Transport Phenomena in Food Processing

Transport Phenomena in Food Processing

2003 by CRC Press LLC

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No claim to original U.S. Government worksInternational Standard Book Number 0-56676-993-0

Library of Congress Card Number 2002073736Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

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Library of Congress Cataloging-in-Publication Data

Transport phenomena in food processing / edited by Jorge Welti-Chanes, Jorge F. Vlez-Ruiz,Gustavo V. Barbosa-Cnovas

p. cm. (Food preservation technology series)Includes bibliographical references and index.ISBN 1-56676-993-0 (alk. paper)1. FoodEffect of heat on. 2. Food industry and trade. I. Welti-Chanes, Jorge. II. Vlez-

Ruiz, Jorge F. III. Barbosa-Cnovas, Gustavo V. IV. Series.

TP371.2.T73 2002664dc21 2002073736


Series Preface

Transport phenomena is one of food process engineerings most important pillars,and this new addition to the CRC Food Preservation Technology Series offers asound combination of the fundamental and applied aspects of this classic engineeringtopic. Transport Phenomena in Food Processing plays an important role in the seriesbecause it offers the tools for quantifying many important operations, classic ornovel, using state-of-the-art calculation approaches.

I am particularly impressed with the analytical depth of the chapters and thewide spectrum of covered topics. For me, this is a very inspiring book that hopefullywill become a key reference. This book is also important because it is the first inthe CRC series to cover fundamental aspects of food processing. In short, I anticipatethat this book will be in good company with upcoming books in the series on otherfundamental aspects of food preservation.

Gustavo V. Barbosa-Cnovas


Preface

The latest applications of new and improved traditional food preservation processeshave generated the need for increased knowledge of the phenomenological andengineering principles that are the basis of the correct application of factors thatproduce stability and maintain the quality of transformed and processed products.This need for knowledge has given the field of food engineering a new identity atboth the research and industrial levels. Understanding the transport phenomena thatgovern the engineering analysis and design of food preservation processes is a keyelement in improving processing conditions and the employment of energy resources,and to increasing the quality of the product.

This book presents the state of the art in the transport phenomena area as appliedto food preservation and transformation. It is divided into four sections containinga total of 33 chapters, each written by prestigious scientists from institutions anduniversities around the world. The first section reviews the fundamental concepts ofmass, heat, and momentum transfer, while the remaining three sections discussspecific applications for a large variety of processes and products where the pre-dominant transfer phenomenon is mass or heat, or processes employing more thanone transport mechanism.

The mass transfer section focuses on phenomena controlling osmotic dehy-dration and hot-air drying processes. However, the themes related to water transferin superficial films placed on foods, and pre-evaporation, ultrasound, and spinningcone columns are also included. The seven chapters that constitute the heat transfersection study the effects of product shape and process equipment on the phenome-nons efficiency. The chapters in the last section deal with the study of the combinationof two or three transfer phenomena in frying, sterilization, and drying processes.

We trust this book on transport phenomena will make a meaningful contributionin facilitating the understanding, design, and implementation of food processing unitoperations that will result in the production of safer, higher quality, and moreconvenient foods.

Jorge Welti-ChanesJorge Vlez-Ruiz

Gustavo V. Barbosa-Cnovas


About the Editors

Jorge Welti-Chanes is a Professor in the Departments of Chemical and FoodEngineering and of Chemistry and Biology at the Universidad de las AmericasPuebla in Mexico. He earned his B.S. degree in Biochemical Engineering and hisM.S. in Food Engineering from the ITESM, Mexico and his Ph.D. from theUniversidad de Valencia, Spain. Dr. Welti-Chanes was president of the InternationalAssociation of Engineering and Food and is a member of the editorial boards of fiveinternational journals. He is the author and co-author of more than 80 scientificpapers and the editor of five books with the food technology, water activity, drying,emerging technologies, and minimal processing areas.

Dr. Jorge F. Vlez-Ruiz was born in Puebla City, Mexico. He received a B.S.in Biochemical Engineering in 1977 and an M.S. in Food Science in 1981, both fromthe Instituto Tecnolgico y de Estudios Superiores de Monterrey (ITESM), Mexico,and a Ph.D. in Food Engineering from Washington State University, Pullman in1993. He began his professional career working in the food industry and in 1979joined the Food Science Department, ITESM. Since 1980, he has been with theDepartment of Chemical and Food Engineering, University of the Amricas, Puebla(UDLA,P), serving as chairman from 1987 to 1990. In 1990, he was recognized asFood Engineering Researcher by the National System of Researchers in Mexico andin 1999 was named titular professor.

Dr. Vlez-Ruizs research activities are focused on rheology of foods; dairyproducts and milk processing; evaporation and dehydration of fluid foods; osmoticconcentration of fruits; heat and mass transfer through the frying process; andphysical properties of foods. He is the author of approximately 45 scientific publi-cations in international journals, more than 90 presentations at national and inter-national professional meetings, and three book chapters, and he is an editor of threefood science and engineering journals.

Gustavo V. Barbosa-Cnovas earned his B.S. in Mechanical Engineering at theUniversity of Uruguay and his M.S. and Ph.D. in Food Engineering at the Universityof MassachusettsAmherst. He then worked as an Assistant Professor at theUniversity of Puerto Rico from 19851990. Next, he went to Washington StateUniversity (WSU), where he is now a Professor of Food Engineering and Directorof the Center for Nonthermal Processing of Food (CNPF). His current research areasare nonthermal processing of foods, physical properties of foods, and food powdertechnology.


Acknowledgments

The editors would like to acknowledge each one of the researchers who kindlyagreed to participate in this project with their contributions.

The support of the Universidad de las Amricas, Puebla, of Washington StateUniversity, and of Texas Christian University is also acknowledged.

For the manuscripts revision and correction process, the editors counted on thevaluable work of M.S. Reyna Len and M.S. Daniela Bermdez, who were supportedby Ing. Luz del Carmen Lpez, to help the complete book. We express our gratitudeto them.


Contributors

A.G. Abdul-GhaniFood Science and Process Engineering

GroupDepartment of Chemical and Materials

EngineeringThe University of AucklandAuckland, New Zealand

E.A.A. AdellDepartamento de Engenharia Qumica

e AlimentosEscola de Engenharia MauInstituto Mau de Tecnologia Praa

MauSo Paulo, Brazil

S.M. AlzamoraDepartamento de IndustriasFacultad de Ciencias Exactas y

NaturalesUniversidad de Buenos AiresBuenos Aires, Argentina

A. AndrsDepartment of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

A. AngersbachDepartment of Food Biotechnology and

Food Process EngineeringBerlin University of TechnologyBerlin, Germany

J. Arul Department of Food Science and

Nutrition and Horticulture Research Center

Laval UniversitySainte-Foy, Quebec, Canada

P.M. AzoubelFaculdade de Engenharia de

AlimentosUNICAMPCampinas, So Paulo, Brazil

E. Azuara-NietoInstituto de Ciencias BsicasUniversidad VeracruzanaXalapa, Veracruz, Mxico

M.O. BalabanFood Science and Human NutritionUniversity of FloridaGainesville, Florida

J. BaratDepartment of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

G.V. Barbosa-CnovasDepartment of Biological Systems

EngineeringWashington State UniversityPullman, Washington

A.F. BaroniDepartamento de Engenharia Qumica

e AlimentosEscola de Engenharia MauInstituto Mau de Tecnologia Praa

MauSo Paulo, Brazil

F.H. BarronDepartment of Packaging ScienceClemson UniversityClemson, South Carolina


J. BeneditoFood Technology DepartmentUniversidad Politcnica de ValenciaValencia, Spain

C.I. Beristain-GuevaraInstituto de Ciencias BsicasUniversidad VeracruzanaXalapa, Veracruz, Mxico

J.M. BunnDepartment of Packaging ScienceClemson UniversityClemson, South Carolina

J. CrcelDepartment of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

F. Castaigne Department of Food Science and

Nutrition and Horticulture Research Center

Laval UniversitySainte-Foy, Quebec, Canada

M.A. CastroDepartamento de Ciencias BiolgicasFacultad de Ciencias Exactas y

NaturalesUniversidad de Buenos Aires Buenos Aires, Argentina

K.V. ChauBiological and Agricultural EngineeringUniversity of FloridaGainesville, Florida

X.D. ChenFood Science and Process Engineering



M.S. ChinnanCenter for Food Safety and Quality

EnhancementDepartment of Food Science and

TechnologyUniversity of Georgia, Griffin

CampusGriffin, Georgia

A. ChiraltDepartment of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

P. CoronelDepartment of Food ScienceNorth Carolina State UniversityRaleigh, North Carolina

J.G. CrespoDepartment of ChemistryFaculdade de Cincias e TecnologaUniversidade Nova de LisboaCaparica, Portugal

S.L. CuppettDepartment of Food Science and

TechnologyUniversity of Nebraska-LincolnLincoln, Nebraska

F. ErdogduDepartment of Food EngineeringUniversity of MersinCiftlikkoy, Mersin, Turkey

M.M. FaridFood Science and Process Engineering




H. FengDepartment of Biological Systems

EngineeringWashington State University Pullman, Washington

P. FitoDepartment of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

A.L. GabasDepartamento de Engenharia de

AlimentosUniversidade Estadual de Campinas Campinas, So Paulo, Brazil

M.A. Garcia-AlvaradoDepartamento de Ingeniera Qumica y

BioqumicaInstituto Tecnolgico de VeracruzVeracruz, Mxico

C. Gonzlez-Martnez Department of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

S. GrabowskiFood Research and Development

CentreAgriculture and Agri-Food CanadaSt. Hyacinthe, Quebec, Canada

G.F. Gutirrez-LpezDepartamento de Graduados e

Investigacin en AlimentosEscuela Nacional de Ciencias

Biolgicas I.P.N.Mxico, Mxico

F. HamouzDepartment of Nutritional Science and

DieteticsUniversity of Nebraska-LincolnLincoln, Nebraska

J.A. Hernndez-Prez Joint Research Unit Food Process

EngineeringCemagref, ENSIA, INAPG, INRAMassy, France

B. HeydJoint Research Unit Food Process


M.V. KarweDepartment of Food ScienceRutgers UniversityNew Brunswick, New Jersey

D. KnorrDepartment of Food Biotechnology and

Food Process EngineeringBerlin University of TechnologyBerlin, Germany

H. KrishnamurthyDepartment of Food Science and

Agricultural ChemistryMcGill University, Macdonald

CampusSte. Anne de Bellevue, Quebec,

Canada

R. Len-CruzDepartamento de Ingeniera Qumica y

AlimentosUniversidad de las Amricas-PueblaSanta Catarina MrtirCholula, Puebla, Mxico

M. MarcotteFood Research and Development

CentreAgriculture and Agri-Food CanadaSt. Hyacinthe, Quebec, Canada


J. Martnez-Monz Department of Food TechnologyUniversidad Politcnica de ValenciaValencia, Spain

F.C. MenegalliDepartamento de Engenharia de


M.R. MenezesDepartamento de Engenharia de


F.J. Molina-CorralGraduate Program in Food Science and

TechnologyUniversity of ChihuahuaChihuahua, Mxico

H. Mjica-PazFacultad de Ciencias QumicasUniversidad Autnoma de ChihuahuaChihuahua, Mxico

A. MuletFood Technology DepartmentUniversidad Politcnica de ValenciaValencia, Spain

F.E.X. MurrFaculdade de Engenharia de Alimentos-

UNICAMPCampinas, So Paulo, Brazil

A. NietoDepartamento de IndustriasFacultad de Ciencias Exactas y

NaturalesUniversidad de Buenos AiresBuenos Aires, Argentina

K. NiranjanSchool of Food BiosciencesThe University of ReadingReading, Berkshire, U.K.

N. NitinDepartment of Food ScienceRutgers UniversityNew Brunswick, New Jersey

R. Olivas-Vargas Advanced Materials Research Center Complejo Industrial ChihuahuaChihuahua, Mxico

E. Ortega-RivasGraduate Program in Food Science and

TechnologyUniversity of ChihuahuaChihuahua, Mxico

K.S. PannuFood Process Engineering DeltaBritish Columbia, Canada

A. Prez-HernndezAdvanced Materials Research CenterComplejo Industrial ChihuahuaChihuahua, Mxico

D.L. PyleSchool of Food BiosciencesThe University of ReadingReading, Berkshire, U.K.

H.S. Ramaswamy Department of Food Science and

Agricultural ChemistryMcGill University, Macdonald CampusSte. Anne de Bellevue, Quebec, Canada

A.L. Raoult-WackCIRAD, Agri-Food Program Montpellier, France


N.K. RastogiDepartment of Food EngineeringCentral Food Technological Research

InstituteMysore, India

E.P. RibeiroDepartamento de Engenharia Qumica

e AlimentosEscola de Engenharia MauInstituto Mau de TecnologiaPraa

MauSo Paulo, Brazil

C. RossellChemistry Department Universitat de les Illes Balears Palma de Mallorca, Spain

S.S. SablaniDepartment of Bioresource and

Agricultural EngineeringSultan Qaboos UniversityMuscat Sultanate of Oman

K.P. SandeepDepartment of Food ScienceNorth Carolina State UniversityRaleigh, North Carolina

T. SchferDepartment of Chemistry-CQFBFaculdade de Cincias e TecnologaUniversidade Nova de LisboaCaparica, Portugal

M. SchnepfDepartment of Nutritional Science and

DieteticsUniversity of Nebraska-LincolnLincoln, Nebraska

D.R. SeplvedaDepartment of Biological Systems

EngineeringWashington State UniversityPullman, Washington

S. SimalChemistry DepartmentUniversitat de les Illes Balears Palma de Mallorca, Spain

R.P. SinghDepartment of Biological and

Agricultural EngineeringUniversity of California, DavisDavis, California

M.E. Sosa-MoralesChemical Engineering and Food

Engineering DepartmentUniversidad de las Amricas-PueblaSanta Catarina Mrtir, CholulaPuebla, Mxico

J. TangDepartment of Biological Systems

EngineeringWashington State University Pullman, Washington

J. Telis-RomeroDepartamento de Engenharia e

Tecnologia de AlimentosUniversidade Estadual PaulistaSo Jos do Rio Preto, So Paulo,

Brazil

R.F. TestinDepartment of Packaging ScienceClemson UniversityClemson, South Carolina

G. TrystramJoint Research Unit Food Process


A. Valdez-FragosoFacultad de Ciencias QumicasUniversidad Autnoma de ChihuahuaChihuahua, Mxico


M.E. Vargas-UgaldeDepartamento de Ingeniera y

TecnologaFacultad de Estudios Superiores

CuautitlnIzcalli, Edo. de Mxico, MxicoSchool of Food BiosciencesThe University of Reading Reading, Berkshire, U.K.Departamento de Graduados e

Investigacin en AlimentosEscuela Nacional de Ciencias

Biolgicas - I.P.N.Mxico, Mxico

J.F. Vlez-RuizChemical Engineering and Food

Engineering DepartmentUniversidad de las Amricas-PueblaSanta Catarina Mrtir, CholulaPuebla, Mxico

P.J. VerganoDepartment of Packaging ScienceClemson UniversityClemson, South Carolina

O. VitracINRA - Food Packaging UnitCentre de Recherches AgronomiquesReims, France

C.L. WellerIndustrial Agricultural Products Center

andDepartment of Biological/Systems

EngineeringUniversity of Nebraska-LincolnLincoln, Nebraska

J. Welti-ChanesDepartamento de Ingeniera Qumica y

AlimentosUniversidad de las Amricas-PueblaSanta Catarina Mrtir, CholulaPuebla, Mxico

S. WichchukitDepartment of Biological and

Agricultural EngineeringUniversity of California, DavisDavis, California

J.L. WilesDepartment of Packaging ScienceClemson UniversityClemson, South Carolina

Y. WuDepartment of Research and

DevelopmentThe Wright GroupCrowley, Louisiana

M.R. ZareifardDepartment of Food Science and

Agricultural ChemistryMcGill University, Macdonald CampusSte. Anne de Bellevue, Quebec, Canada

S.E. ZorrillaInstituto de Desarrollo Tecnolgico para

la Industria Qumica (INTEC)Consejo Nacional de Investigaciones

Cientficas y TcnicasUniversidad Nacional del LitoralSanta Fe, Argentina


Table of Contents

Part I Fundamental Concepts

1 Fundamentals of Mass TransportJ. Welti-Chanes, H. Mjica-Paz, A. Valdez-Fragoso, and R. Len-Cruz

2 Heat Transfer in Food ProductsD.R. Seplveda and G.V. Barbosa-Cnovas

3 Introductory Aspects of Momentum Transfer PhenomenaJ.F. Vlez-Ruiz

Part II Mass Transfer

4 Structural Effects of Blanching and Osmotic DehydrationPretreatments on Air Drying Kinetics of Fruit TissuesS.M. Alzamora, A. Nieto, and M.A. Castro

5 Pretreatment Efficiency in Osmotic Dehydration of CranberriesS. Grabowski and M. Marcotte

6 Mass Transfer Description of the Osmodehydration of Apple SlabsE. Azuara-Nieto, G.F. Gutirrez-Lpez, and C.I. Beristain-Guevara

7 Combined Effect of High Hydrostatic Pressure Pretreatment and Osmotic Stress on Mass Transfer during Osmotic DehydrationN.K. Rastogi, A. Angersbach, and D. Knorr

8 Hydrodynamic Mechanisms in Plant Tissues during Mass Transport OperationsP. Fito, A. Chiralt, J. Martnez-Monz, and J. Barat

9 Effect of Pretreatment on the Drying Kinetics of Cherry Tomato(Lycopersicon esculentum var. cerasiforme)P.M. Azoubel and F.E.X. Murr

10 Determination of Concentration-Dependent Effective Moisture Diffusivity of Plums Based on Shrinkage KineticsA.L. Gabas, F.C. Menegalli, and J. Telis-Romero

11 Modeling Dehydration Kinetics and Reconstitution Properties of Dried Jalapeo PepperR. Olivas-Vargas, F.J. Molina-Corral, A. Prez-Hernndez,

and E. Ortega-Rivas

12 Application of an Artificial Neural Network for Moisture TransferPrediction Considering Shrinkage during Drying of FoodstuffsJ.A. Hernndez-Prez, M.A. Garca-Alvarado, G. Trystram,

and B. Heyd


13 Modeling of Prato Cheese Salting: Fickian and Neural Network ApproachesA.F. Baroni, M.R. Menezes, E.A.A. Adell, and E.P. Ribeiro

14 Influence of Vacuum Pressure on Salt Transport during Brining of Pressed CurdA. Chiralt, P. Fito, C. Gonzlez-Martnez, and A. Andrs

15 Effects of Water Concentration and Water Vapor Pressure on the Water Vapor Permeability and Diffusion of Chitosan FilmsJ.L. Wiles, P.J. Vergano, F.H. Barron, J.M. Bunn, and R.F. Testin

16 Water Vapor Permeability, Water Solubility, and Microstructure of Emulsified StarchAlginateFatty Acid Composite FilmsY. Wu, C.L. Weller, F. Hamouz, S.L. Cuppett, and M. Schnepf

17 Mass Transport Phenomena during the Recovery of VolatileCompounds by PervaporationT. Schfer and J.G. Crespo

18 Ultrasonic Mass Transfer Enhancement in Food ProcessingA. Mulet, J. Crcel, J. Benedito, C. Rossell, and S. Simal

19 Mass Transfer and Residence Time Studies in Spinning Cone ColumnsM.E. Vargas-Ugalde, K. Niranjan, D.L. Pyle, and G.F. Gutirrez-Lpez

Part III Heat Transfer

20 Transport Phenomena during Double-Sided Cooking of Meat PattiesS.E. Zorrilla, S. Wichchukit, and R.P. Singh

21 Thermal Processing of Particulate Foods by Steam Injection. Part 1. Heating Rate Index for Diced VegetablesK.S. Pannu, F. Castaigne, and J. Arul

22 Thermal Processing of Particulate Foods by Steam Injection. Part 2. Convective Surface Heat Transfer Coefficient for SteamK.S. Pannu, F. Castaigne, and J. Arul

23 Modeling of Heat Conduction in Elliptical Cross Sections (Oval Shapes) Using Numerical Finite Difference ModelsF. Erdo du, M.O. Balaban, and K.V. Chau

24 Heat Transfer Coefficient for Model Cookies in a Turbulent Multiple Jet Impingement SystemN. Nitin and M.V. Karwe

25 Flow Dynamics and Heat Transfer in Helical Heat ExchangersP. Coronel and K.P. Sandeep

26 Relating Food Frying to Daily Oil Abuse. Part 1. Determination of Surface Heat Transfer Coefficients with Metal BallsK.S. Pannu and M.S. Chinnan

g


Part IV Combined Transfer Phenomena

27 Relating Food Frying to Daily Oil Abuse. Part 2. A Practical Approach for Evaluating Product Moisture Loss, Oil Uptake,and Heat TransferK.S. Pannu and M.S. Chinnan

28 Heat and Mass Transfer during the Frying Process of DonutsJ.F. Vlez-Ruiz and M.E. Sosa-Morales

29 Influence of Liquid Water Transport on Heat and Mass Transferduring Deep-Fat FryingO. Vitrac, A.L. Raoult-Wack, and G. Trystram

30 Numerical Simulation of Transient Two-Dimensional Profiles of Temperature, Concentration, and Flow of Liquid Foodin a Can during SterilizationA.G. Abdul-Ghani, M.M. Farid, and X.D. Chen

31 Heating Behavior of Canned Liquid/Particle Mixtures during End-over-End Agitation ProcessingS.S. Sablani, H.S. Ramaswamy, and H. Krishnamurthy

32 Dimensionless Correlations for Forced Convection Heat Transferto Spherical Particles under Tube-Flow Heating ConditionsH.S. Ramaswamy and M.R. Zareifard

33 Heat and Mass Transfer Modeling in Microwave and Spouted Bed Combined Drying of Particulate Food ProductsH. Feng and J. Tang


Part I

Fundamental Concepts


Fundamentals of Mass Transport

J. Welti-Chanes, H. Mjica-Paz,A. Valdez-Fragoso, and R. Len-Cruz

CONTENTS

1.1 Introduction1.2 Mass Transfer Variables

1.2.1 Concentration1.2.2 Velocity1.2.3 Flux1.2.4 Flux Relations for Binary Systems

1.3 Mass Transfer by Diffusion1.3.1 Steady State Diffusion1.3.2 Molecular Diffusion in Gases, Liquids, and Solids

1.3.2.1 Molecular Diffusion in Gases1.3.2.2 Molecular Diffusion in Liquids1.3.2.3 Molecular Diffusion in Solids

1.3.3 Unsteady State Diffusion1.3.3.1 Solutions of Fundamental Equations

1.4 Mass Transfer by Convection1.4.1 Film Theory and Mass Transfer Coefficient1.4.2 Two-Film Theory and Mass Transfer Coefficient1.4.3 Dimensionless Numbers for Mass Transfer1.4.4 Transport Analogies1.4.5 Mass Transfer Coefficients and Correlations1.4.6 Mass Transfer Units

NomenclatureReferences

1.1 INTRODUCTION

Mass transfer can be defined as the migration of a substance through a mixture underthe influence of a concentration gradient in order to reach chemical equilibrium.Biochemical and chemical engineering operations, such as absorption, humidifi-cation, distillation, crystallization, and aeration, involve mass transfer principles.

1


In food processing, mass transfer phenomena are present in freeze-drying, osmoticdehydration, salting or desalting, curing and pickling, extraction, smoking, baking,frying, drying of foods, membrane separations, and the transmission of water vapor,gases, or contaminants across a packaging film. Food stability and the preservationof food quality are also affected by mass transfer of environmental components thatcan affect the rate of reactions. Among the components involved in these mass transferprocesses are water, sugars, salt, oils, proteins, acids, flavor and aroma substances,oxygen, carbon dioxide, residual monomers or polymer additives, and toxins or car-cinogens produced by microorganisms. Furthermore, mass transfer phenomena areimportant in the scale-up of processes to pilot- or commercial-scale plants and in thecontrol and optimization of the processes.

This chapter presents the basic principles of mass transfer. First, the variablesthat occur in mass transfer are reviewed. Then the mechanisms of mass transfer,diffusion and convection are discussed. Analogies among momentum, heat, andmass transfer are described; convective mass transfer coefficients and correlationsare derived by analogy with convective heat transfer. Finally, the concept of transferunits is presented.

1.2 MASS TRANSFER VARIABLES

Mass transfer processes involve concentration, velocity, and flux variables, whichare defined and related by a set of basic equations (White, 1988).

1.2.1 CONCENTRATION

The concentration of a mixture and its components may be expressed in terms ofmass and mol. In terms of mass, the mass concentration of the mixture (, kg/m3),the mass concentration of a component i (i, kg/m

3), and the mass fraction ofcomponent i (wi) are given by:

(1.1)

(1.2)

(1.3)

where m and mi are the mass flux of the mixture and component i, respectively.The bulk molar concentration (C, kg mol/m3), the molar concentration of com-

ponent i (Ci, kg mol/m3), and the mole fraction of component i (xi) are defined by:

(1.4)

(1.5)

(1.6)

where n and ni are the mol of the mixture and component i, respectively.

= m V/

i im V= /

w m mi i i= =/ /

C n V= /

C n Vi i= /

x n n C Ci i i= =/ /


According to the previous definitions, it can be easily shown that:

and (1.7)

and (1.8)

and (1.9)

where xi is the mole fraction of component i, and i and Ci are related through themolecular weight of constituent i (Mi, kg/kg mol):

(1.10)

1.2.2 VELOCITY

In mass transfer phenomena, the velocity of a bulk mixture and of its componentscan be measured with respect to fixed coordinates. In addition, the velocity of thecomponents can also be measured relative to the bulk velocity. Figure 1.1 illustratesthese velocities in a binary system of components A and B in the z direction.

FIGURE 1.1 Scheme of individual and bulk velocities of a binary mixture.

m mii

n

==

1

==

ii

n

1

n nii

n

==

1

C Cii

n

==

1

w wii

n

= ==

1

1 x xii

n

= ==

1

1

i i iM C=

vB

v= wA vA + wB vB

vA

vB v = diffusion velocity of B = UB

vA v = diffusion velocity of A = UAz

fixed coordinate


The mass bulk velocity of the mixture (v, m/sec) relative to fixed coordinates isdefined as:

(1.11)

where vi is the velocity of component i with respect to stationary coordinates.In a similar manner, a molar bulk velocity ( , m/sec) measured relative to

stationary coordinates can be defined as:

(1.12)

The velocity of the constituent i relative to the bulk velocity of the mixture is:

(1.13)

(1.14)

where ui (m/sec) and Ui (m/sec) are the mass and molar diffusion velocities, respectively.

1.2.3 FLUX

The mass bulk flux ( kg/m2 sec) and the molar bulk flux ( , kg mol/m2 sec) of amixture relative to fixed coordinates are:

(1.15)

(1.16)

The flux of the components of a mixture can also be expressed relative eitherto fixed coordinates or to the bulk average velocity. The flux of the component irelative to stationary coordinates is:

(1.17)

(1.18)

The diffusion fluxes of the constituents i of the mixture with respect to theaverage bulk velocity are ji (kg/m

2 sec) for the mass flux and Ji (kg mol/m2 sec) for

the molar flux.

(1.19)

(1.20)

v w vm

mv vi

i

n

ii

i

n

ii

i

n

i= = == = =

1 1 1

V

V x Vn

nV

C

CVi

i 1

n

ii

i 1

n

ii

i 1

n

i= = == = =

u v vi i=

U V Vi i=

n, N

n v vii 1

n

i= ==

N CV C Vii 1

n

i= ==

n vi i i=

N C Vi i= i

j u v vi i i i i= = ( )

J C U C (V V)i i i i i= =


The use of concentration, velocities, and fluxes in mass or molar terms is subjectto preferences and convenience. Nevertheless, concentration and flux expressedin molar units and average molar velocity are preferred. In discussing the funda-mentals of mass transfer, systems are frequently assumed as a two-component mixtureto facilitate the understanding of multi-component systems found in engineeringapplications.

1.2.4 FLUX RELATIONS FOR BINARY SYSTEMS

For a binary mixture of components A and B, Equations (1.18) and (1.20) become:

(1.21)

(1.22)

Substituting V from Equation (1.12) into Equation (1.22),

(1.23)

Since and , for component A, Equation (1.23) becomes:

(1.24)

A similar mathematical derivation gives for component B:

(1.25)

Equations (1.24) and (1.25) show that the absolute molar flux (N or NB) resultsfrom a concentration gradient contribution or a molar diffusion flux (JA or JB) anda convective contribution ( or ). The molar diffusion flux is described byFicks law, which for component A is written as:

(1.26)

where DAB is the diffusion coefficient of A through B and dCA/dz is the change ofthe concentration A with respect to the position z.

In terms of mass, the mass fluxes for components A and B are, respectively:

(1.27)

(1.28)

N C VA A A=

J C U C (V V)A A A A A= =

J C V C V C VC

C(C V C V )A A A A A A

AA A B B= = +

N C VA A A= N C VB B B=

N JC

CN NA A

AA B= + +( )

N JC

CN NB B

BA B= + +( )

C VA C VB

J DdC

dzA ABA=

n n n )A AA

A Bj (= + +

n jn

(n n )B BB

A B= + +


Two important simple cases can be considered in Equation (1.24):

1. Diffusion of A through stagnant B, NB = 0

(1.29)

2. Equimolar counter-diffusion NA + NB = 0,

(1.30)

where xA is the mole fraction of component A in the case of dilute systems xA

1.3.2.1 Molecular Diffusion in Gases

The diffusion coefficient of component A through B (DAB) may be estimated forpressures below 20 atm from the formula:

(1.33)

where P is the absolute pressure (atm), T is the absolute temperature (K), AB is thecollision diameter (A), D is the collision integral for molecular diffusion, and MAand MB are the molecular weights of A and B, respectively (Sherwood et al., 1975).

1.3.2.2 Molecular Diffusion in Liquids

The diffusivity of a solute A in a dilute solution B (DAB) can be predicted by thefollowing equation (Treybal, 1981):

(1.34)

where B, B, and MB are an association parameter, the viscosity (centipoises), andthe molecular weight of solvent B, respectively, T is the absolute temperature (K),and VA is the molal volume of A (cm

3/g mol).

1.3.2.3 Molecular Diffusion in Solids

The diffusion of gases and liquids through porous solid materials may occur by acombination of Fick diffusion and Knudsen diffusion.

If the pores are large, the mass transfer within the gas or liquid contained in thepores will be by Fick diffusion. Nevertheless, the diffusivity in the solid is reducedbelow what it would be in a fluid, due to the tortuous nature of the path that amolecule must travel to advance a given distance in the solid and to the restrictedfree cross-sectional area (Sherwood et al., 1975). In such a case, the flux must bedescribed in terms of an effective diffusion coefficient, defined as:

(1.35)

where DAB is the diffusion coefficient in a binary system, is the tortuosity, and isthe fractional void space. Values of , , and Deff must be determined experimentally.

Therefore, when the system is a porous solid that has interconnected voids thataffect the diffusion, for a binary system the molar flow is:

(1.36)

D TM M

PABA B

AB D

= +0 001858 1 13 21 2

2.( / / )/

/

D MTVAB B B B A

= 7 4 10 8 1 2 0 6. ( )/

.

D Deff AB=

N DdC

dzA ABA=


Knudsen diffusion occurs when the pores are small and their size approachesthe mean free path of the gas molecule. A gas molecule now collides predominantlywith the walls of the pores rather than with other molecules. The Knudsen diffusioncoefficient (DK), derived from the kinetic theory of gases (Welty et al., 1984), isgiven by the expression:

(1.37)

where is the density of the solid material, M is the molecular weight of the diffusinggas, and T is the temperature (K).

The molar flux for gas and liquids through a porous solid can be described interms of DK by:

(1.38)

where CA1 and CA2 are the bulk concentrations in the gas and liquid phases, respectively,and pA1 and pA2 are the partial pressures of the gas and liquid phases, respectively.

When Fick and Knudsen diffusions are important, the effective diffusion coef-ficient is defined by:

(1.39)

which applies for equimolar counter-diffusion.Diffusion in liquids is important in separation operations such as liquidliquid

extraction and distillation. Typical examples of diffusion of fluids through solids arewater vapor transport in the drying of porous foods, the diffusion of a gas througha polymer film used in the packaging of foods, and separation processes through amembrane, such as reverse osmosis and ultrafiltration.

1.3.3 UNSTEADY STATE DIFFUSION

The unsteady state diffusion, in which the local concentration change with time isdescribed in one dimension for slab shape and constant DAB, is shown by Fickssecond law:

(1.40)

Analogous equations can be written for diffusion in spherical or cylindricalshapes and two or three dimensions. These equations are used to find the concen-tration of a solute as a function of time and position and are mainly applicable todiffusion in solids and to limited situations in fluids. The analysis of unsteady statesystems, however, is frequently simplified to a one-directional flow.

DS

TMK

=

19400 2

N DC C

zD

p p

RTzA KA A

KA A= = ( ) ( )1 2 1 2

1 1 1D D Deff AB K

= +

=Ct

Dd C

dzA

ABA

2

2


1.3.3.1 Solutions of Fundamental Equations

Solutions to unsteady state diffusion equations can be obtained by analytical, numer-ical, and graphical methods. Analytical solutions for one-dimensional diffusion inan infinite slab, an infinite cylinder, and a sphere have been found by assuming aconstant diffusion coefficient given appropriate boundary conditions (Crank, 1975):

1. Infinite slab:

(1.41)

2. Infinite cylinder:

(1.42)

3. Sphere:

(1.43)

where Co and C are the molar concentration at time t = 0 and t = , respectively.When properties become dependent upon position or concentration, the solutions

become more complicated. In this case, numerical techniques can be used, since theyare more appropriate for complex and real problems. Among the numerical methodsare finite difference and finite element analysis. The finite difference method is basedon the approximation of the difference of a derivative at a point. This analysis is oftenlimited to cases where the body has a simple geometry. The finite element methodof analysis overcomes this problem; its basic concept is that any continuous quantity,such as a concentration, can be approximated by a discrete model composed of a setof piecewise continuous functions defined over a finite number of domains.

For unsteady state heat conduction problems in solids, solutions have beenpresented for simple geometries subject to a given set of boundary and initialconditions. These solutions are presented in charts in terms of dimensionless ratiosand can be translated directly to analogous problems of unsteady state diffusion,allowing the application of heat transfer results to solve mass diffusion problems.The ordinate of these charts represents the fraction of the unaccomplished change[(c1 c)/(c1 c0)] and the abscissa the relative time [DABt /x1

2]. These charts andtheir physical significance, use, and application are presented in several books (Weltyet al., 1984; Geankoplis, 1993). Table 1.1 summarizes the correspondence betweenvariables for unsteady heat and molecular diffusion.

C C

C C4 1

2n 1cos

2n 1 x2

expD(2n 1) t

40

n

n 0

n 2 2

2

= +

+

+

=

=

( ) ( )

C C

C C R

J b r

b J b rDb tn

n nn

n

n

= =

=

0

0

11

22 ( , )

( , )exp( )

C C

C CR

n rn rR

Dn tR

n

n

n

= ( )

+

=

=

0

1

1

2 2

2

2 1 1

sin exp


1.4 MASS TRANSFER BY CONVECTION

Convective mass transport, which occurs in liquids and gases, results from bulkmotion of the fluid imposed by external forces (forced convection) or occurringnaturally, due to the concentration difference or density variations (free convection).The free or forced character is determined by the nature of the motion of the fluid,which may be either laminar or turbulent. In laminar flow, the mechanism is thesame as in a stationary medium, and the transfer occurs by diffusion. In turbulentflow, mass transfer is affected by the irregular motion of small volumes of the fluid.

1.4.1 FILM THEORY AND MASS TRANSFER COEFFICIENT

Convective mass transfer problems in turbulent flow are not always amenable toanalytical methods of solution. Consequently, they are usually approached with theaid of coefficients and empirical relationships. Among several models proposed fora better understanding of mass transfer under turbulent regime, we find the modelbuilt around the film theory. In this theory the interfacial region is treated as ahypothetical stagnant film of thickness x, called the boundary layer, and it is assumedthat all the concentration changes occur in this layer (Figure 1.2) (Welty et al., 1984).

TABLE 1.1Relation between Mass and Heat Transfer Variables for Unsteady State

Mass Transfer

Heat Transfer K = 1 K 1

Y, Unaccomplished change

X, Relative time

m, Relative resistance

n, Relative position

Note: T0 and c0 = temperature and concentration at time t = 0, respectively; T1 and c1 = temperatureand concentration at the position of interest; x1 = distance from the midpoint to the position ofinterest; m = ratio of the convective mass transfer resistance to the internal molecular resistance tothe mass transfer; K = cLi/ci, where cLi = bulk fluid concentration and ci = concentration in the fluidadjacent to the solid surface.

Source: Adapted from Welty, J.R., Wicks, C.E., and Wilson, R.E., Momentum, Heat, and MassTransfer, John Wiley & Sons, New York, 1984 and Geankoplis, C.J., Transport Processes and UnitOperations, 3rd ed., Prentice-Hall, London, 1993. With permission.

T T

T T1

1 0

c c

c c1

1 0

( / )

( / )

c K c

c K c1

1 0

tx1

2

D t

xAB

12

D t

xAB

12

khx1

D

k xAB

m 1

D

Kk xAB

m 1

xx1

xx1

xx1


Thus, the molar flux and the concentration profile of species A (JA) are found from

(1.44)

where DAB/z is the mass transfer coefficient, km, CA1, and CA2 are the bulk concen-trations in the fluid and solid phase, respectively, and

(1.45)

Similarly, the corresponding expression for mass flux is:

(1.46)

where CA is the concentration of the components A, and A1 and A2 are the massconcentration of the fluid and solid, respectively.

The mass transfer coefficient kC or k has units of velocity (m2/sec). It can be

determined either from experimental data and empirical formulas derived from themor with the aid of methods of similitude theory. A rough estimation of km can beattained by assuming km = D/x, provided the effective film thickness and the diffusioncoefficient are known (Sherwood, 1974).

1.4.2 TWO-FILM THEORY AND MASS TRANSFER COEFFICIENT

Mass transfer from a gas and/or a liquid to another fluid is a common engineeringproblem. For interphase transfer, it is convenient to use an overall mass transfercoefficient, the interpretation of which is achieved with the aid of the two-film theory.Figure 1.3 schematically shows the transfer of component A from the gas phase to

FIGURE 1.2 Fluid-solid interfacial region:the film theory.

xz

bulk fluid

CA1

stagnant film

CA2

solid

JD

xC C k C CA

ABA A m A A= = ( ) ( )1 2 1 2

C Cz C C

xA AA A= 1 1 2

( )

j kA A A= ( )1 2


the liquid phase. The two-film theory assumes that the resistance to mass transferlies in each adjacent phase to the interphase and that no resistance is offered to thetransfer of the solute across the interphase (Welty et al., 1984).

At steady state, the fluxes in gas and liquid (J1) phases must be equal:

(1.47)

where kmG and kmL are the mass transfer coefficients in the gas and liquid phases,respectively, CA1i and CA2i are the concentrations of component A at the interface,and CA1 and CA2 are the bulk concentrations in the gas and liquid phases, respectively.

Since the concentrations at the interphase are not easily measurable, it is con-venient to calculate an overall mass transfer coefficient based on an overall potentialgradient between the bulk compositions. The overall driving force is not, however,CA1 CA2, since at the interphase discontinuity of the concentrations exists, and thesolubility in the liquid is not necessarily the same as in the gas. Moreover, the filmthickness x1 and x2 and the diffusivity of the solute may be different in the two phases.

The solubility relationship that governs the equilibrium concentration betweenphases is of the form:

(1.48)

where m is the solubility constant between the two phases, and CG and CL are theconcentrations of the gas and liquid, respectively.

FIGURE 1.3 Mass transfer at a gas liquid interphase: the two-film theory.

1A*

2A mCC = mC

C 2A* 1A =

CA1i

CA2i

interface

CA1

gas phase liquid phase

CA2

x1 2x

layer layer

J k C C k C CmG A A i mL A i A1 1 1 2 2= = ( ) ( )

C mCG L=


Thus, the flux in terms of the total potential gradient in the gas phase is:

(1.49)

where CA2/m = C*A1, which is the concentration in the gas phase that would exist in

equilibrium with CA2, the concentration of species A in the liquid.A similar equation can be obtained for the overall coefficient if the driving force

is based on the concentration in the liquid phase:

(1.50)

where mCA1 is the concentration in the liquid that would exist in equilibrium withC*A1. KmG and KmL represent the overall mass transfer coefficient based on the gasphase and the liquid concentration driving force.

Coefficients KmG and KmL are related to the individual mass transfer coefficientsand the equilibrium constant m of a gasliquid (kmG) or vaporliquid (kmL) systemas follows:

(1.51)

(1.52)

These last two equations show the relationship among the coefficients for theindividual phases and the overall transfer coefficients, expressed as a global resis-tance (1/KmG or 1/KmL) to the transfer of the diffusing component. In studyingperformance separation processes, it is important to determine which individualresistance is the limiting factor.

1.4.3 DIMENSIONLESS NUMBERS FOR MASS TRANSFER

Mass transfer coefficients are expected to vary with hydrodynamic conditions andwith geometrical and physical properties of the fluid. These variables are collectedinto the so-called dimensionless numbers. Dimensionless numbers that arise in masstransfer modeling and scale-up studies are presented in Table 1.2.

For mass transport, the Biot number is defined by the equation Bi = kmd/D. TheBiot number compares the resistance to mass transfer at the surface of a solid to theresistance inside the solid. High Biot numbers can be obtained by increasing the masstransfer coefficient. For instance, in solids drying, the higher the value of Bi, thefaster the drying proceeds; hence, the air humidity increases rapidly.

The Fick number, Fi = Dt/d2, is a nondimensional time parameter. It is importantin studying unsteady state diffusion mass transfer in several simple shapes withcertain restrictive boundary conditions (Gekas, 1992).

J K C C K CC

mmG A A mG AA= =

( )

*1 1 1

2

J K C C K mC CmL A A mL A A= = ( ) ( )*

1 2 1 2

1 1K k

mkmG mG mL

= +

1 1 1K mk kmL mG mL

= +


The Graetz number is defined as the product of the Reynolds and Schmidtnumbers. It includes the effects of forced and free convection.

The Grashof number is a measure of free convection, which will be enhancedby buoyancy forces () and decreased by viscous forces (). The Grashof numberis found in correlations of free convection mass transfer.

The Lewis number, Le = /D, is the ratio of heat to mass diffusivity. It playsan important role in processes where simultaneous convective transfer of energy andmass occurs.

The Peclet number, defined as Pe = vd/D, is the ratio of bulk mass transport todiffusive mass transport.

The Reynolds number, Re = vd/, characterizes the nature of the motion of aflowing gas or liquid, and is interpreted as the ratio of inertial forces to the viscousforces in the flow. When the Re value is below a certain critical value, the flow islaminar; when it is larger, the flow is turbulent.

In the Schmidt number, Sc = /D, the two molecular transport coefficients (and D) are the physical properties of the medium in which the transfer of mass takesplace. The Schmidt number, which represents the ratio of momentum to massdiffusivities, is of great importance in convective mass transfer (Welty et al., 1984).

TABLE 1.2Dimensionless Numbers Related to Mass Transfer

DimensionlessNumber Equation Physical Meaning

BiotMass transfer across the boundary/mass transfer within the solid

FickDimensionless time in the unsteady state regime

Graetz Mass transfer in laminar flow

Grashof Buoyancy forces/viscous forces

Lewis Heat diffusivity/mass diffusivity

Peclet Convection/diffusion

Reynolds Inertial force/viscous force

SchmidtMomentum diffusivity/mass diffusivity

SherwoodMeasure of boundary layer thickness

StantonWall mass transfer/mass transfer by convection

Bik d

Dm=

FiDtd

= 2

ReSc

Grgd T=

3 2

LeD

=

PevdD

=

Re = vd

ScD

=

Shk d

Dm=

Stk

vm=


The Sherwood number is defined as Sh = kmd/D, where km is the mass transfercoefficient, d is a characteristic length dimension, and D is the diffusion coefficient.The Sherwood number is a convenient parameter for the analysis of transfer pro-cesses occurring in a stationary medium in laminar flow. In the case of purelymolecular transfer, the Sherwood number is highly dependent on the geometricalshape of the body.

The Stanton number is interpreted as the ratio of the mass-transfer coefficient(km) to the flow velocity (v). It represents a measure of the mass flux in the directiontransverse to that of the mean flow. The Stanton number is used more convenientlyin turbulent flow.

The dimensionless numbers Sc, Sh, and Le will be encountered in the analysisof convective mass transfer correlations.

1.4.4 TRANSPORT ANALOGIES

The basic laws governing the flux of momentum, heat, and mass transport due tomolecular motion or vibration have the form:

(1.53)

(1.54)

(1.55)

All three processes are quite different from one another at a molecular level.However, certain analogies exist among them. In effect, molecular diffusivitieskinematic viscosity (), thermal diffusivity (), and diffusion (D) have the samedimensions (L2/t). Also, in Ficks law, the molar flux varies with the gradient in molper volume; in the rewritten Fouriers law, the energy flux is proportional to thegradient of energy per volume (CpT); and the momentum flux, given by the rewrittenNewtons law, varies with the gradient of the momentum per volume (v) (Cussler,1984). These analogies are shown in Table 1.3.

The corresponding equations for momentum, heat and mass flux in convectivemotion are:

(1.56)

(1.57)

(1.58)

= =ddz

v vddz

v( ) ( )

qkC

ddz

C Tddz

C Tp

p p= = ( ) ( )

J DdCdzi

=

= =

f v

fvv

12 2

02 ( )

q h ThC

C Tp

p= = ( )

N k Ci m i=


where the transfer coefficient fv/2 is like h/Cp and k. Note that the driving forces inthe momentum, heat, and mass flux are volume concentrations: is expressedin momentum per volume, in energy per volume, and in mol per volume(Table 1.3).

Since the molecular diffusivities have the same dimensions, a ratio of any oftwo of these leads to dimensionless numbers: Pr number for heat transfer, Sc andLe numbers for mass transfer. Likewise, the ratio transfer coefficients to the flowvelocity lead to an [St] number for heat transfer and an [St] number for mass transfer(Tables 1.2 and 1.3).

A useful and simple analogy relating all three types of transport simultaneouslyis the ChiltonColburn analogy, which is written as

(1.59)

The group

is called the jD factor for mass transfer, and

defines the jH factor for heat transfer.

TABLE 1.3Analogies among Momentum, Heat, and Mass Transfer

Analogous Form

Momentum Transfer Heat Transfer Mass Transfer

Variable(momentum/volume) (energy/volume)

C(mol/volume)

Molecular diffusivity (kinematic viscosity)

(thermal diffusivity)

D(diffusion coefficient)

Transfer coefficient f(friction factor)

h(heat transfer coefficient)

km(mass transfer

coefficient)Dimensionlessnumber

v C Tp

Pr =

Sth C

vp=

/

ScD

=

Stk

vm=

LeD

=

( )v 0( )C Tp Ci

k

vSc

h / C

vPr

f2

m 2/3 p 2/3= =

k

vScm 2 3/

h C

vStp

/Pr Pr/ /

2 3 2 3=


The ChiltonColburn analogy agrees well with a wide range of experimentaldata for flow and geometries of different types in forced convection systems. There-fore, when an engineer is concerned with the calculation of heat and mass transfercoefficients, the analogies are of great utility. In this way, when heat transfer andmass transfer occur by the same mechanism, the results of experiments on heattransfer may be used to calculate diffusion processes, or vice versa. Also, informationobtained from a small-scale model can be used to scale up the process, or informationobtained with one substance can be extended to another substance. In very particularcases, Pr = Sc = 1, which means that = = D, and heat and mass transfermeasurements may be used to predict momentum transfer or vice versa (Sherwoodet al., 1975; Treybal, 1981).

1.4.5 MASS TRANSFER COEFFICIENTS AND CORRELATIONS

Although the analogy concept is useful in predicting mass transfer coefficients, incertain turbulent situations correlations based on experimental observations arerequired. A few typical correlations for mass transfer in a variety of geometries andflow conditions are given in Table 1.4 (Treybal, 1981).

TABLE 1.4Typical Mass Transfer Correlations

Situation Equation Remarks

Round tubes Used for turbulent flow

Used for laminar flow in a circular pipe

Flat plates Transfer begins at leading edge

Used for laminar flow over a plate

Evaporation dropsUsed for evaporation drops in spray drying

Stirred dropsUsed in mixing liquid solutions; P/V is important in scale-up

Solid spheres Used for forced convection

Used for natural convection

Bubbles Used for rising bubbles, d > 2.5 mm

Used for rising bubbles, d < 2.5 mm

MembranesUsed in actual or hypothetical membranes

Packed bed of pellets

Re f (particle diameter and superficialvelocity in the bed)

Sh Sc= 0 023 0 83 0 44. Re . .

Shdx

Sc=

1 671 3

. Re/

Sh Sc= 0 664 1 2 1 3. Re / /

Sh Sc= 0 332 1 2 1 3. Re / /

Sh Sc= +2 0 6 1 3 1 2. Re/ /

Shd P V

D=

0 134

3

1 4 1 3

.( / )

/ /

Shdv

D= +

2 0 61 2 1 3

./ /

Shd g

D= + [ ]

2 0 63

2

1 4 1 3

.

/ /

Sh Gr Sc= +2 0 31 1 3 1 2. ( ) ( )/ /

Sh Gr Sc= +0 42 1 3 1 2. ( ) ( )/ /

kdD

= 1

jD =1 17 0 415. Re .


1.4.6 MASS TRANSFER UNITS

The number of transfer units is another approach in the application of pilot plantresults to the design of mass transfer equipment, such as cooling towers, packedcolumns, extractors, or driers.

This method can be illustrated in the determination of the height of a drier (Z),which is obtained as the product:

(1.60)

where HTUc is the height of a transfer unit and NTUD is the number of mass transferunits given by:

(1.61)

Subscripts 1 and 2 refer to inlet and outlet values. The integral is evaluated fromconsiderations of bulk (b) and interphase (i) conditions at any cross-section of thedrier, absorption tower, etc., the former being determined by mass balances and thelatter by equilibrium data.

The height of a transfer unit is given by:

(1.62)

where km is the mass transfer coefficient, a is a specific area, At is a cross-sectionalarea of the drier, g is the density of the gas phase, and G is the gas flow rate.

In practice, it is found that HTU is a property of the type of drier and the materialbeing dried but is independent of size and operating conditions. Therefore, HTUcan be calculated from pilot plant experiments in which the height of the drier andthe number of transfer units are known. Thus, the design of a similar drier can beobtained by scaling up the pilot plant results.

Here again, the Colburn analogy is of great utility. From the arguments used inthe section about transfer analogies, a similar expression for the Colburn analogyhas been proposed (Le Goff, 1980):

(1.63)

where the quantity NEU is the number of energy units, and the Le Goff number(Lf) represents the deviation of the Colburn analogy from unity, the Lf number beingin the range 10.02.

Z HTU NTUC D=

NTUdC

C CD i bC

C

=

1

2

HTUG

k aAm t g=

NTU Sc

Lf

NTU

LfNEUD

D

H

H

2 3 2 3/ /Pr= =


Equations for NTUH and NEU are similar to Equation (1.58) for NTUD:

(1.64)

(1.65)

The number of transfer units is a basic parameter that appears frequently in thedesign of heat and mass exchangers (Van den Bulck et al., 1985; Khodaparast, 1992;Treybal, 1981).

NOMENCLATURE

a Specific area; cm2 or m2

At Cross-sectional area of drier; m2

bn Roots of Bessel functionC, c Molar concentration; kg mol/m3

C* Molar concentration at equilibrium; kg mol/m3

Cp Specific heat capacity; kJ/kg Kd Characteristic dimension of a solid body; m or cmD Diffusion coefficient; m2/sec or cm2/secf Friction factor; dimensionlessg Acceleration constant for gravity, m/sec2

G Flow rate; kg/m2sech Heat transfer coefficient; W/m2KHTUD Height of a mass transfer unit, mj The mass flux of the mixture with respect to the average bulk velocity;

kg/m2secJ Molar diffusion flux of the mixture with respect to the average bulk velocity;

kg mol/m2secJo Bessel function of first kind and order zeroJ1 Bessel function of first kind and order onekm Mass transfer coefficient; m

2/seck Thermal conductivity; W/m KKm Overall mass transfer coefficient based on the gas system concentration

driving forceK Partition coefficient; dimensionlessm Mass of the mixture; kg in Equation (1.1)m Solubility constant between the two phases in Equation (1.48)m Mass flux of the mixture relative to fixed coordinates, kg/m2sec in Equation

(1.11)M Molecular weight; kg/kg moln Mole of the mixture; kg mol in Equation (1.4)n Relative position in Equations (1.7) and (1.12)

NTUdT

T TH i bT

T

=

1

2

NEUP P

v= 1 22


Mass bulk flux of the mixture relative to stationary coordinates, kgmol/m2secMolar bulk flux of the mixture relative to fixed coordinates; kg mol/m2sec

NEU Number of energy unitsNTUD Number of mass transfer unitsp Partial pressure; atmP Pressure; atmq Heat flow rate; Wr Radial coordinateR Radius; cm or m in Equations (1.42) and (1.43)R Gas constant; joules/g mol K in Equation (1.38)S Shape factorT Temperature; Kt Time; secu Mass diffusion velocity of the system relative to the mass bulk velocity; m/secU Molar diffusion velocity of the system relative to the molar bulk velocity; m/secv Mass bulk velocity of the mixture relative to stationary coordinates; m/secV Volume of the mixture, m3 in Equation (1.1)V Molal volume of the system; m3/kg mol in Equation (1.34)

Molar bulk velocity measured with respect to fixed coordinates; m/sec w Mass fraction of the mixturex Rectangular coordinatex Mole fraction of the mixture in Equation (1.29)X Dimensionless timez Rectangular coordinateZ Height of a drier; m

GREEK SYMBOLS

Thermal diffusivity; cm2/sec Coefficient of thermal expansion; 1/K Thickness of a hypothetical stagnant film; cm or m Porosity Viscosity; centipoises Kinematic viscosity; m2/sec Mass concentration of the mixture or fluid density; kg/m3

AB Collision diameter; A TortuosityD Collision integral for molecular diffusion; dimensionless Association parameter

REFERENCES

Crank, J., The Mathematics of Diffusion, 2nd ed., Oxford University Press, London, 1975.Cussler, E.L., How we make mass transfer seem difficult, Chem. Eng. Educ., 18(3), 124127,

149152, 1984.

n

N

V


Geankoplis, C.J., Transport Processes and Unit Operations, 3rd ed., Prentice-Hall., London,1993.

Gekas, V., Transport Phenomena of Foods and Biological Material, CRC Press, Boca Raton,FL, 1992.

Khodaparast, K.A., Predict the number of transfer units for cooling towers, Chem. Eng.Progress, 4, 6768, 1992.

Le Goff, P., Performance energtique des echangeurs de matire et de chaleur: interprtationenergtique des analogies de Reynolds et de Colburn, Chem. Eng. J., 20, 197209, 1980.

Sherwood, T.K., A review of the development of mass transfer theory, Chem. Eng. Educ., 3,204213, 1974.

Sherwood, T.K., Pigford, R.L., and Wilke, C.R., Mass Transfer, McGraw-Hill Kogakusha,Ltd., Tokyo, 1975.

Treybal, R.E., Mass Transfer Operations, 3rd ed., McGraw-Hill International Editions, 1981.Van den Bulck, E., Mitchell, J.W., and Klein, S.A., Design theory for rotary heat and mass

exchangers II. Effectiveness number of transfer units method for rotary heat and massexchangers, Int. J. Heat Mass Transfer, 28(8), 15871595, 1985.

Welty, J.R, Wicks, C.E., and Wilson, R.E., Momentum, Heat and Mass Transfer, 3rd ed, JohnWiley & Sons, New York, 1984.

White, F.M., Heat and Mass Transfer, Addison-Wesley, Reading, MA, 1988.


Heat Transfer in Food Products

D.R. Seplveda and G.V. Barbosa-Cnovas

CONTENTS

2.1 Introduction2.2 General Background

2.2.1 Thermal Properties of Foods2.2.1.1 Specific Heat2.2.1.2 Thermal Conductivity .2.2.1.3 Thermal Diffusivity2.2.1.4 Surface Heat Transfer Coefficient

2.2.2 Heat Transfer by Conduction2.2.2.1 Steady State2.2.2.2 Nonsteady State

2.2.3 Heat Transfer by Convection2.2.3.1 Natural Convection2.2.3.2 Forced Convection

2.2.4 Heat Transfer by Radiation2.3 ConclusionsReferences

2.1 INTRODUCTION

Heating and cooling are common activities in food processing. Several operationsinvolving heating of raw foods are performed for different purposes, such as reduc-tion of the microbial population, inactivation of enzymes, reduction of the amountof water, modification of the functionality of certain compounds, and of course,cooking. On the other hand, heat is removed from foods to reduce the rate of itsdeteriorative chemical and enzymatic reactions and to inhibit microbial growth,extending shelf life by cooling and freezing. Heat transfer plays a central role in allof these operations; therefore, food engineers need to understand it in order to achievebetter control and avoid under- or over-processing, which often results in detrimentaleffects on food characteristics. In practice, heat transfer to or from foods can beattained either by indirect or direct methods. Indirect methods involve the use ofheat exchangers that isolate the product from the medium used as a source or sink

2


of heat. Direct methods allow contact between the food and the heating/coolingmedium. Examples of these methods can be found in Table 2.1.

Indirect heating methods use gases and vapors, such as steam or air, and liquids,such as water or organic compounds, as a source of thermal energy. Chilling byindirect methods involves the use of coolant gases, such as ammonia or Suva, orthe use of coolant liquids, such as water or ethylene glycol. Direct heating can beattained by means of hot air, oil, infrared energy, and dielectric or microwavemethods, among others. Direct chilling can be achieved by the use of cold air or bythe application of the Peltier effect.

2.2 GENERAL BACKGROUND

The first step in understanding heat transfer is to define what heat is and how it diffusesthrough a single body or is transferred from one body to another. Heat is a nonme-chanical form of energy transferred between regions of different temperature. Heattransfer, therefore, is a natural energy transfer process in which energy tends to travelfrom a hotter point to a colder point in order to reach an equilibrium temperature.

Now that we know that the temperature gradient is the driving force in heattransfer processes, and therefore that if a temperature difference exists, energytransference will occur, a couple of questions arise. How fast will this energy transferprocess occur? How will the energy diffuse through foods? The answers to thesequestions are not easy. Great efforts have been made to try to answer these questions,and several models have been developed to describe heat transfer behavior in dif-ferent systems under different conditions.

The heat transfer mode governing the process is defined by the physical state ofthe bodies and their relative position. If a heat gradient exists between two solidbodies in contact, the heat transfer will proceed by conduction. If the same gradientexists between two fluids, or between a fluid and a solid, the energy will be transferredby convection. Finally, any body with a temperature above absolute zero will radiateenergy in the form of electromagnetic waves transferring heat by radiation. Besidesthe physical state or relative position, other physical properties of the bodies involvedin these processes influence the heat transfer rate. Characteristics such as form, size,structure, thermal conductivity, specific heat, density, and viscosity, among others,are of paramount importance in the definition of the behavior of a system.

Early developments in the study of heat transfer in the chemical engineeringfield assumed controlled situations dealing with well-defined substances with fixed

TABLE 2.1Examples of Direct and Indirect Heating and Chilling

Heating Chilling

Direct Frying, boiling, baking, and grilling solid foods

Fluidized bed individual quick freezing (IQF)

Indirect Fluid food pasteurization; canned products sterilization

Fluid food cooling; ice production


physical characteristics and clearly defined heat transfer modes. However, actualsituations in food engineering involve more than one mode of heat transfer simul-taneously, and frequently some of the physical characteristics of food, such asdensity, form, or viscosity, change as heat modifies the chemical structure, affectingthe foods thermal behavior. Furthermore, foods usually have neither regular formnor homogeneous or isotropic behavior. Finally, some particular features of foodbeing heated, such as nonuniform evaporation of water, crust formation, or closingor opening of pores, are of such complexity that they make the modeling of thisprocess difficult or impracticable. Nevertheless, some of these drawbacks have beenovercome, and the modeling of several specific practical situations is possible, mainlydue to the development of knowledge of empirical relations that properly suit thesespecific processes. Present-day analytical techniques, such as the finite elementmethod, allow for the modeling of situations characterized by nonuniform thermalproperties that change with time, temperature, and location, so that great develop-ments can be expected in the modeling of heat transfer processes in foods.

The objective of this chapter is not to describe all of these specific models, but tointroduce the reader to a practical approach to the study of heat transfer in foods. Classicmodels constructed over assumptions of homogeneity and isotropy will be introducedin order to provide a basic knowledge that will enable the modeling of simple systemsand the understanding of more elaborate and specific models. Now we will focus onsome of the most important engineering properties of foods and how they can bemeasured or calculated in order to use them in further modeling of heat transfer processes.

2.2.1 THERMAL PROPERTIES OF FOODS

As stated before, the modeling of heat transfer processes is dependent on some ofthe physical properties of the foods involved. As mathematical techniques becomemore elaborate, a higher accuracy is needed in the measurement or calculation ofproperties such as specific heat, thermal conductivity, thermal diffusivity, and surfaceheat transfer coefficient. These properties will be discussed below.

2.2.1.1 Specific Heat

Specific heat (Cp) is an exclusive property of every substance, and it is defined asthe amount of energy needed to increase the temperature of one kilogram of matterby one degree Celsius; therefore, its units are J/kgC. This property is not dependentupon temperature or mechanical structure (e.g., density). It has been found that astrong correlation exists between a foods composition and its specific heat. Derivedfrom the definition of specific heat, the amount of heat Q required to increase thetemperature of a body with mass m from an initial temperature T1 to a final tem-perature T2 is:

Q = m Cp (T2 T1) (2.1)

In order to determine the Cp value for a specific substance or food, differentialscanning calorimetry (DSC) is used. Comprehensive data have been gathered, andtables containing Cp values for many products are available in the literature (Amer-ican Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1985).


However, as the water content and composition of foods have been found to affectthe Cp value, the data presented in the literature are bound to specific water contentsand formulations, thus reducing their usefulness. For a more practical way to findthe Cp value, several empirical mathematical expressions have been developed.These expressions can vary from product to product and sometimes even from authorto author. Some of these expressions are based only on the water content, havingthe form Cp = C1 + C2W, where W stands for water content and the constants aredefined depending on the situation and product. Examples of these expressions are:

Cp = 0.837 + 3.349W Siebel, 1982 (2.2)

Cp = 1.382 + 2.805W Dominguez et al., 1974 (2.3)

Cp = 1.470 + 2.720W Lamb, 1976 (2.4)

More complex expressions including other food components besides water havebeen developed, and some examples are shown below.

Cp = 2.309Xf + 1.256Xs + 4.187W Charm, 1971 (2.5)

Cp = 1.424Xc + 1.549Xp + 1.675Xf + 0.837Xa + 4.187WHeldman and Singh, 1981 (2.6)

where Xc stands for mass fraction of carbohydrates, Xp for proteins, Xf for fat, Xafor ash, and W for water. Generic expressions for mixtures are also available:

Choi and Okos, 1986a (2.7)

where Ci is the mass concentration of each constituent i and Cpi its corresponding Cp.

2.2.1.2 Thermal Conductivity

Thermal conductivity (k) is a characteristic that tells us how effective a material isas a heat conductor. As stated in Fouriers law for heat conduction, this constant isa proportionality factor needed for calculations of heat conduction transfer processes.This physical property can be directly measured from a food material using athermocouple and a heater, as described in detail by Mohsenin (1980). The suitedunits for this property are W/mC. Some data have been gathered and can be foundin the literature (American Society of Heating, Refrigerating and Air-ConditioningEngineers, Inc., 1985). Thermal conductivity of foods depends mainly on the com-position of the food, but factors such as fiber orientation and void spaces have aninfluence on the heat flow paths through food; therefore, it is important to describethe bulk condition in addition to the composition when reporting a k value. As inthe case of specific heat, some empirical expressions to calculate thermal conduc-tivity values for different foods have been developed. Some examples are:

For fruits and vegetables:

k = 0.148 + 0.493W Sweat, 1974 (2.8)

where W is the mass fraction of water.

Cp C Cpi ii

=( )


For sucrose solutions, fruit juices and milk:

k = 1.73 103(326.8 + 1.0412T 0.00337T2)(0.44 + 0.54W) Riedel, 1949(2.9)

where T is temperature expressed in degrees Celsius.Other applicable more general expressions for mixtures are of the type:

(2.10)

where ki is the thermal conductivity of the ith component and Xi

v its volume fraction.When heat is flowing through wrappings or compound layer materials, a total thermalconductivity coefficient is needed. This expression can be calculated if the individualthermal conductivity coefficients are known.

For heat flowing parallel to two layers, the total thermal conductivity coefficientis given by

kT = k1 (1 c) + k2c Hallstrom et al., 1988 (2.11)

where c is the volume fraction of material two. On the other hand, if the heat flowis perpendicular to the materials layer orientation:

Hallstrom et al., 1988 (2.12)

In the case of material mixtures with random size and orientation particles, thevalue for the total thermal conductivity value will be found between the value forparallel flow and the value for perpendicular flow. For mixtures of more than twocomponents, the same method is followed, taking two materials at a time.

2.2.1.3 Thermal Diffusivity

Thermal diffusivity () is a compounded thermal property of materials and iscalculated from the values found for thermal conductivity, specific heat, and densityof a particular product.

(2.13)

where k is thermal conductivity, Cp specific heat, and mass density. This thermo-physical property links conductivity of the materials with their ability to store heat,thereby showing how heat will diffuse throughout the materials when heated. The unitsfor thermal diffusivity are m2/sec. Although the recommended method to determinethermal diffusivity is a calculation based on experimentally measured values ofthermal conductivity, specific heat, and density, other heat diffusivity measurementmethods have been developed (Choi and Okos, 1983). Expressions based on watercontent and temperature of foods have also been developed. Some examples are:

= 5.7363 108W + 2.8 1010T Martens, 1980 (2.14)

k k Xi iv

i

=( )

kk k

ck c kT=

+ 1 2

1 21( )

= kCp


where W is the mass fraction of water and T is the temperature expressed in degreesKelvin, and

= 8.8 108(1 W) + wW Dickerson, 1969 (2.15)

where w is the thermal diffusivity of water at the studied temperature.As in the cases of other properties, the thermal diffusivity of mixtures can be

calculated with a general expression:

Choi and Okos, 1986b (2.16)

where i is the thermal diffusivity of the ith component and Xi its mass fraction. As

thermal diffusivity can be calculated from other thermal properties, data are notfrequently found in the literature.

2.2.1.4 Surface Heat Transfer Coefficient

The heat transfer coefficient (h) is not a property of materials themselves, but rathera property of convective heat transfer systems involving a solid surface and a fluid.This coefficient is used as a proportionality factor in Newtons law of cooling,adjusting for the characteristics of the system under study.

To define the value of this factor, it is necessary to characterize the convectivemedium and the surface involved in the convective heat transfer process. Some ofthe characteristics involved in the calculation of the heat transfer coefficient are thefluids velocity, viscosity, density, thermal conductivity, and specific heat. The formand surface texture of the solid involved are also important. As can be determinedfrom a dimensional analysis of Newtons cooling law, the units for the heat transfercoefficient are W/m2C. Since the heat transfer coefficient is a property of the systemrather than of the material, its measurement is difficult, and several empirical expres-sions have been developed to overcome this problem. Some of these expressionswill be reviewed below in the section dealing with heat transfer by convection.

A compilation of surface heat transfer coefficient data and empirical expressionsto calculate this coefficient can be found in the 1985 edition of the ASHRAEHandbook of Fundamentals. Further information regarding measurement of the heattransfer coefficient can be found in the literature (Arce and Sweat, 1980).

Now that we have shown methods for determining the most relevant thermalproperties of foods, we will find out how to use them to model heat transfer processes.The particulars of the different heat transfer modes will be dealt with in the followingsections.

2.2.2 HEAT TRANSFER BY CONDUCTION

Heat transfer in solids or highly viscous materials takes place by conduction. In thismode, energy is transferred among particles touching each other with no movementof material. As a solid material is heated on one of its faces, a gradient is establishedbetween the hot face and the opposite cooler face. This gradient is the driving forcepromoting the heat flow from one face to the other. As heat penetrates the body,

=( )i ii

X


the interior temperature changes from point to point and across time. This period oftime is known as the unsteady state period. Later, when heat has traveled all theway across the body and equilibrium in temperatures has been reached, the interiortemperature of each point will remain the same with respect to time and will onlydepend on its relative position inside the body. At this moment, the steady statetransfer regime has been reached, and the body is working as a heat conductor witha determined heat flux going through it.

Thus, the study of heat conduction can be divided into two main areas: the studyof heat conduction in the steady state and the study of heat conduction in thenonsteady state.

2.2.2.1 Steady State

The study of heat conduction in the steady state is useful in modeling the perfor-mance of insulators, heat exchangers, and other equipment used to transfer heat fromone point to another, such as containers, pans, and walls. In these, the initial varia-tions in internal temperature dependent on time have settled, and the temperatureprofile inside the material is stationary. The important issue here is to determine theamount of heat a particular material of a given thickness will allow to flow through it.

This mechanism can be modeled using Fouriers law for heat conduction(Equation 2.17), which establishes that the heat flux Qx transmitted through a solidin the direction x is inversely proportional to the thickness x and directly proportionalto both the perpendicular transmission area A and to the temperature differencebetween its two opposite faces T. The proportionality constant needed by this modelis the thermal conductivity (k), one of the physical characteristics previouslydescribed. In some materials, thermal conductivity may vary with temperature. Inthese situations, the value corresponding to the average temperature should be used.

(2.17)

The negative sign represents the heat flow from the hottest to the coolest surface,thereby rendering a positive value for the heat flux. As can be seen, this equationdescribes heat flow only in one direction. For a complete mathematical description, wewould need to write similar equations for the other two directions in a three-dimensionalsystem and to integrate over the entire volume. However, most steady state processingapplications involve heat conduction in only one direction, as when heat flows acrosswalls or heat exchangers. For practical purposes, these materials can be considered asinfinite slabs, so we do not need to be concerned with a general mathematical solution.

(2.18)

Equation (2.18) is the integrated form of Fouriers law for unidirectional steadystate heat conduction over a path of constant cross-sectional area in a parallelepiped.It can be used directly to calculate the heat flux through a body. Besides heat flowingthrough flat surfaces, another common situation in food engineering is the use of

Q kAT x

xx=

( )

Q kA T

xx=


cylindrical containers or pipes; therefore, an expression for Fouriers law on cylindricalsurfaces is necessary. That expression is shown in Equation (2.19),

(2.19)

where all the terms stand for the same characteristics as in the previous expressionexcept for Alm, which in this case is the logarithmic mean of the inner and outer areas:

(2.20)

Another common situation is the presence of compounded walls. The introduc-tion of the concept of thermal resistance can be of help when studying heat conduc-tion in this kind of system. If thermal resistance R is defined as the ratio of thethickness x over the area A and thermal conductivity k,

(2.21)

Fouriers law takes a form identical to Ohms law for electric current flow, and allmathematical manipulations can be performed as in electrical calculations.

(2.22)

(2.23)

Here, the flow of electric current I is directly proportional to the voltage differ-ence V and inversely proportional to the electric resistance R. In the same way,the heat flow Q is directly proportional to the temperature difference T and inverselyproportional to the thermal resistance R. This analogy shows that in the case ofseveral different layers placed in series (normal heat flow), the overall temperaturedifference is the sum of the individual temperature differences, and the same Q flowsthrough all the resistances. Therefore, the total thermal resistance is simply the sumof the individual resistances.

On the other hand, when there is parallel heat flow through several layers, theanalogy indicates that we have a parallel system. Therefore, the total Q is the sumof the Qx through the individual resistances, and the temperature differences for theindividual resistances are all the same and equal to the overall temperature difference.In this case, the reciprocal of the total thermal resistance is equal to the sum of thereciprocals of the individual resistances. The use of these analogies allows us topredict heat transfer in compound systems.

2.2.2.2 Nonsteady State

The study of heat conduction in the nonsteady state is pertinent when calculatingprocesses in which the focus is to heat or cool a body instead of using it as a heatconduction medium. Such processes include freezing, cooking, and thermal sterilization

Q kA T

xxlm=

AA A

AA

lm =

1 2

1

2ln

Rx

kA=

IV

R=

QT

R=


of foods. In these cases, the main concern is to find out how long it will take thecoldest/hottest point in the body to reach the desired temperature. Moreover, thestudy of this phenomenon allows us to determine the time needed to process foods ata desired temperature or the temperature at a determined point at a given time. Innonsteady state heat transfer processes in which more than one heat transfer modeapplies, conduction usually governs the process, as it is the slowest heat transfer mode.

It is important to consider that heat conduction in foods is frequently a three-dimensional phenomenon, as the food has finite dimensions and sometimes isimmersed in the heat transfer medium, with heat being transferred through all itssurfaces (as in oven baking or deep fat frying). Fouriers second law of heat transferfor three-dimensional nonsteady state heat conduction states that:

(2.24)

where T stands for temperature, t for time, x, y, and z for the distance on the x, yand z axes, respectively, and for thermal diffusivity, which is a physical character-istic of the materials, as previously discussed. To arrive at the analytical solution tothis complex expression can be difficult or impracticable. However, the analysis ofsome practical situations may be simplified through a couple of useful assumptions.

The first assumption supposes that we are dealing with a semi-infinite body, alsoknown as a thick solid. This semi-infinite body is defined as one with infinite width,length, and depth. If a body with these characteristics is immersed in the heatingmedium, we can assume that heat will be transferred just from the surface towardthe interior, following a straight trajectory to the center; therefore, Fouriers law canbe transformed into:

(2.25)

where the x-axis corresponds to any one of the dimensions on a parallelepiped orto the radius on a sphere-like body. The solution to Equation (2.25) becomes simplernow and may be obtained by applying the boundary conditions, which are:

T0 Initial temperature of the body at time 0Tm Heating medium temperature acquired at the surface when immersed in

the fluid.

The solution takes the form of Equation (2.26), where the temperature T at anypoint x, measured from the surface, can be determined using a dimensionless tem-perature ratio:

(2.26)

where erf is the Gauss error function, which can be obtained from Table 2.2, and kis the thermal conductivity.

=

+

+

Tt

Tx

Ty

Tz

2

2

2

2

2

2

=

Tt

Tx

2

2

T T

T Terf

xt

e erfx

th t

km

m

hxk

hk

t

=

+

+

+

0 4 4

2


Since, in this case, the heat is transferred from a fluid to a solid surface, thesurface heat transfer coefficient h is incorporated into the solution. For situationswhere the heat transfer between the fluid and the solid proceeds very efficiently, thatis, if h is infinite, the solid surface takes the medium temperature instantaneously,and Equation (2.26) can be simplified to:

(2.27)

Now that we know how to handle semi-infinite bodies, a problem arises. Thedefinition of a semi-infinite body describes a body with infinite dimensions, whichin reality is not a possible situation. When can a body be considered as a semi-infinitebody for practical purposes? The concept of a semi-infinite body in practice is betterrelated to the systems behavior than to the body dimensions. The surface heattransfer coefficient, along with the thermal conductivity, plays a fundamental rolein determining whether a body will exhibit a thick body response. If a finite bodyof a given thickness is studied in the early steps of heat penetration, it will exhibita semi-infinite body response as heat has only flowed from the outside to the center;therefore, the ratio of thickness to time plays an important role as well. Schneider(1973) found a critical Fourier number to define when a body ceases to exhibit athick body response:

Focritical = 0.00756 Bi0.3 + 0.02 for 0.001 Bi 1000 (2.28)

TABLE 2.2Gauss Error Function

erf () erf () erf ()

0.00 0.00000 0.68 0.66278 1.80 0.989090.04 0.45110 0.72 0.69143 1.90 0.992790.08 0.09008 0.76 0.71754 2.00 0.995320.12 0.13476 0.80 0.74210 2.10 0.997020.16 0.17901 0.84 0.76514 2.20 0.998140.20 0.22270 0.88 0.78669 2.30 0.998860.24 0.25670 0

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