted to these data and the results are summarized in Table 2.5. A different equation was used for viscosity data of water, since the results of the third degree polynomial are not adequate. In the same table an empirical equation from Pakowski et al. (1991) is presented for predicting the water vapor diffusivity in air versus temperature and pressure (see also Figure 2.3). Table 2.5 Empirical Equations for Calculating the Transport Properties of Water, Water Vapor and Air
Saturated Vapor Superheated Vapor Air
7.95E-03 8.07E-03 1.69E-02
4.49E-05 4.04E-05 4.98E-05
a2 -6.13E-08 1.24E-09 -3.19E-08
a3 1.44E-10 -1.21E-12 1.32E-11
Temperature (°C) 0-300 100-700 0-1000
Saturated Water
ao -1.07E+01
ai 1.97E-02
a2 -1.47E-05
a3 1.82E+03
Temperature (°C) 0-350
5.70E-01 1.76E-02 1.77E-02 2.43E-02
1.78E-03 1.05E-04 6.01E-05 7.89E-05
-6.94E-06 -6.7 IE-07 9.5 IE-08 -1.79E-08
2.20E-09 3.07E-09 -3.99E-11 -8.57E-12
Temperature (°C) 0-350 0-300 100-700 0-1000
2.16E-05
1.80E+00
-l.OOEOO
Thermal Conductivity (W/m K)
Saturated Water Saturated Vapor Superheated Vapor Air
Mass Diffusivitv (m2/s) D=ao(T+273)/273)ai P"2
Temperature (°C) Water Vapor in Air
Source: Data from Pakowski et al. (1991)
0-1200
Chapter 2
24
l.E-02
l.E-06 10
100
1000
Temperature (°C)
Figure 2.3 Diffusivity of water vapor in air. (Data from Pakowski et al., 1991.)
Figure 2.4 shows the viscosity of aqueous sucrose solutions as a function of concentration and temperature. Figure 2.5 shows the thermal conductivity of common gases of importance to food processes as a function of temperature.
Transport Properties of Gases and Liquids
25
percentage sucrose by weight = 60
50
100
Temperature (°C)
Figure 2.4 Viscosity of aqueous sucrose solutions. (Adapted from Perry et al., 1984.)
Chapter 2
26
0.070
0.060
0.050
•a a e (J
0.040
0.030
0.020
0.010 -200
200 Temperature (°C)
Figure 2.5 Thermal conductivity of gases. (Adapted from Perry et al., 1984.)
600
Transport Properties of Gases and Liquids
27
REFERENCES
Assael, M.I., Trusler, M.J.P., Tsolakis, T.F. 1996. Thermophysical Properties of Fluids. An Introduction to Their Prediction. London: Imperial College Press. Bird, R.B., Stewart, W.E., Lightfoot, E.N. 1960. Transport Phenomena. New York: John Wiley & Sons. Brodkey, R.S., Hershey, H.C. 1988. Transport Phenomena. A Unified Approach. New York: McGraw-Hill. CEP - Chemical Engineering Progress. 1999. Software Directory. New York: AIChE. Cussler, E.L. 1997. Diffusion Mass Transfer in Fluid Systems. 2nd ed. Cambridge: Cambridge University Press. DIPMIX. 1997. Database of Transport Properties and Related Thermodynamic Data of Binary Mixtures. College Station, TX: Engineering Research Station, Texas A&M University. DIPPR. 1997. Transport Properties and Related Thermodynamic Data of Binary Mixtures. Volumes 1-5. New York: AIChE. DIPPR. 1998. Data Compilation of Pure Compound Properties. New York: Technical Data Services Inc. Eckermann, R. 1983. Information systems for, and prediction of, physical properties of non-food materials. In: Physical Properties of Foods. Jowitt, R., Escher, F., Hallstrom, B., Meffert, H.F. Th., Spiess, W.E.L., Vos, G. eds. London: Applied Science Publ. Geankoplis C.J. 1993. Transport Processes and Unit Operations. 3rd ed. New York: Prentice Hall. Gekas, V. 1992. Transport Phenomena of Foods and Biological Materials. New York: CRC Press. Kisaalita, W.S., Lo, K.V., Staley, L.M. 1986. A Simplified Empirical Expression for Estimating the Viscosity of Steam/Air Mixtures. JFoodEng 5:123-133. Maitland, G.C., Rigby, M., Smith, E.B., and Wakeman, W.A. 1981. Intermolecular Forces. Their Origin and Determination. Oxford: Clarendon Press. Millet, J., Dymond, J.H., Nieto de Castro, C.A. 1996. Transport Properties of Fluids. Their Correlation, Prediction and Estimation. Cambridge: Cambridge University Press. Pakowski, Z., Bartczak, Z., Strumillo, C., Stenstrom, S. 1991. Evaluation of Equations Approximating Thermodynamic and Transport Properties of Water, Steam and Air for Use in CAD of Drying Processes. Drying Technol 9:753773. Perry R.J., Green, J.H., Moloney, J.O. 1984. Perry's Chemical Engineers' Handbook. 6th ed. New York: McGraw-Hill. PPDS-Physical Properties Data Service. 1996. Physical Properties Databases. N.E.L., East Kilbride, U.K. Prausnitz, J.M., Lichtenthaler, R.N., Azevedo, E.G. 1999. Molecular Thermodynamics of Fluid Phase Equilibria. 3rd ed. New York: Prentice Hall.
28
Chapter 2
Reid, R.C., Prausnitz, J.M., Poling, B.E. 1987. The Physical Properties of Gases and Liquids. 4th ed. New York: McGraw-Hill. Smith, J.M., van Ness, B.C. 1987. Introduction to Chemical Engineering Thermodynamics. New York: McGraw-Hill. Syncott, K. 1996. Chemical Engineering Design. In: Coulson and Richardson Chemical Engineering. Vol. 6. pp 274-280. Touloukian. Y. S. 1971. Thermophysical Properties of Matter. Volumes 1-13. New York: IFI/ Plenum.
Food Structure and Transport Properties
I. INTRODUCTION
Food structure at the molecular, microscopic and macroscopic levels, used in the study and evaluation of food texture and food quality, can be applied to the analysis and correlation of the transport properties of foods. Molecular dynamics and molecular simulations, recently developed in polymer science for the study of polymer structure, can be extended to more complex food systems, improving the empirical mechanisms and correlations of transport processes. Food macrostructure, used in engineering and processing applications, can be related to the recent advances of food microstructure at the cellular level. Applications of food structural analysis and experimental data to the transport properties, especially mass transport, will improve food process and product development, and food product quality.
I. MOLECULAR STRUCTURE
A. Molecular Dynamics and Molecular Simulations
The transport properties of simple gases and liquids can be predicted by theoretical and semiempirical models based on molecular dynamics and thermodynamics. Extensive data on the physical and transport properties of gases and liquids are available in the form of tables and data banks, as outlined in Chapter 2 29
30
Chapter 3
(Reid et al., 1987). These correlations and data are of limited use to food systems, since food materials are generally heterogeneous solids, which are difficult to analyze and interpret in terms of pure molecular science. The mechanical and transport properties of polymers can be predicted by the new technique of molecular simulation, which is based on molecular dynamics and uses extensive computer computations. Molecular simulations use statistical analysis and computer computations of a particular material, from which structural, thermodynamic and transport properties are estimated (Theodorou, 1996). Molecular simulations can produce polymer configurations based on equilibrium statistical mechanics, which show the distribution of sizes and shapes of open spaces, formed within the polymer structure, where the penetrant molecules can reside. Application of this technique to polymer science could produce special polymers for specific application, e.g. separation of various molecules.
B. Food Materials Science
The major components of foods are biopolymers (proteins, carbohydrates and lipids) and water. Food materials science is essentially polymer science applied to food materials. Polymer science was developed largely in the synthetic polymer (plastics) industry, but recently it is applied at a growing rate in food science and technology (Slade and Levine, 1991; Levine and Slade, 1992). The transport properties of foods are closely related to the properties of food biopolymers, as shown in the analysis of viscometric properties (see Chapter 4), mass diffusivity (see Chapters 5, 6 and 7) and thermal conductivity/diffusivity (see Chapter 8). The techniques of polymer science are used widely in the determination of mass transport (diffusion) of water and other solutes in food materials (Vieth, 1991). Food biopolymers of importance to transport properties are structural proteins (collagen, keratin, elastin), storage proteins (albumins, globulins, prolamins and glutenins), structural polysaccharides (cellulose, hemicelluloses, pectins, seaweed, plant gums), storage polysaccharides (starch-amylose and amylopectin), and lignin (plant cell walls) (Aguilera and Stanley, 1999).
C. Phase Transitions
Phase transitions, shown in the familiar state diagrams of biopolymers and other food components, have a profound effect on the transport properties of foods, especially mass transport. Extensive data on the major phase transitions of foods, including freezing, glass transitions, gelatinization and crystallization, are presented by Rahman (1995).
Food Structure and Transport Properties
31
Freezing of food materials increases substantially the thermal transport properties (thermal conductivity and diffusivity), as shown in Chapter 8. Glass transition, i.e. change from the glassy to the rubbery state, increases sharply the mass diffusivity of water and other solutes (see Chapters 5, 7). Gelatinization of starch decreases moisture (solute) diffusivity, but it increases thermal conductivity (see Chapters 5, 8). Collapse of the food structures reduces the mass transport properties of freeze-dried and other porous food materials. Collapse temperature is related to the glass transition temperature (Karathanos et al., 1996a). Glass transition of food biopolymers and other components has received special consideration, because of its importance to the mechanical, transport, texture and quality properties of several food materials and products (Roos, 1992; Roos, 1995; Rao and Hartel, 1998). Due to their heterogeneous structure, solid foods show a diffuse but not sharp glass temperature change. The phase changes in food materials are determined usually by differential scanning colorimetry (DSC). The glass transition temperature Tg increases considerably as the moisture content is reduced. The Tg of biopolymers is higher than the oligomers. The effect of temperature on the transport properties (viscosity, diffusivity)
below the glass transition temperature Tg and above (Tg+100K) is described by the familiar Arrhenius equation, while in the range of Tg to (Tg+100K) the WilliamsLandel-Ferry (WLF) equation (see Chapter 7) gives a better representation.
D. Colloid and Surface Chemistry Colloid and surface phenomena are important in the structure of most food materials, especially liquid foods. A detailed analysis of the physicochemical structure of milk, the most important fluid food, is presented by Aguilera and Stanley (1999). Surface phenomena, like emulsions, foams, wetting and adhesion, affect the transport properties, especially viscosity and mass diffusivity of foods. The rheological properties of liquid foods are particularly effected by the colloidal structure of the food components (see Chapter 4). The apparent viscosity of most complex (non-Newtonian) foods decreases considerably, as the shear rate is increased, due to changes in the flow patterns of the components in the suspension. Viscoelastic (both viscous and elastic) phenomena in fluid foods are caused by intramolecular forces during deformation of the material. They are studied by relaxation/creep experiments or by dynamic rheological tests.
32
Chapter 3
III. FOOD MICROSTRUCTURE AND TRANSPORT PROPERTIES
Food microstructure is concerned with the structure of food materials at the microscopic level, and its relation to the processing, storage and quality of food products. Developed initially for the evaluation of food texture and food quality, food microstructure can be applied to the transport properties of food materials, i.e. viscosity, thermal conductivity / diffusivity and mass diffusivity.
A. Examination of Food Microstructure
The microstructure of foods can be measured and evaluated by the following principal techniques (Aguilera and Stanley, 1999; Blonk, 2000): 1. Light microscopy (magnification x 20-500), which includes the compound, the polarizing, the fluorescent, the hot-stage, the computer-assisted and the con focal laser scanning microscopes. 2. Transmission electron microscopy (TEM) with magnification (x200500,000), which includes the scanning transmission electron microscope. 3. Scanning electron microscope (SEM) with magnification (x 20-200,000), which is considered as the best instrument for food microscopical studies and gives the best pictures of the materials.
In addition, the following techniques may be used for special studies/examinations: a) scanning probe microscopy; b) X-ray microscopy; c) light scattering; d) magnetic resonance imaging (MRI); and e) spectroscopy. Image analysis relies on computer technology to recognize, differentiate and quantify images. It involves video cameras, scanners and data processing software. Image analysis is applied to the measurement of particle size and shape (Alien, 1997) and in the control of food processing operations. Fractal analysis is used to measure the irregularity of particle surfaces, which are of importance to food properties.
B. Food Cells and Tissues
Both plant and animal tissues consist of microscopic cells, which characterize each material. The cells contain several components, which are essential in living organisms, such as water, starch, sugars, proteins, lipids and salts. A schematic diagram of a plant parenchyma cell is shown in Figure 3.1.
Food Structure and Transport Properties
33
PLD
CW
CM
ML Figure 3.1 Diagram of a parenchyma plant cell. CW, cell walls; CM, cell membrane (plasmalemma); V, vacuole; N, nucleus; ML, middle lamella; PLD, plasmodesmata; IS, intercellular space; TN, tonoplast; P, protein particles; L, lipid particles.
The microscopic plant cell (size of 2-10 um) consists of the cell walls which contain the cell components in a membrane (plasmalemma) enclosure. The cell contains protein, starch and lipid particles within the cytoplasm. The vacuole, surrounded by the tonoplast, contains water, soluble sugars and salts (Aguilera and Stanley, 1999). The vacuole is responsible for the osmotic pressure and the turgor of the cell. The cell walls contain the middle lamella and they have small channels (plasmodesmata), that allow the flow of cytoplasmic material and water/solutes in and out of the cell. Intercellular spaces may contain water solution or air. The cell walls contain mainly cellulose, hemicelluloses, pectins and glycoproteins. Plant tissues consist of storage or parenchyma cells, phloem for transporting organic materials, xylem for transporting water and protective tissue. The plant cell walls and membranes are of particular importance to food processing and food quality. The turgor (hydrostatic pressure) is lost during dehydration, heating or freezing. The cell wall middle lamella complex is related to food texture.
34
Chapter 3
Animal cells consist of fibrous structures (myofibrils), which have special mechanical properties, characteristic of each animal material.
C. Microstructure and Food Processing
Food structure is preserved in some processes, while in other it is destroyed in order to produce useful processed products, as in refining starch, sugars, oil seeds, grain and milk. Changes in structure occur in freezing, milling (size reduction), crystallization and emulsification. Restructuring of food materials is used in extraction, spinning, margarine and ice cream production. Gels are solidlike structures that contain large amounts of water. They are produced by different mechanisms using gelling substances, such as starch, pectin, gelatin, alginates and various plant gums. The Theological and transport properties of gels are important in the various food processing operations and in food quality. They are also used for investigations of structure-property relationships.
D. Microstructure and Mass Transfer /. Solvent Extraction
Solvent/solid extraction or leaching of food components, such as sugars, lipids and flavor compounds is used in various food processing operations. It can be analyzed and designed by the methods developed in chemical process engineering, i.e. equilibrium-stage or continuous separations (Perry and Green, 1997). The mass diffusivity of solutes in solid substrates Ds is smaller than the diffusivity in the liquid solvent DL due to the complex structure of the material, according to the empirical equation (Aguilera and Stanley, 1999): Ds = Fa,DL
(3-1)
The correction factor Fm varies in the range (0.1-0.9), with the higher values obtained when the cell membranes are destroyed, as by heating in the extraction of sucrose from sugar beets by hot water. Fm becomes very low, approaching zero, in the extraction of high molecular weight components, like proteins, from plant tissues. Extraction is improved by pretreatment of the solid material, e.g. by size reduction, or by flaking, which reduces the diffusion path for both solvent and extracted component. In some applications, extraction can be improved by enzymatic treatment of the plant tissues, which break down, e.g. the pectins of the cell walls.
Food Structure and Transport Properties
35
Values of the diffusivity of food components in various solvents are given by Schwartzberg (1987). Typical DL for sucrose in water is 5xlO' 10 m2/s and for caffeine in water is IxlO" 10 rrr/s (see Table 7.5).
2. Food Dehydration The moisture (water) in heterogeneous solid foods may not be in equilibrium at the microstructural level, although macroscopically the system appears to be in equilibrium. Thus, moisture may be transported between the heterogeneous components of the food system. Thermodynamic analysis of transport processes at the cellular level requires transport and equilibrium properties of the cell components (Rotstein, 1987). Microstructural changes during the drying of food materials include loss of cellular structure, pore formation and shrinkage of the product. These changes affect strongly the transport properties, especially moisture diffusivity (see Chapter 5). Heterogeneous structure may affect moisture diffusivity, like the reduction of the drying rate by the skin of grapes (see Chapter 5). Food microstructure is related to the retention of volatile aroma components during the drying of food materials (see Chapter 7). Microstructure plays an important role in the osmotic dehydration of foods, especially that of fruits and vegetables: The water is transported from the food cells to the osmoactive solution (sugar or salt) and the osmoactive agent is infused into the cellular food (Lewicki and Lenart, 1995). A model of the osmotic dehydration process in cellular foods is presented by Yao and Le Maguer (1996). Mass transfer in plant tissues during osmotic dehydration was analyzed by Spiess and Behsinlian (1998); the plant tissue is considered to consist of a solid matrix, intercellular space, extracellular space and occluded gas. Three pathways of mass transfer may take place at the microstructural level: a) apoplasmatic transport (outside the cell membrane); b) symplasmatic transport through small channels (plasmodesmata) between neighboring cells; and c) transmembrane transport (between the cell and the intercellular space). The function of living plant cells is examined in plant physiology. 3. Microstructure and Frying Frying of foods can be considered as a simultaneous heat and mass transfer process in which water is lost and oil is absorbed by the food product, e.g. potato pieces. Cells in the interior of the fried product can be intact, while surface cells are dehydrated and shrunk. Oil uptake during frying is by a complex mechanism, different from the molecular (Fickian) diffusion, possibly by a capillary or hydrodynamic flow (Aguilera and Stanley, 1999; Aguilera, 2000).
36
Chapter 3
IV. FOOD MACROSTRUCTURE AND TRANSPORT PROPERTIES
The transport properties of solid and semisolid food materials are related to their macroscopic properties, such as density, porosity, particle size and shape. The design, operation and control of food processing operations is based on these properties, which can be measured by simple instruments and techniques. The structure of solid and semisolid foods has been investigated more in relation to food structure and food quality than to transport properties. The structure of fluid foods is usually related to their rheological and viscometric properties, as discussed in Chapter 4. Food macrostructure has received special attention in relation to the dehydration of foods, since significant changes take place during moisture transport in solid and semisolid food materials. Model food materials, based on food biopolymers, such as starch, and fruits and vegetables have been used as experimental materials, since their properties can be related empirically to their structure at the macroscopic, microscopic and molecular levels. Quantitative parameters of physical meaning, i.e. density, porosity and shrinkage, based on three phases (solids, water and air) can be estimated for the characterization of structural changes of foods during processing and storage. Two classes of food materials are examined separately: a) continuous solids, in which shrinkage and porosity develop when water is removed, and b) particulate or granular materials, such as starch granules, in which porosity is a dependent variable that can be controlled, e.g. by compression.
A. Definitions
Assuming moist material to consist of dry solids, water, and air, the following definitions can be considered: m,= ms+mw
(3-2)
where m,, ms, and mw are the total mass and the masses of dry solids and water respectively (kg), while the mass of air is neglected. The total volume of the sample is considered as:
V,= Vs+Vw+Va
(3-3)
where Vs, Vw and Va, are the volumes of dry solids, water and air pores, respectively (m3). The volume of air is referred to the internal pores only.
Food Structure and Transport Properties
37
The apparent density of the food material pb is defined as:
pb = m,/Vt
(3-4)
and the true density pp as:
PP=m,/Vp
(3-5)
where Vp = Vs+Vw is the true (particle) volume, which is the total volume of the sample excluding air pores. Apparent and true density are analogous to the bulk and particle density of granular materials, respectively. The actual densities of dry solids PJ and enclosed water pw can also be defined as:
P, = m,/V,
(3-6)
pw = mw/Vw
(3-7)
The specific volume of the sample u is defined as the total volume per unit mass of dry solids (m3/kg db):
v=V,/ms
(3-8)
The material moisture contention a dry basis (kg water/kg db) is:
X=mw/ms
(3-9)
The volume-shrinkage coefficient ft can be defined by the following equation, which represents the proportion of initial specific volume that shrinks as water is removed: (3.10) A,
where Xi is the initial moisture content of the moist food material, v is the specific volume at material moisture content X, and o,- is the specific volume at X=Xi. The shrinkage coefficient ft varies between 0 (no shrinkage) and 1 (full shrinkage). (See Figure 3. 2.) Assuming that no volume interaction occurs between the water and the solids, combining Eqs. (3-5), (3-6), (3-7) and (3-9) results in:
38
Chapter 3
Pp=
lJrX
A
(3-11)
A..
Equation (3-11) shows the dependence of moisture content on true (particle) density. Combining Eqs. (3-4), (3-7), (3-9) and (3-10) results in:
'iW A,
P-12)
A,.
where pbi is the apparent density &iX=Xi, the initial moisture content. When the zero moisture content is considered as initial moisture content (A)=0), then Eqs. (3-10) and (3-12) are transformed into Eqs. (3-13) and (3-14), respectively:
o = uo+/3— A
(3-13)
where L>O, and pbo are the specific volume and the apparent density at X = 0, respectively. Moreover, in fried products, when oil is considered as one more phase, Eqs. (3-13) and (3-14) are further transformed to Eqs. (3-15) and (3-16), respectively:
( X Y\ o = va+p\ — + — IA
(3-15)
PL)
X +— Y — A»
A
PL.
where pL is the oil density and 7 the oil content Y=mL/ms
(3-17)
Food Structure and Transport Properties
39
where mL is the mass of the oil in the sample of ms dry solids. For fried products Eq. (3-11) is also transformed to Eq. (3-18): +X +
1
_____
X
j ___
P,
P,
(3-18)
Y
I ___
Pi
Volume of water which:
disappears (shrinkage) = p (Xi-X) / pw
remains as water = X/pw remains as air (porosity) = (l-p)(Xi-X)/pw
Initial (Xi)
Final (X)
Figure 3.2 Schematic representation of shrinkage and porosity development, as part of water is removed.
40
Chapter 3
B. Food Macrostructure and Transport Properties The food macrostructure, in calculating the effective transport properties of food materials, is taken into account, using the so-called structural models, some of which are presented for thermal conductivity in Chapter 8 (Table 8.3). Similar structural models for moisture diffusivity of starch materials have been proposed by Vagenas and Karathanos (1991), but their application to food materials remains to be validated. These models require the volume fraction of the food phases (solids, water, air, oil, etc). Thus, the shrinkage and porosity models, described earlier in this chapter, must be combined with the transport properties models. Two examples for continuous (Maroulis et al., 2001; Krokida et al., 200la) and granular foods (Krokida et al., 200Ib) foods follow.
1. Continuous Solids Table 3.1 summarizes the proposed model that combines the thermal conductivity structural models with the density structural models. The corresponding information flow diagram is presented in Figure 3.3. 2. Granular Solids Table 3.2 summarizes the proposed model, which combines the thermal conductivity structural models with the density structural models. The corresponding informational flow diagram is presented in Figure 3.4.
Food Structure and Transport Properties
41
Table 3.1 Effective Thermal Conductivity Generic Model for Continuous Materials Thermal Conductivity Structural Model
(2)
Volume Fraction of the Food Phases /?,=!-—
(4)
* =^
(5)
(6)
Density Structural Model
n
A,
P.
Chapter 3
42
pai, Xi,
O, al, a2, a3
Density of Pure Substances
Shrinkage and Porosity Model Eqs. (7) and (8)
ps, pw
pa, pt
Volume Fraction of Food
Components Eqs. (4), (5) and (6)
Thermal
es, e\v, ea
Conductivity
of Pure Substances
Xs, Xw, Xa
Thermal
Conductivity Structural Model Eqs. (1), (2)
and (3)
bo, bl, b2, b3
JXeff
Figure 3.3 Informational flow diagram for the model of Table 3.1. (Equation numbers refer to those in Table 3.1.)
Food Structure and Transport Properties
43
Table 3.2 Effective Thermal Conductivity Generic Model for Granular Materials Thermal Conductivity Structural Model for Granular Material *«=l-f
-L
f
J-
(2) (3)
Thermal Conductivity Structural Model for Granules (Particles) / yf
(5)
Volume Fractions * 0 =1- A ^
(7)
Particle Density Structural Model
P Pp
= 1l +X X ___ I ___ P.
P.
(9) ^ '
Chapter 3
44
f
fp
aO, al, a2. a3 Particle Density Structural Model Eq. (9)
Density of Pure
Substances
ps, pw
PP
pb Volume
Fractions Eqs. (7) and (8)
Thermal Conductivity of Pure Substances
Xs, Xw
Thermal Conductivity Structural Model
for Granules (Particles) Eqs. (4), (5) and (6) bo, bl, b2, b3 1 e 1
Xp
Xa
Thermal Conductivity Structural Model forGranular Material Eqs. (1), (2) and (3)
Xeffl
Figure 3.4 Informational flow diagram for the model of Table 3.2. (Equation numbers refer to those in Table 3.2.)
Food Structure and Transport Properties
45
C. Determination of Food Macrostructure
The determination of food macrostructure is based on the measurement of mass and volume and the estimation of the various densities and porosity, as shown on the informational flow diagram of the Figure 3.5 (Krokida et al., 1997).
The corresponding methods are described by Rahman (1995) and summarized in Table 3.3.
Mass of Wet Sample m (error 1%)
True Density pp=m,/Vp (error 2%) True Volume Vp (error 1%) Apparent Density pb=mt/Vt (error 3%) Total Volume Vt (error 2%) Specific Volume u = Vt / ms (error 3%) Mass of Dried Sample ms (error 1%)
Figure 3.5 Experimental data evaluation flow diagram for densities and porosity.
46
Chapter 3
Table 3.3 Volume Measurement Techniques 1. 2. 3.
Dimension measurement Buoyant force determination Volume displacement method a. Liquid displacement method
b.
Gas pycnometer method
c.
Solid displacement method
D. Macrostructure of Model Foods
Model foods are useful experimental materials for studying the effect of physical (macro) structure on the transport properties of foods. They consist of a biopolymer matrix, containing water and typical food components, such as carbohydrates, proteins and lipids assembled in the form of a hydrated solid or a solidlike gel. Starch materials, particularly linear amylose and branched amylopectin, have been used in many forms for studies of thermal and mass diffusivity. The physical phenomena of food dehydration have been investigated using various starch materials (Saravacos, 1998). The density of granular starch changes nonlinearly with the moisture content, according to the empirical equation (Marousis and Saravacos, 1990):
pf = 1442 + 837X-3646X2 + 448 LY 3 -1850^ 4
(3-19)
The granular or particle density pp is expressed in (kg/m3) and the moisture content A'in (kg/kg db). Figure 3.6 schematically shows the change of the particle density of granular starch as a function of moisture content. A maximum ofpp is observed near X = 0.15, meaning that at this low moisture content, the water is adsorbed strongly on the biopolymer. At higher moistures, water reduces the particle density by swelling the starch granules. The bulk porosity s of hydrated granular starch increases significantly as the moisture is reduced during air-drying (Figure 3.7). A smaller change of porosity is observed on the porosity of gelatinized starch during the drying process. These significant changes in porosity, estimated by measuring the bulk and particle densities of the material, are related to the changes of moisture diffusivity D during the various drying and rehydration processes (see Chapter 5).
Food Structure and Transport Properties
47
1.60
1.00 0.00
0.20 0.40 0.60 0.80 1 . 0 0 X (kg/kg db)
Figure 3.6 Particle density pp of corn starch as a function of moisture content X.
0.60
0.40
0.20
0.00 0.00
0.20 0.40 0 . 6 0 0 . 8 0 1.00 X (kg/kg db)
Figure 3.7 Bulk porosity s of air-dried granular (GR) and gelatinized (GEL) corn starch as a function of moisture content^
48
Chapter 3
Gelatinization of starch increases significantly the thermal conductivity of the starchy materials, evidently due to the changes in the macromolecular and microscopic structure (see Chapter 8). Changes in porosity of granular starch struc-
tures by the incorporation of sugars, proteins, lipids, and inert particles are reflected in significant changes of the moisture diffusivity. Significant reduction of the porosity of granular starch is obtained by mechanical compression (see Chapter 5). Figure 3.8 shows schematically the macrostructure of dried granular and gelatinized starch materials. The spherical granular starch materials developed radial channels through which water was transported by hydrodynamic flow during drying. The gelatinized starch suffered more shrinkage during drying with irregular cracks in the dried matrix. These changes correspond to changes of moisture diffusivity during the drying processes (Marousis and Saravacos, 1990). Freeze-drying of model food gels affects strongly their macrostructure and the transport properties. The development of macrostructure is the combined result of freezing and drying by sublimation of the ice. Figure 3.9 schematically shows slabs of two different structures, developed by freeze-drying of model gels. The CMC gel developed a fibrous structure, parallel to the flow, which significantly increased the moisture transport rate and thermal conductivity of the material (see Chapters 5 and 8). The freeze-dried starch gel had a uniform microporous structure which had lower moisture and thermal diffusivity than the fibrous material. The pore size distribution in granular, gelatinized and extruded starch materials, measured by mercury porosimetry, shows that the majority of the pores (90%) are larger than 1 urn, and only 10% have smaller pores. Bulk porosity, measured by gas pycnometry, is the most important parameter characterizing the transport properties (Karathanos and Saravacos, 1993). Structural models, developed for thermal conductivity, can be adapted to model the effective moisture diffusivity in granular starch materials (Vagenas and Karathanos, 1991). The pore structure of extruded pasta has significant effect on the moisture sorption and diffusivity of pasta (Xiong et al., 1991). Stress crack formulation in the air-drying of cylindrical samples of hydrated starch is affected by the moisture gradient at the transfer interface (Liu et al., 1997). Crack formation, a problem in pasta drying, is related to the stresses developed in the glassy state of the biopolymer, and it can be prevented by drying at temperatures higher than the glass transition temperature (Willis et al., 1999). Developments of cracks in grain kernels can be prevented by intermittent microwave drying, due to the relaxation of temperature and moisture gradients during the tempering period (Zhang and Mujumdar, 1992). Tailor-made porous solid foods can be prepared using a base of freeze-dried alginate gels (Rassis et al., 1997).
49
Food Structure and Transport Properties
a. Granular
b. Gelatinized
Figure 3.8 Macrostructure of air-dried spherical samples of corn starch (20 mm diameter).
O O Q
a. CMC
O
O
b. Starch
Figure 3.9 Macrostructure of freeze-dried slabs of carboxy methyl cellulose (CMC) and starch gels (20 mm thickness).
50
Chapter 3
E. Macrostructure of Fruit and Vegetable Materials The effect of drying method on bulk density, particle density, specific volume and porosity of banana, apple, carrot and potato at various moisture contents is presented in Figures 3.6-3.9, using a large set of experimental measurements (Krokida and Maroulis 1997). Samples were dehydrated with five different drying methods: conventional, vacuum, microwave, freeze- and osmotic drying. A simple mathematical model, presented in Table 3.4, was used in order to correlate the above properties with the material moisture content. Four parameters with physical meaning were incorporated in the model: the enclosed water density pw, the dry solids density ps, the dry solids bulk density pbo and the volume shrinkage coefficient (3. The effect of drying method on the examined properties was taken into account through its effect on the corresponding parameters (Table 3.5). Only dry solid bulk density was dependent on both material and drying method. Freezedried materials developed the highest porosity, whereas the lowest one was obtained using conventional air drying. The above structural properties for the investigated materials were also examined during rehydration of dehydrated products, using the drying methods. The same dehydrated products did not recover their structural properties after rehydration, due to structural damage that occurred during drying and the hysteresis phenomenon, which took place during rehydration. Porosity of the rehydrated products was higher during rehydration than during dehydration. A structural model of Table 3.4 was also used to describe the structural properties, and of the four parameters that were incorporated (also presented in Table 3.5), only the shrinkage coefficient, which represents volume expansion, changed on rehydration. Apparent density, true density, specific volume and porosity were investigated during deep fat frying of french fries (Krokida et al., 2000). The effect of frying conditions (oil temperature, sample thickness and oil type) on the above properties is presented in Figures 3.10-3.12. Moisture and oil content during deep fat frying and consequently all the examined properties are affected by frying conditions. The results showed that the porosity of french fries increases with increasing oil temperature and sample thickness, and it is higher for products fried with hydrogenated oil. The pore size distribution of fruit and vegetable materials shows three peaks in the ranges of 20 um, 1 um and 0.2-0.04 urn. The pore size of air-dried food materials is much smaller than the pore size of freeze-dried products, due to the collapse of structure during dehydration (Karathanos et al., 1996b).
51
Food Structure and Transport Properties
2.0 -
2.0 T
BANANA A
APPLE
X
Dehydration convective drying vacuum drying microwave drying
freeze drying
osmotic dehydration calculated Rehydration convective drying vacuum drying micro\\ave drying
freeze drying osmotic dehydration
- calculated
0
3 6 Moisture content (kg/kg db)
2.0 -
0
3 6 Moisture content (kg/kg db)
2.0 CARROT
0
3 6 Moisture content (kg/kg db)
0
3 6 Moisture content (kg/kg db)
Figure 3.10 Variation of true density with material moisture content for various drying methods during dehydration and rehydration.
Chapter 3
52
i
2.0 -
o o D
Dehyd(ration convei;tive drying vacuui•n drying micro\vave drying freeze drying osmoti c dehydration ited RehydIration
convet;tive drying
vacuuin drying microvvave drying freeze drying osmoti'c dehydration
0.0
0
3
6
Moisture content (kg/kg db)
Moisture content (kg/kg db)
CARROT
0.0
0
3
Moisture content (kg/kg db)
6
0
3
6
Moisture content (kg/kg db)
Figure 3.11 Variation of apparent density with material moisture content for various drying methods during dehydration and rehydration.
53
Food Structure and Transport Properties
Dehydration 1.0
T
BANANA
•
convective drying
A
vacuumdrying microwavedrying
X
freeze drying osmotic dehydration . calculated Rehydration convective drying vacuumdrying
microwave drying freeze d r y i n g
osmotic dehydration -calculated
Moisture content (kg/ kg db)
Moisture content (kg/ kg db) 1,0 T
CARROT
0
3
Moisture content (kg/kg db)
6
0
3
6
Moisture content (kg/ kg db)
Figure 3.12 Variation of porosity with material moisture content for various drying methods during dehydration and rehydration.
Chapter 3
54
15
Dehydration convective drying vacuum drying microwave drying freeze drying osmotic dehydration
•
A X • •
Rehydration convective drying vacuum drying microwave drying freeze drying osmotic dehydration
o A X 0 D
3
Moisture content (kg/kg db)
6
0
3
6
Moisture content (kg/kg db)
15 -
0
3
Moisture content (kg/kg db)
3
6
Moisture content (kg/kg db)
Figure 3.13 Variation of specific volume with material moisture content for various drying methods during dehydration and rehydration.
Food Structure and Transport Properties
55
Table 3.4 Mathematical Model for Structural Properties of Foods
Structural properties pp pb s v
True density Apparent density Porosity Specific volume
(L/kg db)
Moisture content
(kg/kg db)
(kg/L) (kg/L)
Factors
x
Properties equations 1+ X
Pp
~ J_ A Ps
P.
i +x
pb =• t Pbo
(2) Pw
(3)
8 = 1-*-
Pp
u = — + B' — Pbo
(1)
(4)
Pw
Parameters pw ps pbo /?'
Enclosed water density Dry solid true density Dry solid apparent density Shrinkage or expansion coefficient
Factors affecting parameters Material Drying method
(kg/L) (kg/L) (kg/L)
Chapter 3
56
Table 3.5 Parameter Estimation of the Structural Properties Model Material/Method Ps
M
e
Rehydrat.
'c£•
Apple Convective Vacuum Microwave Freeze Osmotic Convective Vacuum Microwave Freeze Osmotic
P»
• -
P
Pbo
0.99 0.96 1.01 0.34
0.56 0.39 0.56 0.12 0.73 0.56 0.39 0.56 0.12 0.73
'- -g1.31 1.30 0.81 1.22
Banana DC B 'E>
Rehydrat.
Q
it c
'E>
O u
•B >, -C
£
BJD C
'cEu
J>
i
Convective
Vacuum Microwave Freeze Osmotic Carrot Convective Vacuum Microwave Freeze Convective Vacuum Microwave Freeze Potato Convective Vacuum Microwave Freeze Convective Vacuum Microwave Freeze
1.81 0.63 1.79 0.26 1.33 1.07 1.81 0.63 1.10 1.79 1.07 0.26 0.65 1.33 1.07 1.04 0.90 1.05 0.43 1.04
Convective Vacuum Microwave Freeze Osmotic '
L9
°
1.75
L 2
°
1.02 0.99 0.94 0.30 1.02 1.05 1.20 0.22
1.60
1.03 1.03 0.81 0.29 1.02 1.05 1.07 0.74
1.60 0.92 0.53 0.14 12Q 1.60 0.92 0.53 0.14 1.50 1.29 0.44 0.18 10? 1.50 1.29 0.44 0.18
Food Structure and Transport Properties
57
1.5 n
0.9 5
15
20
5
10 Time (min)
15
20
10 15 Time (min)
20
5
10 Time (min)
15
20
10 Time (min)
0.6
5
Figure 3.14
Effect of oil temperature on structural properties of french fries.
3.
1
•a o
s I
01
CO ^
(Q
-
5" x^
K) O
<_ft
I5 "
re
3
d
<-/!
o
O
3
3
3 3 3
(-A
3
•\ •^ •
/
r>
'
O
UJ
0
1
1 I
1 1 1
\ | <
r•
o
LK>
4^
Specific volume (I/kg)
Lft
'
<-*»
o
Porosity
H
5"
3
Apparent density (kg/1) o o
True density (kg/1)
ff
O 39> T3
CO
Food Structure and Transport Properties
59
1.5 -i Concentration of
hydrogenated oil (%) AO
• 50 • 100
5
15
20
10 15 Time (min)
20
10
5
10 15 Time (min)
20
5
10 15 Time (min)
20
Time (min) 0.
5
0
Figure 3.16 Effect of oil type on structural properties of french fries.
60
Chapter 3
REFERENCES
Aguilera, J.M. 2000. Microstructural and Food Product Engineering. Food Techno!54(ll):56-65. Aguilera, J.M., Stanley, D.W. 1999. Microstructural Principles in Food Processing and Engineering. Gaithersburg, MD: Aspen Publishers. Alien, T. 1997. Particle Size Measurement. Vol. 1. 5th ed. London: Chapman and Hall. Blonk, L.C.G. 2000. Viewing Food Microstructure presented at the ICEF 8. Puebla, Mexico. Karathanos, V.T., Anglea, S.A., Karel, M. 1996a. Structure Collapse of Plant Materials during Freeze Drying. J of Texture Analysis 9:204-209. Karathanos, V.T., Kanellopoulos, N.K. and Belessiotis, V.G. 1996b. Development of Porous Structure During Air-Drying of Agricultural Plant Products. J Food Sci 29:167-183. Karathanos, V.T., Saravacos, G. D. 1993. Porosity and Pore Size Distribution of Starch Materials. J Food Eng 18:254-280. Krokida, M.K., Maroulis, Z.B. 1997. Effect of Drying Method on Shrinkage and Porosity. Drying Technol 15 (10):2441-2458. Krokida, M.K., Maroulis, Z.B. 2001a. Structural Properties of Dehydrated Products During Rehydration. International J of Food Sci and Technol 36:1-10. Krokida, M.K., Maroulis, Z.B., Rahman, M.S. 200Ib. A Structural Generic Model to Predict the Effective Thermal Conductivity of Granular Foods. Drying Technol, in press. Krokida, M.K., Oreopoulou, V., Maroulis, Z.B. 2000. Effect of Frying Conditions on Shrinkage and Porosity of Fried Potatoes. J of Food Eng 43:147-154. Krokida, M.K., Zogzas, N.P., Maroulis, Z.B. 1997. Modeling Shrinkage and Porosity During Vacuum Dehydration. International J of Food Sci and Technol 32: 445-458.
Lewicki, P. P., Lenart, A. 1995. Osmotic Dehydration of Fruits and Vegetables. In Handbook of Industrial Drying. 2nd ed. Vol. 1. A.S. Mujumdar, ed. New York: Marcel Dekker, pp. 691-713. Levine, H., Slade, L. 1992. Glass Transitions in Foods. In: Physical Chemistry of Foods, Schwartzberg, H.G. and Hartel, R., eds. New York: Marcel Dekker, pp. 83-221. Liu, H., Zhou, L., Hayakawa, K.I. 1997. Sensitivity Analysis for Hydrostress Crack Formation in Cylindrical Food During Drying. J Food Sci 62:447-450. Maroulis, Z.B., Krokida, M.K., Rahman, M.S. 2001. A Structural Generic Model to Predict the Effective Thermal Conductivity of Fruits and Vegetables. J Food Eng, in press. Marousis, S.N, Karathanos, V.T., Saravacos, G.D. 1991. Effect of Physical Structure of Starch Materials on Water Diffusivity. J Food Proc Preserv 15:183195.
Food Structure and Transport Properties
61
Marousis, S.N., Saravacos, G. D. 1990. Density and porosity in drying starch materials. J Food Sci 55:1367-1372. Perry, R.H., Green, D. 1997. Perry's Chemical Engineers' Handbook. 7th ed. New
York: McGraw-Hill. Rahman, M.S., 1995. Food Properties Handbook. New York: CRC Press. Rao, M.I., Hartel, R.W., eds. 1998. Phase/State Transition of Foods. New York: Marcel Dekker. Rassis, D., Nussinovitch, A., Saguy, I.S. 1997. Tailor-Made Porous Solid Foods. Int J Food Sci Technol 32:271-278. Reid, R.C., Prauznitz, J. M., Poling, B. E. 1987. The Properties of Gases and Liquids. 4th ed. New York: McGraw-Hill. Roos, Y. 1992. Phase Transitions and Transformations in Food Systems. In: Handbook of Food Engineering. Heldman, D.R., Lund, D.B., eds. New York: Marcel Dekker, pp. 145-197. Roos, Y. 1995. Phase Transitions in Foods. New York: Academic Press. Rotstein, E. 1987. The Prediction of Diffusivity and Diffusion-Related Properties in the Drying of Cellular Foods. In: Physical Properties of Foods 2. Jowitt, R., Escher, F., Kent, M., McKenna, B., Roques, M., eds. London: Elsevier Applied Science, pp. 131-145. Saravacos, G.D. 1998. Physical Aspects of Food Dehydration. In: Drying '98 Vol. A. Akritidis, C.A., Marinos-Kouris, D., Saravacos, G.D., eds. Thessaloniki, Greece: Ziti Publ, pp. 35-46. Schwartzberg, H. G. 1987. Leaching Organic Materials. In: Handbook of Separation Process Technology. Rouseau, R.W., ed. New York: Wiley, pp. 540-577. Slade, L, Levine, H. 1991. A Polymer Science Approach to Structure / Property Relationships in Aqueous Food Systems: Non-Equilibrium Behavior of Carbohydrate-Water Systems. In: Water Relationships in Foods, Levine, H., Slade, L., eds. New York: Plenum Press, pp. 29-101. Spiess, W.E.L., Behsinlian, D. 1998. Osmotic Dehydration in Food Processing. Current State and Future Needs. In: Drying '98 Vol. A. Akritidis, C.A., Marinos-Kouris, D., Saravacos, G.D., eds. Thessaloniki, Greece: Ziti Publ, pp. 4756. Theodorou, T.N. 1996. Molecular Simulations of Sorption and Diffusion in Amorphous Polymers. In: Diffusion in Polymers, Neugi, P., ed. New York: Marcel Dekker, pp. 67-142. Vagenas, O.K., Karathanos, V.T. 1991. Prediction of Moisture Diffusivity in Granular Materials with Special Applications to Foods. Biotechnol Progr 7:419-426. Vieth, W.R. 1991. Diffusion In and Through Polymers. Munchen, Germany: Hanser. Willis, B., Okos, M., Campanella, O. 1999. Effect of Glass Transition on Stress Development During Drying of a Shrinking Food System. In: Proceedings of the 6th CoFE '99. Barbosa-Canovas, G.V., Lombardo, S., eds. New York: AIChE, pp. 496-501.
62
Chapter 3
Xiong, X., Narsimhan, G., Okos, M. O. 1991. Effect of Composition and Pore Structure on Binding Energy and Effective Diffusivity of Moisture in Porous Foods. J Food Eng 15:187-208. Yao, Z., Le Maguer, M., 1996. Mathematical Modeling and Simulation of Mass
Transfer in Osmotic Dehydration Processes. 1. Conceptual and Mathematical Models. J Food Eng 29:349-360.
Zhang, D., Mujumdar, A.S. 1992. Deformation and Stress Analysis of Porous Capillary Bodies During Intermittent Volumetric Thermal Drying. Drying Technol 10:421-443.
Rheological Properties of Fluid Foods
I. INTRODUCTION
The viscosity of fluid foods is an important transport property, which is useful in many applications of food science and technology, such as design of food processes and processing equipment, quality evaluation and control of food products, and understanding the structure of food materials. Due to the complex chemical and physical structure of foods, viscosity can not be predicted by theoretical methods, such as molecular dynamics and semi-empirical models, applied to pure fluids, and discussed in Chapter 2 of this book. Therefore, experimental measurements and empirical models of viscosity are necessary for the characterization of fluid foods. Viscosity is part of the wider rheological properties of foods, which cover, in addition to fluids, the solid and semisolid food materials. Foods, in general, can be classified as solids, gels, homogeneous liquids, suspensions in liquid, and emulsions (Rao, 1999). Fluid foods are heterogeneous materials, consisting of dispersions of fibers, cells, protein particles, oil droplets and air bubbles in a continuous phase, like an aqueous solution of sugars, or a vegetable oil (Aguilera and Stanley, 1999). Recent advances in the design and control of food processes, utilizing computer modeling and simulation, require extensive data on the physical and engineering properties of foods. Limited reliable data are available in the literature, particularly in the areas of rheological properties (viscosity) and mass diffusivity of food systems (Saravacos and Kostaropoulos, 1995, 1996; Saravacos, 2000). Food rheology deals with all the phenomena of deformation and flow of food materials due to external forces. Viscometry deals with fluids, which are characterized by mechanical flow, upon the application of an external force. 63
64
Chapter 4
The viscometric properties of fluid foods are discussed here in analogy to the two other basic transport properties, thermal conductivity and mass diffusivity. The rheological properties of solid and semisolid foods (elasticity and viscoelasticity) are discussed in specialized books, such as Bourne (1982) and Rao (1999). Viscosity plays an important role in liquid food texture and in texture-taste interactions (Kokini, 1987). The tasting reaction is controlled by the diffusion of the flavor components through the viscous food layer in contact with the tongue. Fluid viscosity rj is defined by the basic transport Eq. (2-2), which is equivalent to the Newton Eq. (4-1) of shear flow. In most food applications, TJ is synonymous to the shear viscosity in the ^-direction, as shown in Figure 4.1.
T = riy
(4-1)
where T = F/A (applied force / surface area) is the shear stress in (Pa) and
y = (du/dy), change of velocity (Aux) in the y-direction, is the shear rate in (1/s). This book deals mainly with shear viscosity at steady state. However, there are some other types of viscosity, for example extensional viscosity, which is useful in specialized engineering applications. Extensional flows occur when the flow geometry changes abruptly, like in orifice flow, spinning of fibers, extrusion, and impingement (Giesekus, 1983; Padmanabhan, 1995; Rao, 1999). Such flows are important in polymer processing. In the simple uniaxial extensional flow, a cylindrical body is stretched in one direction, while contracting in the other two. In the biaxial extensional flow, the reverse process takes place, i.e. the material is stretched in two directions, while contracting in the other. Finally, in the planar extensional flow, stretching is in one direction, constant in the second, and contraction in the third.
Figure 4.1 Shear viscosity in the x-direction.
Rheological Properties of Fluid Foods
65
In dynamic rheometry, applied to viscoelastic materials and gels, the shear stress-strain relationship is obtained by periodic deformation of the material (Rao and Steffe, 1992; Urbicain and Lozano, 1997). In polymer science, the intrinsic viscosity [ r/J expresses the hydrodynamic volume of the polymer molecule, which is related to the molecular weight and the dimensions of the molecule. The intrinsic viscosity in dilute polymer solutions is defined as the zero-concentration limit of the ratio (rjS[/C), where T]sp=[('n-ris)/riJ is the specific viscosity, r\ and rjs are the viscosity of the solution and the solvent respectively, and C is the concentration (Rao, 1999). The intrinsic viscosity [ t] /of dilute food biopolymer solutions (e.g. gums) has been related to experimental viscosity measurements through empirical models, like the Huggins equation (Tanglertspaibul and Rao, 1987; Yoo et al, 1994): (iv'Q-lvJ + krfrifC
(4-2)
The Huggins constant kt is related to the polymer-polymer interaction and it takes values from 0.3 to 1.0. The low kj values characterize the good solvents, while the high values indicate associations between the macromolecules. Heat treatment of aqueous gum solutions decreases the intrinsic viscosity, presumably due to hydrolysis or breakdown of the biopolymer (Rao, 1995). Real solids are considered as elastic materials, which obey Hooke's law, i.e. a linear relationship between shear stress rand strain /.
T =EY, or T =GY
(4-3
The Young's modulus of elasticity E is used when the applied strain y is the elongation or compression, y = Al /1. The shear modulus G is used when / is the shear strain. Most solid and semisolid foods are not elastic, but they behave as either viscoelastic or viscoplastic. The viscoelasticity of the materials is studied by stress relaxation measurements, i.e. shear stress at constant strain versus time (Rao, 1999). The viscoelastic behavior is characterized by Newton's law (Eq. 4-1), Hooke's law (Eq. 4-3), and Newton's second law (F = m a, where m is the mass and a is the acceleration). Most solid and semisolid foods are considered linear viscoelastic, which allows the adding up of the three elements, i.e. the viscous, elastic, and inertial effects. Neglecting the inertial component, the viscous (dashpot) and the elastic (spring) effects are usually combined into two common models, i.e. the Newton (series model), and the Kelvin-Voigt (parallel model).
66
Chapter 4
II. RHEOLOGICAL MODELS OF FLUID FOODS
The rheological behavior of fluid foods is determined by measurements of shear stress versus shear rate, and representation of the experimental data by viscometric diagrams and empirical equations, as a function of temperature and/or concentration. Molecular dynamics can not predict fluid viscosity, but it can be helpful in understanding the flow mechanism of complex fluid foods. Physical structure plays a decisive role in determining the fluid viscosity. A. Structure and Fluid Viscosity
Simple liquid food materials, like aqueous sugar solutions, clarified juices, and vegetable oils are Newtonian fluids, i.e. the shear stress is linearly proportional to the shear rate, according to Newton's law (Eq. 4-1). Incorporation of polymer molecules and micelles, solid particles, droplets, and gas bubbles into Newtonian fluids changes considerably the rheology of fluid foods, evidenced by the non-linear relationship of shear stress/shear rate. Model fluid suspensions are useful in understanding the flow behavior of the complex fluid food systems. The viscosity of fluid suspensions reflects the complex hydrodynamic flow of the particle / solvent and particle / particle interactions. The particle size, shape and concentration affect strongly the viscometric properties of fluid foods. In a simple suspension of spherical particles, the relative viscosity of the suspension to the viscosity of the continuous phase Tjr is a linear function of the volume fraction of the particles (/), according to the Einstein equation (Giesekus, 1983):
rjr = 1 + 2.5
(4-4)
The Einstein equation holds for dilute concentrations of spherical particles, which do not react with the solvent and with each other. Particle-solvent and particle-particle interactions are reflected by a non-linear relationship of (rjr, (f>). The shape of the particles affects strongly the behavior of suspensions Figure 4.2. Thus, the deviation from linearity (Eq. 4-2) increases when the particle shape changes from spherical to plate and rods. Still, stronger effects are observed with fibers of increasing aspect (L/D, length / diameter) ratios. Attracting forces, acting on the particle surfaces, tend to create aggregates, which significantly increase the viscosity of the suspension. Increasing the shear rate, the aggregates are broken down, resulting in shear-thinning of the suspension. The suspended non-spherical particles give higher shear viscosity, due to the difficulty of rotating (flowing) in the suspension. The aspect (L/D) ratio of rodlike par-
Rheological Properties of Fluid Foods
67
tides (e.g. fibers) is related to the build-up of transient network structures. Shear thinning may be found at low concentrations of rodlike particles (e.g. fibers). Model suspensions of coarse food particles in aqueous CMC solution exhibit Theological properties similar to inorganic models, i.e. their deviation from Newtonian behavior increases when the particle shape becomes elongated, and when the particle concentration is increased (Pordesimo et al., 1994). The non-linearity of suspensions of spherical particles at high concentrations can be described by empirical equations, such as the Frankel-Acrivos and KriegerDougherty models, in which the relative viscosity (t|r) is related to the volume fraction ratio ($/$„), where m is a shear-dependent maximum volume fraction (Rao, 1999). Food particles are usually non-spherical, and they resemble to rough crystals. The empirical Kitano et al. equation (1981) can be applied: TJr=[l-(j/A)]
(4-5)
where A is a constant, A = 0.68 for spheres, and A = 0.44 for crystals.
Figure 4.2 Relative viscosity r]r of suspensions of glass particles as a function of the volume fraction in water. (1) Einstein's equation, (2) spheres, (3) plates, (4) rods. (Adapted from Giesekus, 1983.)
68
Chapter 4
Plots of Eq. (4-5) resemble the plots of Figure 4.2. The Kitano et al. model was applied to reconstituted tomato puree of narrow particle size distribution with average diameters 0.71 and 0.34 mm, and adjusting parameters A = 0.54 and A = 0.44, respectively (Yoo and Rao, 1994). Fluid foods are complex suspensions of hydrophilic particles that interact physically and chemically with the solvent (water), resulting in considerable deviations from Eq. (4-2). The particle size is affected by the various processing operations and storage conditions. Thus, the size of starch granules increases considerably during heating, before gelatinization, resulting in increased relative viscosity of aqueous suspensions (Rao, 1999). Changes in particle size of tomato and other fruit and vegetable products, during pulping and screening operations, will strongly affect the product viscosity. Physicochemical interactions in fluid foods, such as adsorption, emulsification, polymer conformation, de-polymerization, crystallization, and melting have significant effects on viscosity. B. Non-Newtonian Fluids
Fluid foods, containing dissolved macromolecules and suspended particles, deviate considerably from Newtonian behavior (Eq. 4-1). Various empirical models have been used to account for the observed non-linear relationship of shear stress (T)- shear rate (y) of the non-Newtonian fluids (Holdsworth, 1993; Rao, 1987a, 1987b).
1. Time-Independent Viscosity The usual non-Newtonian models apply to time-independent fluid foods, for which the shear stress-shear rate relationship does not change with the time of shearing. The Bingham plastic model (Eq. 4-6) applies to Newtonian fluids that will flow only if a fixed yield stress TO is exceeded:
T = TO + TIY
(4-6)
Many non-Newtonian fluid foods are represented by the power-law or Ostwald-de Waele model (Eq. 4-7): T = Ky"
(4-7)
In the power-law model, two rheological constants (K, n) are required to characterize the flow behavior, K the flow consistency coefficient or index with units (Pa sn), and n the flow behavior index (dimensionless). The K value corresponds to the viscosity of Newtonian fluids. The Herschel-Bulkley model is derived from the power-law model by adding a yield stress term r0:
Rheological Properties of Fluid Foods
69
(4-8)
T = Tn
Most non-Newtonian foods are pseudoplastic materials (nY). The pseudoplastic fluids are also known as shear-thinning fluids, since their apparent viscosity decreases as the shear rate is increased. The power law and Herschel-Bulkley models have been used widely, and extensive rheological data for non-Newtonian fluid foods have been published in the literature (Prentice and Huber, 1983; Okos, 1986; Kokini, 1992; Urbicain and Lozano 1997; Rao, 1999). Figure 4.3 shows typical rheological diagrams, r versus Y, for various non-Newtonian fluids. The apparent viscosity of non-Newtonian fluids is an important property, which can be used in several engineering and technological applications in the place of Newtonian viscosity. The apparent viscosity rja of power-law and Herschey-Bulkley fluids is defined by the equation .n-l
(4-9)
In pseudoplastic (shear-thinning fluids), (n-l) < 0, and therefore the apparent viscosity decreases as the shear rate y is increased.
Shear Rate
Figure 4.3 Shear stress C^-shear rate (y) diagrams for Newtonian (1), Bingham (2), pseudoplastic (3), and dilatant (4) fluids.
70
Chapter 4
Specialized rheological models have been applied to certain fluid foods, such as the Casson model for chocolate:
ras = r^+Kcy0'*
(4-10)
The Casson plastic viscosity rjCa is calculated as the square of the Casson coefficient (Rao, 1999): Ka=(KJ2
(4-11)
The Mizrahi-Berk model (Eq. 4-12) is a modification of the Casson equation, and it has been applied to concentrated orange juices (Mizrahi and Berk, 1972): T™ = *„«*+&?"
(4-12)
The rheological constants (Kc, n) are characteristic for a fluid at a given temperature/concentration. 2. Time-Dependent Viscosity Many non-Newtonian fluid foods of complex structure exhibit timedependent rheology, i.e. their apparent viscosity at a given shear rate changes significantly with time of shearing. The most common time-dependent materials are the thixotropic fluids, for which the apparent viscosity decreases with the time of shearing. The rheopectic fluids exhibit the opposite behavior, i.e. the apparent viscosity increases with the time of shearing. The time-dependent rheological behavior is detected usually by the rheological diagram (T versus y), which forms a characteristic loop, when ascending and descending values of stress rare plotted against the corresponding shear rate y. In thixotropic fluids, the descending line falls below the ascending, while the opposite behavior is observed in the rheopectic fluids (Figure 4.4). Empirical models have been proposed to descibe the thixotropy of fluid foods, in which the shear stress is a function of time, with the initial and equilibrium stresses as constants.
71
Rheological Properties of Fluid Foods
VI !_ « CJ
JS V)
Shear Rate Figure 4.4 Shear stress (r)-Shear rate (y) diagrams (loops) for time-dependent fluids: (1) thixotropic, (2) rheopectic.
C. Effect of Temperature and Concentration
In pure liquids, the effect of temperature on viscosity follows the Arrhenius equation (2-31), which can be derived from the theory of rate processes. The same equation has been applied to the viscosity 77 of Newtonian fluid foods, and to the consistency coefficient K or the apparent viscosity rjaof power-law (nonNewtonian) food materials (Rao, 1999):
= K0exp(Ea/RT)
(4-13)
where K0 is a frequency factor (Pa s"), Ea is the energy of activation for viscous flow (kJ/mol), T is the temperature (K), and R = 8.314 kJ /kmol K is the gas constant.
72
Chapter 4
In Newtonian fluid foods, the energy of activation has been found to increase from 14.4 kJ/mol (water) to more than 60 kJ/mol (concentrated clear juices and sugar solutions). Temperature has a major effect on the consistency coefficient K and the apparent viscosity i]a of the non-Newtonian fluid foods, analogous to the effect on Newtonian viscosity rj. The flow behavior index n is affected only slightly by temperature (a small increase at high temperatures). The energy of activation for flow in non-Newtonian fluids is significantly lower than the corresponding value for Newtonian fluids of the same solids concentration (Saravacos, 1970). In suspensions of fluid foods of high non-soluble solids concentration, like fruit or vegetable pulps, Ea may be lower than the activation energy for viscous flow of water (14.4kJ/mol). The observed strong effect of temperature on the viscosity of viscous Newtonian fluid foods (concentrated clear juices, edible oils) is similar to the effect on the viscosity of nonpolar viscous liquids, such as mineral oils, as is described by the Lewis-Squires Eq. (2-33). Concentration of soluble solids (°Brix) and insoluble solids (e.g. pulp) has a strong non-linear effect on the viscosity of Newtonian fluid foods, the consistency coefficient K, and the apparent viscosity of non-Newtonian foods. Two similar exponential models, one for °Brix and a second for % pulp, were proposed by Vitali and Rao (1984a, 1984b) for concentrated orange juice, of the general form:
K = K0exp(BC)
(4-14)
where K0 is a frequency factor, C is the concentration and B is a constant. The combined effect of temperature and solids concentration can be expressed by combining Eqs. (4-13) and (4-14): K = K0 exp [ (Ea / RT) +BC]
(4-15)
The additive model assumes no temperature-solids concentration interaction. This may not be true when temperature affects the solids structure and particle size in the suspension, e.g. by hydrolysis of macromolecules, coagulation of colloids, and breakdown or buildup of agglomerates. The energy of activation for viscous flow Ea is estimated at a constant shear rate, usually at 100 (1/s) (Saravacos, 1970; Rao, 1999). However, the structure of some food suspensions of high particle (pulp) concentration may be changed due to this relatively high shear rate. For this reason, in such cases, a lower shear rate may be preferable. Prentice and Huber (1983) used a shear rate of 10 (1/s) in evaluating the effect of temperature on the rheology of applesauce.
Rheological Properties of Fluid Foods
73
D. Dynamic Viscosity
The rheological characterization of complex semisolid foods requires, in addition to the shear viscosity 77, the dynamic viscosity 77", which is determined by oscillating instruments at various frequencies. Three basic (constitutive) equations are used to simulate the experimental rheological data, i.e. Newton's, Hooke's, and Maxwell's (Kokini, 1992). In dynamic rheological tests, the stress or strain is changed periodically (sinusoidally) with time at a fixed frequency CD. The complex stresses and strains are related to the storage and loss moduli of the material, G 'and G "respectively. The Bird-Carreau constitutive model, derived for polymer solutions, represents the entire deformation of a material. Thus, the shear and dynamic viscosities (77 and 77') are estimated from the equations:
77'= Y —— ——2
(4-17)
where 77^ is the shear viscosity of p element, y is the shear rate (1/s), co is the frequency of oscillation (1/s), and A/, /L2 are time constants (s). When the shear and dynamic viscosities of food suspensions, like mayonnaise and margarine, are plotted against shear rate or oscillation frequency on loglog scales, parallel straight lines with negative slopes are obtained (Kokini, 1992). In rheology, the dimensionless Deborah number (De) is an index of the fluid or solid behavior of the material. It is defined as the ratio of the time of deformation tD over the time of observation tm and it can be estimated from the viscosity of the continuous phase 77,,, the shear rate y, and the plateau storage modulus G0: De = (tD/t0) = (rlsy/G0)
(4-18)
Liquid-like behavior is characterized by small De numbers, while large De numbers correspond to solidlike materials (Rao, 1999).
74
Chapter 4
III. VISCOMETRIC MEASUREMENTS
A. Viscometers
The shear viscosity of Newtonian fluids and the rheological constants of the non-Newtonian materials are determined using several instruments, which basically measure the shear stress versus shear rate at a given temperature. The external stress (force/area) is applied at steady state, usually stepwise, either in ascending or descending order. The stress takes the form of pressure drop in a capillary tube, or mechanical torque in a rotational system. The resulting rheogram is utilized to extract rheological data for the material under the specific measurement conditions. Three common types of viscometers are used in most fluid food applications, i.e. the tube/capillary, the rotational, and the cone-and-plate viscometers (Figure 4.5). Some other types of instruments are used in special cases, e.g. the parallel plate, the slit, and the falling ball viscometers. Details on viscometers are provided by the manufacturers of scientific instruments. Lists and descriptions of viscometers can be found in the literature of rheology, e.g. van Wazer et al. (1963), Whorlow (1980), Bourne (1982, 1992), and Rao (1999). In food process control, special sensors are used to monitor and control the viscosity of a material, if changes of this property are very important during processing. A typical example is the hot wire viscometer, which can be used to monitor milk clotting in cheese tanks, or enzymatic hydrolysis of starch (Sato et al., 1990). Its operation is based on the change of heat transfer rate on the surface of a hot wire, caused by a viscosity change. 1. Capillary Tube Viscometer The tube viscometer (Figure 4.5a) is based on measuring the flow rate (shear rate) of a fluid in a capillary or small diameter smooth tube at constant pressure drop (shear stress). For a Newtonian fluid, flowing through a straight tube of circular cross section with diameter (D, m) and length (L, m) under a pressure drop (AP, Pa), the Poiseuille equation is applicable: D(AP) /4L = ri(32Q/ic D3)
(4- 1 9)
where Q is the volumetric flow rate (mVs) and rj is the viscosity (Pa s). By comparing Eqs. (4-19) and (4-1) it follows that the shear stress rand shear rate yfor Newtonian fluids are given by the relationships: T = D (AP) /4Landy=32Q/ (n D3)
(4-20)
Rheological Properties of Fluid Foods
75
AP
Q a. Tube Viscometer
b. Rotating Coaxial Viscometer
->
<^ TO
c. Cone-and-Plate Viscometer Figure 4.5 Common viscometers a) capillary/tube; b) rotating coaxial; c) cone-and-plate.
76
Chapter 4
For non-Newtonian fluids, the shear stress is given by the same relationship, but the shear rate is a function of the rheological constants K (Pa sn) and n:
Y = [(l+ 3n) / 4n][ 32Q / (nD3)]
(4-21)
Therefore, the power-law model for capillary tube flow becomes:
D(AP)/4L =K[(l+3n)/4n]"[32Q/(nD3)]"
(4-22)
Equations (4-19) and (4-22) apply to the laminar flow in tubes, i.e. when the Reynolds number (Re) is lower than 2,000. For flow of Newtonian fluids in circular tubes of diameter (D, m), Re = (D u p)/r/, where u is the mean velocity (m/s), p is the density (kg/m3), and 77 is the viscosity (Pa s). For power-law and Herschel-Bulkley fluids, the (Re) number is calculated from the equation Re = [(D"u -'" p)/(8n~1 K)] [ (4n) / (3n + 1)]
(4-23)
The velocity profiles of non-Newtonian flows in a tube viscometer can be measured using nuclear magnetic resonance (McCarthy et al, 1992). The profile of CMC solutions was as predicted by the laminar flow equations, while the profile of tomato juice showed a significant deviation. The capillary tube viscometers require no calibration, since the rheological constants can be determined from fundamental flow equations. They are suitable for both Newtonian and non-Newtonian fluids, and they can handle suspensions of relatively large particles. However, they require more time for preparation and measurement than the rotational viscometers, which are preferred for routine applications. In tube viscometers, very high shear rates at the wall can be obtained, when small tube diameters are used. Slip flow near the wall may be a problem with fluid suspensions of large particles.
2. Rotational Viscometers The most common rotational viscometer is the Couette or concentric cylinder rheometer (Figure 4.5b). For Newtonian fluids the viscosity rj is given by the empirical equation (Rao, 1999) jj = CM/n
(4-24)
where M is the applied torque (N m), /2is the rotational velocity (1/s) and C is the instrument constant (1/m ), which, for the concentric cylinder system, becomes:
Rheological Properties of Fluid Foods
C = (l/4xh)[(l/ri)2-(l/rQ)2J
77
(4-25)
The shear rate of non-Newtonian fluids in rotational viscometers is estimated from empirical equations as a function of the rotational velocity Q, the ratio of the radii (rt/r^, and the flow behavior index n. In concentric cylinder systems, narrow gaps, e.g. (rt / rj = 0.95 are required to insure laminar flow. Fluid foods, containing large suspended particles, cannot be measured in narrow-gap viscometers, and other instruments are more suitable, e.g. capillary tube. Simpler rotational viscometers, e.g. rotating plates, bobs or spindles in relatively large fluid volumes, are used in many routine measurements of Newtonian fluids. Such instruments are not suitable for non-Newtonian fluids, since the applied shear rate cannot be estimated accurately. However, in some systems, empirical relationships of rotational velocity Q and shear rate y can be used to construct the rheogram (T vrs. f) of the non-Newtonian fluid.
3. Cone-and-Plate Viscometer The principle of cone-and-plate viscosimeter, or Weissenberger rheogoniometer, is shown in Figure 4.5c. This instrument is suitable for non-Newtonian fluid suspensions, containing small particles. The shear stress rand shear rate y are calculated from the equations T =3M/r0A
(4-26) (4-27)
where M (N m) is the torque, r0 (m) is the radius of the plate, and A (m2) is the area of the cone. For measurements on non-Newtonian fluids, the cone/plate angle should be small (0 < 5°), usually 2-3°. Small samples should be used with this instrument. In both rotational and cone-and-plate viscometers the shear rate is usually varied stepwise, either in increasing (ascending) or decreasing (descending) order.
78
Chapter 4
B. Measurements on Fluid Foods
In rheological measurements of fluid foods, significant variations are observed on the viscometric data, which may be due to calibration of the instrument or to changes in the structure of the food material. Accurate calibration of instruments is essential for non- Newtonian and time-dependent fluid foods (Sherman, 1984; Bourne, 1992). Standardization of the instrument and the measuring procedure is necessary. Particular attention should be paid to fluid dispersions, whose structure may change remarkably (breakdown or agglomeration) during sample preparation and transfer to the measuring system. Dispersions of fluid foods develop a complex structure, depending on the size, shape and concentration of the suspended particles. When subjected to increasing shear or stress, the suspensions may break down gradually, especially at low shear stresses. These changes are caused by breakdown of particle agglomerates and untangling of non-spherical particles. Many fluid foods do not obey the empirical non-Newtonian models over the entire range of shear rate, available in the commercial instruments. Therefore, it is important to specify the data and model range, which, preferably, should be similar to the range of application in process engineering or product quality. Temperature has a strong effect on the viscosity of Newtonian fluid foods, especially in highly viscous products, such as syrups, honey, and edible oils. Viscous heating during viscometric measurements of thick food fluids may result in erroneous low viscosities. In general, temperature has little effect on the flow behavior index n of non-Newtonian fluid foods. The effect of temperature on the consistency coefficient K is similar to the effect on the viscosity of Newtonian fluids, obeying the Arrhenius equation. However, the energy of activation for viscous flow Ea in non-Newtonian fluids is considerably lower than in Newtonian materials. In several food suspensions and pulps, Ea may be lower than 14.4 kJ/mol, which is the energy of activation for viscous flow of pure water. This is an indication that the consistency coefficient K is affected more by the structure than by the temperature of the suspension The problems of measuring the rheological properties of liquid foods were investigated in the European cooperative research project, COST 90 (Prentice and Huber, 1983). Eight laboratories from different countries participated in the study, using exclusively concentric-cylinder and cone-and-plate viscometers. Some of the results are analyzed and discussed, in relation to literature data, in Sections IV and V of this chapter.
Rheological Properties of Fluid Foods
79
IV. RHEOLOGICAL DATA OF FLUID FOODS
Due to the complexity of food composition and structure, rheological data should be obtained by standardized measurements on each specific material. The technical literature contains a wide variety of such data, and new results continue to be reported in recent years. Collections of rheological data of fluid foods have been published by Steffe et al. (1986), Kokini (1992), Urbicain and Lozano (1997), and Rao (1999). This critical review includes typical viscometric data from a wide variety of fluid foods, classified as follows: edible oils, sugar solutions and clarified juices, suspensions of plant materials, and emulsions and complex suspensions.
A. Edible Oils
The edible or cooking oils and fats are generally considered as Newtonian fluids, if they do not contain particles or emulsions, which may enter the oil phase during processing and use. Collaborative measurements on a commercial cooking oil (mixture of soybean and rapeseed) showed a slight deviation from Newton's law (Prentice and Huber, 1983). Regression analysis of the rheological data, obtained in seven laboratories, using coaxial-cylinder and cone-and-plate viscometers at 25°C, resulted to the following equation:
log (r) = -1.2766 + 0.99673 log (y)
(4-28)
where the rheological constants are K = 0.052 Pa s" and n = 0.99673. According to this equation, the apparent viscosity (r\a = K /"'') of the oil depends slightly on the shear rate, for example r/a = 52 mPa s at 7= 1 (1/s), and r/a = 51mPasat/=1000(l/s). Temperature has a strong effect on viscosity, especially of viscous oils. The Arrhenius equation yielded an energy of activation Ea = 28.3 kJ/mol in the temperature range 25-50°C for the oil described by Eq. (4-28). Empirical equations of viscosity as a function of temperature for specific oils have been proposed in the literature (Rao, 1999). Typical viscosities of edible oils (Kokini, 1992; Rao, 1999) are given in Table 4.1. Table 4.1 Typical Viscosities of Edible Oils Temperature, °C Edible oil 23.9 Corn oil Soybean oil 23.9 Olive oil 40.0 Cocoa butter 25.0
Viscosity, mPa s 52.3 54.3 36.3 124.1
80
Chapter 4
B. Aqueous Newtonian Foods
]. Sugar Solutions and Clarified Juices
Aqueous food fluids, including sugar solutions, honey, and clarified juices (containing no paniculate matter) are considered, in general, Newtonian liquids. However, the COST 90 collaborative measurements on 50°Brix sucrose solution at 25°C (Prentice and Huber, 1983) showed a slight non-Newtonian behavior with an estimated flow behavior index n = 0.986 and a consistency coefficient K = 7.4 mPa s". This corresponds to an apparent viscosity rja = 6.9 mPa s at a shear rate of Sugar solutions and clear juices are characterized by sharp decrease of viscosity at higher temperatures, according to the Arrhenius equation. Concentration has a strong positive effect on viscosity, and the combined temperature/concentration effect can be expressed in most cases by empirical exponential equations, similar to the generalized Eq. (4-15). Figure 4.6 compares the viscosities of clarified apple juice (Saravacos, 1970) and sucrose solutions (Perry et al., 1984) at three different temperatures, 20, 40, and 60°C. It appears that the viscosities of the two fluids are similar, except at concentrations higher than 60°Brix, where the apple juice becomes more viscous, probably because of the increased concentration of nonsugar solids. Figure 4.7 shows that the viscosity of concord grape juice tends to be higher than the viscosity of the corresponding sucrose solution, particularly at high concentrations. This difference may be caused by the precipitation of some compounds of grape juice, which become insoluble at higher concentrations, for example tartrates. These compounds are known to precipitate during the storage of grape juices and wines at low temperatures.
Rheological Properties of Fluid Foods
81
1000 ^±ES:^3^di=Ei^:S3^t
•
apple juice , . ,,,, • sucrose solutions
VI
«
CM
u w
i>
o.i
Figure 4.6 Comparison of viscosities of depectinized apple juice and sucrose solutions at 20-60°C. (From Saravacos, 1970, and Perry et al., 1984.) 1000
~.(
•
grape juice • sucrose solutions
Figure 4.7 Comparison of viscosities of Concord grape juice and sucrose solutions at 40°C. (From Saravacos, 1970, and Perry et al., 1984.)
82
Chapter 4
Figure 4.8 shows the energies of activation for viscous flow of sucrose solutions (Perry et al., 1984), grape juice (Saravacos, 1970), and apple juice (Rao, 1999) as a function of concentration. The energies of activation Ea of the two juices are close to those of sucrose solutions, increasing sharply at higher sugar concentrations. The extrapolated energy of activation at 0°Brix corresponds to the Ea value of water (14.5 kJ/mol), which was estimated from viscosity-temperature data of the literature (Perry et al., 1984) The energy of activation for viscous flow increases as the molecular weight of the dissolved sugar is increased. The viscosity ij of a sugar solution can be expressed by an empirical model, analogous to the Arrhenius equation (Chirife and Buera, 1994): 77 = a exp (EX)
(4-29)
where X is the mole fraction of the sugar, a is a constant close to 1, and £ is a characteristic constant, similar to the Arrhenius activation energy. Calculated values of constant E are: 27.93 for glucose, 57.19 for sucrose, and 94.46 for corn syrup of 35.4 DE (dextrose equivalent).
T'\
™ f-^
sucrose solutions grape juice
•
apple juice
Figure 4.8 Activation energies for viscous flow of sucrose solutions (Perry et al., 1984), grape juice (Saravacos, 1970), and depectinized apple juice (Rao, 1999).
Rheological Properties of Fluid Foods
83
Molasses (thick liquid residues of sugar manufacturing) have high viscosities, e.g. 6600 to 374 mPa s in the temperature range 21-66°C (Hayes, 1987), from which an activation energy of 51.3 kJ/mol was estimated. Clarified cherry juice behaves as a Newtonian fluid in the range 22-74°Brix and 5-70°C with energies of activation 14.4-61 kJ/mol (Giner et al, 1996). The combined effect of temperature and concentration on viscosity is described well with an empirical exponential model similar to Eq. (4-15). Similar activation energies (26.6-64.4 kJ/mol) were obtained in clarified crab (small) apple juice (Cepeda et al., 1999). Clarified orange juice behaves as a Newtonian fluid in the range 3063.5°Brix with energies of activation Ea in the range 17.7-40 kJ/mol (Ibarz et al., 1994). The relatively low Ea values may be due to the presence of colloidal particles in the clarified juice. Similar rheological behavior is shown by other clarified fruit juices: apple and pear juices and concentrates (Ibarz et al., 1987), peach juices and concentrates (Ibarz et al., 1992a), and black currant juices (Ibarz et al., 1992b). Clarified banana juice (20-79.7°Brix) has viscosities ranging from 1 to 6000 mPa s in the range 3070°C, and energies of activation for viscous flow 25-78 kJ/mol (Khalil et al., 1989). 2. Honey and Sugar Extracts The viscosity of honeys depends strongly on the concentration of soluble solids (°Brix), temperature, and plant origin. Most honeys have a concentration between 75-83°Brix, and their viscosity at 20°C is in the range of 4-20 Pa s. Temperature has a strong effect, with corresponding high energies of activation Ea. Utilizing the viscosity-temperature data (10-80°C) of five types of honey, published by Rao (1999), Ea values of 65-70 kJ/mol were obtained, which are in agreement with the generalized diagram of Figure 4.8. Chinese honeys (Junzheng and Changying, 1999) have lower sugar concentrations (71.5-80.2°Brix) and viscosities (0.33-6.30 Pa s at 20°C) with estimated activation energies 60-65 kJ/mol. Australian honeys have high sugar concentrations (82.4-83.3°Brix) with high viscosities (about 100 Pa s at 10°C), and a strong effect of temperature (Bhandari et al., 1999). Licorice extract of soluble solids concentration 3-50°Brix exhibited Newtonian behavior in the temperature range of 10-60°C, with energies of activation 14.4-61 kJ/mol (Maskan, 1999). The viscosity data of honey, presented by Rao (1999), and molasses (Hayes, 1987) are plotted in Figure 4.9, snowing a nearly exponential drop of viscosity with increasing temperature.
Chapter 4
84
0.1
-L
30
40
50
Temperature (°C)
Figure 4.9 Viscosities of honeys and molasses.
3. Other Clear Solutions The viscosity of ethanol at 20°C is 1.2 mPa s, and that of glycerol is 1490 mPa s (Lewis, 1987). The viscosity of wines is higher than that of water, ranging for 1.3-1.67 mPa s for ordinary wines at 25°C, and 1.95-2.45 mPa s for fortified wines, containing glycerol (Rao, 1999). The energy of activation for viscous flow of wines is in the range of 18.8-28.2 kJ/mol, i.e. significantly higher than that of pure water. The viscosity of fermenting musts is higher than that of wines, due to the presence of dissolved sugars and suspended particles in the liquid (Lopez et al., 1989), For white musts, the viscosity varies in the range of 3-4 mPa s at 14-22°C, with activation energy about that of water (14.4 kJ/mol). The viscosity of beer is close to that of wine, i.e. 1.3 mPa s at 20°C (Lewis, 1987). The viscosity of salt (sodium chloride) solutions and brines, depends on the concentration of the ionic species, for example 2.7 mPa s at 22% salt and 2°C (Hayes, 1987).
Rheological Properties of Fluid Foods
85
C. Plant Biopolymer Solutions and Suspensions Plant hydrocolloids (gums and starches) are components of many fluid foods and they are responsible for their non-Newtonian behavior. They are present, in relatively low concentrations, as molecular solutions, colloidal dispersions,
and particulate suspensions. Dispersions of plant gums contain macromolecules of coil configurations, while starch suspensions contain granular particles. Collaborative measurements, within the COST 90 European project, were made on solutions of guar gum, carrageenan gum, xanthan gum, and Karaya gum, using coaxial-cylinder and cone-and-plate viscometers. The results showed high consistency coefficients (K), some yield stresses (TO), and flow behavior indices (n) lower than (1), an indication of shear-thinning (pseudoplastic) behavior. Published shear stress r versus shear rate ^data of 1% gum solutions (Prentice and Huber, 1983) were analyzed statistically, and the resulting regression lines are shown in the logarithmic plots of Figures 4.10 to 4.13. The Theological constants K and n of the power-law model (Eq. 4-8) are shown in Table 4.2: The rheological properties of the gums depend strongly on the concentration and type of macromolecules. Xanthan gums show high consistency and low flow behavior indices, presumably due to the high molecular weight and the large colloidal particles in the solution. Table 4.2 Rheological Constants of 1% Food Gum Solutions at 25°C Food gum Consistency coefficient Flow behavior index _________________K, Pa s"____________n______ Guar 1.24 0.625 Carrageenan 1.40 0.485 Xanthan 9.21 0.183 Karaya_____________1.23____________0.465
Chapter 4
86
1000
0.1
0.01 0.001 0.001
10
1000
100000
Shear Rate (1/s)
Figure 4.10 Rheological data of 1% guar gum solution at 25°C. (Data from Prentice and Huber,1983.) 1000
Carrageenan gum solution
0.1 0.001
0.1
10
1000 100000
Shear Rate (1/s)
Figure 4.11 Rheological data of 1% carrageenan gum solution at 25°C. (Data from Prentice and Huber, 1983.)
Rheological Properties of Fluid Foods
87
1000
0.001
0.1
10
1000100000
Shear Rate (1/s)
Figure 4.12 Rheological data of 1% xanthan gum solution at 25°C. (Data from Prentice and Huber, 1983.) 1000
0.1
0.01 0.001
10
1000
100000
Shear Rate (1/s)
Figure 4.13 Rheological data of 1% karaya gum solution at 25°C. (Data from Prentice and Huber, 1983.)
Chapter 4
88
Tube viscometry of aqueous pectin solutions showed pseudoplastic behavior (Saravacos et al., 1967). Plots of shear rate rversus sear rate /for 2.5% and 5.0% pectin solutions at 25°C (Figure 4.14) show the following rheological properties: K= 3,590 Pa sn and n = 0.767 for the 2.5% solution; K = 57,420 Pa sn and « = 0.822 for the 5.0% solution. The power-law model applies to carboxy-methyl-cellulose (CMC) solutions (0.5-1.5%) in the temperature range 25-125°C, which are of interest to aseptic processing of foods. The rheological properties (K, n) were obtained using a sealed pressure rheometer (Vais et al., 1999). The consistency coefficient of the 1.5% solution decreased from 17 to 3.4 Pa sn, and the flow behavior index n remained nearly constant at 0.43 at temperatures 25-125°C. Starch suspensions and solutions exhibit complex rheological properties, due to the diverse particulate and molecular composition and the physicochemical changes, induced by temperature (gelatinization) and molecular interactions with other food components, like proteins. Rheological characterization of starch systems requires not only shear stress-shear rate data, but also viscoelastic measurements and particle and biopolymer characterization (Rao, 1999).
1000 -, 4 Pectin — 2.5% — 5.0% a a-
-
100
10 10
100
1000
Shear Rate, 1/s
Figure 4.14 Rheological data of aqueous pectin solutions (From Saravacos et al., 1967.)
Rheological Properties of Fluid Foods
89
D. Cloudy Juices and Pulps
The presence of hydrocolloids (gums) and suspended particles (e.g. granules or fibers) in fluid plant foods changes the rheological properties into nonNewtonian, usually pseudoplastic fluids. Aqueous suspensions of dietary fibers, from orange or peach, exhibit strong non-Newtonian behavior, particularly at concentrations higher than 5% (Grigelmo-Miguel et al., 1997). The energy of activation for viscous flow is low, 3-18.4 kJ/mol, and the effect of concentration on the consistency coefficient of the suspension is higher than that of temperature. Rheological measurements on applesauce, using a capillary tube viscometer, yielded the values K = 12.7 Pa sn, and n = 0.28 (Saravacos, 1968). The consistency coefficient K decreased from 12.7 to 0.49 Pa sn and the flow behavior index n increased slightly from 0.28 to 0.35, when the pulp content was reduced by passing the applesauce through an 0.04 inch (1 mm) screen. Similar results on the rheological constants of applesauce were obtained in the COST 90 cooperative measurements (Section V of this chapter). Addition of sugar (glucose) to banana pulp reduces the apparent viscosity and increases the temperature dependence of the juice. The activation energy increased from 8 to 54 kJ/mol when the sugar concentration increased from 21 to 51°Brix (Guerrero and Alzamora, 1997). A similar change of the rheological properties and activation energy by glucose addition was observed on peach, papaya and mango purees (Guerrero and Alzamora, 1998). The combined effect of concentration and temperature follows the generalized exponential model of Eq. (415).
Cloudy apple juice becomes a non-Newtonian fluid at concentrations higher than 40°Brix, with flow behavior indices n = 0.65-0.80, and activation energies lower than the clarified juice, e.g 25.5 kJ/mol versus 35 kJ/mol at 50°Brix (Saravacos, 1970). A similar rheological behavior was observed on raspberry juice, which changed from Newtonian to non-Newtonian above 25°Brix, with « = 0.900.60 (Ibarz and Pagan, 1987). Ultrafiltration of orange juice removes part of the particles and hydrocolloids (pectin), decreasing the consistency coefficient K, and increasing the flow behavior index n and the activation energy (Hernandez et al., 1995). This treatment increases significantly the heat transfer coefficient in the evaporation of orange juice. Shear-thinning food fluids have improved heat transfer characteristics in agitated kettles and heat exchangers, since their apparent viscosity is reduced considerably by agitation (Saravacos and Moyer, 1967). The power-law model of non-Newtonian fluids has been applied to various suspensions of starch and pulp materials: cooked debranned maize flour with flow behavior indices n = 0.258-0.668 (Bhattacharaya and Bhattacharaya, 1996); riceblackgram suspensions (28-44%TS) increased the (n) value from (0.3-0.4) to (0.45-0.50) by fermentation, evidently due to a breakdown of biopolymers (Bhattacharaya and Bhat, 1997).
90
Chapter 4
Enzymatic breakdown of starches and other biopolymers results in lower viscosities and higher n values, such as the enzymatic extrusion of rice starch (Tomas et al., 1997) and the enzyme treatment of mango pulp (Bhattacharaya and
Restogi, 1998). The rheology of tomato juice and tomato concentrates has received special attention due to its importance in processing and in product quality. Typical theological properties are: K = 0.22 to 12.9 Pa sn and n = 0.59 to 0.41 for 5.8 to 25.0% TS (coaxial- cylinder, Harper and El-Sahrigi, 1965); K = 0.22 to 52 Pa s" and n = 0.581 to 0.177 for 5.6 to 32.6°Brix (tube viscometer, Saravacos et al., 1967). The processing method has a significant effect on the rheology of tomato concentrates. The "hot break" product is more viscous than the "cold break" material. Thus, in the range 10-25°Brix, the lvalue of the "cold break" tomato concentrates decreased from (5-80 Pa sn) to (4-30 sn), and the activation energy from 22.7 to 17.0 kJ/mol, compared to the "hot break" product (Fito et al., 1983). The particle size and pulp concentration strongly affect the consistency coefficient K of tomato concentrates (Rao, 1999). An empirical rule is to scale the viscosity by a factor of (total solids)2'5. E. Emulsions and Complex Suspensions
Oil/water (o/w) food emulsions, like mayonnaise and salad dressings, are non-Newtonian fluids with high consistency coefficients K and flow behavior indices n lower than 1. Mayonnaise contains 70-80% oil particles (droplets) in the size range 0.01-10 um, dispersed in an aqueous phase. Food emulsions are stabilized by adsorption of biopolymers (gums, proteins, lecithin) on the particle surfaces (Rao, 1999). Because of their structure, emulsions may exhibit timedependent theological properties, as well as viscoelastic behavior. Addition of sugar to emulsions changes significantly their theological properties. Thus, by adding 8% sugar to sunflower oil-water emulsions, the consistency coefficient K of the power law model decreased from (2.6-3.6) to (0.6-2.2), while the flow behavior index n remained nearly constant (n = 0.49). The activation energy Ea decreased from 31 to 10.7 kJ/mol (Maskan and Gogus, 2000). Milk may be considered as dilute emulsion of fat globules in an aqueous solution of lactose and other components. Single strength (non concentrated) milk behaves as a Newtonian fluid with a viscosity higher than that of water (Kokini, 1992). Literature data on homogenized milk show viscosities decreasing from 3.4 to 0.6 mPa s, in the temperature range 0-80°C (Figure 4.15). Concentration of milk by evaporation above 22.3%TS changes the rheology to non-Newtonian (n = 0.89 at 42.4%TS), with a sharp increase of the activation energy from 10 to 49 kJ/mol (Velez-Ruiz and Barbosa, 1998). The combined effect of temperature and concentration on the consistency coefficient K follows the generalized model of Eq. (4-15). Freeze-concentrated milk showed a similar
91
Rheological Properties of Fluid Foods
rheological behavior (Chang and Hartel, 1997). The power-law model was found applicable, with the flow behavior index decreasing from 1.0 to 0.89, and the activation energy increasing from 20 to 60 kJ/mol in the range of 20-40%TS.
Sodium caseinate suspensions behave as Bingham plastic materials, following an empirical model similar to Eq. (4-15). The activation energy of casein suspensions increased from 14.6 to 37.7 kJ/mol in casein concentrations 10-16%TS. Buttermilk shows a thixotropic (time-dependent) behavior. The Herschel-Bulkley model can be applied, using a structural correction factor (Butler and McNulty, 1995). The flow properties of yogurt can be expressed by the power-law or the Herschel-Bulkley model. However, since yogurt is a complex material, exhibiting both fluid and semisolid (gel) properties, time-dependency (thixotropy) and viscoelasticity should be considered (Afonson and Maia, 1999). A strong degradation of the yogurt may take place at high shear rate. The activation energy for viscous flow at temperatures higher than 25°C increases sharply from 24 to 65 kJ/mol, evidently due to action of thermophilic clotting bacteria (Benezech and Maingonnat, 1994). The rheological properties of salmon surimi can be expressed by either the Herschel-Bulkley or the Casson models, with the flow behavior index n in the range of 0.58-0.75 (Bourami et al., 1997). Table 4.3 shows typical rheological properties of food emulsions. 10 T-
20
40
60
80
100
Temperature (°C)
Figure 4.15 Viscosity of homogenized milk. (Data from Kokini, 1992.)
92
Chapter 4
Table 4.3 Typical Rheological Data of Food Emulsions at 25°C________ Food emulsion Consistency coefficient (K) Flow behavior index (n) Pas" Mayonnaise 0.55 6.4 Mustard 0.39 18.5 0.074 Margarine 297.6 Peanut butter 0.065 501.1 Cream, 40% fat 1.0 0.0069 (40°C) Source: From Kokini, 1992.
V. REGRESSION OF RHEOLOGICAL DATA OF FOODS
Most of the literature data on the viscosity or Theological constants of foods are available in the form of tables or diagrams at specific temperatures and concentrations. More information can be gained by analyzing the data statistically, and extracting a regression line, typical of a food product, or a group of similar products. However, statistical analysis of the literature data is difficult, because the measurements were made with different instruments on different products, and the statistical samples are often small. Rheological data obtained in the COST 90 cooperative measurements (Prentice and Huber, 1983) are suitable for statistical analysis, and some regression lines were presented in Section (IV.C) of this chapter. Additional regression lines are presented in this section for applesauce and chocolate. Approximate representative lines can be obtained by non-linear regression of several data on a food material, obtained by different investigators, provided that the statistical sample is large enough. Details of this regression technique are given in Chapter 6. From a review of the literature, rheological data on the following food products were found suitable for regression analysis: edible oils, applesauce, tomato, orange, pear, and mango juices and concentrates, and chocolate.
A. Edible Oils
Literature data on various edible oils and fats were obtained from Kokini (1992), Noureddini et al. (1992), and Rao (1999). The oils are assumed to be Newtonian fluids. The reported viscosities ranged from 1.6 to 451 mPa s at temperatures from 0 to 121°C. The viscosities are plotted versus the temperature in Figure 4.16. The following form of the Arrhenius equation was applied for the calculation of the activation energy for viscous flow Ea:
Rheological Properties of Fluid Foods
93
(4-30)
77/ = 77 / . exp[—-(— ^ ^ -——)] »
where 77 and TJO are the viscosities at temperatures T and T0, respectively, and R is the gas constant (8.314 J/mol K). For a reference temperature T0 = 25°C, the estimated viscosity of an "average" oil is 7j0 = 55 mPa s, and the activation energy Ea = 45 kJ/mol. These values are comparable to the values (77 = 52 mPa s and Ea = 53.2 kJ/mol) obtained on a typical edible oil, in the cooperative COST 90 project (Prentice and Huber, 1983).
1000 TZ
a.
s
20
40
60
80
100
Temperature (°C)
Figure 4.16 Regression line of viscosity data of edible oils.
120
140
94
Chapter 4
B. Fruit and Vegetable Products
1. Applesauce Rheological data (shear stress versus shear rate) were obtained in seven laboratories on applesauce of 22.5°Brix (screen opening 0.6 mm) at 25°C (Prentice and Huber, 1983). The regression line of the power-law model yielded the following rheological constants: K= 26 A Pa s" and n = 0.286 (Figure 4.17).
1000
I
Apple Sauce
|
1 0.001
0.01 0.1
1
10
100
1000
Shear Rate (1/s) Figure 4.17 Rheological data of applesauce at 25°C. (Data from Prentice and Huber, 1983.)
Rheological Properties of Fluid Foods
95
2. Fruit /Vegetable Juices and Concentrates Rheological data on tomato, orange, pear, and mango juices and concentrates, suitable for regression analysis, were obtained from Harper and El-Sahrigi (1965), Fito (1983), Manohar et al. (1991), and Rao (1999). The fruit and vegetable juices/concentrates, considered in this section, are assumed to behave as non-Newtonian fluids, following the power-law or Herschel-Bulkley models. The consistency coefficient K was estimated from the following form of the generalized Eq. (4-15), Krokida et al. (2000):
(4-31) The flow behavior index n is assumed to be a linear function of concentration and independent of temperature, according to the equation:
n = n0-bC
(4-32)
where K = K0 when C = 0 and T = To, n = n0 when C = 0. C is the concentration
and B, b are constants. The regression lines for the four products are shown in Figures 4.18 to 4.21.
The consistency coefficient K increases exponentially with the concentration, while the flow behavior index n decreases slightly. Concentration has a stronger effect on K of tomato than the other three fruits. The higher activation energy for orange and mango may be due to the higher sugar content.
Table 4.4 Estimated Values of the Parameters of Eqs. (4-31) and (4-32) Material K0. Pa s" b B Em kJ/mol n0 Tomato Orange Pear Mango
1.27 9.28 2.15 1.85
0.149 0.077 0.087 0.089
15.8 35.0 16.1 32.1
0.403 0.950 0.348 0.332
0.0028 0.0034 0.0000 0.0017
o
o'
OQ
o
o_
00
3
C
(Q
n o
o
o
Flow Behavior Index (-)
to
H o
Consistency Coefficient (Pa s")
T3 ff
O
J u
8
C
to
3
(Q C
Flow Behavior Index (-)
3
8-
3
a o
3
o>
O
n o
\ \. \
p o
I c
-a
y \ \
Consistency Coefficient (Pa s") o o
o
a. u>
O o
in O
fl>'
a
CO
TJ
D>
o'
(Q
o o_
Chapter 4
98
c
.-*'^,
M
« 100 -
:
^^^' *^»————
e J>
cu>
^^^
_^*
10 -
*^^-^^^ -^^^^^
e
£Ift
ix^^^"*^
I
-^^
•
.^"^^^^'^ l
^<^c
r :
=— ———
uo
Temperature (°C)
—————— • 40 — ——————— • 60
n1. 20
40
60
Concentration (% solids)
i.o
•8 _o R U
oa
20
40
Concentration (% solids) Figure 4.20 Rheological data of pear juice and concentrates.
60
n
po
k)
3
(Q C
I 8
sa
n
Q
o
P
Flow Behavior Index (-)
1
65
a
o
a
o a
O
Consistency Coefficient (Pas )
O
to
to
Q. (0
-n o o
OL
C
to' to
•o
0)
o'
O
(Q
Chapter 4
100
C. Chocolate
The Theological data on chocolate at 40°C, obtained in the COST 90 project, were analyzed, assuming that the Casson model (Eq. 4-10) is applicable. The regression line, shown in Figure 4.22, gave the following Theological constants: TO = 0.5 27.0 Pa and Kc = 1.68 (Pa s)™. 10000 -p=
I
1000
M
CO
'I
0.001
0.01 0.1
1
10
100
1000
Shear Rate (1/s) Figure 4.22 Rheological data of chocolate at 40°C. (Data from Prentice and Huber, 1983.)
Rheological Properties of Fluid Foods
101
REFERENCES
Afonson, I.M., Maia, J.M. 1999. Rheological Monitoring of Structure Evolution and Development in Stirred Yogurt. J Food Eng 42:183-190. Aguilera, J.M., Stanley, D. W. 1999. Microstructural Principles in Food Processing and Engineering. Gaithersburg, MD: Aspen Publishers. Benezech, T., Maingonnat, J.F. 1994. Characterization of the Rheological Proper-
ties of Yogurt. A Review. J Food Eng 21:447-412. Bhandari, B., D'Arcy, B., Chow, S. 1999. Rheology of Selected Australian Honeys. J Food Eng 41:65-68. Bhattacharaya, S., Bhat, K.K. 1997. Steady Shear Rheology of Rice-Blackgram Suspensions and Suitability of Rheological Models. J Food Eng 32:241-250. Bhattacharaya, S., Bhattacharaya, S. 1996. Rheology of Cooked Debranned Maize Flour Suspensions. J Food Eng 27:97-105. Bhattacharaya, S., Restogi, N.K. 1998. Rheological Properties of Enzyme-Treated Mango Pulp. J Food Eng 36:249-262. Bourami, M.M., Fichtali, J.. Finder, K.L., Nakai, S., Bowen, B.D. 1997. Viscous Properties of Salmon Surimi Paste. J Food Eng 34:441-452. Bourne, M.C. 1982. Food Texture and Viscosity. Concept and Measurement. New York: Academic Press. Bourne, M.C. 1992. Calibration of Rheological Techniques used for Foods. J Food Eng 16:151-163. Butler, F., McNulty, P. 1995. Time-Dependent Rheological Characterization of Buttermilk at 5°C. J Food Eng 25:569-580. Cepeda, E., Villaran, M.C. 1999. Density and Viscosity of Malus Floribunda Juice as a Function of Concentration and Temperature. J Food Eng 41:103-107. Chang, Y.-H., Hartel, R.W. 1997. Flow Properties of Freeze-Concentrated Skim Milk. J Food Eng 31:375-386, Chirife, J., Buera, M.P. 1994. A Simple Model for Predicting the Viscosity of Sugar and Oligosaccharide Solutions. J Food Eng 33:221-226. Fichtali, J. 1993. A Rheological Model for Sodium Caseinate. J Food Eng 19:203211. Fito, P.J. Clemente, G., Sanz, F.J. 1983. Rheological Behavior of Tomato Concentrates (Hot Break and Cold Break). J Food Eng 2:51-62. Giesekus, H. 1983. Disperse Systems: Dependence of Rheological Properties on the Type of Flow with Implications for Food Rheology. In: Physical Properties of Foods. R. Jowitt, F. Escher, B. Halstrom, H.F.Th. Meffert, W.E.L. Spiess, G, Vos, eds. London: Applied Science Publ., pp. 205-220. Giner, J., Ibarz, A., Garza, S., Xhian-Quan, S. 1996. Rheology of Clarified Cherry Juice. J Food Eng 30:147-154. Grigelmo-Miguel, N., Ibarz-Ribas, A., Martin-Belloso, O. 1997. Flow Properties of Orange and Peach Dietary Fiber. IFT 97. Orlando, FL, paper No. 35A-2.
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Chapter 4
Guerrero, S.M., Alzamora, S.M. 1997. Effect of pH, Temperature and Glucose
Addition on Flow Behavior of Fruit Purees. I. Banana Puree. J Food Eng 33:239-256. Guerrero, S.M., Alzamora, S.M. 1998. Effect of Temperature and Glucose Addition on Flow Behavior of Fruit Purees: Peach, Papaya and Mango Purees. J Food Eng 37:77-101. Harper, J.C, El-Sahrigi, A.F. 1965. Viscometric Behavior of Tomato Concentrates. J Food Sci 30:470-476. Hayes, G.D. 1987. Food Engineering Data Handbook. London: Longman Scientific & Technical. Hernandez, E., Chen, C.S., Johnson, J., Carter, R.D. 1995. Viscosity Changes in Orange Juices after Ultrafiltration and Evaporation. J Food Eng 25:387-396. Holdsworth, S.D. 1993. Rheological Models Used for the Prediction of the Flow Properties of Food Products: A Literature Review. Trans Inst Chem Engineers 71 (C): 139-189. Ibarz, A., Pagan, J. 1987. Rheology of Raspberry Juices. J Food Eng 6:269-289. Ibarz, A., Gonzalez, C., Esplugas, S. 1994. Rheology of Clarified Fruit Juices. III. Orange Juices. J Food Eng 21:485-494. Ibarz, A., Gonzales, C., Expulgas, S., Vicente, M. 1992a. Rheology of Clarified Juices. I. Peach Juices. J Food Eng 15:49-61. Ibarz, A., Pagan, J., Miguelsanz, R. 1992b. Rheology of Clarified Juices. II. Blackcurrant Juices. J Food Eng 15:63-73. Ibarz, A., Vicente, M., Gracil, J. 1987. Rheological Behavior of Apple Juice and Pear Juice and their Concentrates. J Food Eng 6:257-267. Junzheng, P., Changying, J. 1999. General Rheological Model for Natural Honeys in China. J Food Eng 36:165-168. Khalil, K.E., Ramakrishna, P., Naujundaswany, P., Patwardhan, M.V. 1989. Rheological Behavior of Clarified Banana Juice: Effect of Temperature and Concentration. J Food Eng 10:231-240. Kitano, T., Kataoka, T., Shirota, T. 1981. An Empirical Equation of the Relative Viscosity of Polymer Melts Filled with Various Inorganic Fillers. Rheological Acta 10:207-209. Kokini, J.L. 1987. The Physical Basis of Liquid Food Texture and Texture-Taste Interactions. J Food Eng 6:51-81. Kokini, J.L. 1992. Rheological Properties of Foods. In: Handbook of Food Engineering. D.R. Heldman, D.R. Lund, eds. New York: Marcel Dekker, pp. 1-38. Krokida, M.K., Maroulis, Z.B., Saravacos, G.D. 2000. Rheological Properties of Fluid Fruit and Vegetable Products: Compilation of Literature Data. Int J Food Properties, in press. Lewis, M.J. 1987. Physical Properties of Foods and Food Processing Systems. London: Ellis Horwood, Lopez, A., Ibarz, A., Pagan, J., Vilavella, M. 1989. Rheology of Wine Musts During Fermentation. J Food Eng 10:155-161.
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Manohar, B., Ramakrishna, P., Udayashankar, K. 1991. Some Physical Properties of Mango Pulp Concentrate. J Texture Studies 21:179-190. Maskan, M. 1999. Rheological Behavior of Liquorice (Glycyrriza Glabra) Extract. J Food Eng 39:389-393. Maskan, M., Gogus, F. 2000. Effect of Sugar on the Rheological Properties of Sunflower Oil-Water Emulsions. J Food Eng 43:173-177. McCarthy, K.L, Kanten, R.J., McCarthy, M.J., Steffe, J.F. 1992. Flow Profiles in a Tube Viscometer Using Magnetic Resonance Imaging. J Food Eng 16:109125. Mizrahi, S., Berk, Z. 1972. Flow Behavior of Concentrated Orange Juice: Mathematical Treatment. J Texture Studies 3:69-79. Noureddini, H., Teoh, B.C., Clements, L.O. 1992. Viscosities of Vegetable Oils and Fatty Acids. J Am Oil Chem Soc 69:189-191. Okos, M.R. ed. 1986. Physical and Chemical Properties of Foods. St. Joseph, MI: ASAE. Padmanabhan, M. 1995. Measurement of Extensive Viscosity of Viscoelastic Liquid Foods. J Food Eng 25:311-327. Perry, R.J., Green, J.H., Maloney, J.O. 1984. Perry's Chemical Engineers' Handbook. 6th ed. New York: McGraw-Hill. Pordesimo, L.O., Zurritz, C.A., Sharma, M.G. 1994. Flow Behavior of Coarse Solid-Liquid Mixtures. J Food Eng 21:495-511. Prentice, J.H., Huber, D. 1983. Results of the Collaborative Study on Measuring Rheological Properties of Foodstuffs. In: Physical Properties of Foods. R. Jowitt, F. Escher, B. Halstrom, H.F.Th. Meffert, W.E.L. Spiess, G, Vos, eds. London: Applied Science Publ, pp. 123-183. Rao, M.A. 1987a. Rheology of Liquid Foods-A Review. J Texture Studies 8:135165. Rao, M.A. 1987b. Measurement of Flow Properties of Fluid Foods- A Review. J Texture Studies 8:257-282. Rao, M.A., Steffe, J.F. eds. 1992. Viscoelastic Properties of Food. London: Elsevier. Rao, M.A. 1995. Rheological Properties of Fluid Foods. In: Engineering Properties of Foods. 2nd ed. M.A. Rao, S.S.H. Rizvi, eds. New York: Marcel Dekker, pp. 1-53. Rao, M.A. 1999. Rheology of Fluid and Semisolid Foods. Gaithersburg, MD: Aspen Publ. Saravacos, G.D. 1968. Tube Viscometry of Fruit Purees and Juices. Food Technol 22:1585-1588. Saravacos, G.D. 1970. Effect of Temperature on Viscosity of Fruit Juices and Purees. J Food Sci 35:122-125. Saravacos, G.D. 2000. Transport Properties in Food Engineering. ICEF 8, Puebla, Mexico. Saravacos, G.D., Kostaropoulos, A.E. 1995. Transport Properties in Processing of Fruits and Vegetables. Food Technol 49(9):99-105.
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Saravacos, G.D., Kostaropoulos, A.E. 1996. Engineering Properties in Food Processing Simulation. Computers Chem Eng 20:8461-8466. Saravacos, G.D., Moyer, J.C. 1967. Heating Rates of Fruit Products in an Agitated Kettle. Food Technol 21(3A):54A-58A. Saravacos, G.D., Oda, Y., Moyer, J.C. 1967. Tube Viscometry of Tomato Concentrates. Report, N.Y. State Agr Exp Station, Cornell University, Geneva, N.Y. Sato, Y., Miyawaki, O., Yano, T. 1990. Application of the Hot-Wire Technique for Monitoring Viscosity of a Food in an Unbaffled Agitated Vessel. J Food Eng 11:93-102. Sherman, P. 1984. Comments on COST 90 Project Relating to the Rheological Properties of Liquid Foods. In: Physical Properties of Foods. R. Jowitt, F. Escher, B. Halstrom, H.F.Th. Meffert, W.E.L. Spiess, G, Vos eds. London: Applied Science Publ, pp. 123-183. Steffe, J.F. 1996. Rheological Methods In Food Process Engineering, 2nd ed. East Lansing, MI: Freeman Press. Steffe, J.F., Mohamaed, I.O., Ford, E,W. 1986. In: Physical and Chemical Properties of Foods. M.R. Okos ed. St. Joseph, MI: ASAE, pp. 1-13. Tanglertspaibul, T., Rao, M.A. 1987. Intrinsic Viscosity of Tomato Serum as Affected by Methods of Determination and Methods of Processing Concentrates. J Food Sci 52:1642-1645, 1688. Tomas, R.L., Oliveira, J.C., McCarthy, K.L. 1997. Rheological Modeling of Enzymatic Extrusion of Rice Starch. J Food Eng 32:167-177. Urbicain, M.J., Lozano, J.E. 1997. Thermal and Rheological Properties of Foodstuffs. In: Handbook of Food Engineering Practice. K.J. Valentas, E. Rotstein, R.P. Singh, eds. New York: CRC Press. Vais, A.E., Papazoglu, T.K., Sandeep, K.P., Daubert, C.R. 1999. Rheological Characterization of CMC under Aseptic Processing Conditions. IFT 99, Chicago, IL, paper No. 22C-13. van Wazer, J.R., Lyons, J.W., Kirn, K.Y., Colwell, R.E. 1963. Viscosity and Flow
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Transport of Water in Food Materials
I. INTRODUCTION
The transport of water in food materials is of fundamental importance to several food processing operations, as well as to various physical, chemical and microbiological changes of food products. Water transport within the food material is the main rate-controlling mechanism in drying operations (Keey, 1972), and various techniques have been developed to increase the drying rate, resulting in reduction of the cost of drying, and improvement of food quality. Water transport to and from food products during storage is important in controlling food preservation and food quality. Although various mechanisms have been proposed to explain water transport in food materials, the diffusion model yields satisfactory results for engineering and technological applications. The water chemical potential model appears to be more appropriate for cellular foods (Gekas, 1992), but the required physical and chemical properties are difficult to determine and they may change during measurement. The mechanisms of water transport in solids are reviewed in Section II, with emphasis on the diffusion in polymers, which constitute the structural backbone of food materials, and for which basic literature is available (Vieth, 1991). The methods of measurement and calculation of mass diffusivity are discussed in detail in Section III. Food structure plays a decisive role in water transport processes within the food materials (Saravacos, 1998). Model food materials, based on granular and gelatinized starch, are convenient experimental materials in studying the mechanism of water transport in various food structures (Section IV). Characteristic moisture diffusivities in the main classes of foods are given in tabular form in Section V. A more detailed and unified regression analysis of the literature data on moisture diffusivity in foods is presented in Chapter 6. 105
106
Chapters
II. DIFFUSION OF WATER IN SOLIDS
The diffusion of water in simple gases and liquids can be analyzed and predicted by molecular dynamics and empirical correlations, as discussed in Chapter 2. The transport of water and other small molecules in solids and semisolids is of particular importance to foods and food processing systems. The mechanism of water transport in solid materials is less well understood than in fluid systems, and empirical approaches are often used to estimate the transport properties. The transport of water is of fundamental importance to the drying of solids (Keey, 1972). In the drying process, liquid water is removed first by a hydrodynamic gradient and capillary forces. As drying progresses, water is removed by vapor diffusion, and finally by desorption from the solids. The transport of water in solids is usually assumed to be controlled by molecular diffusion, i.e. the driving force is a concentration gradient (dC/dz) or the equivalent moisture content gradient (dX/dz). For simplified analysis and calculations, one-dimensional diffusion is considered, and the Pick diffusion equation is applied:
dt
dz
8z
(5-1)
The diffusion coefficient D of water in solids is usually defined as the effective moisture diffusivity, which is an overall transport property, incorporating all transport mechanisms. In addition to diffusion, water may be transported by other mechanisms, such as hydrodynamic and capillary flow, depending on the structure of the solid material. The effective diffusivity D of a molecular species A in a porous solid is much lower than the diffusivity of A in a gas medium B, DAB, according to the equation (Geankoplis, 1993): D = (s/T)DAB
(5-2)
where sis the bulk porosity and ris the tortuosity of the solids (T> 1). The tortuosity is a measure of the tortuous (complex) path of the diffusing molecules, T=Lg/L, where Le is the equivalent length of the diffusion path, and L is the straight-line thickness of the sample. Equation (5-2) is applied to catalysts and other solids of fixed structure, and a reliable tortuosity can be determined experimentally. However, determination of tortuosity in food materials is difficult because food structure changes substantially during food processing and storage. For this reason, the effective moisture diffusivity is determined directly by experimental techniques.
Transport of Water in Food Materials
107
Molecular diffusion is the prevalent mass transport mechanism in most food materials. In molecular diffusion, the mean free path of the diffusing molecules is much shorter than the pore or capillary diameter, and most collisions are between the molecules than with the walls. In Knudsen diffusion, the mean free path of the molecules is near the size of the pore or capillary diameter, and the molecules collide more with the walls than with each other (Brodkey and Hershey, 1988). The Knudsen diffusion coefficient DK is given by the equation
DK = 48.5d0(T/MA)"'2
(5-3)
where d0 is the pore (capillary) diameter, T is the temperature, and MA is the molecular weight of the diffusing species. Equation (5-3) indicates that the Knudsen diffusion coefficient DK in gases is a function of the square root of temperature, in contrast to the molecular diffusivity D, which is proportional to the (3/2) power of the temperature equation (228). Other diffusion mechanisms, which are of minor importance to food systems at normal conditions, are: • Surface diffusion (mass transport by surface concentration gradients) • Molecular effusion (passage of molecules through a small aperture in a thin plate into a vacuum) • Thermal diffusion (mass transport due to a temperature gradient) The capillaries in Knudsen diffusion are much longer than in molecular effusion. Knudsen diffusion may be prevalent in capillary systems under vacuum, as in freeze-drying, where the mean free path of the molecules is very long. A. Diffusion of Water in Polymers
An indication of the mechanism of water transport in polymers is the sorption kinetics test on a sample of the material. Gravimetric sorption data (Section III) in a polymeric sample (usually a film) are compared to the generalized sorption equation (Peppas and Brannon-Peppas, 1994):
(M/Me)=kf
(5-4)
where M and Me are the moisture contents after sorption time t and at equilibrium, respectively, k is a constant, and n is the diffusion index. The diffusion index characterizes the type of diffusion in the material:
• • •
n = 0.5 Fickian diffusion 0.5 < n < 1 non-Fickian diffusion n = 1 type II diffusion
108
Chapters
In the case of Fickian diffusion, the constant k of Eq. (5-4) is related to the diffusivity D and the sample thickness L by Eq. (5-5), which is derived from Eq. (5-9): k = 4(D/nl2)
(5-5)
Thus, sorption kinetics data at a constant temperature can be used to determine D. The effective moisture diffusivity D depends strongly on the physical structure of the polymeric material. Thus, values of D of the order of 10'14 are characteristic of the glassy state of foods, while D increases by nearly 1000 times above the glass transition temperature (rubbery state). As water enters the polymeric network, mechanical stresses are developed, resulting in chain rearrangements and significant changes of the polymer structure. These changes are evidenced by characteristic swelling and relaxation phenomena, which affect substantially the transport mechanism and transport properties of water in the polymer. Swelling reduces the density of the polymer material and the glass transition temperature Tg, resulting in increased moisture diffusivity. The relative importance of relaxation to diffusion is expressed by the diffusional Deborah number De: De = A/t = ZD/L2
(5-6)
where /I is the characteristic relaxation time, t = L2/D is the characteristic diffusion time and L is the film (slab) thickness. The following diffusion types can be indicated by the Deborah number: • De » 1, Fickian diffusion: relaxation time is much higher than diffusion time • De « 1, Fickian diffusion: very fast relaxation • De = 1, Case II diffusion : diffusion is controlled by molecular relaxation Equation (5-6) indicates that the Deborah number is inversely proportional to the square of sample thickness, i.e. the type diffusion mechanism depends strongly on the dimensions of the material. This dependence is evidenced in the determination of moisture diffusivity D by various techniques, when different values of D are obtained for the same material. Thus, smaller D values are obtained by the sorption kinetics technique, which uses thin firms, than by the drying rate method, which uses thicker samples. In the drying method, a higher porosity of the thick sample is obtained, increasing further the moisture diffusivity.
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III. DETERMINATION OF MASS DIFFUSIVITY IN SOLIDS
The determination of mass diffusivity in solid and semisolid materials is essential for the quantitative analysis and control of several mass transfer operations and applications, such as drying, adsorption, extraction, membrane separations, ion exchange, and packaging. The methods of measurement and estimation, discussed in this Section are applicable to the diffusion of all small molecules (gases, vapors, and liquids) in solid or semisolid substrates. Of particular interest to food systems is the measurement and estimation of difrusivity of water, or moisture diffusivity, since water, as a liquid or vapor, is involved in most food processing operations and packaging/storage applications. There is no standard method for evaluating the effective diffusivity of water in food materials, due to the complex physical structure, and the changes that occur to the food samples during the measurement procedure. It should be emphasized that most methods estimate the effective diffusivity D of water in the food material for a well-defined set of conditions. As defined earlier in this chapter, D is an overall transport property, incorporating all transport mechanisms for the particular process, i.e. liquid and vapor diffusion, Knudsen diffusion, capillary flow, hydrodynamic flow, etc. It is assumed that the transport of water is described by the diffusion (Pick) equation (5-1), and the driving force is an overall concentration gradient, a simplification for the actual transport mechanisms, but nevertheless convenient for obtaining quantitative data for such complex systems. The use of chemical potential of water as a driving force, (Eq. 5-35), although thermodynamically sound, is difficult to apply in practice, and there is little literature for food materials (Gekas, 1992). Table 1 lists the most important methods of measurement and estimation of the effective moisture diffusivity, which are subsequently discussed and compared in connection with their applicability to food materials (Zogzas et al., 1994b; Zogzas and Maroulis, 1996; Saravacos, 1995). As a general rule, the method of measurement should be related to the actual intended application, for example the drying method is recommended for applications related to drying, sorption kinetics is applicable to moisture adsorption in food storage, and moisture distribution is related to mass transfer between contacted materials. Some elegant techniques of determination of moisture diffusivity, such as the nuclear magnetic imaging (NMR) and the pulsed field gradient methods, have been proposed in the literature (McCarthy et al., 1991, 1994), but have found little practical application to food materials. Mass diffusivity in fluid food systems (Cussler, 1997) is discussed in Chapter 7.
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Chapters
Table 5.1 Methods of Determination of Mass Diffusivity
1. Sorption Kinetics a. Gravimetric Method
b. Chromatographic Method 2. Permeability a. Time-Lag Method b. Unsteady-State Method 3. Distribution of Diffusant 4. Drying Kinetics a. Constant Diffusivity b. Variable Diffusivity 1. Simplified Methods 2. Simulation Method 3. Numerical Methods 4. Regular Regime Method
A. Sorption Kinetics
The sorption kinetics and permeability methods were developed and are applied extensively in polymer science for the determination of mass diffusivity of gases and vapors in solid materials. Samples in the form of thin slabs (films) are normally used for transient adsorption or desorption at constant gas or vapor pressure and temperature.
1. Gravimetric Method The principle of a gravimetric sorption apparatus is shown in Figure 5.1. The apparatus consists of a constant temperature diffusion chamber, which contains the sample, suspended from a quartz spring or (Cahn) electrobalance. The chamber is first evacuated to a very low pressure to remove all solute from the sample, and then the diffusant is introduced at a fixed pressure. The sorption/diffusion process is followed by recording the sorbed mass (or moisture content) versus time.
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111
B
V
Figure 5.1 Diagram of a gravimetric diffusion apparatus. B, balance; P, pressure; RH, constant relative humidity; T, temperature; V, vacuum.
Moisture sorption kinetics can also be measured by enclosing the sample in a constant humidity chamber at atmospheric pressure, and removing it quickly for weighing. In adsorption or desorption of water vapor, the diffusion chamber is maintained at a fixed constant pressure, using either pure water at a specific temperature, or constant relative humidity solutions, i.e. saturated salt solutions or sulphuric acid of fixed density (Spiess and Wolf, 1983). The sorption process is described by the unsteady state diffusion (Pick) equation, which for one-dimensional diffusion reduces to Eq. (5-1):
dt ~ dz
Bz
(5-7)
The initial and boundary conditions require that the initial concentration of the sample is constant, the surfaces of the sample are kept constant, and the amount of diffusant (solute) is a negligible fraction of the whole. Under these conditions, which can be achieved readily in a well-designed sorption experiment, the diffusion equation for a plane film, assuming a constant diffusivity D yields the solution (Vieth, 1991)
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Chapters
i f
Me
~
2 Z_i /^
. 1\2
~"~i"|
r2
I
>
/
where A/ and Me are, respectively, the amounts of diffusant sorbed after time t (s) and infinity (equilibrium), D is the diffusivity (m2/s), and L is the thickness of the film (m). In gas sorption measurements, the pressure ratio (P/PJ can be used instead of the mass ratio (M/MJ, assuming that the gas law applies. In determinations of moisture diffusivity, the mass ratio (M/MJ is equal to the ratio of moisture contents, Y = (X-Xe)/(X0-X0. For small diffusivities and short times, Eq. (5-8) may be approximated by the simplified equation (Crank, 1975; Vieth, 1991)
M
xL'
(5-9)
Thus, a plot of (M/Mg) versus ft1'2) for the first period of sorption yields a nearly straight line with a slope Qf4(D/7zL2)''2, from which the diffusivity D can be estimated (Vieth, 1991). Alternatively, the sorption data can be plotted as (MM^ versus ft/L2)1'2, and the diffusivity D be determined from the simplified equation (Crank, 1975; Saravacos, 1995)
D = 0.049 [ -4 ]
(5-10)
where (t/L2)i,2 is the "half-equilibrium time" (HE), corresponding to (M/Mg) = 0.5, i.e. 50% sorption, andZ is the thickness of the film. Figure 5.2 shows a typical plot of adsorption of water vapor in a solid food sample.
Transport of Water in Food Materials
113
1.0 M/Me
0.5
0.0
HE
2U/2 (Dt/I/)
Figure 5.2 Adsorption of water vapor in a solid food sample. HE, half-equilibrium time.
In food materials, exhibiting hysteresis, different D values of water vapor are obtained in adsorption and desorption measurements, a mean value may be more representative of the diffusion process (Fish, 1958; Saravacos, 1967). Moisture equilibrium values Xe, at a given temperature, for various food materials can be obtained from the isotherms, if available, of the specific material, from published compilations (Iglesias and Chirife, 1982; Wolf et al., 1985) or from empirical equations (Saravacos, 1995). When diffusivity changes with concentration C, which is the usual case with the diffusivity of water in food materials, the D(C) can be estimated by repeated measurements of D over various ranges of concentration (Crank, 1975). If the critical dimension of the sample, e.g. the thickness L, changes during the sorption measurement, a mean value can be used for the estimation of diffusivity (Saravacos, 1967).
2. Chromatographic Methods Gas chromatography methods, widely used in analytical chemistry, have been proposed for the determination of sorption and diffusion properties of solid materials (Vieth, 1991). Of particular interest to foods is inverse gas phase chromatography (IGPC), which can be used in the study of interaction of polymeric materials with probes (Gilbert, 1984). In IGPC, a polymeric material may be the stationary phase in a gas chromatography (GC) column, and a known probe (solute) is introduced as a mobile phase. A carrier gas transports the probe through the column where it interacts with the polymeric material, which may be coated on a GC support, e.g. diatoma-
114
Chapters
ceous earth. The chromatogram recording reveals physicochemical properties of the polymeric material, such as glass transition points, sorption isotherms, and thermodynamic parameters.
The specific retention volume V°, defined as the net retention volume per unit weight of polymeric material at 0°C, is given by the equation V° = (273/T)¥a(\/Ws)
(5-11)
where Vn is the net retention volume, Ws is the weight of the polymeric material, and T is the temperature (K). The partition coefficient Kp, defined as the ratio between the probe concentration in the polymeric material and in the mobile phase, is given by the equation Kf=(7°psT)/273
(5-12)
where ps is the density of the polymeric material. A technique of determining D of solutes in polymers, using IGPC, is described by Pawlisch and Laurence (1987). Frontal Analysis (FA) may also be used in studying the transport mechanisms between the mobile and stationary phases in a chromatographic column (Vieth, 1991). A mixture of carrier gas and probe vapor is forced through the column at constant flow rate, and the amount adsorbed is obtained from the resulting breakthrough curve. Solute transfer between mobile and stationary phases can be estimated by ideal chromatographic models, particularly the non-equilibrium linear (LNE) model.
B. Permeability Methods The permeability methods are convenient for estimation of mass diffusivity and are applied extensively to samples of polymer materials, which can be prepared in the form of thin films of homogeneous microstructure. They are difficult to apply to solid food materials, which are usually heterogeneous with holes and cracks. Permeability can be used to protective food coatings, which behave like_ polymer films during the measurement (Krochta et al., 1994). The permeability P (kg / m s Pa) of a diffusant in a film is given by the equation: P =——-—— (AP/Az)
(5-13)
Transport of Water in Food Materials
115
where J is the mass flux (kg/m2s), AP is the pressure drop (Pa), and Az is the thickness (m). _ The mass diffusivity D can be estimated from the permeability P, using the following equation: ~
(5-14)
where S is the solubility (kg/m3 Pa) of the diffusant (solute) in the substrate. The solubility S in a gas (vapor)/solid or gas (vapor)/liquid system is defined by the equation: S = CIP
(5-15)
where C is the concentration of the solute in the liquid or solid substrate and P is the partial pressure of the solute in the gas (vapor) phase. Equation (5-15) is a form of the Henry's law, the solubility S being the inverse of the Henry's constant (S = 1/H). The constant S is also related to the dimensionless partition coefficient, defined as the ratio of concentrations of the solute in the two contacting phases. The solubility S of water in food materials can be estimated from the slope (dX/da) of the moisture sorption isotherm at a given temperature, either analyti-
cally (empirical sorption equation), or graphically. The moisture content X (kg moisture / kg dry matter) at a given water activity should be converted to moisture concentration C (kg/m3), using the density of the dry matter, approximately ps 1500 kg/m3. The water activity a should be converted to partial pressure of water P (Pa), i.e. P = aPm where P0 (Pa) is the vapor pressure of water at the given temperature. Thus, the slope of the isotherm is converted to (dC/dP), which has the units of the solubility S (kg/m3Pa).
/. Time-Lag Method The diffusivity of a gas or vapor D and its solubility S in a polymeric film can be estimated simultaneously from permeability measurements, using the timelag method (Crank and Park, 1968; Crank, 1975; Vieth, 1991). The unsteady and steady-state permeation of a solute into a polymeric film is usually measured in a time-lag diffusion cell, shown diagrammatically in Figure 5.3:
Chapter 5
116
I ~m www
AP
T Figure 5.3 Principle of time-lag permeability/diffusion measurement. F, film; G, gas flow; S, perforated support; T, constant temperature; AP, pressure drop.
An elaborate experimental set-up is required, in which the polymeric film is first evacuated to a high vacuum, and a gas stream is passed through, measuring the accumulation of the solute in the film as function of time. After a transient period, a steady-state permeation rate is established, which yields the permeability of the film P , according to Eq. (5-13). The accumulated amount of the solute in the film Q is plotted as a function of time t, from which the diffusivity can be estimated. Under the specified boundary and initial conditions (initially gas-free film, equilibrium at the inlet gas/polymer interface, and zero concentration of gas at the outflow face), and assuming a constant diffusivity D, the Pick equation yields the following solution:
6D
-exp -
(5-16)
where Q is the accumulated amount of penetrant (solute), passing through the film after time t, Cj is the solute concentration in the gas upstream (high pressure), and L is the film thickness. After a short transient period, a steady state is established, and Eq. (5-16) is reduced to:
6D
(5-17)
Transport of Water in Food Materials
117
Figure 5.4 Estimation of the time lag (TL) from a plot of accumulated diffusant Q versus time t.
A plot of (Q, t), after a short initial period, yields a straight line with an intercept (TL, time lag) on the t axis, given by Eq. (5-18), from which the diffusivity D can be estimated. TL--1L
6D
(5-18)
The time required to reach full steady-state diffusion has been found empirically to be equal to 3(TL). Figure 5.4 shows a typical permeability plot (Q, t) for estimation of the time lag.
2. Unsteady-State Method The solubility 5 and diffusivity D of gases and vapors in polymers can be determined also by unsteady-state measurements, using specialized sorption apparatus (Crank, 1975; Vieth, 1991). The polymeric material, usually a thin film of known thickness, is placed into a closed sorption chamber, which is pressurized quickly to a known initial pressure. The pressure in the chamber drops gradually, as the gas sorbs and diffuses into the polymer film until equilibrium is established. The solubility of the gas S in the polymer is estimated from a material balance in the closed system, resulting in the equation:
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Chapters
Fv 273(Pi-Pl S = ——— ———-
r, T( P.
(5-19)
where Vv is the void volume, Vp is the volume of the polymer, T is the temperature (K), and Pk Pe are the initial and equilibrium gas pressures, respectively. A similar expression for the solubility S can be obtained with desorption measurements. The polymer sample is first completely degassed, a fixed gas pressure Pt is applied, the chamber is evacuated by rapid pump down to a lower pressure, and the chamber is left to equilibrate to a final pressure Pe. The desorption of a gas from a polymer film by pump down of a closed chamber is described by the Pick law. Assuming that the required boundary conditions are met, and that diffusivity D is constant, the solution of the diffusion equation yields the same equations of sorption kinetics (5-9) and (5-10), from which the value of D can be calculated.
C. Distribution of Diffusant This method is based on the unsteady-state diffusion of a component in a semi-infinite solid, the contact surface of which is maintained at constant concentration of the diffusant. The concentration of the component in the solid is measured at various distances from the surface as a function of time, obtaining the concentration-distance curve, from which the diffusivity can be extracted (Crank, 1975; Zogzas et al., 1994b; Saravacos, 1995; Kostaropoulos et al, 1994). A common experimental procedure is to use two long cylindrical samples of different uniform concentration, contacted in series, and let diffusion take place along the axis, under the influence of a concentration gradient. The cylindrical sample is maintained at a constant temperature, and after a certain time is removed from the cylinder and sliced into small sections, which are analyzed quickly for component concentration. In measurements of moisture diffusivity, the sliced sections of the sample are analyzed quickly for moisture gravimetrically. The samples may be contained in plastic cylinders (e.g. 13 mm diameter and 100 mm long), which are sliced together with the sample during the analysis (Karathanos et al., 1991). The distance-concentration curve at a specified time is constructed by plotting component concentration versus distance (Figure 5.5).
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119
C,
Figure 5.5 Moisture concentration (C) - distance (z) curves in two contacted cylindrical samples.
The Pick diffusion equation for a semi-infinite solid yields the following solution, assuming constant diffusivity (Crank, 1975): -C C
r = erfc
c-c.
(5-20)
where C0 is the initial concentration of the diffusant in the sample, C is the concentration after time t, Ce is the equilibrium concentration, and z is the distance of penetration. The error functions erf and erfc (= 1-erf) are given in the literature. The equilibrium concentration at the given temperature is taken from the isotherm of the product or from empirical equations. For variable diffusivity, the difrusivity D(Cj) at a specified concentration C/ can be evaluated by the equation: 1
(5-21)
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Chapters
where z is the distance from the interface of the two cylinders, and t is the time at which the concentration profile is determined. The integral and the gradient of z at the specified concentration Cj can be determined by numerical or graphical evaluation of the concentration-distance curve. Thus, the diffusivity may be evaluated as a function of concentration by repeated use of Eq. (5-21). Both Eqs. (5-20) and (5-21) yield mean diffusivity values for both samples of the material. An alternative simulation method can be used to determine two separate diffusivities D for the two samples (Karathanos et al., 1991). Equation (5-20) is applied to both cylinders, and the calculated D values are optimized by minimizing the sum of squares of the differences of predicted and experimental concentrations. The diffusant concentration method is convenient for estimating the diffusivity of small molecules in solids, when diffusion is relatively slow, and thus concentration profiles can be determined within reasonable time intervals. It has the advantage of minimum disruption of the material structure during measurement, and it has been applied to estimate the diffusivities of various components in food materials (see Chapter 7). Application of this method to measurements of moisture diffusivity in food systems presents experimental difficulties, mainly due to changes of moisture content during the analytical procedure (slicing, etc). The method can be used at higher temperatures and pressures, where the usual methods are difficult to apply (Karathanos etal., 1991).
D. Drying Methods
The drying methods are used widely for the determination of moisture (water) diffusivity of food materials, since drying and rehydration are common food processes, and water transport properties are essential in modeling, calculations and control of these operations. Most of the literature data were obtained from drying experiments, and the measurement procedure is relatively simple. Historically, drying was the first method used for the determination of moisture diffusivity in solid materials (Sherwood, 1931). Drying of solids is usually divided into two stages, i.e. the constant rate and the falling rate periods (Perry and Green, 1997). In the constant rate period, water evaporates freely from the surface of the solid, and the drying rate is controlled by the external conditions, i.e. air velocity, temperature, and humidity. In the falling rate period, the main resistance to mass transfer is within the solid material, and the transport of water from the interior to the surface of evaporation is controlled by diffusion and other mechanisms, as discussed earlier in this chapter. Most food materials have short constant rate periods and they dry entirely in the falling rate period.
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121
AIR
11
w
I
1
Figure 5.6 Diagram of an experimental apparatus for determining the air-drying rate. S, sample; W, balance; u, air velocity; T, temperature; RH, relative humidity. Determination of moisture diffusivity during the falling rate period is based on the application of the diffusion equation to a suitable sample (slab, cylinder or sphere) of known basic dimension, assuming that the required initial and boundary conditions are applicable. The sample is dried at controlled temperature, air velocity and relative humidity, and the drying curve is constructed by plotting the weight (moisture content) versus time. Figure 5.6 shows the principal parts of a typical airdrying apparatus for the determination of the drying rate of a solid or semisolid sample. In simple experiments, weighing of the samples may be done quickly outside the dryer. Elaborate measuring apparatus is available, using electronic sensors and a PC to record temperature, humidity, and sample weight during the drying experiment (Marinos-Kouris and Maroulis, 1995). High air velocities (> Im/s) are used in the drying chamber in order to minimize the external resistance to mass transfer from the sample surface to the air stream. For vacuum-drying measurements, the air stream is replaced by a vacuum system with a condenser and a pressure control system. Heat can be supplied by controlled microwave or infrared devices.
/. Constant Diffusivity The diffusion equation (5-1) is solved for the three basic shapes of the solids (slab, infinite cylinder, and sphere), assuming constant diffusivity (D), and appropriate initial and boundary conditions (Sherwood, 1931; Crank, 1975). It is also assumed that the solid is drying entirely in the falling rate period. For a slab or a film of small thickness (compared to the other two dimensions) the diffusion of moisture is described by the equation:
122
Chapters
„ „.
- -
=JLy
2
——;—— p _ v~» • v2 - ~ •
(5.22)
where A' is the mean moisture content after time t, X0 is the initial moisture content, Xe is the equilibrium moisture content, and L is the sample thickness drying from both sides. If the sample is drying from one flat side only, the sample thickness L in the diffusion equation should be substituted by (2L). The moisture content A" is expressed on dry basis, i.e. kg water/kg dry matter. For spherical samples, the diffusion equation yields: (5-23) where r is the sphere's radius. The diffusivity D is estimated from Eqs. (5-22) or Eq. (5-23) by an approximate solution or by a numerical method. The units of D are (m2/s), provided that time is in (s) and sample thickness in (m). Most diffusivities are determined based on a mean value of the thickness L or the radius r of the sample during the drying process. The first reported moisture diffusivities of food materials, obtained from drying data, assumed constant diffusivity (Saravacos and Charm, 1962a). However, it was soon realized that diffusivity was a function of concentration (moisture content), evidently due to the complex structure of the food materials. In some cases, e.g. drying offish muscle, the falling rate period may consist of two distinct parts. The bimodal diffusion was evidenced in a plot of log 7 versus t, which yielded two straight lines with two slopes, Kj > Ki, corresponding to two diffusivities, £>/ > D? (Jason, 1958; Jason and Peters, 1973). The slope K (1/s) of the semi-log plot of the drying curve is actually the drying constant, defined by the equation: X,)
at
(5-24)
If the diffusion equation is applicable, the diffusivity D can be estimated from the drying constant K, for a slab of thickness L using the equation (Moyne et al, 1987; Marinos-Kouris andMaroulis, 1995): (5-25) 7t
The drying constant K refers to specific drying conditions (temperature, air humidity and air velocity) and sample thickness.
Transport of Water in Food Materials
123
2. Variable Diffusivity The moisture diffusivity in drying food materials changes significantly with the moisture content, due to structural changes of the material during drying, and the changes of the drying mechanism and the water-substrate interaction. The changes in difrusivity are evidenced by the non-linearity of the semi-log drying curve (logY versus f). Although mathematical solution of the diffusion equation requires a constant difrusivity, various approximate methods are applied to extract effective moisture diffusivities from experimental drying curves, assuming that water transport by all mechanisms is caused by a concentration gradient.
E. Simplified Methods
Simplified methods are useful for estimating quickly approximate values of variable effective diffusivities D, before a mathematically rigorous method can be applied. They have been used for obtaining D values from irregular drying curves, as in drying of porous solids at low moistures, where moisture is transported by a combination of different mechanisms, and mathematical modeling is difficult. The method of drying constants divides the drying curve (fog7, t) into linear parts and estimates the drying constant (K = dY/dt) at various moisture ratios, from which the effective difrusivity is calculated, using Eq. (5-25) for the corresponding sample thickness. The method of slopes (Perry and Green, 1997; Karathanos et al., 1990; Saravacos, 1995; Uzman and Sahbaz, 2000) is essentially similar to the repeated application of Eq. (5-25). The experimental drying curve (logY, t) is compared to the theoretical diffusion curve (logY, Fo) where Fo = Dt/L2 (Figure 5.7). The slopes of the two curves (dY/dt) ^ and (dY/dFo),h are estimated at the same moisture ratio Y and the effective difrusivity is calculated from the equation
(dY/dt) D = _1———i2L (dY/dFo),,
(5.26)
^
'
Chapter 5
124
logY
th
t o r Fo
Figure 5.7 Comparison of experimental (exp) and theoretical (th) diffusion curves.
F. Simulation Method
The simulation method estimates the effective diffusivity by an optimisation technique, which fits diffusivity values to the experimental drying curve (Karathanos et al., 1990). The input to the computer program is the drying time, the sample shape and dimensions, the initial and equilibrium moisture contents, and the initial guess of the mean diffusivities of the nodes used (e.g. 10). The simulation technique gave similar diffusivity values with the method of slopes, using drying data for granular and gelatinized starch samples.
G. Numerical Methods
Numerical methods assume that the effective diffusivity D is a known function of the moisture content X, and the diffusion equation (5-1) is fitted to the experimental drying data by regression analysis. In a more general approach, a mathematical model is proposed that considers both heat and mass transfer, and the diffusivity is a function of moisture content and temperature (Kiranoudis et al., 1992, 1995). An empirical model for moisture diffusivity is the following exponential expression (Kiranoudis et al., 1994; Marinos-Kouris and Maroulis, 1995):
D = Do exp
X
(5-27)
Transport of Water in Food Materials
125
where A' is the moisture content, Tis the temperature, and X0, T0 are adjustable constants. A large number of drying data is obtained from drying experiments at fixed drying conditions, and fitted simultaneously to the diffusion equation by a nonlinear regression technique. Two iterative methods of calculation can be used, i.e. the finite differences of Crank-Nicolson (Crank, 1975) and the control volume (Patankar, 1980). A numerical method, based on the exponential function of moisture diffusivity, (Eq. 5-27), was applied to the air-drying of potato and carrot. Comparison with the simplified method of drying constants gave acceptable agreement in the low moisture range, where a complex transport mechanism has been evidenced (Kiranoudisetal., 1994). Specialized mathematical models for the effective moisture diffusivity D may describe irregular changes during the drying process. For example, the bell-shaped curve of (D, X) curve, observed in drying porous starch materials at low moistures (X<1) can be represented by the gamma function (Karathanos et al, 1990): 1A-
(5-28)
where /I, j3 are constants, X is the moisture, Xz = X-XRH=O, and F(/3) is the gamma function.
H. Regular Regime Method The regular regime method, developed by Schoeber (1976), is based on the application of mass transfer principles to the experimental drying curve. The method involves a number of calculations, as outlined briefly below (Schoeber and Thijessen, 1977; Gekas, 1992). The drying process involves three periods: the constant rate, the penetration, and the regular regime. During the regular regime (last) period of drying, the moisture profile is moving toward the center of the product, depending on the diffusivity of moisture, but not on the initial moisture content of the product. From the drying curve (X, t) the mass flux per unit external surface of the sample is calculated (dm/dt, kg/m2s), and then the characteristic flux parameter F = (dm /dt) ps (L~/2), kg2/m4s, where ps is the dry solids concentration (kg/m3) and L is the sample thickness (m). The moisture profile in the sample is described by the relationship FY= constant, where Y is the moisture ratio, Y = (X0-X)/(X0-X^>. A curve (F, X) is constructed, which constitutes the regular regime curve for the given material.
126
Chapters
The gradient d(lnF)/d(lnX) is calculated as a function of moisture content X by numerical differentiation, and correlated to the Sherwood number (Sh = kcL/D, where kc is the mass transfer coefficient, m/s). A plot of (2F/Sh) versus A'yields the reduced diffusivity Dr by numerical differentiation. Finally, the mass diffusivity D is calculated from the reduced diffusivity, using the equation, D = Dr/ps2. The regular regime method has found limited applications to food materials (Singh et al, 1984; Tong and Lund, 1990; Sano and Yamamoto, 1990; Inazu and Iwasaki, 2000). The calculated moisture diffusivities are, in general, very close to the diffusivities obtained by other methods for the same food material.
I. Shrinkage Effect
Most food materials undergo significant shrinkage during the drying process, which is reflected in the calculated values of moisture diffusivity, since D is normally proportional to the square of the sample thickness. Shrinkage models and experimental data are presented in Chapter 3 (Zogzas et al., 1994a; Krokida and Maroulis, 1997). In general, shrinkage is a linear function of moisture content, and it should be determined for each material under the appropriate drying conditions. Moisture diffusivity values D, calculated for a mean sample thickness, can be converted to values based on the dry solids of the material Ds, using the following equation (Fish, 1958; Crank, 1975):
Ds = [(p/ps)(l+X)]2/3D
(5-29)
where p and ps are the densities of the sample at moisture content X and dryness, respectively. The volumetric shrinkage is assumed to be isotropic. Gekas and Lamberg (1991) modified the Crank equation (5-29), assuming that shrinkage is not isotropic, and the moisture diffusivity Df follows a fractal relationship:
Dr[(p/Ps)]2/dD where d is the fractal exponent, e.g. for drying of blanched potato d = 1.42.
(5-30)
Transport of Water in Food Materials
127
IV. MOISTURE DIFFUSIVITY IN MODEL FOOD MATERIALS
Model food systems are useful in understanding and predicting the behavior of actual food materials in various heat and mass transport processes. Moisture transport (diffusivity) has been found to vary widely in food materials, due mainly to different physical structure. Starch materials have been used in several simulations of food products, since starch is a basic structural component of various foods of plant origin, e.g. cereal and potato products. Other food biopolymers can also be used in food simulations, like pectin, cellulose and various gums. Starch (usually corn or potato) has the advantage of forming rigid gels when heated, resembling solid food products. The use of food biopolymers in measurements of moisture diffusivity can utilize the experience and advances of polymer science in the area of physical and physicochemical properties.
A. Effect of Measurement Method
The method of measurement may have a profound effect on the value of moisture diffusivity of solid and semisolid foods, due to the changes in the physical structure of the sample during the measurement procedure. In liquid foods, as in pure liquids and gases, the experimental or computed diffusivities are more likely to be constant for a given set of conditions (see Chapter 2). Drying causes significant physical changes in the food sample during measurement (formation of cracks, pores, puffing), strongly affecting the estimated mass transport property (moisture diffusivity). Isothermal adsorption or desorption of water is a milder transport process, with no major structural changes. As a result, the moisture diffusivity D calculated from sorption data is usually lower than the D value obtained from drying data. The method of concentration distribution does not change the physical structure of the material during measurement, and the D values obtained are similar to those of sorption. Although, theoretically, diffusivity should be independent of sample shape and thickness, different results are obtained in practice when changing sample dimensions, especially in drying measurements. Thus, thicker samples yield high D values, evidently due to the formation of larger cracks, pores and channels during drying. Figure 5.8 shows moisture diffusivities D in starch gels at 25°C, obtained by sorption kinetics. Thin films (1 mm) of potato starch gels were used, and the reported diffusivities were the mean values of adsorption and desorption measurements (Fish, 1958). The D values increased exponentially from 1 x 10"13 to 0.5 x 10"10 m2/s when the moisture content^was increased from 0 to 0.2.
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The increase of moisture diffusivity can be explained by the plasticization of the polymer (starch), which facilitates the transport of water through the macromolecular network. The thin samples of starch gel behaved like homogeneous polymer materials, and the small moisture diffusivities are of the same order of magnitude with the D values of small molecules through polymeric materials (Vieth, 1991).
Temperature has a positive effect on moisture diffusivity, following the Arrhenius equation, in a similar way with the activated diffusion in pure liquids, which has been explained by the theory of rate processes (Eq. 2-37). The energy of activation for diffusion ED increases substantially as the moisture content is reduced, e.g. ED = 20 to 41 kJ/mol, at moisture contents X= 0.2 to 0.01, respectively. Higher moisture diffusivities are obtained in starch gels, using drying rate measurements (Saravacos and Raouzeos, 1983). The D values shown in Figure 5.9 were obtained by drying slabs of starch gels 6 mm thick in an airstream at 2 m/s. A maximum of D = 5xlO"10m2/s was observed at nearly X= 2. Incorporation of glucose (50% dry basis) in the starch gel reduced significantly the moisture diffusivity. The higher D values obtained from drying experiments, compared to the sorption data, are the result of significant changes in the structure of the gel during the drying process. Drying experiments require samples much thicker than the thin films of sorption measurements. Drying increases the porosity and creates cracks and channels in the sample, through which water can be transported at a faster rate as a vapor than in the isothermal sorption process (Chapter 3). The presence of small, water-soluble molecules in the gel, like glucose, reduces moisture diffusivity, by decreasing the porosity of the sample during drying. A comparison of moisture diffusivities D obtained from drying and sorption experiments on the same starch material is shown in Figure 5.10. Granular corn starch (high-amylose, HYLON) was used to prepare spherical samples 1 cm in diameter for the drying measurements at 60°C and air velocity 2 m/s. The same starch material, in the form of a slab (film) 1 mm thick was used in the sorption measurements (Leslie et al., 1991; Chung, 1991). In both cases, the diffusivity-moisture content (D versus X) curve goes through a maximum, which is lower moisture content for the sorption data. Higher D values (nearly 3 times) were obtained by the drying method, evidently due to the structural changes in the samples during air-drying.
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Transport of Water in Food Materials
0.001 0.1
0.2
0.3
X (kg/kg dm)
Figure 5.8 Moisture diffusivity in potato starch gels. Sorption kinetics, 25°C. (Data from Fish, 1958.)
0
1
2
3
4
X (kg/kg dm)
Figure 5.9 Effective moisture diffusivity in drying slabs of starch gels at 40°C. S, starch; SG, starch glucose. (Data from Saravacos and Raouzeos, 1983.)
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130
0.4
0.6
0.8
X (kg/kg dm)
Figure 5.10 Comparison of moisture diffusivities in granular starch (AMIOCA), obtained from drying (DR) and adsorption (ADS) at 60°C. (Data from Leslie et al., 1991.)
B. Effect of Gelatinization and Extrusion
Gelatinization causes significant physicochemical and structural changes in starch materials, which considerably affects the heat and mass transport properties (Chapter 3). The moisture diffiisivity is, in general, reduced by gelatinization, due to the disruption of the granular structure and the formation of a homogeneous gel. The effect of gelatinization on diffusivity depends primarily on the chemical and physical composition of the starch material. Gelatinization of high-amylose (linear macromolecules) starch (HYLON) causes a relatively small reduction of moisture diffusiviry, without changing the characteristic shape of the (D, X) curve of porous materials, as shown in Figure 5.11. The gelatinized spherical sample developed a high porosity with several cracks, which resulted in relatively high moisture diffusivity. A completely different effect of gelatinization on moisture diffusivity was observed in highamylopectin (branched macromolecules) starch (AMIOCA), as shown in Figure 5.12(Saravacosetal., 1989).
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The moisture diffusivity D in the gelatinized starch was reduced sharply, especially at low moisture contents. The values of the gelatinized starch increased as
the moisture content X was increased, resembling the (D, X) curve obtained by sorption kinetics of starch gels. The moisture diffusivity curves of Figure 5.12 can be explained by the changes of bulk porosity s as a function of moisture content. In the gelatinized high-amylopectin gel, the porosity increased only slightly during the drying process, contrary to the sharp increase in the granular (non-gelatinized) sample (Figure 3.7). Extrusion cooking of starch material at high temperatures and relatively low moisture contents, produces highly porous products with high moisture diffusivity. Figure 5.13 shows typical moisture diffusivities in high-amylopectin extruded starch, obtained from drying measurements of extruded cylindrical samples.
0.2
0.4
0.6
X (kg/kg dm)
Figure 5.11 Effective moisture diffusivity in drying spherical samples of granular (GR) and gelatinized (GEL) starch (HYLON) at 60°C. (Data from Saravacos et al., 1989.)
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0.2
0.4
0.6
X (kg/kg dm)
Figure 5.12 Effective moisture diffusivity in drying spherical samples of granular (GR) and gelatinized (GEL) starch (AMIOCA) at 60°C. (Data from Saravacos et al., 1989.) 50
40
EX
N 30 o
I 20
a
10 GEL
0.2
0.4
0.6
X (kg/kg dm)
Figure 5.13 Effective moisture diffusivity in drying of extruded (EX) and gelatinized (GEL) starch (AMIOCA) at 60°C. (Data from Marousis et al.,1991.)
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C. Effect of Sugars Incorporation of small water-soluble molecules into the starch materials reduces, in general, the moisture diffusivity. The effect of sugars is more pronounced in granular (non-gelatinized) than in gelatinized starch materials. Figure 5.14 shows typical effects of sugars on he moisture diffusivity D in high-amylose
(HYLON) starch, obtained from drying experiments using spherical samples 1 cm in diameter at 60°C. The D values decreased sharply when water-soluble dextrin was incorporated in the samples. Smaller effects were observed with glucose and sucrose (Marousis et al., 1989). The molecular weight of the water-soluble carbohydrate seems to be proportional to the reduction of moisture diffusivity. It is presumed that the water-soluble molecules precipitate in the starch matrix during drying, reducing significantly the porosity of the drying sample. This explanation is supported by the observed low porosities of osmotically (using sugar) dehydrated fruits, compared to the untreated materials (Chapter 3). The effect of sugars on the moisture diffusivity in gelatinized starches is lower than in granular materials (Figure 5.15). Low molecular weight sugars, like glucose, appear to have a stronger effect because they are more mobile in the gel structure, precipitate in pores created during drying, and reduce moisture diffusivity (see also Figure 5.9). On the other hand, high-molecular weight carbohydrates, like dextrin, are relatively immobile, and they behave like starch, creating high porosity during drying, and increasing moisture diffusivity. Dissolved salts, such as sodium chloride, reduce the moisture diffusivity strongly in hydrated granular starches, but slightly in the gelatinized materials (Uzman and Sahbaz, 2000). Sodium chloride, which is mobile in granular starch, may concentrate and precipitate near the surface of the sample, causing a "casehardening" effect, and reducing substantially the effective moisture diffusivity. It should be noted that the diffusivity of sodium chloride in water is about 12xlO"10 m2/s (Table 2.4).
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134
0.2
0.4
0.6
X (kg/kg dm)
Figure 5.14 Effect of sugars on moisture diffusivity in drying granular starch (HY-
LON) at 60°C. S, starch; SG, starch/glucose; SD, starch/dextrin. (Data from Marousisetal., 1989.)
0.4
0.6
0.8
X (kg/kg dm)
Figure 5.15 Effect of sugar on moisture diffusivity in drying gelatinized starch (HYLON) at 60°C. S, starch; SG, starch/glucose; SD starch/dextrose. (Data from Marousis et al, 1989.)
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D. Effect of Proteins and Lipids Incorporation of proteins in starch materials may substantially reduce the moisture diffusivity D due to physicochemical interactions. Figure 5.16 shows the reduction of D in granular amylopectin (branched macromolecule) by the addition of 25% (by dry weight) gluten (a wheat protein). The D values were estimated from drying measurements on spherical samples at 60°C. A similar effect was observed with the addition of 25% lysozyme (Marousis, 1989). Addition of proteins to amylose (linear macromolecule) reduced the D values to a lesser degree than amylopectin, presumably due to the weaker protein/starch interaction, compared to the stronger interaction of proteins with the branched macromolecules of amylopectin. Proteins reduce moisture diffusivity in gelatinized starches in a similar manner with the granular materials, suggesting a molecular type of interaction. By contrast, addition of sugars reduces the D values of granular starches only, due to significant reduction of porosity (a physical process), but there is little effect in the gelatinized materials. Incorporation of lipids in granular starches may reduce the moisture diffusivity D, but there is little effect in the gelatinized materials (Papantonis, 1991). Figure 5.17 shows a significant reduction in D by the addition of 10% vegetable oil in granular amylopectin starch during drying of spherical samples at 60°C. However, there is no significant change in D of gelatinized starches by the addition of lipids, suggesting a rather physical than chemical interaction. Measurements of moisture diffusivity in model food systems containing starch, proteins and lipids, demonstrate the importance of physical and physicochemical interactions on the transport of water in actual food materials. Proteins appear to have a strong physicochemical effect, reducing the mobility of water molecules. Lipids may reduce moisture diffusivity by physical obstruction of water transport, e.g. by forming hydrophobic films in the starch network.
Chapter 5
136
0.2
0.4
0.6
X (kg/kg dm)
Figure 5.16 Effect of gluten (SP) on the moisture diffusivity (D) of granular amylopectin starch (S). Drying of spherical samples at 60°C. (Data from Marousis, 1989.)
0.2
0,4
0.6
0.8
X (kg/kg dm)
Figure 5.17 Effect of vegetable oil (SO) on the moisture diffusivity (D) of granular amylopectin starch (S). Drying of spherical samples at 60°C. (Data from Papantonis, 1991.)
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Transport of Water in Food Materials
E. Effect of Inert Particles
Inert particles, i.e. solid particles not interacting with water or biopolymers, may support the physical and mechanical structure of starch materials during the drying process. Shrinkage of the samples may be prevented, and the moisture diffusivity may be increased during drying (Leslie et al, 1991). Figure 5.18 shows that the moisture diffusivity of hydrated granular starch (HYLON) increases significantly, when silica particles (25% by weight, dry basis) are incorporated in the spherical samples. A similar effect was observed when carbon black was incorporated in the starch samples. The bulk porosity s of the dried starches, containing inert particles, increased significantly, e.g. for hydrated HYLON starch e increased from 0.45 to 0.52 (silica) and 0.55 (carbon black), and for AMIOCA starch the increase was from 0.45 to 0.50 (silica) and 0.57 (carbon black).
0.4
0.6
0.8
X (kg/kg dm)
Figure 5.18 Effect of inert silica particles (SI) on moisture diffusivity of granular starch S (HYLON). Air drying at 60°C. (Data from Leslie et al., 1991.)
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F. Effect of Pressure
Pressure has a significant effect on the moisture diffusivity of porous materials, such as granular starches. Pressure can be applied in the form of mechanical compression or gas (air) pressure in a closed vessel. As shown in Figure 5.19, the effective moisture diffusivity D in granular starch at moisture content X = 0.5 decreased from about 10xlO~ 10 to 3x10"'° m2/s, when the mechanical pressure was increased from 1 to 40 bar (Marousis et al., 1990). The reduction of D is related directly to the reduction of porosity by the applied mechanical pressure. The effect of pressure on the gelatinized starch materials was relatively smaller, corresponding to smaller changes of porosity. Mechanical pressure reduces the porosity of granular starches, especially at high moisture contents, when the starch granules can be deformed more easily than the dry particles. Figure 5.20 shows that the porosity of granular starch at moisture content X = 0.5 is reduced from about 0.50 to less than 0.10 when the mechanical pressure is increased from 1 to 40 bar. Air pressure applied in a closed vessel to granular starch (HYLON) reduced the moisture diffusivity D in a similar manner with mechanical pressure. Thus, D decreased from about 10xlO"'°to 2xlO"10 m2/s, when the air pressure was increased from 1 to 40 bar (Figure 5.21). The values of D were determined by the moisture distribution method (Karathanos et al., 1991). The effect of gas (air) pressure on diffusivity in porous materials is related to the inverse pressure P - diffusivity D relationship in gas systems at constant temperature, according to the simplified equation, PD = constant
(5-31)
Higher moisture diffusivities D are expected in porous materials at pressure below atmospheric (vacuum), an important characteristic of vacuum and freeze-drying. Thus, the moisture diffusivity in freeze-dried starch gels increased from about O.lxlO' 10 m2/s to 10x10"'° m2/s, when the pressure was reduced from 1 bar to below 1 mbar (Saravacos and Stinchfield, 1965). The diffusion data were obtained by the sorption kinetics method.
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Figure 5.19 Effect of mechanical pressure on the moisture diffusivity of granular starch (HYLON) at moisture X = 0.5. Drying at 60°C. (Data from Marousis et al., 1990.) 0.6
0.4
0.2
0
10
20
30
40
50
P (bar)
Figure 5.20 Effect of mechanical pressure on the porosity e of granular starch (HYLON) at moistureX= 0.5. (Data from Marousis et al., 1990.)
Chapter 5
140
P (bar)
Figure 5.21 Effect of air pressure on the moisture diffusivity of granular starch (AMIOCA) at moisture X= 0.4. Moisture distribution method at 60°C. (Data from Karathanos et al, 1991.)
G. Effect of Porosity
The bulk porosity s of solid materials, estimated from measurements of bulk and solids density (Figure 3.5), is the major parameter affecting mass diffusivity. Temperature and moisture content are also important, but their effect depends strongly on the structure of the material. Regression analysis of several experimental data on granular and gelatinized starch materials has yielded the following equation (Marousis et al., 1991): (5-32)
Figure 5.22 shows that the effective moisture diffusivity increases sharply above e = 0.40. The pore size distribution of dried starch materials is discussed in Chapter 3. The high porosity, developed in the drying of granular starch, is visualized by the formation of flow channels, through which water (liquid and vapor) is
Transport of Water in Food Materials
141
transported to the drying surface. Radial channels are formed in drying spherical samples. Irregular cracks are formed in drying gelatinized starch (Figure 3.8).
The pore shape has a significant effect on the heat and mass transport properties of solids. Freeze-drying experiments have shown that moisture diffusivity is higher in samples with long than small pores (Figure 5.23; Saravacos, 1965). The CMC gel slab, which dried faster, had a fibrous structure with long pores oriented along the diffusion path, while the starch gel sample had small spherical pores, distributed evenly (Figure 3.9).
H. Effect of Temperature
The effect of temperature on moisture diffusivity D depends strongly on the physical structure of the solid material. In porous materials, where vapor diffusion may be controlling, D is proportional to the (3/2) power of temperature, according to the fundamental transport equation (2-28). In nonporous gels, where liquid diffusion may predominate, D may be expressed by the Arrhenius equation (2-37):
D = Aexp(-ED/RT)
(5-33)
The energy of activation for diffusion ED, estimated from diffusivity data at various temperatures, is a good indication of the type of prevailing diffusion mechanism in the material. In general, low ED values indicate a vapor diffusion, while high values suggest liquid (activated) diffusion (Table 5.2). Higher ED values are expected at low moisture contents, due to the stronger water-substrate interaction. Table 5.2 Typical Energies of Activation for Diffusion of Water in Starch Materials__________________________________________ Material____________________Activation energy, kJ / mol____ Granular starch 17.0 Granular starch/sugar 33.5 Gelatinized starch 43.4 Gelatinized starch/sugar 51.4 Granular starch/sodium chloride 61.0
Chapter 5
142
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 5.22 Effect of porosity on moisture diffusivity of starch materials. (Data from Marousis et al, 1991.)
0.01
0.001
Figure 5.23 Freeze drying rates of model food gels. S, potato starch; CMC, carboxy methyl cellulose. Y = (X-Xe)/(Xo-Xe). (Data from Saravacos, 1965.)
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143
I. Drying Mechanisms
Figure 5.24 shows a typical moisture diffusivity-moisture content (D, X) curve for a porous food material, e.g. granular, extruded, or freeze-dried food product. At high moistures (A), e.g. X > 0.5, liquid diffusion and capillary flow may predominate (relatively low D and low porosity). At intermediate moistures (B), water may be transported by capillary flow and vapor diffusion, increasing sharply the D value. In this region, pores, cracks, and channels facilitate the transport of water. At very low moistures (C), strongly bound water is desorbed from the biopolymer solids, reducing sharply the moisture diffusivity. The shape of the (D, X) curve of Figure 5.24 is characteristic of mass transport properties of capillary and porous solids at low concentrations (Toei, 1983). A similar curve is obtained for the change of thermal diffusivity of porous foods in the low moisture region (Figure 8.7), suggesting a heat and mass transport analogy.
40
30
20
10
0.2
0.4
O.f
0.8
X (kg/kg dm)
Figure 5.24 Prevailing mechanisms of water transport in porous food materials. A, liquid diffusion and capillary flow; B, vapor diffusion; C, desorption of sorbed water.
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Chapters
V. WATER TRANSPORT IN FOODS
The transport of water in food materials is an important physical process in the processing, storage, quality and utilization of food products. Of particular importance is the transport of water within the mass of solid and semisolid foods, since external transfer to and from the surrounding environment can be understood and analyzed more readily, based on the principles and applications of interphase heat and mass transfer (Chapter 9). Data and conclusions from model food systems are useful in obtaining approximate values of mass transport properties of broad classes of foods, e.g. porous, gelatinized, and sugar-containing products. However, the structure of real foods of both plant and animal origin is more complex, containing cells, cellular materials, membranes, fibers, etc making difficult the exact physical modeling for analysis of the transport processes. It becomes, therefore, necessary to obtain experimental data on transport properties of broad classes of foods and individual food products. In this section, water transport properties (mainly moisture diffusivity) of various foods are discussed on the basis of physical and physicochemical structure, and some characteristic values are given. A more detailed analysis of the literature data on moisture diffusivity of foods is presented in Chapter 6. Reviews of moisture diffusivity of foods were presented by Bruin and Luyben (1980), Chirife (1983), Saravacos (1995, 1998), Zogzas et al. (1996) and Mittal (1999). Lists of moisture diffusivities are given in some food engineering books (Gekas, 1992; Okos et al., 1992), and in a food properties database (Singh, 1993). A more detailed database was developed in the European Union (European Cooperative Project FAIR "DOPPOF").
A. Mechanisms of Water Transport
Most of the data on moisture diffusivity have been obtained from drying experiments, since mass transfer within the food material is the rate-controlling resistance. Two drying periods are usually observed, the constant rate and the falling rate. The constant rate period is controlled by external conditions of heat and mass transfer, and the interphase transport coefficients are discussed in Chapter 9. In most foods, drying takes place mainly in the falling rate period, and internal mass transport becomes very important. Water is transported within the food materials by a combination of several mechanisms, depending on the physical structure of the product and the external drying conditions. The prevalent mechanisms are molecular diffusion (liquid and vapor), capillary flow, and hydrodynamic flow. Other mechanisms may be also involved, such as Knudsen diffusion, surface diffusion, and thermal diffusion (Soret effect).
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B. Effective Moisture Diffusivity
Molecular diffusion, described by the Pick equation (5-1), is used widely for the estimation of the effective moisture diffusivity D of foods, although water may be transported by mechanisms other than diffusion. It is assumed that the driving force for all water transport is the concentration gradient (dC/dz) or the moisture content gradient (dX/dz). The methods of determination of D are discussed in Section III of this chapter. The drying curve (logY, t), obtained from drying experiments of specified samples under controlled conditions, provides useful information on the mechanism of moisture transport, and it is utilized for the determination of the effective moisture diffusivity. The drying ratio is defined as Y = (X-Xg)/(X0-X' J, where X0, X, and X e, respectively, are the moisture contents (kg/kg dm) at the beginning, after time (t), and at equilibrium. Semi-logarithmic plots of logY versus / may result in straight lines, an indication that the diffusion equation may be applied for the treatment of the drying data. Low (negative) slopes d(logY)/dt of the drying curve indicate external resistance to mass transfer, while high (negative) slopes characterize internal resistance (Mulet, 1994). The applicability of the diffusion equation to the transport of water during drying of foods can be also tested by the following simple techniques: a) Increasing the air velocity should not have a significant effect on the drying rate; b) the Biot number Bi = kcL/D (where kc is the mass transfer coefficient, m/s, L is the sample thickness, m, and D the diffusivity, m2/s) should be very high, e.g. Bi > 1000; and c) the drying time in the falling rate period should be proportional to the square of the sample thickness (Saravacos and Charm, 1962). A constant slope of the drying curve indicates a constant moisture diffusivity D, which can be estimated by analytical or numerical techniques. However, in most foods, the slope is not constant, suggesting that D is a function of the moisture content. In some cases, there are two straight lines with decreasing slopes, from which two D values can be estimated (bimodal diffusion). An example of bimodal diffusion is shown in Figure 5.25, and it refers to the air-drying of codfish fillet (Jason, 1958). The fish slab had a thickness of 15 mm, and it was dried at 35°C and air velocity of 3.7 m/s. Two effective moisture diffusivities were estimated from the two slopes (drying constants), D, = 3.4xlO'10 and D2 = 0.8x10"'° m2/s. The broken drying curve shows that, after some drying period, the moisture diffusivity decreases significantly, evidently due to shrinkage of the fish muscle, without pore formation. The increased resistance to moisture transfer at lower moisture contents is shown from the increase of the energy of activation for diffusion ED from 30 to 37 kJ/mol.
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Chapter 5
0.01
0.001
Figure 5.25 Drying curve (bimodal diffusion) of codfish fillet at 35°C. Slab thickness 15mm. Y=(X-Xe)/(Xo-Xe). (Data from Jason, 1958.)
C. Water Transport in Cellular Foods
Both plant and animal foods consist basically of cellular tissues of various components, including water, biopolymers, sugars, salts, membranes, cell walls, fibers, etc. In food processing, the heat and mass transport processes consider nonliving food materials, i.e. the physiological processes of the living cells are neglected. The physiological processes are normally disrupted by heating, freezing, and dehydration of the food products. Mass transport in cellular foods can be analyzed thermodynamically by the chemical potential approach, instead of the usual concentration gradient of Pick's equation (Rotstein, 1987; Gekas, 1992; Doulia et al., 2000). The chemical potential of water ^ (kJ/mol) is related to the water activity aw by the equation: =RTln(aw)
(5-34)
The effective moisture diffusivity D is related to the chemical potential gradient by the equation:
Transport of Water in Food Materials
D = [Kfa)/pJ
(dn/cK)
147
(5-35)
where K(ju) is the effective mass conductivity based on the chemical potential gradient, with units kg kmol/s kJ, and ps is the solids density, kg/m3. The chemical potential can be estimated from the water activity, using Eq. (5-34). Prediction of water activity is discussed by Rahman (1995). The effective mass conductivity K(p) is the summation of the mass conductivities of all cellular components in all phases, including gas diffusivities and permeabilities. The required values of porosity, tortuosity, and sorption equilibria are estimated from the physical properties of the food system. Although the chemical potential approach is thermodynamically sounder than the concentration gradient (diffusion) method, limited applications have been reported in the literature, evidently due to the involved calculations and the lack of reliable data on structural properties of the food material, such as porosity and tortuosity.
D. Water Transport in Osmotic Dehydration
In osmotic dehydration, water is removed from the food material, due to an osmotic pressure gradient between the contacting osmotic solution and the product. Two transport processes take place simultaneously, i.e. water loss (WL) from the product and sugar gain (SG) into the product. Osmotic solutions used in practice are sugar solutions, e.g. 65°Brix glucose or sucrose, and salt (sodium chloride) solutions. The transport of water and solute in osmotic dehydration at atmospheric pressure is usually modeled as a diffusion process, and the Pick equation is applied to samples of known dimensions at specified boundary conditions (Lazarides et al., 1997). Typical effective moisture diffusivity in apple tissues in contact with a sucrose solution at 25°C is D = 5xlO"10m2/s. The diffusivity of sucrose in the same system is lower as expected: D = Ixl0" 10 m 2 /s. The diffusivity of sodium chloride in osmotic treatments is close to the diffusivity of salt in water at the same temperature and concentration, e.g. D = lOxlO" 10 m2/s (Table 2.4). In vacuum osmotic dehydration (VOD), mass transport (water and solute) is mainly by hydrodynamic flow than by molecular diffusion (Fito, 1994). Hydrodynamic flow, due to mechanical pressure gradients, is facilitated by the existence of intercellular pores in the food material, e.g. apple tissues, which can be evacuated and filled with the osmotic solution by mechanical flow. The osmotic process can be accelerated by pulsed VOD. Water loss WL and sugar gain SG during osmotic dehydration can be modeled by the following empirical kinetic models (Panagiotou et al., 1998a; 1998b):
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Chapters
WL / WLe = 1 - exp(- KWL t)
(5-36)
SG /SGe = 1 - exp(- KSG t)
(5-37)
where WL, and SGe are the equilibrium water loss and sugar gain, respectively (at time t -> ac), and KWL, KSG are the corresponding constants. The rate constants KWL and KSG are related to empirical parameters of sample size, solute concentration and molecular weight, temperature, and flow conditions (speed of agitation). The empirical models of Eqs. (5-36) and (5-37) were applied to the osmotic dehydration of apple, banana, and kiwi. Glucose is a better osmotic agent than sucrose, because of its lower molecular weight and the higher mobility. The osmotic dehydration of fruit with sugars has significant effects on the moisture diffusivity in the dried product. Determination of moisture diffusivity D from airdrying data of osmotically treated apples yielded lower D values, evidently due to the effect of dissolved sugars (Karathanos et al., 1995). The diffusivity - moisture content curve (Figure 5.26) is similar to the curves of sugar containing starch materials (Figure 5.14). The dissolved sugar molecules precipitate in the pores of the fruit during drying, reducing significantly the porosity of the dried product (Chapter 3). Cylindrical apple samples 10 mm diameter were dried at 55°C and 2 m/s air velocity. Sorption kinetics data show a similar effect of sugars on the moisture diffusivity D in osmotically treated apples (Bakalis et al., 1994). Figure 5.27 shows the changes in D during adsorption of water vapor in dried apple samples 8 mm diameter in humidified air. The D values obtained from sorption experiments were significantly lower than the data obtained by the drying method, in a similar manner with the data obtained on model food materials (see Section IV).
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Transport of Water in Food Materials
(S
S
1
2
3
4
5
X (kg/kg dm)
Figure 5.26 Effective moisture diffusivity in air-drying (55°C) of osmo-treated apples. A, untreated apple; AS, osmo-treated apple. (Data from Karathanos et al., 1995.)
0.1
0,2
0.3
0.4
X (kg/kg dm)
Figure 5.27 Effective moisture diffusivity of water vapor in osmo-treated dried apples, obtained from sorption measurements. A, untreated sample; AS, osmotreated sample. (Data from Bakalis et al., 1994.)
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Chapters
E. Effect of Physical Structure As shown in model food systems, the physical and physicochemical structure of foods has a decisive effect on the mechanism of water transport and the effective moisture diffusivity. Figure 5.28 shows the effect of drying method on the moisture diffusivity of apple, using the sorption kinetics technique (Saravacos, 1967). Apple slices were dehydrated by air-drying, explosion-puff drying, and freeze-drying, resulting in products of entirely different structure. Thin slices of the dried materials, 1-2 mm thick, were used in adsorption and desorption measurements in a vacuum sorption apparatus at 30°C and increasing water vapor pressures. The obtained low values of moisture diffusivity are characteristic of sorption measurements on thin samples, resembling the sorption behavior of polymer (starch) films (Fish, 1958). It is evident that the moisture diffusivity D in the freeze-dried material is much higher than in the puff-dried and air-dried samples, over the entire moisture content range. The (D-X) curve for the puff-dried sample resembles the curves of granular starches with an intermediate maximum, a characteristic behavior of porous food materials. By contrast, the moisture diffusivity in the air-dried sample was the lowest, and it increased steadily at higher moisture contents. This behavior characterizes nonporous food materials and polymers (starches), in which diffusion of water is facilitated by moisture adsorption. Similar effects on moisture diffusivity are observed with potato samples, dehydrated by the three different drying methods described previously for the apple (Figure 5.29). A significant increase of moisture diffusivity in oil containing foods is obtained by solvent extraction of the oil. Figure 5.30 shows the moisture diffusivities D of untreated and defatted soybeans, using the sorption technique. The measurements were made on thin slices, 1 mm thick, in a vacuum apparatus at 30°C. The D values of untreated samples increases steadily from O.OlxlO" 10 to 0.03xlO~ 10 m 2 /s as the moisture content is increased from 0.2 to 0.12 kg/kg dm, a characteristic of sorption in low porosity biopolymers. By contrast, a higher D value was obtained in the defatted sample (0.055xlO"10m2/s), which remained constant over the same moisture content range. The effect of shrinkage and porosity on the moisture diffusivity of fruits and vegetables is similar to the effect on model foods (Zogzas et al., 1994a; Krokida andMaroulis, 1997).
151
Transport of Water in Food Materials
o.oi 0.1
0.2
0.3
0.4
X (kg/kg dm)
Figure 5.28 Effect of drying method on the moisture diffusivity in apple. Sorption
kinetics on thin samples, 30°C. AD, air-dried; PD, puff-dried; FD, freeze-dried. (Data from Saravacos, 1967.)
o.oi 0.00
0.05
0.10
0.15
0.20
X (kg/kg dm)
Figure 5.29 Effect of drying method on the moisture diffusivity in potato. Sorption kinetics on thin samples, 30°C. AD, air-dried; PD, puff-dried; FD, freezedried. (Data from Saravacos, 1967.)
Chapter 5
152
(S
E o
0.01 0.00
0.10
0.20
X (kg/kg dm)
Figure 5.30 Moisture diffusivity of full-fat (F) and defatted (DF) soybeans. Sorption kinetics on thin slices at 30°C. (Data from Saravacos, 1967.)
The decrease of moisture diffusivity in freeze-dried, and in porous foods generally, at high moisture contents, is directly related to a collapse of the physical structure (Karel and Flink, 1983).
F. Effect of Physical/Chemical Treatments
Various physical treatments during the processing of foods may change their physical structure, resulting in changes of moisture diffusivity. As discussed in Section III, mechanical compression, gelatinization, coating, and mixing with hydrophilic food components may reduce substantially the moisture diffusivity, due primarily to the reduction of porosity. On the other hand, puffing (extrusion, explosion puffing), vacuum and microwave treatment may increase porosity and, therefore, moisture diffusivity. Starch gelatinization reduces considerably moisture diffusivity in cooking and boiling starch-based foods in water, creating an outside layer of slowly moving water. The water demand of the inner core may be considered as the driving force for water transport inwards. The moisture diffusivity in such food systems can be measured by the NMR technique (Takeuchi et al., 1997; Gomi et al., 1998).
Transport of Water in Food Materials
153
Microwave or dielectric treatment of foods can substantially increase the moisture diffusivity during drying at atmospheric pressure or in vacuum. Microwave energy is absorbed by the water inside the food product, resulting in fast evaporation, and creating a porous structure in the dried material. Pretreatment of fruits and vegetables with microwave energy and subsequent normal airdehydration improves the drying rate. Short-time microwave pretreatment of grapes significantly increases the drying rate of grapes, due to an increase of the moisture permeability of the grape skin. The pretreatment can be applied to the sun drying of grapes (raisins), which is normally very slow, and a long time is required for drying (Kostaropoulos and Saravacos, 1995). Short-time (0.5-3.0 min) microwave pretreatment of starch gels and apples considerably increased the drying rate in subsequent air-drying. The mean effective moisture diffusivity, estimated from the drying curves, increased from 9xlO' 10 to 18xlO"'°m 2 /s in the starch gels and from 6xlO''°to 12xlO'10m2/s in the apples (Saravacos et al., 1997). The skin of some fruits, such as grapes, reduces substantially mass transfer during drying, due to the presence of waxes and other hydrophobic components, that have low moisture permeability and diffusivity. Chemical pretreatments can increase water permeability by dissolving the waxes and breaking down the skin structure. Alkali dips in solutions of sodium hydroxide, containing surfactants (surface active agents), are applied to grapes before sun drying, substantially reducing the drying time. Ethyl oleate, an edible surfactant, considerably increases the drying rate by acting on the skin and the grape berry during the drying process. Figure 5.31 shows the effect of addition of 2% ethyl oleate to a solution of 0.5% NaOH, which was used as pretreatment dip in the air-drying of seedless grapes (Saravacos and Raouzeos, 1986). A similar effect of alkali dips on the drying of grapes was reported by Masi and Riva (1988). The mean moisture diffusivity in the alkalitreated grapes was increased by the addition of ethyl oleate from 0.3x10"10 to 1.0xlO-'°m2/s. Low concentrations of surfactants sharply reduce the surface tension of water. Addition of surfactants can increase the drying rate of porous food materials by increasing the wetting of the drying surfaces during the early stages of drying. This effect has been observed in the drying of porous fruits, such as apples (Saravacos and Charm, 1962b). However, surfactants have little effect on the drying rate of nonporous food materials, such as blanched potato. Similar effects of ethyl oleate were observed in the air-drying of granular (porous) starch (Saravacos et al., 1988). On the other hand, the surfactant treatment had no effect on the drying rate of gelatinized starch, a low-porosity model food material (see Figure 5-32).
154
Chapter 5
Figure 5.31 Effect of alkali/surfactant dip on the air-drying of grapes at 60°C. CL, control (0.5% NaOH); TR, treated (0.5% NaOH +2% ethyl oleate). Y = (X-Xe)/(Xo-Xe). (Data from Saravacos and Raouzeos, 1986.)
Figure 5.32 Effect of ethyl oleate on the drying rate at 60°C of granular starch (HYLON). CL, control; TR, treated 0.2% ethyl oleate. 7 = (X-Xe)f(Xo-Xe). (Data from Saravacos et al., 1988.)
Transport of Water in Food Materials
155
G. Characteristic Moisture Diffusivities of Foods
Characteristic values of moisture diffusivity D of various food products are useful as a first approximation of this important transport property in the design and evaluation of food processes and food storage/quality changes. The wide variation of he literature data on D is due to the complex physical and chemical structure of foods and the different methods of measurement and estimation used. A unified analysis of the literature data on D is presented in Chapter 6. Table 5.3 gives some typical D values for important classes of foods, based on the physical structure and porosity. The effect of structure of foods is similar to the effect of structure on the D of model food materials, as discussed in Section IV. Tables 5.4 to 5.8 give typical D values of various food products, selected from the literature. Table 5.3 Effect of Physical Structure on the Moisture Diffusivity of Food Products Food Structure Porosity Activation energy Diffusivity ______________%_______kJ/mol_________xlQ-'V/s
Highly porous Freeze-dried Puff-dried Fibrous
90
10
50
Porous Granular Vacuum-dried
50
20
20
Low porosity Compressed
20
30
5
Gelatinized
10
45
1
Starch/protein foods
10
50
0.1
Starch/lipid foods
10
60
0.01
Glassy-state foods
0
>60
0.0001
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Chapters
Table 5.4 Typical Moisture Diffusivities of Cereal Grains and Rice at 30°C (typical activation energy ED = 40 kJ/mol) Food material Moisture Diffusivity ___________________kg/kg dm______xlQ-'V/s Corn kernel 0.20 0.40 Corn pericarp 0.20 0.01 Wheat kernel 0.20 0.50 Rice
0.20
0.40
Table 5.5 Typical Moisture Diffusivities of Baked Products and Pasta at 30°C (typical activation energy ED = 40 kJ/mol) Diffusivity Moisture Food material xlO'10m2/s kg/kg dm 5.0 0.40 Dough 2.0 0.30 Bread 0.5 0.15 Cookie 0.3 0.15 Pasta 1.2 Puffed pasta 0.15
Table 5.6 Typical Moisture Diffusivities of Vegetable Products at 30°C (typical activation energy ED = 45 kJ/mol) Food material Potato Carrot Peas Onion Soybeans
Moisture kg/kg dm 0.30 0.30 0.10 0.10 0.20
Diffusivity xlO-'W/s 5.0 2.0 3.0 0.5 0.8
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157
Table 5.7 Typical Moisture Diffusivities of Fruit Products at 30°C (typical activation energy ED = 60 kJ/mol) Food material Moisture Diffusivity _____________________kg/kg dm______xlQ-'°m2/s Apple 0.50 2.0 Apricots 0.40 1.0 Bananas 0.50 2.0 Raisins 0.40 1.5
Table 5.8 Typical Moisture Diffusivities of Meat and Fish at 30°C (typical activation energy ED = 35 kJ/mol) Food material Minced beef Pork sausage Codfish Herring
Mackerel Hulibut
Moisture kg/kg dm
0.60 0.20 0.50 0.50 0.40 0.40
Diffusivity xlO-'W/s 1.0 0.5
2.0 0.8 0.5 0.3
REFERENCES
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Methods of Experimental Determination. A Review. Drying Technol 12:483515 Zogzas, N.P., Maroulis, Z.B., Marinos-Kouris, D. 1996. Moisture Diffusivity Data Compilation in Foodstuffs. Drying Technol 14:2225-2253.
Moisture Diffusivity Compilation of Literature Data for Food Materials
I. INTRODUCTION
There is a wide variation of the reported experimental data of moisture diffusivity of solid food materials, making difficult their utilization in food process and food quality applications. The variation of moisture diffusivity in model and real foods is discussed in Chapter 5. The physical structure of solid foods plays a decisive role not only on the absolute value of moisture diffusivity, but also on the effect of moisture content and temperature on this transport property. Porosity data, obtained from measurements of bulk and particle densities (Chapter 3) at various moisture contents, provide useful data on the type of water transport (liquid or vapor diffusion) and the approximate value of moisture diffusivity. In this chapter, the moisture diffusivity in food materials is approached from a statistical standpoint. Literature data are treated by regression analysis, using an empirical mathematical model. Recently published values of moisture diffusivity in various foods were retrieved from the literature and were classified and analyzed statistically to reveal the influence of material moisture content and temperature. Empirical models relating moisture diffusivity to material moisture content and temperature were fitted to all examined data for each material. The data were screened carefully, using residual analysis techniques. A promising model was proposed based on an Arrhenius-type effect of temperature, which uses a parallel structural model to take into account the effect of material moisture content.
163
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Chapters
II. DATA COMPILATION
Moisture diffusivity data on foods in the literature are scarce because of the effect of the following factors: (a) diverse experimental methods, (b) different methods of analysis used, (c) Variation in composition of the material, and (d) variation of the structure of the material. Literature data for moisture diffusivity in foods materials were selected and presented in the reviews of Bruin and Luyben (1980), Chirife (1983), Gekas (1992), Marinos-Kouris and Maroulis (1995), Zogzas et al. (1996), Mittal (1999), and Doulia et al. (2000). In addition to these reviews, an exhaustive literature search was made in international food engineering and food science journals in recent years, as follows (Panagiotou et al., 2001): • • • • •
Drying Technology, 1983-1999 Journal of Food Science, 1981-1999 International Journal for Food Science and Technology, 1988-1999 Journal of Food Engineering, 1983-1999 Trans. of the ASAE, 1975-1999
A total of 175 papers were retrieved from the above journals according the distribution presented in Figure 6.1. The accumulation of the papers versus the publishing time is presented in Figure 6.2. The search resulted in 1558 data concerning the moisture diffusivity in food materials. The 1558 data retrieved from the above search, plus 16 data from Bruin and Luyben (1980), 58 data from Chirife (1983) and 141 data from Gekas (1992) were organized into a database and analyzed. A total number of 1773 data was obtained. These data are plotted versus moisture and temperature in Figures 6.3 and 6.4, respectively. These figures show a good picture concerning the range of variation of diffusivity, moisture and temperature values. More than 95% of the data are in the ranges: • Diffusivity • Moisture • Temperature
IxlO' 1 2 -IxlO" 6 m 2 /s 0.01 - 15.0 kg/kg db 10-200°C
It should be noted that the lowest possible moisture diffusivity is that in glassy food materials, about IxlO" 14 m2/s, and the highest is the diffusion of water vapor in air at atmospheric pressure, IxlO" 5 m2/s (Table 2.4). Higher diffusivities of water vapor can be obtained in vacuum systems. The self diffusivity of water is near Ixl0" 9 m 2 /s.
Moisture Diffusivity Data Compilation
J. Food Engineering
Drying
Technology
165
J. of Food Science
Trans of the ASAE
Int. J, Food Science and Techn.
Figure 6.1 Number of papers on moisture diffusivity data in food materials published in food engineering and food science journals during recent years. 1000
-1—
1975
1985
1995
2005
Year
Figure 6.2 Accumulation of published papers on moisture diffusivity data for food materials versus time.
Chapter 6
166
l.E-03
l.E-06 - =
•5 l.E-09
l.E-12
l.E-15 0.01
0.1
1
10
100
Moisture (kg/kg db)
Figure 6.3 Moisture diffusivity data for all foods at various moistures. l.E-03
l.E-15 10
100
1000
Temperature (°C)
Figure 6.4 Moisture diffusiviry data for all foods at various temperatures.
Moisture Diffusivity Data Compilation
167
The histogram in Figure 6.5 shows the distribution of the moisture diffusivity values retrieved from the literature. The results obtained are presented in detail in Tables 6.1-6.3. More than 100 food materials are incorporated in these tables. They are classified into 11 food categories. Table 6.1 shows the related publications for every food material. Table 6.2 summarizes the average literature value for each material along with the corresponding average values of corresponding moisture and temperature. Table 6.3 presents the range of variation of moisture diffusivity for each material along with the corresponding ranges of moisture and temperature.
1000
1)
•c
100
I
a •M
a O
B
s Z
10
l.E-15
l.E-12
l.E-09
l.E-06
l.E-03
Moisture Diffusivity Values (mVs)
Figure 6,5 Histogram of observed values of moisture diffusivity in food materials.
Chapter 6
168
Table 6.1 Literature for Moisture Diffusivity Data in Food Materials: References and Number of Data Retrieved Material Reference
Baked Products
Data
33 4
Biscuit Balasubrahmanyam and Datta, 1993 Bread
Zhouetal., 1994
Cookie Gekas, 1992 Lomauroetal, 1985 Crackers
Kirn and Okos, 1999 Dough
Kamthanos et al., 1995 Zanonietal, 1994
4 1 1 2 1 1 18 18 8 4 4
499
Cereal Products
16
Barley Fasina et a!., 1998 Miketinac et al, 1992
Brown rice
Engelsetal, 1986 Gekas, 1992 Hendrickx et al, 1988 Lu and Siebenmorgen, 1992 Steffe and Singh, 1980
7 9 36 9 7 3
Moisture Diffusivity Data Compilation
169
Table 6.1 Continued Material Reference
Corn Bakker-Arkema et a!., 1987 Galan-Domingo and Martinez-Vera, 1996 Gekas, 1992 Harosetal, 1995 Jumah and Mujumdar, 1996 Martinet- Vera eta!., 1 995 Mouradetal, 1996 Muthukumarappan and Gunasekaran, 1994a, b, c Parti and Dugmanics, 1990 Patil, 1988 Shivhare et al, 1992 Syariefetal., 1987 Tolabaetal, 1989 Tolabaetal, 1990 Verma and Prasad, 1999 Waltonetal, 1988 Zahedetai, 1995 Malt Lopezetal, 1997
Milled rice Zhangetal, 1984
Paddy rice Base et al, 1987
Parboiled brown rice Igathinathane and Chattopadhyay, 1999a, b
Parboiled paddy rice Igathinathane and Chattopadhyay, 1999a, b
Parboiled rice Chandra and Singh, 1984 Igathinathane and Chattopadhyay, 1999a, b Pasta Litchfield and Okos, 1992 Meat et al, 1996 Waananen and Okos, 1996 Xiongetal, 1991
Data
168 6 3 3 9 6 5 17 40 8 8 10 9 8 4 7 20 5 4 4 2 2 3 3 6 6 6 6 12 6 6 51 11 4 30 6
Chapter 6
170
Table 6.1 Continued Material Reference
Rice Engelsetal, 1986 Galan-Domingo and Martinez-Vera, 1996 Gekas, 1992 Hendrickx et al, 1986
Hendrickx and Tobback, 1987 Hendrickx etal, 1988 Steffe and Singh, 1980 Zahedetal, 1995 Rough rice
Bakker-Arkema et al, 1987 Chungu and Jindal, 1993 EceandCihan, 1993 Gekas, 1992 Lague andJenkins, 1991 Lu and Siebenmorgen, 1992 Sarkeretal, 1994 Steffe and Singh, 1980 Suarezetal, 1982 Tang and Sokhansanj, 1993 Wheat
Bakker-Arkema et al, 1987 Bruin and Luyben, 1980 Chirife, 1983 Devahastin et al, 1998
Gekas, 1992 Giner and Calvelo, 1987 Gineretal, 1996
Gong etal, 1997 Igathinathane and Chattopadhyay, 1997
Jar os et al, 1992 Jayas et al, 1991 Lomauro et al, 1985 Sapru and Labuza, 1996 Suarezetal, 1982 Sun and Woods, 1994 Zahedetal, 1995
Wild rice
Gekas, 1992
Data
56 6 3 15 3 6 8 10 5 78 3 6 6 14 6 8 21 9 1 4 58 3 2 3 3 4 4 4 4 4 7 9 1 1 1 5 3 3 3
Moisture Diffusivity Data Compilation
171
Table 6.1 Continued
Material Reference
Data
32
Dairy Cheese Change! al, 1998 Desobry and Hardy, 1994 Luna and Chavez, 1992 Dry milk Gekas, 1992 Lomauroetal, 1985 Milk Straatsmaetal., 1999 Skim milk Ferrari et al, 1989 Kerkhof, 1994
9 2 5 2 2 1 1 6 6 15 9 6
63
Fish Catfish Chirife, 1983 Cod
Balaban and Pigott, 1988 Chirife, 1983 Dogfish Bruin and Luyben, 1980 Chirife, 1983 Fish meal Alvarez and Shene, 1996 Blasco and Alvarez, 1999 Haddock Chirife, 1983 Halibut Chirife, 1983 Herring Chirife, 1983 Mackerel Chirife, 1983
1 1 6 4 2 4 1 3 15 8 7 2 2 2 2 7 7 1 1
Chapter 6
172
Table 6.1 Continued Material Reference
Data 12
Shark
Park, 1998 Squid Teixeira and Tobinaga, 1995
Teixeira and Tobinaga, 1998 Swordfish
Chirife, 1983 Whiting
Chirife, 1983
12 6 1 5 4 4 3 3
268
Fruits
64
Apple Bruin and Luyben, 1980 Chirife, 1983 Fuscoetai, 1991 Gekas, 1992 Lazarides et al., 1997 Lomauroetal, 1985
Nieto et al, 1998 Simaletal, 1997 Simaletal, 1998a, b Apricot Abdelhaq and Labuza, 1987
Vagenas and Marinos-Kouris, 1991 Avocado
Chirife, 1983 Gekas, 1992 Banana
Garciaetal, 1988 Gekas, 1992 Johnson et al, 1998 Mauro and Menegalli, 1995 Rastogietal, 1997
Son/cat el al., 1996 Waliszewskietal, 1997
Blueberries
Nsonzi and Ramas\vamy, 1998 Ramaswamy and Nsonzi, 1998
Coconut
Gekas, 1992
3 6 2 14 13 1 6 11 8 10 2 8 11 6 5 55 5 6 3 9 12 11 9 29 11 18 6 6
Moisture Diffusivity Data Compilation
Table 6.1 Continued Material Reference
Grapes Alvarez and Legues, 1986
Gekas, 1992 Mahmutogluetal, 1996 Simaletal, 1996 Vagenasetal, 1990 Mulberry
Mas/can andGogus, 1998 Peach
Gekas, 1992 Pineapple
Azuaraetal, 1992 Beristainetal, 1990
Rastogi and Niranjan, 1998 Raisins
Gekas, 1992 Karathanos et al, 1995 Lomauro et al, 1985 Sapru and Labuza, 1996
Legumes
173
Data
52 18 4 16 8 6 9 9 1 1 20 6 9 5 11 1 8 1 1
29
Broad bean Ptaszniketal, 1990
Fababean Hsu, 1983a, b Lentil Tang and Sokhansanj, 1993
Navy beans
Radajewski et al, 1992
Meat
7 7 2 2 12 12 8 8
53
Beef Gekas, 1992 Huang and Mittal, 1995
4 2 2
174
Chapters
Table 6.1 Continued Material Reference Beef carcass
Data 3
Mallikarjunan and Mittal, 1994
Broiled
McLendon and Gillespie, 1978 Bull
Gekas, 1992
Chicken Ngadi et al, 1997 Ground beef
Gekas, 1992 Hallstrom, 1990 Lomauroetai, 1985 Heifer
3 3 3 3 3 6 6 19 1 17 1 7
Gekas, 1992 Pepperoni
Bruin and Luyben, 1980 Chirife, 1983
Sausage Dincer and Yildiz, 1996
Turkey Chirife, 1983
Gekas, 1992
Model foods
7 5 3 2 1 1 2 1 1
202
Albumin-flour-bran
Strumitto et al, 1996 Amioca
Karathanos et al., 1990 Kostaropoulos and Saravacos, 1997
Marousisetal, 1989 Seowetal., 1999 Vagenas and Karathanos, 1993
Cellulose-oil-water Chirife, 1983 Corn starch Gekas, 1992
Flour
Gekas, 1992 Lomauroetai., 1985
27 27 59 16 4
20 10 9 1 1 2 2 3 1 2
Moisture Diffusivity Data Compilation
175
Table 6.1 Continued_________________________________ Material Reference
Data
Glucose-starch
7 1 6 6
Gekas, 1992
Gluten-starch Xiong et a!., 1991 Hylon-7
Karathanosetal, 1990 Kostaropoulos and Saravacos, 1997
59 9 4
Marousisetal, 1989 Seawetal.,1999 Tsukadaetal, 1991 Vagenas and Karathanos, 1993
18 11 9 8 6
Roques et a!., 1994
6 4 4 23 3 20 5 5
Polyacrylamide gel Potato starch
Gekas, 1992 Rice starch GOBI; e/ a/., 1998 Takeuchi et al, 1997 Starch Geto, 7P92
Nuts_______________________________79 Almond
Beviaetai, 1999 Hazelnuts Lopezetal, 1998
Peanut pods Chinnan and Young, 1977a, b
Peanuts SuarezetaL, 1982
2 2 12 12
64 64 1 1
Other_______________________________45 Canola
Thakoretal., 1999
Chocolate Biquet and Labuza, 1988
24 24 4 4
176
Chapter 6
Table 6.1 Continued Material Reference Coffee
Data 3
Gekas, 1992 Egg
Kincal, 1987
Sunflower seeds Rovedo et al, 1993
Toria Raoetal, 1992
Vegetables
3 6 6 4 4 4 4
470
Beet Chirife, 1983
Broccoli Sanjuanetal, 1999 Simaletal, 1998a, b
Carrot
Cordova-Quirozet al, 1996 Gekas, 1992 Kiranoudis et al, 1992 Kiranoudis et al., 1993
Kompany et al., 1993 Mabrouk and Belghith, 1995 Markov/ski, 1997
Muletetal., 1987 Mulet et al, 1989 Mulet, 1994 Rastogi and Raghavarao, 1997 Stapley et al, 1995
Cassava Chavez-Mendez et al, 1998
Fuscoetal, 1991
Garlic Madambaetal, 1996 Pezzutti and Crapiste, 1997 Pinagaetal, 1984 Vazquezetal, 1999
Okra Gogus and Maskan, 1999
1 1 22 10 12 106 1 4 9 15 10 8 12 7 12 6 12 10 10 2 8 22 5 6 5 6 6 6
Moisture Diffusivity Data Compilation
177
Table 6.1 Continued Material Reference
Data
31
Onion Baroni and Hubinger, 1998 Kiranoudis et al., 1992 Lewicki et al, 1998 LopezetaL, 1995
Paprika Gekas, 1992
Pea Medeiros and Sereno, 1994
Pepper Carbonelletal, 1986 Kiranoudis et al, 1992
Pigeon pea Shepherd and Bhardwaj, 1988
Potato AfzalandAbe, 1998 Bonetal, 1997 Bruin and Luyben, 1980 Chirife, 1983 Costa and Oliveira, 1999 Fuscoetal, 1991 Gekas and Lamberg, 1991 Gekas, 1992 Kiranoudis et al, 1992 Lazarides and Mavroudis, 1996 Lazaridesetal, 1997 Magee and Wilkinson, 1992 Maroulis et al., 1995 McLaughlin and Magee, 1999 McMinn and Magee, 1996 Mishkinetai, 1984 Mulet, 1994 Pinthus et al, 1997 Rice and Gamble, 1989 Rovedo and Viollaz, 1998 Rovedoetal, 1995 Rubnov and Saguy, 1997 Yusheng and Poulsen, 1988 Zhouetal, 1994
9 9 9 4 3 3 9 9 14 5 9 5 5 165 12 6 4 7 4 2 4 12 9 6 1 8 15 8 12 9 8 12 3 3 3 4 12 1
178
Chapter 6
Table 6.1 Continued Material Reference
Soya meal Alvarez and Blasco, 1999 Alvarez and Shene, 1996
Soybean Barrozo et al, 1998
Deshpande et al., 1994 Gekas, 1992 Hsu, 1983a, b Misra and Young, 1980 Oliveira and Haghighi, 1998 Suarezetal, 1982 Sugar beet
Data
18 12 6 19 3 1 1 8 4 1 1 7
Bruin and Luyben, 1980 Chirife, 1983 Fuscoetal, 1991 Tapioca Bruin and Luyben, 1980 Chirife, 1983
Tomato Dincer and Dost, 1995 Hawladeretal., 1991 Karatas and Esin, 1994
Turnip Lomauro et al., 1985
Moreiraetal, 1993 Yam
Hawladeretal, 1999
2 3 2 4 1 3 16 3 8 5 10 1 9 2 2
179
Moisture Diffusivity Data Compilation
Table 6.2 Moisture Diffusivity of Foods Versus Moisture and Temperature:
Average Values of Available Data Material
Diffusivity (m2/s)
Moisture (kg/kg db)
Temperature (°C)
33
Baked Products Biscuit fondant coated Bread Cookie oatmeal Crackers Dough -
4 5.26E-09
0.12
85
5.00E-08
0.67
80
3.99E-12
0.18
25
6.08E-10
0.08
63
4.89E-10
0.48
122
4 1 1 2 2 18 18 8 8
492
Cereal Products Barley kernel Brown rice bran endosperm kernel testa Corn dent endosperm flint germ grains hard endosperm kernel pericarp semident shelled soft endosperm without pericarp
No. of Data
9 1.85E-10
0.23
49
2.64E-10 2.59E-11 6.68E-11 4.36E-11 4.46E-11
0.19 0.22 0.24 0.24 0.24
40 45 40 40 40
3.27E-07 3.40E-10 1.08E-10 3.20E-11 1.31E-10 6.04E-10 1.72E-11 4.49E-11 1.09E-11 4.54E-11 1.01E-10 2.96E-11 4.06E-13
0.34 0.28 0.20 0.50 0.14 0.28 0.09 0.35 0.19 0.50 0.19 0.06 0.21
41 60 40 55 35 48 33 49 38 55 71 33 33
9 36 18 9 3 3 3 168 31 15 3 3 11 28 4 25 13 3 20 4 8
Chapter 6
180
Table 6.2 Continued Material
Malt Milled rice Paddy rice grains
Diffusivity (m2/s)
Moisture (kg/kg db)
Temperature (°C)
8.73E-08
0.45
50
No. of Data 4 4
2 1.31E-10
0.32
60
1.53E-11
0.11
50
Parboiled brown rice
2 3 3 6 6 6 6 12 6
bran
3.52E-11
0.33
75
Parboiled paddy rice husk Parboiled rice
1.01E-IO
0.30
75
endosperm long grain short grain
3.21E-10 6.05E-11 1.37E-10
0.52 0.50 0.50
75 60 60
1.94E-11 4.76E-11 1.21E-10
0.16 0.13 0.12
63 81 68
21 18 12 56
4.49E-1 1 4.13E-09 9.94E-11 1.05E-10 1.20E-11 1.21E-10 4.43E-11
0.26
37 130 39 61 45 40 40
13
3.71E-09 8.68E-12 1.73E-10 1.30E-11 1.06E-11 2.56E-11 1.62E-08
0.19 0.15 0.15 0.15 0.23 0.15 0.28
Pasta dense porous Rice cooking endosperm grains hull kernel testa
Rough rice bran endosperm grains
hull husk
kernel
3 3 51
0.23 0.26 0.23
49 43 47 41 45 45 60
2 21 3
2 12 3 78 35
8 6 1 9 7 12
181
Moisture Diffusivity Data Compilation
Table 6.2 Continued Material
Wheat bran endosperm flakes grains hard
kernel shredded soft Wild rice broken unprocessed
whole
Dairy Cheese Dry milk nonfat Milk powder Skim milk -
Diffusivity (m2/s)
1.45E-10 1.73E-10 1.91E-10 8.33E-14 3.69E-11 2.02E-09 6.54E-11 5.53E-12 1.51E-10
Moisture (kg/kg db)
0.23 0.35 0.50 0.50 0.15 0.17 0.16 0.11 0.15
Temperature
No. of
(°C)
Data
49 30 30 25 66 20 63 25 50 25 20 20
7.00E-13 4.00E-13 2.00E-13
32 9 2.02E-08
0.58
9
2.12E-11
0.12
25
6.58E-10
0.30
40
1.36E-10
0.56
46
9 2 2 6 6 15 15
30
63 1 1
Fish Catfish Cod muscle Dogfish Fish meal -
Haddock muscle
58 26 2 2 1 10 2 11 1 3 3 1 1 1
8.00E-1 1 2.78E-10 3.40E-10
3.00
30
1.48E-10 7.97E-10 6.00E-11 3.30E-10
42 30
0.40
118 30 30
6 5 1 4 4 15 15 2 1 1
Chapter 6
182
Table 6.2 Continued Material
Halibut muscle Herring Mackerel Shark muscle Squid mantle Swordflsh salted Whiting muscle
Diffusivity
Moisture
Temperature
No. of
(m2/s)
(kg/kg db)
(°C)
Data
5.80E-11 2.50E-10
30 30
6.53E-11
30
3.50E-11
30
1.80E-10
1.31
30
8.91E-11
1.50
34
3.45E-10 2.95E-10
48 48
4.80E-11 1.76E-10
30 28
Fruits
2 1 1 7 7 1 1 12 12 6 6 4 2 2 3 1 2
268
Apple -
6.64E-10
2.80
47
Apricot -
1.39E-07
2.88
53
64 10 10
50
11 11
Avocado Banana plantain ripe
Blueberries Coconut Grapes red seedless Mulberry -
64
6.35E-10 1.4 IE-09 6.51E-10 1.43E-09
1.63 0.90 3.00
53 55 60
2.12E-10
1.75
45
9.77E-10
0.60
83
1.37E-10 1.79E-10 2.03E-10
1.49
53 60 60
1.18E-09
1.50
70
55 49 3 3 29 29 6 6 52 18 2 32 9 9
183
Moisture Diffusivity Data Compilation
Table 6.2 Continued Material
Diffusivity
(mVs)
Peach Pineapple Raisins -
Moisture (kg/kg db)
Temperature (°C)
30
8.00E-12 1.47E-09
4.50
40
1.67E-10
0.37
37
Legumes 6.53E-07
0.26
30
Fababean -
1.78E-07
0.75
30
Lentil cotyledons hilum seedcoat
2.25E-11 7.22E-09 1.57E-12
0.13 0.15 0.18
40 40 40
Navy beans •
4.56E-08
0.28
50
Meat Beef
raw
Beef carcass bone fat muscle Broiled waste Bull -
Ground beef heat treated raw
Data
1 1 20 20 11 11
29
Broad bean seeds
meatball
No. of
1.40
30 140 30
9.81E-06
0.30
21
7.40E-1 1
0.76
27
3.03E-11 1.48E-10 8.61E-11
0.16 0.80 1.13
25 51 43
5.56E-10 3.20E-10 l.OOE-11 5.48E-12 3.07E-11 5.83E-10
7 7 2 2 12 3 3 6 8 8
53 4 1 2 1 3 1 1 1 3 3 3 3 19 2 8 9
184
Chapter 6
Table 6.2 Continued Material
Heifer heat treated raw
Pepperoni sausage Sausage Turkey -
Diffusivity
Moisture
(m2/s)
(kg/kg db)
(°C)
Data
1.69E-10 8.33E-11
1.00 1.00
51 43
5.20E-11 5.33E-11
0.19
12 12
1.31E-07
0.32
180
8.00E-15
0.04
22
4 3 5 2 3 1 1 2 2
Temperature
7
Model foods Albumin-flour-bran mixture! mixture 2 mixtures Amioca gel hydrated Cellulose-oil-water Corn starch Flour Glucoose-starch Gluten-starch gelatinized ungelatinized Hylon-7 gel hydrated Polyacrylamide gel Potato starch .
No. of
202 1.45E-09 1.0 IE-09 7.70E-10
0.43 0.49 0.49
92 92 92
2.26E-09 8.20E-10 1.90E-09
0.36 0.33 0.33
74 60 60
3.10E-09
68
2.25E-10
30
2.26E-1 1
0.12
25
2.27E-10
0.60
39
3.33E-11 2.67E-11
0.14 0.14
74 74
2.09E-09 2.06E-09 2.27E-09
0.30 0.44 0.33
63 60 60
1.52E-10
0.90
40
6.91E-12
25.28
25
27 9 9 9 59 49 5 5 1 1 2 2 3 3 7
7 6 3 3 59 48 6 5 6 6 4 4
185
Moisture Diffusivity Data Compilation
Table 6.2 Continued Material
Rice starch full heated
Diffusivity
Moisture
(m2/s)
(kg/kg db)
Temperature
nonheated
9.75E-10 2.1 IE-09
1.63 1.50
50 57
Starch -
4.23E-10
0.60
42
2.32E-12
0.05
281
shelled unshelled
4.03E-09 6.16E-09
0.15 0.15
55 55
hull kernel
4.69E-11 7.28E-11
0.60 0.60
35 35
Peanuts -
4.00E-11
0.10
50
Peanut pods
embryo kernel
3.58E-09 5.09E-09
0.02 0.02
80 80
Chocolate dark Coffee
1.03E-13
2.00
20
extract
1.08E-10
50
1.44E-11 1.61E-11
1.00 0.80
36 36
4.40E-10 1.20E-10
0.07 0.07
45 45
9.80E-11
0.10
65
Sunflower seeds
hull kernel Toria seeds
2 2 12 6 6 64 32 32 1 1
45
Other Canola
incubated
23 13 10 5 5
79
Nuts Almond Hazelnuts
Egg fresh
No. of
(°C)Data
24 12 12 4 4 3 3 6 3 3 4 2 2 4 4
186
Chapter 6
Table 6.2 Continued Material
Diffusivity (mVs)
Moisture (kg/kg db)
No. of Temperature Data (°C)
Vegetables Beet -
1.50E-09
Broccoli stems
1.29E-09
8.80
62
Carrot Cassava
2.05E-09
4.60
53
roots
6.30E-10
0.63
67
Garlic -
1.74E-10
0.80
50
2.24E-09
2,00
70
1.0 IE-09
1.65
64
Okra Onion Paprika Pea -
Pepper redpo-wder Pigeon pea kernel Potato restructured product tissue Soya meal Soybean grains Sugar beet roots Tapioca roots
65
48
2.17E-10 2.74E-10
0.97
48
6.22E-09 2.09E-10
3.70 0.06
70 49
5.07E-11
0.20
70
1.32E-09 2.02E-09 1.67E-09
3.17 1.85
58 105 65
1.16E-08
0.10
162
9.0 IE-08 1.12E-09
0.56 0.60
47 48
6.59E-10
2.60
57
6.00E-10
1.05
78
467 1 1 22 22 106 106 10 10 22 22 6 6 31 31 3 3 9 9 14 9 5 5 5 165 148 16 1 18 18 16 13 2 7 7 4 4
Moisture Diffusivity Data Compilation Table 6.2 Continued Material
Tomato concentrate droplets Turnip Yam _
Diffusivity (m2/s)
187
Moisture (kg/kg db)
Temperature
(°C)
7.57E-10 1.87E-09
10.00 0.50
64 77
1.64E-09
6.33
57
1.27E-09
0.10
45
No. of Data
16 11 5 10 10 2 2
188
Chapter 6
Table 6.3 Moisture Diffusivity of Foods Versus Moisture and Temperature: Variation Range of Available Data Temperature
Material
min
Diffusivity (m2/s) max
Baked Products
3.97E-I2
5.00E-08
0.03
0.67
15
203
4.06E-09 4.06E-09 5.00E-08 5.00E-08 3.97E-12 3.97E-12 1.40E-11 1.40E-11 1.30E-10 1.30E-10
6.29E-09 6.29E-09 5.00E-08 5.00E-08 4.00E-12 4.00E-12 1.81E-09 1.81E-09 l.OOE-09 l.OOE-09
0.03 0.03
0.39
77 77 80 80 25 25 40 40 15 15
91 91 80 80 25 25 90 90 203 203
8.33E-14
4.04E-06 6.52E-10 6.52E-10 3.94E-09 3.94E-09 4.07E-11 9.64E-11 7.25E-11 9.31E-11 4.04E-06 4.04E-06 1.48E-09 2.01E-10 4.50E-11 8.42E-10 6.1 IE-09 2.06E-1 1 1.14E-10 7.37E-11 6.80E-11 2.23E-10 3.72E-11 5.50E-13
Biscuit fondant coated Bread Cookie oatmeal Crackers Dough -
Cereal Products Barley kernel Brown Rice bran endosperm kernel testa Corn dent endosperm flint germ grains hard endosperm kernel pericarp semident shelled soft endosperm without pericarp
1.31E-11 1.31E-11 1.81E-12 1.81E-12 1.48E-11 4.36E-11 2.00E-11 1.25E-11 9.72E-14 2.33E-11 2.39E-11 3.61E-11 1.90E-11 5.28E-12 5.24E-11 1.29E-11 1.11E-11 9.72E-14 2.50E-11 2.16E-11 2.12E-11 2.56E-13
Moisture (db)
(°C)
min
max
min
max
0.67 0.67 0.18 0.18 0.03 0.03 0.20 0.20
0.39 0.67 0.67 0.18 0.18 0.14 0.14 0.60 0.60
0.02
0.56
5
150
0.10
0.27 0.27 0.25 0.25 0.24 0.24 0.24 0.24 0.56 0.50 0.50 0.30 0.50 0.30 0.40 0.10 0.56 0.30 0.50 0.19 0.07 0.23
30 30 12 12 35 30 30 30 10 10 30 40 45 25 10 25 25 25 45 38 25 25
70 70 120 120 55 50 50 50 120 120 90 40 65 40 120 40 90 60 65 104 40 40
0.10 0.16 0.16 0.21 0.24 0.24 0.24 0.06 0.10 0.10 0.10 0.50 0.10 0.10 0.07 0.12 0.10 0.50 0.19 0.06 0.19
189
Moisture Diffusivity Data Compilation
Table 6.3 Continued Diffusivity
Material
min
Malt Milled Rice Paddy Rice grains Parboiled Brown Rice bran Parboiled Paddy Rice husk Parboiled Rice endosperm long grain short grain
Pasta dense porous
Rice cooking endosperm grains hull kernel testa Rough Rice bran endosperm grains hull husk kernel
1.11E-08 1.1 IE-08 8.33E-11 8.33E-11 2.28E-12 2.28E-12 1.15E-11 1.15E-11 3.43E-11 3.43E-11 2.38E-11 2.12E-10 2.62E-11 2.38E-11 1.55E-12 1.55E-12 9.40E-12 2.43E-11 3.30E-12 3.30E-12 1.92E-09 4.40E-11 6.67E-11 4.00E-12 5.19E-11 1.20E-11 7.56E-13 7.56E-13 4.03E-12 9.86E-11 1.30E-11 4.08E-12 1.11E-11 4.75E-12
(m2/s) max
2.14E-07 2.14E-07 1.78E-10 1.78E-10 3.44E-11 3.44E-11 6.26E-11 6.26E-11 1.80E-10 1.80E-10 4.84E-10 4.84E-10 1.02E-10 2.98E-10 3.42E-10 4.84E-1 1 1.06E-10 3.42E-10 6.33E-09 1.17E-10 6.33E-09 1.98E-10 1.48E-10 2.00E-11 2.28E-10 9.30E-11 1.68E-07 2.64E-08 2.21E-11 2.20E-10 1.30E-11 3.59E-11 4.41E-11 1.68E-07
Moisture (db)
min
max
0.45 0.45 0.13 0.13 0.11 0.11 0.30 0.30 0.28 0.28 0.50 0.51 0.50 0.50 0.02 0.02 0.05 0.04 0.10 0.10
0.45 0.45 0.50 0.50 0.11 0.11 0.36 0.36 0.33 0.33 0.53 0.53 0.50 0.50 0.32 0.32 0.23 0.21 0.35 0.35
0.20 0.22
0.25 0.30
0.23
0.24
0.05 0.10 0.15 0.15 0.15 0.22 0.15 0.05
0.50 0.33 0.15 0.15 0.15 0.24 0.15 0.50
Temperature (°C) min max
20 20 60 60 40 40 50 50 50 50 40 50 40 40 40 40 40 40 8 20 110 8 61 35 30 30 12 12 25 30 41 35 30 38
80 80 60 60 60 60 100 100 100 100 100 100 80 80 122 85 122 105 150 55 150 55 61 55 50 50 120 120 60 60 41 55 60 82
Chapter 6
190
Table 6.3 Continued Diffusivity
Moisture
(m2/s)
Material
Wheat bran endosperm flakes grains hard kernel shredded soft Wild Rice broken unprocessed whole
Dairy Cheese Dry Milk nonfat Milk powder Skim Milk
Fish Catfish Cod muscle
(db)
Temperature (°C) max min
min
max
min
max
8.33E-14 6.07E-12 1.68E-10 1.91E-10 8.33E-14 1.50E-11 3.30E-IO 1.15E-11 5.53E-12 2.98E-11 2.00E-13 7.00E-13 4.00E-13 2.00E-13
3.70E-09 5.30E-10 1.78E-10 1.92E-10 8.33E-14 7.94E-1 1 3.70E-09 1.43E-10 5.53E-12 3.19E-10 7.00E-13 7.00E-13 4.00E-13 2.00E-13
0.10 0.17 0.35 0.50 0.50 0.10 0.13 0.10 0.11 0.15
0.50 0.30 0.35 0.50 0.50 0.20 0.20 0.30 0.11 0.15
5 5 30 30 25 40 20 40 25 30 20 25 20 20
86 85 30 30 25 86 20 80 25 70 25 25 20 20
2.10E-11 5.60E-11
9.00E-08
0.12
0.80
0
9.00E-08
0.80
9.00E-08 2.13E-11 2.13E-11 1.83E-09 1.83E-09 2.56E-10 2.56E-10
0.80 0.12 0.12 0.40 0.40 0.80 0.80
0 0 25 25 10 10 30 30
70 13
5.60E-11 2.10E-11 2.10E-11 3.50E-11 3.50E-11 2.51E-11 2.51E-11
0.35 0.35 0.12 0.12 0.20 0.20 0.20 0.20
1.30E-11
1.89E-09
0.33
3.00
20
8.00E-11 8.00E-11 8.10E-11 8.10E-11 3.40E-10
8.00E-11 8.00E-11 5.13E-10 5.13E-10 3.40E-10
3.00 3.00
3.00 3.00
13 25 25 70 70 70 70
30
170 30
30 30 30 30
30 60 60 30
191
Moisture Diffusivity Data Compilation
Table 6.3 Continued Material Dogfish Fish Meal -
Haddock muscle
Halibut muscle Herring Mackerel -
Shark muscle Squid
mantle Swordfish salted Whiting muscle
Fruits Apple Apricot Avocado -
Banana plantain ripe
min
Diffusivity (m2/s) max
8.30E-11 8.30E-11 1.95E-11 1.95E-11 6.00E-11 6.00E-1 1 3.30E-10 5.80E-11 5.80E-11 2.50E-10 1.30E-11 1.30E-11 3.50E-11 3.50E-11 8.70E-11 8.70E-11 8.30E-11 8.30E-11 2.60E-10 3.00E-10 2.60E-10 4.80E-11 4.80E-11 8.20E-11
2.20E-10 2.20E-10 1.89E-09 1.89E-09 3.30E-10 6.00E-11 3.30E-10 2.50E-10 5.80E-11 2.50E-10 1.90E-10 1.90E-10 3.50E-11 3.50E-11 2.85E-10 2.85E-10 1.09E-10 1.09E-10 3.90E-10 3.90E-10 3.30E-10 2.70E-10 4.80E-11 2.70E-10
4.00E-13
Moisture (db)
min
max
Temperature (°C) max min 30 30 65 65 30 30 30 30 30 30 30 30 30 30 20 20 34 34 40 40 40 25 30 25
30 30
0.33 0.33
0.55 0.55
1.18 1.18 0.50 0.50
1.42 1.42 2.50 2.50
6.10E-07
0.00
8.70
15
110
4.00E-12
6.40E-09
8.70
4.00E-12 l.OOE-11 l.OOE-11 1.10E-10 1.10E-10 1.60E-10 1.60E-10 3.16E-10 5.50E-10
6.40E-09 6.10E-07 6.10E-07 1.80E-09 1.80E-09 3.40E-09 3.40E-09 1.15E-09 2.66E-09
0.00 0.00 0.50 0.50
0.25 0.25 0.90 3.00
3.00 3.00 0.90 3.00
20 20 40 40 31 31 25 25 40 60
90 90 80 80 60 60 110 110 70 60
8.70 3.48 3.48
170
170 30 30 30 30 30 30 30 30 30 30 40 40 34 34 55 55 55 30 30 30
Chapter 6
192
Table 6.3 Continued Material
Blueberries Coconut -
Grapes red seedless Mulberry Peach Pineapple Raisins ~
Legumes Broad Bean seeds
Fababean Lentil cotyledons
hilutn seedcoat Navy Beans
min
Diffusivity (mVs) max
3.80E-11
5.10E-10
3.80E-11 4.60E-10 4.60E-10 4.83E-11 4.85E-11 5.80E-11 4.83E-11 2.32E-10 2.32E-10 8.00E-12 8.00E-12 5.38E-10 5.38E-10 4.00E-13 4.00E-13
5.10E-10 1.28E-09 1.28E-09 9.28E-10 4.20E-10 3.00E-10 9.28E-10 2.76E-09 2.76E-09 8.00E-12 8.00E-12 2.64E-09 2.64E-09 4.80E-10 4.80E-10
2.06E-14 3.66E-07 3.66E-07 1.28E-07 1.28E-07 2.06E-14 1.43E-11 4.58E-09 2.06E-14 3.33E-08 3.33E-08
5.48E-12 l.OOE-11
1.17E-05 5.56E-10
5.56E-10 2.50E-10 l.OOE-11
5.56E-10 3.90E-10 l.OOE-11
Moisture
Temperature
(db)
(°C)
min
max
min
max
0.50 0.50 0.60 0.60 0.39
4.00
37
60
4.00 0.60 0.60 2.35
0.39 0.50 0.50
2.35 3.00 3.00
3.80 3.80 0.15 0.15
5.00 5.00 0.60 0.60
37 45 45 30 50 50 30 60 60 30 30 30 30 15 15
60 110 110 75 70 70 75 80 80 30 30 50 50
1.07E-06
0.12
1.00
20
65
1.07E-06 1.07E-06 2.27E-07 2.27E-07 1.02E-08 3.17E-11 1.02E-08 4.03E-12 5.56E-08 5.56E-08
0.17
0.37
20
40
0.17 0.50 0.50 0.12 0.13 0.15 0.12 0.15 0.15
0.37 1.00 1.00 0.24 0.13 0.15 0.24 0.40 0.40
20 30 30 30 30 30 35 35
40 30 30 50 50 50 50 65 65
0.04
2.45
10
180
1.40
1.40
140
1.40
1.40
30 30 140 30
30
70
70
"
Meat Beef meatball
raw
30 140 30
193
Moisture Diffusivity Data Compilation
Table 6.3 Continued Material Beef Carcass bone fat muscle Broiled waste Bull Ground Beef heat treated raw Heifer heat treated raw Pepperoni sausage Sausage Turkey
Model Foods Albumin-Flour-Bran mixture! mixlure2 mixtures Amioca gel hydrated Cellulose-oil-water Corn Starch .
min
Diffusivity (mVs) max
5.48E-12 5.48E-12 3.07E-11 5.83E-10 8.06E-06 8.06E-06 6.30E-11 6.30E-11 3.00E-11 3.00E-11 7.50E-11 4.00E-11 5.40E-11 1.30E-10 5.40E-11 4.70E-11 4.70E-11 4.70E-1 1 1.3 IE-07 1.3 IE-07 8.00E-15 8.00E-15
5.83E-10 5.48E-12 3.07E-11 5.83E-10 1.17E-05 1.17E-05 8.20E-11 8.20E-11 2.30E-10 3.07E-11 2.30E-10 1.70E-10 2.14E-10 2.14E-10 1.20E-10 5.70E-11 5.70E-11 5.70E-11 1.3 IE-07 1.3 IE-07 8.00E-15 8.00E-15
l.OOE-14
2.25E-08
5.80E-10 1.15E-09 8.00E-10 5.80E-10 6.13E-11 6.13E-11 5.50E-10 1.40E-09 3.10E-09 3.10E-09 1.89E-10 1.89E-10
1.85E-09 1.85E-09 1.26E-09 1.05E-09 7.30E-09 7.30E-09 1.20E-09 2.40E-09 3.10E-09 3.10E-09 2.60E-10 2.60E-10
Moisture (db)
Temperature
min
max
min
0.30 0.30 0.76 0.76 0.16 0.16 0.60 0.60 1.00 1.00 1.00 0.19
0.30 0.30 0.76 0.76 1.60 0.16 1.00 1.60 1.00 1.00 1.00 0.19
10
0.19 0.32 0.32 0.04 0.04
0.19 0.32 0.32 0.04 0.04
10 20 20 25 25 30 30 30 30 30 12 12 12 180 180 22 22
0.02 0.08
80.00
20
0.83
0.08 0.13 0.10 0.02 0.02 0.07 0.05
0.82 0.83 0.80 0.93 0.93 0.75 0.75
75 75 75 75 20 20 60 60 68 68 30
30
(°C) max
32 32 30 30 75 25 75 60 75 75 60 12 12 12 180 180 22 22
105 105 105 105 105 100 100 60 60 68 68 30 30
194
Chapter 6
Table 6.3 Continued Material
Flour Glucoose-Starch Gluten-Starch gelatinized ungelatinized Hylon-7 gel hydrated Polyacrylamide Gel Potato Starch -
Rice Starch full heated
non-heated Starch ~
Nuts Almond Hazelnuts shelled unshelled Peanut Pods hull kernel
Peanuts .
min
Diffusivity (m2/s) max
3.86E-12 3.86E-12 6.60E-11 6.60E-11 1.90E-11 2.20E-1 1 1.90E-11 2.48E-10 2.48E-10 8.00E-10 1.70E-09 8.77E-12 8.77E-12 l.OOE-14 l.OOE-14 8.48E-11 8.48E-11 3.92E-10 1.50E-10 1.50E-10
Moisture
(db)
Temperature
min
max
min
(°C) max
3.20E-11 3.20E-11 5.90E-10 5.90E-10 4.00E-11 4.00E-1 1 3.20E-11 2.25E-08 2.25E-08 3.70E-09 2.70E-09 3.00E-10 3.00E-10 2.40E-11 2.40E-1 1 4.62E-09 2.84E-09 4.62E-09 6.90E-10 6.90E-10
0.07 0.07 0.60 0.60 0.09 0.09 0.09 0.03 0.03 0.05 0.05 0.10 0.10 0.80 0.80 0.67 0.67 0.67 0.60 0.60
0.17 0.17 0.60 0.60 0.20 0.20 0.20 1.00 0.90 1.00 0.75 2.00 2.00 80.00 80.00 4.00 4.00 2.33 0.60 0.60
25 25 30 30 74 74 74 20 20 60 60 40 40 25 25 25 25 25 30 30
25 25 50 50 74 74 74 100 100 60 60 40 40 25 25 80 80 80 50 50
4.36E-13
1.24E-08
27
281
2.39E-12 2.39E-12 1.24E-08 8.39E-09 1.24E-08 2.36E-10 1.17E-10 2.36E-10 4.00E-11 4.00E-11
0.05 0.05
0.60
2.24E-12 2.24E-12 1.48E-09 1.48E-09 3.65E-09 4.36E-13 4.36E-13 8.52E-12 4.00E-11 4.00E-11
281 281 30 30 30 27 27 27 50 50
281 281 80 80 80 43 43 43 50 50
0.05 0.15 0.15 0.15 0.60 0.60 0.60 0.10 0.10
0.05 0.05 0.15 0.15 0.15 0.60 0.60 0.60 0.10 0.10
Moisture Diffusivity Data Compilation
195
Table 6.3 Continued Diffusivity
Moisture
(mVs) Material
(db)
min
max
min
max
0.00 0.00
2.00
0.00 0.00 2.00 2.00
0.05 0.05 2.00 2.00
8.20E-14
6.60E-09
Canola
2.10E-09
6.60E-09
embryo kernel
2.10E-09 3.70E-09 8.20E-14 8.20E-14 5.00E-11 5.00E-11 1.03E-11 1.03E-11 1.39E-11 7.00E-11 1.70E-10 7.00E-1 1 2.85E-11 2.85E-11
5.20E-09 6.60E-09 1.33E-13 1.33E-13 1.65E-10 1.65E-10 2.03E-11 2.03E-11 1.90E-11 7.10E-10 7.10E-10 1.70E-10 2.08E-10 2.08E-10
0.80 1.00 0.80 0.07 0.07 0.07 0.08 0.08
2.20E-12
3.05E-07
0.03
1.50E-09
1.50E-09
1.50E-09 1.78E-10 1.78E-10 2.20E-12 2.20E-12 1.07E-10 1.07E-10 1.14E-11 1.14E-11 4.28E-10 4.28E-10 1.38E-11 1.38E-11 5.80E-11 5.80E-11 1.10E-10 1.10E-10
1.50E-09 3.41E-09 3.41E-09 7.46E-09 7.46E-09 2.15E-09 2.15E-09 4.18E-10 4.18E-10 6.80E-09 6.80E-09 6.60E-09 6.60E-09 4.08E-10 4.08E-10 4.40E-10 4.40E-10
Other
Chocolate dark Coffee extract Egg fresh incubated Sunflower Seeds hull kernel Toria seeds
Vegetables Beet Broccoli stems Carrot -
Cassava roots Garlic Okra Onion -
Paprika Pea .
0.05
1.00 1.00 0.80 0.07 0.07 0.07 0.12 0.12
15.00
13.93
2.00 2.00 0.10 0.10 0.60 0.60 0.10 0.10 2.00 2.00 0.10 0.10
13.93 15.00 15.00 0.64 0.64 1.50 1.50 2.00 2.00 10.00 10.00
0.50 0.50
1.50 1.50
Temperature (°C) max min 20 80
80 80
80 80 20 20 30 30 33 33 33 40 40 40 50 50
80 80 20 20 70 40 40 40 50 50 50 80 80
20
300
65
65
65 25 25 20 20 55 55 22 22 60 60 40 40 25 25 30 30
65 90 90 100 100 90 90 90 90 80 80 80 80 70 70 65 65
70
Chapter 6
196
Table 6.3 Continued Moisture (db)
Diffusivity Material Pepper redpowder Pigeon Pea kernel Potato restructured product tissue
Soya Meal Soybean grains
Sugar Beet roots Tapioca roots Tomato concentrate droplets Turnip Yam .
min 5.86E-11 1.85E-09 5.86E-11 2.88E-11 2.88E-11 8.00E-12 8.00E-12 7.30E-10 1.67E-09 1.47E-10 1.47E-10 9.30E-11 9.30E-11 9.30E-11 1.96E-10 1.96E-10 3.50E-10 3.50E-10 1.52E-10 1.52E-10 1.69E-10 7.61E-12 7.61E-12 7.30E-10 7.30E-10
(m2/s) max
1.16E-08 1.16E-08 4.08E-10 7.70E-11
7.70E-11 1.25E-08 1.25E-08 4.52E-09 1.67E-09 4.01E-08 4.01E-08 1.17E-05 3.05E-07 2.15E-09 1.30E-09 1.30E-09 9.00E-10 9.00E-10 6.46E-09 2.36E-09 6.46E-09 3.62E-09 3.62E-09 1.81E-09 1.81E-09
Temperature
CO
min
max
min
max
0.06 0.10 0.06 0.20 0.20 0.10 0.10
10.00 10.00 0.06 0.20 0.20 10.00 10.00
1.85 0.03 0.03 0.19 0.19 0.20 2.21 2.21 1.05 1.05 0.50 10.00 0.50 0.31 0.31 0.06 0.06
1.85 0.22 0.22 1.00 1.00 1.00 3.00 3.00 1.05 1.05 10.00 10.00 0.50 7.00 7.00 0.14 0.14
25 60 25 50 50 24 24 105 65 65 65 20 20 25 40 40 55 55 40 40 60 20 20 40 40
80 80 70 90 90 185 185 105 65 300 300 95 95 72 81 81 97 97 100 80 100 100 100 50 50
Moisture Diffusivity Data Compilation
197
III. MOISTURE DIFFUSIVITY OF FOODS AS A FUNCTION OF MOISTURE CONTENT AND TEMPERATURE
Moisture diffusivity of foods depends strongly on moisture, temperature and structure of the material. In porous materials, the porosity and the pore structure distribution significantly affect diffusivity. Limited information concerning the effect of structure on diffusivity is available in the literature (Chapter 5). On the other hand, the effect of moisture and temperature on diffusivity has been studied more extensively. Nevertheless, general models describing the effect of moisture content and temperature on diffusivity of foods do not exist. A large number of empirical equations are summarized and analyzed in detail by Marinos-Kouris and Maroulis (1995), Zogzas et al. (1996), and Mittal (1999). According to the most popular consideration, the effect of temperature and moisture content is introduced into the Arrhenius model. A concept proposed by Maroulis et al. (2001) is adopted here and applied to obtain an integrated and uniform analysis of the available moisture diffusivity data. The concept was applied simultaneously to all the data of each material, regardless of data sources. Thus, the results are not based on the data of only one author, and consequently they are of elevated accuracy. Assume that a material of intermediate moisture content consists of a uniform mixture of two different materials: (a) a dried material and (b) a wet material with infinite moisture. Moisture diffusivity is, generally, different for each material. The diffusivity of the mixture could be estimated using a two-phase structural model:
l +X
^
"
l +X
(6-1)
where D (m2/s) the effective moisture diffusivity, DXo (m2/s) the moisture diffusivity of the dried material (phase a), DXj (m2/s) the moisture diffusivity of the wet material (phase b), X (kg/kg db) the material moisture content, and T (°C) the material temperature. Assume that the diffusivities of both phases depend on temperature by an Arrhenius-type model: = u exp - — R(T
T
(6-2)
198
Chapters
x,
, -r
R
,
T
T
(6-3)
where Tr= 60°C a reference temperature, R = 0.0083143 kJ/mol K, the ideal gas constant, and D0,DI,E0,EI are adjustable parameters of the proposed model. The reference temperature of 60°C was chosen as a typical temperature of air-drying of foods. Thus, the moisture diffusivity for every material is characterized and described by four parameters with physical meaning:
• • • •
D0 (m2/s) Dj (m2/s) E0 (kJ/mol) Ei (kJ/mol)
diffusivity at moisture X = 0 and temperature T = Tr diffusivity at moisture X = oo and temperature T - Tr activation energy for diffusion in dry material at X = 0 activation energy for diffusion in wet material at X = oo
The resulting model is summarized in Table 6.4 and can be fitted to data using a nonlinear regression analysis method. The model is fitted to all literature data for each material and the estimates of the model parameters are obtained. Then the
residuals are examined and the data with large residuals are rejected. The procedure is repeated until an acceptable standard deviation between experimental and calculated values is obtained (Draper and Smith, 1981). Among the available data only 19 materials have more than 10 data, which come from more than 3 publications. The procedure is applied to these data and the results of parameter estimation are presented in Table 6.5 and in Figure 6.6. It is clear that moisture diffusivity is larger in wet materials.
Moisture Diffusivity Data Compilation
199
Figures 6.7-6.29 present retrieved moisture diffusivities from the literature and model-calculated values for selected food materials as a function of moisture content and temperature. Moisture diffusivity D tends to increase with the moisture content X and the temperature T. The maxima of the curves observed at low moisture contents in the model foods (see Chapter 5) are smoothed out by the statistical treatment of the data. It must be noted that the regression procedure was applied simultaneously to all the data of each material, regardless of the data sources. Thus, the results are not based on the data of only one author and consequently they are of higher accuracy and general applicability. The diffusivity parameters D0 and Z)/ of the proposed model, shown in Figure 6.6, vary in the range of 10~10 to 10~8 m2/s. It should be noted that the selfdiffusivity of water is approximately 10"9 m2/s, and the moisture diffusivity in bone-dry food material should be lower (in our analysis, by a factor of 100). Low moisture diffusivities are found in nonporous and sugar-containing foods, while higher values of moisture diffusivity characterize porous food materials. Diffusivities higher than the self-diffusivity of water are indicative of vapor diffusion in porous solids. The moisture diffusivity increases, in general, with increasing moisture content. Temperature has a positive effect, which depends strongly on the food material. The energy of activation for diffusion E of water is, in general, higher in the dry food materials. Some observed exceptions may be explained by the prevailing type of diffusion. Thus, lower values of activation energy for diffusion are expected for porous foods, where vapor diffusion is important. In general, temperature has a stronger effect on diffusivity in the liquids and solids than in the gas state (see Chapter 2).
Chapter 6
200
Table 6.4 Mathematical Model for Calculating Moisture Diffusivity in Foods as a Function of Moisture Content and Temperature
Proposed Mathematical Model X
£> = •
Do exp
where
D (m2/s) X (kg/kg db) r(°C) Tr = 60°C R = 0.0083143 kJ/mol K
l +X
RT T
\+x
t
exp
RT
T
the moisture diffusivity, the material moisture content, the material temperature, a reference temperature, and the ideal gas constant.
Adjustable Model Parameters
• • • •
D0 (m2/s) D, (m2/s) E0 (kJ/mol) EI (kJ/mol)
diffusivity at moisture X = 0 and temperature T = Tr diffusivity at moisture X = oo and temperature T = T. activation energy for diffusion in dry material at X = 0 activation energy for diffusion in wet material at X = oo
Moisture Diffusivity Data Compilation
201
Table 6.5 Parameter Estimates of the Proposed Mathematical Model Material
No. of No. of Papers Data
Di (m2/s)
Do Ei Eo (mVs) (kJ/mol) (kJ/mol)
sd (m2/s)
Cereal products Corn dent grains kernel pericarp
4 3 3 4 3
26 15 28 25 13
4.40E-09 1.19E-08 1.15E-09 5.87E-10 1.13E-09
O.OOE+00 O.OOE+00 6.66E-11 5.32E-10 O.OOE+00
0.0 49.4 10.2 0.0 10.0
10.4 73.1 57.8 33,8 5.0
1.48E-10 3.30E-10 3.17E-10 1.88E-11 2.34E-11
Pasta -
3
21
1.39E-09
O.OOE+00
16.2
2.0
7.71E-12
3
12
9.75E-09
O.OOE+00
12.5
2.0
5.52E-11
7
35
2.27E-09
O.OOE+00
12.7
0.7
3.66E-11
6
22
1.94E-09
1.30E-09
0.0
46.3
9.53E-11
8
39
7.97E-10
1.16E-10
56.6
1.92E-10
4
34
2.03E-09
4.66E-10
9.9
4.6
1.77E-10
3
32
5.35E-09
O.OOE+00
34.0
10.4
1.45E-10
3
10
8.11E-10
1.05E-10
21.4
50.1
6.88E-11
4
49
1.52E-08
1.52E-08
0.0
33.3
1.02E-09
5
48
1.96E-08
1.96E-08
0.0
24.2
3.87E-09
9
90
2.47E-09
1.54E-09
13.9
11.3
1.69E-09
4
22
5.33E-10
1.68E-11
15.4
7.1
7.43E-11
4
31
1.45E-08
O.OOE+00
70.2
10.4
1.58E-09
16
106
1.57E-09
4.31E-10
44.7
76.9
4.02E-10
Rice kernel Rough rice Wheat -
Fruits Apple Banana Grapes seedless Raisins -
16.7
Model foods Amioca Hvlon-7 -
Vegetables Carrot
Garlic Onion Potato -
Chapter 6
202
l.E-06
• Moisture - infinite l.E-07
Q Moisture - zero
S" l.E-08 S,
•I" l.E-09 IS l.E-10 l.E-11 l.E-12
8 1 11 £
o-
3 |
JJ
T S .a a a = t •= •! a i a o o ,2
«
100
• Moisture = infinite 13 Moisture = zero
o
E i-: ^ >.
£*
I .
* o
Figure 6.6 Parameter estimates of the proposed mathematical model.
I 1 o S.
Moisture Diffusivity Data Compilation
l.E-06
l.E-07
Moisture (kg/kg db)
Figure 6.7 Predicted values of moisture diffusivity of model foods at 25°C.
203
204
Chapter 6
.E-06
H
Model foods :ratui•e (°C) = 60
.E-07
|——
-f —
l.E-08
Hylon-7
—
.|" l.E-09
Amioca
— i
1
l.E-10
.E-ll
l.E-12
0.1
1 Moisture (kg/kg db)
Figure 6.8 Predicted values of moisture diffusivity of model foods at 60°C.
10
Moisture Diffusivity Data Compilation
l.E-06
l.E-07
l.E-08
f l.E-09 4
l.E-10
l.E-1
l.E-12
Moisture (kg/kg db)
Figure 6.9 Predicted values of moisture diffusivity of fruits at 25°C.
205
206
Chapter 6
l.E-06
Temperature (°C) = 60 -4
l.E-07
Moisture (kg/kg db)
Figure 6.10 Predicted values of moisture diffusivity of fruits at 60°C.
Moisture Diffusivity Data Compilation
207
l.E-06 1
H
Vegetables
Tempera ture i °C) = 25 l.E-07
l.E-08
l.E-12 10
0.1
Moisture (kg/kg db)
Figure 6.11 Predicted values of moisture diffusivity of vegetables at 25°C.
Chapter 6
208
l.E-06
l.E-07
l.E-12 0.1
1 Moisture (kg/kg db)
Figure 6.12 Predicted values of moisture diffusivity of vegetables at 60°C.
10
Moisture Diffusivity Data Compilation
209
l.E-06
l.E-07
l.E-12
0.1
10
Moisture (kg/kg db)
Figure 6.13 Predicted values of moisture diffusivity of corn at 25°C.
Chapter 6
210
l.E-06
-
h-
Jtr 60 f
——|Cere al products (corn)
r Temperature (°C) =
.E-07
l.E-12 0.1
Moisture (kg/kg db)
Figure 6.14 Predicted values of moisture diffusivity of corn at 60°C.
Moisture Diffusivity Data Compilation
211
Garlfc.E-06 real products ——————— | C(
t~r
Temperature (°C) = 25 i
i
.E-07 -
-
——
.E-08 -
.E-09 -
— Rice kernel Corn
^^ •"
.E-10 -j
r^j i Vheat •"^^ j
Lx^' Rough rice
i
!
'
x
X ^|x.E-ll -——^^^r —————— F asta ^^^
Pact n
F.n a 0.1
1
10
Moisture (kg/kg db)
Figure 6.15 Predicted values of moisture diffusivity of cereal products at 25°C.
Chapter 6
212
l.E-06 Cereal products 1^«™-™-™M™«»™I»™
Temperature (°C) = 60
.E-07
l.E-08
£ £ l.E-09
' Rice kernel
l.E-10 -I
wheat
Rough rice Pasta l.E-11
l.E-12 0.1
1
10
Moisture (kg/kg db)
Figure 6.16 Predicted values of moisture diffusivity of cereal products at 60°C.
Moisture Diffusivity Data Compilation
Fruits
213 Apple
Total Number of Papers Total Experimental Points
8 64
Points Used in Regression Analysis Standard Deviation (sd, rti'/s)
36
Relative Standard Deviation (rsd, %) Parameter Estimates Di (m2/s) Do (mj/s) Ei (kJ/mol) Eo (kJ/mol)
(56%)
1.92E-10
457 7.97E-10 1.16E-10 16.7 56.6
.E-06
Temperature (°C) — 140
— «60 l.E-07
A 80
l.E-08
•I" l.E-09
l.E-10
l.E-11
l.E-12 0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.17 Moisture diffusivity of apple at various temperatures and moisture contents.
214
Chapter 6
Fruits
Grapes
Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, rrrVs) Relative Standard Deviation (rsd, %)
3 32 20 1.45E-10 1 31
seedless (63%)
Parameter Estimates Di(m 2 /s) Do (m2/s) Ei (kJ/mol) Eo(kJAnol)
5.35E-10 O.OOE+00 34.0 10.4
1 F-Ofi ————————————————— ;——————
—
— Tem perat ure (°C)
l.E-07 -
l.E-08 -
• 40 • 60 A 80
————— ———————— ————————
—
_L
1 ——————————
h-
•f l.E-09 - —————————— —————— 1
-1
^>•
1 i
l.E-10 1
^^
=^ ~~~~ •* ^ !• •• * 1 -_ =1IE I J r^ ~*—» ———
^^
i
^
01
'-
l.E-11 -
l.E-12 ———————————————————————————————————— 0.1 1.0 10.0 Moisture (kg/kg db)
Figure 6.18 Moisture diffusivity of grapes at various temperatures and moisture contents.
215
Moisture Diffusivity Data Compilation
Fruits
Banana
Total Number of Papers
4
Total Experimental Points
49
Points Used in Regression Analysis Standard Deviation (sd, rrvVs) Relative Standard Deviation (rsd, %) Parameter Estimates Di (m'Vs) Do(m'Vs) Ei (kJ/mol)
15 1.77E-10 15
(31%)
2.03E-09 4.66E-10 9.9
Eo (kJ/mol)
4.6
l.F-06 ——————————— —— Tern perature (°C) — ' • 60 -4-
l.E-07 -
A 80 ———
——— ———
(i Sm.
m
MD
o
^
h=Hs_^M
^s. BH
Si
•
|—
-1——————f ——— I " 0
hn
^-
Diffusivity (m2/s)
l.E-08 -
* —— ———
l.E-11 -
l.E-12 01
1.0
10.0
Moisture (kg/kg db)
Figure 6.19 Moisture diffusivity of banana at various temperatures and moisture contents.
Chapter 6
216
Vegetables
Potato
Total Number of Papers Total Experimental Points Points Used in Regression Analysis
13 148 66
(45%)
Standard Deviation (sd, m2/s) 4.02E-10 Relative Standard Deviation (rsd, %)______122 Parameter Estimates Di (m2/s) 1.57E-09 Do (m2/s) 4.31E-10 Ei (kJ/mol) 44.7 Eo (kJ/mol) 76.9 l.E-06
Temperature ( C) • 40 • 60 A SO
l.E-07
l.E-12 0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.20 Moisture difrusivity of potato at various temperatures and moisture contents.
Moisture Diffusivity Data Compilation
217
Vegetables Total Number of Papers
Carrot 12
Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m2/s)
106 98 1.69E-09
(92%)
Relative Standard Deviation (rsd, %)_____18699 Parameter Estimates Di (m"/s) 2.47E-09 1.54E-09 Do(rrrVs)
Ei (kJ/mol) Eo(kJ/mol)
13.9 11.3
I.F-Ofi
l.E-07
l.E-08
•
l.E-09
l.E-10
l.E-11
l.E-12 0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.21 Moisture diffusivity of carrot at various temperatures and moisture contents.
Chapter 6
218
Vegetables
Points Used in Regression Analysis Standard Deviation (sd, m'Vs) Relative Standard Deviation (rsd, %)
Onion 4 31 22 1.58E-09 575
Parameter Estimates Di (m"/s) Do (mVs) Ei (kJ/mol) Eo (kJ/mol)
1.45E-09 O.OOE+00 70.2 10.4
Total Number of Papers
Total Experimental Points
(71%)
1.E-06
l.E0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.22 Moisture diffusivity of onion at various temperatures and moisture contents.
Moisture Diffusivity Data Compilation
219
Vegetables
Garlic
Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m'Vs) Relative Standard Deviation (rsd, %) Parameter Estimates Di (m"/s) Do(nWs) Ei(kJ/moI) Eo(kJ/mol)
4 22 19 7.43E-1 1 385
(86%)
5.33E-10 1.68E-11 15.4 7.1
l.E-06 -
—j —
— Tern perat ure(°C) = • 40 • 60 A 80
l.E-07 -
re =P
l.E-08 W5
"E, •f l.E-09 -
la 5
rr*~m 3 i~~*
«• ~ *•? *—• ^ -} «• l.E-10 , •if* •* 1 *~\fff* ^*^ —— ———— •—————
^
l.E-11 -
l.E-12 0.1
1 1
—— — — - - - - -
1.0
10.0
Moisture (kg/kg db)
Figure 6.23 Moisture diffusivity of garlic at various temperatures and moisture
contents.
Chapter 6
220
Cereal Products
Wheat
Total Number of Papers Total Experimental Points Points Used in Regression Analysis
Standard Deviation (sd, m2/s)
5 26 15
(58%)
9.53E-1 1
Relative Standard Deviation (rsd, %)
54
Parameter Estimates Di(m"/s) Do(mVs)
Ei (kJ/mol) Eo (kJ/mol)
1.94E-10 1.30E-10
0.0 46.3
1 ,F,-06 i————— i—————
)
!
———— B40 ———
• 60 A 80 ———
l.E-07 -
j
tfi
i
i
l.E-08 1
\
i
•1" l.E-09 V)
———4, ———*-• k
S
A I\
-
\
»
l.E-10 -
^fc —f— J 1 — •1
l.E-11 -
1 1
1
———— I EE^ 1——
l.E-12 0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.24 Moisture diffusivity of wheat at various temperatures and moisture contents.
Moisture Diffusivity Data Compilation
Cereal Products
221
Corn
Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, rrrVs) Relative Standard Deviation (rsd, %)
dent
3 15 15 3.30E-10 343
(100%)
Parameter Estimates Di(rrrVs) Do (m"/s) Ei (kJ/mol)
1.19E-09 O.OOE+00 49.4
Eo(kJ/mol)
73.1
l.E-06 peral ure CC) • 40 • 60 A80
l.E-07 -
i l.E-08 5" =
+* _>1
^^
\
•f l.E-09 -- .^T —* M **?—-———— < k— ^* \±* a L****T l.E-10
J
i
•^M
;\^^j^^
i 1 1
—« t11
l.E-11 -
0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.25 Moisture diffusivity of corn (dent) at various temperatures and moisture contents.
Chapter 6
222
Cereal Products
Corn
Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m'Vs) Relative Standard Deviation (rsd, %)
grains
3 28 26 3 . 1 7E- 1 0 1 53
(93%)
Parameter Estimates Di(m
1.15E-09 6.66E-11 10.2 57.8
l.F-06 —————————————————————————————————— ,—
u
T A m n o r n + i i f o ("f~
)
^
-i
———— «40 ——— - • 60 —————
l.E-07 -
A 80 —p-
l.E-08 k— •—
"E
1
•f l.E-09 1
r^
^•^
li
I
Jj
1 l.E-10 -
\
• "*
|
=>
1
3
l.E-11 T
0.1
1.0
10.0
Moisture (kg/kg db)
Figure 6.26 Moisture diffiisivity of corn (grains) at various temperatures and moisture contents.
Moisture Diffusivity Data Compilation
Cereal Products
223
Corn
Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, nWs) Relative Standard Deviation (rsd, %)
Parameter Estimates Di(m'Vs) Do(m'Vs) Ei (kJ/mol) Eo (kJ/mol)
kernel
3 25 21 1.88E-1 1 32
(84%)
5.87E-11 5.32E-11 0.0 33.8
I P-flfi ————————————————— r-i————————————————————————
tE
CO =E ~ q ———— B40 —
• 60 ———— A 80 —
l.E-07 -
l.E-08 -
•f l.E-09 -
ia
4 l.E-10 - -f-
^^It 4=^
=^1 • J l.E-11 -
l.E-12 0.1
_•
1.0
10.0
Moisture (kg/kg db)
Figure 6.27 Moisture diffusivity of corn (kernel) at various temperatures and moisture contents.
Chapter 6
224
Cereal Products
Corn
Total Number of Papers
3
Total Experimental Points Points Used in Regression Analysis
13 12
Standard Deviation (sd, nTVs)
pericarp (92%)
2.34E-1 1
Relative Standard Deviation (rsd, %)
7558
Parameter Estimates Di(m 2 /s) Do (nrVs) Ei (kJ/mol) Eo (kJ/mol)
r
l.E-06 -
1.13E-10 O.OOE+00 10.0 5.0
=
i
^
—— Temperat ure(°C) =g= q - ———— «40 • 60 -p——— A 80 —
l.E-07 -
i
l.E-08 :
ff> (S
•f l.E-09 -
•a
1
'
Q
l.E-10 -
i————— •
——i l.E-11 '| ^
*
i
l.E-12 01
1.0
10.0
Moisture (kg/kg db)
Figure 6.28 Moisture diffusivity of corn (pericarp) at various temperatures and moisture contents.
225
Moisture Diffusivity Data Compilation
Cereal Products Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m2/s) Relative Standard Deviation (rsd, %)
Pasta 3 21 17 7.71E-12 36
(81%)
Parameter Estimates Di(m 2 /s)
1.39E-10
Do(m 2 /s)
-1.6 IE-21
Ei(kJ/mol) Eo (kJ/mol)
16.2 2.0
i F-Ofi ——————— ~ 1_ f
—— Tern perat ure(°C) =t4= • 40 • 60 A 80
l.E-07 -
_ . ^rj^
l.E-08 Sfl
ts
•f l.E-09 '&
3 l.E-10 -jfr-
^**^&k *—
=^TJ
l.E-ll ^^ ' •————
l.E-12 01
1.0
10.0
Moisture (kg/kg db)
Figure 6.29 Moisture diffusivity of pasta at various temperatures and moisture
contents.
226
Chapter 6
REFERENCES
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Diffusivity and Permeability of Small Solutes in Food Systems
I. INTRODUCTION
The diffusion of small molecules in food and food packaging materials is important in food processing operations, food product development, and food process control. It involves the transport of small molecules, such as sugars, organic acids, flavor components, salts, preservatives, and gases (e.g. oxygen, carbon dioxide). In addition, the transport of larger molecules, such as lipids, is of interest to food processing, such as oil extraction. The transport of solutes is of fundamental importance to the physical separation processes, such as solvent extraction (e.g. sugar/water), ion exchange (e.g. de-acidification and de-bittering of juices), reverse osmosis, and ultrafiltration. Mass diffusivity is the basic component of permeability of packaging films and food coatings, which are used as barriers to water, oxygen, and carbon dioxide transport in food materials. Theoretical prediction of solute diffusivity in solid and semisolid food materials is not feasible, and experimental measurements and data are necessary. The experimental methods and theoretical analysis of water transport in food materials (Chapter 5) are utilized in the diffusion of small solutes in food systems. A. Diffusivity of Small Solutes
Mass transfer of small molecules can be analyzed by either the diffusion (Pick) model or the mass transfer coefficient concept (Cussler, 1997). The diffusion model, in general, is used in modeling mass transport of water and small solutes within foods, since solid and semisolid food materials have complex structures that strongly effect mass diffusivity. Mass transfer coefficients are used in 237
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designing processing equipment where mass is transported between phases, as in drying and separation processes, such as extraction and membrane processing. The transport of small solutes in simple gases and liquids can be predicted by molecular dynamics (see Chapter 2), and reliable data on mass diffusivity is available in the literature. Prediction of diffusivity is difficult in complex fluid foods and in solid/semisolid foods, where experimental measurements and empirical correlations are essential. Although small solutes can be transported in food systems by different mechanisms, molecular diffusion is generally accepted as the basic transport process, in a similar manner with the transport of water (see Chapter 5). Thus, the effective diffusivity, an overall transport coefficient, can be defined, assuming that the driving force is a concentration gradient (dC/dz), and applying the diffusion (Pick) equation: (8C/&) = (d/dz) [D(cC/&)}
(7-1)
Table 7.1 shows some typical values of diffusivity in gases, liquids, and solids (see also Table 2-4). The diffusivity of solutes in fluid foods is of the order of 1x10"9 m2/s, while in solid and semisolid foods the diffusivity varies widely in the range 10'14 to 10~8 m2/s, due to heterogeneous structure, which involves diffusion in gas and liquid phases. Diffusion in polymers varies from 10"18 to 10'10 m2/s, due to different structures (rubbery and glassy states). Table 7.1. Typical Diffusivities of Small Solutes Diffusivity in
D, m / s
Gases Liquids Polymers (rubbery) Polymers (glassy)
IxlO' 5 IxlO' 9 IxlO' 14 - IxlO' 10 IxlO' 18 - IxlO' 12
Diffusivity and Permeability of Small Solutes in Food Systems
239
B. Measurement of Diffusivity
The measurement of diffusivities of small molecules in solid food materials is discussed in Chapter 5, in connection with the transport of water (moisture). Some of these methods are also used for the determination of the diffusivities of small solutes in polymers and food solids, especially the sorption kinetics, the permeability and the concentration-distribution techniques. Additionally, two more experimental methods, used for the measurement of diffusivities of solutes in gases, liquids, and membranes, are important for mass transfer in food systems, i.e. the diaphragm cell and the Taylor dispersion method. 1. The Diaphragm Cell The Stokes diaphragm cell is used for the measurement of the diffusion coefficients in gases, liquids, and across membranes, with an accuracy of up to 0.2% (Cussler, 1997). The method is based on the diffusion of the solute between two compartments separated by either a fritted glass surface or by a porous membrane, as shown diagrammatically in Figure 7.1. The two compartments are kept at different but constant solute concentrations, using magnetic stirring. The fritted glass diaphragm may be replaced by a piece of filter paper for faster measurement.
A
magnetic
stirrer glass "frit stirring
B
bars
Figure 7.1 Stokes diaphragm cell.
240
Chapter 7
After a certain time t of diffusion, the contents of the two compartments are analyzed for the concentration of the solute, and the diffusivity D is estimated from the equation:
D = (I/ft) [(C, o - C2 o)/(Cj - C2)]
(7-2)
where C/o, C?ft C/ C2 are, respectively, the initial and final solute concentrations in compartments 1 and 2, and jBis the calibration constant, related to the dimensions of the cell:
P=(Alt)(\IV,+ \/V2)
(7-3)
where A is the area available for diffusion, / is effective thickness of the diaphragm, and F; and V2 are the volumes of the two cell compartments. The two compartments are placed vertically, so that the diaphragm surface for diffusion is horizontal. Usually, 1 and 2 indicate the bottom and top cell compartments, respectively. The horizontal position of the diaphragm is necessary to assure uniform concentration gradient and prevent free convection, which might develop in a vertical or inclined position. The accuracy of the method depends on the accuracy of determination of the concentration differences between the compartments, and not on the concentrations themselves.
2. The Taylor Dispersion Method The Taylor dispersion method, used for both gases and liquids, is based on the dispersion by diffusion of a solute injected in a stream of a carrying fluid (Cussler, 1997). A sharp pulse on the solute is injected into a fluid moving in laminar flow in a long tube. The flow exiting the tube is analyzed for the solute (by differential refractometry) over a period of time and the concentration profile is determined. The diffusivity D is estimated from the concentration profile C at time t, representing the decay of a pulse, using the equation: C = (Ml nr ') exp[-(z - u t)2 / 4 E t] I (4 nEi )1/2
(7-4)
where Mis the total solute injected, r is the tube radius, z is the diffusion distance, and £ is a dispersion coefficient given by the equation: E = (u r)21 (48 D)
(7-5)
where u is the average velocity of the flowing solvent. Since the refractive index is a linear function of concentration, the refractive index profile can be used for the determination of the diffusivity.
Diffusivity and Permeability of Small Solutes in Food Systems
241
Very high accuracies in the determination of diffusivity of solutes in fluids can be obtained by interferometers, such as the Gouy, the Mach-Zehnder, and the Rayleigh instruments. The interferometers are based on measuring an unsteadystate profile of the refractive index of two solutions in a transparent system, diffusing into each other.
II. DIFFUSIVITY IN FLUID FOODS
The diffusivity of solutes in dilute aqueous solutions is of importance to food systems, since most food components are present in foods at low concentrations (infinitely dilute solutions). A. Dilute Solutions
Table 7.2 shows some typical diffusivities of solute gases in dilute aqueous solutions (Cussler, 1997) (see also Table 2-4). The diffusivity of water and oxygen in dilute ethanol solutions at 25°C is 1.24 x 1Q"9 and 2.64 x 10~9m2/s, respectively. Table 7.3 shows some typical diffusivities of solutes in dilute water solutions, which are of interest to food systems (Cussler, 1997; Schwartzberg and Chao, 1982). The diffusivity of low-molecular weight solutes is in the same range with the self-diffusivity of water (1 x 1Q"9 m2/s). The diffusivity, in general, decreases as the molecular size of the solute is increased. High-molecular weight food components, such as proteins and polysaccharides, have diffusivities close to that of water in a solid starch/sugar gel (see Chapter 5). Table 7.2 Diffusivities of Gases in Dilute Water Solutions at 25°C Solute________________Ax 10'9m2/s Air 2.00 Oxygen
2.10
Nitrogen
1.90
Chlorine
1.25
Carbon dioxide
1.90
Ethylene
1.87
Hydrogen
4.50
Methane Ammonia
1.49 1.64
242
Chapter 7
Table 7.3 Diffusivities of Solutes in Dilute Water Solutions at 25°C Solute
D, x 10-'°m2/s
Ethanol
8.40
Acetic acid
12.1
Butyric acid Glycine Sucrose Glucose/fructose Maltose Glycerol Hemoglobin Fibrinogen Lactoglobulin Ovalbumin
9.20 10.6 5.40
6.90 4.80 9.20 0.69 0.20 0.70 0.78
The diffusivity of solutes in dilute water solutions can be predicted by empirical equations based on molecular dynamics and hydrodynamics, like the Wilke-Chang equation (2-34) and the Stokes-Einstein equation (7-6):
D = (kBr>/(6xtiBr)
(7-6)
where r is the particle radius, rjB is the viscosity of the solvent (water), T is the absolute temperature, and kB = 1.38xlO"22 J/molecule K is the Boltzmann constant. The Stokes-Einstein equation is based on hydrodynamic and not molecular forces, and it is applicable to solutes of molecular size five times larger than the solvent. For smaller molecules, the Wilke-Chang equation gives better prediction (Cussler, 1997). In both equations, the diffusivity is inversely proportional to the viscosity of the solution. In very viscous solutions, the diffusivity becomes independent of viscosity, e.g. the D of sugar in a gel is nearly equal to the D in water. B. Concentrated Solutions
The diffusivity of solutes in liquids D varies considerably with the concentration, sometimes with maximum or minimum values at certain concentrations. The D can be estimated from the diffusivity at infinite dilution Dm using a correction factor to account for the effect of chemical activity on the transport rate (Reid etal., 1987; Cussler, 1997):
D = D0(l+dlna/dlnC)
(7-7)
Diffusivity and Permeability of Small Solutes in Food Systems
243
where a is the activity and C is the concentration of the solute in the solution. The diffiisivity of the mixture at infinite dilution D0 can be estimated from the diffusivities at infinite dilution of the solute and the solvent, and the corresponding mole fractions (x/ and *?):
A, = [A,(x,= i)]MZ) 0 fe=im
(7-8
The correction factor (d lnor/9 InQ represents the molecular and hydrodynamic interactions in the concentrated solution, and it is negative in nonideal solutions (Cussler, 1997). Thus, D of the solute in a mixture becomes lower than D0 at both extreme concentrations (xlt x 2 = 1), with a minimum at an intermediate concentration.
III. DIFFUSION IN POLYMERS
The sorption and transport of small molecules (solutes) in polymeric materials are the basic physical phenomena of several important applications, such as separation processes, barrier films, and controlled release. Most of the research and theory in this area concerns synthetic polymers of known composition and structure, but the available knowledge can be applied to natural polymers, which are the basic structural components of most food materials. Molecular (Fickian) diffusion is assumed as the main mass transport mechanism, although in some cases other mechanisms may be involved. Solution of the diffusion equation (7-1) forms the basis of mathematical analysis of the experimental data. Most of the diffusivity data of solutes in polymers have been obtained using the sorption and/or the permeability methods (Chapter 5). The physical and transport properties of polymers are affected strongly by the size and shape (linear, branched, cross-linked) of the molecules (van Krevelen, 1990; Bicerano, 1996). Polymer materials can change their size (molecular weight) and microstructure during processing, changing their thermodynamic and transport properties, such as phase equilibria and diffusion coefficients. These changes should be considered in modeling and simulations of industrial processing and applications of polymers (Bokis et al., 1999). The polymer structure is defined by the chemical constitution, set by synthesis (or biosynthesis) and the morphology (microstructure), set by processing (Theodorou, 1996). Quantitative relations can be established between polymer structure and transport properties (diffusivity, permeability), based mainly on experimental measurements and phenomenological correlations from various systems (Petropoulos, 1994). Theoretical predictions and computer simulations, based on molecular science, are still at the development stage, and they could find useful applications in the future.
244
Chapter 7
A. Diffusivity of Small Solutes in Polymers The transport of small solutes (penetrants) normally obeys the Pick diffusion equation, and an effective diffusivity D can be estimated, assuming that the driving force is the concentration gradient. The Fickian diffusion is applicable to low concentrations (infinite dilution) of the solute, which is the case of most applications in polymer and food systems. In some biological systems, the thermodynamic diffusivity DT is used, based on the chemical potential gradient, which is related to the normal diffusivity D by the equation.
D = DT(d\na/d\nQ
(7-9)
where a is the chemical activity of the species at concentration C (Frisch and Stern, 1983). In most food-related applications, the concentration of the solutes in the polymer matrix is low, and the two coefficients become equal (D = D-f). Sorption kinetics and permeability measurements (see Chapter 5) can be used for the determination of diffusivity D of solutes in polymeric materials (Vieth, 1991) Solid polymers are amorphous materials, which exist in two nonequilibrium states, i.e. glassy and rubbery, with transition between the states at the glass transition temperature (Tg). The glassy state is characterized by a dense, tough, and low porosity (2-8%) structure. The diffusivity of small solutes in the glassy state is very low, e.g. IxlO" 18 to IxlO" 10 m2/s, depending on the polymer structure and the molecular size and concentration of the penetrant. The solute diffusivity increases substantially at higher solute concentrations, by plasticization of the polymer matrix. The activation energy for diffusion is much higher than in the rubbery state, and it increases near the glass transition temperature. In the rubbery state, polymers are flexible, elastic materials, with relatively large free volume, which facilitates molecular diffusion. Crystallization or stretching (induced orientation) of the polymers can reduce solute diffusivity. Liquids and vapors may cause swelling of the glass polymeric matrix, facilitating the diffusion process. Normal diffusion in the glassy and rubbery state is Fickian, i.e. the diffusion rate is proportional to the square root of time, according to Eq. (5-4) (Peppas and Brannon-Peppas, 1994). In glassy polymers (T < Tg), non-Fickian or anomalous diffusion of solutes may take place, since diffusion and polymer relaxation are comparable. Case II diffusion is also possible, when the diffusion rate is much faster than the relaxation of polymer molecules.
Diffusivity and Permeability of Small Solutes in Food Systems
l.E-09
245
T
l.E-14 30
35
40
Temperature (°C)
Figure 7.2 Arrhenius plots of diffusivity of solutes (water and carbon dioxide) in a polymeric material showing breaks at the glass transition temperature (Tg).
Most of the research and development in polymer science and engineering is directed to the design of specific polymer structures of known barrier properties (membranes), which can be used in separation processes of various molecular species. Separations based on molecular or particle size include reverse osmosis, gas separation, and ultrafiltration.
246
Chapter 7
B. Glass Transition
Mass transport (diffusion) of solutes in polymers is affected strongly by the thermodynamic state of the material. The molten polymer is a viscous fluid of non-Newtonian characteristics, which upon cooling forms two amorphous solid states, the rubbery, and, at lower temperature, the glassy state. The glass transition temperature Tg, a second-order transformation, is an important characteristic of the polymeric materials (Roos, 1992). The nonequilibrium rubbery and glassy states are affected strongly by the presence of solutes, such as gases, water and organic solvents, which reduce, in general, the glass transition temperature. The mechanical and transport properties of polymers at temperatures below and much above Tg, are affected by the temperature, following the familiar Arrhenius equation. However, in the temperature range Tg to (rg+100°C) the Williams-Landel-Ferry (WLF) equation is more appropriate (Levine and Slade, 1992): log (ar) = [-C, (T - T,)} I [ C , + (T- Tg)]
(7-10)
where aT is a scaling parameter, or the property ratio at T and Tgi e.g. relaxation time, viscosity, or diffusivity, and C/ and C2 are characteristic parameters of the WLF equation, determined experimentally. In normal systems the values C\ = 17.44 and C2 = 51.6 are used. The WLF equation predicts a sharp change of the scaling factor as the temperature is increased immediately above Tg, e.g. the viscosity decreases 3 to 5 orders of magnitude at temperatures 20-3 0°C above Tg. The WLF equation can be used to nonpolymer systems, which exhibit a glass transition temperature, such as sugar solutions, which are of interest to foods (Roos, 1992) The effect of water on the glass transition temperature of polymers and other food components, exhibiting glass transition, is of particular importance to food processing and food quality. The Tg of dry food components is relatively high but it decreases continuously even below 0°C as the moisture content is increased. Figure 7.3 schematically shows the change of Tg of a food biopolymer as a function of moisture content (Roos, 1992).
Diffusivity and Permeability of Small Solutes in Food Systems
247
!
—————-
!
C
V >w
o
\
0
\
n
X X.
x^
3
cmpcraturc (°C)
\.
x^ ^\^^
%v 0 -
0
^i —— r
5
10
15
-
2
Moisture (%)
Figure 7.3 Change of glass transition temperature Tg of maltodextrin with water content.
C. Clustering of Solutes in Polymers
Clustering of solute molecules in polymeric materials is of importance to the sorption and diffusion properties of the system. The clustering of water is of particular interest to food systems. The clustering theory of Zimm and Lundberg is based on the statistical mechanics of fluctuations, and a simplified version of clustering of water in polymers is presented by Vieth (1991). The theory interprets the sorption isotherm over the entire range of penetrant activities. The clustering function CF is a characteristic quantity that enables the calculation of the tendency of the (water) molecules to cluster in the given polymer matrix. The clustering function is defined as the ratio CF = Gu/V,, where G// is the cluster integral, calculated from the molecular pair distribution, and V\ is the partial molecular volume of the solute (e.g. water). The cluster function varies normally from -1 to above 2. Positive CF means that the solute increases the free volume of the polymer matrix, increasing the sorption capacity, diffusivity, and permeability (high relative humidity RH). Negative CF means that the solute molecules are attached to specific sites dispersed throughout the polymer matrix, reducing the sorption and transport properties (low RH). Clustering of water can occur even at low RH by cross-linking of the polymer, or by the addition of plasticizers, like polyols.
248
Chapter 7
D. Prediction of Diffusivity
The experimental data of diffusivity of small solutes in polymers are often correlated by empirical equations as a function of concentration and temperature, in a similar manner with the data on moisture diffusivity (see Chapters 5 and 6). Although satisfactory prediction is presently not feasible, some theoretical approaches have been used for this purpose, i.e. the dual-sorption model, the freevolume model, and the molecular simulation method. 1. Dual-Sorption Model
This model has been applied to the sorption and diffusion of small molecules (mainly gases) in glassy polymers. The glassy matrix is assumed to contain some microcavities or "holes", created when the polymer melt or rubber is quenched (cooled rapidly). The solute is dissolved in the glassy polymer by two parallel mechanisms, i.e. dissolution in the polymer mass according to the Henry law, and filling of the "holes" according to the Langmuir model (Frisch and Stern, 1983; Vieth, 1991). The Henry law for dissolution is written in the form: CD = SDp
(7-11)
where CD is the concentration of the solute in the polymer, p is the partial pressure of the solute (gas), and SD is the solubility, which is equal to \/H, where H is the Henry constant. The Lagmuir equation for filling the holes takes the form:
CH = (C'bp)l(\+bp)
(7-12)
where C'is a "hole saturation" constant, and b is a "hole affinity constant", representing the ratio of rate constants of gas adsorption and desorption in microcavities. The two populations are assumed to be in local equilibrium, and the overall solubility Sp, derived from the last equation, is given by:
S p = C / P = SD + ( C ' b ) / ( l + bp)
(7-13)
The effective diffusivity D and the solubility S of the solute in the polymer are determined experimentally from sorption and permeability measurements (see Chapter 5). The effective diffusivity D is related to the diffusivities in the dissolved state DD and in the holes DH by the overall flux equation: J= - D (dCI dz) = - DD(dCDl dz) - DH (dCHl dz)
(7-14)
Diffusivity and Permeability of Small Solutes in Food Systems
249
The dissolved solute can diffuse readily, while only part of the solute in the "holes" is available for diffusion, i.e. DD > DH (partial-immobilization model). 2. Free- Volume Model Free-volume models have been proposed for the prediction of transport properties in liquids and solids, based on the availability of elements of free volume within the material, through which the solute molecules can be transported (Frisch and Stern, 1983: Petropoulos, 1994). For polymeric materials, the Vrentas and Duda model, which can be used for both the glassy and the rubbery state, is discussed briefly here (Duda and Zielinski, 1 996). The self-diffusion coefficient of a molecule (1) in a binary mixture is an exponential function of the ratio of the volume required for diffusion of one mole V \ to the total free ("hole") volume per diffusing mole VFH. The diffusion coefficient DI of a solute (1) in a binary polymer (2) mixture, in the rubbery state, is given by the equation: D, = Do exp(- E I RT) exp { - [ y(a>, V* , + w^ V\}\ I VFH }
(7- 1 5)
where D0 is a constant, E is the activation energy, R is the gas constant, T is the absolute temperature, coj and ca2 are the mass fractions of 1 and 2, f=F*; MjV^M^ and MI and A/? are the molecular weights of 1 and 2. The accommodation factor y is taken between 0.5 and 1 .0. The specific free- volume VFH is calculated from the equation: VFH= (o,K,, (K2, + T- Tgl) + co2KI2 (K22 + T- Tg2)
(7-16)
where Tgt, Tg2 are the glass transition temperatures of 1 and 2, and Klh K2i, KI2 and K22 are free-volume parameters of 1 and 2, determined experimentally. The diffusivity (D = £>;) of trace amounts of a solute (1) in a glassy polymer (2) is given by the simplified equations: D, = Do exp(-E/RT) exp [ -(yco2 £ V'2) I VFm ]
(7-17)
and Tg2)}
(7-18)
where /L= 1 - (a2-a2g), and a2wd a2g are the thermal expansion coefficients of the rubbery and glassy states of the polymer. The free-volume theory predicts the following changes of diffusion coefficient (Duda and Zielinski, 1996): Strong effect of temperature and concentration
250
Chapter 7
near the glass transition temperature; increase with the size of solute molecule; plasticizers increase the available free volume, decrease the Tg, and increase the diffusivity; addition of impermeable fillers reduces D by increasing the tortuosity of the diffusing solute. Yildiz and Kokini (1999) modified the free-volume theory to account for the effect of temperature and water activity on the retention and release of flavor compounds in food polymers. The diffusivity of hexanol, hexanal, and octanoic acid in uncooked soy flour was predicted to decrease sharply as the temperature is reduced in the rubbery state until the Tg, leveling-off at lower temperatures (glassy state). The diffusivity of flavor compounds in gliadin was predicted to increase sharply from about 1 x 10~18 m2/s to 1 x 10~10 m2/s, as the water activity was increased from 0.2 to 0.8 (at 25°C). Cross-linking of food polymers, e.g. by cooking of soy flour, predicts significant increase of diffusivity (i.e. reduced retention) of flavor compounds (e.g. hexanal).
3. Molecular Simulation Molecular simulations can describe sorption and diffusion phenomena in polymer systems, based on chemical constitution of the components. Most of the simulation work is related to simple amorphous rubbery and glassy systems, in which solute transport is assumed to follow the solution-Fickian diffusion mechanism of mass transport (Theodorou, 1996). Molecular simulations are essentially solutions of the statistical mechanics of a model of given molecular geometry and interaction parameters. They involve the generation of configurations of the system, from which structural, thermodynamic and transport properties can be extracted. Molecular dynamics (MD) assumes that the penetrant (solute) moves into channels of the sorption sites, created by small fluctuations in the polymer configuration. Transition state theory (TST) provides a more approximate treatment of the penetrant diffusion process, assuming a jumplike transport mechanism. The computer time required for the extensive computations can be reduced by certain approximations, which are less severe than the ones used in the dual-sorption and free-volume models. Computer calculations involve the estimation of the Henry constant, the geometric characteristics of the accessible volume in the polymer matrix, and its distribution and rearrangement with thermal action, using Monte Carlo algorithms. Molecular dynamics simulations have successfully predicted the selfdiffusion coefficient in glassy and rubbery polymers, interacting with penetrant solutes. The objective of molecular simulations is to develop the field of applied "molecular engineering of materials" for producing materials with tailored separation and barrier properties.
Diffusivity and Permeability of Small Solutes in Food Systems
251
IV. DIFFUSION OF SOLUTES IN FOODS
The diffusivity of solutes and other molecules in food materials depends primarily on the size of the diffusing molecule and the food structure. The needed experimental measurements of diffusivity in solid and semisolid foods are usually based on the concentration-distribution method, described in Chapter 5 (Naessens et al., 1981, 1982; Giannakopoulos and Guilbert, 1986). Diffusivity data on salts, organic and flavor components are of particular interest to food processing and food quality. A. Diffusivity of Salts
Table 7.4 shows typical diffusivities of sodium chloride in model food gels and food materials. The diffusivity depends strongly on the physical structure of the food material. The diffusivity D of salt in dilute gels (Gros and Ruegg, 1987) is very close to the D of salt in aqueous solutions, i.e. 12.5 * 10"10 m2 / s (see Table 2.4). Similar high diffusivities are observed in high-moisture foods of gel structure, like pickles (Pflug et al., 1975). Evidently, the salt ions can migrate in such gels at rates similar to the diffusion in liquid water. The salt diffusivity in Swiss cheese (Gros and Ruegg, 1987) is considerably lower than in gels (1.9x 10"10 m2/s), evidently due to the higher solids concentration and the presence of fat globules in the material. Higher salt diffusivity values D were reported by Pajonk et al. (2000) in brining Swiss cheese. The D value decreased from about 7 x 10"10 to 2 x 10"10m2/s when the brine concentration was increased from 0 to 20% NaCl. The diffusivity of salt in white feta cheese was determined as 2.3 x 10"10 m2/s (Yanniotis et al., 1994).
Table 7.4 Diffusivities D of Sodium Chloride in Food Materials (20°C) Material
D, x 10"'° m2/s
Agar gel, 3 % solids
12.0
Pickles
11.0
Swiss cheese
1.90
Meat muscle, fresh Meat muscle, thawed
4.00
Herring
2.30
Green olives, fresh
0.38 1.95
Green olives, treated
2.20
252
Chapter 7
The salt diffusivity in fresh meat muscle is 2.2 x 10"'° m2/s, while it is considerably higher (4.0x 10~10 m2/s) in meat flesh that has been frozen and then thawed (Dussap and Gros, 1980; Fox, 1980). The relatively low D of salt in the meat is caused by the resistance of the cellular structure to mass transfer. The salt diffusivity in fish is, in general, similar to the D in meat, e.g. 2.3 x 10"10 m2/s in herring (Rodger et al, 1984). The diffusivity of salt in fresh green olives is quite low (0.38 x 10"'° m2/s), evidently due to the presence of skin and to high oil concentration. Treatment of the olives with 1.8% caustic soda increases the D value to 1.95 x 10"10m2/s (Drusasetal., 1988). The diffusivity of sodium hydroxide in tomato skin, measured with a modified diffusion cell (Figure 7.1), was found to be 0.02 x 10"10m2/s (Floras et al., 1989). A higher value was found for the diffusivity of the same alkali in the skin of pimiento pepper (0.055 x 10~10m2/s). Diffusivities of other salts of interest to foods (chlorides, nitrites, nitrates, etc.) are similar to the D values of sodium chloride. A bibliography on the diffusivity of salt in foods was prepard by Ruegg and Schar (1985).
B. Diffusivity of Organic Components The diffusivity of organic solutes in food materials is important in food processing operations, like extraction (sugars, lipids, flavors), and in food quality (e.g. sugar taste, volatile flavor retention). The diffusivity D of organics in liquid foods is related closely to the viscosity 77 of the solution, through the relation r/D/T= constant Eq. (2-36). Organoleptic flavor perception is related to the diffusivity of the flavor component (e.g. sugar) and the viscosity of the liquid food (Kokini et al., 1982; Kokini, 1987). The flavor of highly viscous pseudoplastic foods is enhanced by shearing, which reduces considerably the apparent viscosity, increasing at the same time the diffusivity of the flavor component(s). For large molecules in food liquids, like peroxydase, the Stokes-Einstein equation (7-6) can be applied, while for smaller solutes in sugar solutions (e.g. nicotidamine) the Wilke-Chang equation (2-34) has been found applicable (Loncin, 1980; Stahl and Loncin, 1979). The diffusivity of nicotinamide in fructose solutions decreases from about 8 x 10~10 to 0.5 x 10~10 m2/s, when the sugar concentration is increased from 0 to 60%. In the same range of fructose concentration, the diffusivity of peroxidase decreases from 1.0 x 10"10 to 0.1 x IQ"10 m2/s. The activation energy for diffusion of both species increases sharply from 20 to 45 kJ/mol in the same sugar concentration range. The prediction models for the diffusivity of solutes in polymers, discussed earlier in this chapter, are difficult to apply in solid and semisolid foods, due mainly to the heterogeneous physical structure of the food materials. The presence
Diffusivity and Permeability of Small Solutes in Food Systems
253
of significant open space in food solids, such as pores, cracks, and channels, complicates the diffusion process, since a portion of the solutes can diffuse quickly in the gas phase, while the rest diffuses very slowly from the sorbed or trapped state. The diffusivity in the gas phase is about five orders of magnitude (x 105) higher than in the solid phase. The free-volume model, suggested for the prediction of diffusivities in polymers, was applied by Yildiz and Kokini (1999) for the prediction of diffusivity of flavor components in solid foods. Application of this model assumes that the food material behaves as a homogeneous polymer material of low porosity, such as uniform protein, carbohydrate or lipid films. The molecular simulation model (Theodorou, 1996), requiring extensive computer calculations, when developed and applied further in the polymer field, could be adapted to food materials in the future. Table 7.5 shows some typical diffusivities of organic solutes in food materials, which are useful in calculations involving solvent extraction (leaching) and liquid infusion operations (Schwartzberg and Chao, 1982). The diffusivity of sugars in gels (e.g. agar) is similar to the diffusivity in water solutions, Table 7.3 (Warin et al., 1997). The diffusivity of solutes in solid foods D is considerably lower than in dilute water solutions, shown in Tables 7.2 and 7.3, due to blockage of diffusion paths, occlusion (trapping), and sorption by the food biopolymers. The D in solids is related to the diffusivity of the solutes in water Dw by an empirical relation analogous to Eq. (5-2): D = (ew/r)Dw
(7-19)
where ew is the volume fraction of free water in the solid (analogous to porosity), and T is the tortuosity of the diffusion path.
Table 7.5 Diffusivities of Solutes in Solid Foods Solid
Solute
Solvent
Sugar beets
Sucrose
Water
Sugar cane
Sucrose
Water
Apple slices
Sugars
Water
Coffee beans
Coffee solubles
Water
Soybean flakes
Soybean oil
Cottonseed oil
Cottonseed oil
Hexane Hexane
Peanuts
Peanut oil
Hexane
r,°c 65 75 75 98 69 69 25
7,
D, x 10'10 m /s
6.80 2.00 11.5 1.00 1.00 0.27 0.006
254
Chapter 7
The free water fraction in the solid can be estimated from the moisture content and the sorption isotherm, but the tortuosity factor must be estimated indirectly from the measured D. Both parameters are not constant during food processing and storage, due to the significant changes of the food structure. The effect of solids content on the diffusivity of organic compounds in foods, is illustrated by the diffusivity of cyclohexanol in potato, which decreases from 6 x 10'10 to 2 x 10~10 m2/s in high solids potato (Loncin, 1980). The activation energy for diffusion is analogous to that of water in potato, 35.7 kJ/mol. The diffusivity of a solute may be reduced significantly by the presence of another solute, diffusing simultaneously in a solid food material (multicomponent diffusion). Thus, the individual diffusivity of citric acid (1) in prepeeled potato is reduced from /)/ = 4.3 x 10~10 to D12 = 6.6 x 10"" m2/s in the presence of ascorbic acid (2), diffusing simultaneously. At the same time, the diffusivity of ascorbic acid is reduced from D2 = 5.4 x IQ' 10 to D2, = 8.3 x IQ"11 m2/s (Lombardi et al, 1996). The diffusivities of the two solutes in dilute water solutions (w) are D!w = 6.6 x 10'10andZ)2w= 8.4 x I(r10m2/s.
C. Volatile Flavor Retention The diffusion of volatile flavor (aroma) components in foods is important in food processing operations, such as evaporation and drying, and in storage and quality of food products. Most aroma components are very volatile in aqueous solutions, since they form highly nonideal mixtures with water. The volatility of these components at thermodynamic equilibrium is characterized by the activity coefficient and the relative volatility, which are the basic elements of the vaporliquid equilibria (VLB). Calculation of (VLB) is required for the analysis of any vapor-liquid separation or interaction (Prausnitz et al., 1986; Reid et al., 1987; Le Maguer, 1992). The relative volatility of an aroma compound A in dilute water solution aAw is defined by the equation (Saravacos, 1995)
aAw=yApAo/Pwo
(7-20)
where yA is the activity coefficient of A, and pAO, pHO are the vapor pressures of A and water, respectively, at the given temperature. The activity coefficient of a component YA is related to the concentration Q and the chemical activity aA by the equation: aA = /ACA
(7-21)
Diffusivity and Permeability of Small Solutes in Food Systems
255
The activity coefficients of aroma components in water and aqueous foods are very high, especially for partially soluble organic flavor components, like esters and higher alcohols. They are estimated by computer-aided techniques using empirical models, like the UNIQUAC and the UNIFAC (Reid et al., 1987). The presence of sugars in aqueous solutions, like in food materials, increases considerably the activity coefficient (Saravacos et al., 1990; Sancho and Rao, 1997). Table 7.6 shows some typical relative volatilities of volatile flavor compounds in dilute water solutions (Saravacos, 1995; Chandraskaren and King, 1972). The relative volatility of these compounds in aqueous solutions of 60% sucrose is 20 to 10 times higher than in water, due to the strong interactions of the 3component system (Saravacos et al., 1990). The volatile flavors (aromas) are normally recovered during the evaporation of fruit juices and other aqueous systems by stripping and distillation processes (Saravacos, 1995; Karlsson and Tragardh, 1997). Maximum removal of a volatile from the liquid phase is obtained when vapor-liquid equilibrium (VLB) is established. However, establishment of true equilibrium requires infinite time, so evaporation and distillation are actually nonequilibrium processes with partial removal of volatiles. Diffusion of the flavor components from the interior to the surface of food particles is reduced sharply in the presence of sugars and other solids. Evaporation from falling liquid films (Lazarides et al., 1990) or from mechanically agitated films (Marinos-Kouris and Saravacos, 1974) can increase the stripping efficiency of volatiles.
Table 7.6 Typical Relative Volatilities of Aroma Compounds in Aqueous Solutions aAw at Infinite Dilution (25°C) Volatile compound
aAw
Methyl anthranilate
3.90
Methanol
8.30
Ethanol
8.60
1 -Propanol
9.50
1-Butanol
14.1
n-Amyl alcohol
23.0
Hexanol 2-Butanone Diethyl ketone Ethyl acetate Ethyl butyrate
31.0 76.0 77.0 205 643
256
Chapter 7
The retention of volatile flavors during food dehydration depends primarily on the presence of sugars and other solids, which reduce the aroma diffusivity in the food material. Contrary to aroma recovery processes, aroma retention is a highly nonequilibrium process, utilizing conditions that will prevent the flavor compounds from reaching the evaporation surface, such as fast surface drying (Rulkens and Thijssen, 1972). The loss of volatile flavors depends on the evaporation or drying rate of water. The mass transport of volatiles should be considered as a ternary diffusion process, with three binary diffusivities, i.e. water/solids, volatile/water and volatile/solids (Coumans et al., 1994a). Figure 7.4 shows the loss of a very volatile flavor compound, ethyl butyrate (relative volatility in water aAw = 643), as function of % water evaporated in aqueous solutions and during vacuum- or freeze-drying (Saravacos and Moyer, 1968a, b). The loss of the volatile ester from the water solution is very rapid, e.g. 90% loss by evaporation of 30% water. The presence of pectin in the water solution reduces the volatile loss and increases its retention. A higher retention is obtained by freeze-drying. Volatile retention during spray drying depends not only on the relative volatility but also on the interaction of the compound with the nonvolatile components of the food liquid. Thus, in spray drying of food emulsions containing flavors, ethyl butyrate is retained only by 20%, while limonene may be retained almost quantitatively (Furuta et al., 2000). The retention of volatile flavors during food dehydration is a very important consideration in the selection of drying processes and equipment for optimum product quality. Flavor retention is related to the reduction of diffusivity of flavor compounds by sugars and other food solids. Figure 7.5 shows that the diffusivity of diacetyl in water solutions is reduced by almost 100 times, when the sugar concentration is increased from 0 to 70% (Voilley and Simatos, 1980; Voilley and Roques, 1987).
Diffusivity and Permeability of Small Solutes in Food Systems
50
257
100
Evaporation (%)
Figure 7.4 Retention of ethyl butyrate: W, evaporation of water; VD, vacuum-drying of pectin solution; FD, freeze-drying of pectin solution.
Thermodynamic and transport phenomena analysis indicate that flavor retention is a diffusion-controlled process (Kerkhof, 1975; Bruin and Luyben, 1980). Fast drying processes, like spray drying, improve volatile retention by trapping the solute in the solid matrix. A selective diffusion mechanism may explain the volatile retention in spray- and freeze-drying (Coumans et al., 1994b). Atomization and evaporation of water/volatiles from drops control flavor retention in spraydried particles (King, 1994; Hecht and King, 2000). Retention of charactertistic aroma during storage of dried fruits can be improved by using low relative humidities (Rff) and low temperatures. Moisture sorption of stored fruits increases sharply at RH > 60%, resulting in a strong rise of flavor diffusivity and subsequent loss of aroma (Saravacos et al., 1988).
258
Chapter 7
l.E-09
\
l.E-10
l.E-11
l.E-12 20
40
80
Sugar(%)
Figure 7.5 Diffusivity of diacetyl in sucrose solutions.
D. Flavor Encapsulation Encapsulation and controlled release of solutes is used widely in pharmaceuticals, medicinal products, flavors, and pesticides. Controlled release is based on relaxation-controlled dissolution of the coating material, which consists usually of a glassy polymer (Cussler, 1997). Encapsulation of flavors, acidulants (citric and ascorbic acid), salts, and enzymes is used to prevent or control the diffusion of the solutes in various food processing and food utilization operations (Karel, 1990). Encapsulation can be achieved by entrapment in glassy polymers or in sugar crystals, in fat-based matrices, or by incorporation in liposomes (e.g. lecithin). Release of encapsulated solutes is achieved by temperture and moisture control, enzymatic release, grinding etc. The role of glass transition temperature Tg to solute release is important, since diffusivity rises sharply above Tg. The WLF equation (7-10) relates the diffusivitiy to the temperature and the Ts. The "collapse
Diffusivity and Permeability of Small Solutes in Food Systems
259
temperature" is related to T& and both temperatures decrease as the moisture content is increased. Spray- and freeze-drying are used to encapsulate flavor solutes in polymer matrices, using high initial drying rates to form a dried polymer layer, which reduces diffusivity.
V. PERMEABILITY IN FOOD SYSTEMS
The transport of small solutes, such as water, oxygen, and carbon dioxide through polymer films and protective coatings is of fundamental importance to food packaging and food processing. The permeability of these materials is based on the principles of diffusion of solutes in polymer systems. The permeability of synthetic membranes is important to separation processes used in food processing, such as reverse osmosis, gas separation, and ultrafiltration. The structural and physicochemical factors, which affect the diffusivity of solutes in polymers, are also important in characterizing the performance of packaging films and food coatings. Control of such factors as glassy/rubbery state, cross-linking, and polymer orientation, can determine the permeability of these materials. A. Permeability The permeability P of a film or thin layer of thickness z is related to the diffusivity D and the solubility S of the penetrant (solute) in the material, according to the equation: J = P (Aplz) = DS (Aplz)
(7-22)
where J is the mass transfer rate (kg/m2s), Aplz is the pressure gradient (Pa/m), and 5 is the gas/liquid equilibrium constant, S = C/p where C is the concentration (kg/m3) and p the pressure (Pa). The solubility S is equal to the inverse of the Henry constant (S=1/H), and it has units (kg/m3Pa); it can be determined as the slope of the sorption isotherm (C versus p). From equations (7-22) it follows that: P = DS
(7-23)
The permeability has SI units (kg/m s Pa) or (g/m s Pa), but various other units are used in packaging, reflecting the measuring technique or the particular food/package application (Hernandez, 1997; Donhowe and Fennema, 1994).
260
Chapter 7
The permeability P is related to the permeance PM or transmission rate 77? (=PM) and the water vapor transfer rate WVTR by the equation (McHugh and Krochta, 1994): P = PMz = WVTR I Ap
(7-24)
The units of permeance (kg/m2 s Pa) are identical to the units of the mass transfer coefficient kp. The units of WVTR are (kg/m s) (Saravacos, 1997). The SI units are useful in relating and comparing the literature data on P and WVTR to the fundamental mass transport property of diffusivity D (m2/s). The permeability of polymer films and coatings can be determined by measurements of sorption kinetics and diffusion, discussed in Chapter 5, in relation to water transport. Conversion of solubility S and diffusivity D data to permeability P though Eq. (7-23) is possible, when the material behaves like a homogeneous medium and Fickian diffusion can be assumed. Simplified permeability measurement methods are used for packaging and coating films (barriers), and most of the literature data are reported in units related to the special methods used (ASTM, 1990, 1994). The measured permeabilities represent an overall transport property of the material, based on the applied pressure gradient. Since the polymer film may have structural inhomogeneities, such as pores, channels, cracks, and pinholes, mass transport may involve, in addition to molecular diffusion, Knudsen diffusion and hydrodynamic or capillary flow (Hernardez, 1997). In such cases, the simplified relationship between diffusivity and permeability Eq. (7-23) is not applicable. Permeability is affected significantly by environmental condition, such as air relative humidity (RH), which may increase sharply the permeability of most packaging and coating films. The total permeability PT of a multilayer laminate is related to the permeabilities and the thicknesses of the individual films (P, z/) by the equation (Cookseyetal., 1999) /V=[(Sr,)]/[W/)]
(7-25)
The total permeance PMT or transmission rate TRT can be calculated from the equation: PMT=l/I,(z,/Pd
(7-26)
Temperature increases permeability P according to the Arrhenius equation in a similar manner with the effect of temperature on diffusivity D and solubility S:
= P0 exp(-£//?7), D = D0 exp(-ED/RT), S = S0 exp(-Es /RT)
(7-27)
Diffusivity and Permeability of Small Solutes in Food Systems
261
Table 7.7 Conversion Factors to SI Permeability Units (g/m s Pa) Conversion from / to (g / m s Pa) 3
2
cm (STP) mil /100 in day atm 3
2
Multiplying factor 6.42 x 10'17
cm (STP) mil / m day atm
4.14* 10"18
cm3(STP) urn / m2 day kPa
1.65 x 1Q'17
g)im/m 2 daykPa
1.16X10' 1 4
2
1.16x10""
gmm/m daykPa 2
g mil/m day atm
2.90 x 10"15
g mil/m2 day (mm Hg)
2.20 x 10'12
2
g mil/m day (90 % RH, 100 °F) g mil/100 in2 day (90 % RH, 100 °F)
4.50 x 10'14 7.00 x 10'13
perm (ASTM, 1990)____________________1.45 x 1Q-9 STP = standard temperature and pressure. 1 mil = 0.001 inch = 2.54xl0 0 m. Pressure drop of water vapor across the film at 90/0% RH and 100°F, AP = 6560 Pa. 1 mm Hg = 133.3 Pa.
The energy of activation for permeability Ep may vary, depending on the type of polymer and the temperature, in the wide range of 10 to 80 kJ/mol (Hernandez, 1997). Table 7.7 shows the conversion factors from the various literature units of permeability to SI units (g/m s Pa).
B. Food Packaging Films Synthetic polymer firms are used as barriers to the transport of water vapor, oxygen, carbon dioxide, and food components, like aroma/flavor compounds and lipids, from or to the packaged food product. Food packaging films are made of special polymeric materials, like polyethylene, both low density (LDPE) and high density (HDPE), polypropylene (PP), polyvinyl chloride (PVC), polystyrene (PS), polyethylene terephthalate (PET) and polyamides (nylon) (Hernandez, 1997; Miltz, 1992). Permeability, mechanical properties, food compatibility (nontoxicity), and cost are the main characteristics in selecting the proper material (Brody and Marsh, 1997; Hanlon et al., 1998). Permeability, like diffusivity, is affected significantly by polymer microstructure, solute-polymer interactions, solute concentration (especially moisture content or RH), and temperature. Some typical permeabilities of common packaging films to water vapor and oxygen are shown in Table 7.8 at 25°C (Miltz, 1992; Hernandez, 1997).
262
Chapter 7
Table 7.8 Permeabilities of Packaging Films to Water Vapor and Oxygen (25°C) Permeability, x 1 0 - 1 2 g / m s P a Packaging film Water vapor Oxygen 1.40 0.031 LDPE 0.007 0.20 HDPE 1.00 0.010 PP 3.00 0.005 PVC 12.0 0.018 PS 1.40 0.0006 PET 0.002 0.0004 Nylon LDPE = low density polyethylene, HOPE = high density polyethylene, PP = polypropylene, PVC = polyvinyl chloride, PS = polystyrene, PET = polyethylene terephthalate, Nylon = polyamide.
The permeability of nylon to oxygen at various moisture contents has been analyzed by the dual-sorption model (Hernandez, 1994). Although the water diffusivity increases at higher moistures, the solubility and the permeability decrease sharply at the beginning, leveling-off at water activities above 0.2. The permeability of polymer firms and food coatings to carbon dioxide is important in food packaging and storage. Typical values of permeability of carbon dioxide at 25°C are: LDPE, 1.6 x 10'13; HDPE, 5.3 x 10'15 g/m s Pa. C. Food Coatings
The permeability of edible food coatings to water is of particular interest to food quality, since their primary function is to act as barriers to moisture transport
during storage. Food coatings can also control the transport of gases (mostly oxygen), flavor components and lipids in food systems. Edible coatings, used as barriers in foods, include proteins (wheat gluten, caseinates, whey protein, corn zein), polysaccharides (starch, dextrins), pectins, lipids, and chocolate. Composite coatings, containing a food biopolymer (e.g. protein) and a hydrophobic material, like lipid, fatty acid, chocolate, and beeswax, usually have very low water permeabilities. The food coatings are prepared as solutions or dispersions/emulsions of the primary biopolymer in solvents (ethanol, alkalis, or acids). They contain various plasticizers, such as glycerine and sorbitol, which improve the physical and mechanical properties of the coating. They are applied to the various fresh and processed foods, like fruits and vegetables by dipping in an emulsion, spraying or foaming and brushing. Table 7.9 shows typical water permeabilities of food coatings (McHugh and Krochta, 1994):
Diffusivity and Permeability of Small Solutes in Food Systems
263
Table 7.9 Water Vapor Permeabilities P of Food Coatings (25°C) P, xlO- 1 0 g/msPa Food coating 6.10 Gluten-glycerine 7.20 Whey protein-sorbitol 1.00 Zein-glycerine 4.20 Sodium casemate 0.12 Chocolate 0.006 Beeswax
D. Permeability/Diffusivity Relation The simple permeability/diffusivity/solubility relation of Eq. (7-23) is useful for estimating the permeability P from diffusivity D and solubility S data of polymer films, and for comparison of P and D data. This relation applies to systems behaving as homogeneous materials, in which solute transport is by Fickian molecular diffusion. It does not hold for heterogeneous materials, consisting of pores, channels and capillaries, in which a significant portion of mass transfer takes place by mechanisms other than molecular diffusion. Table 7.10 shows some typical diffusivity and permeability data for packaging films and food coatings. The comparison is facilitated by using consistent (SI) units (Saravacos, 2000). A typical application of the permeability-diffusivity relation is given for chocolate film, using published data of Biquet and Labuza (1988): Typical permeability P and diffusivity D values for a chocolate coating about 0.6 mm thick at 20 0 C:P = 0.11 x 10-'°g/msPaandZ)=l x 10-13m2/s. The solubility S of water in the chocolate material can be estimated from the sorption isotherm at 20°C. It is defined by the Henry equation, C = S p, where C is the concentration (kg/m3) in the material andp is the partial pressure of water (Pa). Thus, the solubility is equal to the slope of the isotherm (S = C/p). Considering the initial sorption stage, water activity a,v 0 to 0.1, S = (1.7 kg water/100 kg solids)/Ap where Ap = a,vp0 or Ap = 0.1 PO, and PO is the vapor pressure of water at 20°C (p0 = 2340 Pa), and Ap = 234Pa. The concentration of water in the chocolate material is converted to consistent (SI) units, as follows: Assume density of dry chocolate 1600 kg/m3; therefore, the volume of 100 kg dry material will be 100/1600 = 0.0625 m3. The water concentration in the chocolate becomes C = (1.7/0.0625) = 27.2 kg/m3, and the solubility S = 27.2/232= 0.116 kg/m3 Pa. Using the measured diffusivity of the system (D = 1 x 10~13 m2/s), the permeability of the chocolate film according to Eq. (7-23) will be P = D S = 0.116 x 10"13kg/ms Pa, or P = 0.116 x 10"'°g/ms Pa, which is very close to the measured permeability.
264
Chapter 7
Table 7.10 Typical Water Vapor Permeabilities and Diffusivities Film or coating P, x 10-'°g/msPa A xio-'°m 2 /s 0.005 0.002 HDPE LDPE 0.010 0.014 PP 0.010 0.010 0.041 0.050 PVC 1.00 Cellophane 3.70 Protein films 0.100 0.10-10.0 0.100 0.10-1.00 Polysaccharide films 0.010 Lipid films 0.003-0.100 Chocolate 0.001 0.11 Gluten 1.00 5.00 Com pericarp 0.10 1.60 LPDE = low density polyethylene, HDPE = high density polyethylene, PP = polypropylene, PVC = polyvinyl chloride
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Saravacos, G.D. 2000. Transport Properties in Food Engineering. In: Engineering and Food for the 21st Century. J. Welti-Chanes and G. Barbosa-Canova, eds. Lancaster, PA: Technomic Press, in press. Saravacos, G.D., Moyer, J.C 1968a. Volatility of Some Aroma Compounds during Vacuum-Drying of Fruit Juices. Food Terchnol. 22:89-93. Saravacos, G.D., Moyer, J.C 1968b. Volatility of Some Flavor Compounds during Freeze-Drying of Foods. Chem. Eng. Progress Symposium Series Vol. 64 No 86, pp. 37-42. Saravacos, G.D., Tsami, E., Marinos-Kouris, D. 1988. Effect of Water Activity on Volatile Flavors of Dried Fruits. In: Frontiers of Flavor. G. Charalambous ed. Amsterdam: Elsevier, pp. 347-356. Saravacos, G.D., Karahanos, V.T., Marinos-Kouris, D. 1990. Volatility of Fruit Aroma Compounds in Sugar Solutions. In: Flavors and Off-Flavors '89. G. Charalambous ed. Amsterdam: Elsevier, pp. 729-738. Schwartzberg, H.G. and Chao, R.Y. 1982. Solute Diffusivities in the Leaching Processes. Food Technol. 36:73-86. Stahl, R., Loncin, M. 1979. Prediction of Diffusion in Solid Foodstuffs. J. Food Proc. Preserv. 3:313-320. Theodorou, D.N. 1996. Molecular Simulations of Sorption and Diffusion in Amorphous Polymers. In: Diffusion in Polymers. P. Neogi, ed. New York: Marcel Dekker, pp. 67-142. Van Krevelen, D.W. 1990. Properties of Polymers 3rd ed. Amsterdam: Elsevier. Vieth, W. R. 1991. Diffusion in and Through Polymers. Munich, Germany: Hanser Publ. Voilley, A. and Roques, M.A. 1987. Diffusivity of Volatiles in Water in the Presence of a Third Substance. In: Physical Properties of Foods - 2. R. Jowitt, F. Escher, M. Kent, B. McKenna, M. Roques, eds. London: Elsevier Applied Science, pp. 109-121. Voilley, A. and Simatos, D. 1980. Retention of Aroma During Freeze- and AirDrying. In: Food Process Engineering Vol. 1, P. Linko, Y. Malkki, J. Olkku, J. Larinkari, eds. London: Applied Science, pp. 371-384. Warin, F., Gekas, V., Dejmek, J. 1997. Sugar Diffusivity in Agar Gel / Milk Billayer Systems. J. Food Sci. 62:454-456. Yanniotis, S., Zarmpoutis, J., Anifantakis, E. 1994. Diffusion of Salt in Dry-Salted Feta Cheese. In: Developments in Food Engineering Part 1. T. Yano, R. Matsuno, K. Nakamura, eds. London: Blackie Academic and Professional, pp. 358-360. Yildiz, M.E., Kokini, J.L. 1999. Development of a Predictive Methodology to Determine the Diffusion of Small Molecules in Food Polymers. In: Proceedings of 6th Conference of Food Engineering COFE'99. G.V.Barbosa-Canovas and S.P.Lombardo eds. New York: AIChE, pp.99-105.
8 Thermal Conductivity and Diffusivity of Foods
I. INTRODUCTION
The thermal transport properties, thermal conductivity and thermal diffusivity of simple gases and liquids can be predicted by molecular dynamics and semiempirical correlations, and numerous tables and data banks are available in the literature (Chapter 2). Experimental measurements are necessary for the thermal transport properties of foods, due to their complex physical structure. Empirical models have been proposed for the correlation of experimental data and the possible explanation of the heat transport mechanisms. The thermal conductivity (X) of a material is a measure of its ability to conduct heat and is defined by the basic transport equation (2-3), which is integrated to give: q/A=A(TrT2)/x
(8-1)
where qlA is the heat flux (W/m), x is the thickness of the material (m), T, and T2 are the two surface temperatures of the material, and A is the surface of the material normal to the direction of heat flow (m2). The S.I. units of A are W/mK. Equation (8-1) is the basis for the direct measurement of A (guarded hot-plate method). The thermal diffusivity a of a material can be estimated from the thermal conductivity A using the equation:
a = JJpCp
(8-2)
where p is the density (kg/m3) and Cp is the specific heat (J/kgK) of the material. The S.I. units of a are m2/s. 269
270
Chapter 8
The thermal conductivity of foods depends of the chemical composition, the physical structure, the moisture content, and the temperature of the material. The A of unfrozen foods varies between the A, of air (0.020 W/mK) and water (0.62 W/mK). Higher A values characterize the frozen foods (about 1.5 W/mK). The thermal diffusivity of foods does not change substantially, because any changes of A are compensated by changes of the density of the material Eq. (8-2). Typical values of a for unfrozen food are 1.3xlO"7 m2/s and for frozen food 4xlO"7 m2/s. The thermal conductivity of solid foods is a strong function of the porosity of the material. This variation is about one order of magnitude, compared to the very wide variation of mass diffusivity in porous foods. The changes in heat and mass transport properties of porous foods reflect the differences in /I and D of gases and liquids, according to the approximate relations:
A(gas)//l(liquid)=l/10
(8-3)
£>(gas)/£>(liquid) = 10000/1
(8-4)
Empirical models of thermal conductivity, analogous to the models of electrical conductivity, can be used to correlate the experimental data. The literature data on X can be analyzed statistically, using correlations analogous to the models of moisture diffusivity (see Chapter 6).
II. MEASUREMENT OF THERMAL CONDUCTIVITY AND DIFFUSIVITY
The measurement of the thermal transport properties of foods is described by several authors in the literature, notably by Mohsenin (1980), Nesvadba (1982), Sweat (1995), Rahman (1995), and Urbicain and Lozano (1997). A comprehensive study of the subject was undertaken within the collaborative research project COST 90 in the European Union (Meffert, 1983; Kent et al., 1984). A. Thermal Conductivity
Two experimental methods are normally used for the measurement of the thermal conductivity (X), i.e. the guarded hotplate and the heated probe. Other methods, suggested for food materials, are the Fitch method and its modifications, the thermal comparator method, and the temperature history method (Rahman, 1995).
Thermal Conductivity and Diffusivity of Foods
271
1. Guarded Hot Plate The guarded hot-plate method is based on the Fourier equation for steadystate heat flux (8-1). The experimental apparatus is diagrammatically shown in Figure 8.1. Details of the apparatus are given by Drouzas and Saravacos (1985). The apparatus consists of two circular brass plates, between which the sam-
ple material is placed. The upper plate is heated electrically and the lower cold plate is maintained at a constant temperature. Unidirectional flow of heat is assured by two guard rings around the plates. After establishment of steady state, the heat flow is measured with an electrical meter and the thermal conductivity (X) is determined from equation (8-1). Although the guarded hot plate is an accurate method, it requires special precautions, like uniform sample thickness, good contact with the plates, and relatively long time to reach steady state, which may change the moisture content of the material. 2. Heated Probe The heated probe method is faster and it requires less sample material. For these reasons it is used more widely than the guarded hot-plate method. The hot probe is a transient method, based on the measurement of the sample temperature as a function of time, while the sample is heated by a known line heat source. Assuming that the line heat source is an infinite medium, and that the heat flow is radial, the temperature T at a point very close to the line source, after a time t will be (Rahman, 1995):
hot plate
quard rings
••f'H.4.-^ *-:; -a.-^:-; "'. s-y "£• ' %-'•-"-'"- •<<.':£ ":-
81_ AAAAAA^
/ , sample / cold plate Jjpyjij*
iffigam
Figure 8.1 Diagram of a guarded hot plate apparatus.
272
Chapter 8
(8-5)
where ft is a constant, and q is the heat flow per unit length of the line source (W/m). The temperature difference (ArP=T2-T1) after times t} and tt of heating will be:
Thus the thermal conductivity A can be obtained from the slope (qttnX) of a semilog plot of T versus t. The heated probe method requires short measurement times (less than a minute), and relatively small samples.
heated wire
thermocouple seal
stainless steel needle
filling material seal
Figure 8.2 Principle of heated probe.
Thermal Conductivity and Diffusivity of Foods
273
Figure 8.2 shows schematically a heated probe used for thermal conductivity (Drouzas and Saravacos, 1988; Sweat, 1995). A stainless steel needle (or syringe) is used to house the line source (a heated constantan wire) and the thermocouple (chromel-constantan). Time-temperature data of the probe, inserted in the sample, are used to calculate /I from Eq. (8-7), which is derived directly from (8-6): A = [23I2R logfo/?;)]/[4^)7]
(8-7)
where / is the electrical current (A) and R' is the specific electrical resistance of the heating wire (H/m). The heated probe is calibrated with glycerin solutions or with known insulating materials.
B. Thermal Diffusivity The thermal diffusivity a is usually estimated indirectly from the thermal conductivity A, the density p and the specific heat Cp of the material, according to Eq. (8-2).
Direct measurement of a can be made using various methods, discussed by Rahman (1995). All methods are based to some simplified solutions of the unsteady-state heat conduction equation (Fourier): (8-8)
/. Dickerson Method
The simplest experimental method is the transient method of Dickerson (1965), modified by Poulsen (1982), and used by Drouzas and Saravacos (1985). The principle of the method is to obtain time-temperature data in a cylindrical sample of material, as shown in the diagram of Figure 8.3. The transient heating time t in the cylindrical container is given by the semiempirical equation (Ball and Olson, 1957): t=f\Qg\j(Ts-T0)l(Ts-T)}
(8-9)
where T0 is the initial temperature, Ts is the outside surface temperature, and T is the temperature after time t. The linear semilog relationship is defined by the lag factory and the reciprocal of the slope/ Equation (8-9) is used in thermal process (sterilization) calculations of the formula method.
274
Chapter 8
thermocouples
constant temperature"
sample
Figure 8.3 Transient method for thermal diffusivity.
For a long cylindrical container with high surface heat transfer coefficients h, i.e. for Biot numbers Bi > 40 (Bi = MX), the parameter/of the heating source is given by the equation: /= 0.398(r2/a)
(8-10)
where r is the radius of the container. Thus, the thermal diffUsivity (a) can be obtained from Eq. (8-10). 2, Modified Heated Probe The thermal diffusivity (a) can be obtained experimentally, using a modified heated probe (Mohsenin, 1980; Drouzas et al., 1991). The second thermocouple is inserted in the basic probe (Figure 8.2) at a fixed short distance r from the initial thermocouple. The temperature rise AT at the distance r at the heating time t is given by the following solution of the unsteady-state heat conduction in a cylindrical solid: AT= (q/4aX) [-0.58/2 -
where j3 = r/2(af)1'2
! - (34/4x2!
(8-11)
Thermal Conductivity and Diffusivity of Foods
275
Thus, a can be calculated from Eq. (8-11), using the corresponding A value from a parallel measurement with the normal probe. The exact distance r of the two thermocouples is critical in the modified probe method and a calibration of the system may be required. A comparison of the direct probe method with the indirect calculation of a indicated a better accuracy of the indirect method (Drouzas et al., 1991).
III. THERMAL CONDUCTIVITY AND DIFFUSIVITY DATA OF FOODS
The literature contains several data on the thermal transport properties of foods in the form of tables, empirical models, regression equations, and data banks. A. Unfrozen Foods Table 8.1 shows typical values of thermal conductivity of foods: Kostaropoulos (1971), Mohsenin (1980), Rahman (1995), Sweat (1995), and Singh (1995). Sections IV and V of this chapter contain some mathematical models and statistical compilations of thermal conductivity and diffusivity. Table 8.1 Thermal Conductivity K of Unfrozen Foods (25°C) % Moisture Food material A, W/mK 0.620 100 Water 0.567 Sucrose solution 85 0.110 Granular starch 15 0.547 Gelatinized starch 85 0.533 82 Potato 0.513 85 Apple 0.550 65 Tomato paste 0.452 74 Beef 0.460 80 Fish 0.530 Milk 85 0.180 Vegetable oil 0.237 18 Dried fruit 0.041 Freeze-dried gel 4
276
Chapter 8
The thermal conductivity A of foods decreases, in general, as the moisture content of the material is reduced. The A of dried food materials decreases sharply as the porosity is increased, e.g. by freeze-drying or puffing. The /I of fibrous dry foods is higher parallel than across the fiber (Chapter 3). Thus, the /I of freeze-dried cellulose gum (fibrous structure) was 0.0627 W/mK compared to 0.0410 W/mK of porous freeze-dried starch gel (Saravacos and Pilsworth, 1965). An analogous effect was noticed in the drying rates of the two materials (see Chapter 5). Figure 8.4 shows the effect of air pressure on the thermal conductivity of a freeze-dried gel (Saravacos and Pilsworth, 1965). The A decreases sharply from 0.041 to about 0.010 W/mK as the air pressure is reduced from atmospheric to below 1 mbar. Similar effects of pressure was observed in freeze-dried fruits (Harper, 1962) and milk and orange pulp (Fito et al., 1984). The thermal diffusivity (a) of unfrozen foods does not change appreciably with the moisture and temperature, ranging from about l.OxlO' 7 to 1.5xlO"7 m2/s. Detailed tables of (a) for various food materials are presented by Rahman (1995). B. Frozen Foods
The thermal conductivity of frozen foods is significantly higher than the A. of unfrozen materials, due to the higher /I of ice. The /I of a frozen food of 85% moisture content decreases gradually from about 1.5 W/mK at -40°C to 0.5 W/mK at 0°C, as shown in the diagram of Figure 8.5. A sharp drop of A is observed near 0°C, due to the melting of ice. An analogous variation of the thermal diffusivity a of the frozen foods is observed below the freezing point (Figure 8.6). The a of a typical frozen food decreases from about 1x10'6 m2/s at -40°C to 1x10'7 m2/s above 0°C. The sharp drop near the freezing point is due to the sharp increase of cp, which incorporates the heat of melting, near 0°C (Singh, 1995). C. Analogy of Heat and Mass Diffusivity
The heat and mass transport analogy in gases (Chapter 2) has been observed in porous foods at low moisture content (Kostaropoulos and Saravacos, 1997). The analogy is based on the transport mechanisms in the gas phase, which becomes important in highly porous materials, like granular, freeze-dried, and extrusioncooked foods. As in the case of moisture diffusivity (Figure 5.24), the thermal diffusivity of granular starch shows a maximum near 15% moisture, and it increases gradually at moistures higher than 35% (Figure 8.7).
277
Thermal Conductivity and Diffusivity of Foods
0.050
0.000
10
0.1
100
1000
Pressure (mbar)
Figure 8.4 Effect of air pressure on thermal conductivity of porous food material.
-40
-30
-20
-10
0
10
Temperature (°C)
Figure 8.5 Thermal conductivity of frozen food material.
20
30
40
Chapter 8
278
l.OE-06
-40
-30
-20
-10
0
10
20
30
40
Temperature (°C)
Figure 8.6 Thermal diffusivity of frozen food material.
10
20
30
40
50
60
Moisture Content (%)
Figure 8.7 Thermal diffusivity versus moisture content of porous food material.
Thermal Conductivity and Diffusivity of Foods
279
D. Empirical Rules
A preliminary checking of thermal transport property data can be made, using the empirical rules, suggested by Kostaropoulos (1981): I. Thermal Conductivity
i. Food with moisture content (Xw) > 30-40% Xi~ 0.40- 0.58 W/mK ii. Frozen foods, Xw > 30-40% XK ~ 2.5 A, j
iii. Dry food
iv. Fats and oils X i v ~ 0.25- 0.50
2. Thermal Diffusivity
i Foods, X w >30% ctj^ 1.4xlO" 7 r ii. Frozen food iii. Dry food
iv Fats and oils
280
Chapter 8
IV. MODELING OF THERMAL TRANSPORT PROPERTIES
A. Composition Models
Several composition models have been proposed in the literature, most of which are summarized by Miles et al. (1983), Sweat (1995), and Rahman, (1995). The most promising seems to be the model proposed of Sweat (1995):
A = Q.5SXv + Q.l55Xp+Q.25Xc+Q.l6Xf+Q.135Xa
(8-12)
where Xm Xp, Xf and Xa are the mass fractions of water, protein, fat, and ash, respectively. The above model was fitted to more than 430 liquids and solid foods with satisfactory results. It is not accurate for porous foods containing air, for which structural models are needed. The thermal conductivity of water in the above equation was fitted to about 0.58W/mK which is less than the thermal conductivity of pure water, 0.605W/mK. Either the selected data are biased, or they indicate that the effective thermal conductivity of water in foods is less than the thermal conductivity of pure water (Sweat, 1995). The key to the accuracy of the above equation is having accurate values for the thermal conductivity of "pure" components. This is easy for the water and oil fractions but very difficult for the other fractions. In fact, the thermal conductivity of proteins and carbohydrates probably varies according to their chemical and physical form. However, it is not needed to find more accurate additive composition models, because of the inherent inaccuracy in the composition models, which they don't take into account the geometry of the component mixing. As in the case of air-containing foods, structural models must be used. The temperature effect is not included in the above equation. Thus, it is valid at the fitting region approximately at 25°C. The temperature effects of the major food components are summarized by Rahman (1995) in Table 8.2 and in Figure 8.8.
281
Thermal Conductivity and Diffusivity of Foods
Table 8.2 Effect of Temperature on the Major Food Components X=bo+biT+b2T2+b3T:1
bo
b,
b2
b,
Air
2.43E-02
7.89E-05
-1.79E-08
-8.57E-12
Protein
1.79E-01
1.20E-03
-2.72E-06
Gelatin
3.03E-01
1.20E-03
-2.72E-06
Ovalbumin
2.68E-01
2.50E-03
Carbohydrate
2.01E-01
1.39E-03
Starch
8.7 IE-02
9.36E-04
Gelatinized Starch
3.22E-01
4.10E-04
Sucrose
3.04E-01
9.93E-04
Fat
1.81E-01
2.76E-03
-1.77E-07
Fiber
1.83E-01
1.25E-03
-3.17E-06
Ash
3.30E-01
1.40E-03
-2.91E-06
Water
5.70E-01
1.78E-03
-6.94E-06
Ice
2.22E+00
-6.25E-03
1.02E-04
-4.33E-06
2.20E-09
Chapter 8
282
0.75
I u 3
•U B O
U
"5
E 0.25
50
Temperature (°C)
Figure 8.8 Effect of temperature on the major food components.
100
Thermal Conductivity and Diffusivity of Foods
283
B. Structural Models
For heterogeneous foods, the effect of geometry must be considered using structural models. Utilizing Maxwell's and Eucken's work in the field of electricity, Luikov et al. (1968) initially used the idea of an elementary cell, as representative of the model structure of materials, in order to calculate the effective thermal conductivity of powdered systems and solid porous materials. In the same paper, a method is proposed for the estimation of the effective thermal conductivity of mixtures of powdered and solid porous materials.
Since then, a number of structural models have been proposed, some of which are given in Table 8.3. The series model assumes that heat conduction is
perpendicular to alternate layers of the two phases, while the parallel model assumes that the two phases are parallel to heat conduction. In the random model, the two phases are assumed to be randomly mixed. The Maxwell model assumes that one phase is continuous, while the other phase is dispersed as uniform spheres. Several other models have been reviewed by Rahman (1995), among others.
In the mixed model (also called and Krischer model) heat conduction is assumed to take place by a combination of parallel and perpendicular heat flow. This model recognizes that there are two extremes in thermal conductivity values, one being derived from the parallel model and the other from the series model, whilst the real value of thermal conductivity should be somewhat in between these two extremes. A conceptual diagram is shown in Figure 8.9. The distribution factor/is a weighting factor between these extremes. It characterizes the structure of the material and it should be independent of material moisture content and temperature,
I-/
Parallel Structure
/
Series Structure
Figure 8.9 The mixed model of thermal conductivity.
284
Chapter 8
Granular (particulate) materials consist of granules (particles) and air, randomly packed (Figure 8.10). The induvidual particles consist of solids and water (Figure 8.10). The use of some of these structural models to calculate the thermal conductivity of a hypothetical porous material is presented in Figure 8.11. The parallel model gives the largest value for the effective thermal conductivity, while the series model gives the lowest. All other models predict values in between. Figure 8.12 represents the mixed model for various values of the distribution factor/ as a function of the void fraction (porosity). A systematic general procedure for selecting suitable structural models, even in multiphase systems, has been proposed by Maroulis et al. (1990). The method is based on a model discrimination procedure. If a component has unknown thermal conductivity, the method estimates the dependence of the temperature on the unknown thermal conductivity, and the suitable structural models simultaneously. An excellent example of applicability of the above is in the case of starch, an important component of plant foods. The granular starch consists of two phases, the wet granules and the air/vapor mixture in the intergranular space. The starch granule also consists of two phases, the dry starch and the water. Consequently, the thermal conductivity of the granular starch depends on the thermal conductivities of pure materials (that is, dry pure starch, water, air, and vapor, all functions of temperature) and the structures of granular starch and the starch granule. It has been shown that the parallel model is the best model for both the granular starch and the starch granule (Maroulis et al., 1990). These results led to simultaneous experimental determination of the thermal conductivity of dry pure starch versus temperature. Dry pure starch is a material that cannot be isolated for direct measurement.
Thermal Conductivity and Diffusivity of Foods
GRANULAR MATERIAL
Particles
GRANULE (PARTICLE)
Figure 8.10 Schematic model of granular materials.
285
286
Chapter 8
Table 8.3 Structural Models for Thermal Conductivity
Series 1 \-e)
g
Random
Mixed (Krischer )
=
287
Thermal Conductivity and Diffusivity of Foods
0.125
Parallel
Maxwell (Continuous phase 1) Random
Maxwell (Continuous phase 2) Series
0.025 0.00
0.25
0.50
Void Fraction
Figure 8.11 Structural models for porous materials.
0.75
Chapter 8
288
0.125
Mixed Model (Krischer) Distribution Factor =
0.00 (Parallel) 0.25 0.50 0.75 1.00 (Series)
0.025 0.00
0.25
0 . 5 0 0.75
1.00
Void Fraction
Figure 8.12 The mixed (Krischer) model for various values of distribution factor.
Thermal Conductivity and Diffusivity of Foods
289
V. COMPILATION OF THERMAL CONDUCTIVITY DATA OF FOODS
There is a wide variation of the reported experimental data of thermal conductivity of solid food materials, making difficult their utilization in food process and food quality applications. The variation of thermal conductivity in model and real foods is discussed in Section III of this chapter. The physical structure of solid foods plays a decisive role not only on the absolute value of thermal conductivity, but also on the effect of moisture content and temperature on this transport property. In this section, the thermal conductivity in food materials is approached from a statistical standpoint. Literature data are treated by regression analysis, using the parallel structural model. Recently published values of thermal conductivity in various foods were retrieved from the literature, and they were classified and analyzed statistically to reveal the influence of material moisture content and temperature. Structural models, relating thermal conductivity to material moisture content and temperature were fitted to all examined data for each material. The data were screened carefully, using residual analysis techniques. The most promising model was proposed, which is based on an Arrhenius-type effect of temperature and it uses a parallel structural model to take into account the effect of material moisture content. Thermal conductivity data in the literature show a wide variation due to the effect of the following factors: (a) diverse experimental methods, (b) variation in composition of the material, (c) variation of the structure of the material. Thermal conductivity depends strongly on moisture, temperature and structure of the material. An exhaustive literature search was made in international food engineering and food science journals in recent years, as follows (Krokida et al., 2001): • • • • • •
Drying Technology, 1983-1999 Journal of Food Science, 1981-1999 International Journal for Food Science and Technology, 1988-1999 Journal of Food Engineering, 1983-1999 Transactions of the ASAE, 1975-1999 International Journal of Food Properties, 1998-2000
A total number of 146 papers were retrieved from the above journals according to the distribution presented in Figure 8.13. The accumulation of the papers versus the publishing time is presented in Figure 8.14. The search resulted in 1210 data concerning the thermal conductivity in food materials.
Chapter 8
290
J. Food Engineering
J, of Food Science
Drying Technology
Trans of the Int. J. Food International ASAE
Science &
Journal of
Techn.
Food Properties
Figure 8.13 Number of papers on thermal conductivity data in food materials published in food engineering and food science journals during recent years. 160
120 o L. 1
sa
Z o
1970
1980
1990
2000
year
Figure 8.14 Accumulation of published papers on thermal conductivity data for food materials versus time.
291
Thermal Conductivity and Diffusivity of Foods
0.001
0.01 0.1
1
10
100
Moisture (kg/kg db)
Figure 8.15 Thermal conductivity data for all foods at various moistures.
0.01
0.1
1
10
100
1000
Temperature (oQ
Figure 8.16 Thermal conductivity data for all foods at various temperatures.
Chapter 8
292
These data are plotted versus moisture and temperature in Figures 8.15 and 8.16, respectively. These figures show a good picture concerning the range of variation of thermal conductivity, moisture and temperature values. More than 95% of the data are in the ranges:
• Thermal Conductivity 0.03 - 2.0 W/mK 0.01-65 kg/kg db • Moisture -43 -160 °C • Temperature The histogram in Figure 8.17 shows the distribution of the thermal conductivity values retrieved from the literature. Most of the K values are between 0.1 and 1.0 W/mK. Thermal conductivities higher than that of water (0.62 W/mK at 25°C) are characteristic of frozen foods of high moisture content, since the thermal conductivity of ice is about 2 W/mK. The results obtained are presented in detail in Tables 8.4-8.6. More than 100 food materials are incorporated in the tables. They are classified into 11 food categories. Table 8.4 shows the related publications for every food material. Table 8.5 summarizes the average literature value for each material along with the corresponding average values of corresponding moisture and temperature. Table 8.6 presents the range of variation of thermal conductivity for each material along with the corresponding ranges of moisture and temperature.
1000 •a I '=
100
o L.
£
10
0.01
0.03
0.10
3.00 1.00 0.30
Thermal Conductivity Values (W/mK)
Figure 8.17 Histogram of observed values of thermal conductivity in food materials.
Thermal Conductivity and Diffusivity of Foods
293
Table 8.4 Literature for Thermal Conductivity Data in Food Materials: References and Number of Data Retrieved Material
Reference
Data
60
Baked products
14
Bread Zanonietal, 1995 Zanonietal., 1994
Goedeken et a!., 1998 Dough Bouvier et al, 1987
Zanonietal, 1995 Griffith etal., 1985 Soy flour
Maroulisetal.,1990 Wallapapan et al., 1982
Cake Zanonietal., 1995 Yellow batter Baiketal, 1999
Cup batter Baiketal., 1999
Cereal products
5 3 6 20 3 8 9 11 7 4 2 2 1 1 12 12
76 9
Barley
Alagusundaram et al., 1991 Corn
Bekeetal, 1994 Changetal, 1980 Okos etal, 1986 Rice
Okos etal, 1986 Ramesh, 2000
Wheat Changetal., 1980 Okos etal., 1986 Corn meal
Laietal, 1992 Kumaretal, 1989
9 21 9 3 9 13 4 9 10 3 7 7 4 3
Chapter 8
294
Table 8.4 Continued Material
Reference
Data 4
Iclli batter
Murthyetal, 1997
Maize Halltdayetal, 1995 Tolabaetal.,1988
Oat Okosetal.,1986
4 11 9 2 1 1
136
Dairy Cheese
Lunaetal., 1985 Tavmanetal., 1999 Milk
Duaneetal., 1992 Duaneetal, 1993 Duaneetal, 1994 Me Proud eta!., 1983 Hori, 1983 Zieglereta!., 1985 Ready etal, 1993 Tavmanetal, 1999 Okosetal.,1986
Cream Duaneetal, 1998 Butter
Tavmanetal, 1999 Okosetal.,1986 Yogurt
Kirn etal, 1997 Tavmanetal., 1999 Whey
Okosetal.,1986
23 1 22 84 1 1 1 1 6 3 9 9 53 1 1 5 2 3 19 9 10 4 4
Thermal Conductivity and Diffusivity of Foods
295
Table 8.4 Continued Material
Reference
Fish
Data
83
Cod
Sam et a!., 1987
Mackerel Sametal., 1987 Squid
Rahman et al, 1991 Rahman, 1991 Carp
Hung el al., 1983 Surimi
Wangetal., 1990 AbuDaggaetal, 1997 Cake Borquezetal, 1999 Shrimp
Karunakar et al, 1998
Calamari Rahman, 1991
Salmon Sametal., 1987
5 5 5 5 16 12 4 2 2 30 21 9 1 1 13 13 2 2 9 9
143
Fruits Apple Ramaswamy elal, 1981 Mattea et al., 1989 Telis-Romero et al, 1998 Rahman, 1991 Constenlaetal, 1989 Bhumblaetal, 1989 Ziegleretal, 1985 Mattea etal, 1986 Madambaetal, 1995 Sheen etal, 1993 Buhrietal, 1993 Okos etal, 1986 Chenetal, 1998
Banana Njieetal, 1998
82 11 3 3
2 9 25 3
3 2 1 1 10
9 1 1
Chapter 8
296
Table 8.4 Continued Material
Reference
Peach Okosetal.,1986
Strawberry
Delgado et al, 1997 Bhumbla et al., 1989 Okosetal.,1986 Raspberry Bhumbla etal, 1989
Okosetal.,1986
Grape Bhumbla et al, 1989 Okosetal.,1986 Plantain
Njieetal, 1998 Raisin
Vagenas et al., 1990
Pear Matteaetal, 1989 Rahman, 1991 Dincer, 1997 Mattea et al., 1986 Okosetal.,1986 Orange
Telis-Romeroetal, 1998 Bhumbla etal, 1989 Ziegler et al, 1985 Okosetal.,1986
Bilberry Bhumbla et al, 1989 Okosetal.,1986 Cherry
Bhumbla etal, 1989 Okosetal.,1986
Data 1 1 5 3 1 1 2 1 1 8 1 7 6 6 4 4 15 3 2 1 3 6 15 9 1 4 1 2 1 1 2 1 1
9
Legumes
9
Lentils Alagusundaram et al, 1991
9
Thermal Conductivity and Diffusivity of Foods
297
Table 8.4 Continued Material
Reference
Meat
Data 134
Beef Hung etai, 1983 Marinos-Kouris et al, 1995 Me Proud et al., 1983 Perezetal, 1984 Rahman, 1991 Baghe-Khandan et al, 1982 Sanzetal, 1987 Califano et al, 1997 Chicken
Rahman, 1991 Sanzetal, 1987
Sausage Sheen etal, 1990 Ziegleretal, 1987 Akterian, 1997 Turkey
Sanzetal, 1987 Mutton
Sanzetal, 1987
75 4 2 2 9 2 30 25 1 9 2 7 13 2 10 1 12 12 10 10
11
Pork
Sanzetal, 1987 Pork/soy
Muzillaetal, 1990
Model foods
11 4 4
281
Amioca
Maroulis etal, 1990 Laietal, 1992 Drouzasetal, 1991 Maroulis et al, 1991 Drouzasetal, 1988 Hylon-7
Maroulis etal, 1990 Laietal, 1992 Maroulis et al, 1991 Drouzasetal, 1988
51 7 18 8
9 9 43 9 19 9 6
Chapter 8
298
Table 8.4 Continued Material
Reference
Potato starch
Okosetal.,1986 Starch
Renaudetal, 1991 Njieetal, 1998 Maroulis et al, 1991 Morley et al, 1997
Wangetal.,1993 Lanetal, 2000
Sucrose Renaudetal., 1991 Ziegleretai., 1985
Gelatin Renaudetal, 1991 Okosetal.,1986 Ovalbumin
Renaudetal, 1991 Cornillon et al, 1995
Tylose
Phametal, 1990
Agar-water Delgado et al, 1997 Barringer et al, 1995 Bentonite-water
Sheen et al, 1993 Gelatin-water
HalUdayetal.,1995 Amylose Voudouris et al, 1995
Cellulose gum Saravacos et al,1965 Pectin 5% Saravacos et al.,1965
Pectin 10% Saravacos el al.,1965
Pectin 5%-glucose 5% Saravacos et al.,1965
Gelatin-sucrose-water Hallidayetal, 1995
Glycerin
Ryniecki et al, 1993
Data 2 2 61 24 1 6 6 18 6 33 30 3 26 24 2 36 24 12 6
6 2 1 1 1 1 1 1 4 4 2 2 2 2 2 2 2 2 2 2 3 3
299
Thermal Conductivity and Diffusivity of Foods
Table 8.4 Continued Material
Reference
Nuts Macadamia
Rahman, 1991
Data
1 1 1
134
Other Coconut
Duane et a!., 1995
Chenetal, 1998 Coffee
Sagaraetal., 1994
Soybean Okos et al, 1986
Palm kernel Duane etal, 1996
Lard
10 1 9 10 10 12 12 1 1 1
Duane et al., 1997
Agar-water Wang etal, 1992
Water-NaCI Lucas etal, 1999
Water-sucrose Lucas etal, 1999
Rapeseed Bilanskietal, 1976 Moyseyetal, 1977 Okos eta!., 1986 Tobacco
Casadaetal, 1989 Sorghum
Changetal, 1980 Okos etal. ,1986 Sugar
Okos etal, 1986 Albumen
Okos etal, 1986 NaCl
Okos etal, 1986
1 52 18 10 24 3 3 7 3 4 15 15 2 2 3 3
Chapter 8
300
Table 8.4 Continued Material
Reference
Honey Okos et al.,1986 Albumine
Okos eta!., 1986
Vegetables
Data 12 12 3 3
154
Carrot
Niesteruk, 1998 Njie et al, 1998 Rahman, 1991 Buhri et al, 1993
Cassava Njieetal, 1998
Garlic Madamba et al, 1995
Onion
Rapusasetal, 1994 Pea
Sastry et al, 1983 Alagusundarametal, 1991
Potato Niesteruk, 1996 Niesteruk, 1997 Niesteruk, 1998
Hungetal.,1983 Luelal, 1999 Njieetal, 1998
WangetaL, 1992 Rahman, 1991
Matteaetal, 1986 Hallidayetal, 1995 Madamba eta!., 1995 Buhri eta!., 1993 Cratzeketal.,1993
Sugar beet
5 1 1 2 1 6 6 3 3 7 7 12 3 9 45 1 1 1 2 1 2 16 2 3 9 2 1 4 7
Niesteruk, 1998 Okos etal, 1986
Turnip Buhri et al, 1993
4 3 1 1
Thermal Conductivity and Diffusivity of Foods
Table 8.4 Continued Material
Reference
Yam Njieetal., 1998 Beetroot Niesteruk, 1998
Parsley Niesteruk, 1998
Celery
Niesteruk, 1998
Tomato Dincer, 1997 Choietd.,1983 Filkovaetal, 1987
Okos eta!., 1986 Drouzasetal., 1985
Cucumber
Dincer, 1997 Spinach
Delgado et al, 1997 Mushrooms Shrivastavaetal, 1999
Rutabagas Buhri et al, 1993 Radish
Buhrietal, 1993
Parsnip Buhri etai, 1993 Kidney bean
Zuritz et al, 1989
301
Data 6 6 2 2 1 1 1 1 31 1 9 3 9 9 1 1 10 10 9 9 1 1 1 1 1 1 4 4
Chapter 8
302
Table 8.5 Thermal Conductivity of Foods Versus Moisture and Temperature: Average Values of Available Data Temperature
(W/mK)
Moisture (kg/kg db)
(°C)
Data
Baked products
0.34
0.57
46
60
Conductivity Material
Bread
0.23
0.39
44
14
-
0.27
0.35
47
9
Crust
0.06
0.00
68
2
Crumb
0.26
0.76
17
3
Dough
0.34
0.89
60
20
Wheat bread
0.41
0.78
23
3
Rye bread
0.47
1.06
20
3
Biscuit
0.40
0.07
20
2
Soy
0.35
0.33
150
3
Soy flour
0.22
0.21
27
11
Defatted
0.44
0.36
25
4
Dry defatted
0.12
0.00
40
3
Cup cake batter
0.17
0.63
15
2
-
0.17
0.63
15
2
Yellow cake batter
0.22
0.71
20
1
-
0.22
0.71
20
1
Cake
0.25
0.56
51
12
0.25
0.56
51
12
Cereal products
0.29
0.67
40
76
"
Barley
0.20
0.18
0
9
Seeds
0.20
0.18
0
9
Corn
0.39
1.55
36
21
Dent
0.16
0.20
36
3
Shelled
0.55
0.73
30
9
Dust
0.09
0.15
22
3
Syrup
0.43
4.16
52
6
Thermal Conductivity and Diffusivity of Foods
303
Table 8.5 Continued Conductivity Material
(W/mK)
Moisture (kg/kg db)
Temperature (°C)
Data 13
Rice
0.15
0.22
48
Paddy
0.15
0.22
48
13
Wheat
0.26
0.13
26
10
Dust
0.07
0.15
22
3
Hard red spring
0.16
0.17
-3
2
Soft white
0.13
0.13
16
2
Flour
0.59
0.10
56
3
Corn meal
0.36
0.27
88
7
-
0.36
0.27
88
7
Idli batter
0.45
1.71
16
4
-
0.45
1.71
16
4
Maize
0.27
0.32
62
11
Kernel
0.17
0.16
50
2
Grits
0.29
0.36
65
9
Oat
0.13
0.14
27
1
White
0.13
0.14
27
1
Dairy
0.45
3.78
38
136
Cheese
0.42
1.20
22
23
Cheddar
0.35
0.56
23
2
Mozzarella
0.38
0.80
23
2
Cuartirolo Argentina
0.37
1.20
15
1
Hamburger
0.39
0.69
23
2
Old Kashkaval
0.38
0.69
23
2
Tulum
0.38
0.69
23
2
Fresh Kashkaval
0.40
0.78
23
2
Buffet Kashkaval
0.41
0.99
23
2
Fresh cream
0.43
1.29
23
2
Spreadable cheese
0.49
1.54
23
2
Labne
0.47
2.24
23
2
Low fat labne
0.55
2.94
23
2
Chapter 8
304
Table 8.5 Continued (W/mK)
Moisture (kg/kg db)
Temperature (°C)
Data
Milk
0.46
4.02
46
84
-
0.46
4.00
20
3
Conductivity
Material
Fresh
0.57
9.00
23
1
Powder
0.11
25.33
25
3
Whole
0.46
3.04
53
18
Skim
0.57
6.33
36
15
Concentrated
0.41
0.92
50
9
Condensed
0.49
4.01
54
9
Half-half
0.54
5.11
40
9
Baby food
0.55
0.03
50
6
Powdered
0.30
0.05
54
11
Butter
0.22
0.20
21
5
-
0.23
0.18
23
2
Fat
0.21
0.21
20
3
Yogurt
0.45
3.60
31
19
-
0.56
6.25
21
2
Plain
0.33
2.05
40
9
Strained
0.54
2.88
23
2
Pasterized
0.58
4.71
23
2
Light
0.58
4.54
23
2
Extra light
0.59
6.58
23
2
Whey
0.59
9.00
40
4
-
0.59
9.00
40
4
Cream
0.13
44.00
25
1
Powder
0.13
44.00
25
1
Fish
0.79
3.29
8
83
Cod
1.23
4.88
-10
5
Perpendicular
1.23
4.88
-10
5
Mackerel
0.80
3.42
0
5
Perpendicular
0.80
3.42
0
5
Thermal Conductivity and Diffusivity of Foods
305
Table 8.5 Continued Temperature
(W/mK)
Moisture (kg/kg db)
(°C)
Data
Conductivity
Material Squid
0.35
2.36
26
16
Fresh
0.50
5.04
30
Mantle
0.50
3.83
15
3 2
Dried
0.24
0.87
30
9
Tentacle, arrow
0.48
3.56
15
1
Tentacle
0.50
3.56
15
1
Carp
1.21
0.83
0
2
-
1.21
0.83
0
2
Surimi
0.78
4.02
14
30
-
0.85
4.08
-2
7
6% cryoprotectant cone
0.87
4.08
-2
7
12% cryoprotectant cone
0.86
4.08
-2
7
Pacific whiting
0.59
3.88
53
9
Cake
0.10
0.00
15
1
Pressed
0.10
0.00
15
1
Shrimp
1.03
3.24
-8
13
peeled and head removed
1.03
3.24
-8
13
Calamari
0.51
4.04
15
2
mantle
0.51
4.04
15
2
Salmon
1.06
2.40
-12
9
Perpendicular
1.06
2.40
-12
9
Fruits
0.45
3.73
30
143
Apple
0.45
3.44
30
82
-
0.32
2.44
20
25
Red
0.51
5.60
15
1
Green
0.41
0.14
45
1
Golden delicious
0.41
4.89
18
4
Granny Smith
0.19
2.17
25
3
37
47
Juice
0.53
3.81
Sauce
0.59
10.11
1
Chapter 8
306
Table 8.5 Continued Moisture
Conductivity (W/mK)
(kg/kg db)
Temperature (°C)
0.48
3.12
20
Dessert
0.48
3.12
20
Peach
0.04
Freeze-dried
0.04
35
1 1 1 1
Plantain
0.37
0.98
30
6
Fruits
0.37
0.98
30
6
Pear -
0.49
3.22
34
15
0.47
2.85
24
4
Material Banana
35
Data
Green
0.52
7.41
15
2
Juice
0.51
2.60
50
6
Williams
0.45
2.17
25
3
Orange
0.41
4.28
27
15
Juice
0.41
4.28
27
15
Bilberry
0.55
8.52
18
2
Juice
0.55
8.52
18
2
Cherry
0.55
6.52
18
2
Juice
0.55
6.52
18
2
Grape
0.52
4.08
42
8
-
0.52
3.60
50
6
Juice
0.55
5.54
18
2
Raspberry
0.55
7.70
18
2
Juice
0.55
7.70
18
2
Strawberry
0.63
8.44
14
5
Juice
0.57
11.05
18
2
Tioga
0.67
6.70
11
3
Raisin
0.23
1.35
45
4
-
0.23
1.35
45
4
Legumes
0.22
0.18
2
9
Lentils
0.22
0.18
2
9
Seeds
0.22
0.18
2
9
Thermal Conductivity and Diffusivity of Foods
307
Table 8.5 Continued Material
Conductivity
Moisture
Temperature
(W/mK)
(kg/kg db)
(°C)
Data
Meat
0.71
2.40
16
134
Beef
0.63
2.22
28
75
-
0.54
2.43
1
2
Fat
0.28
0.14
7
3
Lean
1.03
0.63
0
2
Ground
1.01
0.75
0
2
Minced
0.56
1.96
11
5
Muscle semitendinosus
0.31
0.99
20
5
Dryfiber
0.21
1.03
25
4
Boneless
1.03
3.31
-4
20
Ground round
0.51
2.53
60
3
Whole round
0.49
2.32
60
3
Ground shank
0.51
2.45
60
3
Ground brisket
0.44
2.38
60
3
Whole rib steak
0.50
1.84
60
3
Ground sirloin tip
0.49
2.35
60
3
Whole sirloin tip
0.48
2.27
60
3
Ground rib
0.41
1.11
60
3
Ground s\viss steak
0.51
2.90
60
3
Whole swiss steak
0.49
2.82
60
3
Loaf, uncooked
0.40
2.58
15
1
Loaf heated
0.47
1.96
60
1
Chicken
1.05
3.46
-9
9
Boneless
0.97
3.00
-4
7
White
1.33
5.10
-25
2
Chapter 8
308
Table 8.5 Continued Conductivity
Material
(W/mK)
Sausage
0.42
Moisture (kg/kg db)
Temperature (°C)
Data
1.02
18
13
-
0.33
1.24
22
4
Italian
0.93
0.64
0
2
Salami cooked
0.37
1.70
22
1
Lebanon bologna
0.36
1.63
22
1
Salami cotto
0.37
1.33
22
1
Thuringer
0.35
0.96
22
1
Salami Genoa
0.30
0.56
22
1
Salami hard
0.32
0.52
22
1
Pepperoni
0.28
0.37
22
1
Turkey
1.18
2.85
-12
12
Boneless
1.18
2.85
-12
12
Mutton
0.86
2.66
-3
10
Boneless
0.86
2.66
-3
10
Pork Boneless
0.93
3.25
-4
0.93
3.25
-4
11 11
Pork/soy
0.05
3.41
25
4
Unprocessed
0.05
3.17
25
2
Processed
0.05
3.64
25
2
Model foods
0.63
5.26
30
281
Amioca
0.32
3.35
52
51
-
0.25
2.86
58
20
Gelatinized
0.51
1.23
52
16
Powder
0.13
10.37
48
9
Granular
0.34
0.12
40
6
Thermal Conductivity and Diffusivity of Foods
309
Table 8.5 Continued Conductivity (W/mK)
Moisture (kg/kg db)
Hylon-7
0.33
5.08
60
43
Gelatinized
0.22
5.99
85
16
0.53
1.90
46
15
Powder
0.13
15.55
48
6
Granular
0.34
0.12
40
6
Material
Temperature (°C) Data
Potato starch
0.04
0.08
41
2
Gel
0.04
0.08
41
2
Starch
0.68
4.71
42
61
-
0.10
0.00
25
1
Gel
1.02
8.80
4
29
Gelatinized
0.34
0.10
50
3
Hydrated
0.38
0.28
47
3
Granular
0.09
0.20
45
7
Gels
0.50
1.65
100
18
Sucrose
0.85
3.57
-1
33
-
0.48
3.72
20
3
Gel Gelatin
0.89
3.56
-4
30
0.96
7.92
0
26
Gel
0.96
7.92
0
26
Ovalbumin
0.99
7.59
-5
36
-
0.88
4.23
-7
12
Gel
1.05
9.28
-4
24
Xylose
0.99
3.35
5
6
Gel
0.99
3.35
5
6
Agar-water
0.61
36.95
25
2
Gel Gelatin-water -
0.61
36,95
25
2
0.59
65.67
25
1
0.59
65.67
25
1
Amylose
0.53
3.50
30
4
Gel
0.53
3.50
30
4
Chapter 8
310
Table 8.5 Continued Conductivity Material
(W/mK)
Moisture (kg/kg db)
Temperature (°C)
Data
Cellulose gum
0.06
0.08
41
2
Freeze-dried gel
0.06
0.08
41
2
Pectin 5%
0.04
0.08
41
2
Freeze-dried gel
0.04
0.08
41
2
Pectin 10%
0.05
0.08
41
2
Freeze-dried gel
0.05
0.08
41
2
Pectin 5%-glucose 5%
0.05
0.08
41
2
Freeze-dried gel
0.05
0.08
41
2
Glycerin
0.47
3.79
20
3
-
0.47
3.79
20
3
0.44
1.44
15
2
0.44
1.44
15
2
Nuts
0.22
0.02
15
1
Macadamia
0.22
0.02
15
1
Integrifolia
0.22
0.02
15
1
Vegetables
0.43
3.81
39
154
Carrot
0.48
5.85
22
5
-
0.45
3.82
27
3
Large
0.52
8.91
15
2
Cassava
0.47
1.22
30
6
Roots
Gelatin-sucrose-water
"
0.47
1.22
30
6
Garlic
0.36
0.80
15
3
-
0.36
0.80
15
3
Onion .
0.42
2.05
32
7
0.42
2.05
32
7
311
Thermal Conductivity and Diffusivity of Foods
Table 8.5 Continued Material
Conductivity
Moisture
Temperature
(W/mK)
(kg/kg db)
(°C)
Data
Pea
0.22
0.18
2
9
Seeds
0.22
0.18
2
9
Potato
0.45
2.35
49
45
-
0.42
2.74
57
25
Mashed
1.22
0.72
0
2
Flesh
0.54
4.54
20
1
Granule
0.35
0.64
62
10
White
0.53
4.55
18
4
Spunta
0.46
2.17
25
3
Sugar beet
0.53
3.38
22
7
25
3
20
4
-
0.56
4.22
Roots
0.52
2.75
Turnip
0.48
0.08
45
1
-
0.48
0.08
45
1
Yam
0.47
1.45
30
6
Tubers
0.47
1.45
30
6
Beetroot
0.56
9.10
20
2
-
0.56
9.10
20
2
Parsley
0.17
2.30
20
1
-
0.17
2.30
20
1
Celery
0.15
2.30
20
1
-
0.15
2.30
20
1
Tomato
0.51
6.23
68
31
-
0.61
15.60
21
1
Juice
0.48
7.71
83
21
Paste
0.55
1.73
40
9
Cucumber
0.62
24.00
22
1
-
0.62
24.00
22
1
Spinach
0.38
11.01
-2
10
Fresh
0.37
13.66
-2
5
Blanched
0.39
8.35
-2
5
Chapter 8
312
Table 8.5 Continued (W/mK)
Moisture (kg/kg db)
Temperature (°C)
Data
Mushrooms
0.37
3.27
55
9
Pleurotusflorida
0.37
3.27
55
9
Rutabagas
0.45
0.08
45
1
-
0.45
0.08
45
1
Radish
0.50
0.06
45
1
-
0.50
0.06
45
1
Parsnip
0.39
0.21
45
1
-
0.39
0.21
45
1
Kidney bean
0.15
0.24
20
4
-
0.15
0.24
20
4
Other
0.23
2.06
25
134
Coconut
0.15
3.08
37
10
Milkpowder
0.15
3.08
37
10
Coffee
0.21
1.62
6
10
Solutions
Conductivity
Material
0.21
1.62
6
10
Soybean
0.09
0.14
34
12
Powder
0.08
0.10
36
3
Whole
0.11
0.13
36
3
Crushed
0.10
0.11
36
3
Flour
0.05
0.22
26
3
Palm kernel
0.10
26.00
25
1
Milkpowder
0.10
26.00
25
1
Lard
0.12
32.00
25
1
Milkpowder
0.12
32.00
25
1
Water-Nad
0.46
4.00
10
1
Solution
0.46
4.00
10
1
Water-sucrose
0.32
0.67
10
1
Solution
0.32
0.67
10
1
Thermal Conductivity and Diffusivity of Foods
313
Table 8.5 Continued Material
Conductivity (W/mK)
Moisture (kg/kg db)
Temperature (°C)
Data
Rapeseed
0.11
0.10
14
52
mole
0.13
0.11
17
21
Ground
0.07
0.11
16
9
Torch
0.10
0.10
-4
9
Midas
0.09
0.01
19
1
Crushed
0.13
0.11
18
12
Agar-water
0.62
19.90
30
1
Gel
0.62
19.90
30
1
Tobacco
0.06
0.26
15
3
-
0.06
0.26
15
3
Sugar
0.52
4.32
42
15
Glucose
0.54
4.80
44
6
Cane sugar
0.51
4.00
40
9
Albumen
0.04
0.08
41
2
Freeze-dried gel
0.04
0.08
41
2
Sorghum
0.24
0.17
21
7
Rs610
0.14
0.22
5
2
NC+RS66
0.56
0.16
36
2
Grain dust
0.09
0.15
22
3
NaCl
0.61
4.00
43
3
Solution
0.61
4.00
43
3
Honey
0.53
4.83
36
12
Albumine
0.53
4.83
36
12
0.41
0.67
60
3
Solution
0.41
0.67
60
3
314
Chapter 8
Table 8.6. Thermal Conductivity of Foods Versus Moisture and Temperature: Variation Range of Available Data Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Max Min Min Max Min Max 15
150
0.82
15
120
0.79
25
100
0.00
0.00
15
120
0.72
0.82
15
18
0.04
1.17
20
150
0.00
1.17
0.055
0.650 0.530
0.00
0.080
0.530
0.05
Crust
0.055
0.066
Crumb
0.232
0.298
Dough
0.230
0.600
Baked products
0.048
Bread -
Wheat bread
0.327
0.500
0.72
0.82
20
28
Rye bread
0.396
0.600
0.85
1.17
20
20
Biscuit
0.390
0.405
0.04
0.09
20
20
Soy
0.230
0.488
0.10
0.60
150
150
Soy flour
0.106
0.650
0.00
0.64
20
60
defatted
0.180
0.650
0.10
0.64
25
25
dry defatted
0.106
0.143
0.00
0.00
20
60
Cup cake barter
0.121
0.223
0.55
0.71
15
15
-
0.121
0.223
0.55
0.71
15
15
Yellow cake batter
0.223
0.223
0.71
0.71
20
20
-
0.223
0.223
0.71
0.71
20
20
Cake
0.048
0.356
0.11
1.22
20
103
-
0.048
0.356
0.11
1.22
20
103
Cereal products
0.067
0.740
0.01
8.09
-28
160
Barley
0.167
0.225
0.11
0.26
-28
29
Seeds
0.167
0.225
0.11
0.26
-28
29
Corn
0.085
0.740
0.01
8.09
10
77
Dent
0.142
0.175
0.01
0.42
36
36
Shelled
0.371
0.740
0.40
1.00
10
50
Dust
0.085
0.101
0.10
0.20
22
22
Syrup
0.347
0.513
0.23
8.09
27
77
315
Thermal Conductivity and Diffusivity of Foods
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Max Min Max Min Max Min 0.366
0.11
0.082
0.366
0.067
0.689
Dust
0.067
Hard red spring
0.144
Soft white
0.118
Rice
0.082
Paddy Wheat
70
0.43
20
0.11
0.43
20
70
0.01
0.29
-3
66
0.073
0.10
0.20
22
22
0.166
0.05
0.29
-3
-3
0.140
0.01
0.25
15
16
Flour
0.450
0.689
0.10
0.10
43
66
Corn meal
0.270
0.464
0.18
0.43
20
160
-
0.270
0.464
0.18
0.43
20
160
Idli batter
0.395
0.493
1.00
2.33
15
20
-
0.395
0.493
1.00
2.33
15
20
Maize
0.067
0.525
0.11
0.59
35
95
Kernel
0.156
0.174
0.11
0.20
50
50
Grits
0.067
0.525
0.16
0.59
35
95
Oat
0.130
0.130
0.14
0.14
27
27
White
0.130
0.130
0.14
0.14
27
27
Dairy
0.039
0.686
0.02
44.00
1
90
Cheese
0.345
0.548
0.56
2.94
15
30
Cheddar
0.345
0.351
0.56
0.56
15
30
Mozzaretta Cuartirolo ArgenTino
0.380
0.383
0.80
0.80
15
30
0.372
0.372
1.20
1.20
15
15
Hamburger
0.381
0.398
0.69
0.69
15
30
Old Kashkaval
0.368
0.384
0.69
0.69
15
30
Tulum
0.377
0.379
0.69
0.69
15
30
0.78
15
30
Fresh Kashkaval
0.403
0.403
0.78
Buffet Kashkaval
0.406
0.409
0.99
0.99
15
30
Fresh cream
0.433
0.434
1.29
1.29
15
30
Spreadable cheese
0.476
0.494
1.54
1.54
15
30
Labne
0.463
0.486
2.24
2.24
15
30
Low fat labne
0,542
0.548
2.94
2.94
15
30
Chapter 8
316
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min
Max
Min
Max
Min
Max
Milk
0.112
0.686
0.02
30.00
5
90
-
0.325
0.576
1.00
9.00
20
20
Fresh
0.570
0.570
9.00
9.00
23
23
Powder
0.112
0.115
22.00
30.00
25
25
mole
0.280
0.629
0.39
9.00
5
90
Skim
0.481
0.646
1.50
19.00
5
75
Concentrated
0.325
0.498
0.43
1.50
35
65
Condensed
0.325
0.634
1.00
9.00
23
79
Half-half
0.471
0.634
2.33
9.00
5
75
Baby food
0.405
0.686
0.03
0.04
35
65
Powdered
0.182
0.538
0.02
0.14
54
54
Butter
0.093
0.345
0.02
0.42
15
30
-
0.227
0.233
0.18
0.18
15
30
Fat
0.093
0.345
0.02
0.42
20
20
Yogurt
0.039
0.639
0.06
6.58
1
55
-
0.525
0.603
6.25
6.25
1
40
Plain
0.039
0.639
0.06
5.66
25
55
Strained
0.539
0.540
2.88
2.88
15
30
Pasterized
0.571
0.593
4.71
4.71
15
30
Light
0.571
0.583
4.54
4.54
15
30
Extra light
0.584
0.596
6.58
6.58
15
30
Whey
0.547
0.642
9.00
9.00
7
87
-
0.547
0.642
9.00
9.00
7
87
Cream
0.127
0.127
44.00
44.00
25
25
Powder
0.127
0.127
44.00
44.00
25
25
Fish
0.040
1.720
0.00
5.25
-40
80
Cod
0.549
1.543
4.88
4.88
-22
3
Perpendicular
0.549
1.543
4.88
4.88
-22
3
Mackerel Perpendicular
0.409
1.428
3.42
3.42
-20
20
0,409
1.428
3.42
3.42
-20
20
317
Thermal Conductivity and Diffusivity of Foods
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Max Max Min Max Min
Squid
0.040
0.507
0.10
Fresh
0.490
0.500
Mantle
0.483
0.507
Dried
0.040
0.440
Tentacle, arrow
0.475
0.475
Tentacle
0.501
0.501
5.20
15
30
4.75
5.20
30
30
3.83
3.83
15
15
0.10
2.86
30
30
3.56
3.56
15
15
3.56
3.56
15
15
Carp
0.700
1.720
0.83
0.83
-15
15
-
0.700
1.720
0.83
0.83
-15
15
Surimi
0.477
1.508
2.85
5.25
-40
80
-
0.487
1.473
4.08
4.08
-40
30
6% cryoprotectant
0.477
1.508
4.08
4.08
-40
30
12% cryoprotectant
0.489
1.465
4.08
4.08
-40
30
Pacific whiting
0.524
0.708
2.85
5.25
30
80
Cake
0.100
0.100
0.00
0.00
15
15
Pressed
0.100
0.100
0.00
0.00
15
15
Shrimp Peeled and head removed
0.490
1.600
1.00
4.20
-30
30
0.490
1.600
1.00
4.20
-30
30
4.04
4.04
15
15
Calamari
0.508
0.517
Mantle
0.508
0.517
4.04
4.04
15
15
Salmon
0.497
1.245
2.03
2.70
-24
5
Perpendicular
0.497
1.245
2.03
2.70
-24
5
Fruits
0.043
2.270
0.14
19.00
-40
90
Apple
0.070
2.270
0.14
19.00
-40
90
-
0.070
1.510
0.25
5.99
-40
45
Red
0.513
0.513
5.60
5.60
15
15
Green
0.405
0.405
0.14
0.14
45
45
Golden delicious
0.401
0.412
4.88
4.89
15
20
Granny Smith
0.090
0.296
0.50
4.00
25
25
Juice
0.230
2.270
0.25
19.00
-7
90
Sauce
0.591
0.591
10.11
10.11
318
Chapters
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Max Min Min Max Min Max
Banana
0.481
0.481
3.12
3.12
20
20
Dessert
0.481
0.481
3.12
3.12
20
20
Peach
0.043
0.043
35
35
Freeze-dried
0.043
0.043
35
35
Plantain
0.130
0.520
0.16
2.00
30
30
Fruits
0.130
0.520
0.16
2.00
30
30
Pear
0.340
0.629
0.50
7.41
15
80
-
0.340
0.557
0.50
4.90
23
25
Green
0.514
0.533
7.41
7.41
15
15
Juice
0.402
0.629
0.64
5.67
20
80
Williams
0.359
0.505
0.50
4.00
25
25
Orange
0.290
0.560
0.64
19.00
1
62
Juice
0.290
0.560
0.64
19.00
1
62
Bilberry
0.553
0.554
8.52
8.52
16
20
Juice
0.553
0.554
8.52
8.52
16
20
Cherry
0.553
0.554
6.52
6.52
16
20
Juice
0.553
0.554
6.52
6.52
16
20
Grape
0.396
0.639
0.59
8.09
16
80
-
0.396
0.639
0.59
8.09
20
80
Juice
0.537
0.556
5.54
5.54
16
20
Raspberry
0.544
0.553
7.70
7.70
16
20
Juice
0.544
0.553
7.70
7.70
16
20
Strawberry
0.520
0.935
6.70
11.05
-15
28
Juice
0.571
0.571
11.05
11.05
16
20
Tioga
0.520
0.935
6.70
6.70
-15
28
Raisin
0.126
0.392
0.16
4.00
45
45
-
0.126
0.392
0.16
4.00
45
45
Legumes
0.187
0.253
0.11
0.26
-21
28
Lentils
0.187
0.253
0.11
0.26
-21
28
Seeds
0.187
0.253
0.11
0.26
-21
28
Thermal Conductivity and Diffusivity of Foods
319
Table 8.6. Continued Material Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)
Meat
Min
Max
Min
Max
Min
Max
0.049
1.660
0.01
5.10
-40
90
Beef
0.095
1.650
0.01
3.69
-30
90
Fat Lean
0.454
0.622
2.28
2.57
-18
20
0.264
0.311
0.10
0.16
-10
15
0.510
1.550
0.63
0.63
-15
15
Ground
0.400
1.620
0.75
0.75
-15
15
Minced Muscle semitendinosus
0.360
0.844
1.11
3.44
-5
30
0.01
2.84
20
20 25 30
0,095
0.490
Dry fiber
0,140
0.243
0.38
2.30
25
Boneless
0.429
1.650
2.92
3.69
-30
Ground round
0.452
0.590
1.99
2.94
30
90
Whole round
0.475
0.504
1.50
2.94
30
90
Ground shank
0.442
0,598
1.58
2.92
30
90
Ground brisket
0.436
0.458
1.36
3.05
30
90
Whole rib steak
0.459
0.552
1.07
2.32
30
90
Ground sirloin tip
0.460
0.518
1.61
2.92
30
90
Whole sirloin tip
0.467
0.494
1.30
2.92
30
90 90
Ground rib
0.368
0.450
0.78
1.37
30
Ground Swiss steak
0.467
0.575
2.16
3.44
30
90
Whole swiss steak
0.467
0.508
1.84
3.44
30
90
Loaf, uncooked
0.400
0.400
2.58
2.58
15
15
Loaf, heated
0.470
0.470
1.96
1.96
60
60
Chicken
0.490
1.452
2.91
5.10
-25
20
Boneless
0.490
1.452
2.91
3.22
-20
20
White
1.268
1.387
5.10
5.10
-25
-25
320
Chapter 8
Table 8.6. Continued
Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Min Max Max Max Min
Sausage
0.275
1.380
0.37
1.86
-10
22
-
0.283
0.367
0.40
1.86
20
22
Italian
0.470
1.380
0.64
0.64
-10
10
Salami cooked
0.370
0.370
1.70
1.70
22
22
Lebanon bologna
0.355
0.355
1.63
1.63
22
22
Salami cotto
0.365
0.365
1.33
1.33
22
22
Thuringer
0.345
0.345
0.96
0.96
22
22
Salami genoa
0.295
0.295
0.56
0.56
22
22
Salami hard
0.315
0.315
0.52
0.52
22
22
Pepperoni
0.275
0.275
0.37
0.37
22
22
Turkey
0.490
1.660
2.85
2.85
-24
4
Boneless
0.490
1.660
2.85
2.85
-24
4
Mutton
0.391
1.510
2.45
2.80
-40
24
Boneless
0.391
1.510
2.45
2.80
-40
24
Pork
0.480
1.450
3.15
3.31
-30
30
Boneless
0.480
1.450
3.15
3.31
-30
30
Pork/soy
0.049
0.055
3.08
3.75
25
25
Unprocessed
0.049
0.051
3.08
3.25
25
25
Processed
0.053
0.055
3.54
3.75
25
25
Model Foods
0.038
2.330
0.00
65.67
-43
150
Amioca
0.080
0.661
0.00
20.00
20
150
Gelatinized
0.432
0.661
0.01
3.00
20
135
Powder
0.080
0.195
0.00
20.00
25
70
Granular
0.227
0.454
0.01
0.23
20
60
321
Thermal Conductivity and Diffusivity of Foods
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)
Min
Max
Min
Max
Min
Max
Hylon-7
0.100
0.661
0.00
Gelatinized
0.442
0.661
0.01
20.00
20
150
4.00
20
70
Powder
0.100
0.160
11.10
Granular
0.227
20.00
25
70
0.01
0.23
20
0.454
Potato starch
0.039
0.041
60
0.02
0.14
41
41
Gel
0.039
0.041
0.02
0.14
41
41
Starch
0.061
2.100
0.00
24.00
-42
120
-
0.100
0.100
0.00
0.00
25
25
Gel
0.480
2.100
1.78
24.00
-42
50
Gelatinized
0.330
0.355
0.10
0.10
20
80
Hydrated
0.364
0.388
0.28
0.28
10
80
Granular
0.061
0.125
0.05
0.30
15
75
Gels
0.436
0.567
0.66
3.00
80
120
Sucrose
0.350
1.770
0.67
9.00
-41
32
-
0.405
0.566
0.67
9.00
20
20
Gel
0.350
1.770
1.00
9.00
-41
32
Gelatin
0.039
2.070
0.02
19.00
-41
41
Gel
0.039
2.070
0.02
19.00
-41
41
Ovalbumin
0.450
2.330
2.30
19.00
-43
26
-
0.470
1.750
2.30
6.40
-43
20
Gel
0.450
2.330
3.20
19.00
-42
26
Tylose
0.483
1.530
3.35
3.35
-30
50
Gel
0.483
1.530
3.35
3.35
-30
50
Agar-water
0.600
0.622
24.90
49.00
20
30
Gel
0.600
0.622
24.90
49.00
20
30
Gelatin-water
0.594
0.594
65.67
65.67
25
25
-
0.594
0.594
65.67
65.67
25
25
Amylose
0.515
0.551
3.00
4.00
30
30
Gel
0,515
0.551
3.00
4.00
30
30
Chapter 8
322
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Max Max Min Min Max
Cellulose gum
0.056
Freeze-dried gel
0.056
0.063
Pectin 5%
0.038
0.039
0.063
0.14
41
41
0.02
0.14
41
41
0.02
0.14
41
41
0.02
Freeze-dried gel
0.038
0.039
0.02
0.14
41
41
Pectin 10%
0.044
0.047
0.02
0.14
41
41
Freeze-dried gel
0.044
0.047
0.02
0.14
41
41
0.04S
0.050
0.02
0.14
41
41 41
Pectin 5%-glucose 5%
Freeze-dried gel
0.048
0.050
0.02
0.14
41
Glycerin
0.450
0.490
3.35
4.26
20
20
0.450
0.490
3.35
4.26
20
20
Gelatin-sucrosewater
0.396
0.487
0.65
2.22
15
15
-
0.396
0.487
0.65
2.22
15
15
Nuts
0.224
0.224
0.02
0.02
15
15
Macadamia
0.224
0.224
0.02
0.02
15
15
Integrifolia
0.224
0.224
0.02
0.02
15
15
Vegetables
0.103
0.670
0.06
24.00
-29
150
Carrot
0.182
0.605
0.15
9.00
15
45
9.00
15
45
8.91
15
15
-
0.182
0.605
0.15
Large
0.509
0.532
8.91
Cassava
0.160
0.570
0.22
2.33
30
30
Roots
0.160
0.570
0.22
2.33
30
30
Garlic
0.230
0.448
0.08
1.65
15
15
-
0.230
0.448
0.08
1.65
15
15
Onion _
0.290
0.520
0.32
4.15
31
33
0.520
0.32
4.15
31
33
0.290
Thermal Conductivity and Diffusivity of Foods
323
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Max Min Max Min Max
Pea
0.181
0.256
0.11
4.50
-29
28
Seeds
0.181
0.256
0.11
0.26
-21
28
Potato -
0.120
0.643
0.11
7.33
-15
130
0.209
0.643
0.34
7.33
24
130
Flesh
0.536
0.536
4.54
4.54
20
20
Granule
0.120
0.579
0.11
1.44
30
95
White
0.519
0.536
4.54
4.55
15
20
Spunta
0.331
0.550
0.50
4.00
25
25
Sugar beet
0.448
0.589
1.50
5.67
20
25
-
0.535
0.585
3.00
5.67
25
25 20 45
Roots
0.448
0.589
1.50
4.00
20
Turnip
0.480
0.480
0.08
0.08
45
45 30
-
0.480
0.480
0.08
0.08
45
Yam
0.160
0.600
0.19
3.76
30
Tubers
0.160
0.600
0.19
3.76
30
30
Beetroot
0.549
0.572
6.90
11.30
20
20
-
0.549
0.572
6.90
11.30
20
20
Parsley
0.170
0.170
2.30
2.30
20
20
-
0.170
0.170
2.30
2.30
20
20
Celery
0.147
0.147
2.30
2.30
20
20
-
0.147
0.147
2.30
2.30
20
20
Tomato
0.230
0.670
0.25
19.83
20
150
-
0.611
0.611
15.60
15.60
21
21
Juice
0.230
0.670
0.25
19.83
20
150
2.40
30
50
22
22
Paste
0.460
0.660
1.16
Cucumber
0.621
0.621
24.00
24.00
-
0.621
0.621
24.00
24.00
22
22
Spinach
0.347
0.434
8.35
13.66
-20
21
Fresh
0.347
0.400
13.66
13.66
-20
21
Blanched
0.356
0.434
8.35
8.35
-20
16
Chapter 8
324
Table 8.6. Continued Material
Mushrooms
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Min Max Max Min Max 0.218
0.520
0.11
8.69
40
70
Pleurotusflorida
0.218
0.520
0.11
8.69
40
70
Rutabagas
0.447
0.447
0.08
0.08
45
45
-
0.447
0.447
0.08
0.08
45
45
Radish
0.499
0.499
0.06
0.06
45
45
-
0.499
0.499
0.06
0.06
45
45
Parsnip
0.392
0.392
0.21
0.21
45
45
-
0.392
0.392
0.21
0.21
45
45
20
20
Kidney bean
0.103
0.201
0.12
0.41
0.103
0.201
0.12
0.41
20
20
Other
0.039
0.656
0.01
32.00
-26
90
Coconut
0.115
0.217
0.19
26.00
25
50
Milkpowder
0.115
0.217
0.19
26.00
25
50
Coffee
0.153
0.277
1.22
2.51
-14
26
Solutions
0.153
0.277
1.22
2.51
-14
26
Soybean
0.040
0.133
0.05
0.40
10
66
Powder
0.066
0.104
0.10
0,10
10
66
Whole
0.095
0.133
0.13
0.13
10
66
Crushed
0.085
0.126
0.11
0.11
10
66
Flour
0.040
0.061
0.05
0.40
26
26
Palm kernel
0.102
0.102
26.00
26.00
25
25
Milkpowder
0.102
0.102
26.00
26.00
25
25
Lard
0.120
0.120
32.00
32.00
25
25
Milkpowder
0.120
0.120
32.00
32.00
25
25
Water-NaCl
0.460
0.460
4.00
4.00
10
10
Solution
0.460
0.460
4.00
4.00
10
10
Water-sucrose
0.320
0.320
0.67
0.67
10
10
Solution
0.320
0.320
0.67
0.67
10
10
Thermal Conductivity and Diffusivity of Foods
325
Table 8.6. Continued Material
Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)
Min
Max
Min
Max
Min
Max
Rapeseed
0.060
0.155
0.01
0.24
-26
32
Whole
0.108
0.155
0.06
0.15
4
32
Ground
0.062
0.088
0.07
0.15
4
32
Torch
0.086
0.120
0.01
0.24
-26
19
Midas
0.092
0.092
0.01
0.01
19
19
Crushed
0.060
0.080
0.07
0.15
4
32
Agar-water
0.617
0.617
19.90
19.90
30
30
Gel
0.617
0.617
19.90
19.90
30
30
Tobacco
0.055
0.070
0.20
0.32
15
15
-
0.055
0.070
0.20
0.32
15
15
Sugar
0.382
0.637
0.67
9.00
0
80
Glucose
0.450
0.637
1.50
8.09
2
80
Cane sugar
0.382
0.637
0.67
9.00
0
80
Albumen
0.039
0.042
0.02
0.14
41
41
Freeze-dried gel
0.039
0.042
0.02
0.14
41
41
Sorghum
0.084
0.150
0.01
0.30
5
36
Rs6lO
0.130
0.150
0.15
0.28
5
5
Grain dust
0.084
0.094
0.10
0.20
22
22
NaCI
0.568
0.656
4.00
4.00
10
80
Solution
0.568
0.656
4.00
4.00
10
80
Honey
0.440
0.618
1.50
9.00
2
71
-
0.440
0.618
1.50
9.00
2
71
Albumine
0.382
0.425
0.67
0.67
27
90
Solution
0.382
0.425
0.67
0.67
27
90
Note: Thermal conductivities higher than that of water (0.62 W/mK at 25°C) are characteristic of frozen foods of high moisture content, since the thermal conductivity of ice is about 2 W/mK
Chapter 8
326
VI. THERMAL CONDUCTIVITY OF FOODS AS A FUNCTION OF MOISTURE CONTENT AND TEMPERATURE
A concept proposed by Maroulis et al. (2001) is adopted here and applied to obtain an integrated and uniform analysis of the available data. The concept was applied simultaneously to all the data of each material, regardless the data sources. Thus, the results are not based on the data of only one author and consequently they are of elevated accuracy. A simplified analysis is presented in Chapter 6 for the moisture diffusivity. Assume that a material of intermediate moisture content consists of a uniform mixture of two different materials: (a) a dried material and (b) a wet material with infinite moisture. The thermal conductivity is, generally, different for each material. The thermal conductivity of the mixture could be estimated using a two phase structural model: 1
X A, Y { 1 / ~r
:(T)
x
"
(8-13)
where /I (W/mK) the effective thermal conductivity, Ax (W/mK) the thermal conductivity of the dried material (phase a), Axi (W/mK) the thermal conductivity of the wet material (phase b), X (kg/kg db) the material moisture content, and T (°C) the material temperature. Assume that the thermal conductivities of both phases depend on temperature by an Arrhenius-type model:
= A0 exp
exp
R(T
T
(8-14)
(8-15)
where Tr =60°C a reference temperature, R = 0.0083\43kJImolKthe ideal gas constant, and A0, /l(., E0, Et are adjustable parameters of the proposed model. The reference temperature of 60°C was chosen as a typical temperature of air-drying of foods. Thus, the thermal conductivity for every material is characterized and described by four parameters with physical meaning:
Thermal Conductivity and Diffusivity of Foods
327
/l0(W/mK) thermal conductivity at moisture X = 0 and temperature T = Tr At (WlmK} thermal conductivity at moisture X = oo and temperature T = Tr Ea (kJ I moT) Activation Energy for heat conduction in dry material at X = 0 Et (kJ I mol) Activation Energy for heat conduction in wet material at X — co
The resulting model is summarized in Table 8.7 and can be fitted to data using a nonlinear regression analysis method. The model is fitted to all literature data for each material and the estimates of the model parameters are obtained. Then the residuals are examined and the data with large residuals are rejected. The procedure is repeated until an accepted standard deviation between experimental and calculated values is obtained (Draper and Smith, 1981). Among the available data only 13 materials have more than 10 data, which come from more than 3 publications. The procedure is applied to these data and the results of parameter estimation are presented in Table 8.8 and in Figure 8.18. It is clear that thermal conductivity is larger in wet materials. Figures 8.19-8.36 present retrieved thermal conductivities from the literature and model-calculated values for selected food materials as a function of moisture content and temperature. Thermal conductivity A, tends to increase with the moisture content X and the temperature T. The thermal conductivity parameters /10 and A/, shown in Figure 8.18, vary in the range of 0.05 to 1.0 W/mK. It should be noted that the thermal conductivity of air is about 0.026 W/mK, while that of water is 0.60 W/mK. Values of thermal conductivity of foods higher than 0.60 W/mK are normally found in frozen food materials (Aice=2 W/mK). The thermal conductivity increases, in general, with increasing moisture content. Temperature has a positive effect, which depends strongly on the food material. The energy of activation for heat conduction E is, in general, higher in the dry food materials.
328
Chapter 8
Table 8.7 Mathematical Model for Calculating Thermal Conductivity in Foods as a Function of Moisture Content and Temperature
Proposed Mathematical Model
X0exp
where
RT
T,
X . +——X.exp l +X
RT T
/i (W/mK) the thermal conductivity, X (kg/kg db) the material moisture content, T(°C) the material temperature, Tr = 60°C a reference temperature, and R = 0.0083143 kJ/mol K the ideal gas constant.
Adjustable Model Parameters
• • • •
Ka(W /mK) thermal conductivity at moisture X = 0 and temperature T = Tr "k.(W/ mK) thermal conductivity at moisture X = °o and temperature T = Tr E/U / mol) activation energy for heat conduction in dry material at X = 0 E,(kJ/ mol) activation energy for heat conduction in wet material at X = oo
329
Thermal Conductivity and Diffusivity of Foods
Table 8.8. Parameter Estimates of the Proposed Mathematical Model 4, E; E0 (W/mK) (kJ/mol) (kJ/mol)
sd (W/mK)
7.2
5.0
0.047
0.287 0.106 0.270
2.4 1.3 2.4
11.7 0.0 1.9
0.114 0.007 0.016
0.718 0.623 0.800
0.120 0.243 0.180
3.2 0.3 9.9
14.4 0.4
0.037 0.006 0.072
37 28
0.611 0.680
0.049 0.220
0.0 0.2
47.0 5.0
0.059 0.047
5
33
0.665
0.212
1.7
1.9
0.005
Beef
6
37
0.568
0.280
2.2
3.2
0.017
Other Rapeseed
3
35
0.239
0.088
3.6
0.6
0.023
3
15
0.800
0.273
2.7
0.0
0.183
Papers
No. of Data
3
15
1.580
0.070
12 4 5
68 13 15
0.589 0.642 0.658
5 4 3
29 24 21
12 5
No. of
Material
4 (W/mK)
Cereal products Corn
Fruits Apple Orange Pear
Model foods Amioca
Starch Hylon
Vegetables Potato Tomato
Dairy Milk
Meat
Baked products Dough
Chapter 8
330
I Moisture=infinite
Moist ure=zero
.t!
0,1
•I u
•a a o
U
0.01
100
• Moisture=infinite 0 Moisture=zero
O
S
e 10
&
a
.2
13
i
•^ u
o.i
o Figure 8.18 Parameter estimates of the proposed mathematical model.
Thermal Conductivity and Diffusivity of Foods
Moisture (kg/kg db)
Figure 8.19 Predicted values of thermal conductivity of fruits at 25°C.
331
Chapter 8
332
Moisture (kg/kg db)
Figure 8.20 Predicted values of thermal conductivity of fruits at 60°C.
Thermal Conductivity and Diffusivity of Foods
Moisture (kg/kg db)
Figure 8.21 Predicted values of thermal conductivity of vegetables at 25°C.
333
Chapter 8
334
Moisture (kg/kg db)
Figure 8.22 Predicted values of thermal conductivity of vegetables at 60°C.
Thermal Conductivity and Diffusivity of Foods
1.0
335
10.0
Moisture (kg/kg db)
Figure 8.23 Predicted values of thermal conductivity of miscellaneous foods at 25°C.
336
Chapter 8
Moisture (kg/kg db)
Figure 8.24 Predicted values of thermal conductivity of miscellaneous foods at 60°C.
Thermal Conductivity and Diffusivity of Foods
Fruits Total Number of Papers Total Experimental Points Points Used in Regression Analysis
Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %) Parameter Estimates
APPLE 12 73 68 0.11 142
Xi (W/mK) Xo (W/mK) Ei (kJ/mol) _____Eo (kJ/mol)
337
(93%)
0.59 0,29 2.45 11.7
1.0
10.0
Moisture (kg.kg db)
Figure 8.25 Thermal conductivity of apple at various temperatures and moisture contents.
Chapter 8
338
Fruits Total Number of Papers
ORANGE 4
Total Experimental Points Points Used in Regression Analysis
15 13
Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %) Parameter Estimates
(87%)
0.01 2
Xi (W/mK)
0.64
/u>(W/mK.) Ei(kJ/mol)
0.11 1.26
_____Eo (kJ/mol)______0.0
0.1
1.0
10.0
Moisture (kg.kg db)
Figure 8.26 Thermal conductivity of orange at various temperatures and moisture contents.
Thermal Conductivity and Diffusivity of Foods
Fruits Total Number of Papers
339
PEAR 5
Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %) Parameter Estimates Xi (W/mK) Xo (W/mK)
15 15 (100%) 0.02 10 0.66 0.27
Ei (kJ/mol) Eo(kJ/mol)
2.45 1.9
1 ————————————————————— ——— Temperature °C • 40
i
^
A A
^
**
V s^ +* s+ i ££
E I
»60 A80
,——L i ^ •• ••
<:&? u-E & ->
0
i
"5
a •a
o i
\
0.1 - ————— ———— —— — —— — -
0.1
- -r —— — -
1.0
-
-
-
-
10.0
Moisture (kg.kg db)
Figure 8.27 Thermal conductivity of pear at various temperatures and moisture contents.
340
Chapter 8
Vegetables
POTATO 12 45 37 (82%) Standard Deviation (sd, W/mK) 0.06 Relative Standard Deviation (rsd, %)_____2209 Parameter Estimates W(W/mK) 0.61 Xo (W/mK) 0.05 Ei (kJ/mol) 0.00 Eo (kJ/mol) 47.0 Total Number of Papers Total Experimental Points Points Used in Regression Analysis
0.1
1.0
10.0
Moisture (kg.kg db)
Figure 8.28 Thermal conductivity of potato at various temperatures and moisture contents.
341
Thermal Conductivity and Diffusivity of Foods
Vegetables
TOMATO
Total Number of Papers
5
Total Experimental Points
31
Points Used in Regression Analysis Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %)
28 (90%) 0.05 25
Parameter Estimates Xi (W/mK)
0.68
Xo (W/mK) Ei(kJ/mol)
0.22 0.17
____Eo (kJ/mol)______5.0
1.0
10.0
Moisture (kg.kg db)
Figure 8.29 Thermal conductivity of tomato at various temperatures and moisture contents.
Chapter 8
342
Model Foods Total Number of Papers
AMIOCA 5
Total Experimental Points 51 Points Used in Regression Analysis 29 (57%) Standard Deviation (sd, W/mK) 0.04 Relative Standard Deviation (rsd, %)______219 Parameter Estimates Xi (W/mK) 0,72 Xo(W/mK) 0.12
Ei (kJ/mol)
3.22
_____Eo (kJ/mol)______14.4
0.1
1.0
10.0
Moisture (kg.kg db)
Figure 8.30 Thermal conductivity of amioca (starch) at various temperatures and moisture contents.
Thermal Conductivity and Diffusivity of Foods
Model Foods
343
HYLON
Total Number of Papers 3 Total Experimental Points 43 Points Used in Regression Analysis 21 (49%) Standard Deviation (sd, W/mK) 0.07 Relative Standard Deviation (rsd, %) 9 Parameter Estimates Xi (W/mK) 0.80 Xo(W/mK) 0.18 Ei (kJ/mol) 9.90 _____Eo (kJ/mol)______0.0
Temperature °C -j • 40
1.0
10.0
Moisture (kg.kg db)
Figure 8.31 Thermal conductivity of hylon (starch) at various temperatures and moisture contents.
Chapter 8
344
Model Foods
STARCH
Total Number of Papers
4
Total Experimental Points
55
Points Used in Regression Analysis Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %)
24 (44%) 0.01 0
Parameter Estimates Xi (W/mK) Xo (W/mK) Ei (kJ/mol)
0.62 0.24 0.32
____Eo (kJ/mol)_____0.4 Temperature °C • 40 M 1
0.1
• 60
1.0
IT
10.0
Moisture (kg.kg db)
Figure 8.32 Thermal conductivity of starch at various temperatures and moisture contents.
Thermal Conductivity and Diffusivity of Foods
345
Dairy MILK Total Number of Papers 5 Total Experimental Points 84 Points Used in Regression Analysis 33 (39%) Standard Deviation (sd, W/mK) 0.01 Relative Standard Deviation (rsd, %)______6 Parameter Estimates Xi (W/mK) Xo(W/mK) Ei(kJ/mol)
0.67 0.21 1.73
_____Eo (kJ/mol)______1.9
0.1
1.0
10.0
Moisture (kg.kg db)
Figure 8.33 Thermal conductivity of milk at various temperatures and moisture contents.
Chapter 8
346
Cereal Products
CORN
Total Number of Papers Total Experimental Points Points Used in Regression Analysis
3 28 15
Standard Deviation (sd, W/mK)
(54%)
0.05
Relative Standard Deviation (rsd, %)______77
Parameter Estimates Xi (W/mK) Xo(W/mK)
0.47 0.31
Ei (kJ/mol)
0.00
____Eo (kJ/mol)_____9.0
0.1
1.0
10.0
Moisture (kg.kg db)
Figure 8.34 Thermal conductivity of corn (grains) at various temperatures and moisture contents.
Thermal Conductivity and Diffusivity of Foods
Baked Products
347
DOUGH
Total Number of Papers 3 Total Experimental Points 20 Points Used in Regression Analysis 15 (75%) Standard Deviation (sd, W/mK) 0.18 Relative Standard Deviation (rsd, %)_______0 Parameter Estimates Xi (W/mK) 0.80 Xo (W/mK) 0.27 Ei(kJ/mol) 2.71
____Eo (kJ/mol)______0.0
0.1
1.0
10.0
Moisture (kg.kg db)
Figure 8.35 Thermal conductivity of dough at various temperatures and moisture contents.
Chapter 8
348
_______________Meat____BEEF Total Number of Papers 6 Total Experimental Points 75 Points Used in Regression Analysis 37 (49%) Standard Deviation (sd, W/mK) 0.02 Relative Standard Deviation (rsd, %)______15
Parameter Estimates Xi (W/mK) 0.57 Xo (W/mK) 0.28 Ei(kJ/mol) 2.15 _____Eo (kJ/mol)______3.2
1.0
10.0
Moisture (kg.kg db)
Figure 8.36 contents.
Thermal conductivity of beef at various temperatures and moisture
Thermal Conductivity and Diffusivity of Foods
349
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Wallapapan, K., Sweat, V.E., Diehl, K.C., and Engler, C.R. 1986. Thermal Properties of Porous Solids. In: Physical and Chemical Properties of Foods, M.R. Okos, ed. St. Joseph, MI: ASAE, pp. 77-119. Wang, D.Q., Kolbe, E. 1990. Thermal Conductivity of Surimi - Measurement and Modeling. Journal of Food Science 55:1217-1221. Wang, J., Hayakawa, K.I. 1993. Thermal Conductivity of Starch Gels at High Temperatures Influenced by Moisture. Journal of Food Science 58:884-887. Wang, N., Brennan J.G. 1992. Thermal Conductivity of Potato as a Function of Moisture Content. Journal of Food Engineering 17:153-160. Willix, J., Lovatt, S.J., Amos, N.D. 1998. Additional Thermal Conductivity Values of Foods Measured by a Guarded Hot Plate. Journal of Food Engineering 37:159-174. Woinet, B., Andrieu, J., Laurent, M. 1998. Experimental and Theoretical Study of Model Food Freezing. Part I. Heat Transfer Modeling. Journal of Food Engineering 35:381-393. Xie, G., Sheard, M.A. 1995. Estimation of Confidence Interval of Pasteurizing Values of Conduction-heated Sous Vide food in a Combenation Oven. International Journal of Food Science and Technology 30:745-755. Xu, Y., Burfoot, D. 1999. Predicting Condensation in Bulk of Foodstuffs. Journal of Food Engineering 40:121-127. Zanoni, B., Peri, C.and Gianotti, R. 1995. Determination of the Thermal Diffusivity of Bread as a Function of Porosity. Journal of Food Engineering 26:497510. Zanoni, B., Pierucci, S., Peri, C. 1994. Study of the Bread Baking Process-II. Mathematical Modelling. Journal of Food Engineering 23:321-336. Zheng, M, Huang, Y.W., Nelson, S.O., Bartley, P.O., Gates, K.W. 1998. Dielectric Properties and Thermal Conductivity of Marinated Shrimp and Channel Catfish. Journal of Food Science 63:668-672. Zhou, L., Puri, V.M., Anantheswaran, R.C. 1994. Effect of Temperature Gradient on Moisture Migration during Microwave Heating. Drying Technology 12:777-798. Ziegler, G.R., Rizvi, S.S.H. 1985. Thermal Conductivity of Liquid Foods by the Thermal Comparator Method. Journal of Food Science 50:1458-1462. Ziegler, G.R., Rizvi, S.S., Acton, J.C. 1987. Relationship of Water Content to Textural Characteristics, Water Activity and Thermal Conductivity of Some Commercial Sausages. Journal of Food Science 52:901-906. Zuritz, C.A., Sastry, S.K, McCoy, S.C., Murakami, E.G., Blaisdell, J.L. 1989. A Modified Fitch Device for Measuring the Thermal Conductivity of Small Food Particles. Transactions of the ASAE 32:711-718.
Heat and Mass Transfer Coefficients in Food Systems
I. INTRODUCTION
Heat and mass transfer coefficients are used in the design, optimization, operation and control of several food processing operations and equipment. They are related to the basic heat and mass transport properties of foods (thermal conductivity and mass diffusivity), and they depend strongly on the food/equipment interface and the thermophysical properties of the system. Table 9.1 shows some important heat transfer operations, which are used in food processing. In all of these operations, heat must be supplied to or removed from the food material with an external heating or cooling medium, through the interface of some type of processing equipment. Some operations, such as evaporation, involve mass transfer, but the controlling transfer mechanism is heat transfer (Heldman and Lund, 1992; Valentas etal., 1997). Table 9.2 shows some mass transfer operations that are applied to food processing. They are characterized by the removal or separation of a component of the food material by the application of heat, e.g. drying, or other driving potential, such as osmosis, reverse osmosis, adsorption, or absorption (King, 1971; Saravacos, 1995). Heat and mass transfer coefficients are empirical transfer constants that characterize a given operation from theoretical principles, but they are either obtained experimentally or correlated in empirical equations applicable to particular transfer operations and equipment. Heat transfer coefficients and heat transfer, in general, are used more extensively than mass transfer data in most food processing operations. In many cases, mass transfer correlations are similar to correlations developed earlier in heat transfer. In some operations, simultaneous heat and mass transfer may control the process, e.g. in the drying of solids. 359
360
Chapter 9
Table 9.1 Heat Transfer Operations in Food Processing Operations____________Objective__________________ Blanching Enzyme inactivation Pasteurization Inactivation of microorganisms and enzymes Sterilization Inactivation of microorganisms Evaporation Concentration of liquid foods Refrigeration Preservation of fresh foods Freezing Food preservation Frying______________Food preparation______________ Table 9.2 Mass Transfer Operations in Food Processing Operations_____________Objective_____________ Drying Food preservation Extraction Recovery of components Distillation Recovery of volatiles Adsorption Removal/recovery of components Absorption Absorption/removal of gases Reverse osmosis Concentration, desalting Crystallisation___________Purification of components____
The parallel treatment of heat and mass transfer coefficients is important, since there is an analogy of the two transfer processes, evident in some systems, e.g. air/water, which is based on the transport phenomena.
II. HEAT TRANSFER COEFFICIENTS
A. Definitions
The heat transfer coefficient h (W/m2K) at a solid/fluid interface is given by the equation: q/A=h(AT)
(9-1)
where qlA is the heat flux (W/m2) and /IT is the temperature different (°C or K). A similar definition is applicable to liquid/fluid interfaces. Heat transfer is considered to take place by heat conduction through a film of thickness L of thermal conductivity /I, according to the equation:
Heat and Mass Transfer Coefficients in Food Systems
q/A = QJL)(AT)
361
(9-2)
Thus, the heat transfer coefficient is equivalent to h=UL. However, Eq. (9-2) is difficult to apply, since the film thickness L cannot be determined accurately because it varies with the conditions of flow at the interface. The overall heat transfer coefficient U (W/m2K) between two fluids separated by a conducting wall is given by the equation q/A = UAT
(9-3)
where AT is the overall temperature difference (K). The coefficient U is related to the heat transfer coefficients hi and h2 of the two sides of the wall and the wall heat conduction x/X by the equation: \/U=\/h,+x/X+l/h2
(9-4)
where x is the wall thickness (m), and X is the wall thermal conductivity (W/mK). In industrial heat exchangers, the thermal resistance of fouling deposits must be added in series to the resistances of Eq. (9-4). The overall heat transfer coefficients are specific for each processing equipment and fluid system, and it is determined usually from experimental measurements. B. Determination of Heat Transfer Coefficients
The heat transfer coefficient h at a given interface can be determined experimentally by various methods (Rahman, 1995). In the constant heating (steady state) method, the heat flux q/A is measured (e.g. by electrical measurement) at a given temperature difference AT, and the coefficient h is calculated from Eq. (9-1). In the quasi-steady state method, the heat transfer coefficient is determined from the slope of the heating line of a high conductivity solid, which is assumed to heat uniformly. The heat transfer coefficient can be estimated from the analytical or numerical solution of the heat conduction (Fourier) equation:
dt
dX1
(9-5)
where a is the thermal diffusivity of the material. The solution of Eq. (9-5) involves the Biot number for heat transfer, BiH = (hL/X), from which the heat transfer coefficient can be estimated. The heat transfer coefficient h at the interface of processing equipment can be measured by the heat flux sensors method, which simultaneously measures the surface temperature and the heat flux (Karwe and Godavarti, 1997). The sensors consist of a differential
362
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thermopile of thermal resistance with two inserted thermocouples. They are mounted on the heat transfer surface by a high thermal conductivity paste. Approximate values of h can be estimated indirectly by measuring the parameters of some physical processes, which involve heat transfer, such as the freezing time of a material (Plank's equation) or the evaporation rate of a liquid in a flat surface at a given temperature difference A T. Special experimental arrangements are required for the estimation of the heat transfer coefficients between particles and a liquid, both in motion, as in aseptic processing of food suspensions. The particle temperature may be measured by a moving thermocouple or estimated from the change of color of special materials, such as liquid crystals coated on acrylic spherical particles and observed through a transparent flow tube. C. General Correlations of the Heat Transfer Coefficient
Correlations of heat transfer data are useful for estimating the heat transfer coefficient h in various processing equipment and operating conditions. These correlations contain, in general, dimensionless numbers, characteristic of the heat transfer mechanism, the flow conditions, and the thermophysical and transport properties of the fluids. Table 9.3 lists the most important dimensionless numbers used in both heat and mass transfer operations. The Reynolds number (Re=uL/v) is used widely in almost all correlations. In this number, the velocity u is in (m/s), the length I is in (m) and the kinematic viscosity or momentum diffusivity (v=rj/p~) is in (m2/s). The length L can be the internal diameter of the tube, the equivalent diameter of the noncircular duct, the diameter of a spherical particle or droplet, or the thickness of a falling film. Some dimensionless numbers, used in both heat and mass transfer correlations, are denoted by the subscripts H and M respectively, i.e. Bin Bi^, Stn, St^, JH andy^/Table 9.4 shows some heat transfer correlations of general applications. For natural convection, the parameters a and m characterize the various shapes of the equipment and the conditions of the fluid (McAdams, 1954; Perry and Green, 1984; Geankoplis, 1993; Rahman, 1995). The ratio of tube diameter to tube length D/L is important in the laminar flow (Re < 2100), but it becomes negligible in the tubular flow in long tubes (L/D > 60). For shorter tubes, the ratio D/L should be included in the correlation. The viscosity ratio r\/r]w refers to the different viscosity in the bulk of the fluid 77 and at the tube wall t]w. This ratio becomes important in highly viscous fluids, like oils, in which the viscosity drops sharply at the high wall temperatures, increasing the heat transfer coefficients. Several other correlations have been proposed in the literature for different heat transfer of fluid systems, like flow outside tubes and flow in packed beds.
Heat and Mass Transfer Coefficients in Food Systems
363
The heat transfer coefficients of condensing vapors have been correlated to the geometry of the tubes and the properties of the liquid film or droplets. Very high heat transfer coefficients are obtained by drop-wise condensation.
Table 9.3 Dimensionless Numbers in Heat and Mass Transfer Calculations Number Applications Flow processes Re=uL/v Reynolds Heat transfer Nu=hL/A Nusselt Heat transfer Pr=v/a Prandtl Free convection Gr=L3g(Ap/p)/v2 Grashof Graetz Heat transfer Gz=GACpM Heat transfer BiH=hm Biot Mass transfer Sh=kcL/D Sherwood Diffusion processes Sc=v/D Schmidt Heat transfer Stanton StH=h/G Cp Stanton Mass transfer StM=kc/u Heat/mass transfer Le=a/D Lewis Flow/diffusion Pe=uL/D Peclet Mass transfer BiM=kcL/X Biot Heat transfer JH=StHPr2'^ Heat Transfer Factor Mass transfer Mass Transfer Factor jM=StMSc /s A, interfacial area (m2); L, length (m); a thermal diffusivity (m2/s); Cp, specific heat (J/kg K); D mass diffusivity (m2/s); g acceleration of gravity (m2/s); G=up, mass flow rate kg/m2s; h, heat transfer coefficient (W/m2K); kc, mass transfer coefficient (m/s); 77 viscosity (Pas); p, density (kg/m3); u, velocity (m/s)
Table 9.4 General Heat Transfer Correlations____________________________ Transfer System________________Correlation_____________ Natural convection Nu = a (Gr Pr)m Laminar inside tubes Nu = l.B6[RePr(D/L)f\rj/rjwfu Turbulent inside long tubes Nu = O.Q23Re°*Pr1'3 (rj/tjw}°M Parallel to flat plate (laminar) Nu = 0.664/?e°5 Pr113 Parallel to flat plate (turbulent) Nu = 0.0366#e°8 Pr1'3 Past single sphere_______________Nu = 2.0 +0.60Re°'5 Pr113_______ Dimensionless numbers defined in Table 9.3. a and m, parameters of natural convection characteristic of the system (Perry and Green, 1984); L, D length and diameter of tube. Long tubes L/D>60
364
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D. Simplified Equations for Air and Water The heat transfer coefficients of air and water in some important operations can be estimated from simplified dimensional equations, applicable to specific equipment geometries and system conditions (Perry and Green, 1984; Geankoplis, 1993):
a. Natural convection of air: Horizontal tubes, h = 1 .42 (A T/d0) 1/4 Vertical tubes, h=\A2 (AT/L)W
(9-6) (9-7)
b. Air in drying (constant rate):
Parallel flow, h = 0.0204G0'8 Perpendicular flow, h=l.ll G°'37
(9-8) (9-9)
c. Falling films of water: /j = 9150r 1/3
(9-10)
d. Condensing water vapors: Horizontal tubes, h = 10800 / [(Nd0f\AT)l/3] Vertical tubes, h = 13900 / [Lw(AT)m]
(9-11) (9-12)
where AT is the temperature difference (K), d0 is the outside diameter (m), L is the length (m), G is the mass flow rate (kg/m2s), F is the irrigation flow rate of the films (kg/m s) and N is the number of horizontal tubes in a vertical plane. III. MASS TRANSFER COEFFICIENTS
A. Definitions
Mass transfer in industrial and other applications is usually expressed by phenomenological mass transfer equations, instead of the basic mass diffusion model. The mass transfer equations use lumped parameters and average concentration, while the diffusion model has distributed parameters for the dependent variable (concentration), which can vary with the independent variables of distance and time (Cussler, 1997). The mass transfer coefficients are functions of the mass diffusivity, the viscosity, the velocity of the fluid, and the geometry of the transfer systems. The mass diffusivity, in the diffusion model, is a fundamental property based on molecular interactions and on the physical structure of the material. The mass transfer coefficient kc (m/s) in a process is defined in an analogous manner with the heat transfer coefficient: J = kcAC
(9-13)
Heat and Mass Transfer Coefficients in Food Systems
365
where J is the mass flux (kg/m2s) and AC is the concentration difference (kg/m3). In contrast to heat transfer where the driving force is the temperature difference AT, m mass transfer the driving force can be expressed by the concentration difference AC, the difference of mass fraction AY, or the pressure difference AP. Thus, three mean mass transfer coefficients can be defined by the following equation (Saravacos, 1997): J = kcAC = kYAY = kpAP
(9-14)
The units of the three mass transfer coefficients depend on the units of AC, AY and AP and they are usually kc (m/s), kY (kg/m2s) and kp (kg/m2sPa). In food engineering and especially in drying calculations, the symbol hM is used instead of kY, with the same units (kg/m2s). In an analogy with the overall heat transfer coefficient K, the overall mass transfer coefficient is used to express mass transfer through the interface of two fluids, according to the equation: l/K=\fkci+\/kc2
(9-15)
where kcl and kC2 are the mass transfer coefficients of the two contacting fluids. It should be noted that in mass transfer there is no wall resistance and the two fluids at the interface are assumed to be in thermodynamic equilibrium. Volumetric mass transfer coefficients (kcv) may be used in some industrial operations, defined by the equation: kcv=a.kc
(9-16)
where a = A/Vis the specific surface of the transfer system (m2/m3). Thus the units of kcv will be (1/s) and of h B. Determination of Mass Transfer Coefficients The mass transfer coefficients can be determined by direct or indirect measurement of the mass transfer rates in a controlled experimental system. The wetted wall column has been used to determine ^-values in liquid/gas and liquid/vapor systems, like absorption of gas in aqueous solutions (Sherwood et al., 1975; Brodkey and Hershey, 1988). The mass flux is measured at a given driving force (AC, AY or AP) and the corresponding coefficients (kc, kY or kp) are determined. The mass transfer coefficients (kc or hM) during the constant rate period of drying can be estimated from the drying rate of a known sample at well-defined
366
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drying conditions. As an illustration, the mass transfer coefficient in the air drying of spherical starch samples 21 mm diameter at 60°C, 10% RHand 2 m/s air velocity was determined as kc= 34 mm/s (Saravacos et al, 1988). It should be noted that the drying rate of wet high moisture samples is close to the evaporation of water from a free surface. However, in drying food materials, some resistance to mass transfer is usually present at the interface and in the interior of the product, resulting in significantly lower drying rates. Thus, the mass transfer coefficient in drying grapes is lower, e.g. 7 mm/s or 13 mm/s, depending on skin resistance to moisture transfer. The mass transfer coefficient during drying kY or hM can be estimated simultaneously with the heat transfer coefficient h and the moisture diffusivity D from drying data (Marinos-Kouris and Maroulis, 1995). The experimental drying data are fitted by regression analysis to a heat and mass transfer model, assuming certain empirical relationships. The results, obtained for the heat and mass transfer coefficients, are much lower than the values of evaporation of water from free surfaces, since during drying the heat and mass transfer interface moves inside the porous solid food material, becoming much larger than the outside surface of the material. C. Empirical Correlations
Tables 9.5 and 9.6 show some empirical correlations of the mass transfer coefficient (kc) in fluid/solid and fluid/fluid systems. Fluid/solid systems are common in drying of solids, solvent extraction of solids and adsorption operations. Fluid/liquid interfaces are important in aeration, de-aeration, and carbonation/decarbonation of liquid foods. Table 9.5 Mass Transfer Correlations for Fluid/Solid Interfaces Transfer system_________________Correlation_______ Membrane Sh = 1 Laminar inside tubes Sh = 1.62 (cfuILD)1/3 Turbulent inside tubes Sh = 0.026 Re°8Sc1'3 Parallel to flat plate (laminar) Sh = 0.646 Re0'5 Sc>/3 Past single sphere Sh = 2.0 + 0.60 Re°'5Scl/3 Packed beds Sh=\.ll Re°A2 (1 /Sc)2/3 Spinning disc__________________Sh = Q.62Re°'5Sc1'3 Dimensionless numbers defined in Table 9.3.
Heat and Mass Transfer Coefficients in Food Systems
367
Table 9.6 Mass Transfer Correlations for Fluid/Fluid Interfaces Transfer system____________Correlation________________ Gas bubbles in unstirred tank Sh = OA2 Gr1/45c"3 Gas bubbles in stirred tank Sh=l.62 [(P/V) cflpP3]1/4 5c1/3 Small liquid drops in unstirred solution Sh = 1.13 (dulD)°'% Falling films______________Sh = 0.69 (zu/Df5_____________ Dimensionless numbers defined in Table 9.3; d, drop diameter (m); z, position along film (m); P/V stirrer power per volume.
D. Theories of Mass Transfer
The empirical mass transfer data, used in various correlations can be interpreted in terms of approximate or exact theories of mass transfer. The mass transfer theories were developed mainly for fluid/fluid systems. The most important theories are briefly the following (Cussler, 1997). 1. Film Theory The mass transfer coefficient kc is a function of the first power of the diffusion coefficient £>: hc =D/L
(9-17)
where L (m) is the film thickness, which is difficult to determine accurately, since it is a function of the flow conditions, the geometry of the system, and the physical properties of the fluid. 2. Penetration Theory The mass transfer coefficient kc is a function of the square root of the mass diffusivity D:
kc=2(Du/nLf2
(9-18)
where L is the depth of penetration (m) and u is the velocity (m/s) of penetration. The contact time between the diffusivity components and the fluid is defined as u/L, and it is difficult to determine experimentally. 3. Surface Renewal Theory The mass transfer coefficient kc is a function of the square root of mass diffusivity D, in a similar manner with the penetration theory:
368
Chapter 9
kc=(Drf2
(9-19)
where T is the average time for a fluid element in the interface region. 4. Boundary Layer Theory The boundary layer theory, applied primarily in fluid mechanism and heat transfer, gives a more accurate correlation of the mass transfer coefficient kc in the laminar flow. The kc is a function of the 2/3 power of mass diffusivity D. The average mass transfer coefficient kc, past a flat plate of length L, is given by the following empirical equation, which is analogous to the corresponding heat transfer relationship:
kc = 0.00646 (D/L)RelK So213
(9-20)
where the Reynolds number is defined as Re=Lu/v. The heat and mass transfer analogies are useful in evaluating the heat/mass transfer mechanisms and in estimating and inter converting the heat and mass transfer coefficients. The Chilton Colburn (or Colburn) analogy for heat and mass transfer indicates that in fluid systems, under certain conditions, the heat and mass transfer factors are equal (Geankoplis, 1993; Saravacos, 1997): JH=JM
(9-21)
where jH= 5^/'r2/3,yw= StMSc2n and StH= h/upCp, StM= kc/u or StM= h^up The Colburn analogy in air/water mixtures (applications in drying and air conditioning) is simplified, since the Pr and Sc are approximately equal (Pr = Sc = 0.8). Therefore, we may have StH = StM or h/upCp = kC/u or h/pCp =kc, In terms of the mass transfer coefficient hM, the last relationship becomes: h/Cp = hM
(9-22)
The specific heat of atmospheric air at ambient conditions is approximately Cp = 1000 J/kgK. Therefore, Eq. (9-22) yields h = 1000hM, where h is in W/m2K and hM in kg/m2s. If the units of hM are taken as g/m2s, the last relationship is written as (Saravacos, 1997): Atmospheric air, h (W/m2K) = HM (g/m2s)
(9-23)
A similar relationship is obtained between the coefficients h and kc'.
Atmospheric air, h (W/m2K) = kc (mm/s)
(9-24)
Heat and Mass Transfer Coefficients in Food Systems
369
IV. HEAT TRANSFER COEFFICIENTS IN FOOD SYSTEMS
The heat and mass transfer coefficients in food systems are determined experimentally or correlated empirically from pilot plant and industrial data. They are specific for each food process and processing equipment and are related to the physical structure of the food materials. Most of the literature data refer to heat transfer coefficients, since heat transfer is the rate controlling mechanism in many processing operations. Mass transfer coefficients can be related to heat transfer in some important operations, like drying, using the Colburn analogy of heat and mass transfer. Typical values of heat transfer coefficients are shown in Table 9.7 (Hallstrom et al., 1988; Perry and Green, 1997; Rahman, 1995; Saravacos, 1995). Detailed data and empirical correlations for both transfer coefficients are presented in sections VI and VII of this chapter. A. Heat Transfer in Fluid Foods
Heat transfer in viscous non-Newtonian fluids in laminar flow in tubes is expressed by a correlation analogous to the equation for Newtonian fluids: (9-25)
where the Graetz number Gz = GrCp/AL, and G is the mass flow rate (kg/m2s). Table 9.7 Typical Heat Transfer Coefficient h and Overall Coefficients U in Food Processing Operations_____________________ ______ Heat Transfer System h, W/m2K Air/process equipment, natural convention 5 - 20a Baking ovens 20 - 80a Air drying, constant rate period 30 - 200a Air drying, falling rate period 20 - 60 Water, turbulent flow 1000 - 3000 Boiling water 5000 - 10000 Condensing water vapor 5000 - 50000 Refrigeration, air cooling 20 - 200 Canned foods, retorts 150 - 500 Aseptic processing, particles 500 - 3000 Freezing, air/refrigerants 20-500 Frying, oil/solids 250 - 1000 Heat exchangers (tubular/plate) 500 - 3500 (overall U) Evaporators____________________500 - 3000 (overall U)
' Similar numerical values for the mass transfer coefficients kc (mm/s) or hM (g/m2s), applying the Colbum analogy.
Chapter 9
370
The apparent viscosities at the bulk of the fluid and at the wall tja and ^ are determined for the given shear rate y using the Theological constants K and n of the fluid for a mean temperature. Heat transfer in agitated vessels is expressed by the empirical correlation (Saravacos and Moyer, 1967): = CRe°'66Pr 1/3 Ola/Tlaw)',0.14
(9-26)
where the coefficient C = 0.55 for Newtonian and C = 1.474 for non-Newtonian fluids. The Reynolds number is estimated as Re = (d2Np)lrja where d is the diameter of the impeller, and rja is the apparent viscosity estimated at the agitation speed TV as r\a = Ky"'1 where K and n are the Theological constants of the fluid at the mean temperature. The shear rate y for the pilot-scale agitated kettle, described in this reference (0.40 m diameter, anchor agitator), was calculated from the empirical relation 7= 13N. The heat transfer coefficients h at the internal interface of the vessel for a sugar solution and for applesauce increased linearly with speed of agitation (RPM), as shown in Figure 9.1. Figure 9.2 shows that the overall heat transfer coefficient U in the agitated kettle decreases almost linearly when the flow consistency coefficient K is increased. 10000
1000 --
Figure 9.1 Heat transfer coefficients in agitated kettle. S, sucrose solution 40° Brix; A, applesauce; RPM, 1/min
371
Heat and Mass Transfer Coefficients in Food Systems
1600
1300
1000
10 K (Pa s")
Figure 9.2. Overall heat transfer coefficient (U) of fruit purees in agitated kettle. K, flow consistency coefficient.
B. Heat Transfer in Canned Foods Several heat transfer correlations for canned foods are presented by Rahman (1995). In most cases of heating/cooling of cans, the product heat transfer coefficient ht is controlling the transfer process, since the outside (heating/cooling medium) coefficient and the heat conductance of the wall l/x are generally high (metallic or glass containers). However, heat transfer in plastic containers may be controlled by the wall thermal resistance, due to the low thermal conductivity and the high wall thickness of the plastic material Eq. (9-4). The Reynolds numbers for Newtonian fluids is estimated as Re = where d is the can diameter and N is the speed of rotation (1/s) of the can. For nonNewtonian fluids, the dimensionless numbers used are the following (Rao, 1999): Re = -
4n l3n + l
(9-27)
372
Chapter 9
4n Gr
22
-2
V
;
where K and n are the rheological constants of the fluid at a mean temperature, and fi = (A V/AT)IV, 1/K (natural convection). Heat transfer in cans in an agitated retort (Steritort) is considered as the sum of the contributions of both natural and forced convection:
Nu = A[(Gr}(Prf + C^Re\Pr\D / L)]D
(9-30)
where, for Newtonian fluids, A = 0.135, B = 0.323, C = 3.91xlO"3, and D = 1.369 and for non-Newtonian fluids, A = 2.319, B = 0.218, C = 4.1xlO'7, and D = 1.836 In end-over-end agitated cans the following correlations were obtained (Rao, 1999): Nu = 2 .9 Re°A36 Pr°2*7 for Newtonian fluids
(9-3 1 )
Nu = Re°ABS Pr°'361 for non-Newtonian fluids
(9-32)
Non-Newtonian biopolymers, when subjected to extreme heat treatment, suffer significant losses in apparent viscosity. C. Evaporation of Fluid Foods
Heat transfer controls the evaporation rate of fluid foods and high heat transfer coefficients are essential in the various types of equipment. Prediction of the heat transfer coefficients in evaporators is difficult, and experimental values of the overall heat transfer coefficient U are used in practical applications. The overall heat transfer coefficient is a function of the two surface heat transfer coefficients //,• and h0, the wall thermal conductance MX, and the fouling resistance Eq. (9-33): -1 = 1 + - + — + FR U h k h,
(9-33)
The fouling resistance 7-7? becomes important in the evaporation of liquid foods containing colloids and suspensions, which tend to deposit on the evaporator walls, reducing significantly the heat transfer rate.
Heat and Mass Transfer Coefficients in Food Systems
373
10000
Figure 9.3 Overall heat transfer coefficients U in evaporation of clarified CL and unfiltered UFT apple juice at 55°C.
Falling film evaporators are used extensively in the concentration of fruit juices and other liquid foods because they are simple in construction and they have high heat transfer coefficients. Figure 9.3 shows overall heat transfer coefficients U for apple juices in a pilot plant falling film evaporator, 5 cm diameter and 3 m long tube (Saravacos and Moyer, 1970). Higher U values were obtained in the evaporation of depectinized (clarified) apple juice (1200 to 2000 W/m2K) than the unfiltered (cloudy) juice, which tended to foul the heat transfer surface as the concentration was increased. The U value for water, under the same conditions was higher as expected: U= 2300 W/m2K. D. Improvement of Heat / Mass Transfer
Jet impingement ovens and freezers operate at high heat transfer rates, due to the high air velocities at the air/food interface. Heat transfer coefficients of 250350 W/m2K can be obtained in ovens, baking cookies, crackers and cereals (Nitin andKarwe, 1999). Ultrasounds can substantially improve the air-drying rate of porous foods, like apples (acoustically-assisted drying). Ultrasound of 155-163 db increased the moisture diffusivity at 60°C from 7xlO' 10 to 14xlO'10 m2/s (Mulet et al., 1999).
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V. HEAT TRANSFER COEFFICIENTS IN FOOD PROCESSING: COMPILATION OF LITERATURE DATA
Recently reported heat transfer coefficient data in food processing were retrieved from the following journals (Krokida et al., 200 la): • • • • • •
Drying Technology, 1983-1999 Journal of Food Science, 1981-1999 International Journal for Food Science and Technology, 1988-1999 Journal of Food Engineering, 1983-1999 Transactions of the ASAE, 1975-1999 International Journal of Food Properties, 1998-2000
A total number of 54 papers were retrieved from the above journals. The data refer to 7 different processes (Table 9.8) and include about 40 food materials (Table 9.9). Most of the data were available in the form of empirical equations using dimensionless numbers. All available empirical equations were transformed in the form of heat transfer factor versus Reynolds number (jH = aRe"). This equation was also fitted to all data for each process and the resulting equations characterize the process, since they are based on the data from all available materials. The results are classified by process and material and are presented in Table 9.10. All the equations are presented in Figure 9.4 to define the range of variation of they'// and Re. The range of variation by process is also sketched in Figure 9.5. The above results are presented analytically for each process in Figures 9.6-9.11. The effect of food material is obvious in these diagrams. The results of fitting the equation to all data for each process is summarized in Table 9.11 and in Figure 9.12. Heat transfer coefficient values for process design can be obtained easily from the proposed equations and graphs. The range of variation of this uncertain coefficient can also be obtained in order to carry out valuable process sensitivity analysis. Estimations for materials not included in the data can also be made using similar materials or average values. It is expected that the resulting equations are more representative and predict more accurately the heat transfer coefficients.
Heat and Mass Transfer Coefficients in Food Systems
375
Table 9.8 Number of Available Equations for each Food Process
1
Process Baking
1
2
Forced convection Blanching Steam
1
3
Cooling Forced convection
4
Fluidized bed Rotary
4 6
Storage Forced convection
7
16 1
Freezing Forced convection
6
9
Drying Convective
5
No. of equations
4
Sterilization Aseptic
9
Retort
3
Total No. of equations
54
376
Chapter 9
Table 9.9 Number of Available Equations for each Food Material
1
Material
Apples 2 Apricots 3 Barley 4 Beef 5 Cakes 6 Calcium alginate gel
1 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Canola seeds Carrot Corn Corn starch Figs
Fish Grapes Green beans Hamburger Maize Malt Meat carcass Model food Newtonian liquids Non-food material Particulate liquid foods Peaches Potatoes Raspberries Rice
Soya Soybean Strawberries Sugar Wheat Spherical particles Tomatoes Corn cream Rapeseed Meatballs
Total No. of equations
No. of Equations 1 1 2 1 1 1 1 1
2 1 1 1 3 1 2 1 1 1 4 1 3 3 1 2 1 1 2 1 1 1 3 1 1 1 1 1 54
Heat and Mass Transfer Coefficients in Food Systems
377
Table 9.10 Parameters of the Equation jH = aRe" for each Process and each Material „ mm Re Process/product/reference a max Re
Baking Cakes Baiketal., 1999
0.801
-0.390
40
3,000
Green Beans Zhangetal., 1991 0.00850
-0.443
150
1,500
Cooling Apples Fikiinetal., 1999
0.0304
-0.286
4,000
48,000
0.114
-0.440
2,000
25,000
8.39
-0.492
3,500
9,000
0.472
-0.516
1,300
17,000
2.93
-0.569
2,000
12,000
0.186
-0.500
3,700
43,000
0.0293
-0.320
1,300
16,000
0.136
-0.440
1,900
25,000
0.267
-0.550
1,000
24,000
Blanching
Apricots
Fikiinetal., 1999 Figs Dincer, 1995 Grapes Fikiinetal., 1999 Model Food Alvarez et al., 1999 Peaches Fikiinetal., 1999 Raspberries Fikiinetal., 1999 Strawberries Fikiinetal., 1999 Tomatoes Dincer, 1997
Chapter 9
378
Table 9.10 Continued Process/product/reference Drying
a
n
minRe
ma\Re
3.26
-0.650
20
1,000
0.458
-0.241
30
50
0.692
-0.486
500
5,000
1.06 4.12
-0.566 -0.650
400 20
1,100 1,000
0.665 0.741
-0.500 -0.430
8 1,000
50 3,000
11.9
-0.901
150
1,500
0.196
-0.185
60
80
0.224
-0.200
2,000
11,000
4.12
-0.650
20
1,000
2.48
-0.523
200
1,500
149 3.26
-0.340 -0.650
50 20
100 1,000
0.101 -0.355
3,200
13,000
Convective Barley Sokhansanj, 1987 Canola Seeds Langetal., 1996 Carrot Mulct etal., 1989 Corn Fortes etal., 1981 Torrezetal., 1998 Graves Ghiausetal, 1997 Vagenas et al, 1990 Maize Mourad et al., 1997 Malt Lopezetal., 1997 Potatoes Wangetal., 1995 Rice Torrezetal., 1998 Soybean Taranto et al., 1997 Wheat Langetal., 1996 Sokhansanj, 1987
Fluidized bed Corn Starch Shu-De etal., 1993
379
Heat and Mass Transfer Coefficients in Food Systems
Table 9.10 Continued n
min Re
max Re
-0.258
80
300
-0.587 -0.258
10 20
100
-0.528
1,500
17,000
0.650
-0.418
80
25,000
48.6
-0.535
300
600
8.87 4.67
-0.672
-0.645
7,500 9,000
150,000 73,000
0.228
-0.269
1,800
20,000
0.536
-0.485
3,400
28,000
0.658
-0.425
70
90
0.0136
-0.196
Process/product/reference
Rotary Fish Sheneetal, 1996 0.00160 Soya Alvarez et al., 1994 0.00960 Sheneetal., 1996 0.000300 Susar Wangetal., 1993 0.805 Freezing Beet Heldman, 1980 Calcium alsinate sel Sheng, 1994 Hamburser Floresetal, 1988 Toccietal., 1995 Meat carcass Mallikarjunan et al., 1994 Meatballs Toccietal., 1995
80
Storage Potatoes Xuetal., 1999 Wheat Changetal, 1993
1,500 10,000
Chapter 9
380
Table 9.10 Continued Process/product/reference
//
min Re
max Re
0.500 0.448 3.42
-0.507 -0.519 -0.687
5,000 2,400 2,000
20,000 45,000 11,000
0.748 0.662 0.517
-0.512 -0.508 -0.441
3,000 3,000 3,000
85,000 85,000 85,000
0.225 0.0493
-0.400 -0.199
140 1,800
1,500 5,200
2.26
-0.474
4,300
13,000
2.74
-0.562
11,000
400,000
0.564
-0.403
30
0.108
-0.343 130,000
Sterilization
Aseptic Model food Balasubramaniam et al, 1994 Sastryetal., 1990 Zuritzetal, 1990 Non-food material Kramers, 1946 Ranzetal.,1952 Whitaker, 1972 Paniculate liquid foods Mankadetal., 1997 Sannervik et al., 1996 Spherical particles Astrometal., 1994 Retort Newtonian liquids Anantheswaran et al., 1985 Particulate liquid foods Sablanietal, 1997 Corn cream Zamanetal., 1991
1,600 1,100,000
Heat and Mass Transfer Coefficients in Food Systems
381
Table 9.11 Parameters of the Equation jH = a Re" for each Process
Process
a
«
mm Re
max Re
Baking
0.80
-0.390
40
3,000
Blanching
0.0085
-0.443
150
1,500
Cooling
0.143
-0.455
1,000
48,000
Drying /convective
1.04
-0.455
8
11,000
Drying /fluidized bed
0.10
-0.354
3,200
13,000
Drying /rotary
0.001
-0.161
10
300
Freezing
1.00
-0.486
80
150,000
Storage
0.259
-0.387
70
10,000
Sterilization /aseptic
0.357
-0.450
140
45,000
Sterilization /retort
1.034
-0.499
30
110,000
The data of Tables 9.10 and 9.11 demonstrate the importance of the flow conditions (Reynolds number, Re) and the type of food process and product on the heat transfer characteristics (heat transfer factor, jH). As expected from theoretical considerations and experience in other fields, the heat transfer factor, jH decreases with a negative exponent of about -0.5 of the Re. The highest jH values are obtained in drying and baking operations, while the lowest values are in storage and blanching. Granular food materials, such as corn and wheat appear to have better heat transfer characteristics than large fruits (apples). Regression analysis of published mass transfer data show the similarity between the heat transfer factory'// and the mass transfer factor jM (see section VI of this chapter).
Chapter 9
382
JH 0.01
0.001
0.0001
0.00001 1
10
100
1000
10000
100000
1000000 10000000
Re
Figure 9.4 Heat transfer factor jH versus Reynolds number Re for all the examined processes and materials.
Heat and Mass Transfer Coefficients in Food Systems
383
JH 0.01
0.001
0.0001 10
1 000
10 000
100 000
1 000 000
Re
Figure 9.5 Ranges of variation of the heat transfer factor^ versus Reynolds number Re for all the examined processes.
384
Chapter 9
0.001
1 000
10000
Re
100000
Figure 9.6 Heat transfer factory'// versus Reynolds number Re for cooling process and various materials.
Heat and Mass Transfer Coefficients in Food Systems
385
JH 0.1
0.01 10
100
1000
10000
Re
Figure 9.7 Heat transfer factor jH versus Reynolds number Re for convective drying
process and various materials.
386
Chapter 9
J H 0.l
0.01
0.001
100 000
Figure 9.8 Heat transfer factor jH versus Reynolds number Re for freezing process and various materials.
Heat and Mass Transfer Coefficients in Food Systems
387
Storage i
0.1
JH
0.01
WlhesiT
0.001 10
100
1000
10000
100000
Re
Figure 9.9 Heat transfer factory'// versus Reynolds number Re for storage process and various materials.
Chapter 9
388
0.1
Sterilization Aseptic
Spherical Parti i :les
JH
0.01
Non-Foa4 Matcri
Partiqulatg
Upii
Fo )di
V Moi lei
F( od
0.001 1000
10000
100 000
Re
Figure 9.10 Heat transfer factory'// versus Reynolds number Re for sterilization aseptic process and various materials.
Heat and Mass Transfer Coefficients in Food Systems
389
0.1
Sterilization Retort
\ Part culat? Lii Fi oils
JH 0.01
0.001
100
1000
10000 Re
100000
1000000
Figure 9.11 Heat transfer factory// versus Reynolds number Re for sterilization retort process and various materials.
Chapter 9
390
JH
0.01
0.001
0.0001
10
100
1000
10000
100000
1000000
Re
Figure 9.12 Estimated equations of heat transfer factory'// versus Reynolds number Re for all the examined processes.
Heat and Mass Transfer Coefficients in Food Systems
391
VI. MASS TRANSFER COEFFICIENTS IN FOOD PROCESSING: COMPILATION OF LITERATURE DATA
Recently reported mass transfer coefficient data in food processing were retrieved from literature following the same procedure described in Section V for heat transfer coefficient data (Krokida et al., 2001b). A total number of 15 papers were retrieved from the above journals. The data refer to 4 different processes (Table 9.12) and include about 9 food materials (Table 9.13). All available empirical equations were transformed in the form of mass transfer factor versus Reynolds number (JM = aRen). The results are classified by process and material and are presented in Tables 9.14 and 9.15. All the equations are presented in Figure 9.13 to define the range of variation of the jM and Re. The range of variation by process is sketched in Figure 9.14. The above results are presented for convective drying process in Figure 9.15. The effect of food material is obvious in this diagram. The results of fitting the equation to all data for each process is summarized in Table 9.14 and in Figure 9.16. Mass transfer coefficient values for process design can be obtained easily form the proposed equations and graphs. The range of variation of this uncertain coefficient can also be obtained in order to carry out valuable process sensitivity analysis. Table 9.12 Number of Available Equations for each Food Process
_____Process____________No. of Equations 1 Drying Convective
6
Spray
1
2 Freezing Forced Convection
6
3 Storage Forced Convection
1
4 Sterilization ______Forced Convection________________1^
_____Total No. of Equations__________15
392
Chapter 9
Table 9.13 Number of Available Equations for each Food Material
Material
1 2 3 4 5 6 7 8 9
No. of Equations 1
Com Grapes Maize Meat Model food Potatoes Rice Carrots
2 1 6 1 1 1 1 1
Milk Total No. of Equations
15
Table 9.14 Parameters of the Equation/^ = aRe" for each Process Process Drying/convective
Drying/spray
a
n
mm Re
max Re
23.5 -0.882
5
5,000
2.95
-0.889
1
2
Freezing
0.10 -0.268
2,500
70,000
Storage
0.67 -0.427
50
55
Sterilization
11.2 -1.039
6,500
26,000
393
Heat and Mass Transfer Coefficients in Food Systems
Table 9.15 Parameters of the Equation jM = aRe" for each Process and each Material
/;
min Re
5.15-0.575
20
-0.462 0.004 0.741-0.430
10 900
40 3,000
Mouradetal., 1997
34.6 -1.000
5
15
Rice Torrezetal., 1998 Carrot
5.15-0.575
20
Muletetal., 1987
0.69 -0.486
500
5,000
Process/product/reference
a
max Re
Drying Convective Corn Torrezetal., 1998 Graves Ghiausetal., 1997 Vagenasetal, 1990 Maize
1,000
1,000
Spray Milk Straatsma et al., 1999
2.947
-0.890
1
2
Meat Toccietal., 1995
2.496
-0.495
2,500
70,000
Storage Potatoes Xuetal.,1999
0.667
-0.428
50
55
11.220
-1.039
6,500
26,000
Freezing
Sterilization Model food Fuetal.,1998
394
Chapter 9
10 ^
V
V
\\ Sr— "*^
N^ 5fc s »
% \
s,,v
0.1
k
Nt,""• 0
'V
JM-1.llRe' '
ss
S
ss
*^ x^ •^
^
N.
JM
;., ^ »' ^"V
1s ^
0.01
— S iir-5* !l
^*;-
\__ 5 _ * «^s ^> sS
*','^n
^
\
0.001
_ __
. , __
-^
\
S V
0.0001
1
10
1 000
100
10 000
100 000
Re Figure 9.13 Mass transfer factory^ versus Reynolds number Re for all the examined processes and materials.
Heat and Mass Transfer Coefficients in Food Systems
395
10 TJfyfitg Con-wclti
\
\
0.1
nezjir
JM
0.01
0.001
izat 0.0001 10
100
1000
10 000
100 000
Re
Figure 9,14 Ranges of variation of the mass transfer factor JM versus Reynolds number Re for all the examined processes.
396
Chapter 9
10
Convective Drying \Miize
0.1
5
JM
Gra«s
0.01
0.001 10
100 Re
1000
10000
Figure 9.15 Mass transfer factor jM versus Reynolds number Re for convective drying process and for various materials.
Heat and Mass Transfer Coefficients in Food Systems
397
10 1
\s
I\ t
sTa
I>
ing
j
H -J
It II BCtlV 5 )i ing
1
^.
^1 s, V——
\
i
\ St
0.1 1
JM
=
nv>r
a
i
1
j
s.
^> j
ST _ . ——— ——
:
1 1
^1 i j
1
\
\V
i
0.01
1
J
*"*"-^ =f-
F •e izil 2(
S
1
1
1
0.001
V" \
1
X
y I
1
..tod
\'n ttrJ
0.0001
1
10
100
1000 10000
100000
Re
Figure 9.16 Estimated equations of mass transfer factory^ versus Reynolds number Re for all the examined processes.
398
Chapter 9
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Appendix: Notation A a Bi b cp cv C
transport area, m constant of Redlich-Kwong Eq. (2-12) Biot number constant of Redlich-Kwong Eq. (2-12) heat capacity at constant pressure kJ/kmol K heat capacity at constant volume kJ/kmol K concentration, kg/rrr3
D
mass diffusivity, m2/s
D d De DPM E E ED Ea F Fo G G' G " G Gz h h JA JH JM K K Kp kB kc L M M M MA N Nu n
diameter, m diameter, m Deborah number dipole moment, debye modulus of elasticity, Pa activation energy, kJ/kmol energy of activation for diffusion, kJ/mol activation energy for viscous flow, kJ/mol force, N Fourier number shear modulus, Pa storage modulus, Pa loss modulus, Pa mass flow rate, kg/m2s Graetz number height, m heat transfer coefficient, W/m K mass flux of A, kg/m2s or kmol/m2s heat transfer factor mass transfer factor flow consistency coefficient, Pa sn drying constant, 1/s partition coefficient Boltzmann constant, kB= R/N= 1.38xlO"23 J/molecule K mass transfer coefficient, m/s length, m mass, kg torque, N m molecular weight, kg/kmol molecular weight of A Avogadro's number, 6.022xl023 molecules/mol Nusselt number flow behavior index 403
Appendix: Notation
404
n P P PM Pr
Q Q q r
R Re rt r0 5
Sh t T f
index pressure, Pa or bar permeability, kg / m s Pa permeance, kg/ m2s Pa Prandtl number volumetric flow rate, m3/s accumulated quantity, kg/m2 heat transport rate, W radius, m gas constant, 8.314 kJ/kmol K Reynolds number inside radius, m outside radius, m solubility, kg/m3Pa Sherwood number time, s temperature, K, C
Tg U M u(r) V V W WVTR
kBT/s glass transition temperature, K, °C velocity, m/s velocity, m/s potential energy (Lennard-Jones potential), J molar volume, cm3/mol, m3/mol volume, m3 weight, kg water vapor transmission rate, kg/m s
X
transport property
X
moisture content, kg/kg dm compressibility factor
Greek a a Y
F
thermal diffusivity, m2/s relative volatility activity coefficient film flow rate, kg/m s 3.141
Y Y
S S" AP s
shear rate, 1/s strain (relative deformation) generalized transport coefficient dimensionless dipole moment pressure drop, Pa interaction energy parameter, J
Appendix: Notation
s rj rj 77' T;,, 77,. 9 9 /I Am ju v p W co or /2 /3
porosity viscosity, Pa s shear viscosity, Pa s dynamic viscosity, Pa s apparent viscosity, Pa s relative viscosity collision frequency, 1/s angle of cone/plate thermal conductivity, W/m K mean free path, m chemical potential, kJ/mol momentum diffusivity (kinematic viscosity), m2/s density kg/m3 or mol/m3 collision diameter, m tortuosity shear stress, Pa yield stress, Pa interaction parameter volume fraction generalized transport rate acentric factor frequency of oscillation. 1/s collision integral rotational velocity, 1/s
Subscripts A B
b c D e G
id K L 0
P r res V
component A(diffusant) component B (medium) boiling critical diffusion equilibrium gas ideal Knudsen liquid dilute, initial particle reduced residual viscosity
405
Index Ash, 281 Avocado, 172, 182, 191 Absorption, 228, 234, 360 Acentric factor, 11-12 Acetic acid, 242, 267 Activation energy, 72, 82-84, 91, 95, 198,249,253 Adsorption, 113,228,231,360 Agar, 298, 299, 309, 312, 321, 324 Agitated kettle, 370, 371 Albumen, 313, 325 Albumin, 184, 193 Albumine, 313, 325 Almond, 175, 185, 194,226 Amioca, see Amylopectin Ammonia, 241 Amylopectin (Amioca), 128-132, 140,184, 193,201-204,308, 320, 342 Amylose (Hylon), 128-132, 134, 137,154,184,193,201-204, 308, 320, 343 Apparent viscosity, 69,79, 370 Apparent density, 50, 55 Apple, 50-56, 149-151,182, 191, 201,213,253,275,305,317, 329, 337, 377 Apple juice, 80-82, 373 Apple sauce, 72, 89, 92, 94 Apple slices, 253 Apricots, 157, 172, 182, 191, 377 Arrhenius equation, 16, 71, 78-80, 82,93, 128, 141,245,260 Aseptic processing,369, 375, 381
B Baking, 373, 377, 381 Baking ovens, 369 Banana, 50-56, 182, 191, 201, 215, 306,318 Barley, 179, 188, 302, 314, 378 Beef, 183, 192, 275, 307, 319, 329, 348, 354, 379 Beef carcass, 183, 192 Beer, 84 Beeswax, 263 Beet, 186, 195,300,311,323 Bentonite, 298, 355 Bilberry, 296, 306, 318 Bingham plastic, 68-69, 91 Biopolymers, 30, 150, 254, 372 Biot number, 145, 273, 363 Bird-Carreau model, 73 Biscuit, 168, 179, 188,302,314 Blanching, 360, 375,377, 381 Bluebenies, 172, 182, 191 Boiling water, 369 Boltzmann constant, 9, 17, 241 Boundary layer theory, 368 Bread, 156, 168, 179, 188, 293, 302, 314 Brine, 252 Broad bean, 173, 183, 192 Broccoli, 176, 186, 195 Broiled meat, 173, 183, 192 Brown rice, 168,169, 179,180, 188-189 Bull, 174, 183, 192 407
408
Buoyant Force, 46 Butanol, 255 Butanone, 255 Butter, 92, 294, 304, 316 Buttermilk, 91,101 Butyric acid, 242
c Cake, 295,302, 314, 377 Calamari, 305, 317
Calcium alginate gel, 376, 379 Canned foods, 369, 371 Canola, 185, 194,378 Capillary tube vicometer, 74-77 Carbohydrate, 281 Carbon dioxide, 12-13, 20, 241 Carp, 295,305, 317 Carrageenan gum, 85-86 Carrot, 50-56, 186, 195, 201, 217, 310,322,378,393 Cassava, 186, 195,310,322 Catfish, 181 Celery, 300,311,323 Cellophane, 264
Cellulose gum, 298, 309, 321 Cellulose-oil-water, 174, 184, 193 Cereal products, 168, 179, 188,201, 293,302,314,329 Chapman-Enskog equation, 12, 15 Cheese, 171, 181, 190,227,230, 251,303,315 Chemical potential, 105, 109,146147, 244 Cherry, 83, 306, 318 Chicken, 174, 231, 297, 307, 319 Chlorine, 241 Chocolate, 70, 92, 100, 175, 185, 194,226,262-264 Chromatographic method, 110, 113 Clustering of solutes, 247 Coating, 258-260, 262-264 Cocoa, 79 Coconut, 182, 191,312,324 Cod, 181, 190,304,316
Index
Codfish fillet, 145-146, 157 Coffee, 194,253,312,324 Colburn analogy, 368, 369 Collapse, 31,50, 60, 152,259 Collision, 9, 12, 13, 15 Colloid / Surface Chemistry, 31 Compressibility factor, 11 Condensing water vapor, 364, 369 Condensing water vapors, 364 Controlled release, 258 Cookie, 156, 168, 179, 188 Cooling, 3, 227, 245, 350-351, 359, 371,375,377,381,384 Corn, 156, 179, 188, 201, 209-210, 221-224, 264, 302-303, 314-315, 329, 346, 378, 380, 393, Corn cream, 376, 380 Corn meal, 303,315 Corn oil, 79 Corn pericarp, 156, 234, 264 Corn Starch, 127, 162, 174, 184, 193,353,376,378 Cottonseed oil, 253 Crackers, 179, 188 Cracks, 47, 114, 127-128, 130, 141, 143,253,260 Cream, 92, 303-304, 315-316 Critical conditions, 11-12,14-15 Crystallization, 30, 33, 360 Cucumber, 301,311, 323 & Dairy products, 181, 190,303,315, 329 Data banks, 7, 19, 29, 269, 275 Database, 27, 144, 161, 164 Deborah number, 73, 108 Diaphragm cell, 238-239 Dickerson method, 273, 351 Dietary fibers, 89 Diethyl ketone, 255 Diffusivity, mass determination, 109-123 sorption kinetics, 110
Index
[Diffusivity, mass] permeability methods, 114 distribution of diffusant, 118 drying methods, 120 flavors, 254-258 in fluid foods, 241-243 in polymers. 243 organic components, 252 salts, 251 small solutes, 237 Diffusivity, moisture baked products, 168, 179, 188 cereal products, 168,179,188, 201, 209-212 dairy products, 171,181, 190 fish, 171,181, 190 fruits, 172, 182, 191, 201, 205, 206 legumes, 173, 183, 192 meat, 173, 183, 192 model foods,127-143,174, 184, 193,201,203,204 nuts, 175, 185, 194 other, 175, 185, 194 vegetables, 176, 186, 195,201 Dilatant fluid, 69 Dimensional equations, 363 Dimensionless numbers, 14, 362, 363,366-367,371,374 Distillation, 266, 360 Distribution of diffusant, 110, 118 Dogfish, 171, 181, 190 Dough, 168, 179, 188,229,293, 302,314,329,347,352-353, 356 Dried fruit, 275 Dry milk, 181,190 Dry solids apparent density, 55 Dry solids true density, 55 Drying Kinetics, 110, 159-160, 162, 227, 230-232, 234 Dual-sorption model, 248, 262 Dynamic viscosity, 73
409
Edible oils, 72, 78-79, 92-93 Egg, 175, 185, 194,229 Einstein equation, 66 Elastic materials, 65, 244 Empirical rules, 279 Emulsions, 90, 92, 262 Enclosed water density, 50, 55 Enzyme inactivation, 360 Ethanol, 12, 13, 17, 20, 84, 241-242, 255 Ethyl acetate, 255 Ethyl butyrate, 255 Ethyl oleate, 153, 154 Ethylene, 12-13, 20, 241 Eucken factor, 15 Evaporation, 254-257, 360, 372 Evaporators, 369 Excess contributions, 14 Extensional viscosity, 64 Extraction, 33-34, 252-253, 360 Extrusion, 130-131 Eyring theory, 16, 18 Fababean, 173, 183, 192 Falling films, 364, 367 Falling rate period, 120-122, 144145,369 Fat, 232-233, 281, 304, 307, 316, 319,355 Fiber, 101,281 Fibrinogen, 242 Pick diffusion equation, 8, 106, 118119,145-147,237,243 Fickian diffusion, 107-108, 243, 250,260 Figs, 376-377 Fish, 122, 145,157,181,190,252, 275,304,316,379 Flavor, 254-259, 261-262 Flavor encapsulation, 258
Index
410
Flour, 184, 193, 302-303, 314-315, 324 Flow behavior index, 68, 72, 76, 78, 80, 85, 88-92, 95 Flow consistency coefficient, 68, 370-371 Fluidisedbeds, 375, 378, 381 Food coatings, 262-263 Food Materials Science, 2, 30 Food preparation, 360 Food preservation, 360 Food rheology, 3, 63 Free convection, 363 Free-volume model, 249 Freeze-dried gel, 275 Freezing, 379,381,391-393 Fructose, 242, 266 Fruits, 50, 94, 146-152,157, 182, 191,201,205-206,305-306, 317-318,329,331-332,373 Frying, 35, 360, 369
G Gamma function, 125 Garlic, 186, 195, 201, 219, 310,322, Gas bubbles, 367 Gas constant, 16, 71, 93, 198, 200, 249, 326, 328 Gas pycnometer method, 46 Gelatin, 281, 298, 309-310, 321-322 Gelatinized starch,130, 141, 275, 281 Gelatin, 298, 309-310, 321-322 Glass transition, 30-31, 244-247 Glucose, 242, 312, 324 Gluten, 174, 184, 193,263-264 Glycerin, 242, 263, 298, 310, 322 Glycine, 242 Graetz number, 363, 369 Granular materials, 36, 61, 133, 135, 285 Granular starch, 130, 133-135, 137141, 150, 154,275,284
Grapes, 191,201,296,306,318, 377-378, 393 Grashof number, 363 Gravimetric method, 110 Green beans, 376-377 Green olives 251 Ground beef, 174, 183, 192 Guar gum, 85-86 Guarded hot plate, 270-271
H Haddock, 171, 181, 190 Halibut, 171, 182, 191 Hamburger, 303, 315, 376, 379 Hazelnuts, 175, 185, 194 HOPE, 261-262, 264 Heat capacity, 8, 15 Heat exchangers, 369 Heat transfer coefficients, determination, 361 baking, 375, 377, 381 blanching, 375, 377, 381 cooling, 375, 377, 381,384 drying, 375, 378, 381,385 freezing, 375, 379,381,387 storage, 375, 379, 381, 387 sterilization, 375, 380, 381, 388389 Heat transfer correlations, 362-363, 371 Heat transfer factor, 363, 374, 382, 384-389 Heated probe, 270-274 Heifer, 184, 193 Hemoglobin, 242 Herring, 157, 182,191,251 Herschel-Bulkley equation, 68-69, 75,91,95 Hexane, 253 Hexanol, 255 Honey, 78, 80, 83, 299, 313, 325 Horizontal tubes, 364 Huggins equation, 65 Hydrodynamic flow, 147
411
Index
Hydrogen, 241 Hylon, see Amylose I Ice, 33, 47, 276, 281, 292, 325, 327
Idli Batter, 293, 303, 315 J Juices, 80-83, 89-90, 92, 95-99, 305306,311,317-318,323,373
K Karaya gum, 85, 87 Kidney bean, 301,311,323 Kinematic viscosity, 8, 362
L Lactoglobulin, 242 Lard, 299, 312, 324 LDPE, 261-262, 264 Legumes, 173, 183, 192, 296, 306 Lentils, 183, 192,306,318 Lewis number, 363 Lewis-Squires equation, 16, 72 Licorice extract, 83 Lipid films, 264 Liquid diffusion, 141, 143 Liquid displacement method, 46
M Macadamia, 298, 310, 322 Mackerel, 157, 182, 191, 304, 316 Macrostructure, 2-4, 29, 35, 40, 4547, 49-50 Maize, 303, 315, 378, 393 Malt, 180, 189,230,376,378 Maltose, 242 Mango, 89, 90, 92, 95, 99, 101-102 Margarine, 33, 73, 92 Mass transfer coefficients determination, 365 drying, 392-394, 397, 397 freezing, 391-393, 397 storage, 391-393, 397 sterilization, 391-393, 397
Mass transfer correlations, 359, 362, 366-367 Mass transfer factor, 363, 368, 391, 395-398 Mass transfer operations, 109, 359360, 362 Mayonnaise, 90, 92 Meat, 157,183, 192, 251, 307, 319, 329, 379, 393 Meat Carcass, 376, 379 Meat Muscle, 251 Meatballs, 376, 379 Membrane, 366 Methane, 241 Methanol, 255 Method of slopes, 123-124 Methyl anthranilate, 255 Microstructure, 2, 5, 29, 31-35, 60, 114,243,261 Milk, 90-91, 181, 190, 275-276, 304,312,316,324,329,345, 393 Milled Rice, 169, 180, 189 Mixed model, 283-284 Mizrahi-Berk model, 70 Model food, 346-49,142, 203 Molecular diameter, 9, 12 Molecular diffusion, 4, 106-107, 144, 147, 238, 244, 260, 263 Molecular effusion, 107 Molecular simulation, 2, 5,29- 30, 61,248,250,253,268 Mulberry, 182, 191,231 Mushrooms, 301, 311, 323, 357 Mustard, 92 Mutton, 297, 308, 320
N Natural convection, 363-364, 369 Navy beans, 183, 192,232 Nernst-Haskel equation, 18 Newton equation, 8, 64, 65, 73 Newtonian foods, 69, 72, 380 Nitrogen, 12-13,20,241
Index
412
Non-Newtonian foods, 68, 372 Nuclear magnetic resonance, 75 Numerical methods, 124-125 Nusselt number, 351, 363, 399 Nuts, 175,185, 194, 298, 310, 322
o Oat, 294, 303, 315 Okra, 176, 186, 195,228 Olive oil, 79 Onion, 156, 186, 195, 201, 218, 310, 322 Orange, 83, 89, 95, 97, 306, 318, 329,338 Osmotic dehydration, 35, 147 Ovalbumin, 242, 281, 309, 321 Overall heat transfer coefficient, 371-373 Oxygen, 12-13,20,241,262 P
Packaging, 259-264 Packed beds, 366 Paddy Rice, 180, 189 Palm kernel, 299, 312, 324 Paprika, 176, 186, 195, 227 Parallel flow, 364 Parallel model, 65, 283, 284 Parboiled Rice, 169, 180, 189,227 Parsley, 300, 311,323 Parsnip, 301,311,323 Past single sphere, 363, 366 Pasta, 156, 169, 180, 189, 201, 225 Pasteurization, 360 Peas, 177, 186, 195, 300, 310, 322 Peaches, 183, 192, 306, 318, 377 Peanuts, 175, 185, 194,253 Peanut butter, 92 Peanut oil, 253 Peanut pods, 175, 185, 194,227 Pear, 95, 98, 102, 306, 318, 329, 339 Peclet number, 363 Pectin, 33, 88-89, 127, 256-257, 298,309-310,321-322
Penetration theory, 367 Pepper, 177, 186, 195 Pepperoni, 174, 184, 193, 308, 320 Permeability, 110, 114-117,237238, 243-244, 248-249, 259-264 Permeance, 260 Perpendicular flow, 364 Phase transition, 30, 61 Pickles, 251,267 Pigeon pea, 177, 186, 195,233 Pineapple, 173, 183, 192, 226, 232 Plant cells, 32-33 Plant hydrocolloids, 85 Plantain, 306, 318 Plasticization, 128, 244 Poiseuille equation, 74 Polyacrylamide gel, 175, 184, 193 Polymer Science, 30, 65,110, 244 Polysaccharide films, 264 Pork, 157, 265, 297, 308, 320, 355 Potatoes, 50, 54, 150, 151, 156, 184, 186,193, 195,201,216,275, 297,310,322,329,340,378379, 392-393 Potato starch, 158, 175, 184, 193, 297, 309, 321 Power-law model, 68-69, 71, 75, 85, 88-89, 91, 94-95 Prandtl number, 363 Propanol, 255 Protein films, 264 Proteins, 135-136, 155,262-264, 280-281 Pseudoplastic fluids, 69, 85, 88-89, 252 Puffing, 127,152,231,275 PVC, 261-262, 264
R Radish, 301,311,323 Raisins, 157, 183, 192, 201, 306 318 Rapeseed, 312, 324, 329 Raspberries, 376-377 Raspberry, 89, 102, 295, 306, 318
Index
Real gases, 11-12, 15 Recovery of volatiles, 360 Refrigeration, 360, 369 Regular regime theory, 125-126 Retorts, 375, 380-381 Reverse Osmosis, 360 Rheological properties (viscosity) aqueous Newtonian, 80 chocolate, 100 cloudy juices/pulps, 89 edible oils, 79 emulsions, 90 fruit and vegetable juices, 95 plant biopolymer solutions, 85 Rheology, 3, 5, 63, 66, 70, 72-74, 90, 101-103 Rice, 156, 179-180, 185, 188-189, 201,303,315,378,393 Rice Starch, 185, 194 Rotary drying, 375, 379, 381 Rotational viscometer, 75-76 Rough Rice, 180, 189, 201, 233 Rubbery state, 108, 238, 244 Rutabagas, 301,311
5 Salad, 90 Salmon, 91, 101,295,305,317 Salt, 18, 20, 84, 133, 146-147, 237, 251-252 Sausage, 157,184, 193, 308, 320 Scanning microscopes, 31-32 Schmidt number, 363 Shark, 171, 182, 191 Shear rate, 31, 64-66, 68-76, 78-80, 85,88,91,94,370 Shear stress, 8, 64-66, 68-71 74-76, 78, 85, 88, 94 Sherwood number, 120-121, 126, 161,363,365 Shrimp, 305,317 Shrinkage, 36-37, 39-40, 47, 50, 55, 60, 126, 137, 150 Shrinkage coefficient, 55
413 Simplified methods, 110, 123, 125 Simulation, 29-30, 124, 232, 250, 253 Skim milk, 101, 171, 181, 190, 352 Sodium caseinate, 101, 263 Solid displacement method, 46 Solubility, 115, 117-118,248-249, 259-260, 262-263 Sorbitol, 263 Sorghum, 299, 313, 325 Sorption kinetics, 107-111, 127, 129, 148,150-152,244 Soy flour, 293, 302, 314 Soya, 177, 186, 195,376,379 Soya meal, 177, 186, 195 Soybean, 152, 156, 186, 195,312, 324, 378 Soybean flakes, 253 Soybean oil, 79,253 Soybeans, 150, 152, 156,161 Specific volume, 55 Spinach, 301, 311,323, 351 Spinning disc, 366 Squid, 191,305,317 Stanton number, 363 Starch gel, 49, 127-129, 131 Steam, 19, 27, 226, 352, 375 Sterilization, 360, 380-381, 388389, 392-394 Stokes diaphragm cell, 239 Stokes-Einstein equation, 17,241, 242, 252 Storage, 30-32, 257, 379, 381 387, 391-393 Strawberries, 295, 306, 318, 351, 376-377 Structural models, 40, 47, 50, 163, 197,280,283,284,287,326 Structural properties, 55 Sucrose, 24-25, 80-83, 133, 147148,242,253,258,281,309310,312,321-322,324,371 Sucrose solution, 275
414
Sugar Beets, 178, 186, 195,253, 300,311,323 Sugar solutions, 80 Sunflower seeds, 185, 194 Surface renewal theory, 367 Surfactants, 153 Surimi, 305, 317 Swiss cheese, 251,267 Swordfish, 172, 182, 191
Index
Transport coefficients, 144 Transport gradient, 8 Transport Phenomena, 5, 7, 27, 158, 228, 257, 265, 354 True density, 55 Turbulent flow, 363, 366, 369 Turkey, 184, 308, 320 Turnip, 187, 196,311,323 Tylose, 309,321
T
u
Tailor-made porous solid foods, 47, 60 Tapioca, 178, 186, 195 Taylor dispersion method, 238, 240 Texture, 3, 29, 31, 33, 60, 64, 101, 102-104,266 Thermal conductivity of foods determination, 270-273 baked products, 293, 302, 314, 329 cereal products, 293, 302, 314, 329 dairy products, 294, 303, 314, 329 fish, 294, 304, 316 fruits, 295, 305, 317, 329, 331-332 legumes, 296, 306, 318 meat, 296, 307, 319,329 model foods, 297, 308, 320, 329 nuts, 298, 310, 322 other, 299, 312, 324, 329 vegetables, 300, 310, 322, 329, 333-334 Thermal diffusion, 107, 145 Thermal diffusivity, 269, 273-279 Thermodynamics, 7, 10, 27, 29, 158, 266-267 Thixotropic fluids, 70-71, 91 Time lag method, 110, 117 Tobacco, 299, 312,324, 350 Tomato, 67-68, 90, 95-96, 187, 196, 252,311,323,329,341,377 Tomato paste, 275 Torque, 74, 76 Tortuosity, 106, 147, 250, 254
Unsteady-state method, 110, 117118,273-274
V Vapor diffusion, 106, 109, 141, 143, 163, 199 Variable Diffusivity, 110, 123, 226 Vegetable oil, 275 Vegetables, 35, 50, 95, 150, 156, 176,186, 195,201,207-208, 333-334 Vertical tubes, 364 Viscometers capillary tube, 74 rotational, 75 cone-and-plate, 76 Viscosity, see Rheological properties Volatile compounds, 255, 266 Volume displacement method, 46
w
Weissenberger rheogoniometer, 76 Wheat, 135, 156, 181, 190,201, 220,302-303,315,378-379 Whey, 263, 294, 304,316 Whiting, 172, 182, 191 Wild rice, 170, 181, 190 Wilke-Chang equation, 241-242, 252 Williams-Landel-Ferry equation, 31, 245 Wine, 84, 102
Index
415
x
y
Xanthan gum, 85, 87
Yam, 187, 196, 311, 323, 355 Yellow batter, 293 Yogurt, 91, 101, 294, 304, 316, 353
