food processing operations modeling design and analysis

2 PREDICTION OF SPECIFIC HEATThe speci®c heat of a food is de®ned as the quantity of thermal energyassociated with a unit mass of the food and a unit of change in temperature.This thermo

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Food Processing Operations Modeling

Design and Analysis

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The Pennsylvania State University University Park, Pennsylvania

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FOOD SCIENCE AND TECHNOLOGY

A Series of Monographs, Textbooks, and Reference Books

EDITORIAL BOARD

Senior Editors

Owen R Fennema University of Wisconsin–Madison

Y.H Hui Science Technology System

Marcus Karel Rutgers University (emeritus)

Pieter Walstra Wageningen University

John R Whitaker University of California–Davis

Additives P Michael Davidson University of Tennessee–Knoxville

Dairy science James L Steele University of Wisconsin–Madison

Flavor chemistry and sensory analysis John H Thorngate III University

of California–Davis

Food engineering Daryl B Lund University of Wisconsin–Madison

Food proteins/food chemistry Rickey Y Yada University of Guelph

Health and disease Seppo Salminen University of Turku, Finland

Nutrition and nutraceuticals Mark Dreher Mead Johnson Nutritionals

Phase transition/food microstructure Richard W Hartel University of

Wisconsin–Madison

Processing and preservation Gustavo V Barbosa-C á novas Washington

State University–Pullman

Safety and toxicology Sanford Miller University of Texas–Austin

1 Flavor Research: Principles and Techniques, R Teranishi, I stein, P Issenberg, and E L Wick

Horn-2 Principles of Enzymology for the Food Sciences, John R Whitaker

3 Low-Temperature Preservation of Foods and Living Matter, Owen R Fennema, William D Powrie, and Elmer H Marth

4 Principles of Food Science

Part I: Food Chemistry, edited by Owen R Fennema Part II: Physical Methods of Food Preservation, Marcus Karel, Owen

R Fennema, and Daryl B Lund

5 Food Emulsions, edited by Stig E Friberg

6 Nutritional and Safety Aspects of Food Processing, edited by Steven

9 Handbook of Tropical Foods, edited by Harvey T Chan

10 Antimicrobials in Foods, edited by Alfred Larry Branen and P Michael Davidson

11 Food Constituents and Food Residues: Their Chromatographic

Determination, edited by James F Lawrence

12 Aspartame: Physiology and Biochemistry, edited by Lewis D Stegink and L J Filer, Jr.

13 Handbook of Vitamins: Nutritional, Biochemical, and Clinical Aspects,

edited by Lawrence J Machlin

14 Starch Conversion Technology, edited by G M A van Beynum and J.

21 Food Biotechnology, edited by Dietrich Knorr

22 Food Texture: Instrumental and Sensory Measurement, edited by Howard R Moskowitz

23 Seafoods and Fish Oils in Human Health and Disease, John E Kinsella

24 Postharvest Physiology of Vegetables, edited by J Weichmann

25 Handbook of Dietary Fiber: An Applied Approach, Mark L Dreher

26 Food Toxicology, Parts A and B, Jose M Concon

27 Modern Carbohydrate Chemistry, Roger W Binkley

28 Trace Minerals in Foods, edited by Kenneth T Smith

29 Protein Quality and the Effects of Processing, edited by R Dixon Phillips and John W Finley

30 Adulteration of Fruit Juice Beverages, edited by Steven Nagy, John A Attaway, and Martha E Rhodes

31 Foodborne Bacterial Pathogens, edited by Michael P Doyle

32 Legumes: Chemistry, Technology, and Human Nutrition, edited by Ruth H Matthews

33 Industrialization of Indigenous Fermented Foods, edited by Keith H Steinkraus

34 International Food Regulation Handbook: Policy · Science · Law,

edited by Roger D Middlekauff and Philippe Shubik

35 Food Additives, edited by A Larry Branen, P Michael Davidson, and Seppo Salminen

36 Safety of Irradiated Foods, J F Diehl

37 Omega-3 Fatty Acids in Health and Disease, edited by Robert S Lees and Marcus Karel

38 Food Emulsions: Second Edition, Revised and Expanded, edited by

K å re Larsson and Stig E Friberg

39 Seafood: Effects of Technology on Nutrition, George M Pigott and Barbee W Tucker

40 Handbook of Vitamins: Second Edition, Revised and Expanded,

edited by Lawrence J Machlin

41 Handbook of Cereal Science and Technology, Klaus J Lorenz and Karel Kulp

42 Food Processing Operations and Scale-Up, Kenneth J Valentas, Leon Levine, and J Peter Clark

43 Fish Quality Control by Computer Vision, edited by L F Pau and R Olafsson

44 Volatile Compounds in Foods and Beverages, edited by Henk Maarse

45 Instrumental Methods for Quality Assurance in Foods, edited by Daniel Y C Fung and Richard F Matthews

46 Listeria, Listeriosis, and Food Safety, Elliot T Ryser and Elmer H Marth

47 Acesulfame-K, edited by D G Mayer and F H Kemper

48 Alternative Sweeteners: Second Edition, Revised and Expanded, ited by Lyn O'Brien Nabors and Robert C Gelardi

ed-49 Food Extrusion Science and Technology, edited by Jozef L Kokini, Chi-Tang Ho, and Mukund V Karwe

50 Surimi Technology, edited by Tyre C Lanier and Chong M Lee

51 Handbook of Food Engineering, edited by Dennis R Heldman and Daryl B Lund

52 Food Analysis by HPLC, edited by Leo M L Nollet

53 Fatty Acids in Foods and Their Health Implications, edited by Ching Kuang Chow

54 Clostridium botulinum: Ecology and Control in Foods, edited by Andreas H W Hauschild and Karen L Dodds

55 Cereals in Breadmaking: A Molecular Colloidal Approach,

Ann-Charlotte Eliasson and K å re Larsson

56 Low-Calorie Foods Handbook, edited by Aaron M Altschul

57 Antimicrobials in Foods: Second Edition, Revised and Expanded,

edited by P Michael Davidson and Alfred Larry Branen

58 Lactic Acid Bacteria, edited by Seppo Salminen and Atte von Wright

59 Rice Science and Technology, edited by Wayne E Marshall and James I Wadsworth

60 Food Biosensor Analysis, edited by Gabriele Wagner and George G Guilbault

61 Principles of Enzymology for the Food Sciences: Second Edition, John

R Whitaker

62 Carbohydrate Polyesters as Fat Substitutes, edited by Casimir C Akoh and Barry G Swanson

63 Engineering Properties of Foods: Second Edition, Revised and

Expanded, edited by M A Rao and S S H Rizvi

64 Handbook of Brewing, edited by William A Hardwick

65 Analyzing Food for Nutrition Labeling and Hazardous Contaminants,

edited by Ike J Jeon and William G Ikins

66 Ingredient Interactions: Effects on Food Quality, edited by Anilkumar

69 Nutrition Labeling Handbook, edited by Ralph Shapiro

70 Handbook of Fruit Science and Technology: Production, Composition,

Storage, and Processing, edited by D K Salunkhe and S S Kadam

71 Food Antioxidants: Technological, Toxicological, and Health

Perspec-tives, edited by D L Madhavi, S S Deshpande, and D K Salunkhe

72 Freezing Effects on Food Quality, edited by Lester E Jeremiah

73 Handbook of Indigenous Fermented Foods: Second Edition, Revised

and Expanded, edited by Keith H Steinkraus

74 Carbohydrates in Food, edited by Ann-Charlotte Eliasson

75 Baked Goods Freshness: Technology, Evaluation, and Inhibition of

Staling, edited by Ronald E Hebeda and Henry F Zobel

76 Food Chemistry: Third Edition, edited by Owen R Fennema

77 Handbook of Food Analysis: Volumes 1 and 2, edited by Leo M L Nollet

78 Computerized Control Systems in the Food Industry, edited by Gauri

S Mittal

79 Techniques for Analyzing Food Aroma, edited by Ray Marsili

80 Food Proteins and Their Applications, edited by Srinivasan daran and Alain Paraf

Damo-81 Food Emulsions: Third Edition, Revised and Expanded, edited by Stig

E Friberg and K å re Larsson

82 Nonthermal Preservation of Foods, Gustavo V Barbosa-C á novas, Usha R Pothakamury, Enrique Palou, and Barry G Swanson

83 Milk and Dairy Product Technology, Edgar Spreer

84 Applied Dairy Microbiology, edited by Elmer H Marth and James L Steele

85 Lactic Acid Bacteria: Microbiology and Functional Aspects: Second

Edition, Revised and Expanded, edited by Seppo Salminen and Atte von Wright

86 Handbook of Vegetable Science and Technology: Production,

Composition, Storage, and Processing, edited by D K Salunkhe and

90 Dairy Technology: Principles of Milk Properties and Processes, P Walstra, T J Geurts, A Noomen, A Jellema, and M A J S van Boekel

91 Coloring of Food, Drugs, and Cosmetics, Gisbert Otterst ä tter

92 Listeria, Listeriosis, and Food Safety: Second Edition, Revised and Expanded, edited by Elliot T Ryser and Elmer H Marth

93 Complex Carbohydrates in Foods, edited by Susan Sungsoo Cho, Leon Prosky, and Mark Dreher

94 Handbook of Food Preservation, edited by M Shafiur Rahman

95 International Food Safety Handbook: Science, International

Regula-tion, and Control, edited by Kees van der Heijden, Maged Younes, Lawrence Fishbein, and Sanford Miller

96 Fatty Acids in Foods and Their Health Implications: Second Edition,

Revised and Expanded, edited by Ching Kuang Chow

97 Seafood Enzymes: Utilization and Influence on Postharvest Seafood

Quality, edited by Norman F Haard and Benjamin K Simpson

98 Safe Handling of Foods, edited by Jeffrey M Farber and Ewen C D Todd

99 Handbook of Cereal Science and Technology: Second Edition,

Re-vised and Expanded, edited by Karel Kulp and Joseph G Ponte, Jr.

100 Food Analysis by HPLC: Second Edition, Revised and Expanded,

edited by Leo M L Nollet

101 Surimi and Surimi Seafood, edited by Jae W Park

102 Drug Residues in Foods: Pharmacology, Food Safety, and Analysis,

Nickos A Botsoglou and Dimitrios J Fletouris

103 Seafood and Freshwater Toxins: Pharmacology, Physiology, and

Detection, edited by Luis M Botana

104 Handbook of Nutrition and Diet, Babasaheb B Desai

105 Nondestructive Food Evaluation: Techniques to Analyze Properties

and Quality, edited by Sundaram Gunasekaran

106 Green Tea: Health Benefits and Applications, Yukihiko Hara

107 Food Processing Operations Modeling: Design and Analysis, edited

110 Applied Dairy Microbiology: Second Edition, Revised and Expanded,

edited by Elmer H Marth and James L Steele

111 Transport Properties of Foods, George D Saravacos and Zacharias

B Maroulis

112 Alternative Sweeteners: Third Edition, Revised and Expanded, edited

by Lyn O ’ Brien Nabors

113 Handbook of Dietary Fiber, edited by Susan Sungsoo Cho and Mark

116 Food Additives: Second Edition, Revised and Expanded, edited by A Larry Branen, P Michael Davidson, Seppo Salminen, and John H Thorngate, III

117 Food Lipids: Chemistry, Nutrition, and Biotechnology: Second Edition,

Revised and Expanded, edited by Casimir C Akoh and David B Min

118 Food Protein Analysis: Quantitative Effects on Processing, R K Owusu-Apenten

119 Handbook of Food Toxicology, S S Deshpande

120 Food Plant Sanitation, edited by Y H Hui, Bernard L Bruinsma, J Richard Gorham, Wai-Kit Nip, Phillip S Tong, and Phil Ventresca

121 Physical Chemistry of Foods, Pieter Walstra

122 Handbook of Food Enzymology, edited by John R Whitaker, Alphons

G J Voragen, and Dominic W S Wong

123 Postharvest Physiology and Pathology of Vegetables: Second Edition,

Revised and Expanded, edited by Jerry A Bartz and Jeffrey K Brecht

124 Characterization of Cereals and Flours: Properties, Analysis, and

Ap-plications, edited by G ö n ü l Kaletun ç and Kenneth J Breslauer

125 International Handbook of Foodborne Pathogens, edited by Marianne

D Miliotis and Jeffrey W Bier

Additional Volumes in Preparation Handbook of Dough Fermentations, edited by Karel Kulp and Klaus Lorenz

Extraction Optimization in Food Engineering, edited by Constantina Tzia and George Liadakis

Physical Principles of Food Preservation: Second Edition, Revised

and Expanded, Marcus Karel and Daryl B Lund Handbook of Vegetable Preservation and Processing, edited by Y H Hui, Sue Ghazala, Dee M Graham, K D Murrell, and Wai-Kit Nip Food Process Design, Zacharias B Maroulis and George D Saravacos

Considering that our readership spans various disciplines and grams, our primary goal is to engage the audience with an array of topicsbased on fundamental engineering principles Because of this diversity, thebook begins with a brief review of the physical properties of food materialsand an introduction to modeling After that introduction, the book proceeds

pro-with discussions of applications involving basic to complex problemsencountered in processing and handling of food materials.

Each chapter ®rst addresses the theory behind the process, and thendiscusses a complex case study that demonstrates how to obtain the model,numerical formulation, and solution For each case study, the discussionexplains the thermophysical properties involved and takes into account themodeling complexity and any nonlinearity in the material properties of thesystem The many phenomena addressed include heat and mass transfer,

¯uid ¯ow, electromagnetics, and stochastic processes Operations discussed

in the course of the book are drying, microwave heating, infrared heating,frying, electric resistance heating, aseptic processing, the neural networkapproach to modeling and process control, and stochastic process modeling

of heat and mass transfer in food

Food Processing Operations Modeling applies a variety of theories tosolve practical problems relevant to research in and teaching of food processengineering Unfortunately, this book cannot offer a complete catalog ofmodeling for the numerous operations used in food processing However,working through the case studies provided, the reader will learn a concep-tual framework that will enable him or her to understand and solve diverseproblems that emerge in food processing operations

I wish to acknowledge my parents for their encouragement and support

in a variety of ways in making this book possible I would also like to expressspecial thanks to Mr Hong Yang for his help in preparing the index

Joseph Irudayaraj

Preface

1 Prediction Models for Thermophysical Properties of FoodsDennis R Heldman

2 Introduction to Modeling and Numerical Simulation

K P Sandeep and Joseph Irudayaraj

3 Aseptic Processing of Liquid and Particulate Foods

K P Sandeep and Virendra M Puri

4 Modeling Moisture Diffusion in Food Grains During AdsorptionKasiviswanathan Muthukumarappan and Sundaram Gunasekaran

5 Deep-Fat Frying of Foods

Rosana G Moreira

6 Mathematical Modeling of Microwave Processing of Foods:

An Overview

Ashim K Datta

7 Infrared Heating of Biological Materials

Oladiran O Fasina and Robert Thomas Tyler

8 Modeling Electrical Resistance (``Ohmic'') Heating of FoodsPeter J Fryer and Laurence J Davies

9 Stochastic Finite-Element Analysis of Thermal Food ProcessesBart M NicolãÈ, Nico Scheerlinck, Pieter Verboven, andJosse De Baerdemaeker

10 Neural Networks Approach to Modeling Food ProcessingOperations

Vinod K Jindal and Vikrant Chauhan

Peter J Fryer, Ph.D School of Chemical Engineering, University ofBirmingham, Birmingham, United Kingdom

Sundaram Gunasekaran, Ph.D Department of Biological Systems ineering, University of Wisconsin±Madison, Madison, Wisconsin

Eng-Dennis R Heldman, Ph.D Department of Food Science, Rutgers ± TheState University of New Jersey, New Brunswick, New Jersey

Joseph Irudayaraj, Ph.D Department of Agricultural and Biological ineering, The Pennsylvania State University, University Park, PennsylvaniaVinod K Jindal, Ph.D Processing Technology Program, Asian Institute ofTechnology, Bangkok, Thailand

Eng-Rosana G Moreira, Ph.D Department of Agricultural Engineering, TexasA&M University, College Station, Texas

Kasiviswanathan Muthukumarappan, Ph.D Department of Agriculturaland Biosystems Engineering, South Dakota State University, Brookings,South Dakota

Bart M NicolãÈ Department of Agricultural and Applied BiologicalSciences, Katholieke Universiteit Leuven, Leuven, Belgium

Virendra M Puri, Ph.D Department of Agricultural and BiologicalEngineering, The Pennsylvania State University, University Park,Pennsylvania

K P Sandeep, Ph.D Department of Food Science, North Carolina StateUniversity, Raleigh, North Carolina

Nico Scheerlinck, M.D Department of Agro-Engineering and -Economics,Katholieke Universiteit Leuven, Leuven, Belgium

Robert Thomas Tyler, Ph.D Department of Applied Microbiology andFood Science, University of Saskatchewan, Saskatoon, Canada

Pieter Verboven, Ph.D Department of Agro-Engineering and -Economics,Katholieke Universiteit Leuven, Leuven, Belgium

1 Prediction Models for Thermophysical Properties of Foods

Dennis R Heldman

Rutgers ± The State University of New Jersey, New Brunswick,New Jersey

1 INTRODUCTIONProperties of food and food ingredients are critical parameters in thedesign of a process used in the manufacturing of a food product.Although property magnitudes may be estimated based on published valuesfor similar materials, improvements in process ef®ciency and the design ofequipment used to perform the process, depend on more accurate propertymagnitudes Thermophysical properties are unique and in¯uence the design

of any thermal process; a food manufacturing process involving changes intemperature of the ingredients and product Thermophysical properties nor-mally include speci®c heat, density, and thermal conductivity Individually,these properties may have in¯uence on process evaluation and design.For example, speci®c heat and density are important components of ananalysis involving mass and/or energy balances Thermal conductivity isthe key property in the quanti®cation of thermal energy transfer within amaterial by conduction The combination of the three properties is thermaldiffusivity, a key property in the analysis of unsteady-state heat transfer

Thermophysical properties of food and food ingredients have beeninvestigated for several years, with much of the emphasis on the measure-ment of property magnitudes for many foods as a function of temperatureand composition Although these published properties have and continue tocontribute to the improved design of selected processes, the properties arenot in a format for ideal input to most process design situations All thermalprocesses involve changes in product temperature, and many involvechanges in product composition Most published thermophysical propertiesdata do not accommodate situation in which property magnitudes changeduring the process.

More recently, prediction models for thermophysical properties based

on product composition have evolved Many of these models are based

on property magnitudes for the basic compositional components offoods; proteins, fat, carbohydrates, ash, and water Knowledge of the prop-erties of these basic components as a function of temperature provides theopportunity to develop prediction models that will accommodate theneeds of process design models These thermophysical property modelsrepresent a signi®cant opportunity to improve the ef®ciency of thermalprocesses for food and the ultimate design of equipment used for processing

of foods

The overall objective of the information to be presented is to discussmodels for the prediction of thermophysical properties of food and foodingredients based on composition The following are more speci®c objec-tives:

To present and discuss models for the prediction of the speci®cheat of foods based on the composition of foods and food ingre-dients, with emphasis on the application of models to processdesign

To present and discuss models for the prediction of density offoods based on the composition of foods and food ingredients,with emphasis on the in¯uence of physical structure of the pro-duct

To present and discuss models for the prediction of the thermalconductivity of foods as a function of composition and tempera-ture, with emphasis on the use of models that incorporate thein¯uence of physical structure of the product

As suggested, the emphasis will be on models and steps needed to usethe models in process design Reference to the appropriate thermo-physical property data obtained from experimental measurements will beillustrated

2 PREDICTION OF SPECIFIC HEATThe speci®c heat of a food is de®ned as the quantity of thermal energyassociated with a unit mass of the food and a unit of change in temperature.This thermophysical property is often referred to as heat capacity and is anessential component of a thermal energy analysis on a food product, athermal process, or processing equipment used for heating or cooling of afood A prediction model provides the opportunity to conduct analyses overde®ned ranges of temperature and composition for a given process used for

a food product

The in¯uence of composition on the speci®c heat of foods is obvious inthe earliest of prediction models from Siebel [1]:

cpˆ 0:837 ‡ 0:034…moisture content; %† …1†This empirical expression is based on experimental data for high-moisturefoods, and it is anticipated that the coef®cients within the expression willvary with temperature Several similar models have summarized recently bySweat [2], An obvious dependence of the magnitude of the speci®c heat of ahigh-moisture food on moisture content is evident

A relationship expanding on the dependence of the speci®c heat of afood on composition was suggested by Leninger and Beverloo [3] as follows:

cpˆ …0:5Mf ‡ 0:3Ms‡ Mw†4:18 …2†This equation contains the mass fractions of fat (Mf), nonfat solids (Ms),and water (Mw) and references the speci®c heat of water (4.18 kJ/kg) at208C A very similar relationship was suggested by Charm [4]:

cpˆ 2:094Mf ‡ 1:256Ms‡ 4:187Mw …3†The Charm equation uses the speci®c heat of water at 758C and the co-ef®cients for the fat and nonfat solids are the same as Eq (2) when thetemperature is adjusted The magnitude for speci®c heat of fat is 2.094 kJ/

kg, when the fat is in a liquid state at 758C A value for solid fat would be1.675 kJ/kg (at the appropriate lower temperature) An additional dimen-sion of the dependence of speci®c heat on composition was suggested byHeldman and Singh [5]:

cpˆ 1:424Mc‡ 1:549Mp‡ 1:675Mf ‡ 0:837Ma‡ 4:187Mw …4†

In this expression, the coef®cients represent the speci®c heats of hydrates, proteins, fat, and ash at 208C or less As suggested earlier, themagnitude in the equation is for fat in the solid state

carbo-As is evident, the models presented up to this point are entirely cal or lack reference to the speci®c temperature for applications A very

empiri-general model for the prediction of speci®c heat of food was suggested byvan Beek [6]:

cpˆX…cpiMi† …5†The general model indicates that the speci®c heat of a food can be predictedfrom knowledge of the composition and the speci®c heat of each compo-nent More speci®cally, the predicted value is the summation of the product

of the mass fraction of component (i) and the speci®c heat of component (i).The successful use of the general model depends on two key inputs: Composition information on the food or food ingredient beingconsidered These types of data for an array of food productsare found in USDA Handbook No 8 [7] The composition of aningredient is likely to be established or measured, as would be thecase for new product formulations For processes where the com-position changes during the process, information may be limited,but models of the type being presented should encourage monitor-ing of these changes during a process

Data on the speci®c heat of the key compositional components(proteins, carbohydrates, fat, ash) of food It must be emphasizedthat the property magnitudes are for moisture-free components It

is recognized that foods contain many different types of proteins,carbohydrates, and fats Data published to date suggests thatdifferences in speci®c heat magnitudes for different proteins,carbohydrates, or fats are relatively small These differences may

be smaller than changes in speci®c heat magnitudes due to aphase change for the same component It is very important forthe speci®c heat data for these compositional components to

be available over a range of temperatures associated with typicalthermal processes for food To date, the best and most completedata were published by Choi and Okos [8]

The data presented by Choi and Okos [8] are based on an extensive studyand analysis of speci®c heat data for many liquid foods with different com-positions and generally over a temperature range of 20±1008C The resultsare summarized inTable 1

The best approach to illustrating the use of the general speci®c heatmodel is in the form of an exarnple The example will describe use of thegeneral model [Eq (5)] to predict changes in speci®e heat of the productduring a process in which both the product temperature and compositionare changing in a de®ned manner during the process

2.1 EXAMPLE

A liquid food, with a composition of 3.5% protein, 4.9% carbohydrate,3.9% fat, 0.7% ash, and 87% water, is heated from 208C to 1008C, andthe concentration of product solids increases to 30% during the process Theprocess requires 40 min when the heating medium temperature is 1058C Thechanges in concentration and temperature as a function of time (t) aredescribed by the following relationships:

%TS ˆ %TS0exp…0:021t†

where %TS is the percentage of total product solids, or total of protein,carbohydrate, fat, and ash within the product and expressed as a percentage,and

T TM

T0 TMˆ exp… 0:07t†

where T is temperature at any time (t), T0 is the initial product temperature(at t ˆ 0), and TM is the heating medium temperature Predict the speci®cheat of the product, as a function of time, during the process

SolutionThe results of the solution will be presented in the form of a tablewith predicted speci®c heat values at time increments during the process

TABLE1 Speci®c Heat Relationships for FoodProduct Components

Standard Standard

4:8008  10 6T2Carbohydrate cp ˆ 1:54884 ‡ 1:9625  10 3T 0.0986 5.96

‡ 9:9516  10 4T2Water …> 08C) cp ˆ 4:1762 9:0864  10 5T 0.0159 0.38

‡ 5:4731  10 6T2

Source: Ref 8.

The solution will include steps required for prediction of each speci®cmagnitude.

1 At t ˆ 0: The speci®c heat of each product component is puted at 208C

com-For protein,

cpˆ 2:0082 ‡ …1:2089  10 3†…20† …1:3129  10 6…2†2

ˆ 2:032 kJ=kgFor carbohydrate

cpˆ 1:5488 ‡ …1:9625  10 3…20† …5:9399  10 6…20†2

ˆ 1:587 kJ=kgFor fat,

cpˆ 1:9842 ‡ …1:4733  10 3…20† …4:8008  10 6…20†2

ˆ 2:012 kJ=kgFor ash,

cpˆ 1:0926 ‡ …1:8896  10 3…20† …3:6187  10 6…20†2

ˆ 1:129 kJ=kgFor water,

cpˆ 4:1762 ‡ …9:0864  10 5…20† ‡ …5:4731  10 6…20†2

ˆ 4:176 kJ=kgThe speci®c heat of the product at the beginning of the process is

At 40 min, the water content has decreased to 70% and themass fractions of all other components have increased Based onthe adjusted composition and temperature change, the speci®c

heat of the product becomes

The results of the example illustrate that the speci®c heat of theproduct decreases during the process as product temperature increasesand water content of the product decreases The predictions indicatethat the speci®c heat of product components increase with temperature.The in¯uence of this increase is smaller than the in¯uence associated withthe change in product composition As the concentration of productsolids increases, the amount of water in the product decreases Becausethe speci®c heat of the product solids is much lower than the speci®c heat ofwater, the higher mass fractions of the lower-speci®c-heat components result

in a lower speci®c heat of the product at the end of the process

3 PREDICTION OF DENSITYThere are only a limited number of models for predicting the density of afood product based on composition The suggestions by Heldman [9,10]illustrate the in¯uence of freezing on the density of a high-moisture food(Figure 1) These models are similar to the general model for density pre-diction as proposed by Choi and Okos [8]:

The use of Eq (6) is similar to speci®c heat and involves the use of productcomposition (Mi) for protein, fat, carbohydrate, ash, and water and thedensity (moisture-free) for each component (i) The density data to beused as inputs to the proposed model were published by Choi and Okos[8] These data are summarized inTable 3.

The proposed model predicts the bulk density of a high-moisturefood from typical composition information [7] and the density relationships

FIGURE1 In¯uence of phase change on the density of strawberries (From Ref 9.)

TABLE3 Density Relationships for FoodProduct Components

Standard Standard

Protein  ˆ 1:3299  103 5:184  10 1T 39.9501 3.07Carbohydrate  ˆ 1:59919  103 3:1046  10 1T 93.1249 5.98

presented in Table 3 The output density magnitudes can re¯ect changes incomposition and temperature as might occur during a food manufacturingprocess There would be a lower limit on the proposed application based onmoisture content The exact magnitude of this limiting moisture contentwould be product dependent and will be discussed in more detail whendiscussing the prediction of density of low-moisture foods.

The density of a dry food is directly dependent on the structure of theproduct, with a gas phase (air) having a signi®cant in¯uence on the magni-tude of the property magnitude This relationship to a gas phase depends onmany external factors (packing, pressure, etc.) and prevents the use of pre-diction models, before some type of reference measurement is accomplished.The following relationships provide the opportunity to predict the productdensity of a low-moisture food as a function of temperature and moisturecontent

3.1 Dry Particle FoodsMany dry foods are in the form of particles created by the manufacturingprocess For these types of product, the bulk density is dependent on particledensity, as well as the magnitude of void space around the particles Particledensity is the mass of the particle per unit of particle volume At a productmoisture content of zero, the particle is a two-phase system and can bedescribed in terms of volume fractions (ei) as follows:

es‡ eaˆ 1 …7†indicating that the particle volume is composed of product solids and air(gas phase) In addition, the particle density can be predicted by

pˆ ses‡ aea …8†with the density of product solids and air as inputs It should be noted thatthe density of product solids could be predicted from the relationship, based

on compositional components presented previously Equations (7) and (8)can be used to obtain the following:

esˆp a

indicating that the volume fraction of solids (and for air) can be determinedafter measurement of the particle density of product at a moisture content ofzero

Based on the concept proposed by Sarma and Heldman (11), the initialaddition of moisture to the low-moisture food results in the replacement ofair space within the particle, and the volume fraction of solids is constant.The concept is illustrated in Figure 2 As the increase in moisture content

continues, the volume fraction of water increases until the magnitude isequal to the volume fraction of air at moisture content equal to zero.Within the low-moisture-content range, the particle density increases line-arly with moisture content until the air space within the particle is replaced

by water The moisture content when the air space within the particle isreplaced by water can be de®ned as the critical moisture content When themoisture content of the product is increased above the critical moisturecontent, the particle density remains constant The linear increase in theparticle density and the maximum particle density at the critical moisturecontent has been illustrated for starch particles [11] A particle density of

1476 kg/m3 occurred at a dry-basis moisture content of 0.2 kg water/kg drysolids Later, Sabliov and Heldman [12] have shown that the magnitude forcasein particles was 1279 kg/m3 at 0.25 kg water/kg dry solids

Above the critical moisture content, the particle is a two-phasesystem; water and product solids An increase in moisture content causes

an expansion of particle volume Over a range of moisture contents (above20±25% dry basis), the particle density can be predicted from

pˆws…1 ‡ M†

w‡ Ms …10†

FIGURE2 Proposedrelationship of physical structure of foodparticles to ure content for the range from zero to greater than 0.6 kg water/kg dry solids.(From Ref 11.)

moist-where M is the dry-basis moisture content Even though the particle volumeexpands with increasing moisture content, the volume fraction of water (ascompared to the volume fraction of solids) increases The moisture contentwhen the two volume fractions are equal (at 0.5) can be predicted as illu-strated by Sabliov and Heldman [12] inFigure 3.

When considering the prediction of bulk density, the total mass andvolume of the product must be considered By considering the product as atwo-phase system [a particle phase and an air (gas) phase], the followingrelationship would apply:

Ep‡ Eaˆ 1 …11†where the volume fraction for the particle (Ep) and the volume fraction ofair (Ea) are the total volume of product Because the bulk density can bede®ned as

bˆ Epp‡ Eaa …12†the volume fraction of the particle becomes

Epˆb a

These relationships should be applied to nonparticulate dry foods by nizing the product solids phase would replace the particle phase in theparticle system In these situations, the volume phase of product solids

recog-FIGURE3 In¯uence of moisture content on volume fractions of water duct solids within a food particle (From Ref 12.)

andpro-would be predicted by the previous equation It should be noted that thesolid phase for the nonparticle food would be composed of dry productsolids and water.

For both of the previous situations, the bulk density of the dry foodwould be predicted as a function of moisture content and temperature using

bˆ Ep…p a† ‡ a …14†For such predictions, the volume fraction of particles (or solids) would bebased on the measurement of bulk density at the critical moisture content

At moisture comments above the critical moisture content (20±25% db), therelationship would use inputs from the previous particle analysis to obtainthe volume fraction of particles (Ep) and the particle density (p) as a func-tion of moisture content and temperature The expression for volume frac-tion can be used to predict the volume fraction of the particle (or solid) andfor air, as a function of moisture content As moisture content increases, thevolume fraction of the particle increases and the volume fraction of airdecreases At a de®ned moisture content or bulk density, the two fractionsbecome equal at 0.5 A typical relationship for the bulk fraction of starchhas been presented by Sabliov and Heldman [12], as illustrated inFigure 4

An additional analysis of the relationship between the volume fraction

of particles (or solids) and moisture content indicates that the volume

FIGURE4 In¯uence of bulk density on volume fractions of air and productparticles within a foodsystem (From Ref 12)

fraction of air decreases to zero at a relatively high moisture content At thisupper limit of the moisture content of the product, the product becomes atwo-phase system: product solids and water Although the exact signi®cance

of this upper-limit moisture content, it would seem appropriate to use thegeneral prediction models for foods above the upper-limit moisture content

An analysis of starch particles has been completed by Sabliov and Heldman[12] and indicates that the dry-basis moisture content at the upper limit isapproximately 1.5 kg water/kg dry solids It is anticipated that the changes

in the various parameters would be modi®ed slightly when the moisturecontent is decreased

Based on the observations presented, six moisture content ranges can

be de®ned The approach to the prediction of the bulk density of the foodproduct would vary with moisture content range These ranges may bedescribed in the following manner:

1 Very low moisture content: less than 10% db, or where thevolume fraction of water equals volume fraction of air, withinthe particle

2 Low moisture content: approximately 10% to 20% db, or themoisture content when the volume fraction of air within the par-ticle becomes zero

3 Low intermediate moisture content: 20% to 65% db, or themoisture content when the volume fraction of water exceeds thevolume fraction of product solids within the particle

4 Medium intermediate moisture content: 65% to 90% db, or themoisture content when the volume fraction of particles exceeds thevolume fraction of air within the product system

5 High intermediate moisture content: 90% to 135% db, or themoisture context when the volume fraction of air become zero

6 High moisture content: above 135% db, or the moisture contentwhen the volume fraction of air is zero

The moisture content ranges identi®ed are based on studies of starch andcasein and will vary with food product and composition, as well as theprocess used to manufacture the product In addition, the bulk density is

a function of many factors associated with handling and packaging of thedry product

4 PREDICTION OF THERMAL CONDUCTIVITYThermal conductivity is a basic thermophysical property of any material,the magnitude expressing the rate of thermal energy transfer within thematerial Because the complexity of the physical structure of food, the

prediction models for the thermal conductivity of food must account forcharacteristics of the physical structure The development of predictionmodels for thermal conductivity of food should assist in the improvement ofprocess design and the equipment used to accomplish the process Becausetemperature and moisture content have a signi®cant in¯uence on thermalconductivity, the magnitude of the property changes signi®cantly during themany processes used for the manufacturing of foods.

Historically, the models for the prediction of the thermal conductivity

of foods have been empirical and based on experimental data One of theearliest models was proposed by Riedel [13]:

k ˆ …326:58 ‡ 1:0412T ‡ 0:00337T2†…0:46 ‡ 0:54Mw†…1:73  10 3†

…15†This empirical expression is based on experimental data for fruit juices,sugar solutions and milk over a temperature range from 08C to 1808C.Additional empirical models have evolved and have been summarized

by Sweat [2] These expressions include a model for the prediction of thethermal conductivity for fruits and vegetables, with water content above60% [14]:

k ˆ 0:148 ‡ 0:439Mw …16†Later, Sweat [15] suggested a similar type of empirical model for meat, withmoisture content between 60% and 80% (wet basis) and temperaturesbetween 08C and 608C:

k ˆ 0:08 ‡ 0:52Mw …17†These empirical models do not account for the in¯uence of thermal con-ductivity differences among the other compositional components in foods.Based on a detailed statistical analysis of thermal conductivity data forliquid foods, Choi and Okos [16] proposed the following prediction model:

k ˆ 0:2051Mc‡ 0:2Mp‡ 0:175Mf ‡ 0:135Ma‡ 0:61Mw …18†where the coef®cients for each of the compositional components wouldrepresent the thermal conductivity of that component A similar analysiswas completed by Sweat [2] to create the following model:

k ˆ 0:25Mc‡ 0:155Mp‡ 0:161mf ‡ 0:135Ma‡ 0:58Mw …19†

As is evident, the coef®cients for these models are similar, but the outputfrom the model is in¯uenced by the input data for the analysis The pre-diction accounts for the temperature used during the collection of the

experimental thermal conductivity A more general model was proposedChoi and Okos [8]:

k ˆXkiEi …20†where the volume fraction (Ei) is estimated for each compositional compo-nent by

EiˆXMi=i

…Mi=i† …21†The general model uses compositional data from measurements or theUSDA Handbook No 8 [7] and the thermal conductivity inputs from therelationships presented inTable 4

The general prediction model is adequate for the prediction of thethermal conductivity of food as a function of temperature and productcomposition The limits on the use of the model are moisture content andphysical structure of the product Because the experimental data used togenerate the relationships in Table 4 were for high moisture foods, a lowerlimit on moisture content for the application of the relationships must beconsidered Alternately, the best applications of the general model would

be high-moisture foods, where water is the predominant component orcontinuous phase within the product Physical structure is not considered

in the general model and the in¯uence of the orientation of product

com-TABLE4 Thermal Conductivity Relationships for Components of Food

Standard Standard

Protein k ˆ 1:7881  10 1‡ 1:1958  10 3T 0.012 5.91

2:7178  10 6T2Carbohydrate k ˆ 2:0141  10 1‡ 1:3874  10 3T 0.0134 5.42

4:3312  10 6T2

1:7749  10 7T2Ash k ˆ 3:2962  10 1‡ 1:4011  10 3T 0.0083 2.15

2:9069  10 6T2Water k ˆ 5:7109  10 1‡ 1:7625  10 3T 0.0028 0.45

6:7036  10 6T 2

‡ 1:0154  10 4T2

Source: Ref 8.

ponents on the magnitude of the thermal conductivity is not a part of themodel.

The in¯uence of physical structure on the thermal conductivity diction has been considered in several different models These modelshave been reviewed by Sweat [2]; the ®rst type of model is for thermalconductivity within an isotropic system with a discontinuous componenthomogeneously dispersed within a continuous second component of theproduct The model derived and proposed by Kopelman [17] for the situa-tion when the thermal conductivity magnitude of the continuous component

pre-is similar to the magnitude for the dpre-iscontinuous component pre-is

k ˆ kc 1 Ed2=3

1 E2=3d …1 Ed1=3†

!

…23†

The application of this second model to foods is limited

In many foods, the discontinuous component has a de®ned orientationwithin the continuous component For this situation, Sweat [2] suggested thefollowing simple model, when heat transfer is parallel to the orientation ofthe compositional components:

k ˆ Eckc‡ Edkd …24†The more complex model for the anisotropic model was derived and pre-sented by Kopelman [17] as follows:

Kopelman [17] derived and presented the following model for heat tion perpendicular to the orientation of compositional components:

conduc-k ˆ conduc-kc 1 Ed1=2‰1 …kd=kc†Š

1 E1=2d ‰1 …kd=kc†Š…1 Ed1=2†

!

…27†

The use of the models to account for the in¯uence of component orientation

on thermal conductivity is illustrated in Figure 5, where the in¯uence ofphase change and muscle ®bers in beef are evident The in¯uence of frozenwater on the thermal conductivity of beef is dramatic The thermal conduc-tivity prediction for heat transfer parallel to the direction of muscle ®bers ishigher than for heat transfer perpendicular to the ®bers and is consistentwith experimental data

Kopelman [17] derived and described a third model for prediction ofthermal conductivity within an anisotropic, two-component system, wherethe two components are in layers within the system The physical structuresfor all three of the models developed by Kopelman [7] are illustrated in

Figure 6 For heat transfer parallel to the two component layers,

The corresponding expression for heat transfer perpendicular to the ponent layers is

and the thermal conductivities of the two components in the food system.The importance of de®ning the discontinuous phase is evident in all of themodels Application of the models to food systems requires careful de®ni-tion of the components within the product It is appropriate to refer toproperty magnitudes for food products as ``effective thermal conductivities''

to differentiate the property from the magnitudes for the components of thefood

The various models for prediction of effective thermal conductivitymust be used in combination with the general model [Eq (20)] to adequatelydescribe heat transfer in typical food products The key factors that differ-entiate among food products are physical structure and composition Thesefactors, in addition to temperature, are essential to predicting the changes ineffective thermal conductivity of the product during a typical manufacturingprocess The following steps have been demonstrated far the prediction ofthe thermal conductivity of food products at various moisture contents andtemperatures

A general component in application of the prediction models for tive thermal conductivity is the general model [Eq (20)], in combinationwith the relationships in Table 4 These relationships are used to predict thethermal conductivity of the product solids component within the food struc-ture Overall, the in¯uence of the product solids on the effective thermalconductivity of the food system is represented by the general model I andthe relationships in Table 4 The in¯uence of other components (water and/

effec-or air) is described by the various models inceffec-orpeffec-orating physical structure.The moisture content has the most signi®cant in¯uence on the use of thevarious models

4.1 Very-Low-Moisture Foods

At moisture contents below the critical moisture content, as de®ned in cussions related to density prediction, the primary component in¯uencingeffective thermal conductivity of the product is the gas phase or air Withinthis moisture content range, food particles will contain three components(product solids, water, air) Within the lower portion of the moisture con-tent range [0 to 10% (db)], air will be the continuous component and thecombination of water and product solids will be the discontinuous compo-nent The homogeneous physical structure model [Eq (22)] can be used topredict the thermal conductivity of the particle The next step would be theprediction of the effective thermal conductivity using the same model, withair as the continuous component and product particles as the discontinuouscomponent

dis-4.2 Low-Moisture Foods

At the higher-moisture-content end of the range below the critical moisturecontent, the volume fraction of water within the particle exceeds the volumefraction of air within the particle It follows that the homogeneous model[Eq (22)] is used to predict the thermal conductivity of particles, with theparticles {product solids and water) as the continuous component The ®nalstep in the prediction of the effective thermal conductivity of the product isthat it would be the same as for the very low-moisture food

4.3 Low-Intermediate-Moisture Foods

As indicated during the analysis of density of dry particle foods, the particlebecomes a two-component (product solids and water) system in the moist-ure content range immediately above the critical moisture content[20% (db)] Within the lower portion of this moisture content range, thethermal conductivity of the particle is predicted by the homogeneous model[Eq (22)] The product solids are the continuous component and water is thediscontinuous component The effective thermal conductivity of the product

is predicted using the homogeneous model (or appropriate model, based onthe orientation of components) with air as the continuous component andproduct particles as the discontinuous component

4.4 Medium-Intermediate-Moisture Foods

As the moisture content increases, the volume fraction of water within theparticle exceeds the volume fraction of the product solids The point at whichthe volume fractions are equal has been de®ned as the particle transitionmoisture content At moisture contents higher than the particle transitionmoisture content, the thermal conductivity of the particle is predicted by thehomogeneous model [Eq (22)], with water as the continuous componentand product solids as the discontinuous component The effective thermalconductivity of the product is predicted using the homogeneous model (orappropriate alternative model), with air as the continuous component andproduct particles as the discontinuous component The upper limit of thismoisture content range occurs when the volume fraction of particles exceedsthe volume fraction of air within the product system This point has beende®ned as the transition bulk density

4.5 High-Intermediate-Moisture Foods

At moisture contents above the transition bulk density, when the volumefraction of particle exceeds the volume fraction of air in the product, thethermal conductivity of the particle is predicted by using the homogeneous

model Within the model, the volume fraction of water becomes the tinuous component and product solids are the discontinuous component Asthe moisture content increases, the volume fraction of particle increases andthe volume fraction of air decreases Within this moisture content range, theeffective thermal conductivity of the product is predicted by using theappropriate model, with product particles as the continuous componentand air as the discontinuous component The upper limit of this moisturecontent range occurs when the volume fraction of air becomes zero.4.6 High-Moisture Foods

con-At moisture contents and bulk densities when the volume fraction of air iszero, the product becomes a two-component system of water and productsolids The model for the prediction of thermal conductivity depends on theorientation of product solids within the water component The effectivethermal conductivity is predicted by using water as the continuous compo-nent and the product solids as the discontinuous component in the appro-priate model The thermal conductivity of the product solids would bepredicted from the composition of protein, carbohydrate, fat, and ash andthe appropriate relationship for each compositional component

4.7 SummaryThe six moisture content ranges clearly illustrate the relationships betweendensity of product and the appropriate models for the prediction of theeffective thermal conductivity of the food product The results inFigure 7

illustrate the agreement between the predicted effective thermal conductivityand experimental measurements for a signi®cant range of moisture contents.The experimental veri®cation of the proposed models was accomplished bySabliov and Heldman [19] During these investigations, the bulk density wasmeasured as a part of the experimental determination of thermal conduc-tivity This experimental bulk density was used as an input to the predictionmodels Although the application of the models have been for situationswhen moisture content is increasing, it is proposed that the same modelswould be used for decreasing moisture content The differences in applica-tion would be created by differences in the physical structure of the productoccurring during moisture removal from the product

5 SUMMARYThe prediction of speci®c heat, density, and thermal conductivity offoods and food ingredients is accomplished based on composition anduse of appropriate models to account for property magnitudes for each

component, temperature, and physical structure The models for the tion of speci®c heat are based entirely on composition and require knowl-edge of the speci®c heat of the moisture-free components of the food, as well

predic-as the composition

The prediction models for density of a food product are equally forward at high-moisture contents When considering lower-moisture con-tents when a gas phase or air is introduced into the product structure, theprediction models used must accommodate the in¯uence of the low-densityair on the product density When the dry product structure includes particles,the prediction models incorporate the prediction of particle density.The models for prediction of thermal conductivity of a food productare the most complex, due to the need to consider the in¯uence of physicalstructure at all moisture contents In general, the prediction models considerthe in¯uence of the orientation of the components on heat conductionwithin the product structure At lower moisture contents when air becomes

straight-a component of the product structure, the prediction models straight-are developed

to accommodate the in¯uence of a thermal conductivity component At lowmoisture contents, the distribution of water and air within the physicalstructure of the product requires an understanding of the components repre-senting the continuous and discontinuous components of the food productsystem

FIGURE7 Experimental veri®cation of the predicted effective thermal conductivity

of casein at 258C, over intermediate moisture content ranges (From Ref 19.)

1 JE Siebel Speci®c heat of various products Ice Refrig 2:256, 1892

2 VE Sweat Thermal conductivity of foods In: MA Rao, SSH Rizvi, eds.Engineering Properties of Foods, 2nd ed New York: Marcel Dekker, 1995,

6 G van Beek, Vade mecum, Koeltechnik-Klimaatregeling 3.1.1±3.3.4, 1976

7 BK Watt, AL Merrill Compositian of Foods Agriculture Handbook No 8.Washington, DC: US Department of Agriculture, 1975

8 YChoi, MR Okos Effects of temperature and composition on the thermalproperties of foods In: M LaMaguer, P Jelen, eds Food Engineering andProcess Applications, Vol 1, Transport Phenomenon, New York: Elsevier,1986

9 DR Heldman Food properties during freezing Food Technol 36(2):92, 1982

10 DR Heldman Food Freezing In: DR Heldman, DB Lund, eds Handbook ofFood Engineering, New York: Marcel Dekker, 1992, pp 277±315

11 SC Sarma, DR Heldman An improved approach to prediction of thermalconductivity of a granular food Unpublished Paper University of Missouri,Columbia, 1996

12 CS Sabliov, DR Heldman Factors in¯uencing thermal conductivity of a foodover the range of moisture content from 20 to 120% (db) J Food Sci (in press)

13 L Riedel Measurement of the thermal conductivity of sugar solutions fruitjuices and milk Chem Ing-Tech 21:340±341, 1949

14 VE Sweat Experimental values of thermal conductivity of selected fruits andvegetables J Food Sci 39:1980±1083, 1974

15 VE Sweat Modeling the thermal conductivity of meats ASAE Trans 18(3):564±568, 1975

16 YChoi, MR Okos The thermal properties of tomato juice concentrate ASAETrans 26(1):305±311, 1983

17 IJ Kopelman Transient heat transfer and thermal properties in food system.PhD dissertation, Michigan State University, East Lansing, 1966

18 DR Heldman, DP Gorby Prediction of thermal conductivity in frozen food.ASAE Trans 18:156, 1975

19 SC Sabilov, DR Heldman A model for prediction of thermal conductivity offood as a function of moisture content and temperature J Food Process Engr(in press)

2 Introduction to Modeling and Numerical Simulation

on the safety and quality of a process Parametric analyses can be conducted

to understand the relative effects of different parameters

The use of approximate methods to solve problems described by partialdifferential equations has been employed for various reasons, including butnot limited to the lack of availability of analytical solutions or empiricalcorrelations, simplicity of solution technique, ability to quickly performparametric analyses, and because it serves as a means for quickly honing

in on the range of parameters to be used in experimental studies or fordesign purposes

There are three main categories into which mathematical modelingfalls: differential method, integral method, and stochastic method The

®nite-difference method falls under the category of differential method.Under the integral method, we have the variational method, ®nite-volumemethod, and method of weighted residuals The method of weightedresiduals can be further divided into four categories: collocation method,subdomain method, Galerkin's method, and least squares method; the

®nite-volume (or control-volume) method can be categorized into twogroups: cell-centered schemes and nodal point schemes The variationalmethod and the method of weighted residuals form the basis for the

®nite-element method The boundary element method is a subset of the

®nite-element method in that it uses a similar approach, but for the surface

or boundary under consideration It can be used in conjunction with the

®nite-element or ®nite-volume method The Monte Carlo method fallsunder the stochastic method This is a computationally intensive and prob-abilistic method used primarily when the number of independent variablesare large

The ®nite-element method and the ®nite-difference methods are themost popular techniques used to solve problems associated with food pro-cessing Relatively simple problems can be tackled with ease by commer-cially available software Complicated problems require either modi®cation

of commercial codes or writing the code from scratch

2 CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONSPartial differential equations (PDEs) are classi®ed as linear or nonlineardepending on whether there is a product of two terms containing eitherthe dependent variable or its derivatives If a PDE is linear in its highest-order derivative, but nonlinear in one or more of the lower-orderderivatives, it is called a quasilinear PDE The order of a PDE is the highestpower of the derivative in the equation

Consider the following second-order PDE:

A@2

@x2 ‡ B@x @y@2 ‡ C@2

@y2 ‡ D@@x‡ E @@y ‡ F ‡ G ˆ 0The coef®cients A, B, C, D, E, F, and G can be functions of x, y, or .The above PDE is said to be elliptic if B2 4AC < 0, parabolic if

B2 4AC ˆ 0, and hyperbolic if B2 4AC > 0 at all points in the domain.Auxiliary variables are usually introduced to convert second-orderPDEs to ®rst-order PDEs, at least for the purpose of classi®cation Thisformulation may then be used for solving the system of equations too

A PDE is said to be in conservative form (or conservation form

or conservation-law form or divergence form) if the coef®cients of all thederivative terms in the equation are either constant or, if variable, theirderivatives do not appear anywhere in the equation The schemes that main-tain the discretized version of the conservation statement exactly (except forround-off errors) for any grid size over any region in the domain for anynumber of grid points is said to have the conservative property

The nonconservative form of the continuity equation is

@u@x‡ @v@y‡ u @@x‡ v@@yˆ 0The conservative form of the same equation is

@

@x…u† ‡

@

@y…v† ˆ 0 or r
…V† ˆ 0Equilibrium problems (or jury problems) are problems for which thesolution of the PDE is required in a closed domain for a given set ofboundary conditions Equilibrium problems are boundary-value problemsand are governed by elliptic PDEs

Marching (or propagation) problems are transient or appear to betransient problems and the solution of the PDE is required in an opendomain for a given set of initial and boundary conditions Problems inthis category are either initial-value or initial-boundary-value problems.Marching problems are governed by hyperbolic or parabolic PDEs

3 NUMERICAL FORMULATIONNumerical formulations are based on the classi®cation of the governingequation When dealing with the non-steady-state heat equation or thescalar (linear or nonlinear) Burger's equation, formulations applicable

to parabolic equations are used When dealing with the wave equation, mulations for hyperbolic equations are used, and when dealing with Laplace'sequation, formulations for elliptic equations are used Formulations for alltypes of equations can be explicit or implicit Explicit formulations are simple,but the number of computations and the instability of the formulation(addressed in the next section) are some of its drawbacks

for-The Navier±Stokes equations are hyperbolic in the inviscid regionand parabolic in the viscous region For steady-state conditions, they arehyperbolic in the inviscid region and elliptic in the viscous region The scalarequations which are similar to the Navier±Stokes equations, are the Burger'sequations (linear and nonlinear) Thus, the starting point for solving the