Transport Properties of Foods

Transport Properties of Foods George D. Sanauacos Rutgers University New Brunswick, New Jersey and National Technical U...

Author: George D. Saravacos

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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

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

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

x

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

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

IV. DIFFUSION OF SOLUTES IN FOODS A. Diffusivity of Salts

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

251 251

269 269 270 270 273

275

Contents

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

xi

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):

K) O

T3

CO

o'

(Q

o o_

Chapter 4

98

c

.-*'^,

M

« 100 -

:

cu>

r :

^^^' *^»————

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^^^

_^*

10 -

*^^-^^^ -^^^^^

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ix^^^"*^

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=— ——— .^"^^^^'^ l

^ 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.


109

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.

110

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.


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)

112

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.


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)


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

~m www

I 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)


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:

118

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).


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)

120

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.


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):

7t

(5-25)

The drying constant K refers to specific drying conditions (temperature, air humidity and air velocity) and sample thickness.


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)


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. 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)


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.

128

Chapters

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.

129


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.)

130

Chapter 5

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).


131

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.)

132

Chapter 5

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.)


133

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).

134

Chapter 5

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.)


135

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.)

137


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.)

138

Chapters

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.


139

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


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.)


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.

144

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).


145

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.

146

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:


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):

148

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).

149


(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.)

150

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


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).


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.)


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

156

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


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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160

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Marousis, S.N., Vagenas, O.K., Saravacos, G.D. 1990. Water Difftisivity in Compressed Granulated Starches. Proceedings International Drying Symposium (IDS 7). Prague. Masi, P. Riva, M. 1988. Modeling of Grape Drying Kinetics. In: Preconcentration and Drying of Food Materials. S. Bruin, ed. Amsterdam: Elsevier, pp. 203214. McCarthy, M.J., Lasseux, D., Maneval, IE. 1994. NMR Imaging in the Study of Diffusion of Water in Foods. J Food Eng 22:211-234. McCarthy, M.J., Perez, E., Ozilgan, M. 1991. Model for Transient Moisture Profiles of a Drying Apple Slab Using Data Obtained with Magnetic Resonance Imaging. Biotechnol Progr 7:540-543. Mittal, G.S. 1999. Mass Diffusivity of Food Products. Food Rev Int 15 (l):19-66. Moyne, C., Roques, M., Wolf, W. 1987. Cooperative Experiment on Drying Beds of Glass Spheres. In: Physical Properties of Foods 2. R. Jowitt, F. Escher, M. Kent, B. McKenna, M. Roques, eds. London: Applied Science Publ, pp. 2754. Mulet, A. 1994. Drying Modeling of Water Diffusivity in Carrots and Potatoes. J Food Eng 22:324-348. Okos, M.R., Narsimham, G., Singh, R.K., Weitnauer, A.C. 1992, Food Dehydration. In: Hanbook of Food Engineering. Heldman, D.R. Lund, D.B. eds. New York: Marcel Dekker, pp 437-562. Panagiotou, N.M., Karathanos, V.T., Maroulis, Z.B. 1998a. Modeling of Osmotic Dehydration of Fruits. In: Drying '98 Vol A. C.B. Akritidis, D. MarinosKouris, G.D. Saravacos, eds. Thessaloniki, Greece: Ziti Editions, pp. 954961. Panagiotou, N.M., Karathanos, V.T., Maroulis, Z.B. 1998b. Mass Transfer Modeling of the Osmotic Dehydration of Some Fruits. Int J Food Sci Technol 33:267-284. Papantonis, N. 1991. Effect of Edible Oils on the Water Diffusivity of Starch Materials. M.S. Thesis. Rutgers University. Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow, New York: McGraw-Hill. Pawlisch, C.A. Laurence, R.L. 1987. Solute Diffusivity in Polymers by Inverse Gas Chromatography. Macromolecules 20:1564-1578. Peppas, N.A. Brannon-Peppas, L. 1994. Water Diffusion in Amorphous Macromolecular Systems and Foods. J Food Eng 22:189-210. Perry, R.H. Green, D.W. 1997. Perry's Chemical Engineers' Handbook. 7th ed. New York: McGraw-Hill. Rahman, S. 1995. Food Properties Handbook. New York: CRC Press. Raouzeos, G.S. Saravacos, G.D. 1983. Air-Drying Characteristics of Starch Gels. In: Proceedings of the International Drying Symposium IDS 3 Vol 1. J.C. Ashworth, ed. Birmingham, UK, pp. 91-99. Rotstein, E. 1987. The Prediction of Diffusivity and Diffusion-Related Transport Properties in the Drying of Cellular Foods. In: Physical Properties of Foods 2.


161

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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

164

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.

165

Moisture Diffusivity Data Compilation

J. Food Engineering

Drying

Technology

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.


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


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


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


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


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



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


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


177


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


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


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



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

(m2/s)

Moisture

Temperature

No. of

(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

268

Apple -

6.64E-10

2.80

47

Apricot -

1.39E-07

2.88

53

Avocado Banana plantain ripe

Blueberries Coconut Grapes red seedless Mulberry -

2 1 1 7 7 1 1 12 12 6 6 4 2 2 3 1 2

64

6.35E-10

50

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

64 10 10

11 11

55 49 3 3 29 29 6 6 52 18 2 32 9 9

183



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

Temperature

(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

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

(m2/s)

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



Rice starch full heated

Diffusivity

(m2/s)

Moisture

(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


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


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

190

Chapter 6


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


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


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


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


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

195



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

196

Chapter 6

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


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.


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 £> = •

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

\+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


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.


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

Chapter 6

204

.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


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.


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


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.

210

Chapter 6

l.E-06

-

h-

r

Jtr 60 f

——|Cere al products (corn) 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.


211

Garlfc.E-06 real products ——————— | C(

t~r

Temperature (°C) = 25 i

i

.E-07 -

-

——

.E-08 -

.E-09 -

— Rice kernel

^^ •"

.E-10 -j

Corn

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.

212

Chapter 6

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.


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


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 ———

——— ———

m

MD

o

(i Sm.

h=Hs_^M

0

hn

^-

Diffusivity (m2/s)

l.E-08 -

^s. BH

^

Si

•

|—

-1——————f ——— I "

* —— ———

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

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

(45%)

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.


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.

218

Chapter 6

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



(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.


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~~*

1

«• ~ *•? *—• ^ -} «• l.E-10 , •if* •* 1 *~\fff* ^*^ —— ———— •—————

^

l.E-11 -

l.E-12 0.1

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


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 l.E-07 -

A 80 ———

j

tfi

i

i

l.E-08 1

\

i

•1" l.E-09 V)

———4, ———*-• k

S

A I\

-

\

»

^fc —f— J 1

l.E-10 -

— •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.


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/(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.


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).


247

!

—————-

C

V

!

>w

o

\ \

n

0

X X.

x^

x^ ^\^^

3

cmpcraturc (°C)

\.

%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)


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.


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


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)


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).


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


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)


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):


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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268

Chapter 7

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 "£• ' %-'•-"-'"- • 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


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


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.


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


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.


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

1990

1980

2000

year

Figure 8.14 Accumulation of published papers on thermal conductivity data for food materials versus time.

291


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.

292

Chapter 8

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.


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


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


295


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


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

297



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 Pork

Sanzetal, 1987 Pork/soy

Muzillaetal, 1990

Model foods

75 4 2 2 9 2 30 25 1 9 2 7 13 2 10 1 12 12 10 10

11

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


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



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


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

301



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

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


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


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 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

Moisture

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


307


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


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



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


313


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


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


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



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


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


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



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


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

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

Gelatin-sucrosewater

-

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


323


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


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


325


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 :(T)

A, Y { 1 / ~r

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:


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


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

Material

Papers

No. of

No. of Data

4 (W/mK)

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

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

& (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

^ •• ••

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.

Chapter 8

340

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


Vegetables

TOMATO


5


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.


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


4


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.


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


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.


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


349

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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

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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:


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.


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

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Chapter 9

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 b. Air in drying (constant rate):

Parallel flow, h = 0.0204G0'8 Perpendicular flow, h=l.ll G°'37

(9-6) (9-7)

(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)


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

Chapter 9

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.


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)


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


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.


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).

374

Chapter 9

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.


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


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


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


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.


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.

Chapter 9

384

0.001

1 000

10000

Re

100000

Figure 9.6 Heat transfer factory'// versus Reynolds number Re for cooling process and various materials.


385

JH 0.1

0.01 10

100

1000

Re

10000

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.


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

Moi lei

V

F( od

0.001 1000

10000

Re

100 000

Figure 9.10 Heat transfer factory'// versus Reynolds number Re for sterilization aseptic process and various materials.


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

Re

10000

100000

1000000

Figure 9.12 Estimated equations of heat transfer factory'// versus Reynolds number Re for all the examined processes.


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

1 2 3 4 5 6 7 8 9

Material

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


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

Chapter 9

394

10 ^

V

V

\\ Sr— "*^

N^ %

0.1

s,,v

\

5fc s » k

Nt,""• 0

'V

JM-1.llRe' '

ss

S

ss

*^ x^

•^

^

N.

JM

;., ^ »' ^"V

1s ^

0.01

— S iir-5* !l

^*;-

\__ 5 _ * «^s ^> sS

ing

j

H -J

It II BCtlV 5 )i ing

1

^.

^1 s, V——

\

St

0.1 1

JM

i

0.01

1

1

=

nv>r

i

\ a

i

:

1 1

1

j

s.

^> j

ST _ . ——— ——

^1 i j

1

\

\V

=f-

F •e izil 2(

S

1

0.001

V" \

1

I

0.0001

1

1

J

*"*"-^

10

X

1

100

..tod

1000 10000

y \'n ttrJ

100000

Re

Figure 9.16 Estimated equations of mass transfer factory^ versus Reynolds number Re for all the examined processes.

398

Chapter 9

REFERENCES

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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

Greek a a Y

F

moisture content, kg/kg dm compressibility factor

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

Subscripts A B

b c D e G

id K L 0

P

r res V

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 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

T 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

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

u

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

x

Xanthan gum, 85, 87

415

y

Yam, 187, 196, 311, 323, 355 Yellow batter, 293 Yogurt, 91, 101, 294, 304, 316, 353

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