Transport Properties of Foods – G. D. Sanauacos & Z. B. Maroulis

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

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Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

To our wives Katie G. Saravacos andRena Z. Maroulis for their encouragement and support

Preface The basic transport properties of momentum (flow), heat and mass are an important part of the engineering properties of foods, which are essential in the design, operation, and control of food processes and processing equipment. They are also useful in the quantitative analysis and evaluation of food quality and food safety during processing, packaging, storage and distribution of foods. The engineering properties are receiving increasing attention recently due to the need for more efficient processes and equipment for high quality and convenient food products, under strict environmental and economic constraints. The fundamentals of transport properties were developed in chemical engineering for simple gases and liquids, based on molecular dynamics and thermodynamics. However, the complex structure of solid, semi-solid, and fluid foods prevents the direct use of molecular dynamics for the prediction of the transport properties of foods. Thus, experimental measurements and empirical correlations are essential for the estimation of these important food properties. The need for reliable experimental data on physical properties of foods, especially on transport properties, was realized by the development of national and international research programs, like the European cooperative projects COST 90 and COST 90 bis, which dealt with such properties as viscosity, thermal conductivity, and mass diffusivity of foods. One outcome of these projects was the importance of context (relevancy) of the measurement and sample conditions. This explains the wide variation of the food transport properties, particularly mass diffusivity. Statistical analysis of compiled literature data may yield general conclusions and certain empirical "constants", which characterize the transport property (thermal conductivity or moisture diffusivity) of a given food or food class. All transport properties are structure-sensitive at the three levels, i.e. molecular, microstructural, and macrostructural. Correlation of food macrostructure to transport properties is relatively easy by means of measurements of density, porosity, and shrinkage. Correlation to molecular and microstructural (cellular) structure, although more fundamental, is difficult and requires further theoretical and applied work before wider application in food systems. The material of this book is arranged in a logical order: The introduction, Chapter 1, summarizes the contents of the book, emphasizing the need for a unified approach to the transport properties based on certain general principles. Chapter 2 introduces the fundamental transport properties as applied to simple gases and liquids. The three levels of food structure, molecular, micro- and macrostruc-

vi

Preface

ture, as related to transport properties are reviewed in Chapter 3. A unified treatment of the rheological properties of fluid foods is presented in Chapter 4. The theory, measurement and experimental data of moisture diffusivity are discussed in Chapter 5, while a statistical treatment of the literature data on moisture diffusivity is presented in Chapter 6. The diffusion of solutes in food systems is discussed in Chapter 7, with special reference to flavor retention and food packaging films and coatings. Thermal conductivity and thermal diffusivity are discussed in Chapter 8. Finally, heat and mass transfer coefficients are treated together in Chapter 9. We wish to acknowledge the contributions and help of several persons to our efforts over the years to prepare and utilize the material used in this book: Our colleagues, associates, and graduate students D. Marinos-Kouris, A. Drouzas, C. Kiranoudis, M. Krokida, N. Panagiotou, N. Zogzas of the National Technical University, Athens; M. Solberg, M. Karel, J. Kokini, K. Hayakawa, V. Karathanos, S. Marousis, K. Shah, and N. Papantonis of CAFT and Rutgers University; M. Bourne and A. Rao of Cornell University, Geneva, NY; A. Kostaropoulos of the Agricultural University of Athens; and V. Gekas of the Technical University of Crete. We also appreciate the discussions with the members of the European groups of cooperative projects COST 90 and COST 90 bis, especially R. Jowitt and W. Spiess. Special thanks are due to Dr. Magda Krokida for her substantial contributions in compilation and statistical analysis of the extensive literature data on transport properties of foods, and her continued help in preparing the illustrations and typing the manuscript. Finally, we wish to thank the staff of the publisher Marcel Dekker, Inc., especially Maria Allegra and Theresa Dominick, for their help and encouragement. We hope that this book will help the efforts to develop and establish food engineering as a basic discipline in the wide area of food science and technology. We welcome any comments and criticism from the readers. We regret any errors in the text that may have escaped our attention.

GeorgeD. Saravacos Zacharias B. Maroulis

Contents Preface

1. Introduction

1

I. RHEOLOGICAL PROPERTIES II. THERMAL TRANSPORT PROPERTIES III. MASS TRANSPORT PROPERTIES

3 3 4

2. Transport Properties of Gases and Liquids

7

I. INTRODUCTION II. ANALOGIES OF TRANSPORT PROCESSES III. MOLECULAR BASIS OF TRANSPORT PROCESSES A. Ideal Gases B. Thermodynamic Quantities C. Real Gases IV. PREDICTION OF TRANSPORT PROPERTIES OF FLUIDS A. Real Gases B. Liquids C. Comparison of Liquid/Gas Transport Properties D. Gas Mixtures V. TABLES AND DATA BANKS OF TRANSPORT PROPERTIES

7 8 9 9 10 12 14 15 16 18 19 19

3. Food Structure and Transport Properties I. INTRODUCTION II. MOLECULAR STRUCTURE A. Molecular Dynamics and Molecular Simulations

29 29 29 29 vii

viii

Contents

B. Food Materials Science C. Phase Transitions D. Colloid and Surface Chemistry III. FOOD MICROSTRUCTURE AND TRANSPORT PROPERTIES A. Examination of Food Microstructure B. Food Cells and Tissues C. Microstructure and Food Processing D. Microstructure and Mass Transfer IV. FOOD MACROSTRUCTURE AND TRANSPORT PROPERTIES A. Definitions B. Food Macrostructure and Transport Properties C. Determination of Food Macrostructure D. Macrostructure of Model Foods E. Macrostructure of Fruit and Vegetable Materials

30 30 31 32 32 32 34 34 36 36 40 45 46 50

4. Rheological Properties of Fluid Foods

63

I. INTRODUCTION II. RHEOLOGICAL MODELS OF FLUID FOODS A. Structure and Fluid Viscosity B. Non-Newtonian Fluids C. Effect of Temperature and Concentration D. Dynamic Viscosity III. VISCOMETRIC MEASUREMENTS A. Viscometers B. Measurements on Fluid Foods IV. RHEOLOGICAL DATA OF FLUID FOODS A. Edible Oils B. Aqueous Newtonian Foods C. Plant Biopolymer Solutions and Suspensions D. Cloudy Juices and Pulps E. Emulsions and Complex Suspensions V. REGRESSION OF RHEOLOGICAL DATA OF FOODS A. Edible Oils B. Fruit and Vegetable Products C. Chocolate

63 66 66 68 71 73 74 74 78 79 79 80 85 89 90 92 92 94 100

Contents

5. Transport of Water in Food Materials

ix

105

I. INTRODUCTION II. DIFFUSION OF WATER IN SOLIDS A. Diffusion of Water in Polymers III. DETERMINATION OF MASS DIFFUSIVITY IN SOLIDS A. Sorption Kinetics B. Permeability Methods C. Distribution of Diffusant D. Drying Methods E. Simplified Methods F. Simulation Method G. Numerical Methods H. Regular Regime Method I. Shrinkage Effect IV. MOISTURE DIFFUSIVITY IN MODEL FOOD MATERIALS A. Effect of Measurement Method B. Effect of Gelatinization and Extrusion C. Effect of Sugars D. Effect of Proteins and Lipids E. Effect of Inert Particles F. Effect of Pressure G. Effect of Porosity H. Effect of Temperature I. Drying Mechanisms V. WATER TRANSPORT IN FOODS A. Mechanisms of Water Transport B. Effective Moisture Diffusivity C. Water Transport in Cellular Foods D. Water Transport in Osmotic Dehydration E. Effect of Physical Structure F. Effect of Physical/Chemical Treatments G. Characteristic Moisture Diffusivities of Foods

105 106 107 109 110 114 118 120 123 124 124 12 5 126 127 127 13 0 133 13 5 137 138 140 141 143 144 144 145 146 147 150 152 155

6. Moisture Diffusivity Compilation of Literature Data for Food Materials

163

I. INTRODUCTION II. DATA COMPILATION

163 164

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):

T3

CO



o'

(Q

o o_

Chapter 4

98

c

.-*'^,

M

« 100 -

:

cu>

r :

^^^' *^»————

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^^^

_^*

10 -

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

e

ix^^^"*^

-^^

£Ift

I

•

=— ——— .^"^^^^'^ l

^ = •

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

Moisture Diffusivity Data Compilation

201

Table 6.5 Parameter Estimates of the Proposed Mathematical Model Material

No. of No. of Papers Data

Di (m2/s)

Do Ei Eo (mVs) (kJ/mol) (kJ/mol)

sd (m2/s)

Cereal products Corn dent grains kernel pericarp

4 3 3 4 3

26 15 28 25 13

4.40E-09 1.19E-08 1.15E-09 5.87E-10 1.13E-09

O.OOE+00 O.OOE+00 6.66E-11 5.32E-10 O.OOE+00

0.0 49.4 10.2 0.0 10.0

10.4 73.1 57.8 33,8 5.0

1.48E-10 3.30E-10 3.17E-10 1.88E-11 2.34E-11

Pasta -

3

21

1.39E-09

O.OOE+00

16.2

2.0

7.71E-12

3

12

9.75E-09

O.OOE+00

12.5

2.0

5.52E-11

7

35

2.27E-09

O.OOE+00

12.7

0.7

3.66E-11

6

22

1.94E-09

1.30E-09

0.0

46.3

9.53E-11

8

39

7.97E-10

1.16E-10

56.6

1.92E-10

4

34

2.03E-09

4.66E-10

9.9

4.6

1.77E-10

3

32

5.35E-09

O.OOE+00

34.0

10.4

1.45E-10

3

10

8.11E-10

1.05E-10

21.4

50.1

6.88E-11

4

49

1.52E-08

1.52E-08

0.0

33.3

1.02E-09

5

48

1.96E-08

1.96E-08

0.0

24.2

3.87E-09

9

90

2.47E-09

1.54E-09

13.9

11.3

1.69E-09

4

22

5.33E-10

1.68E-11

15.4

7.1

7.43E-11

4

31

1.45E-08

O.OOE+00

70.2

10.4

1.58E-09

16

106

1.57E-09

4.31E-10

44.7

76.9

4.02E-10

Rice kernel Rough rice Wheat -

Fruits Apple Banana Grapes seedless Raisins -

16.7

Model foods Amioca Hvlon-7 -

Vegetables Carrot

Garlic Onion Potato -

Chapter 6

202

l.E-06

• Moisture - infinite l.E-07

Q Moisture - zero

S" l.E-08 S,

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8 1 11 £

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Figure 6.6 Parameter estimates of the proposed mathematical model.

I 1 o S.

Moisture Diffusivity Data Compilation

l.E-06

l.E-07

Moisture (kg/kg db)

Figure 6.7 Predicted values of moisture diffusivity of model foods at 25°C.

203

Chapter 6

204

.E-06

H

Model foods :ratui•e (°C) = 60

.E-07

|——

-f —

l.E-08

Hylon-7

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.|" l.E-09

Amioca

— i

1

l.E-10

.E-ll

l.E-12

0.1

1 Moisture (kg/kg db)

Figure 6.8 Predicted values of moisture diffusivity of model foods at 60°C.

10

Moisture Diffusivity Data Compilation

l.E-06

l.E-07

l.E-08

f l.E-09 4

l.E-10

l.E-1

l.E-12

Moisture (kg/kg db)

Figure 6.9 Predicted values of moisture diffusivity of fruits at 25°C.

205

206

Chapter 6

l.E-06

Temperature (°C) = 60 -4

l.E-07

Moisture (kg/kg db)

Figure 6.10 Predicted values of moisture diffusivity of fruits at 60°C.

Moisture Diffusivity Data Compilation

207

l.E-06 1

H

Vegetables

Tempera ture i °C) = 25 l.E-07

l.E-08

l.E-12 10

0.1

Moisture (kg/kg db)

Figure 6.11 Predicted values of moisture diffusivity of vegetables at 25°C.

Chapter 6

208

l.E-06

l.E-07

l.E-12 0.1

1 Moisture (kg/kg db)

Figure 6.12 Predicted values of moisture diffusivity of vegetables at 60°C.

10

Moisture Diffusivity Data Compilation

209

l.E-06

l.E-07

l.E-12

0.1

10

Moisture (kg/kg db)

Figure 6.13 Predicted values of moisture diffusivity of corn at 25°C.

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.

Moisture Diffusivity Data Compilation

211

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

t~r

Temperature (°C) = 25 i

i

.E-07 -

-

——

.E-08 -

.E-09 -

— Rice kernel

^^ •"

.E-10 -j

Corn

r^j i Vheat •"^^ j

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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.

Moisture Diffusivity Data Compilation

Fruits

213 Apple

Total Number of Papers Total Experimental Points

8 64

Points Used in Regression Analysis Standard Deviation (sd, rti'/s)

36

Relative Standard Deviation (rsd, %) Parameter Estimates Di (m2/s) Do (mj/s) Ei (kJ/mol) Eo (kJ/mol)

(56%)

1.92E-10

457 7.97E-10 1.16E-10 16.7 56.6

.E-06

Temperature (°C) — 140

— «60 l.E-07

A 80

l.E-08

•I" l.E-09

l.E-10

l.E-11

l.E-12 0.1

1.0

10.0

Moisture (kg/kg db)

Figure 6.17 Moisture diffusivity of apple at various temperatures and moisture contents.

214

Chapter 6

Fruits

Grapes

Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, rrrVs) Relative Standard Deviation (rsd, %)

3 32 20 1.45E-10 1 31

seedless (63%)

Parameter Estimates Di(m 2 /s) Do (m2/s) Ei (kJ/mol) Eo(kJAnol)

5.35E-10 O.OOE+00 34.0 10.4

1 F-Ofi ————————————————— ;——————

—

— Tem perat ure (°C)

l.E-07 -

l.E-08 -

• 40 • 60 A 80

————— ———————— ————————

—

_L

1 ——————————

h-

•f l.E-09 - —————————— —————— 1

-1

^>•

1 i

l.E-10 1

^^

^^

=^ ~~~~ •* ^ !• •• * 1 -_ =1IE I J r^ ~*—» ———

i

01

^

'-

l.E-11 -

l.E-12 ———————————————————————————————————— 0.1 1.0 10.0 Moisture (kg/kg db)

Figure 6.18 Moisture diffusivity of grapes at various temperatures and moisture contents.

215

Moisture Diffusivity Data Compilation

Fruits

Banana

Total Number of Papers

4

Total Experimental Points

49

Points Used in Regression Analysis Standard Deviation (sd, rrvVs) Relative Standard Deviation (rsd, %) Parameter Estimates Di (m'Vs) Do(m'Vs) Ei (kJ/mol)

15 1.77E-10 15

(31%)

2.03E-09 4.66E-10 9.9

Eo (kJ/mol)

4.6

l.F-06 ——————————— —— Tern perature (°C) — ' • 60 -4-

l.E-07 -

A 80 ———

——— ———

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.

Moisture Diffusivity Data Compilation

217

Vegetables Total Number of Papers

Carrot 12

Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m2/s)

106 98 1.69E-09

(92%)

Relative Standard Deviation (rsd, %)_____18699 Parameter Estimates Di (m"/s) 2.47E-09 1.54E-09 Do(rrrVs)

Ei (kJ/mol) Eo(kJ/mol)

13.9 11.3

I.F-Ofi

l.E-07

l.E-08

•

l.E-09

l.E-10

l.E-11

l.E-12 0.1

1.0

10.0

Moisture (kg/kg db)

Figure 6.21 Moisture diffusivity of carrot at various temperatures and moisture contents.

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

Total Number of Papers

Total Experimental Points

(71%)

1.E-06

l.E0.1

1.0

10.0

Moisture (kg/kg db)

Figure 6.22 Moisture diffusivity of onion at various temperatures and moisture contents.

Moisture Diffusivity Data Compilation

219

Vegetables

Garlic

Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m'Vs) Relative Standard Deviation (rsd, %) Parameter Estimates Di (m"/s) Do(nWs) Ei(kJ/moI) Eo(kJ/mol)

4 22 19 7.43E-1 1 385

(86%)

5.33E-10 1.68E-11 15.4 7.1

l.E-06 -

—j —

— Tern perat ure(°C) = • 40 • 60 A 80

l.E-07 -

re =P

l.E-08 W5

"E, •f l.E-09 -

la 5

rr*~m 3 i~~*

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

Total Number of Papers Total Experimental Points Points Used in Regression Analysis

Standard Deviation (sd, m2/s)

5 26 15

(58%)

9.53E-1 1

Relative Standard Deviation (rsd, %)

54

Parameter Estimates Di(m"/s) Do(mVs)

Ei (kJ/mol) Eo (kJ/mol)

1.94E-10 1.30E-10

0.0 46.3

1 ,F,-06 i————— i—————

)

!

———— B40 ———

• 60 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.

Moisture Diffusivity Data Compilation

Cereal Products

221

Corn

Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, rrrVs) Relative Standard Deviation (rsd, %)

dent

3 15 15 3.30E-10 343

(100%)

Parameter Estimates Di(rrrVs) Do (m"/s) Ei (kJ/mol)

1.19E-09 O.OOE+00 49.4

Eo(kJ/mol)

73.1

l.E-06 peral ure CC) • 40 • 60 A80

l.E-07 -

i l.E-08 5" =

+* _>1

^^

\

•f l.E-09 -- .^T —* M **?—-———— < k— ^* \±* a L****T l.E-10

J

i

•^M

;\^^j^^

i 1 1

—« t11

l.E-11 -

0.1

1.0

10.0

Moisture (kg/kg db)

Figure 6.25 Moisture diffusivity of corn (dent) at various temperatures and moisture contents.

Chapter 6

222

Cereal Products

Corn

Total Number of Papers Total Experimental Points Points Used in Regression Analysis Standard Deviation (sd, m'Vs) Relative Standard Deviation (rsd, %)

grains

3 28 26 3 . 1 7E- 1 0 1 53

(93%)

Parameter Estimates Di(m/(6xtiBr)

(7-6)

where r is the particle radius, rjB is the viscosity of the solvent (water), T is the absolute temperature, and kB = 1.38xlO"22 J/molecule K is the Boltzmann constant. The Stokes-Einstein equation is based on hydrodynamic and not molecular forces, and it is applicable to solutes of molecular size five times larger than the solvent. For smaller molecules, the Wilke-Chang equation gives better prediction (Cussler, 1997). In both equations, the diffusivity is inversely proportional to the viscosity of the solution. In very viscous solutions, the diffusivity becomes independent of viscosity, e.g. the D of sugar in a gel is nearly equal to the D in water. B. Concentrated Solutions

The diffusivity of solutes in liquids D varies considerably with the concentration, sometimes with maximum or minimum values at certain concentrations. The D can be estimated from the diffusivity at infinite dilution Dm using a correction factor to account for the effect of chemical activity on the transport rate (Reid etal., 1987; Cussler, 1997):

D = D0(l+dlna/dlnC)

(7-7)

Diffusivity and Permeability of Small Solutes in Food Systems

243

where a is the activity and C is the concentration of the solute in the solution. The diffiisivity of the mixture at infinite dilution D0 can be estimated from the diffusivities at infinite dilution of the solute and the solvent, and the corresponding mole fractions (x/ and *?):

A, = [A,(x,= i)]MZ) 0 fe=im

(7-8

The correction factor (d lnor/9 InQ represents the molecular and hydrodynamic interactions in the concentrated solution, and it is negative in nonideal solutions (Cussler, 1997). Thus, D of the solute in a mixture becomes lower than D0 at both extreme concentrations (xlt x 2 = 1), with a minimum at an intermediate concentration.

III. DIFFUSION IN POLYMERS

The sorption and transport of small molecules (solutes) in polymeric materials are the basic physical phenomena of several important applications, such as separation processes, barrier films, and controlled release. Most of the research and theory in this area concerns synthetic polymers of known composition and structure, but the available knowledge can be applied to natural polymers, which are the basic structural components of most food materials. Molecular (Fickian) diffusion is assumed as the main mass transport mechanism, although in some cases other mechanisms may be involved. Solution of the diffusion equation (7-1) forms the basis of mathematical analysis of the experimental data. Most of the diffusivity data of solutes in polymers have been obtained using the sorption and/or the permeability methods (Chapter 5). The physical and transport properties of polymers are affected strongly by the size and shape (linear, branched, cross-linked) of the molecules (van Krevelen, 1990; Bicerano, 1996). Polymer materials can change their size (molecular weight) and microstructure during processing, changing their thermodynamic and transport properties, such as phase equilibria and diffusion coefficients. These changes should be considered in modeling and simulations of industrial processing and applications of polymers (Bokis et al., 1999). The polymer structure is defined by the chemical constitution, set by synthesis (or biosynthesis) and the morphology (microstructure), set by processing (Theodorou, 1996). Quantitative relations can be established between polymer structure and transport properties (diffusivity, permeability), based mainly on experimental measurements and phenomenological correlations from various systems (Petropoulos, 1994). Theoretical predictions and computer simulations, based on molecular science, are still at the development stage, and they could find useful applications in the future.

244

Chapter 7

A. Diffusivity of Small Solutes in Polymers The transport of small solutes (penetrants) normally obeys the Pick diffusion equation, and an effective diffusivity D can be estimated, assuming that the driving force is the concentration gradient. The Fickian diffusion is applicable to low concentrations (infinite dilution) of the solute, which is the case of most applications in polymer and food systems. In some biological systems, the thermodynamic diffusivity DT is used, based on the chemical potential gradient, which is related to the normal diffusivity D by the equation.

D = DT(d\na/d\nQ

(7-9)

where a is the chemical activity of the species at concentration C (Frisch and Stern, 1983). In most food-related applications, the concentration of the solutes in the polymer matrix is low, and the two coefficients become equal (D = D-f). Sorption kinetics and permeability measurements (see Chapter 5) can be used for the determination of diffusivity D of solutes in polymeric materials (Vieth, 1991) Solid polymers are amorphous materials, which exist in two nonequilibrium states, i.e. glassy and rubbery, with transition between the states at the glass transition temperature (Tg). The glassy state is characterized by a dense, tough, and low porosity (2-8%) structure. The diffusivity of small solutes in the glassy state is very low, e.g. IxlO" 18 to IxlO" 10 m2/s, depending on the polymer structure and the molecular size and concentration of the penetrant. The solute diffusivity increases substantially at higher solute concentrations, by plasticization of the polymer matrix. The activation energy for diffusion is much higher than in the rubbery state, and it increases near the glass transition temperature. In the rubbery state, polymers are flexible, elastic materials, with relatively large free volume, which facilitates molecular diffusion. Crystallization or stretching (induced orientation) of the polymers can reduce solute diffusivity. Liquids and vapors may cause swelling of the glass polymeric matrix, facilitating the diffusion process. Normal diffusion in the glassy and rubbery state is Fickian, i.e. the diffusion rate is proportional to the square root of time, according to Eq. (5-4) (Peppas and Brannon-Peppas, 1994). In glassy polymers (T < Tg), non-Fickian or anomalous diffusion of solutes may take place, since diffusion and polymer relaxation are comparable. Case II diffusion is also possible, when the diffusion rate is much faster than the relaxation of polymer molecules.

Diffusivity and Permeability of Small Solutes in Food Systems

l.E-09

245

T

l.E-14 30

35

40

Temperature (°C)

Figure 7.2 Arrhenius plots of diffusivity of solutes (water and carbon dioxide) in a polymeric material showing breaks at the glass transition temperature (Tg).

Most of the research and development in polymer science and engineering is directed to the design of specific polymer structures of known barrier properties (membranes), which can be used in separation processes of various molecular species. Separations based on molecular or particle size include reverse osmosis, gas separation, and ultrafiltration.

246

Chapter 7

B. Glass Transition

Mass transport (diffusion) of solutes in polymers is affected strongly by the thermodynamic state of the material. The molten polymer is a viscous fluid of non-Newtonian characteristics, which upon cooling forms two amorphous solid states, the rubbery, and, at lower temperature, the glassy state. The glass transition temperature Tg, a second-order transformation, is an important characteristic of the polymeric materials (Roos, 1992). The nonequilibrium rubbery and glassy states are affected strongly by the presence of solutes, such as gases, water and organic solvents, which reduce, in general, the glass transition temperature. The mechanical and transport properties of polymers at temperatures below and much above Tg, are affected by the temperature, following the familiar Arrhenius equation. However, in the temperature range Tg to (rg+100°C) the Williams-Landel-Ferry (WLF) equation is more appropriate (Levine and Slade, 1992): log (ar) = [-C, (T - T,)} I [ C , + (T- Tg)]

(7-10)

where aT is a scaling parameter, or the property ratio at T and Tgi e.g. relaxation time, viscosity, or diffusivity, and C/ and C2 are characteristic parameters of the WLF equation, determined experimentally. In normal systems the values C\ = 17.44 and C2 = 51.6 are used. The WLF equation predicts a sharp change of the scaling factor as the temperature is increased immediately above Tg, e.g. the viscosity decreases 3 to 5 orders of magnitude at temperatures 20-3 0°C above Tg. The WLF equation can be used to nonpolymer systems, which exhibit a glass transition temperature, such as sugar solutions, which are of interest to foods (Roos, 1992) The effect of water on the glass transition temperature of polymers and other food components, exhibiting glass transition, is of particular importance to food processing and food quality. The Tg of dry food components is relatively high but it decreases continuously even below 0°C as the moisture content is increased. Figure 7.3 schematically shows the change of Tg of a food biopolymer as a function of moisture content (Roos, 1992).

Diffusivity and Permeability of Small Solutes in Food Systems

247

!

—————-

C

V

!

>w

o

\ \

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)

Diffusivity and Permeability of Small Solutes in Food Systems

249

The dissolved solute can diffuse readily, while only part of the solute in the "holes" is available for diffusion, i.e. DD > DH (partial-immobilization model). 2. Free- Volume Model Free-volume models have been proposed for the prediction of transport properties in liquids and solids, based on the availability of elements of free volume within the material, through which the solute molecules can be transported (Frisch and Stern, 1983: Petropoulos, 1994). For polymeric materials, the Vrentas and Duda model, which can be used for both the glassy and the rubbery state, is discussed briefly here (Duda and Zielinski, 1 996). The self-diffusion coefficient of a molecule (1) in a binary mixture is an exponential function of the ratio of the volume required for diffusion of one mole V \ to the total free ("hole") volume per diffusing mole VFH. The diffusion coefficient DI of a solute (1) in a binary polymer (2) mixture, in the rubbery state, is given by the equation: D, = Do exp(- E I RT) exp { - [ y(a>, V* , + w^ V\}\ I VFH }

(7- 1 5)

where D0 is a constant, E is the activation energy, R is the gas constant, T is the absolute temperature, coj and ca2 are the mass fractions of 1 and 2, f=F*; MjV^M^ and MI and A/? are the molecular weights of 1 and 2. The accommodation factor y is taken between 0.5 and 1 .0. The specific free- volume VFH is calculated from the equation: VFH= (o,K,, (K2, + T- Tgl) + co2KI2 (K22 + T- Tg2)

(7-16)

where Tgt, Tg2 are the glass transition temperatures of 1 and 2, and Klh K2i, KI2 and K22 are free-volume parameters of 1 and 2, determined experimentally. The diffusivity (D = £>;) of trace amounts of a solute (1) in a glassy polymer (2) is given by the simplified equations: D, = Do exp(-E/RT) exp [ -(yco2 £ V'2) I VFm ]

(7-17)

and Tg2)}

(7-18)

where /L= 1 - (a2-a2g), and a2wd a2g are the thermal expansion coefficients of the rubbery and glassy states of the polymer. The free-volume theory predicts the following changes of diffusion coefficient (Duda and Zielinski, 1996): Strong effect of temperature and concentration

250

Chapter 7

near the glass transition temperature; increase with the size of solute molecule; plasticizers increase the available free volume, decrease the Tg, and increase the diffusivity; addition of impermeable fillers reduces D by increasing the tortuosity of the diffusing solute. Yildiz and Kokini (1999) modified the free-volume theory to account for the effect of temperature and water activity on the retention and release of flavor compounds in food polymers. The diffusivity of hexanol, hexanal, and octanoic acid in uncooked soy flour was predicted to decrease sharply as the temperature is reduced in the rubbery state until the Tg, leveling-off at lower temperatures (glassy state). The diffusivity of flavor compounds in gliadin was predicted to increase sharply from about 1 x 10~18 m2/s to 1 x 10~10 m2/s, as the water activity was increased from 0.2 to 0.8 (at 25°C). Cross-linking of food polymers, e.g. by cooking of soy flour, predicts significant increase of diffusivity (i.e. reduced retention) of flavor compounds (e.g. hexanal).

3. Molecular Simulation Molecular simulations can describe sorption and diffusion phenomena in polymer systems, based on chemical constitution of the components. Most of the simulation work is related to simple amorphous rubbery and glassy systems, in which solute transport is assumed to follow the solution-Fickian diffusion mechanism of mass transport (Theodorou, 1996). Molecular simulations are essentially solutions of the statistical mechanics of a model of given molecular geometry and interaction parameters. They involve the generation of configurations of the system, from which structural, thermodynamic and transport properties can be extracted. Molecular dynamics (MD) assumes that the penetrant (solute) moves into channels of the sorption sites, created by small fluctuations in the polymer configuration. Transition state theory (TST) provides a more approximate treatment of the penetrant diffusion process, assuming a jumplike transport mechanism. The computer time required for the extensive computations can be reduced by certain approximations, which are less severe than the ones used in the dual-sorption and free-volume models. Computer calculations involve the estimation of the Henry constant, the geometric characteristics of the accessible volume in the polymer matrix, and its distribution and rearrangement with thermal action, using Monte Carlo algorithms. Molecular dynamics simulations have successfully predicted the selfdiffusion coefficient in glassy and rubbery polymers, interacting with penetrant solutes. The objective of molecular simulations is to develop the field of applied "molecular engineering of materials" for producing materials with tailored separation and barrier properties.

Diffusivity and Permeability of Small Solutes in Food Systems

251

IV. DIFFUSION OF SOLUTES IN FOODS

The diffusivity of solutes and other molecules in food materials depends primarily on the size of the diffusing molecule and the food structure. The needed experimental measurements of diffusivity in solid and semisolid foods are usually based on the concentration-distribution method, described in Chapter 5 (Naessens et al., 1981, 1982; Giannakopoulos and Guilbert, 1986). Diffusivity data on salts, organic and flavor components are of particular interest to food processing and food quality. A. Diffusivity of Salts

Table 7.4 shows typical diffusivities of sodium chloride in model food gels and food materials. The diffusivity depends strongly on the physical structure of the food material. The diffusivity D of salt in dilute gels (Gros and Ruegg, 1987) is very close to the D of salt in aqueous solutions, i.e. 12.5 * 10"10 m2 / s (see Table 2.4). Similar high diffusivities are observed in high-moisture foods of gel structure, like pickles (Pflug et al., 1975). Evidently, the salt ions can migrate in such gels at rates similar to the diffusion in liquid water. The salt diffusivity in Swiss cheese (Gros and Ruegg, 1987) is considerably lower than in gels (1.9x 10"10 m2/s), evidently due to the higher solids concentration and the presence of fat globules in the material. Higher salt diffusivity values D were reported by Pajonk et al. (2000) in brining Swiss cheese. The D value decreased from about 7 x 10"10 to 2 x 10"10m2/s when the brine concentration was increased from 0 to 20% NaCl. The diffusivity of salt in white feta cheese was determined as 2.3 x 10"10 m2/s (Yanniotis et al., 1994).

Table 7.4 Diffusivities D of Sodium Chloride in Food Materials (20°C) Material

D, x 10"'° m2/s

Agar gel, 3 % solids

12.0

Pickles

11.0

Swiss cheese

1.90

Meat muscle, fresh Meat muscle, thawed

4.00

Herring

2.30

Green olives, fresh

0.38 1.95

Green olives, treated

2.20

252

Chapter 7

The salt diffusivity in fresh meat muscle is 2.2 x 10"'° m2/s, while it is considerably higher (4.0x 10~10 m2/s) in meat flesh that has been frozen and then thawed (Dussap and Gros, 1980; Fox, 1980). The relatively low D of salt in the meat is caused by the resistance of the cellular structure to mass transfer. The salt diffusivity in fish is, in general, similar to the D in meat, e.g. 2.3 x 10"10 m2/s in herring (Rodger et al, 1984). The diffusivity of salt in fresh green olives is quite low (0.38 x 10"'° m2/s), evidently due to the presence of skin and to high oil concentration. Treatment of the olives with 1.8% caustic soda increases the D value to 1.95 x 10"10m2/s (Drusasetal., 1988). The diffusivity of sodium hydroxide in tomato skin, measured with a modified diffusion cell (Figure 7.1), was found to be 0.02 x 10"10m2/s (Floras et al., 1989). A higher value was found for the diffusivity of the same alkali in the skin of pimiento pepper (0.055 x 10~10m2/s). Diffusivities of other salts of interest to foods (chlorides, nitrites, nitrates, etc.) are similar to the D values of sodium chloride. A bibliography on the diffusivity of salt in foods was prepard by Ruegg and Schar (1985).

B. Diffusivity of Organic Components The diffusivity of organic solutes in food materials is important in food processing operations, like extraction (sugars, lipids, flavors), and in food quality (e.g. sugar taste, volatile flavor retention). The diffusivity D of organics in liquid foods is related closely to the viscosity 77 of the solution, through the relation r/D/T= constant Eq. (2-36). Organoleptic flavor perception is related to the diffusivity of the flavor component (e.g. sugar) and the viscosity of the liquid food (Kokini et al., 1982; Kokini, 1987). The flavor of highly viscous pseudoplastic foods is enhanced by shearing, which reduces considerably the apparent viscosity, increasing at the same time the diffusivity of the flavor component(s). For large molecules in food liquids, like peroxydase, the Stokes-Einstein equation (7-6) can be applied, while for smaller solutes in sugar solutions (e.g. nicotidamine) the Wilke-Chang equation (2-34) has been found applicable (Loncin, 1980; Stahl and Loncin, 1979). The diffusivity of nicotinamide in fructose solutions decreases from about 8 x 10~10 to 0.5 x 10~10 m2/s, when the sugar concentration is increased from 0 to 60%. In the same range of fructose concentration, the diffusivity of peroxidase decreases from 1.0 x 10"10 to 0.1 x IQ"10 m2/s. The activation energy for diffusion of both species increases sharply from 20 to 45 kJ/mol in the same sugar concentration range. The prediction models for the diffusivity of solutes in polymers, discussed earlier in this chapter, are difficult to apply in solid and semisolid foods, due mainly to the heterogeneous physical structure of the food materials. The presence

Diffusivity and Permeability of Small Solutes in Food Systems

253

of significant open space in food solids, such as pores, cracks, and channels, complicates the diffusion process, since a portion of the solutes can diffuse quickly in the gas phase, while the rest diffuses very slowly from the sorbed or trapped state. The diffusivity in the gas phase is about five orders of magnitude (x 105) higher than in the solid phase. The free-volume model, suggested for the prediction of diffusivities in polymers, was applied by Yildiz and Kokini (1999) for the prediction of diffusivity of flavor components in solid foods. Application of this model assumes that the food material behaves as a homogeneous polymer material of low porosity, such as uniform protein, carbohydrate or lipid films. The molecular simulation model (Theodorou, 1996), requiring extensive computer calculations, when developed and applied further in the polymer field, could be adapted to food materials in the future. Table 7.5 shows some typical diffusivities of organic solutes in food materials, which are useful in calculations involving solvent extraction (leaching) and liquid infusion operations (Schwartzberg and Chao, 1982). The diffusivity of sugars in gels (e.g. agar) is similar to the diffusivity in water solutions, Table 7.3 (Warin et al., 1997). The diffusivity of solutes in solid foods D is considerably lower than in dilute water solutions, shown in Tables 7.2 and 7.3, due to blockage of diffusion paths, occlusion (trapping), and sorption by the food biopolymers. The D in solids is related to the diffusivity of the solutes in water Dw by an empirical relation analogous to Eq. (5-2): D = (ew/r)Dw

(7-19)

where ew is the volume fraction of free water in the solid (analogous to porosity), and T is the tortuosity of the diffusion path.

Table 7.5 Diffusivities of Solutes in Solid Foods Solid

Solute

Solvent

Sugar beets

Sucrose

Water

Sugar cane

Sucrose

Water

Apple slices

Sugars

Water

Coffee beans

Coffee solubles

Water

Soybean flakes

Soybean oil

Cottonseed oil

Cottonseed oil

Hexane Hexane

Peanuts

Peanut oil

Hexane

r,°c 65 75 75 98 69 69 25

7,

D, x 10'10 m /s

6.80 2.00 11.5 1.00 1.00 0.27 0.006

254

Chapter 7

The free water fraction in the solid can be estimated from the moisture content and the sorption isotherm, but the tortuosity factor must be estimated indirectly from the measured D. Both parameters are not constant during food processing and storage, due to the significant changes of the food structure. The effect of solids content on the diffusivity of organic compounds in foods, is illustrated by the diffusivity of cyclohexanol in potato, which decreases from 6 x 10'10 to 2 x 10~10 m2/s in high solids potato (Loncin, 1980). The activation energy for diffusion is analogous to that of water in potato, 35.7 kJ/mol. The diffusivity of a solute may be reduced significantly by the presence of another solute, diffusing simultaneously in a solid food material (multicomponent diffusion). Thus, the individual diffusivity of citric acid (1) in prepeeled potato is reduced from /)/ = 4.3 x 10~10 to D12 = 6.6 x 10"" m2/s in the presence of ascorbic acid (2), diffusing simultaneously. At the same time, the diffusivity of ascorbic acid is reduced from D2 = 5.4 x IQ' 10 to D2, = 8.3 x IQ"11 m2/s (Lombardi et al, 1996). The diffusivities of the two solutes in dilute water solutions (w) are D!w = 6.6 x 10'10andZ)2w= 8.4 x I(r10m2/s.

C. Volatile Flavor Retention The diffusion of volatile flavor (aroma) components in foods is important in food processing operations, such as evaporation and drying, and in storage and quality of food products. Most aroma components are very volatile in aqueous solutions, since they form highly nonideal mixtures with water. The volatility of these components at thermodynamic equilibrium is characterized by the activity coefficient and the relative volatility, which are the basic elements of the vaporliquid equilibria (VLB). Calculation of (VLB) is required for the analysis of any vapor-liquid separation or interaction (Prausnitz et al., 1986; Reid et al., 1987; Le Maguer, 1992). The relative volatility of an aroma compound A in dilute water solution aAw is defined by the equation (Saravacos, 1995)

aAw=yApAo/Pwo

(7-20)

where yA is the activity coefficient of A, and pAO, pHO are the vapor pressures of A and water, respectively, at the given temperature. The activity coefficient of a component YA is related to the concentration Q and the chemical activity aA by the equation: aA = /ACA

(7-21)

Diffusivity and Permeability of Small Solutes in Food Systems

255

The activity coefficients of aroma components in water and aqueous foods are very high, especially for partially soluble organic flavor components, like esters and higher alcohols. They are estimated by computer-aided techniques using empirical models, like the UNIQUAC and the UNIFAC (Reid et al., 1987). The presence of sugars in aqueous solutions, like in food materials, increases considerably the activity coefficient (Saravacos et al., 1990; Sancho and Rao, 1997). Table 7.6 shows some typical relative volatilities of volatile flavor compounds in dilute water solutions (Saravacos, 1995; Chandraskaren and King, 1972). The relative volatility of these compounds in aqueous solutions of 60% sucrose is 20 to 10 times higher than in water, due to the strong interactions of the 3component system (Saravacos et al., 1990). The volatile flavors (aromas) are normally recovered during the evaporation of fruit juices and other aqueous systems by stripping and distillation processes (Saravacos, 1995; Karlsson and Tragardh, 1997). Maximum removal of a volatile from the liquid phase is obtained when vapor-liquid equilibrium (VLB) is established. However, establishment of true equilibrium requires infinite time, so evaporation and distillation are actually nonequilibrium processes with partial removal of volatiles. Diffusion of the flavor components from the interior to the surface of food particles is reduced sharply in the presence of sugars and other solids. Evaporation from falling liquid films (Lazarides et al., 1990) or from mechanically agitated films (Marinos-Kouris and Saravacos, 1974) can increase the stripping efficiency of volatiles.

Table 7.6 Typical Relative Volatilities of Aroma Compounds in Aqueous Solutions aAw at Infinite Dilution (25°C) Volatile compound

aAw

Methyl anthranilate

3.90

Methanol

8.30

Ethanol

8.60

1 -Propanol

9.50

1-Butanol

14.1

n-Amyl alcohol

23.0

Hexanol 2-Butanone Diethyl ketone Ethyl acetate Ethyl butyrate

31.0 76.0 77.0 205 643

256

Chapter 7

The retention of volatile flavors during food dehydration depends primarily on the presence of sugars and other solids, which reduce the aroma diffusivity in the food material. Contrary to aroma recovery processes, aroma retention is a highly nonequilibrium process, utilizing conditions that will prevent the flavor compounds from reaching the evaporation surface, such as fast surface drying (Rulkens and Thijssen, 1972). The loss of volatile flavors depends on the evaporation or drying rate of water. The mass transport of volatiles should be considered as a ternary diffusion process, with three binary diffusivities, i.e. water/solids, volatile/water and volatile/solids (Coumans et al., 1994a). Figure 7.4 shows the loss of a very volatile flavor compound, ethyl butyrate (relative volatility in water aAw = 643), as function of % water evaporated in aqueous solutions and during vacuum- or freeze-drying (Saravacos and Moyer, 1968a, b). The loss of the volatile ester from the water solution is very rapid, e.g. 90% loss by evaporation of 30% water. The presence of pectin in the water solution reduces the volatile loss and increases its retention. A higher retention is obtained by freeze-drying. Volatile retention during spray drying depends not only on the relative volatility but also on the interaction of the compound with the nonvolatile components of the food liquid. Thus, in spray drying of food emulsions containing flavors, ethyl butyrate is retained only by 20%, while limonene may be retained almost quantitatively (Furuta et al., 2000). The retention of volatile flavors during food dehydration is a very important consideration in the selection of drying processes and equipment for optimum product quality. Flavor retention is related to the reduction of diffusivity of flavor compounds by sugars and other food solids. Figure 7.5 shows that the diffusivity of diacetyl in water solutions is reduced by almost 100 times, when the sugar concentration is increased from 0 to 70% (Voilley and Simatos, 1980; Voilley and Roques, 1987).

Diffusivity and Permeability of Small Solutes in Food Systems

50

257

100

Evaporation (%)

Figure 7.4 Retention of ethyl butyrate: W, evaporation of water; VD, vacuum-drying of pectin solution; FD, freeze-drying of pectin solution.

Thermodynamic and transport phenomena analysis indicate that flavor retention is a diffusion-controlled process (Kerkhof, 1975; Bruin and Luyben, 1980). Fast drying processes, like spray drying, improve volatile retention by trapping the solute in the solid matrix. A selective diffusion mechanism may explain the volatile retention in spray- and freeze-drying (Coumans et al., 1994b). Atomization and evaporation of water/volatiles from drops control flavor retention in spraydried particles (King, 1994; Hecht and King, 2000). Retention of charactertistic aroma during storage of dried fruits can be improved by using low relative humidities (Rff) and low temperatures. Moisture sorption of stored fruits increases sharply at RH > 60%, resulting in a strong rise of flavor diffusivity and subsequent loss of aroma (Saravacos et al., 1988).

258

Chapter 7

l.E-09

\

l.E-10

l.E-11

l.E-12 20

40

80

Sugar(%)

Figure 7.5 Diffusivity of diacetyl in sucrose solutions.

D. Flavor Encapsulation Encapsulation and controlled release of solutes is used widely in pharmaceuticals, medicinal products, flavors, and pesticides. Controlled release is based on relaxation-controlled dissolution of the coating material, which consists usually of a glassy polymer (Cussler, 1997). Encapsulation of flavors, acidulants (citric and ascorbic acid), salts, and enzymes is used to prevent or control the diffusion of the solutes in various food processing and food utilization operations (Karel, 1990). Encapsulation can be achieved by entrapment in glassy polymers or in sugar crystals, in fat-based matrices, or by incorporation in liposomes (e.g. lecithin). Release of encapsulated solutes is achieved by temperture and moisture control, enzymatic release, grinding etc. The role of glass transition temperature Tg to solute release is important, since diffusivity rises sharply above Tg. The WLF equation (7-10) relates the diffusivitiy to the temperature and the Ts. The "collapse

Diffusivity and Permeability of Small Solutes in Food Systems

259

temperature" is related to T& and both temperatures decrease as the moisture content is increased. Spray- and freeze-drying are used to encapsulate flavor solutes in polymer matrices, using high initial drying rates to form a dried polymer layer, which reduces diffusivity.

V. PERMEABILITY IN FOOD SYSTEMS

The transport of small solutes, such as water, oxygen, and carbon dioxide through polymer films and protective coatings is of fundamental importance to food packaging and food processing. The permeability of these materials is based on the principles of diffusion of solutes in polymer systems. The permeability of synthetic membranes is important to separation processes used in food processing, such as reverse osmosis, gas separation, and ultrafiltration. The structural and physicochemical factors, which affect the diffusivity of solutes in polymers, are also important in characterizing the performance of packaging films and food coatings. Control of such factors as glassy/rubbery state, cross-linking, and polymer orientation, can determine the permeability of these materials. A. Permeability The permeability P of a film or thin layer of thickness z is related to the diffusivity D and the solubility S of the penetrant (solute) in the material, according to the equation: J = P (Aplz) = DS (Aplz)

(7-22)

where J is the mass transfer rate (kg/m2s), Aplz is the pressure gradient (Pa/m), and 5 is the gas/liquid equilibrium constant, S = C/p where C is the concentration (kg/m3) and p the pressure (Pa). The solubility S is equal to the inverse of the Henry constant (S=1/H), and it has units (kg/m3Pa); it can be determined as the slope of the sorption isotherm (C versus p). From equations (7-22) it follows that: P = DS

(7-23)

The permeability has SI units (kg/m s Pa) or (g/m s Pa), but various other units are used in packaging, reflecting the measuring technique or the particular food/package application (Hernandez, 1997; Donhowe and Fennema, 1994).

260

Chapter 7

The permeability P is related to the permeance PM or transmission rate 77? (=PM) and the water vapor transfer rate WVTR by the equation (McHugh and Krochta, 1994): P = PMz = WVTR I Ap

(7-24)

The units of permeance (kg/m2 s Pa) are identical to the units of the mass transfer coefficient kp. The units of WVTR are (kg/m s) (Saravacos, 1997). The SI units are useful in relating and comparing the literature data on P and WVTR to the fundamental mass transport property of diffusivity D (m2/s). The permeability of polymer films and coatings can be determined by measurements of sorption kinetics and diffusion, discussed in Chapter 5, in relation to water transport. Conversion of solubility S and diffusivity D data to permeability P though Eq. (7-23) is possible, when the material behaves like a homogeneous medium and Fickian diffusion can be assumed. Simplified permeability measurement methods are used for packaging and coating films (barriers), and most of the literature data are reported in units related to the special methods used (ASTM, 1990, 1994). The measured permeabilities represent an overall transport property of the material, based on the applied pressure gradient. Since the polymer film may have structural inhomogeneities, such as pores, channels, cracks, and pinholes, mass transport may involve, in addition to molecular diffusion, Knudsen diffusion and hydrodynamic or capillary flow (Hernardez, 1997). In such cases, the simplified relationship between diffusivity and permeability Eq. (7-23) is not applicable. Permeability is affected significantly by environmental condition, such as air relative humidity (RH), which may increase sharply the permeability of most packaging and coating films. The total permeability PT of a multilayer laminate is related to the permeabilities and the thicknesses of the individual films (P, z/) by the equation (Cookseyetal., 1999) /V=[(Sr,)]/[W/)]

(7-25)

The total permeance PMT or transmission rate TRT can be calculated from the equation: PMT=l/I,(z,/Pd

(7-26)

Temperature increases permeability P according to the Arrhenius equation in a similar manner with the effect of temperature on diffusivity D and solubility S:

= P0 exp(-£//?7), D = D0 exp(-ED/RT), S = S0 exp(-Es /RT)

(7-27)

Diffusivity and Permeability of Small Solutes in Food Systems

261

Table 7.7 Conversion Factors to SI Permeability Units (g/m s Pa) Conversion from / to (g / m s Pa) 3

2

cm (STP) mil /100 in day atm 3

2

Multiplying factor 6.42 x 10'17

cm (STP) mil / m day atm

4.14* 10"18

cm3(STP) urn / m2 day kPa

1.65 x 1Q'17

g)im/m 2 daykPa

1.16X10' 1 4

2

1.16x10""

gmm/m daykPa 2

g mil/m day atm

2.90 x 10"15

g mil/m2 day (mm Hg)

2.20 x 10'12

2

g mil/m day (90 % RH, 100 °F) g mil/100 in2 day (90 % RH, 100 °F)

4.50 x 10'14 7.00 x 10'13

perm (ASTM, 1990)____________________1.45 x 1Q-9 STP = standard temperature and pressure. 1 mil = 0.001 inch = 2.54xl0 0 m.

Pressure drop of water vapor across the film at 90/0% RH and 100°F, AP = 6560 Pa. 1 mm Hg = 133.3 Pa.

The energy of activation for permeability Ep may vary, depending on the type of polymer and the temperature, in the wide range of 10 to 80 kJ/mol (Hernandez, 1997). Table 7.7 shows the conversion factors from the various literature units of permeability to SI units (g/m s Pa).

B. Food Packaging Films Synthetic polymer firms are used as barriers to the transport of water vapor, oxygen, carbon dioxide, and food components, like aroma/flavor compounds and lipids, from or to the packaged food product. Food packaging films are made of special polymeric materials, like polyethylene, both low density (LDPE) and high density (HDPE), polypropylene (PP), polyvinyl chloride (PVC), polystyrene (PS), polyethylene terephthalate (PET) and polyamides (nylon) (Hernandez, 1997; Miltz, 1992). Permeability, mechanical properties, food compatibility (nontoxicity), and cost are the main characteristics in selecting the proper material (Brody and Marsh, 1997; Hanlon et al., 1998). Permeability, like diffusivity, is affected significantly by polymer microstructure, solute-polymer interactions, solute concentration (especially moisture content or RH), and temperature. Some typical permeabilities of common packaging films to water vapor and oxygen are shown in Table 7.8 at 25°C (Miltz, 1992; Hernandez, 1997).

262

Chapter 7

Table 7.8 Permeabilities of Packaging Films to Water Vapor and Oxygen (25°C) Permeability, x 1 0 - 1 2 g / m s P a Packaging film Water vapor Oxygen 1.40 0.031 LDPE 0.007 0.20 HDPE 1.00 0.010 PP 3.00 0.005 PVC 12.0 0.018 PS 1.40 0.0006 PET 0.002 0.0004 Nylon LDPE = low density polyethylene, HOPE = high density polyethylene, PP = polypropylene, PVC = polyvinyl chloride, PS = polystyrene, PET = polyethylene terephthalate, Nylon = polyamide.

The permeability of nylon to oxygen at various moisture contents has been analyzed by the dual-sorption model (Hernandez, 1994). Although the water diffusivity increases at higher moistures, the solubility and the permeability decrease sharply at the beginning, leveling-off at water activities above 0.2. The permeability of polymer firms and food coatings to carbon dioxide is important in food packaging and storage. Typical values of permeability of carbon dioxide at 25°C are: LDPE, 1.6 x 10'13; HDPE, 5.3 x 10'15 g/m s Pa. C. Food Coatings

The permeability of edible food coatings to water is of particular interest to food quality, since their primary function is to act as barriers to moisture transport

during storage. Food coatings can also control the transport of gases (mostly oxygen), flavor components and lipids in food systems. Edible coatings, used as barriers in foods, include proteins (wheat gluten, caseinates, whey protein, corn zein), polysaccharides (starch, dextrins), pectins, lipids, and chocolate. Composite coatings, containing a food biopolymer (e.g. protein) and a hydrophobic material, like lipid, fatty acid, chocolate, and beeswax, usually have very low water permeabilities. The food coatings are prepared as solutions or dispersions/emulsions of the primary biopolymer in solvents (ethanol, alkalis, or acids). They contain various plasticizers, such as glycerine and sorbitol, which improve the physical and mechanical properties of the coating. They are applied to the various fresh and processed foods, like fruits and vegetables by dipping in an emulsion, spraying or foaming and brushing. Table 7.9 shows typical water permeabilities of food coatings (McHugh and Krochta, 1994):

Diffusivity and Permeability of Small Solutes in Food Systems

263

Table 7.9 Water Vapor Permeabilities P of Food Coatings (25°C) P, xlO- 1 0 g/msPa Food coating 6.10 Gluten-glycerine 7.20 Whey protein-sorbitol 1.00 Zein-glycerine 4.20 Sodium casemate 0.12 Chocolate 0.006 Beeswax

D. Permeability/Diffusivity Relation The simple permeability/diffusivity/solubility relation of Eq. (7-23) is useful for estimating the permeability P from diffusivity D and solubility S data of polymer films, and for comparison of P and D data. This relation applies to systems behaving as homogeneous materials, in which solute transport is by Fickian molecular diffusion. It does not hold for heterogeneous materials, consisting of pores, channels and capillaries, in which a significant portion of mass transfer takes place by mechanisms other than molecular diffusion. Table 7.10 shows some typical diffusivity and permeability data for packaging films and food coatings. The comparison is facilitated by using consistent (SI) units (Saravacos, 2000). A typical application of the permeability-diffusivity relation is given for chocolate film, using published data of Biquet and Labuza (1988): Typical permeability P and diffusivity D values for a chocolate coating about 0.6 mm thick at 20 0 C:P = 0.11 x 10-'°g/msPaandZ)=l x 10-13m2/s. The solubility S of water in the chocolate material can be estimated from the sorption isotherm at 20°C. It is defined by the Henry equation, C = S p, where C is the concentration (kg/m3) in the material andp is the partial pressure of water (Pa). Thus, the solubility is equal to the slope of the isotherm (S = C/p). Considering the initial sorption stage, water activity a,v 0 to 0.1, S = (1.7 kg water/100 kg solids)/Ap where Ap = a,vp0 or Ap = 0.1 PO, and PO is the vapor pressure of water at 20°C (p0 = 2340 Pa), and Ap = 234Pa. The concentration of water in the chocolate material is converted to consistent (SI) units, as follows: Assume density of dry chocolate 1600 kg/m3; therefore, the volume of 100 kg dry material will be 100/1600 = 0.0625 m3. The water concentration in the chocolate becomes C = (1.7/0.0625) = 27.2 kg/m3, and the solubility S = 27.2/232= 0.116 kg/m3 Pa. Using the measured diffusivity of the system (D = 1 x 10~13 m2/s), the permeability of the chocolate film according to Eq. (7-23) will be P = D S = 0.116 x 10"13kg/ms Pa, or P = 0.116 x 10"'°g/ms Pa, which is very close to the measured permeability.

264

Chapter 7

Table 7.10 Typical Water Vapor Permeabilities and Diffusivities Film or coating P, x 10-'°g/msPa A xio-'°m 2 /s 0.005 0.002 HDPE LDPE 0.010 0.014 PP 0.010 0.010 0.041 0.050 PVC 1.00 Cellophane 3.70 Protein films 0.100 0.10-10.0 0.100 0.10-1.00 Polysaccharide films 0.010 Lipid films 0.003-0.100 Chocolate 0.001 0.11 Gluten 1.00 5.00 Com pericarp 0.10 1.60 LPDE = low density polyethylene, HDPE = high density polyethylene, PP = polypropylene, PVC = polyvinyl chloride

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McHugh, T.H., Krochta, J.M. 1994. Permeability Properties of Edible Films. In: Edible Coatings and Films to Improve Food Quality. J.M. Krochta, E.A. Baldwin, M. Nisperos-Carriedo, eds. Lancaster, PA: Technomic Publ., pp. 139-187. Miltz, J. 1992. Food Packaging. In: Handbook of Food Engineering. D.R. Heldmanan and D.B. Lund, eds. New York: Marcel Dekker, pp. 667-718. Naessens, W., Bresseleers, G., Tobback, P. 1981. A Method for the Determination of Diffusion Coefficients of Food components in Low and Intermediate Moisture Systems. J. Food Sci. 46:1446-1449. Naessens, W., Bresseleers, G., Tobback, P. 1982. Diffusional Behavior of Tripalmitin in a Freeze-Dried Model System of Different Water Activities. J. Food Sci. 47:1245-1249 Pajonk, A.S., Saurel, R., Blank, D., Laurent, P., Andrieu, J. 2000. Experimental Study and Modeling of Effective NaCl Diffusion Values During Swiss Cheese Brining. Proceedings of 12th Int. Drying Symposium, IDS 2000, Noordwijk, NL, paper No. 425. Peppas, N.A., Brannon-Peppas, L. 1994. Water Diffusion in Amorphous Macromolecular Systems and Foods. J. Food Eng. 22:189-210. Petropoulos, J.H. 1994. In: Polymeric Gas Separation Membranes. D.R. Paul and Y.P. Yampolski. eds. New York: CRC Press. Pflug, I. J. Fellers, P.J., Gurevitz, D. 1975. Diffusion of Salt in the Desalting of Pickles. Food Technol. 21:1634-1638. Prausnitz, J.M., Lichtenhlater, R., Azevedo, E.G. 1986. Molecular Thermodynamics of Fluid Phase Equilibria. Englewood Cliffs, NJ: Prentice-Hall. Reid, R.C., Prausnitz, J.M., Poling, B.E. 1987. The Physical Properties of Gases and Liquids. 4th ed. New York: McGraw-Hill. Rodger, G., Hastings, R., Cryne, C., Bailey, J. 1984. Diffusion Properties of Salt and Acetic Acid into Herring. J. Food Sci. 49:714-720. Roos, Y. H. 1992. Phase Transitions and Transformations in Food Systems. In: Handbook of Food Engineering. D.R. Heldman and D.B. Lund, eds. New York: Marcel Dekker, pp. 145-197. Ruegg, M., Schar, W. 1985. Diffusion of Salt in Food-Bibliography and Data. Liebefeld, Berne: Swiss Federal Dairy Research Institute. Rulkens, W.H. and Thijssen, H.A.C. 1972. The Retention of Organic Volatiles in Spray Drying Aqueous Carbohydrate Solutions. J. Food Technol. 7:95-105. Sancho, M.F., Rao, M.A. 1997. Infinite Dilution Activity Coefficients of Apple Juice Aroma Compounds. J. Food Eng. 34:145-158. Saravacos, G.D. 1995. Mass Transfer Properties of Foods. In: Engineering Properties of Foods 2nd ed. M.A. Rao and S.S.H. Rizvi, eds. New York: Marcel Dekker, pp. 169-221. Saravacos, G.D. 1997. Moisture Transport Properties of Foods. In: Advances in Food Engineering CoFE 4. G. Narsimham, M.R. Okos, S. Lombardo, eds. West Lafayette IN: Purdue University, pp. 53-57.

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

Thermal Conductivity and Diffusivity of Foods

Table 8.2 Effect of Temperature on the Major Food Components X=bo+biT+b2T2+b3T:1

bo

b,

b2

b,

Air

2.43E-02

7.89E-05

-1.79E-08

-8.57E-12

Protein

1.79E-01

1.20E-03

-2.72E-06

Gelatin

3.03E-01

1.20E-03

-2.72E-06

Ovalbumin

2.68E-01

2.50E-03

Carbohydrate

2.01E-01

1.39E-03

Starch

8.7 IE-02

9.36E-04

Gelatinized Starch

3.22E-01

4.10E-04

Sucrose

3.04E-01

9.93E-04

Fat

1.81E-01

2.76E-03

-1.77E-07

Fiber

1.83E-01

1.25E-03

-3.17E-06

Ash

3.30E-01

1.40E-03

-2.91E-06

Water

5.70E-01

1.78E-03

-6.94E-06

Ice

2.22E+00

-6.25E-03

1.02E-04

-4.33E-06

2.20E-09

Chapter 8

282

0.75

I u 3

•U B O

U

"5

E 0.25

50

Temperature (°C)

Figure 8.8 Effect of temperature on the major food components.

100

Thermal Conductivity and Diffusivity of Foods

283

B. Structural Models

For heterogeneous foods, the effect of geometry must be considered using structural models. Utilizing Maxwell's and Eucken's work in the field of electricity, Luikov et al. (1968) initially used the idea of an elementary cell, as representative of the model structure of materials, in order to calculate the effective thermal conductivity of powdered systems and solid porous materials. In the same paper, a method is proposed for the estimation of the effective thermal conductivity of mixtures of powdered and solid porous materials.

Since then, a number of structural models have been proposed, some of

which are given in Table 8.3. The series model assumes that heat conduction is

perpendicular to alternate layers of the two phases, while the parallel model assumes that the two phases are parallel to heat conduction. In the random model, the two phases are assumed to be randomly mixed. The Maxwell model assumes that one phase is continuous, while the other phase is dispersed as uniform spheres. Several other models have been reviewed by Rahman (1995), among others.

In the mixed model (also called and Krischer model) heat conduction is assumed to take place by a combination of parallel and perpendicular heat flow. This model recognizes that there are two extremes in thermal conductivity values, one being derived from the parallel model and the other from the series model, whilst the real value of thermal conductivity should be somewhat in between these two extremes. A conceptual diagram is shown in Figure 8.9. The distribution factor/is a weighting factor between these extremes. It characterizes the structure of the material and it should be independent of material moisture content and temperature,

I-/

Parallel Structure

/

Series Structure

Figure 8.9 The mixed model of thermal conductivity.

284

Chapter 8

Granular (particulate) materials consist of granules (particles) and air, randomly packed (Figure 8.10). The induvidual particles consist of solids and water (Figure 8.10). The use of some of these structural models to calculate the thermal conductivity of a hypothetical porous material is presented in Figure 8.11. The parallel model gives the largest value for the effective thermal conductivity, while the series model gives the lowest. All other models predict values in between. Figure 8.12 represents the mixed model for various values of the distribution factor/ as a function of the void fraction (porosity). A systematic general procedure for selecting suitable structural models, even in multiphase systems, has been proposed by Maroulis et al. (1990). The method is based on a model discrimination procedure. If a component has unknown thermal conductivity, the method estimates the dependence of the temperature on the unknown thermal conductivity, and the suitable structural models simultaneously. An excellent example of applicability of the above is in the case of starch, an important component of plant foods. The granular starch consists of two phases, the wet granules and the air/vapor mixture in the intergranular space. The starch granule also consists of two phases, the dry starch and the water. Consequently, the thermal conductivity of the granular starch depends on the thermal conductivities of pure materials (that is, dry pure starch, water, air, and vapor, all functions of temperature) and the structures of granular starch and the starch granule. It has been shown that the parallel model is the best model for both the granular starch and the starch granule (Maroulis et al., 1990). These results led to simultaneous experimental determination of the thermal conductivity of dry pure starch versus temperature. Dry pure starch is a material that cannot be isolated for direct measurement.

Thermal Conductivity and Diffusivity of Foods

GRANULAR MATERIAL

Particles

GRANULE (PARTICLE)

Figure 8.10 Schematic model of granular materials.

285

286

Chapter 8

Table 8.3 Structural Models for Thermal Conductivity

Series 1 \-e)

g

Random

Mixed (Krischer )

=

287

Thermal Conductivity and Diffusivity of Foods

0.125

Parallel

Maxwell (Continuous phase 1) Random

Maxwell (Continuous phase 2) Series

0.025 0.00

0.25

0.50

Void Fraction

Figure 8.11 Structural models for porous materials.

0.75

Chapter 8

288

0.125

Mixed Model (Krischer) Distribution Factor =

0.00 (Parallel) 0.25 0.50 0.75 1.00 (Series)

0.025 0.00

0.25

0 . 5 0 0.75

1.00

Void Fraction

Figure 8.12 The mixed (Krischer) model for various values of distribution factor.

Thermal Conductivity and Diffusivity of Foods

289

V. COMPILATION OF THERMAL CONDUCTIVITY DATA OF FOODS

There is a wide variation of the reported experimental data of thermal conductivity of solid food materials, making difficult their utilization in food process and food quality applications. The variation of thermal conductivity in model and real foods is discussed in Section III of this chapter. The physical structure of solid foods plays a decisive role not only on the absolute value of thermal conductivity, but also on the effect of moisture content and temperature on this transport property. In this section, the thermal conductivity in food materials is approached from a statistical standpoint. Literature data are treated by regression analysis, using the parallel structural model. Recently published values of thermal conductivity in various foods were retrieved from the literature, and they were classified and analyzed statistically to reveal the influence of material moisture content and temperature. Structural models, relating thermal conductivity to material moisture content and temperature were fitted to all examined data for each material. The data were screened carefully, using residual analysis techniques. The most promising model was proposed, which is based on an Arrhenius-type effect of temperature and it uses a parallel structural model to take into account the effect of material moisture content. Thermal conductivity data in the literature show a wide variation due to the effect of the following factors: (a) diverse experimental methods, (b) variation in composition of the material, (c) variation of the structure of the material. Thermal conductivity depends strongly on moisture, temperature and structure of the material. An exhaustive literature search was made in international food engineering and food science journals in recent years, as follows (Krokida et al., 2001): • • • • • •

Drying Technology, 1983-1999 Journal of Food Science, 1981-1999 International Journal for Food Science and Technology, 1988-1999 Journal of Food Engineering, 1983-1999 Transactions of the ASAE, 1975-1999 International Journal of Food Properties, 1998-2000

A total number of 146 papers were retrieved from the above journals according to the distribution presented in Figure 8.13. The accumulation of the papers versus the publishing time is presented in Figure 8.14. The search resulted in 1210 data concerning the thermal conductivity in food materials.

Chapter 8

290

J. Food Engineering

J, of Food Science

Drying Technology

Trans of the Int. J. Food International ASAE

Science &

Journal of

Techn.

Food Properties

Figure 8.13 Number of papers on thermal conductivity data in food materials published in food engineering and food science journals during recent years. 160

120 o L.

1

sa

Z o

1970

1990

1980

2000

year

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

291

Thermal Conductivity and Diffusivity of Foods

0.001

0.01 0.1

1

10

100

Moisture (kg/kg db)

Figure 8.15 Thermal conductivity data for all foods at various moistures.

0.01

0.1

1

10

100

1000

Temperature (oQ

Figure 8.16 Thermal conductivity data for all foods at various temperatures.

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.

Thermal Conductivity and Diffusivity of Foods

293

Table 8.4 Literature for Thermal Conductivity Data in Food Materials: References and Number of Data Retrieved Material

Reference

Data 60

Baked products

14

Bread Zanonietal, 1995 Zanonietal., 1994

Goedeken et a!., 1998 Dough Bouvier et al, 1987

Zanonietal, 1995 Griffith etal., 1985 Soy flour

Maroulisetal.,1990 Wallapapan et al., 1982

Cake Zanonietal., 1995 Yellow batter Baiketal, 1999

Cup batter Baiketal., 1999

Cereal products

5 3 6 20 3 8 9 11 7 4 2 2 1 1 12 12

76 9

Barley

Alagusundaram et al., 1991 Corn

Bekeetal, 1994 Changetal, 1980 Okos etal, 1986 Rice

Okos etal, 1986 Ramesh, 2000

Wheat Changetal., 1980 Okos etal., 1986 Corn meal

Laietal, 1992 Kumaretal, 1989

9 21 9 3 9 13 4 9 10 3 7 7 4 3

Chapter 8

294

Table 8.4 Continued Material

Reference

Data 4

Iclli batter

Murthyetal, 1997

Maize Halltdayetal, 1995 Tolabaetal.,1988

Oat Okosetal.,1986

4 11 9 2 1 1

136

Dairy Cheese

Lunaetal., 1985 Tavmanetal., 1999 Milk

Duaneetal., 1992 Duaneetal, 1993 Duaneetal, 1994 Me Proud eta!., 1983 Hori, 1983 Zieglereta!., 1985 Ready etal, 1993 Tavmanetal, 1999 Okosetal.,1986

Cream Duaneetal, 1998 Butter

Tavmanetal, 1999 Okosetal.,1986 Yogurt

Kirn etal, 1997 Tavmanetal., 1999 Whey

Okosetal.,1986

23 1 22 84 1 1 1 1 6 3 9 9 53 1 1 5 2 3 19 9 10 4 4

Thermal Conductivity and Diffusivity of Foods

295

Table 8.4 Continued Material

Reference

Fish

Data

83

Cod

Sam et a!., 1987

Mackerel Sametal., 1987 Squid

Rahman et al, 1991 Rahman, 1991 Carp

Hung el al., 1983 Surimi

Wangetal., 1990 AbuDaggaetal, 1997 Cake Borquezetal, 1999 Shrimp

Karunakar et al, 1998

Calamari Rahman, 1991

Salmon Sametal., 1987

5 5 5 5 16 12 4 2 2 30 21 9 1 1 13 13 2 2 9 9

143

Fruits Apple Ramaswamy elal, 1981 Mattea et al., 1989 Telis-Romero et al, 1998 Rahman, 1991 Constenlaetal, 1989 Bhumblaetal, 1989 Ziegleretal, 1985 Mattea etal, 1986 Madambaetal, 1995 Sheen etal, 1993 Buhrietal, 1993 Okos etal, 1986 Chenetal, 1998

Banana Njieetal, 1998

82 11 3 3

2 9 25 3

3 2 1 1 10

9 1 1

Chapter 8

296

Table 8.4 Continued Material

Reference

Peach Okosetal.,1986

Strawberry

Delgado et al, 1997 Bhumbla et al., 1989 Okosetal.,1986 Raspberry Bhumbla etal, 1989

Okosetal.,1986

Grape Bhumbla et al, 1989 Okosetal.,1986 Plantain

Njieetal, 1998 Raisin

Vagenas et al., 1990

Pear Matteaetal, 1989 Rahman, 1991 Dincer, 1997 Mattea et al., 1986 Okosetal.,1986 Orange

Telis-Romeroetal, 1998 Bhumbla etal, 1989 Ziegler et al, 1985 Okosetal.,1986

Bilberry Bhumbla et al, 1989 Okosetal.,1986 Cherry

Bhumbla etal, 1989 Okosetal.,1986

Data 1 1 5 3 1 1 2 1 1 8 1 7 6 6 4 4 15 3 2 1 3 6 15 9 1 4 1 2 1 1 2 1 1

9

Legumes

9

Lentils Alagusundaram et al, 1991

9

297

Thermal Conductivity and Diffusivity of Foods

Table 8.4 Continued Material

Reference

Meat

Data 134

Beef Hung etai, 1983 Marinos-Kouris et al, 1995 Me Proud et al., 1983 Perezetal, 1984 Rahman, 1991 Baghe-Khandan et al, 1982 Sanzetal, 1987 Califano et al, 1997 Chicken

Rahman, 1991 Sanzetal, 1987

Sausage Sheen etal, 1990 Ziegleretal, 1987 Akterian, 1997 Turkey

Sanzetal, 1987 Mutton

Sanzetal, 1987 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

Table 8.4 Continued Material

Reference

Potato starch

Okosetal.,1986 Starch

Renaudetal, 1991 Njieetal, 1998 Maroulis et al, 1991 Morley et al, 1997

Wangetal.,1993 Lanetal, 2000

Sucrose Renaudetal., 1991 Ziegleretai., 1985

Gelatin Renaudetal, 1991 Okosetal.,1986 Ovalbumin

Renaudetal, 1991 Cornillon et al, 1995

Tylose

Phametal, 1990

Agar-water Delgado et al, 1997 Barringer et al, 1995 Bentonite-water

Sheen et al, 1993 Gelatin-water

HalUdayetal.,1995 Amylose Voudouris et al, 1995

Cellulose gum Saravacos et al,1965 Pectin 5% Saravacos et al.,1965

Pectin 10% Saravacos el al.,1965

Pectin 5%-glucose 5% Saravacos et al.,1965

Gelatin-sucrose-water Hallidayetal, 1995

Glycerin

Ryniecki et al, 1993

Data 2 2 61 24 1 6 6 18 6 33 30 3 26 24 2 36 24 12 6

6 2 1 1 1 1 1 1 4 4 2 2 2 2 2 2 2 2 2 2 3 3

299

Thermal Conductivity and Diffusivity of Foods

Table 8.4 Continued Material

Reference

Nuts Macadamia

Rahman, 1991

Data

1 1 1

134

Other Coconut

Duane et a!., 1995

Chenetal, 1998 Coffee

Sagaraetal., 1994

Soybean Okos et al, 1986

Palm kernel Duane etal, 1996

Lard

10 1 9 10 10 12 12 1 1 1

Duane et al., 1997

Agar-water Wang etal, 1992

Water-NaCI Lucas etal, 1999

Water-sucrose Lucas etal, 1999

Rapeseed Bilanskietal, 1976 Moyseyetal, 1977 Okos eta!., 1986 Tobacco

Casadaetal, 1989 Sorghum

Changetal, 1980 Okos etal. ,1986 Sugar

Okos etal, 1986 Albumen

Okos etal, 1986 NaCl

Okos etal, 1986

1 52 18 10 24 3 3 7 3 4 15 15 2 2 3 3

Chapter 8

300

Table 8.4 Continued Material

Reference

Honey Okos et al.,1986 Albumine

Okos eta!., 1986

Vegetables

Data 12 12 3 3

154

Carrot

Niesteruk, 1998 Njie et al, 1998 Rahman, 1991 Buhri et al, 1993

Cassava Njieetal, 1998

Garlic Madamba et al, 1995

Onion

Rapusasetal, 1994 Pea

Sastry et al, 1983 Alagusundarametal, 1991

Potato Niesteruk, 1996 Niesteruk, 1997 Niesteruk, 1998

Hungetal.,1983 Luelal, 1999 Njieetal, 1998

WangetaL, 1992 Rahman, 1991

Matteaetal, 1986 Hallidayetal, 1995 Madamba eta!., 1995 Buhri eta!., 1993 Cratzeketal.,1993

Sugar beet

5 1 1 2 1 6 6 3 3 7 7 12 3 9 45 1 1 1 2 1 2 16 2 3 9 2 1 4 7

Niesteruk, 1998 Okos etal, 1986

Turnip Buhri et al, 1993

4 3 1 1

301

Thermal Conductivity and Diffusivity of Foods

Table 8.4 Continued Material

Reference

Yam Njieetal., 1998 Beetroot Niesteruk, 1998

Parsley Niesteruk, 1998

Celery

Niesteruk, 1998

Tomato Dincer, 1997 Choietd.,1983 Filkovaetal, 1987

Okos eta!., 1986 Drouzasetal., 1985

Cucumber

Dincer, 1997 Spinach

Delgado et al, 1997 Mushrooms Shrivastavaetal, 1999

Rutabagas Buhri et al, 1993 Radish

Buhrietal, 1993

Parsnip Buhri etai, 1993 Kidney bean

Zuritz et al, 1989

Data 6 6 2 2 1 1 1 1 31 1 9 3 9 9 1 1 10 10 9 9 1 1 1 1 1 1 4 4

Chapter 8

302

Table 8.5 Thermal Conductivity of Foods Versus Moisture and Temperature: Average Values of Available Data Temperature

(W/mK)

Moisture (kg/kg db)

(°C)

Data

Baked products

0.34

0.57

46

60

Conductivity Material

Bread

0.23

0.39

44

14

-

0.27

0.35

47

9

Crust

0.06

0.00

68

2

Crumb

0.26

0.76

17

3

Dough

0.34

0.89

60

20

Wheat bread

0.41

0.78

23

3

Rye bread

0.47

1.06

20

3

Biscuit

0.40

0.07

20

2

Soy

0.35

0.33

150

3

Soy flour

0.22

0.21

27

11

Defatted

0.44

0.36

25

4

Dry defatted

0.12

0.00

40

3

Cup cake batter

0.17

0.63

15

2

-

0.17

0.63

15

2

Yellow cake batter

0.22

0.71

20

1

-

0.22

0.71

20

1

Cake

0.25

0.56

51

12

0.25

0.56

51

12

Cereal products

0.29

0.67

40

76

" Barley

0.20

0.18

0

9

Seeds

0.20

0.18

0

9

Corn

0.39

1.55

36

21

Dent

0.16

0.20

36

3

Shelled

0.55

0.73

30

9

Dust

0.09

0.15

22

3

Syrup

0.43

4.16

52

6

Thermal Conductivity and Diffusivity of Foods

303

Table 8.5 Continued Conductivity Material

(W/mK)

Moisture (kg/kg db)

Temperature (°C)

Data 13

Rice

0.15

0.22

48

Paddy

0.15

0.22

48

13

Wheat

0.26

0.13

26

10

Dust

0.07

0.15

22

3

Hard red spring

0.16

0.17

-3

2

Soft white

0.13

0.13

16

2

Flour

0.59

0.10

56

3

Corn meal

0.36

0.27

88

7

-

0.36

0.27

88

7

Idli batter

0.45

1.71

16

4

-

0.45

1.71

16

4

Maize

0.27

0.32

62

11

Kernel

0.17

0.16

50

2

Grits

0.29

0.36

65

9

Oat

0.13

0.14

27

1

White

0.13

0.14

27

1

Dairy

0.45

3.78

38

136

Cheese

0.42

1.20

22

23

Cheddar

0.35

0.56

23

2

Mozzarella

0.38

0.80

23

2

Cuartirolo Argentina

0.37

1.20

15

1

Hamburger

0.39

0.69

23

2

Old Kashkaval

0.38

0.69

23

2

Tulum

0.38

0.69

23

2

Fresh Kashkaval

0.40

0.78

23

2

Buffet Kashkaval

0.41

0.99

23

2

Fresh cream

0.43

1.29

23

2

Spreadable cheese

0.49

1.54

23

2

Labne

0.47

2.24

23

2

Low fat labne

0.55

2.94

23

2

Chapter 8

304

Table 8.5 Continued (W/mK)

Moisture (kg/kg db)

Temperature (°C)

Data

Milk

0.46

4.02

46

84

-

0.46

4.00

20

3

Conductivity

Material

Fresh

0.57

9.00

23

1

Powder

0.11

25.33

25

3

Whole

0.46

3.04

53

18

Skim

0.57

6.33

36

15

Concentrated

0.41

0.92

50

9

Condensed

0.49

4.01

54

9

Half-half

0.54

5.11

40

9

Baby food

0.55

0.03

50

6

Powdered

0.30

0.05

54

11

Butter

0.22

0.20

21

5

-

0.23

0.18

23

2

Fat

0.21

0.21

20

3

Yogurt

0.45

3.60

31

19

-

0.56

6.25

21

2

Plain

0.33

2.05

40

9

Strained

0.54

2.88

23

2

Pasterized

0.58

4.71

23

2

Light

0.58

4.54

23

2

Extra light

0.59

6.58

23

2

Whey

0.59

9.00

40

4

-

0.59

9.00

40

4

Cream

0.13

44.00

25

1

Powder

0.13

44.00

25

1

Fish

0.79

3.29

8

83

Cod

1.23

4.88

-10

5

Perpendicular

1.23

4.88

-10

5

Mackerel

0.80

3.42

0

5

Perpendicular

0.80

3.42

0

5

Thermal Conductivity and Diffusivity of Foods

305

Table 8.5 Continued Temperature

(W/mK)

Moisture (kg/kg db)

(°C)

Data

Conductivity

Material Squid

0.35

2.36

26

16

Fresh

0.50

5.04

30

Mantle

0.50

3.83

15

3 2

Dried

0.24

0.87

30

9

Tentacle, arrow

0.48

3.56

15

1

Tentacle

0.50

3.56

15

1

Carp

1.21

0.83

0

2

-

1.21

0.83

0

2

Surimi

0.78

4.02

14

30

-

0.85

4.08

-2

7

6% cryoprotectant cone

0.87

4.08

-2

7

12% cryoprotectant cone

0.86

4.08

-2

7

Pacific whiting

0.59

3.88

53

9

Cake

0.10

0.00

15

1

Pressed

0.10

0.00

15

1

Shrimp

1.03

3.24

-8

13

peeled and head removed

1.03

3.24

-8

13

Calamari

0.51

4.04

15

2

mantle

0.51

4.04

15

2

Salmon

1.06

2.40

-12

9

Perpendicular

1.06

2.40

-12

9

Fruits

0.45

3.73

30

143

Apple

0.45

3.44

30

82

-

0.32

2.44

20

25

Red

0.51

5.60

15

1

Green

0.41

0.14

45

1

Golden delicious

0.41

4.89

18

4

Granny Smith

0.19

2.17

25

3

37

47

Juice

0.53

3.81

Sauce

0.59

10.11

1

Chapter 8

306

Table 8.5 Continued 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

Thermal Conductivity and Diffusivity of Foods

307

Table 8.5 Continued Material

Conductivity

Moisture

Temperature

(W/mK)

(kg/kg db)

(°C)

Data

Meat

0.71

2.40

16

134

Beef

0.63

2.22

28

75

-

0.54

2.43

1

2

Fat

0.28

0.14

7

3

Lean

1.03

0.63

0

2

Ground

1.01

0.75

0

2

Minced

0.56

1.96

11

5

Muscle semitendinosus

0.31

0.99

20

5

Dryfiber

0.21

1.03

25

4

Boneless

1.03

3.31

-4

20

Ground round

0.51

2.53

60

3

Whole round

0.49

2.32

60

3

Ground shank

0.51

2.45

60

3

Ground brisket

0.44

2.38

60

3

Whole rib steak

0.50

1.84

60

3

Ground sirloin tip

0.49

2.35

60

3

Whole sirloin tip

0.48

2.27

60

3

Ground rib

0.41

1.11

60

3

Ground s\viss steak

0.51

2.90

60

3

Whole swiss steak

0.49

2.82

60

3

Loaf, uncooked

0.40

2.58

15

1

Loaf heated

0.47

1.96

60

1

Chicken

1.05

3.46

-9

9

Boneless

0.97

3.00

-4

7

White

1.33

5.10

-25

2

Chapter 8

308

Table 8.5 Continued Conductivity

Material

(W/mK)

Sausage

0.42

Moisture (kg/kg db)

Temperature (°C)

Data

1.02

18

13

-

0.33

1.24

22

4

Italian

0.93

0.64

0

2

Salami cooked

0.37

1.70

22

1

Lebanon bologna

0.36

1.63

22

1

Salami cotto

0.37

1.33

22

1

Thuringer

0.35

0.96

22

1

Salami Genoa

0.30

0.56

22

1

Salami hard

0.32

0.52

22

1

Pepperoni

0.28

0.37

22

1

Turkey

1.18

2.85

-12

12

Boneless

1.18

2.85

-12

12

Mutton

0.86

2.66

-3

10

Boneless

0.86

2.66

-3

10

Pork Boneless

0.93

3.25

-4

0.93

3.25

-4

11 11

Pork/soy

0.05

3.41

25

4

Unprocessed

0.05

3.17

25

2

Processed

0.05

3.64

25

2

Model foods

0.63

5.26

30

281

Amioca

0.32

3.35

52

51

-

0.25

2.86

58

20

Gelatinized

0.51

1.23

52

16

Powder

0.13

10.37

48

9

Granular

0.34

0.12

40

6

Thermal Conductivity and Diffusivity of Foods

309

Table 8.5 Continued Conductivity (W/mK)

Moisture (kg/kg db)

Hylon-7

0.33

5.08

60

43

Gelatinized

0.22

5.99

85

16

0.53

1.90

46

15

Powder

0.13

15.55

48

6

Granular

0.34

0.12

40

6

Material

Temperature (°C) Data

Potato starch

0.04

0.08

41

2

Gel

0.04

0.08

41

2

Starch

0.68

4.71

42

61

-

0.10

0.00

25

1

Gel

1.02

8.80

4

29

Gelatinized

0.34

0.10

50

3

Hydrated

0.38

0.28

47

3

Granular

0.09

0.20

45

7

Gels

0.50

1.65

100

18

Sucrose

0.85

3.57

-1

33

-

0.48

3.72

20

3

Gel Gelatin

0.89

3.56

-4

30

0.96

7.92

0

26

Gel

0.96

7.92

0

26

Ovalbumin

0.99

7.59

-5

36

-

0.88

4.23

-7

12

Gel

1.05

9.28

-4

24

Xylose

0.99

3.35

5

6

Gel

0.99

3.35

5

6

Agar-water

0.61

36.95

25

2

Gel Gelatin-water -

0.61

36,95

25

2

0.59

65.67

25

1

0.59

65.67

25

1

Amylose

0.53

3.50

30

4

Gel

0.53

3.50

30

4

Chapter 8

310

Table 8.5 Continued Conductivity Material

(W/mK)

Moisture (kg/kg db)

Temperature (°C)

Data

Cellulose gum

0.06

0.08

41

2

Freeze-dried gel

0.06

0.08

41

2

Pectin 5%

0.04

0.08

41

2

Freeze-dried gel

0.04

0.08

41

2

Pectin 10%

0.05

0.08

41

2

Freeze-dried gel

0.05

0.08

41

2

Pectin 5%-glucose 5%

0.05

0.08

41

2

Freeze-dried gel

0.05

0.08

41

2

Glycerin

0.47

3.79

20

3

-

0.47

3.79

20

3

0.44

1.44

15

2

0.44

1.44

15

2

Nuts

0.22

0.02

15

1

Macadamia

0.22

0.02

15

1

Integrifolia

0.22

0.02

15

1

Vegetables

0.43

3.81

39

154

Carrot

0.48

5.85

22

5

-

0.45

3.82

27

3

Large

0.52

8.91

15

2

Cassava

0.47

1.22

30

6

Roots

Gelatin-sucrose-water

"

0.47

1.22

30

6

Garlic

0.36

0.80

15

3

-

0.36

0.80

15

3

Onion .

0.42

2.05

32

7

0.42

2.05

32

7

311

Thermal Conductivity and Diffusivity of Foods

Table 8.5 Continued Material

Conductivity

Moisture

Temperature

(W/mK)

(kg/kg db)

(°C)

Data

Pea

0.22

0.18

2

9

Seeds

0.22

0.18

2

9

Potato

0.45

2.35

49

45

-

0.42

2.74

57

25

Mashed

1.22

0.72

0

2

Flesh

0.54

4.54

20

1

Granule

0.35

0.64

62

10

White

0.53

4.55

18

4

Spunta

0.46

2.17

25

3

Sugar beet

0.53

3.38

22

7

25

3

20

4

-

0.56

4.22

Roots

0.52

2.75

Turnip

0.48

0.08

45

1

-

0.48

0.08

45

1

Yam

0.47

1.45

30

6

Tubers

0.47

1.45

30

6

Beetroot

0.56

9.10

20

2

-

0.56

9.10

20

2

Parsley

0.17

2.30

20

1

-

0.17

2.30

20

1

Celery

0.15

2.30

20

1

-

0.15

2.30

20

1

Tomato

0.51

6.23

68

31

-

0.61

15.60

21

1

Juice

0.48

7.71

83

21

Paste

0.55

1.73

40

9

Cucumber

0.62

24.00

22

1

-

0.62

24.00

22

1

Spinach

0.38

11.01

-2

10

Fresh

0.37

13.66

-2

5

Blanched

0.39

8.35

-2

5

Chapter 8

312

Table 8.5 Continued (W/mK)

Moisture (kg/kg db)

Temperature (°C)

Data

Mushrooms

0.37

3.27

55

9

Pleurotusflorida

0.37

3.27

55

9

Rutabagas

0.45

0.08

45

1

-

0.45

0.08

45

1

Radish

0.50

0.06

45

1

-

0.50

0.06

45

1

Parsnip

0.39

0.21

45

1

-

0.39

0.21

45

1

Kidney bean

0.15

0.24

20

4

-

0.15

0.24

20

4

Other

0.23

2.06

25

134

Coconut

0.15

3.08

37

10

Milkpowder

0.15

3.08

37

10

Coffee

0.21

1.62

6

10

Solutions

Conductivity

Material

0.21

1.62

6

10

Soybean

0.09

0.14

34

12

Powder

0.08

0.10

36

3

Whole

0.11

0.13

36

3

Crushed

0.10

0.11

36

3

Flour

0.05

0.22

26

3

Palm kernel

0.10

26.00

25

1

Milkpowder

0.10

26.00

25

1

Lard

0.12

32.00

25

1

Milkpowder

0.12

32.00

25

1

Water-Nad

0.46

4.00

10

1

Solution

0.46

4.00

10

1

Water-sucrose

0.32

0.67

10

1

Solution

0.32

0.67

10

1

Thermal Conductivity and Diffusivity of Foods

313

Table 8.5 Continued Material

Conductivity (W/mK)

Moisture (kg/kg db)

Temperature (°C)

Data

Rapeseed

0.11

0.10

14

52

mole

0.13

0.11

17

21

Ground

0.07

0.11

16

9

Torch

0.10

0.10

-4

9

Midas

0.09

0.01

19

1

Crushed

0.13

0.11

18

12

Agar-water

0.62

19.90

30

1

Gel

0.62

19.90

30

1

Tobacco

0.06

0.26

15

3

-

0.06

0.26

15

3

Sugar

0.52

4.32

42

15

Glucose

0.54

4.80

44

6

Cane sugar

0.51

4.00

40

9

Albumen

0.04

0.08

41

2

Freeze-dried gel

0.04

0.08

41

2

Sorghum

0.24

0.17

21

7

Rs610

0.14

0.22

5

2

NC+RS66

0.56

0.16

36

2

Grain dust

0.09

0.15

22

3

NaCl

0.61

4.00

43

3

Solution

0.61

4.00

43

3

Honey

0.53

4.83

36

12

Albumine

0.53

4.83

36

12

0.41

0.67

60

3

Solution

0.41

0.67

60

3

314

Chapter 8

Table 8.6. Thermal Conductivity of Foods Versus Moisture and Temperature: Variation Range of Available Data Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Max Min Min Max Min Max 15

150

0.82

15

120

0.79

25

100

0.00

0.00

15

120

0.72

0.82

15

18

0.04

1.17

20

150

0.00

1.17

0.055

0.650 0.530

0.00

0.080

0.530

0.05

Crust

0.055

0.066

Crumb

0.232

0.298

Dough

0.230

0.600

Baked products

0.048

Bread -

Wheat bread

0.327

0.500

0.72

0.82

20

28

Rye bread

0.396

0.600

0.85

1.17

20

20

Biscuit

0.390

0.405

0.04

0.09

20

20

Soy

0.230

0.488

0.10

0.60

150

150

Soy flour

0.106

0.650

0.00

0.64

20

60

defatted

0.180

0.650

0.10

0.64

25

25

dry defatted

0.106

0.143

0.00

0.00

20

60

Cup cake barter

0.121

0.223

0.55

0.71

15

15

-

0.121

0.223

0.55

0.71

15

15

Yellow cake batter

0.223

0.223

0.71

0.71

20

20

-

0.223

0.223

0.71

0.71

20

20

Cake

0.048

0.356

0.11

1.22

20

103

-

0.048

0.356

0.11

1.22

20

103

Cereal products

0.067

0.740

0.01

8.09

-28

160

Barley

0.167

0.225

0.11

0.26

-28

29

Seeds

0.167

0.225

0.11

0.26

-28

29

Corn

0.085

0.740

0.01

8.09

10

77

Dent

0.142

0.175

0.01

0.42

36

36

Shelled

0.371

0.740

0.40

1.00

10

50

Dust

0.085

0.101

0.10

0.20

22

22

Syrup

0.347

0.513

0.23

8.09

27

77

315

Thermal Conductivity and Diffusivity of Foods

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Max Min Max Min Max Min 0.366

0.11

0.082

0.366

0.067

0.689

Dust

0.067

Hard red spring

0.144

Soft white

0.118

Rice

0.082

Paddy Wheat

70

0.43

20

0.11

0.43

20

70

0.01

0.29

-3

66

0.073

0.10

0.20

22

22

0.166

0.05

0.29

-3

-3

0.140

0.01

0.25

15

16

Flour

0.450

0.689

0.10

0.10

43

66

Corn meal

0.270

0.464

0.18

0.43

20

160

-

0.270

0.464

0.18

0.43

20

160

Idli batter

0.395

0.493

1.00

2.33

15

20

-

0.395

0.493

1.00

2.33

15

20

Maize

0.067

0.525

0.11

0.59

35

95

Kernel

0.156

0.174

0.11

0.20

50

50

Grits

0.067

0.525

0.16

0.59

35

95

Oat

0.130

0.130

0.14

0.14

27

27

White

0.130

0.130

0.14

0.14

27

27

Dairy

0.039

0.686

0.02

44.00

1

90

Cheese

0.345

0.548

0.56

2.94

15

30

Cheddar

0.345

0.351

0.56

0.56

15

30

Mozzaretta Cuartirolo ArgenTino

0.380

0.383

0.80

0.80

15

30

0.372

0.372

1.20

1.20

15

15

Hamburger

0.381

0.398

0.69

0.69

15

30

Old Kashkaval

0.368

0.384

0.69

0.69

15

30

Tulum

0.377

0.379

0.69

0.69

15

30

0.78

15

30

Fresh Kashkaval

0.403

0.403

0.78

Buffet Kashkaval

0.406

0.409

0.99

0.99

15

30

Fresh cream

0.433

0.434

1.29

1.29

15

30

Spreadable cheese

0.476

0.494

1.54

1.54

15

30

Labne

0.463

0.486

2.24

2.24

15

30

Low fat labne

0,542

0.548

2.94

2.94

15

30

Chapter 8

316

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min

Max

Min

Max

Min

Max

Milk

0.112

0.686

0.02

30.00

5

90

-

0.325

0.576

1.00

9.00

20

20

Fresh

0.570

0.570

9.00

9.00

23

23

Powder

0.112

0.115

22.00

30.00

25

25

mole

0.280

0.629

0.39

9.00

5

90

Skim

0.481

0.646

1.50

19.00

5

75

Concentrated

0.325

0.498

0.43

1.50

35

65

Condensed

0.325

0.634

1.00

9.00

23

79

Half-half

0.471

0.634

2.33

9.00

5

75

Baby food

0.405

0.686

0.03

0.04

35

65

Powdered

0.182

0.538

0.02

0.14

54

54

Butter

0.093

0.345

0.02

0.42

15

30

-

0.227

0.233

0.18

0.18

15

30

Fat

0.093

0.345

0.02

0.42

20

20

Yogurt

0.039

0.639

0.06

6.58

1

55

-

0.525

0.603

6.25

6.25

1

40

Plain

0.039

0.639

0.06

5.66

25

55

Strained

0.539

0.540

2.88

2.88

15

30

Pasterized

0.571

0.593

4.71

4.71

15

30

Light

0.571

0.583

4.54

4.54

15

30

Extra light

0.584

0.596

6.58

6.58

15

30

Whey

0.547

0.642

9.00

9.00

7

87

-

0.547

0.642

9.00

9.00

7

87

Cream

0.127

0.127

44.00

44.00

25

25

Powder

0.127

0.127

44.00

44.00

25

25

Fish

0.040

1.720

0.00

5.25

-40

80

Cod

0.549

1.543

4.88

4.88

-22

3

Perpendicular

0.549

1.543

4.88

4.88

-22

3

Mackerel Perpendicular

0.409

1.428

3.42

3.42

-20

20

0,409

1.428

3.42

3.42

-20

20

317

Thermal Conductivity and Diffusivity of Foods

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Max Max Min Max Min

Squid

0.040

0.507

0.10

Fresh

0.490

0.500

Mantle

0.483

0.507

Dried

0.040

0.440

Tentacle, arrow

0.475

0.475

Tentacle

0.501

0.501

5.20

15

30

4.75

5.20

30

30

3.83

3.83

15

15

0.10

2.86

30

30

3.56

3.56

15

15

3.56

3.56

15

15

Carp

0.700

1.720

0.83

0.83

-15

15

-

0.700

1.720

0.83

0.83

-15

15

Surimi

0.477

1.508

2.85

5.25

-40

80

-

0.487

1.473

4.08

4.08

-40

30

6% cryoprotectant

0.477

1.508

4.08

4.08

-40

30

12% cryoprotectant

0.489

1.465

4.08

4.08

-40

30

Pacific whiting

0.524

0.708

2.85

5.25

30

80

Cake

0.100

0.100

0.00

0.00

15

15

Pressed

0.100

0.100

0.00

0.00

15

15

Shrimp Peeled and head removed

0.490

1.600

1.00

4.20

-30

30

0.490

1.600

1.00

4.20

-30

30

4.04

4.04

15

15

Calamari

0.508

0.517

Mantle

0.508

0.517

4.04

4.04

15

15

Salmon

0.497

1.245

2.03

2.70

-24

5

Perpendicular

0.497

1.245

2.03

2.70

-24

5

Fruits

0.043

2.270

0.14

19.00

-40

90

Apple

0.070

2.270

0.14

19.00

-40

90

-

0.070

1.510

0.25

5.99

-40

45

Red

0.513

0.513

5.60

5.60

15

15

Green

0.405

0.405

0.14

0.14

45

45

Golden delicious

0.401

0.412

4.88

4.89

15

20

Granny Smith

0.090

0.296

0.50

4.00

25

25

Juice

0.230

2.270

0.25

19.00

-7

90

Sauce

0.591

0.591

10.11

10.11

318

Chapters

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Max Min Min Max Min Max

Banana

0.481

0.481

3.12

3.12

20

20

Dessert

0.481

0.481

3.12

3.12

20

20

Peach

0.043

0.043

35

35

Freeze-dried

0.043

0.043

35

35

Plantain

0.130

0.520

0.16

2.00

30

30

Fruits

0.130

0.520

0.16

2.00

30

30

Pear

0.340

0.629

0.50

7.41

15

80

-

0.340

0.557

0.50

4.90

23

25

Green

0.514

0.533

7.41

7.41

15

15

Juice

0.402

0.629

0.64

5.67

20

80

Williams

0.359

0.505

0.50

4.00

25

25

Orange

0.290

0.560

0.64

19.00

1

62

Juice

0.290

0.560

0.64

19.00

1

62

Bilberry

0.553

0.554

8.52

8.52

16

20

Juice

0.553

0.554

8.52

8.52

16

20

Cherry

0.553

0.554

6.52

6.52

16

20

Juice

0.553

0.554

6.52

6.52

16

20

Grape

0.396

0.639

0.59

8.09

16

80

-

0.396

0.639

0.59

8.09

20

80

Juice

0.537

0.556

5.54

5.54

16

20

Raspberry

0.544

0.553

7.70

7.70

16

20

Juice

0.544

0.553

7.70

7.70

16

20

Strawberry

0.520

0.935

6.70

11.05

-15

28

Juice

0.571

0.571

11.05

11.05

16

20

Tioga

0.520

0.935

6.70

6.70

-15

28

Raisin

0.126

0.392

0.16

4.00

45

45

-

0.126

0.392

0.16

4.00

45

45

Legumes

0.187

0.253

0.11

0.26

-21

28

Lentils

0.187

0.253

0.11

0.26

-21

28

Seeds

0.187

0.253

0.11

0.26

-21

28

Thermal Conductivity and Diffusivity of Foods

319

Table 8.6. Continued Material Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)

Meat

Min

Max

Min

Max

Min

Max

0.049

1.660

0.01

5.10

-40

90

Beef

0.095

1.650

0.01

3.69

-30

90

Fat Lean

0.454

0.622

2.28

2.57

-18

20

0.264

0.311

0.10

0.16

-10

15

0.510

1.550

0.63

0.63

-15

15

Ground

0.400

1.620

0.75

0.75

-15

15

Minced Muscle semitendinosus

0.360

0.844

1.11

3.44

-5

30

0.01

2.84

20

20 25 30

0,095

0.490

Dry fiber

0,140

0.243

0.38

2.30

25

Boneless

0.429

1.650

2.92

3.69

-30

Ground round

0.452

0.590

1.99

2.94

30

90

Whole round

0.475

0.504

1.50

2.94

30

90

Ground shank

0.442

0,598

1.58

2.92

30

90

Ground brisket

0.436

0.458

1.36

3.05

30

90

Whole rib steak

0.459

0.552

1.07

2.32

30

90

Ground sirloin tip

0.460

0.518

1.61

2.92

30

90

Whole sirloin tip

0.467

0.494

1.30

2.92

30

90 90

Ground rib

0.368

0.450

0.78

1.37

30

Ground Swiss steak

0.467

0.575

2.16

3.44

30

90

Whole swiss steak

0.467

0.508

1.84

3.44

30

90

Loaf, uncooked

0.400

0.400

2.58

2.58

15

15

Loaf, heated

0.470

0.470

1.96

1.96

60

60

Chicken

0.490

1.452

2.91

5.10

-25

20

Boneless

0.490

1.452

2.91

3.22

-20

20

White

1.268

1.387

5.10

5.10

-25

-25

320

Chapter 8

Table 8.6. Continued

Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Min Max Max Max Min

Sausage

0.275

1.380

0.37

1.86

-10

22

-

0.283

0.367

0.40

1.86

20

22

Italian

0.470

1.380

0.64

0.64

-10

10

Salami cooked

0.370

0.370

1.70

1.70

22

22

Lebanon bologna

0.355

0.355

1.63

1.63

22

22

Salami cotto

0.365

0.365

1.33

1.33

22

22

Thuringer

0.345

0.345

0.96

0.96

22

22

Salami genoa

0.295

0.295

0.56

0.56

22

22

Salami hard

0.315

0.315

0.52

0.52

22

22

Pepperoni

0.275

0.275

0.37

0.37

22

22

Turkey

0.490

1.660

2.85

2.85

-24

4

Boneless

0.490

1.660

2.85

2.85

-24

4

Mutton

0.391

1.510

2.45

2.80

-40

24

Boneless

0.391

1.510

2.45

2.80

-40

24

Pork

0.480

1.450

3.15

3.31

-30

30

Boneless

0.480

1.450

3.15

3.31

-30

30

Pork/soy

0.049

0.055

3.08

3.75

25

25

Unprocessed

0.049

0.051

3.08

3.25

25

25

Processed

0.053

0.055

3.54

3.75

25

25

Model Foods

0.038

2.330

0.00

65.67

-43

150

Amioca

0.080

0.661

0.00

20.00

20

150

Gelatinized

0.432

0.661

0.01

3.00

20

135

Powder

0.080

0.195

0.00

20.00

25

70

Granular

0.227

0.454

0.01

0.23

20

60

321

Thermal Conductivity and Diffusivity of Foods

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)

Min

Max

Min

Max

Min

Max

Hylon-7

0.100

0.661

0.00

Gelatinized

0.442

0.661

0.01

20.00

20

150

4.00

20

70

Powder

0.100

0.160

11.10

Granular

0.227

20.00

25

70

0.01

0.23

20

0.454

Potato starch

0.039

0.041

60

0.02

0.14

41

41

Gel

0.039

0.041

0.02

0.14

41

41

Starch

0.061

2.100

0.00

24.00

-42

120

-

0.100

0.100

0.00

0.00

25

25

Gel

0.480

2.100

1.78

24.00

-42

50

Gelatinized

0.330

0.355

0.10

0.10

20

80

Hydrated

0.364

0.388

0.28

0.28

10

80

Granular

0.061

0.125

0.05

0.30

15

75

Gels

0.436

0.567

0.66

3.00

80

120

Sucrose

0.350

1.770

0.67

9.00

-41

32

-

0.405

0.566

0.67

9.00

20

20

Gel

0.350

1.770

1.00

9.00

-41

32

Gelatin

0.039

2.070

0.02

19.00

-41

41

Gel

0.039

2.070

0.02

19.00

-41

41

Ovalbumin

0.450

2.330

2.30

19.00

-43

26

-

0.470

1.750

2.30

6.40

-43

20

Gel

0.450

2.330

3.20

19.00

-42

26

Tylose

0.483

1.530

3.35

3.35

-30

50

Gel

0.483

1.530

3.35

3.35

-30

50

Agar-water

0.600

0.622

24.90

49.00

20

30

Gel

0.600

0.622

24.90

49.00

20

30

Gelatin-water

0.594

0.594

65.67

65.67

25

25

-

0.594

0.594

65.67

65.67

25

25

Amylose

0.515

0.551

3.00

4.00

30

30

Gel

0,515

0.551

3.00

4.00

30

30

Chapter 8

322

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Max Max Min Min Max

Cellulose gum

0.056

Freeze-dried gel

0.056

0.063

Pectin 5%

0.038

0.039

0.063

0.14

41

41

0.02

0.14

41

41

0.02

0.14

41

41

0.02

Freeze-dried gel

0.038

0.039

0.02

0.14

41

41

Pectin 10%

0.044

0.047

0.02

0.14

41

41

Freeze-dried gel

0.044

0.047

0.02

0.14

41

41

0.04S

0.050

0.02

0.14

41

41 41

Pectin 5%-glucose 5%

Freeze-dried gel

0.048

0.050

0.02

0.14

41

Glycerin

0.450

0.490

3.35

4.26

20

20

0.450

0.490

3.35

4.26

20

20

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

Thermal Conductivity and Diffusivity of Foods

323

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Max Min Max Min Max

Pea

0.181

0.256

0.11

4.50

-29

28

Seeds

0.181

0.256

0.11

0.26

-21

28

Potato -

0.120

0.643

0.11

7.33

-15

130

0.209

0.643

0.34

7.33

24

130

Flesh

0.536

0.536

4.54

4.54

20

20

Granule

0.120

0.579

0.11

1.44

30

95

White

0.519

0.536

4.54

4.55

15

20

Spunta

0.331

0.550

0.50

4.00

25

25

Sugar beet

0.448

0.589

1.50

5.67

20

25

-

0.535

0.585

3.00

5.67

25

25 20 45

Roots

0.448

0.589

1.50

4.00

20

Turnip

0.480

0.480

0.08

0.08

45

45 30

-

0.480

0.480

0.08

0.08

45

Yam

0.160

0.600

0.19

3.76

30

Tubers

0.160

0.600

0.19

3.76

30

30

Beetroot

0.549

0.572

6.90

11.30

20

20

-

0.549

0.572

6.90

11.30

20

20

Parsley

0.170

0.170

2.30

2.30

20

20

-

0.170

0.170

2.30

2.30

20

20

Celery

0.147

0.147

2.30

2.30

20

20

-

0.147

0.147

2.30

2.30

20

20

Tomato

0.230

0.670

0.25

19.83

20

150

-

0.611

0.611

15.60

15.60

21

21

Juice

0.230

0.670

0.25

19.83

20

150

2.40

30

50

22

22

Paste

0.460

0.660

1.16

Cucumber

0.621

0.621

24.00

24.00

-

0.621

0.621

24.00

24.00

22

22

Spinach

0.347

0.434

8.35

13.66

-20

21

Fresh

0.347

0.400

13.66

13.66

-20

21

Blanched

0.356

0.434

8.35

8.35

-20

16

Chapter 8

324

Table 8.6. Continued Material

Mushrooms

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C) Min Min Max Max Min Max 0.218

0.520

0.11

8.69

40

70

Pleurotusflorida

0.218

0.520

0.11

8.69

40

70

Rutabagas

0.447

0.447

0.08

0.08

45

45

-

0.447

0.447

0.08

0.08

45

45

Radish

0.499

0.499

0.06

0.06

45

45

-

0.499

0.499

0.06

0.06

45

45

Parsnip

0.392

0.392

0.21

0.21

45

45

-

0.392

0.392

0.21

0.21

45

45

20

20

Kidney bean

0.103

0.201

0.12

0.41

0.103

0.201

0.12

0.41

20

20

Other

0.039

0.656

0.01

32.00

-26

90

Coconut

0.115

0.217

0.19

26.00

25

50

Milkpowder

0.115

0.217

0.19

26.00

25

50

Coffee

0.153

0.277

1.22

2.51

-14

26

Solutions

0.153

0.277

1.22

2.51

-14

26

Soybean

0.040

0.133

0.05

0.40

10

66

Powder

0.066

0.104

0.10

0,10

10

66

Whole

0.095

0.133

0.13

0.13

10

66

Crushed

0.085

0.126

0.11

0.11

10

66

Flour

0.040

0.061

0.05

0.40

26

26

Palm kernel

0.102

0.102

26.00

26.00

25

25

Milkpowder

0.102

0.102

26.00

26.00

25

25

Lard

0.120

0.120

32.00

32.00

25

25

Milkpowder

0.120

0.120

32.00

32.00

25

25

Water-NaCl

0.460

0.460

4.00

4.00

10

10

Solution

0.460

0.460

4.00

4.00

10

10

Water-sucrose

0.320

0.320

0.67

0.67

10

10

Solution

0.320

0.320

0.67

0.67

10

10

Thermal Conductivity and Diffusivity of Foods

325

Table 8.6. Continued Material

Conductivity (W/mK) Moisture (Kg/Kg db) Temperature (°C)

Min

Max

Min

Max

Min

Max

Rapeseed

0.060

0.155

0.01

0.24

-26

32

Whole

0.108

0.155

0.06

0.15

4

32

Ground

0.062

0.088

0.07

0.15

4

32

Torch

0.086

0.120

0.01

0.24

-26

19

Midas

0.092

0.092

0.01

0.01

19

19

Crushed

0.060

0.080

0.07

0.15

4

32

Agar-water

0.617

0.617

19.90

19.90

30

30

Gel

0.617

0.617

19.90

19.90

30

30

Tobacco

0.055

0.070

0.20

0.32

15

15

-

0.055

0.070

0.20

0.32

15

15

Sugar

0.382

0.637

0.67

9.00

0

80

Glucose

0.450

0.637

1.50

8.09

2

80

Cane sugar

0.382

0.637

0.67

9.00

0

80

Albumen

0.039

0.042

0.02

0.14

41

41

Freeze-dried gel

0.039

0.042

0.02

0.14

41

41

Sorghum

0.084

0.150

0.01

0.30

5

36

Rs6lO

0.130

0.150

0.15

0.28

5

5

Grain dust

0.084

0.094

0.10

0.20

22

22

NaCI

0.568

0.656

4.00

4.00

10

80

Solution

0.568

0.656

4.00

4.00

10

80

Honey

0.440

0.618

1.50

9.00

2

71

-

0.440

0.618

1.50

9.00

2

71

Albumine

0.382

0.425

0.67

0.67

27

90

Solution

0.382

0.425

0.67

0.67

27

90

Note: Thermal conductivities higher than that of water (0.62 W/mK at 25°C) are characteristic of frozen foods of high moisture content, since the thermal conductivity of ice is about 2 W/mK

Chapter 8

326

VI. THERMAL CONDUCTIVITY OF FOODS AS A FUNCTION OF MOISTURE CONTENT AND TEMPERATURE

A concept proposed by Maroulis et al. (2001) is adopted here and applied to obtain an integrated and uniform analysis of the available data. The concept was applied simultaneously to all the data of each material, regardless the data sources. Thus, the results are not based on the data of only one author and consequently they are of elevated accuracy. A simplified analysis is presented in Chapter 6 for the moisture diffusivity. Assume that a material of intermediate moisture content consists of a uniform mixture of two different materials: (a) a dried material and (b) a wet material with infinite moisture. The thermal conductivity is, generally, different for each material. The thermal conductivity of the mixture could be estimated using a two phase structural model: 1

X :(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:

Thermal Conductivity and Diffusivity of Foods

327

/l0(W/mK) thermal conductivity at moisture X = 0 and temperature T = Tr At (WlmK} thermal conductivity at moisture X = oo and temperature T = Tr Ea (kJ I moT) Activation Energy for heat conduction in dry material at X = 0 Et (kJ I mol) Activation Energy for heat conduction in wet material at X — co

The resulting model is summarized in Table 8.7 and can be fitted to data using a nonlinear regression analysis method. The model is fitted to all literature data for each material and the estimates of the model parameters are obtained. Then the residuals are examined and the data with large residuals are rejected. The procedure is repeated until an accepted standard deviation between experimental and calculated values is obtained (Draper and Smith, 1981). Among the available data only 13 materials have more than 10 data, which come from more than 3 publications. The procedure is applied to these data and the results of parameter estimation are presented in Table 8.8 and in Figure 8.18. It is clear that thermal conductivity is larger in wet materials. Figures 8.19-8.36 present retrieved thermal conductivities from the literature and model-calculated values for selected food materials as a function of moisture content and temperature. Thermal conductivity A, tends to increase with the moisture content X and the temperature T. The thermal conductivity parameters /10 and A/, shown in Figure 8.18, vary in the range of 0.05 to 1.0 W/mK. It should be noted that the thermal conductivity of air is about 0.026 W/mK, while that of water is 0.60 W/mK. Values of thermal conductivity of foods higher than 0.60 W/mK are normally found in frozen food materials (Aice=2 W/mK). The thermal conductivity increases, in general, with increasing moisture content. Temperature has a positive effect, which depends strongly on the food material. The energy of activation for heat conduction E is, in general, higher in the dry food materials.

328

Chapter 8

Table 8.7 Mathematical Model for Calculating Thermal Conductivity in Foods as a Function of Moisture Content and Temperature

Proposed Mathematical Model

X0exp

where

RT

T,

X . +——X.exp l +X

RT T

/i (W/mK) the thermal conductivity, X (kg/kg db) the material moisture content, T(°C) the material temperature, Tr = 60°C a reference temperature, and R = 0.0083143 kJ/mol K the ideal gas constant.

Adjustable Model Parameters

• • • •

Ka(W /mK) thermal conductivity at moisture X = 0 and temperature T = Tr "k.(W/ mK) thermal conductivity at moisture X = °o and temperature T = Tr E/U / mol) activation energy for heat conduction in dry material at X = 0 E,(kJ/ mol) activation energy for heat conduction in wet material at X = oo

329

Thermal Conductivity and Diffusivity of Foods

Table 8.8. Parameter Estimates of the Proposed Mathematical Model 4, E; E0 (W/mK) (kJ/mol) (kJ/mol)

sd (W/mK)

7.2

5.0

0.047

0.287 0.106 0.270

2.4 1.3 2.4

11.7 0.0 1.9

0.114 0.007 0.016

0.718 0.623 0.800

0.120 0.243 0.180

3.2 0.3 9.9

14.4 0.4

0.037 0.006 0.072

37 28

0.611 0.680

0.049 0.220

0.0 0.2

47.0 5.0

0.059 0.047

5

33

0.665

0.212

1.7

1.9

0.005

Beef

6

37

0.568

0.280

2.2

3.2

0.017

Other Rapeseed

3

35

0.239

0.088

3.6

0.6

0.023

3

15

0.800

0.273

2.7

0.0

0.183

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

Thermal Conductivity and Diffusivity of Foods

Vegetables

TOMATO

Total Number of Papers

5

Total Experimental Points

31

Points Used in Regression Analysis Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %)

28 (90%) 0.05 25

Parameter Estimates Xi (W/mK)

0.68

Xo (W/mK) Ei(kJ/mol)

0.22 0.17

____Eo (kJ/mol)______5.0

1.0

10.0

Moisture (kg.kg db)

Figure 8.29 Thermal conductivity of tomato at various temperatures and moisture contents.

Chapter 8

342

Model Foods Total Number of Papers

AMIOCA 5

Total Experimental Points 51 Points Used in Regression Analysis 29 (57%) Standard Deviation (sd, W/mK) 0.04 Relative Standard Deviation (rsd, %)______219 Parameter Estimates Xi (W/mK) 0,72 Xo(W/mK) 0.12

Ei (kJ/mol)

3.22

_____Eo (kJ/mol)______14.4

0.1

1.0

10.0

Moisture (kg.kg db)

Figure 8.30 Thermal conductivity of amioca (starch) at various temperatures and moisture contents.

Thermal Conductivity and Diffusivity of Foods

Model Foods

343

HYLON

Total Number of Papers 3 Total Experimental Points 43 Points Used in Regression Analysis 21 (49%) Standard Deviation (sd, W/mK) 0.07 Relative Standard Deviation (rsd, %) 9 Parameter Estimates Xi (W/mK) 0.80 Xo(W/mK) 0.18 Ei (kJ/mol) 9.90 _____Eo (kJ/mol)______0.0

Temperature °C -j • 40

1.0

10.0

Moisture (kg.kg db)

Figure 8.31 Thermal conductivity of hylon (starch) at various temperatures and moisture contents.

Chapter 8

344

Model Foods

STARCH

Total Number of Papers

4

Total Experimental Points

55

Points Used in Regression Analysis Standard Deviation (sd, W/mK) Relative Standard Deviation (rsd, %)

24 (44%) 0.01 0

Parameter Estimates Xi (W/mK) Xo (W/mK) Ei (kJ/mol)

0.62 0.24 0.32

____Eo (kJ/mol)_____0.4

Temperature °C • 40 M

1

0.1

• 60

1.0

IT

10.0

Moisture (kg.kg db)

Figure 8.32 Thermal conductivity of starch at various temperatures and moisture contents.

Thermal Conductivity and Diffusivity of Foods

345

Dairy MILK Total Number of Papers 5 Total Experimental Points 84 Points Used in Regression Analysis 33 (39%) Standard Deviation (sd, W/mK) 0.01 Relative Standard Deviation (rsd, %)______6 Parameter Estimates Xi (W/mK) Xo(W/mK) Ei(kJ/mol)

0.67 0.21 1.73

_____Eo (kJ/mol)______1.9

0.1

1.0

10.0

Moisture (kg.kg db)

Figure 8.33 Thermal conductivity of milk at various temperatures and moisture contents.

Chapter 8

346

Cereal Products

CORN

Total Number of Papers Total Experimental Points Points Used in Regression Analysis

3 28 15

Standard Deviation (sd, W/mK)

(54%)

0.05

Relative Standard Deviation (rsd, %)______77

Parameter Estimates Xi (W/mK) Xo(W/mK)

0.47 0.31

Ei (kJ/mol)

0.00

____Eo (kJ/mol)_____9.0

0.1

1.0

10.0

Moisture (kg.kg db)

Figure 8.34 Thermal conductivity of corn (grains) at various temperatures and moisture contents.

Thermal Conductivity and Diffusivity of Foods

Baked Products

347

DOUGH

Total Number of Papers 3 Total Experimental Points 20 Points Used in Regression Analysis 15 (75%) Standard Deviation (sd, W/mK) 0.18 Relative Standard Deviation (rsd, %)_______0 Parameter Estimates Xi (W/mK) 0.80 Xo (W/mK) 0.27 Ei(kJ/mol) 2.71

____Eo (kJ/mol)______0.0

0.1

1.0

10.0

Moisture (kg.kg db)

Figure 8.35 Thermal conductivity of dough at various temperatures and moisture contents.

Chapter 8

348

_______________Meat____BEEF Total Number of Papers 6 Total Experimental Points 75 Points Used in Regression Analysis 37 (49%) Standard Deviation (sd, W/mK) 0.02 Relative Standard Deviation (rsd, %)______15

Parameter Estimates Xi (W/mK) 0.57 Xo (W/mK) 0.28 Ei(kJ/mol) 2.15 _____Eo (kJ/mol)______3.2

1.0

10.0

Moisture (kg.kg db)

Figure 8.36 contents.

Thermal conductivity of beef at various temperatures and moisture

Thermal Conductivity and Diffusivity of Foods

349

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352

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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:

Heat and Mass Transfer Coefficients in Food Systems

q/A = QJL)(AT)

361

(9-2)

Thus, the heat transfer coefficient is equivalent to h=UL. However, Eq. (9-2) is difficult to apply, since the film thickness L cannot be determined accurately because it varies with the conditions of flow at the interface. The overall heat transfer coefficient U (W/m2K) between two fluids separated by a conducting wall is given by the equation q/A = UAT

(9-3)

where AT is the overall temperature difference (K). The coefficient U is related to the heat transfer coefficients hi and h2 of the two sides of the wall and the wall heat conduction x/X by the equation: \/U=\/h,+x/X+l/h2

(9-4)

where x is the wall thickness (m), and X is the wall thermal conductivity (W/mK). In industrial heat exchangers, the thermal resistance of fouling deposits must be added in series to the resistances of Eq. (9-4). The overall heat transfer coefficients are specific for each processing equipment and fluid system, and it is determined usually from experimental measurements. B. Determination of Heat Transfer Coefficients

The heat transfer coefficient h at a given interface can be determined experimentally by various methods (Rahman, 1995). In the constant heating (steady state) method, the heat flux q/A is measured (e.g. by electrical measurement) at a given temperature difference AT, and the coefficient h is calculated from Eq. (9-1). In the quasi-steady state method, the heat transfer coefficient is determined from the slope of the heating line of a high conductivity solid, which is assumed to heat uniformly. The heat transfer coefficient can be estimated from the analytical or numerical solution of the heat conduction (Fourier) equation:

dt

dX1

(9-5)

where a is the thermal diffusivity of the material. The solution of Eq. (9-5) involves the Biot number for heat transfer, BiH = (hL/X), from which the heat transfer coefficient can be estimated. The heat transfer coefficient h at the interface of processing equipment can be measured by the heat flux sensors method, which simultaneously measures the surface temperature and the heat flux (Karwe and Godavarti, 1997). The sensors consist of a differential

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Chapter 9

thermopile of thermal resistance with two inserted thermocouples. They are mounted on the heat transfer surface by a high thermal conductivity paste. Approximate values of h can be estimated indirectly by measuring the parameters of some physical processes, which involve heat transfer, such as the freezing time of a material (Plank's equation) or the evaporation rate of a liquid in a flat surface at a given temperature difference A T. Special experimental arrangements are required for the estimation of the heat transfer coefficients between particles and a liquid, both in motion, as in aseptic processing of food suspensions. The particle temperature may be measured by a moving thermocouple or estimated from the change of color of special materials, such as liquid crystals coated on acrylic spherical particles and observed through a transparent flow tube. C. General Correlations of the Heat Transfer Coefficient

Correlations of heat transfer data are useful for estimating the heat transfer coefficient h in various processing equipment and operating conditions. These correlations contain, in general, dimensionless numbers, characteristic of the heat transfer mechanism, the flow conditions, and the thermophysical and transport properties of the fluids. Table 9.3 lists the most important dimensionless numbers used in both heat and mass transfer operations. The Reynolds number (Re=uL/v) is used widely in almost all correlations. In this number, the velocity u is in (m/s), the length I is in (m) and the kinematic viscosity or momentum diffusivity (v=rj/p~) is in (m2/s). The length L can be the internal diameter of the tube, the equivalent diameter of the noncircular duct, the diameter of a spherical particle or droplet, or the thickness of a falling film. Some dimensionless numbers, used in both heat and mass transfer correlations, are denoted by the subscripts H and M respectively, i.e. Bin Bi^, Stn, St^, JH andy^/Table 9.4 shows some heat transfer correlations of general applications. For natural convection, the parameters a and m characterize the various shapes of the equipment and the conditions of the fluid (McAdams, 1954; Perry and Green, 1984; Geankoplis, 1993; Rahman, 1995). The ratio of tube diameter to tube length D/L is important in the laminar flow (Re < 2100), but it becomes negligible in the tubular flow in long tubes (L/D > 60). For shorter tubes, the ratio D/L should be included in the correlation. The viscosity ratio r\/r]w refers to the different viscosity in the bulk of the fluid 77 and at the tube wall t]w. This ratio becomes important in highly viscous fluids, like oils, in which the viscosity drops sharply at the high wall temperatures, increasing the heat transfer coefficients. Several other correlations have been proposed in the literature for different heat transfer of fluid systems, like flow outside tubes and flow in packed beds.

Heat and Mass Transfer Coefficients in Food Systems

363

The heat transfer coefficients of condensing vapors have been correlated to the geometry of the tubes and the properties of the liquid film or droplets. Very high heat transfer coefficients are obtained by drop-wise condensation.

Table 9.3 Dimensionless Numbers in Heat and Mass Transfer Calculations Number Applications Flow processes Re=uL/v Reynolds Heat transfer Nu=hL/A Nusselt Heat transfer Pr=v/a Prandtl Free convection Gr=L3g(Ap/p)/v2 Grashof Graetz Heat transfer Gz=GACpM Heat transfer BiH=hm Biot Mass transfer Sh=kcL/D Sherwood Diffusion processes Sc=v/D Schmidt Heat transfer Stanton StH=h/G Cp Stanton Mass transfer StM=kc/u Heat/mass transfer Le=a/D Lewis Flow/diffusion Pe=uL/D Peclet Mass transfer BiM=kcL/X Biot Heat transfer JH=StHPr2'^ Heat Transfer Factor Mass transfer Mass Transfer Factor jM=StMSc /s

A, interfacial area (m2); L, length (m); a thermal diffusivity (m2/s); Cp, specific heat (J/kg K); D mass diffusivity (m2/s); g acceleration of gravity (m2/s); G=up, mass flow rate kg/m2s; h, heat transfer coefficient (W/m2K); kc, mass transfer coefficient (m/s); 77 viscosity (Pas); p, density (kg/m3); u, velocity (m/s)

Table 9.4 General Heat Transfer Correlations____________________________ Transfer System________________Correlation_____________ Natural convection Nu = a (Gr Pr)m Laminar inside tubes Nu = l.B6[RePr(D/L)f\rj/rjwfu Turbulent inside long tubes Nu = O.Q23Re°*Pr1'3 (rj/tjw}°M Parallel to flat plate (laminar) Nu = 0.664/?e°5 Pr113 Parallel to flat plate (turbulent) Nu = 0.0366#e°8 Pr1'3 Past single sphere_______________Nu = 2.0 +0.60Re°'5 Pr113_______ Dimensionless numbers defined in Table 9.3. a and m, parameters of natural convection characteristic of the system (Perry and Green, 1984); L, D length and diameter of tube. Long tubes L/D>60

364

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)

Heat and Mass Transfer Coefficients in Food Systems

365

where J is the mass flux (kg/m2s) and AC is the concentration difference (kg/m3). In contrast to heat transfer where the driving force is the temperature difference AT, m mass transfer the driving force can be expressed by the concentration difference AC, the difference of mass fraction AY, or the pressure difference AP. Thus, three mean mass transfer coefficients can be defined by the following equation (Saravacos, 1997): J = kcAC = kYAY = kpAP

(9-14)

The units of the three mass transfer coefficients depend on the units of AC, AY and AP and they are usually kc (m/s), kY (kg/m2s) and kp (kg/m2sPa). In food engineering and especially in drying calculations, the symbol hM is used instead of kY, with the same units (kg/m2s). In an analogy with the overall heat transfer coefficient K, the overall mass transfer coefficient is used to express mass transfer through the interface of two fluids, according to the equation: l/K=\fkci+\/kc2

(9-15)

where kcl and kC2 are the mass transfer coefficients of the two contacting fluids. It should be noted that in mass transfer there is no wall resistance and the two fluids at the interface are assumed to be in thermodynamic equilibrium. Volumetric mass transfer coefficients (kcv) may be used in some industrial operations, defined by the equation: kcv=a.kc

(9-16)

where a = A/Vis the specific surface of the transfer system (m2/m3). Thus the units of kcv will be (1/s) and of h B. Determination of Mass Transfer Coefficients The mass transfer coefficients can be determined by direct or indirect measurement of the mass transfer rates in a controlled experimental system. The wetted wall column has been used to determine ^-values in liquid/gas and liquid/vapor systems, like absorption of gas in aqueous solutions (Sherwood et al., 1975; Brodkey and Hershey, 1988). The mass flux is measured at a given driving force (AC, AY or AP) and the corresponding coefficients (kc, kY or kp) are determined. The mass transfer coefficients (kc or hM) during the constant rate period of drying can be estimated from the drying rate of a known sample at well-defined

366

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.

Heat and Mass Transfer Coefficients in Food Systems

367

Table 9.6 Mass Transfer Correlations for Fluid/Fluid Interfaces Transfer system____________Correlation________________ Gas bubbles in unstirred tank Sh = OA2 Gr1/45c"3 Gas bubbles in stirred tank Sh=l.62 [(P/V) cflpP3]1/4 5c1/3 Small liquid drops in unstirred solution Sh = 1.13 (dulD)°'% Falling films______________Sh = 0.69 (zu/Df5_____________ Dimensionless numbers defined in Table 9.3; d, drop diameter (m); z, position along film (m); P/V stirrer power per volume.

D. Theories of Mass Transfer

The empirical mass transfer data, used in various correlations can be interpreted in terms of approximate or exact theories of mass transfer. The mass transfer theories were developed mainly for fluid/fluid systems. The most important theories are briefly the following (Cussler, 1997). 1. Film Theory The mass transfer coefficient kc is a function of the first power of the diffusion coefficient £>: hc =D/L

(9-17)

where L (m) is the film thickness, which is difficult to determine accurately, since it is a function of the flow conditions, the geometry of the system, and the physical properties of the fluid. 2. Penetration Theory The mass transfer coefficient kc is a function of the square root of the mass diffusivity D:

kc=2(Du/nLf2

(9-18)

where L is the depth of penetration (m) and u is the velocity (m/s) of penetration. The contact time between the diffusivity components and the fluid is defined as u/L, and it is difficult to determine experimentally. 3. Surface Renewal Theory The mass transfer coefficient kc is a function of the square root of mass diffusivity D, in a similar manner with the penetration theory:

368

Chapter 9

kc=(Drf2

(9-19)

where T is the average time for a fluid element in the interface region. 4. Boundary Layer Theory The boundary layer theory, applied primarily in fluid mechanism and heat transfer, gives a more accurate correlation of the mass transfer coefficient kc in the laminar flow. The kc is a function of the 2/3 power of mass diffusivity D. The average mass transfer coefficient kc, past a flat plate of length L, is given by the following empirical equation, which is analogous to the corresponding heat transfer relationship:

kc = 0.00646 (D/L)RelK So213

(9-20)

where the Reynolds number is defined as Re=Lu/v. The heat and mass transfer analogies are useful in evaluating the heat/mass transfer mechanisms and in estimating and inter converting the heat and mass transfer coefficients. The Chilton Colburn (or Colburn) analogy for heat and mass transfer indicates that in fluid systems, under certain conditions, the heat and mass transfer factors are equal (Geankoplis, 1993; Saravacos, 1997): JH=JM

(9-21)

where jH= 5^/'r2/3,yw= StMSc2n and StH= h/upCp, StM= kc/u or StM= h^up The Colburn analogy in air/water mixtures (applications in drying and air conditioning) is simplified, since the Pr and Sc are approximately equal (Pr = Sc = 0.8). Therefore, we may have StH = StM or h/upCp = kC/u or h/pCp =kc, In terms of the mass transfer coefficient hM, the last relationship becomes: h/Cp = hM

(9-22)

The specific heat of atmospheric air at ambient conditions is approximately Cp = 1000 J/kgK. Therefore, Eq. (9-22) yields h = 1000hM, where h is in W/m2K and hM in kg/m2s. If the units of hM are taken as g/m2s, the last relationship is written as (Saravacos, 1997): Atmospheric air, h (W/m2K) = HM (g/m2s)

(9-23)

A similar relationship is obtained between the coefficients h and kc'.

Atmospheric air, h (W/m2K) = kc (mm/s)

(9-24)

Heat and Mass Transfer Coefficients in Food Systems

369

IV. HEAT TRANSFER COEFFICIENTS IN FOOD SYSTEMS

The heat and mass transfer coefficients in food systems are determined experimentally or correlated empirically from pilot plant and industrial data. They are specific for each food process and processing equipment and are related to the physical structure of the food materials. Most of the literature data refer to heat transfer coefficients, since heat transfer is the rate controlling mechanism in many processing operations. Mass transfer coefficients can be related to heat transfer in some important operations, like drying, using the Colburn analogy of heat and mass transfer. Typical values of heat transfer coefficients are shown in Table 9.7 (Hallstrom et al., 1988; Perry and Green, 1997; Rahman, 1995; Saravacos, 1995). Detailed data and empirical correlations for both transfer coefficients are presented in sections VI and VII of this chapter. A. Heat Transfer in Fluid Foods

Heat transfer in viscous non-Newtonian fluids in laminar flow in tubes is expressed by a correlation analogous to the equation for Newtonian fluids: (9-25)

where the Graetz number Gz = GrCp/AL, and G is the mass flow rate (kg/m2s). Table 9.7 Typical Heat Transfer Coefficient h and Overall Coefficients U in Food Processing Operations_____________________ ______ Heat Transfer System h, W/m2K Air/process equipment, natural convention 5 - 20a Baking ovens 20 - 80a Air drying, constant rate period 30 - 200a Air drying, falling rate period 20 - 60 Water, turbulent flow 1000 - 3000 Boiling water 5000 - 10000 Condensing water vapor 5000 - 50000 Refrigeration, air cooling 20 - 200 Canned foods, retorts 150 - 500 Aseptic processing, particles 500 - 3000 Freezing, air/refrigerants 20-500 Frying, oil/solids 250 - 1000 Heat exchangers (tubular/plate) 500 - 3500 (overall U) Evaporators____________________500 - 3000 (overall U)

' Similar numerical values for the mass transfer coefficients kc (mm/s) or hM (g/m2s), applying the Colbum analogy.

Chapter 9

370

The apparent viscosities at the bulk of the fluid and at the wall tja and ^ are determined for the given shear rate y using the Theological constants K and n of the fluid for a mean temperature. Heat transfer in agitated vessels is expressed by the empirical correlation (Saravacos and Moyer, 1967): = CRe°'66Pr 1/3 Ola/Tlaw)',0.14

(9-26)

where the coefficient C = 0.55 for Newtonian and C = 1.474 for non-Newtonian fluids. The Reynolds number is estimated as Re = (d2Np)lrja where d is the diameter of the impeller, and rja is the apparent viscosity estimated at the agitation speed TV as r\a = Ky"'1 where K and n are the Theological constants of the fluid at the mean temperature. The shear rate y for the pilot-scale agitated kettle, described in this reference (0.40 m diameter, anchor agitator), was calculated from the empirical relation 7= 13N. The heat transfer coefficients h at the internal interface of the vessel for a sugar solution and for applesauce increased linearly with speed of agitation (RPM), as shown in Figure 9.1. Figure 9.2 shows that the overall heat transfer coefficient U in the agitated kettle decreases almost linearly when the flow consistency coefficient K is increased. 10000

1000 --

Figure 9.1 Heat transfer coefficients in agitated kettle. S, sucrose solution 40° Brix; A, applesauce; RPM, 1/min

371

Heat and Mass Transfer Coefficients in Food Systems

1600

1300

1000

10 K (Pa s")

Figure 9.2. Overall heat transfer coefficient (U) of fruit purees in agitated kettle. K, flow consistency coefficient.

B. Heat Transfer in Canned Foods Several heat transfer correlations for canned foods are presented by Rahman (1995). In most cases of heating/cooling of cans, the product heat transfer coefficient ht is controlling the transfer process, since the outside (heating/cooling medium) coefficient and the heat conductance of the wall l/x are generally high (metallic or glass containers). However, heat transfer in plastic containers may be controlled by the wall thermal resistance, due to the low thermal conductivity and the high wall thickness of the plastic material Eq. (9-4). The Reynolds numbers for Newtonian fluids is estimated as Re = where d is the can diameter and N is the speed of rotation (1/s) of the can. For nonNewtonian fluids, the dimensionless numbers used are the following (Rao, 1999): Re = -

4n l3n + l

(9-27)

372

Chapter 9

4n Gr

22

-2

V

;

where K and n are the rheological constants of the fluid at a mean temperature, and fi = (A V/AT)IV, 1/K (natural convection). Heat transfer in cans in an agitated retort (Steritort) is considered as the sum of the contributions of both natural and forced convection:

Nu = A[(Gr}(Prf + C^Re\Pr\D / L)]D

(9-30)

where, for Newtonian fluids, A = 0.135, B = 0.323, C = 3.91xlO"3, and D = 1.369 and for non-Newtonian fluids, A = 2.319, B = 0.218, C = 4.1xlO'7, and D = 1.836 In end-over-end agitated cans the following correlations were obtained (Rao, 1999): Nu = 2 .9 Re°A36 Pr°2*7 for Newtonian fluids

(9-3 1 )

Nu = Re°ABS Pr°'361 for non-Newtonian fluids

(9-32)

Non-Newtonian biopolymers, when subjected to extreme heat treatment, suffer significant losses in apparent viscosity. C. Evaporation of Fluid Foods

Heat transfer controls the evaporation rate of fluid foods and high heat transfer coefficients are essential in the various types of equipment. Prediction of the heat transfer coefficients in evaporators is difficult, and experimental values of the overall heat transfer coefficient U are used in practical applications. The overall heat transfer coefficient is a function of the two surface heat transfer coefficients //,• and h0, the wall thermal conductance MX, and the fouling resistance Eq. (9-33): -1 = 1 + - + — + FR U h k h,

(9-33)

The fouling resistance 7-7? becomes important in the evaporation of liquid foods containing colloids and suspensions, which tend to deposit on the evaporator walls, reducing significantly the heat transfer rate.

Heat and Mass Transfer Coefficients in Food Systems

373

10000

Figure 9.3 Overall heat transfer coefficients U in evaporation of clarified CL and unfiltered UFT apple juice at 55°C.

Falling film evaporators are used extensively in the concentration of fruit juices and other liquid foods because they are simple in construction and they have high heat transfer coefficients. Figure 9.3 shows overall heat transfer coefficients U for apple juices in a pilot plant falling film evaporator, 5 cm diameter and 3 m long tube (Saravacos and Moyer, 1970). Higher U values were obtained in the evaporation of depectinized (clarified) apple juice (1200 to 2000 W/m2K) than the unfiltered (cloudy) juice, which tended to foul the heat transfer surface as the concentration was increased. The U value for water, under the same conditions was higher as expected: U= 2300 W/m2K. D. Improvement of Heat / Mass Transfer

Jet impingement ovens and freezers operate at high heat transfer rates, due to the high air velocities at the air/food interface. Heat transfer coefficients of 250350 W/m2K can be obtained in ovens, baking cookies, crackers and cereals (Nitin andKarwe, 1999). Ultrasounds can substantially improve the air-drying rate of porous foods, like apples (acoustically-assisted drying). Ultrasound of 155-163 db increased the moisture diffusivity at 60°C from 7xlO' 10 to 14xlO'10 m2/s (Mulet et al., 1999).

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.

Heat and Mass Transfer Coefficients in Food Systems

375

Table 9.8 Number of Available Equations for each Food Process

1

Process Baking

1

2

Forced convection Blanching Steam

1

3

Cooling Forced convection

4

Fluidized bed Rotary

4 6

Storage Forced convection

7

16 1

Freezing Forced convection

6

9

Drying Convective

5

No. of equations

4

Sterilization Aseptic

9

Retort

3

Total No. of equations

54

376

Chapter 9

Table 9.9 Number of Available Equations for each Food Material

1

Material

Apples 2 Apricots 3 Barley 4 Beef 5 Cakes 6 Calcium alginate gel

1 8 9 10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Canola seeds Carrot Corn

Corn starch Figs

Fish

Grapes Green beans Hamburger Maize Malt Meat carcass Model food Newtonian liquids Non-food material Particulate liquid foods Peaches Potatoes Raspberries Rice

Soya

Soybean Strawberries Sugar Wheat Spherical particles Tomatoes Corn cream Rapeseed Meatballs

Total No. of equations

No. of Equations 1 1 2 1 1 1 1 1

2 1 1 1 3 1 2 1 1 1 4 1 3 3 1 2 1 1 2 1 1 1 3 1 1 1 1 1 54

Heat and Mass Transfer Coefficients in Food Systems

377

Table 9.10 Parameters of the Equation jH = aRe" for each Process and each Material „ mm Re Process/product/reference a max Re

Baking Cakes Baiketal., 1999

0.801

-0.390

40

3,000

Green Beans Zhangetal., 1991 0.00850

-0.443

150

1,500

Cooling Apples Fikiinetal., 1999

0.0304

-0.286

4,000

48,000

0.114

-0.440

2,000

25,000

8.39

-0.492

3,500

9,000

0.472

-0.516

1,300

17,000

2.93

-0.569

2,000

12,000

0.186

-0.500

3,700

43,000

0.0293

-0.320

1,300

16,000

0.136

-0.440

1,900

25,000

0.267

-0.550

1,000

24,000

Blanching

Apricots

Fikiinetal., 1999 Figs Dincer, 1995 Grapes Fikiinetal., 1999 Model Food Alvarez et al., 1999 Peaches Fikiinetal., 1999 Raspberries Fikiinetal., 1999 Strawberries Fikiinetal., 1999 Tomatoes Dincer, 1997

Chapter 9

378

Table 9.10 Continued Process/product/reference Drying

a

n

minRe

ma\Re

3.26

-0.650

20

1,000

0.458

-0.241

30

50

0.692

-0.486

500

5,000

1.06 4.12

-0.566 -0.650

400 20

1,100 1,000

0.665 0.741

-0.500 -0.430

8 1,000

50 3,000

11.9

-0.901

150

1,500

0.196

-0.185

60

80

0.224

-0.200

2,000

11,000

4.12

-0.650

20

1,000

2.48

-0.523

200

1,500

149 3.26

-0.340 -0.650

50 20

100 1,000

0.101 -0.355

3,200

13,000

Convective Barley Sokhansanj, 1987 Canola Seeds Langetal., 1996 Carrot Mulct etal., 1989 Corn Fortes etal., 1981 Torrezetal., 1998 Graves Ghiausetal, 1997 Vagenas et al, 1990 Maize Mourad et al., 1997 Malt Lopezetal., 1997 Potatoes Wangetal., 1995 Rice Torrezetal., 1998 Soybean Taranto et al., 1997 Wheat Langetal., 1996 Sokhansanj, 1987

Fluidized bed Corn Starch Shu-De etal., 1993

379

Heat and Mass Transfer Coefficients in Food Systems

Table 9.10 Continued n

min Re

max Re

-0.258

80

300

-0.587 -0.258

10 20

100

-0.528

1,500

17,000

0.650

-0.418

80

25,000

48.6

-0.535

300

600

8.87 4.67

-0.672

-0.645

7,500 9,000

150,000 73,000

0.228

-0.269

1,800

20,000

0.536

-0.485

3,400

28,000

0.658

-0.425

70

90

0.0136

-0.196

Process/product/reference

Rotary Fish Sheneetal, 1996 0.00160 Soya Alvarez et al., 1994 0.00960 Sheneetal., 1996 0.000300 Susar Wangetal., 1993 0.805 Freezing Beet Heldman, 1980 Calcium alsinate sel Sheng, 1994 Hamburser Floresetal, 1988 Toccietal., 1995 Meat carcass Mallikarjunan et al., 1994 Meatballs Toccietal., 1995

80

Storage Potatoes Xuetal., 1999 Wheat Changetal, 1993

1,500 10,000

Chapter 9

380

Table 9.10 Continued Process/product/reference

//

min Re

max Re

0.500 0.448 3.42

-0.507 -0.519 -0.687

5,000 2,400 2,000

20,000 45,000 11,000

0.748 0.662 0.517

-0.512 -0.508 -0.441

3,000 3,000 3,000

85,000 85,000 85,000

0.225 0.0493

-0.400 -0.199

140 1,800

1,500 5,200

2.26

-0.474

4,300

13,000

2.74

-0.562

11,000

400,000

0.564

-0.403

30

0.108

-0.343 130,000

Sterilization

Aseptic Model food Balasubramaniam et al, 1994 Sastryetal., 1990 Zuritzetal, 1990 Non-food material Kramers, 1946 Ranzetal.,1952 Whitaker, 1972 Paniculate liquid foods Mankadetal., 1997 Sannervik et al., 1996 Spherical particles Astrometal., 1994 Retort Newtonian liquids Anantheswaran et al., 1985 Particulate liquid foods Sablanietal, 1997 Corn cream Zamanetal., 1991

1,600 1,100,000

Heat and Mass Transfer Coefficients in Food Systems

381

Table 9.11 Parameters of the Equation jH = a Re" for each Process

Process

a

«

mm Re

max Re

Baking

0.80

-0.390

40

3,000

Blanching

0.0085

-0.443

150

1,500

Cooling

0.143

-0.455

1,000

48,000

Drying /convective

1.04

-0.455

8

11,000

Drying /fluidized bed

0.10

-0.354

3,200

13,000

Drying /rotary

0.001

-0.161

10

300

Freezing

1.00

-0.486

80

150,000

Storage

0.259

-0.387

70

10,000

Sterilization /aseptic

0.357

-0.450

140

45,000

Sterilization /retort

1.034

-0.499

30

110,000

The data of Tables 9.10 and 9.11 demonstrate the importance of the flow conditions (Reynolds number, Re) and the type of food process and product on the heat transfer characteristics (heat transfer factor, jH). As expected from theoretical considerations and experience in other fields, the heat transfer factor, jH decreases with a negative exponent of about -0.5 of the Re. The highest jH values are obtained in drying and baking operations, while the lowest values are in storage and blanching. Granular food materials, such as corn and wheat appear to have better heat transfer characteristics than large fruits (apples). Regression analysis of published mass transfer data show the similarity between the heat transfer factory'// and the mass transfer factor jM (see section VI of this chapter).

Chapter 9

382

JH 0.01

0.001

0.0001

0.00001 1

10

100

1000

10000

100000

1000000 10000000

Re

Figure 9.4 Heat transfer factor jH versus Reynolds number Re for all the examined processes and materials.

Heat and Mass Transfer Coefficients in Food Systems

383

JH 0.01

0.001

0.0001 10

1 000

10 000

100 000

1 000 000

Re

Figure 9.5 Ranges of variation of the heat transfer factor^ versus Reynolds number Re for all the examined processes.

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.

Heat and Mass Transfer Coefficients in Food Systems

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.

Heat and Mass Transfer Coefficients in Food Systems

387

Storage i

0.1

JH

0.01

WlhesiT

0.001 10

100

1000

10000

100000

Re

Figure 9.9 Heat transfer factory'// versus Reynolds number Re for storage process and various materials.

Chapter 9

388

0.1

Sterilization Aseptic

Spherical Parti i :les

JH

0.01

Non-Foa4 Matcri

Partiqulatg

Upii

Fo )di

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.

Heat and Mass Transfer Coefficients in Food Systems

389

0.1

Sterilization Retort

\ Part culat? Lii Fi oils

JH 0.01

0.001

100

1000

10000 Re

100000

1000000

Figure 9.11 Heat transfer factory// versus Reynolds number Re for sterilization retort process and various materials.

Chapter 9

390

JH

0.01

0.001

0.0001

10

100

1000

Re

10000

100000

1000000

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

Heat and Mass Transfer Coefficients in Food Systems

391

VI. MASS TRANSFER COEFFICIENTS IN FOOD PROCESSING: COMPILATION OF LITERATURE DATA

Recently reported mass transfer coefficient data in food processing were retrieved from literature following the same procedure described in Section V for heat transfer coefficient data (Krokida et al., 2001b). A total number of 15 papers were retrieved from the above journals. The data refer to 4 different processes (Table 9.12) and include about 9 food materials (Table 9.13). All available empirical equations were transformed in the form of mass transfer factor versus Reynolds number (JM = aRen). The results are classified by process and material and are presented in Tables 9.14 and 9.15. All the equations are presented in Figure 9.13 to define the range of variation of the jM and Re. The range of variation by process is sketched in Figure 9.14. The above results are presented for convective drying process in Figure 9.15. The effect of food material is obvious in this diagram. The results of fitting the equation to all data for each process is summarized in Table 9.14 and in Figure 9.16. Mass transfer coefficient values for process design can be obtained easily form the proposed equations and graphs. The range of variation of this uncertain coefficient can also be obtained in order to carry out valuable process sensitivity analysis. Table 9.12 Number of Available Equations for each Food Process

_____Process____________No. of Equations 1 Drying Convective

6

Spray

1

2 Freezing Forced Convection

6

3 Storage Forced Convection

1

4 Sterilization ______Forced Convection________________1^

_____Total No. of Equations__________15

392

Chapter 9

Table 9.13 Number of Available Equations for each Food Material

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

Heat and Mass Transfer Coefficients in Food Systems

Table 9.15 Parameters of the Equation jM = aRe" for each Process and each Material

/;

min Re

5.15-0.575

20

-0.462 0.004 0.741-0.430

10 900

40 3,000

Mouradetal., 1997

34.6 -1.000

5

15

Rice Torrezetal., 1998 Carrot

5.15-0.575

20

Muletetal., 1987

0.69 -0.486

500

5,000

Process/product/reference

a

max Re

Drying Convective Corn Torrezetal., 1998 Graves Ghiausetal., 1997 Vagenasetal, 1990 Maize

1,000

1,000

Spray Milk Straatsma et al., 1999

2.947

-0.890

1

2

Meat Toccietal., 1995

2.496

-0.495

2,500

70,000

Storage Potatoes Xuetal.,1999

0.667

-0.428

50

55

11.220

-1.039

6,500

26,000

Freezing

Sterilization Model food Fuetal.,1998

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