optimization in food engineering

Optimization inFood Engineering

2008 by Taylor & Francis Group, LLC.

Contemporary Food Engineering

Series Editor

Professor Da-Wen Sun, DirectorFood Refrigeration & Computerized Food Technology

National University of Ireland, Dublin(University College Dublin)

Dublin, Irelandhttp://www.ucd.ie/sun/

Optimization in Food Engineering, edited by Ferruh Erdogdu (2009)

Advances in Food Dehydration, edited by Cristina Ratti (2009)

Optical Monitoring of Fresh and Processed Agricultural Crops, edited by Manuela Zude (2009)

Food Engineering Aspects of Baking Sweet Goods, edited by Servet Glm Sumnu and Serpil Sahin (2008)

Computational Fluid Dynamics in Food Processing, edited by Da-Wen Sun (2007)


Optimization inFood Engineering

Edited by

Ferruh Erdogdu

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York


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2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

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Dedication

To my mother, Aynur for all her love, support,

patience and efforts giving me the best education possible,

to my wife Belgin and my sister Aylin for their love,

support and patience and to the memories of my father, 2008 by Taylor & Francis Group, LLC.Feran and my uncle, Seref.

ContSeries Editor s PrefacePrefaceSeries Editor

PART I Modeling: Signi cance, Fundamentals,

Chapter 2 Anal ytical So lutions in Conducti on Heat Transfer Proble ms

Chapter 3 Num erical Solutions : Finit e Di fference Meth ods

PART II OptimizationChapter 5 Opt imization: An Introducti on

Ferruh Erdogdu

Chapter 6 Stati stical Opt imization: Response Surface Meth odolog y .

Kun-Nan Chen and Ming-Ju ChenChapter 4 Num erical Solutions : Finit e Eleme nt and Finit eVol ume Meth ods

Rui C. Martins, Vitor V. Lopes, Antnio A. Vicente,and Jos A. Teixeira 2008 by Taylor &T. Koray Palazoglu and Ferruh ErdogduFerruh Erdogdu and Mahir TurhanQuang Tri Ho, Hibru Kelemu Mebatsion, Bart Nicola,and Pieter VerbovenChapter 1 Signi cance of Mathemat ical Model ing and Simu lationfor Optimizat ionand MethodsEditorContributor s entsFrancis Group, LLC.

Chapte r 7

Chapte r 8

Chapte r 9

Chapte r 10 Neur al Net works and Gene tic Algo rithms

in Food Process ing

and Appl ications

Chapte r 14 Eig envalue Optim ization Techniqu es for Nonline ar

Chapte r 15

Chapte r 16 Philip Doganis and Haralambos Sarimveis 2008 by Taylor &Mixe d Integer Linear Program ming Schedul ingin the Fo od Indust ryFerruh Erdogdu and Murat O. BalabanCom plex Meth od Opt imizationLuis G. Matallana, Anbal M. Blanco,and J. Alberto BandoniDyna mic Anal ysis and Des ignMekapati Srinivas and Gade Pandu RangaiahChapte r 13 Tab u Search: Deve lopment, Algori thm, Performance ,J. Ricardo Prez-Correa, Claudio A. Gelmi,and Lorenz T. BieglerChapte r 12 Dyna mic Optim izationFerruh ErdogduChapte r 11 Com putational Fluid Dynami cs for Optimizat ionYang Meng and Hosahalli S. RamaswamyAndrey V. KuznetsovAppl ications o f the Minimu m Princi ple of Pontryag infor Solvi ng Opt imal Con trol Proble msCheah Keen Seng and Gade Pandu RangaiahMult i-Obje ctive Optimizat ion in Food Engineer ingShuryo Nakai, Yasumi Horimoto, Jinglie Dou,and Roxana A. VerdiniRan dom-Ce ntroid Optim izationFrancis Group, LLC.

Chapter 17 Mixe d Integer Nonline ar Programmi ng: Appl icationsto Food Dehydrati on and Deep Chilling

PART III Optimization Studies for Different

the Performance of Batch Reactors

Measu reme nt and Optim ization

Chapter 20 Opt imization of Freeze-Dry ing Pr ocess Applied to Fo od

Chapter 21 Opt imization of Spray Dry ing of Sugar-Ri ch Foods

Bev erage Con tainers

Chapter 24 Rea l-Time Nonline ar Optimal Control of Refrige rationProce sses 2008 by Taylor &Ioan Cristian TreleaChapter 23 Opt imization for Con tinuous Shorte st Pathsin Transpor tation

J. Miguel Daz-BezKoetsu Yamazaki, Jing Han, and Sadao NishiyamaChapter 22 Struct ural Optim ization Techniqu es for Deve lopingVinh TruongMichle Marin, Stphanie Passot, Fernanda Fonseca,and Ioan Cristian Treleaand Biolog ical Product s: From Response SurfaceMeth odolog ies to an Inter active ToolSundaram GunasekaranChapter 19 Pu lsed Microw ave Heating of Foods: Tempera tureIqbal M. MujtabaChapter 18 Opt imization and Control Strat egy to ImproveFood ProcessesPanagiotis P. Repoussis and Christos T. KiranoudisFrancis Group, LLC.

Chapte r 25 Opt imization of Apple Juice Extract ion

Mara Teresa Gonzlez and Martn Juan Urbicain

Chapte r 26 Opt imization of Canned Food Process ing

Ricardo Simpson and Arthur A. Teixeira

Chapte r 28

Chapte r 29

Chapte r 30

Chapte r 31 2008 by Taylor &Chapte r 32 Opt imizin g the Manage ment of Curing Cha mbers

Jose Bon and Antonio MuletNoem C. Petracci, and J. Alberto Bandoni

Guillermo L. Masini, Anbal M. Blanco,Opt imal Operational Planni ng in the Fruit Indust rySu pply Cha inMuhiddin Can and A. Burak EtemogluOpt imization of the Arrays of Impingi ng JetsReinaldo Morabito and Vitria PurezaKevin Cronin and Philippe Baucour

Loa ding Opt imizationin Thermal Process ing of Pac kaged Fo ods

Pro cess Optim ization Strategies to Reduce Var iabilityChapte r 27 Opt imal Design of Contin uous Thermal Process ingwi th Plate Heat Exc hangers

Jorge Andrey Wilhelms Gut and Jos Maurcio PintoFrancis Group, LLC.

Series Editors PrefaceCONTEMPORARY FOOD ENGINEERING

Food engineering is the multidisciplinary eld of applied physical sciences combinedwith the knowledge of product properties. Food engineers provide the technologicalknowledge transfer essential to the cost-effective production and commercialization offood products and services. In particular, food engineers develop and design processesand equipment in order to convert raw agricultural materials and ingredients into safe,convenient, and nutritious consumer food products. However, food engineering topicsare continuously undergoing changes to meet diverse consumer demands, and thesubject is being rapidly developed to reect market needs.

In the development of food engineering, one of the many challenges is to employmodern tools and knowledge, such as computational materials science and nano-technology, to develop new products and processes. Simultaneously, improving foodquality, safety, and security remain critical issues in food engineering study. Newpackaging materials and techniques are being developed to provide more protectionto foods, and novel preservation technologies are emerging to enhance food securityand defense. Additionally, process control and automation regularly appear amongthe top priorities identied in food engineering. Advanced monitoring and controlsystems are developed to facilitate automation and exible food manufacturing.Furthermore, energy saving and minimization of environmental problems continueto be an important food engineering issue and signicant progress is being made inwaste management, efcient utilization of energy, and reduction of efuents andemissions in food production.

The Contemporary Food Engineering series, consisting of edited books,attempts to address some of the recent developments in food engineering. Advancesin classical unit operations in engineering applied to food manufacturing are coveredas well as such topics as progress in the transport and storage of liquid and solidfoods; heating, chilling, and freezing of foods; mass transfer in foods; chemicaland biochemical aspects of food engineering and the use of kinetic analysis;dehydration, thermal processing, nonthermal processing, extrusion, liquid food con-centration, membrane processes, and applications of membranes in food processing;shelf-life, electronic indicators in inventory management, and sustainable technolo-gies in food processing; and packaging, cleaning, and sanitation. The books areaimed at professional food scientists, academics researching food engineeringproblems, and graduate level students.

The books editors are leading engineers and scientists from many parts of theworld. All the editors were asked to present their books to address the market needand pinpoint the cutting-edge technologies in food engineering.

Furthermore, all contributions are written by internationally renowned expertswho have both academic and professional credentials. All authors have attempted to 2008 by Taylor & Francis Group, LLC.

provide critical, comprehensive, and readily accessible information on the art andscience of a relevant topic in each chapter, with reference lists for further informa-tion. Therefore, each book can serve as an essential reference source to students andresearchers in universities and research institutions.

Da-Wen Sun, Series Editor 2008 by Taylor & Francis Group, LLC.

PrefaceFood engineering has gained more and more signicance in the last couple ofdecades. Mathematical models have been used to better understand and improvefood processing operations, and in this concept, various optimization approacheshave played a signicant role. As a result, there has been a dramatic increase in theefciency and reliability of optimization methods for different problem categories.

Optimization methods can be easily applied in food processing as long as thechanges during a process can be predicted mathematically. This case, of course,depends on the presence of mathematical models. Since heat, mass, and momentumtransfers are major mechanisms in food processing, mathematical models describingthese phenomena are also required for further mathematical-based optimizationprocedures. Within this context, mathematical optimization plays an important rolein optimizing different food processing operations.

Excellent text and reference books are available for educational and researchpurposes in the eld of optimization and food processing. It will therefore be quitesignicant to combine the advantages in this eld for further optimization strategiesto improve the quality and safety of food processes and optimal operating policies inthe food industry.

Based on this concept, Optimization in Food Engineering has been divided intothe following sections to serve as a reference for professional food scientists, foodengineers, academicians, and graduate level students working in the eld of foodengineering and processing.

This book consists of three parts. In the rst part, the signicance of modeling,fundamentals, and methods are covered for analytical and numerical procedures,since an optimization procedure depends on the presence of an effective mathemat-ical model. It is a known fact that knowledge of mathematical modeling techniquesprovides signicant information for further research and developments in foodprocessing. In addition, the changes predicted by a model in a given process arerequired if the given process is described other than by trial-and-error physicalexperiments.

In the second part, optimization and different optimization techniques are pre-sented. This part begins with statistical optimization techniques and continues withPontryagins method, multi-objective and dynamic optimization techniques, andmixed integer linear and nonlinear programming methodologies. In addition, limi-tations and possibilities of using neural networks and genetic algorithms and com-putational uid dynamics programming approaches are presented with tabu search,complex method, and Eigenvalue optimization techniques. Finally, in the last part,optimization studies for different food processes are discussed. This part covers abroad area for different processes starting from the optimization strategies to improvethe performance of batch reactors to the optimization of conventional thermalprocessing, microwave heating, freeze drying, spray drying, and refrigerationsystems. Different food processing areas are presented for optimization purposes, 2008 by Taylor & Francis Group, LLC.

and structural optimization techniques for developing beverage containers are dis-cussed. Loading optimization, optimization approaches for impingement processing,and optimal operational planning methodologies are also covered. In each chapter,the required parameters for the given process are presented in detail along with theoptimization procedures that need to be applied.

Ferruh Erdogdu 2008 by Taylor & Francis Group, LLC.

Series Editor

ally and delivered keynote speeches aauthority in food engineering, he hasprofessorships from 10 top universitiShanghai Jiaotong University, HarbinUniversity, South China University orecognition of his signicant contribuhis outstanding leadership in the eld,Engineering (CIGR) awarded him the Cand the Institution of Mechanical Enghim Food Engineer of the Year 2004,Award in recognition of his distinguAgricultural Engineering scientists around the world. 2008 by Taylor & Francis Group, LLC.t international conferences. As a recognizedbeen conferred adjunct=visiting=consulting

es in China including Zhejiang University,Institute of Technology, China Agriculturalf Technology, and Jiangnan University. Intion to food engineering worldwide and forthe International Commission of AgriculturalIGR Merit Award in 2000 and again in 2006ineers based in the United Kingdom namedin 2008 he was awarded CIGR Recognitionished achievements as top one percent ofProfessorDa-Wen Sunwas born in SouthernChina and is a world authority on food engin-eering research and education. His mainresearch activities include cooling, drying,and refrigeration processes and systems; qual-ity and safety of food products; bioprocesssimulation and optimization; and computervision technology. His innovative studies onvacuum cooling of cooked meats, pizza qual-ity inspection by computer vision, and ediblelms for shelf-life extension of fruits and

vegetables have been widely reported in national and international media. Results ofhis work have been published in over 180 peer-reviewed journal papers and more than200 conference papers.

Professor Sun received rst class BSc honors and MSc in mechanical engineer-ing, and a PhD in chemical engineering in China before working in various univer-sities in Europe. He became the rst Chinese national to be permanently employed inan Irish university when he was appointed college lecturer at National University ofIreland, Dublin (University College Dublin), Ireland, in 1995, and was then con-tinuously promoted in the shortest possible time to senior lecturer, associate profes-sor, and full professor. Sun is now professor of Food and Biosystems Engineeringand director of the Food Refrigeration and Computerized Food Technology ResearchGroup in University College Dublin.

As a leading educator in food engineering, Sun has contributed signicantly tothe eld of food engineering. He has trained many PhD students, who have madetheir own contributions to the industry and academia. He has also, on a regular basis,given lectures on advances in food engineering in academic institutions internation-

He is a fellow of the Institution of Agricultural Engineers. He has also receivednumerous awards for teaching and research excellence, including the PresidentsResearch Fellowship, and has received the Presidents Research Award from Uni-versity College Dublin on two occasions. He is a member of the CIGR executiveboard and honorary vice president of CIGR; editor-in-chief of Food and BioprocessTechnologyAn International Journal (Springer); series editor of ContemporaryFood Engineering (CRC Press=Taylor & Francis); former editor of Journal of FoodEngineering (Elsevier); and editorial board member for Journal of Food Engineering(Elsevier), Journal of Food Process Engineering (Blackwell), Sensing and Instru-mentation for Food Quality and Safety (Springer), and Czech Journal of FoodSciences. He is also a chartered engineer registered in the U.K. Engineering Council. 2008 by Taylor & Francis Group, LLC.

EditorDr. Ferruh Erdogdu is an associate professor of food engineering at the Universityof Mersin, Mersin, Turkey. He was born in Eregli, Turkey, and graduated from theDepartment of Food Engineering at Hacettepe University in Ankara in 1992 withhonors and the highest GPA. In 1994, he succeeded in a nationwide exam by theMinistry of National Education of Turkey to pursue masters and PhD degrees in foodengineering in the United States.Dr. Erdogdu received his master of engineering degree in 1996 and PhD in 2000at the University of Florida, Gainesville, Florida. While working with Dr. MuratO. Balaban at the University of Florida, he maintained a status of distinguishedscholar. He received outstanding academic achievement awards from College ofEngineering (19972000) and College of Agriculture (1999), and won the studentpaper competition hosted by the food engineering division of the Institute of FoodTechnologists (IFT). After receiving his PhD, he conducted his postdoctoral work atthe University of California, Davis, California, with Dr. R. Paul Singh.

In 2001, Dr. Erdogdu joined the faculty of food engineering at the University ofMersin where he has been teaching undergraduate- and graduate-level courses ontopics in food engineering. In 2007, he was appointed holder of a scholarship withinthe SwedishTurkish Programme by the Swedish Institute for studies=research workat Lund University, Lund, Sweden. Ferruh is the author or coauthor of more than 30research papers published in internationally known peer-reviewed journals, 4 bookchapters, and more than 50 presentations. He is the coauthor of the books VirtualExperiments in Food Processing, published in 2004, and Industrial Scale FoodFreezing Simulation Software published by the World Food Logistics Organization.

Dr. Erdogdu is a professional member of the IFT. He has been serving on theeditorial board of the Journal of Food Process Engineering since 2003 and assistingin review processes for Journal of Food Engineering; Journal of Food and Biopro-cess Technologies; Journal of Food Technology and Biotechnology; InternationalJournal of Engineering; Computers and Chemical Engineering; and ChemicalProcess Engineering.

His current research interests include mathematical modeling and optimizationof heat, mass, and momentum transfer operations in food processing. 2008 by Taylor & Francis Group, LLC.

Contributors

Murat O. BalabanFishery Industrial Technology CenterUniversity of Alaska FairbanksFairbanks, Alaska

J. Alberto BandoniPlanta Piloto de Ingeniera QuimicaBaha Blanca, Argentina

Philippe BaucourInstitut FEMTO-ST, Department ofCREST

Belfort, France

Lorenz T. BieglerDepartment of Chemical EngineeringCarnegie Mellon UniversityPittsburgh, Pennsylvania

Anbal M. BlancoPlanta Piloto de Ingeniera QuimicaBaha Blanca, Argentina

Jose BonResearch Group Analysis andSimulation of Agro-food Processes

Food Technology DepartmentPolytechnic University of ValenciaValencia, Spain

Muhiddin CanUludag UniversityFaculty of Engineering and ArchitectureMechanical Engineering DepartmentGorukle CampusBursa, Turkey

Ming-Ju ChenDepartment of Animal Science andTechnology

National Taiwan UniversityTaipei, Taiwan

Kevin CroninDepartment of Process and ChemicalEngineering

University College CorkCork, Ireland

J. Miguel Daz-BezUniversidad de SevillaDepartament de Matemtica Aplicada IIEscuela Superior de IngenierosSevilla, Spain

Philip DoganisNational Technical Universityof Athens,

School of Chemical Engineering,Zografou Campus,Athens, Greece

Jinglie DouUniversity of British ColumbiaFood, Nutrition and HealthVancouver, British Columbia, Canada

Ferruh ErdogduDepartment of Food EngineeringUniversity of Mersiniftlikky-Mersin, Turkey

A. Burak EtemogluUludag University, Faculty ofKun-Nan ChenDepartment of Mechanical EngineeringTungnan UniversityTaipei, Taiwan

Engineering and ArchitectureMechanical Engineering DepartmentGorukle CampusBursa, Turkey 2008 by Taylor & Francis Group, LLC.

Fernanda Fonseca Christos T. Kiranoudis

INRA School of Chemical Engineering

Joint Research Unit Gnie etMicrobiologie des ProcdsAlimentaires AgroParisTech, INRA

ThivervalGrignon, France

Claudio A. GelmiDepartment of Chemical andBioprocess Engineering

Ponticia Universidad Catlicade Chile

Santiago, Chile

Mara Teresa GonzlezPlanta Piloto de Ingeniera QumicaBaha Blanca, Argentina

Sundaram GunasekaranBiological Systems EngineeringDepartment

University of Wisconsin-MadisonMadison, Wisconsin

Jorge Andrey Wilhelms GutDepartment of ChemicalEngineeringEscolaPolitcnica

University of So PauloSo Paulo, Brazil

Jing HanUniversal Can CorporationShizuoka, Japan

Quang Tri HoBIOSYST-MeBioS, K.U.LeuvenLeuven, Belgium

Yasumi HorimotoUniversity of British ColumbiaFood, Nutrition and HealthVancouver, British Columbia, Canada

Department of Process Control andPlant Design

National Technical Universityof Athens

Athens, Greece

Andrey V. KuznetsovDepartment of Mechanical andAerospace Engineering

North Carolina State UniversityRaleigh, North Carolina

Vitor V. LopesInstitute of Systems and RoboticsTechnical University of LisbonLisbon, Portugal

Michle MarinAgroParisTech, INRAJoint Research Unit Gnie etMicrobiologie des ProcdsAlimentaires AgroParisTech, INRA


Rui C. MartinsBioInformaticsMolecular andEnvironmental Research Centre

University of MinhoBraga, Portugal

Guillermo L. MasiniFacultad de IngenieraDepartamento de Mecnica AplicadaUniversidad Nacional del ComahueNeuqun, Argentina

Luis G. MatallanaPlanta Piloto de Ingeniera QumicaBaha Blanca, Argentina


Hibru Kelemu Mebatsion T. Koray Palazoglu

BIOSYST-MeBioS, K.U. Leuven Department of Food Engineering

Leuven, Belgium

Yang MengDepartment of Food ScienceMcGill University Macdonald CampusSte-Anne-de-Bellevue, Quebec, Canada

Reinaldo MorabitoDepartment of Production EngineeringUniversidade Federal de So CarlosSo Paulo, Brazil

Iqbal M. MujtabaSchool of Engineering, Design andTechnology

University of BradfordBradford, England

Antonio MuletResearch group Analysis andSimulation of Agro-food Processes

Food Technology DepartmentPolytechnic University of ValenciaValencia, Spain

Shuryo NakaiUniversity of British ColumbiaFood, Nutrition and HealthVancouver, British Columbia, Canada

Bart NicolaBIOSYST-MeBioS, K.U. LeuvenLeuven, Belgium

Sadao NishiyamaUniversal Can CorporationTokyo, Japan

University of Mersiniftlikky-Mersin, Turkey

Stphanie PassotAgroParisTechJoint Research Unit Gnie etMicrobiologie des ProcdsAlimentaires AgroParisTech, INRA


J. Ricardo Prez-CorreaDepartment of Chemical and BioprocessEngineering

Ponticia Universidad Catlica de ChileSantiago, Chile

Noem C. PetracciPlanta Piloto de Ingeniera QumicaBaha Blanca, Argentina

Jos Maurcio PintoAdvanced Control and OperationsResearch Technology Group

Praxair, Inc.Danbury, Connecticut

Vitria PurezaDepartment of ProductionEngineering

Universidade Federal de So CarlosSo Paulo, Brazil

Hosahalli S. RamaswamyDepartment of Food ScienceMcGill University MacdonaldCampus

Ste-Anne-de-Bellevue,Qubec, Canada


Gade Pandu RangaiahDepartment of Chemical andBiomolecular Engineering

National University of SingaporeSingapore, Republic of Singapore

Panagiotis P. RepoussisSchool of Chemical EngineeringDepartment of Process Control andPlant Design

National Technical Universityof Athens

Athens, Greece

Haralambos SarimveisNational Technical Universityof Athens

School of Chemical EngineeringAthens, Greece

Cheah Keen SengDepartment of Chemical andBiomolecular Engineering


Ricardo SimpsonDepartamento de Procesos QumicosBiotecnolgicos, y AmbientalesUniversidad Tcnica FedericoSanta Mara

Valparaso, Chile

Mekapati SrinivasDepartment of Chemical andBiomolecular Engineering


Arthur A. TeixeiraDepartment of Agricultural andBiological Engineering

Institute of Food and AgriculturalSciences

University of FloridaGainesville, Florida

Jos A. TeixeiraInstitute for Biotechnology andBioEngineering

Centro de Engenharia Biolgica,University of MinhoBraga, Portugal

Ioan Cristian TreleaAgroParisTechJoint Research Unit Gnie etMicrobiologie des ProcdsAlimentaires AgroParisTech, INRA


Vinh TruongDepartment of Chemical EngineeringNong Lam UniversityHo Chi Minh, Vietnam

Mahir TurhanDepartment of Food EngineeringUniversity of Mersiniftlikky-Mersin, Turkey

Martn Juan Urbicain (Deceased)Planta Piloto de IngenieraQumica

Baha Blanca, Argentina

Pieter VerbovenBIOSYST-MeBioS, K.U. LeuvenLeuven, Belgium


Roxana A. VerdiniInstituto de DesarrolloTechnolgico para la IndustriaQumica

Santa Fe, Argentina

Antnio A. VicenteInstitute for Biotechnology andBioEngineering

Centro de Engenharia BiolgicaUniversity of MinhoBraga, Portugal

Koetsu YamazakiDivision of Innovative Technologyand Science

Graduate School of NaturalScience and Technology

Kanazawa UniversityKanazawa, Ishikawa, Japan


Part I

Modeling: Signicance,Fundamentals, and Methods 2008 by Taylor & Francis Group, LLC.

1 Signicance of

1.4.2 Kinetics Parameters ........................................................................... 11

1.5 Solution Methods ......................................................................................... 14

1.5.1 General Considerations ..................................................................... 141.5.2 Case Study: Solution of Three-Dimensional Gas Exchange

and Respiration in Pear Fruit ............................................................ 141.6 Towards Food Process Modeling at Different Scales:

Multiscale Modeling .................................................................................... 151.7 Conclusion ................................................................................................... 16Acknowledgments................................................................................................... 16Nomenclature .......................................................................................................... 16

Greek Letters ................................................................................................ 17References ............................................................................................................... 17Mathematical Modelingand Simulation forOptimization

Quang Tri Ho, Hibru Kelemu Mebatsion,Bart Nicola, and Pieter Verboven

CONTENTS

1.1 Introduction .................................................................................................... 31.2 Heat and Mass Transfer Modeling ................................................................ 4

1.2.1 General Considerations ....................................................................... 41.2.2 Case Study: PermeationDiffusionReaction Model

of Gas Exchange in Pear Fruit ............................................................ 61.3 Kinetics Modeling.......................................................................................... 7

1.3.1 General Considerations ....................................................................... 71.3.2 Case Study: Model for the Respiration of Fruit ................................. 9

1.4 Model Parameters ........................................................................................ 111.4.1 Thermophysical Properties ................................................................ 11 2008 by Taylor & Francis Group, LLC.

The merits and limitati ons of food proces s simulation are demon strated by meansof an illustrativ e case study that is explor ed in all its model ing facet s throu ghout the

chapter. We demon strate model ing to the appli cation of ultr a low oxygen stor age ofpears. Pears are typi cally stored under a contr olled atmospher e with reduced O2 andincreased CO2 level s to extend their commerci al stor age lif e, whic h can be as long as9 mont hs. The exact opti mal gas condit ions depend on factors such as cult ivar,origin, growing condition s, and picking date of the frui t. At too low-oxyg en con-centratio n, anoxia may occur eventu ally leadi ng to cell d eath and loss of theproduct. Ot her fruit such as apples are considerabl y less sensitiv e to varia tion s inlow o xygen condit ions. This is probably related to diff erences in ga s concent rationgradients resul ting from diff erences in tissue diffusivity and respirato ry acti vity.There is littl e informat ion about such gas gradi ents in fruit. Know ledge on inte rnalgas exchange woul d be, nevert heless, very valuab le to guide commerci al storagepractices since disorde rs un der controlle d atmospher e related to ferm entation are aprime cause of concern . Opt imal stor age condit ions of new cultivars are generallydetermin ed by tedi ous e xperiment al trials that shoul d cover severa l grow ing years.Modeling will help bett er understand the proces ses of gas exchange and kinet ics ofrespiration associated with frui t storage potential and will allow perfor ming numer-ical experiment s to deter mine optimal stor age condition s.

This chapter is subdivided as follow s. In Secti on 1.2, mathemat ical modelingof heat and mass trans fer is introduced. In Se ction 1.3, model ing of kinet ics isdescribed. Secti on 1.4 dea ls with model param eters. Solution metho ds are mentionedin Sectio n 1.5 and in Se ction 1.6 a future perspectiv e is given in terms of multisca lemodeling of food proces ses. Finally some con clusions are draw n in Sec tion 1.7. Thechapter is intended to give the reader a framework and avor for the followingdetailed chapters.

1.2 HEAT AND MASS TRANSFER MODELING

1.2.1 GENERAL CONSIDERATIONS

In general, heat and mass transport occurs by diffusion as well as convectivemechanisms. In its physical denition, diffusion is due to the spontaneous netmovement of particles from high to low concentration. For heat transfer, the con-duction term is more often used and refers to the transfer of thermal energy froma region of higher temperature to a region of lower temperature. However, diffusionis often used as an apparent mechanism encompassing more complex micro- and1.1 INTRODUCTION

This ch apter introduces fundam ental mecha nisms of heat mass transfer and math-ematical basic s for modeling a spects involving k inetics (such as bioche micalchanges , qu ality, and safet y). Math ematical aspects of form ulating and solvingmodels that describ e time-dep endent spatial and couple d phenom ena in food pro-cesses are outl ined and explained. Model ing issues related to thermop hysicalproperties and model param eters, such as varia bility and parameter estimat ion, arementioned , and tool s for food process simulat ion at different scale s are presen ted. 2008 by Taylor & Francis Group, LLC.

nanoscale phenomena such as pressure driven ow, Knudsen ow, and capillaryow. The driving force behind convective transport is a pressure gradient in thecase of forced convection (e.g., due to a pump), or density differences because of,e.g., temperature gradients. For simplicity, the discussion in this chapter is restrictedto a single Newtonian system. This means that the materials for which there is alinear relationship between shear stress and velocity gradient, such as water orair will be considered. More complicated uids such as ketchup, starch solutionsetc., are so-called non-Newtonian uids, and the reader is referred to standard bookson rheology for more details. In case of solid materials, convection will not besignicant.

Applying the conservation principle to a xed innitesimal control volumedx1dx2dx3 one obtains the mass continuity, momentum and energy, and mass fractionequations, written in index notation for Cartesian coordinates xi (i 1, 2, 3 for thex-, y-, and z-direction, respectively), and whenever an index appears twice in anyterm, summation over the range of that index is implied (for example, @ruj@xj becomes@ru1@x1 @ru2@x2

@ru3@x3

in reality). This model can be applied to any such system

@r

@t @ruj

@xj 0 (1:1)

@rui@t

@rujui@xj

@@xj

h@ui@xj @uj@xi

@@xi

p 23h@uj@xj

fi (1:2)

@rH

@t @rujH

@xj @

@xjk@T

@xj

@p

@t Q (1:3)

@rXa@t

@@xj

rujXa @@xi

rDa@

@xiXa ra (1:4)

For a full derivation of these equations we refer to any text book on uid mechanics.The system of at least ve equations (three equations for the velocity componentsplus the continuity and the energy equation), and added with a mass fractionequation for each component of interest) contains at least eight variables (u1, u2,u3, p, H, T, Xa, r). Therefore, additional equations to close the system are required.Thermodynamic equation of state gives the relation between density r and pressure pand temperature T. The constitutive equation relates the enthalpy h to the pressureand the temperature by means of the heat capacity c @H@T

p. There are no conclu-

sive general rules for implementation of boundary conditions for the NavierStokesequations to have a well-posed problem because of their complex mathematicalnature. For incompressible and weakly compressible ows, it is possible to deneDirichlet boundary conditions (xed values of the variables mostly for an incomingow), Neumann boundary conditions (xed gradients, mostly for an outgoing ow),and wall boundary conditions (a wall function reecting the system behavior at thesolid boundaries at the edge of the system considered). Initial values must beprovided for all variables. 2008 by Taylor & Francis Group, LLC.

1.2.2 CASE STUDY: PERMEATIONDIFFUSIONREACTION MODELOF GAS EXCHANGE IN PEAR FRUIT

The tissue structure of pear fruit is considered to contain mainly two phases:intracellular liquid phase of the cells and air-lled intercellular space. Assuminglocal equilibrium at a certain concentration of the gas component i in the gas phaseCi,g (mol m

3), concentration of the compound in the liquid phase of fruit tissuenormally follows Henrys law. If the tissue has a porosity , the volume-averagedconcentration Ci,tissue (mol m

3) of species i is then dened as

Ci,tissue Ci,g (1 ) R T Hi Ci,g (1:5)

with Hi is Henrys constant of component i (i is O2, CO2, or N2). From this denition,following expression for the gas capacity (ai) of the component i of the tissue isderived

ai (1 ) R T Hi Ci,tissueCi,g (1:6)

A permeationdiffusionreaction model was constructed describing the diffusionand permeation processes in pear tissue for the three major atmospheric gases O2,CO2, and N2. Equations for transport of O2, CO2, and N2 were established byHo et al. (2008)

ai@Ci@tr (uCi) r DirCi Ri (1:7)

with boundary conditions at the external surface of the pear:

Ci Ci,1 (1:8)

whereRi is the production term of the gas component i related to O2 consumptionor CO2 production

r (m1) is the gradient operator

The index1 refers to the gas concentration of the ambient atmosphere. The rst termin Equation 1.7 represents the accumulation of gas i, the second term permeationtransport driven by an overall pressure gradient, the third term molecular diffusiondue to a partial pressure gradient, and the last term consumption or production of gasi because of respiration or fermentation. If, for example, oxygen is consumed in thefruit center, it creates a local partial pressure gradient, which drives molecular diffu-sion. However, if the rates of transport of different gasses are different, overallpressure gradients may build up and cause permeation transport. Permeation throughthe barrier of tissue by the pressure gradient is described by Darcys law (Geankoplis,1993), which in fact is an apparent form of the momentum equation outlined above 2008 by Taylor & Francis Group, LLC.

82C6u KmrP K R T

mr

XCi

(1:9)

The relation between gas concentration and pressure was assumed to follow the

800.5

4

0 000 0.02 0.020.020.02x (m) x (m)x (m)x (m)

FIGURE 1.1 Finite element mesh of pear geometry and simulated gas partial pressuredistribution in pear intact fruit using permeationdiffusionreaction model. Simulation wascarried out at 18C, 20 kPa O2, 0 kPa CO2 at the ambient atmosphere and applied to an axi-symmetrical pear shape. Parameters were taken from Ho, Q.T. et al., PLoS. Comput. Biol., 4,e1000023, 2008.92

90

88

86

84

3.5

3

2.5

2

1.5

1 N2

parti

al p

ress

ure

(kPa)

O2

parti

al p

ress

ure

(kPa)

O2

parti

al p

ress

ure

(kPa)

20

18

16

14

12

10

8ideal gas law (PCRT). A typical model simulation result for gas exchange inpear tissue is shown in Figure 1.1. An axi-symmetric geometry model was createdfor the pear. Simulation showed that, due to the respiration of the tissue, the O2 gaspartial pressure decreased from surface to the pear center while CO2 decreased in theopposite direction. An increase of N2 from surface to the pear center was also found.The proles are strongly dependent on the tissue properties. In the past, it wasassumed that gas transfer barriers were restricted to the skin layers of fruit. Here,it was demonstrated that the resistance to gas exchange of the eshy part of fruit isalso signicant and could lead to oxygen deciency.

1.3 KINETICS MODELING


Many food processes are associated with a kinetic aspect, an attribute that changeswith time (e.g., microbial activity, active components of disinfectants in cool rooms,color changes), and the resulting kinetic reaction must be solved. Therefore, thereactions rates, property changes, and heat releases must be calculated as a part of thesolution. Consider the following reaction

A B! C (1:10)where the reaction rate Rc (mol s

1) is dened


d d d n m o p q rRc dt

[A] dt

[B] dt

[C] kf [A] [B] [C] kb[A] [B] [C] (1:11)

with kf the forward rate constant and kb the backward rate constant. The rateconstants can be modeled by the following Arrhenius-like expression

kf ,b aTbeER

1T 1Tref

(1:12)

wherea and b are the empirical constantsE is the empirical activation energyTref is the reference temperature

The heat of reaction can be calculated from the heats of formation of the species anddepends on temperature. The reaction leads to sources=sinks in the conservationand energy equations.

The most widely studied kinetics is that of enzymes. They are involved in manyaspects of food quality, catalyzing the complex underlying biochemical reactionsthat result in quality changes in such attributes as taste, odor, color, and many more.In its simplest form, in an enzymatic reaction the substrate (S) is converted intoa product (P) with the help of enzyme (E):

S !E P (1:13)

The rate of reaction rP can be expressed in terms of either the change of substrateconcentration, CS or the product concentration, CP

rP dCSdt dCP

dt(1:14)

It is important to know how the reaction rate is inuenced by reaction conditionssuch as substrate, product, and enzyme concentration if you want to understandthe effectiveness and characteristics of an enzymatic reaction. If the initial reactionat different levels of substrate and enzyme concentrations are measured, weoften obtain a series of characteristic curves, where the reaction rate is proportionalto the substrate concentration (rst-order reaction) at low values of substrateconcentration, and does not depend on the substrate concentration (zero-orderreaction) at high values of substrate concentration which means the reactiongoes gradually from rst- to zero-order as the concentration of the substrate isincreased. The maximum reaction rate, Vmax is proportional to the enzyme concen-tration. This was what Henri observed in 1902, and he proposed the followingrate equation 2008 by Taylor & Francis Group, LLC.

rP VmaxCSKM CS (1:15)

whereVmax (mol m

3 s1)KM (mol m

3) are kinetic parameters which need to be experimentallydetermined

KM is the substrate concentration required for an enzyme to reach half of its maximumvelocity. This equation describes many experimental results well. A quantitativetheory exists to support the observed enzyme kinetics and is still widely used todayunder the name MichaelisMenten kinetics.

1.3.2 CASE STUDY: MODEL FOR THE RESPIRATION OF FRUIT

Respiration is one of the most important processes in fruits. Extended MichaelisMenten kinetics is widely used as a semiempirical model to describe the relationshipof the respiration to the O2 and CO2 concentration, and the whole respirationpathway is assumed to be determined by one rate-limiting enzymatic reaction(Chevillotte, 1973). A noncompetitive inhibition model (Peppelenbos et al., 1996;Chang, 1981; Lammertyn et al., 2001) can be used to describe consumption of O2 byrespiration as formulated by

RO2 Vm,O2 PO2

(Km,O2 PO2 ) 1 PCO2Kmn,CO2 (1:16)

whereVm,O2 (mol m

3 s1) is the maximum oxygen consumption rateP (kPa) is the partial pressure for O2 and CO2Km (kPa) is the MichaelisMenten constant for O2 consumption andnoncompetitive CO2 inhibition

RO2 (mol m3 s1) is the O2 consumption rate of the sample

The equation for production rate of CO2 consists of an oxidative respiration part anda fermentative part (Peppelenbos et al., 1996)

RCO2 rq,ox RO2 Vm, f ,CO2

1 PO2Km, f ,O2 (1:17)

whereVm,f,CO2 (mol m

3 s1) is the maximum fermentative CO2 production rateKm,f,O2 (kPa) is the MichaelisMenten constant of O2 inhibition on fermentativeCO2 production

rq,ox is the respiration quotient at high O2 partial pressureRCO2 (mol m

3 s1) is the CO2 production rate of the sample 2008 by Taylor & Francis Group, LLC.

The effect of temperature was described by Arrhenius law (Hertog et al., 1998)

Vm,O2 Vm,O2,ref expEa,VmO2

R

1Tref

1T

(1:18)

Vm, f ,CO2 Vm, f ,CO2,ref expEa,VmfCO2

R

1Tref

1T

(1:19)

whereVm,O2,ref and Vm, f,CO2,ref (mol m

3s1) are the maximal O2 consumption andmaximal fermentative CO2 production rate at Tref 2938K, respectively

Ea,Vm (kJ mol1) is the activation energy for O2 consumption and fermentative

CO2 production

Typical respiration rates of pear tissue are given in Figure 1.2. The estimated param-eters for Vm,O2 and Vm,f,CO2 of cortex tissue were (2.39 0.14) 104 mol m3 s1and (1.61 0.13) 104 mol m3 s1, respectively. Km,O2, a measure for thesaturation of respiration with respect to O2 was relatively small and equal to(1.00 0.23) kPa. A signicant but low inhibition effect of CO2 on O2 consumptionof pear cortex tissue was found (Kmn,CO2 66.4 21.3 kPa). The respiration quotientrq,ox was 0.97 0.04 and showed that the O2 consumption was about the same as

the oxidative CO2 production. The value Km,f,O2 is a measure of the extent to whichfermentation can be inhibited by O2. The estimated value of 0.28 0.14 kPa impliesthat fermentation was already inhibited at very low levels of O2 concentration.

O2 partial pressure (kPa) O2 partial pressure (kPa)50 10 0

0

11

0

22

2 4 6 8 10

104104

RO

2 (m

ol m

3 s

1 )

RCO

2 (m

ol m

3 s

1 )

FIGURE 1.2 O2 consumption and CO2 production rate in pear tissue disks at 208C; Solidlines () and dashed lines (- -) indicate the respiration model at 0 and 10 kPa CO2 while thesymbols () and (o) indicate the experiment at 0 and 10 kPa CO2. (Adapted from Ho, Q.T.et al., PLoS. Comput. Biol., 4, e1000023, 2008.) 2008 by Taylor & Francis Group, LLC.

1.4 MODEL PARAMETERS

1.4.1 T HERMOPHYSICAL PROPERTIES

The therm ophysical properties k, r, and c may be tem perature dependen t (due toinsuf cient probl em dec omposit ion, i.e., the underl ying physi cochemical changesare not modeled expli citly) so that the problem becom es no nlinear. Thermo physicalproperties of vario us agric ultural and food product s are compi led in variou s refer-ence books (e.g., the compilation b y ASHR AE). Further, equations have beenpublished, whi ch relate the thermophys ical properties of agric ultural product s andfood materials to thei r c hemical compo sition. In general, both he at capaci ty anddensity can be calculated with suf cient accuracy , but the model s for thermalconductivi ty requi re some assum ptions about the orien tation of d ifferent mai nchemical constitu ents with respect to the direction of heat ow.

Determ ination of material proper ties, including diff usivity of certa in compo n-ents, is a task that often requi res experi ments beca use most ly one is inte rested indiffusivity as an apparen t proper ty for a particula r mat erial in particula r condition s,and one cannot rely on fundam ental equati ons to calculate the proper ties. For thecase study under consi deration here, one should meas ure gas concent ration as afunction of time and space, to whi ch the mass trans fer model is tted by o ptimizingthe apparen t diff usivity value. This is usual ly achiev ed by an iter ative least square sprocedu re. Althoug h one c an careful ly desig n experi ments to imp rove the q uality ofthe tting, in many cases, variability due to the food material composition andstructure must be investigated by, e.g., analysis of variance to reveal signicanteffects (Ho et al., 2006a,b). Estimated diffusivities of pear tissue are given inTable 1.1. Hi gh variation of the estimated value was found in the meas urement.Lowest diffusivity was reported for the skin, and anisotropic diffusivity was found inthe axial and radial directions. The higher diffusivity in the axial direction comparedto that along the radial direction is probably due to the fact that vascular bundles maybe not fully lled with sap during storage of the fruit and facilitate gas exchange.Higher diffusivity of CO2 compared to O2 and N2 is probably due to the largersolubility of CO2 in water than that of O2 and N2. In addition, while O2 and N2 wouldbe transported mostly through the apoplast, CO2 would also diffuse through thecytoplasm.

1.4.2 KINETICS PARAMETERS

The kinetics parameters may be determined by tting the proposed kinetics model tothe experimental data of the observed changes (e.g., quality changes) using a non-linear regression program (Ho et al., 2008). For example in the case study ofrespiration of pears, the data on O2 consumption and CO2 production rates arepooled, and the same weight can be attributed to both gases. Accuracy of theestimated kinetics parameters reecting the variability and experimental error struc-ture can be expressed by condence intervals, and asymptotic condence intervalscan be calculated from the asymptotic covariance matrix C of the parameters

C (JT J)1s2 (1:20) 2008 by Taylor & Francis Group, LLC.

TABLE 1.1Gas Transport Properties of Pear Tissue

Diffusivity(m2 s1)

Estimated Values Basedon Tissue Measurement

DO2,skin (1.86 0.70) 1010 aDCO2,skin (5.06 3.3) 1010 aDN2,skin (1.06 0.29) 1010 bDO2,r (2.8 1.59) 1010 bDCO2,r (2.32 0.41) 109 aDN2,r (2.67 1.62) 1010 bDO2,z (1.10 0.40) 109 bDCO2,z (6.97 2.19) 109 aDN2,z (1.06 0.66) 109 b

Note: 95% condence limits. Indices skin, r, and z referto the position of the skin, along the radial direction

and along the vertical axis of pear, respectively.a Indicate values measured by Ho, Q.T. et al., Post. Biol.

Tech., 41, 113, 2006a.b Indicate values measured by Ho, Q.T. et al., J. Exp. Bot.,

57, 4215, 2006b.whereJ is the Jacobi an mat rix with respec t to the estimat ed param eterss 2 is the mean square d error

The asym ptot ic (1 a)% con de nce interval on the i th parameter estimat e Pi wascalculated from

Pi t 1 a2 , n p

Ci, ip ( 1:21 )wheret is the Studen t t -distributi onn is the numbe r of meas urementsp is the numbe r of param etersCi,i is the ith diagon al elem ent of C

Correla tion coef cients ri,j of estimat ed model param eters i and j indi cate thestrength and direc tion of the relations hip betw een estimated model param eters iand j . These coef cient s can be compu ted from

ri, j Ci , jCi , i Cj, j

p ( 1:22 )The correl ation coef cients of kinet ics parameter s of pear tissue respiration fromFigure 1.2 are given in Tab le 1.2. The correl ation coef cient s are all smaller than0.71 suggesting that the model is not over-parameterized.


TABLE 1.2Correlation Coefcient Table of Estimated Parameters on Pear Tissue Respiration

Parameters Vm,O2,tissue Km,O2 Kmn,CO2 rq,ox Km,f,O2 Vm,f,CO2,tissue Ea,VmO2 Ea,VmfCO2

Vm,O2 ,tissue 1 0.70 0.50 0.56 0.38 0.02 0.20 0.01Km,O2 0.70 1 0.22 0.26 0.58 0.03 0.12 0.01Kmn,CO2 0.50 0.23 1 0.02 0.04 0.00 0.10 0.00rq,ox 0.56 0.26 0.02 1 0.43 0.05 0.11 0.02Km,f,O2 0.38 0.58 0.04 0.43 1 0.30 0.06 0.10Vm,f ,CO2 ,tissue 0.02 0.03 0.00 0.05 0.30 1 0.00 0.34Ea,VmO2 0.20 0.12 0.10 0.11 0.06 0.00 1 0.00Ea,VmfCO2 0.01 0.01 0.00 0.02 0.10 0.34 0.00 1

2008

byTaylor

&Francis

Group,L

LC.

1.5 SOLUTION METHODS


Unless simplications are made, the models presented in this chapter cannot besolved by analytical means. In many industrial applications, however, simplicationsare very well possible. At a rst simplication level, the geometry can be simplied.For simple shapes such as cylinders, spheres, and blocks one can nd, with certainconditions on the model parameters, (usually they have to be constants!), analyticalsolutions as a function of time and spatial coordinates. At a second level, overallbalances can sometimes be made excluding the spatial dimension. One then typicallyends up with ordinary differential equations that can be solved quite efciently withthe latest numerical solvers. At the last simplication level, one is also able toexclude the time dimension and overall balances leading to the algebraic equations.If one does have to rely on numerical means to solve mathematical models, thereare a number of efcient methods available. For this purpose the problem is rstreduced signicantly by requiring a solution only for a discrete number of points (theso-called grid) rather than for each point of the space-time continuum through whichthe heat and mass transfer proceed. The original governing partial differentialequations are accordingly transformed into a system of difference equations andsolved by simple mathematical manipulations such as addition, subtraction, multi-plication, and division, which can easily be automated using a computer program.However, as a consequence of the discretization, the obtained solution is no longerexact, but only an approximation of the exact solution. Fortunately, the approxima-tion error can be decreased substantially by increasing the number of discretizationpoints at the expense of additional computing time.

Various discretization methods have been used in the past for the numericalsolution of heat and mass transfer problems arising in food technology. Among themost commonly used are nite difference method, nite element method, and nitevolume method. It must be emphasized that, particularly in the case of nonlinear heattransfer problems, the numerical solution must always be validated. It is very wellpossible that a plausible, convergent but incorrect solution is obtained. At least a griddependency study must be carried out to verify whether the solution basicallyremains the same when the computational grid is rened.

1.5.2 CASE STUDY: SOLUTION OF THREE-DIMENSIONAL GAS EXCHANGEAND RESPIRATION IN PEAR FRUIT

Numerical solution can be applied to solve the governing partial differential equa-tions of heat and mass transfer using the nite element method. In the axi-symmetriccase study, 2719 quadratic nite elements with triangular shape were used andrequired less than 5 min of CPU time on a desktop PC. For the mass transfermodel involving a kinetic reaction term, for example consumption of O2 inside thefruit, mathematical equation of the reaction may not exclude negative concentrations.Numerical problems may then be expected when the concentration approaches zeroresulting in nonphysical negative results. Here is an example of two alternative

approaches to solve the problem for O2 exchange in intact fruit involving respiration:


1.

Mpomquinrean

liktybewram

Hence, the mass transfer equation for O2 transforms into

(exp ( UO2 ) a O2 )@ UO2@ t

r (u exp ( UO2 )) r ( DO2 exp ( UO2 )) r UO 2 RO2 ( 1:24)

At the boundary UO2,r ln (CO2,1). Similarly, exponential transformationof the O2 concentration was applied in the other equations. Both methodsavoided nonrealistic errors in the computations of gas exchange in fruit(Figure 1.1).

6 TOWARDS FOOD PROCESS MODELING AT DIFFERENTSCALES: MULTISCALE MODELING

any problems related to mathematical modeling of foods and food processes is theor understanding of the microscopic and nanoscopic mechanisms that affect theacroscopic behavior of the food or process that is being modeled. As a conse-ence, apparent material properties that have to be expressed in complex equationsrelation to other variables are used, and sophisticated experiments to nd theselationships are required. Hence, more then often the validity range is quite limited,d variability is large.For the case of biological materials like the pear fruit, the macroscopic properties

ely depend on various microscopic histological and cellular features such as tissuepes, geometric properties of the cell, presence of an adhesive middle lamellatween individual cells, cellular water potential, mechanical properties of the cellall, presence of intercellular spaces, and many more. These features cover a widenge of spatial scales, from nanoscopic (plasmodesmata, plasma membranes), overicroscopic (cell wallmiddle lamella complex, cell geometry), to macroscopic1. To ensure that the O2 concent ration cann ot becom e negativ e due to O2consumpti on ( RO2 0), the respi ration term in the perm eation diff usionreaction model was modi ed for O2 and CO 2 in the solution .

If CO2,g < 0 then RO 2 0 and RCO2 Vm ,f ,CO2 .If CO2,g 0 then R O2 and RCO2 are descri bed by thei r original equationsabove. Anal ytically , there is no O2 consumpti on when O2 reaches zero.Therefor e, the O2 concent ration shoul d never becom e negative. The solu-tion, therefore, will be physi cally consistent .

2. Another met hod was based o n the exponent ial transform ation of the vari-able in the model equations in such a way that the solut ion is guarant eed tobe positive . For examp le, exponent ial transform ation of the mai n variab lewas used to impose positive values for the O2 concent ration.

CO2 , g exp ( UO 2 ) (1:23)2008 by Taylor & Francis Group, LLC.

material behavior at different spatial scales in such a way that the submodels are

However, this amounts to writing effective equations for the macroscale that

modeling in food applications could have a large contribution to a better understand-

ing of the complex food composition and behavior of foods in industrial processes.

1.7 CONCLUSION

Basic mathematical models for describing food processes in terms of the transportphenomena and kinetic changes taking place were outlined, and that the mainproblem for mathematical modeling of food processes is poor understanding of thebiochemical and microscopic mechanisms that cause macroscopic changes inappearance and appreciation of foods was demonstrated. As a result, considerableuncertainty in food process simulation and optimization was to be managed.

ACKNOWLEDGMENTS

The authors wish to acknowledge nancial support by the Flanders Fund forScientic Research (FWO-Vlaanderen) (project G.0200.02) and the K.U. Leuven(project IDO=00=008 and OT 04=31, IRO PhD scholarship for Q.T. Ho). PieterVerboven is Fellow of the Industrial Research Fund at the K.U. Leuven.

NOMENCLATURE

c Specic heat J kg1 8C1

fi External body forces, including thegravitational force N m3

2 1account for lower scales, and it is a difcult task. Alternatively, equations for thene scale itself can be solved. The up-scaling of ne scale solutions to a macroscalesolution is known as homogenization. Homogenization has been dened as acollection of methods for extracting or constructing equations for the coarse scale(macroscale) behavior of materials and systems, which incorporate many smaller(nano-, micro- and meso-) scales. The main objective of such an approach is to usesimpler ne scale equations that are considerably less expensive to solve, and whosesolutions have the same coarse scale properties. While still in its infancy, multiscaleinterconnected. As a result, investigation of the microstructure becomes a prerequi-site to understand transitional theoretical frameworks and modeling techniques tobridge the gap between length scale extremes. Multiscale modeling may involvechallenging physical processes such as transport phenomena. Sometimes it is suf-cient to nd the solution of the coarser scale by including procedures to construct theequations on the coarser scale that account for the contribution of ner scales.(actual geometry of the material). The material properties of the continuum model,such as diffusion properties incorporate both actual physical material constants suchas the diffusivity of water and air but also the microscale geometry of the tissue andintracellular space (Mebatsion et al., 2008).

Multiscale models are basically a hierarchy of submodels, which describe theD Apparent diffusion coefcient, diffusivity m s


H Static enthalpy J kg1

Hi Henrys constant of component I mol m3 kPa1

:.,

.

,

.REFERENCES

Chang, R., Physical Chemistry with Applications to Biological Systems, MacMillanPublishers, New York, 1981.

Chevillotte, P., Relation between the reaction cytochrome oxidase-oxygen and oxygen uptakein cells in vivo, J. Theor. Biol., 39, 277, 1973.

Geankoplis, J.C., Transport Processes and Unit Operations, Prentice-Hall Inc., EnglewoodCliffs, New Jersey, 1993.

Hertog, M.L.A. et al., A dynamic and generic model on the gas exchange of respiring producethe effects of oxygen, carbon dioxide and temperature, Post. Biol. Tech., 14, 335, 1998

Ho, Q.T. et al., Gas diffusion properties at different positions in the pear, Post. Biol. Tech., 41113, 2006a.

Ho, Q.T. et al., A permeationdiffusionreaction model of gas transport in cellular tissue ofplant materials, J. Exp. Bot., 57, 4215, 2006b.

Ho, Q.T. et al., A continuum model for metabolic gas exchange in pear fruit, PLoS ComputBiol., 4(3): e1000023, 2008. doi: 10.1371=journal.pcbi.1000023.

Lammertyn, J., Comparative study of the O2, CO2 and temperature effect on respirationbetween Conference pear cells in suspension and intact pears, J. Exp. Bot., 521769, 2001.

Mebatsion, H.K. et al., Modelling fruit (micro)structures, why and how, Trends Food SciTech., 19, 59, 2008.

Peppelenbos, H.W. et al., Modelling oxidative and fermentative carbon dioxide production offruit and vegetables, Post. Biol. Tech., 9, 283,1996.k Thermal conductivity W m1 8C1

K Permeation coefcient m2

P Pressure PaQ Heat source or sink W m3

ra Source or sink of component a of thematerial kg m3 s1

R Unversal gas constant 8.314 J mol1 K1

Ri Production term of the gas component mol m3 s1

T Temperature 8C, Ku Apparent velocity vector m s1

ui (i 1, 2, 3) Cartesian components of the velocityvector U(u1, u2, u3) m s

1

Xa Mass fraction of a component aof the material

t Time s

GREEK LETTERS

m, h Viscosity Pa sr Density kg m3 2008 by Taylor & Francis Group, LLC.

Subscripts ..................................................................................................... 28

testing the models for convergence of the optimization algorithms. In this chapter,

the exact solutions, mostly preferred in the food engineering literature, for theconduction heat transfer problems are reviewed.

A general heat transfer problem encountered in food process engineering area isto determine the steady or unsteady (transient) state temperature distribution in solidfood products where the initial temperature distribution and the boundary conditionsare specied. An unsteady temperature distribution includes not only the temperaturevariation from point to point in the medium but also with time (Kakac and Yener,1993). This problem includes nding exact solution of the governing diffusionequation for different geometries or different coordinate systems. The simplestReferences ............................................................................................................... 28

2.1 INTRODUCTION

In process optimization studies, the rst step generally would be to have a math-ematical model for the given process. Exact solutions of the differential equationsdescribing the process and numerical solutions (nite difference and nite elementsolutions) are preferred for this objective. Since all the variables affecting the processand the physicalchemical changes occurring in the medium can be dened in anumerical model, applying these solutions in the optimization models would resultin longer run-time solutions. Therefore, exact solutions are sometimes preferred for2 Analytical Solutionsin Conduction HeatTransfer Problems

Ferruh Erdogdu and Mahir Turhan

CONTENTS

2.1 Introduction .................................................................................................. 192.2 Analytical Solutions..................................................................................... 212.3 Application and Use of Analytical Solutions .............................................. 262.4 Conclusion ................................................................................................... 27Nomenclature .......................................................................................................... 27

Greek Letters ................................................................................................ 28 2008 by Taylor & Francis Group, LLC.

case of diffusion equation with constant and isotropic thermophysical properties isgiven as follows:

r2T 1a @T@t

(2:1)

where a (thermal diffusivity) is given by

a kr cp (2:2)

The Laplacian (r2) of temperature in various coordinate systems is as follows(Kakac and Yener, 1993):

Rectangular (Cartesian):

r2T @2T

@x2 @

2T

@y2 @

2T

@z2(2:3)

Cylindrical:

r2T 1r @@r

r @T@r

1

r2 @

2T

@f2 @

2T

@z2(2:4)

Spherical:

r2T 1r2 @@r

r2 @T@r

1r2 @@m

(1 m2) @T@m

1

r2 (1 m2) @2T

@f2(2:5)

where m cos u.Physical signicance of thermal diffusivity is associated with the speed of heat

propagation into the solid product (Ozisik, 1993). The higher the thermal diffusivitythe faster the heat transfer rate is the general belief for thermal diffusivity. In a recentpublication, Palazoglu (2006) reported that the speed of heat penetration was afunction of thermal diffusivity and heat transfer coefcient combination rather thanthe thermal diffusivity itself.

The exact solution, called analytical solution, of Equation 2.1 for Cartesian (forinnite and nite slab shaped geometries), cylindrical (for innite and nite cylindergeometries), and spherical (for sphere) coordinate systems are generally used in theliterature to especially verify numerical solutions and to develop numerical schemesand grid generation methods (Cai et al., 2006). The rst step in obtaining theanalytical solution is to choose the orthogonal coordinate system of which thecoordinate surfaces will be coinciding with the boundary surfaces of the solidproduct (Ozisik, 1993). For example, Cartesian coordinate system is used forrectangular bodies while cylindrical coordinate system is used for cylinder shapesand spherical coordinate system is used for sphere shapes. 2008 by Taylor & Francis Group, LLC.

the constant thermophysical (thermal conductivity, specic heat and density) and

constant boundary conditions with uniform initial temperature distribution are usedand to develop numerical schemes and grid generation methods (Cai et al., 2006).Analytical solutions give a concise parametric solution in ideal problems ofacademic interest or to check numerical simulations as explained above. Theanalytical solutions are generally restricted by the following assumptions:

. Solid object conforming to a regular geometry (i.e., slab, cylinder, orsphere) with the exception of lumped system analysis

. Constant thermal properties and physical dimensions

If the initial temperature distribution and surrounding medium temperatures are notconstant and are given by certain functions, an analytical solution can still beobtained, but the results might be complex compared to the constant temperatureTo simplify the analytical solutions, generally one-dimensional solution repre-senting an innite slab for Cartesian coordinates, an innite cylinder for cylindricalcoordinates, and sphere for spherical coordinates are applied assuming all thesurfaces facing equivalent boundary conditions. An additional solution for thegiven coordinates requiring the temperature change in only one dimension mightbe a semi-innite solution approach. Lumped system solution is another method-ology independent of any coordinate system.

Even though the situations described by analytical solutions represent a smallproportion in heat transfer analysis, these solutions can be applied on the basisleading to check the solutions given, for example, by numerical solutions. Therefore,the objective of this chapter is to summarize the usefulness of analytical solutionswith their applications in variety of situations encountered in food processing.

2.2 ANALYTICAL SOLUTIONS

The exact solutions for the given cases above are generally called analytical solutions,and they play a signicant role in heat transfer simulations for design and optimizationpurposes where the solid food products can be approximated by regular shapes of slab,cylinder, or sphere (Ramaswamy et al., 1982). The solutions are available in theliterature to obtain the transient temperature distribution in such shaped foods.

These solutions are obtained using some analytical techniques including Laplacetransform and the method of separation of variables. Separation of variables has beenwidely used in solving the heat conduction problems where the homogeneousequation systems are readily handled (Ozisik, 1993). This method is based onexpanding a function in terms of Fourier series. In this method, the dependentvariable (temperature, T in the given cases here) is assumed to be the product ofindependent variables (location, x and time, t). This method is applied when thegoverning equation and differential equations representing the boundary and initialconditions are homogeneous and linear. For the cases of nonhomogeneous condi-tions where there is more than one nonhomogeneous condition, the superpositiontechniques are applied to split the problem into simpler problems (Cengel, 2007).

One typical use of analytical solutions is to validate numerical solutions where 2008 by Taylor & Francis Group, LLC.

cases. This statement is also true when any of the thermal properties is not constant(e.g., when the thermal conductivity is a function of temperature).

In a more general format for one-dimensional heat transfer, the Equation 2.1 canbe modied for innite slab (n 0), innite cylinder (n 1), and sphere (n 2)geometries as follows:

1a @T@t 1

xn @@x

xn @T@x

(2:6)

Equation 2.6 can then be written for a one-dimensional heat transfer in an innitecylinder and sphere, respectively

1a @T@t 1

x @T@x @

2T

@x2(2:7)

1a @T@t 2

x @T@x @

2T

@x2(2:8)

The case of @T@t 0 is the simplest approach in a heat transfer analysis leading to thesteady state condition where the knowledge of two boundary (since a double integra-tion in the space variable is involved) is required. When the steady-state assumption isnot valid, integration in time with requirement of an initial condition must be used.

Boundary conditions across the surfaces usually encountered in conduction heattransfer are prescribed surface temperature, prescribed heat ux, and convectionboundary:

. Prescribed surface temperature: Surface temperature as constant or functionof space and time

. Prescribed heat ux: Heat ux across the boundaries specied at constant oras a function of space and time

. Convection boundary condition: Equivalence of convection over theboundary to the conduction towards the solid geometry

k @T@x

s

h (T js T1) (2:9)

The analytical solutions are obtained using convection boundary present across thesurface boundary and a symmetry condition at the center (special case of theprescribed heat ux where q00 and therefore dTdx is equal to 0) with a constant uniforminitial temperature distribution by applying separation of variables solution method-ology to Equations 2.3 through 2.5

T(x,t) T1Ti T1

X1n1

Cn(x) exp m2na tL2

h i(2:10)

where mn and Cn(x) and are given for regular shaped geometries of slab, cylinder,

and sphere, respectively


Slab:

NBi m tan (m) (2:11)

Cn(x) 2 sin (mn)mn sin (mn) cos (mn)

cos mn x

L

(2:12)

Cylinder:

NBi m J1(m)J0(m) (2:13)

Cn(x) 2 J1(mn)mi J20 (mn) J21 (mn)

J0 mn xL

(2:14)

Sphere:

NBi 1 mtan (m)

(2:15)

Cn(x) 2 [ sin (mn) mn cos (mn)]mn sin (mn) cos (mn)

sin mn xL

mn xL

(2:16)

whereJ0 and J1 are the rst kind, 0th and 1st order Bessel functionsL is half-thickness for slab and radius for cylinder and spherex is the distance from the center (0 x L)NBi is the Biot number

These equations were further simplied by Ramaswamy et al. (1982) to easilydetermine the transient temperature change in solid foods with convective boundarycondition at the surface in the Biot number range of 0.02200 with Fouriernumber being greater than 0.2. The reference by Carslaw and Jaeger (1959) issuggested for different situations where numerous initial and boundary conditionsare applied.

The given solutions are further simplied using the rst term (C1) of the givenseries where Fourier number NFo a tL2

is greater than 0.2. There has been some

argument in the literature on limiting value of NFo. McCabe et al. (1987) reportedthat only the rst term of the series analytical solutions is signicant and other termscan be neglected when the value of NFo is greater than about 0.1. Kee et al. (2002)stated that analytical solutions at the center can be approximated by an exponentialdecay when NFo is greater than 0.15.

Since all the regular shaped geometries cannot be modeled in a one-dimensionalway (an innite plate, innite cylinder and sphere), combinations or intersections ofthe given geometries can be modeled for determining the temperature changes intwo- or three-dimensional geometries. For the case of two- (nite cylinder) and 2008 by Taylor & Francis Group, LLC.

three-dimensional (nite slab) geometries, given solutions can be combined usingthe superimposition technique (Newman, 1936)

T T1Ti T1

finite slab

T T1Ti T1

slab,depth

T T1Ti T1

slab,width

T T1Ti T1

slab,height

(2:17)

T T1Ti T1

finite cylinder

T T1Ti T1

cylinder

T T1Ti T1

slab

(2:18)

For the case of volume average temperature change, the problem becomes todetermine the transient volume average temperature change in the solid body.Therefore, the solution given with Equation 2.7 is integrated throughout the wholevolume

1V

V0 T(x,t) dV 1V

x0 T(x,t) A dx

resulting in the Cn values as

independent of the location

Slab:

Cn 2mn sin

2 (mn)

mn sin (mn) cos (mn)(2:19)

Cylinder:

Cn 4m2n 2 J

21 (mn)

J20 (mn) J21 (mn)(2:20)

Sphere:

Cn 6m3n [ sin (mn) mn cos (mn)]

2

mn sin (mn) cos (mn)(2:21)

For this case, it should be noted that the measurement of average transient tempera-ture change becomes a challenging task since the location of the average temperaturemight not be constant.

In addition to these coordinate systems, the elliptical coordinate system is appliedfor the case of elliptical cylinders. Application of this latter case is rather limited inthe literature. McLahlan (1945) rst published the equations governing the heatconduction problem in an innite elliptical cylinder subjected to a medium of inniteheat transfer coefcient. Kirkpartrick and Stokey (1959) described the numericalsolution of McLahlans solution for required zeros of the modied Mathieu functions(solution of the differential equation d

2ydx2 [a 2 q cos (2x)] y 0 that arises

during the solution in elliptical coordinate system) necessary for solution in ellipticalcoordinate systems for different eccentricities from 0 (indicating an innite circular 2008 by Taylor & Francis Group, LLC.

cylinder) to 1 (indicating an innite plate). The Laplacian (r2) in elliptical coordinatesystem is given by

r2T 1a2 ( sinh2 u sin2 v)

@2T

@u2 @

2T

@v2

(2:22)

wherex a cosh u cos xy a sinh u sin v

For different geometries of anomalous shapes, Smith et al. (1968) presented a uniedsystem of charts and graphs for use over a wide range of conditions. The basis ofthe system was on the concept of representing a given shape by a geometry index,and equations, based in the rst term of the innite series solutions for transientconduction heat transfer, given to determine these parameters.

The analytical solutions for slab, cylinder, and sphere have also been reduced torelatively simple charts (Heisler charts), where the center temperature ratio is plottedas a function of NFo and NBi. These charts, in any heat transfer book, are given in theNFo range of 0100 and NBi range of 0.011. Schneider (1963) also presents a totalof 120 charts covering temperature response of different simple geometries (semi-innite solids, one- and two-dimensional slabs, cylinder, cylindrical cavity andshells, sphere, spherical cavity and shells, and ellipse and ellipsoids) under a varietyof boundary conditions and for a wide range of Fourier and Biot numbers.

Other than these given analytical solutions, another solution is obtained when asemi-innite body (a body with innite depth, width, and length) assumption is heldfor the heat transfer medium. It is generally accepted that the semi-innite mediumapproach would hold with the following criteria is satised (Hagen, 1999)

L0 a tp (2:23)where L0 is the thickness of the given semi-innite region. Solution for semi-innitemedium approach with a very high heat transfer coefcient assumption is given as

T(x,t) TsTi Ts erf

x

2 a tp

(2:24)

erf is the error function.Under some special circumstances of conduction heat transfer, spatial tempera-

ture distribution within the solid during processing can also be ignored. Lumpedsystem approach provides a great simplication in the conduction heat transferanalysis (Ozisik, 1993) even though the range of applicability is restricted. For thisapproach to hold, thermal conductivity value of the solid must be really high toenable the uniform temperature distribution inside the body

limk!1

NBi limk!1

h Lk

! 0 (2:25) 2008 by Taylor & Francis Group, LLC.

Validity of lumped system solution holds when N 0.1 where the characteristicBidimension is used as VA. Lumped system analysis is commonly used to experimentallydetermine the convective heat transfer coefcient (Erdogdu et al., 1998; Erdogdu,2005).

For the cases of nonuniform initial or medium temperature cases, analyticalsolutions can be obtained using the separation of variables technique, but resultswill denitely be complicated compared to the given cases. Analytical solutions forinnite slab, innite cylinder, and sphere were also reduced to simple to use chartswhere the temperature ratio at the center was plotted as a function of Fourier and Biotnumbers. These charts, in any heat transfer book, are given in the Fourier numberrange of 0100 and Biot number range of 0.011.

Unlike conduction, convection heat transfer is extremely difcult to obtainsimple to use analytical solutions since simultaneous solution for energy, momen-tum, and continuity equations are required. Cai and Zhang (2003) gave a detailedexplanation for explicit analytical solutions of two-dimensional laminar naturalconvection along a vertical porous plate and between two vertical plates. Complexityof these solutions obviously increases in cylindrical and spherical coordinates, andgenerally a computational uid dynamics (CFD) numerical solution is preferred.

2.3 APPLICATION AND USE OF ANALYTICAL SOLUTIONS

A major concern in the use of analytical solutions is to decide on the number of termsof the innite series to obtain a correct a solution. A general approach is to applyonly the rst term when the Fourier number is greater than 0.2 where the temperature

ratio ln T(x,t) T1Ti T1

is accepted to be linear. As long as thermal diffusivity for the

case of heat transfer, slope of the temperature ratio versus time curve m21 aL2

maybe easily used to determine the thermal diffusivity value with the knowledge of heattransfer coefcient of the heat transfer medium. The knowledge of heat transfercoefcient (with the thermal conductivity of the product) leads to knowing the m1value where it can be used to determine the thermal diffusivity. As it can be realizedfrom the slopes not being a function of location, this method will not require theknowledge of location in the material where temperature change is recorded whenWith this condition, temperature would only vary with time, and this situation iscalled lumped system methodology. After applying an energy balance for thiscondition to a given geometry

@T

@t h Ar V cp (T T1) (2:26)

For this case, only initial condition is required. When Equation 2.26 is solved byapplying the uniform initial temperature condition (T(t 0) Ti)

T T1Ti T1 exp

h Ar V cp t

exp [NBi NFo] (2:27) 2008 by Taylor & Francis Group, LLC.

of).en5

depending on the geometry) and therefore the heat transfer coefcient assuming

accumulated errors for assumptions of approximation a nite geometry as an innite

one during unsteady-state heat and mass transfer processes. These assumptionswere based on that area of one of the surfaces if too large compared to the area ofthe other surfaces of a given geometry. The results of these studies concluded thesignicant error accumulations were functions of Biot and Fourier numbers.

The lumped system methodology is another preferred method to determine theheat transfer coefcient using a highly conductive material (e.g., copper, aluminum,etc.) of the same shape with the solid object. In addition, Anderson and Singh (2002)used the semi-innite medium approach to determine the heat transfer coefcientsolving an inverse heat conduction problem.

2.4 CONCLUSION

Analytical solutions of the diffusion equation play a signicant role in heat transfersimulations for design and optimization purposes where the solid food products canbe approximated by a regular shape to obtain the transient temperature distribution insuch food products. Another typical use of analytical solutions is to validate thenumerical solutions and to develop numerical schemes and grid generation methods.In addition, they give a concise parametric solution in ideal problems of academicinterest or to check numerical simulations. They are also useful for applications inthe optimization algorithms instead of numerical solutions to test the convergence ofthe algorithm and to save the processing time.

NOMENCLATURE

A Surface area m2

cp Specic heat J=kg-Kh Convective heat transfer coefcient W=m2-KJ 1st kind 0th order Bessel functionthermal conductivity of the solid object is known. Erdogdu (2005) gives a detailedanalysis of this situation. Yldz et al. (2007) determined the effective heat (heattransfer coefcient) and mass transfer (mass transfer and diffusion coefcient) coef-cients during frying of two-dimensional nite slab shaped potato slices. In this study,Equation 2.17 was applied (accepting the third dimension is innitely long, and thetemperature or mass ratio is 1) assuming the linear portion of the temperature andmassconcentration curves started when the Fourier number was greater than 0.1. The resultsand conclusions of this study deserved to be mention since the analytical solutionswere used when the phase change was also present in the heat transfer medium.

Turhan and Erdogdu (2003, 2004) and Erdogdu and Turhan (2006) usedthe analytical solutions for nite and innite slab and cylinders to demonstrate thethe heat transfer coefcient is known. At any location, after a certain amountprocessing time, slope of the temperature ratio would be the same (Erdogdu, 2005

The heat transfer coefcient, on the other hand, can be determined from thintercept of the temperature ratio curve where the intercept is a function of locatioandm value. The known m value leads to Biot number (Equations 2.11, 2.13, and 2.10


m Root of the characteristic equation of innite slab,

ject.

3-D06.Int.

ess,

ntie06.ntal

ientkey,innite cylinder or spheref Azimuthal angle in spherical coordinatesr Density kg=m3

u Radial direction in cylindrical coordinates and polar anglein spherical coordinates

SUBSCRIPTS

i Initials Surfaces Surface boundary1 Medium

REFERENCES

Anderson, B. and Singh, R.P., Air impingement heat transfer on a cylindrically shaped obIFT Annual Meeting and Food Expo. Presentation no: 91C-22, 2002.

Cai, R., Gou, C., and Li, H., Algebraically explicit analytical solutions of unsteadynonlinear non-Fourier (hyperbolic) heat conduction. Int. J. Thermal Sci., 45, 893, 20

Cai, R. and Zhang, N., Explicit analytical solutions of 2-D laminar natural convection.J. Heat Mass Transfer, 26, 931, 2003.

Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids. Oxford University PrLondon UK, 1959.

Cengel, Y., Heat Transfer: A Practical Approach. McGraw Hill Inc., New York, 2007.Erdogdu, F. and Turhan, M., Analysis of dimensional ratios of regular geometries for in

geometry assumptions in conduction heat transfer problems. J. Food Eng., 77, 818, 20Erdogdu, F., Mathematical approaches for use of analytical solutions in experime

determination of heat and mass transfer parameters. J. Food Eng., 68, 233, 2005.Erdogdu, F., Balaban, M.O., and Chau, K.V., Automation of heat transfer coefc

determination: development of a Windows-based software tool. Food Tech. Tur10, 66, 1998.J1 1st kind 1st order Bessel functionk Thermal conductivity W=m-KNBi Biot numberNFo Fourier numberL Half-thickness of a slab or radius of a cylinder or sphere m

Characteristic dimension in lumped system analysis (V=A) mL0 Thickness of a semi-innite region mr Radial distance, distance from the center mt Time sT Temperature 8C, KV Volume m3

x Distance from the center m

GREEK LETTERS

a Thermal diffusivity m2=s 2008 by Taylor & Francis Group, LLC.

Hagen, K.D., Heat Transfer with Applications, Prentice-Hall, Inc., Upper Saddle River,New Jersey, 1999.

Kakac, S. and Yener, Y., Heat Conduction. Taylor & Francis, Washington DC, 1993.Kee, W.L., Ma, S., and Wilson, D.I., Thermal diffusivity measurements of petfood.

Int. J. Food Prop., 5, 145, 2002.Kirkpatrick, E.T. and Stokey, W.F., Transient heat conduction in elliptical plate and cylinders.

J. Heat Transfer, 80, 54, 1959.McCabe, W.L., Smith, J.C., and Harriot, P., Unit Operations in Chemical Engineering.

4th edn., p. 268, McGraw-Hill, Inc., New York, 1987.McLahlan, N.W., Heat conduction in elliptical cylinder and an analogous electromagnetic

problem, Philosophical Magazine, 36, 600, 1945.Newman, A.B., Heating and cooling rectangular and cylindrical solids, Ind. Eng. Chem., 28,

545, 1936.Ozisik, M.N., Heat Conduction. John Wiley and Sons,