Optimization in
Food Engineering
Contemporary Food Engineering
Series Editor
Professor Da-Wen Sun, Director
Food Refrigeration & Computerized Food Technology National University of Ireland, Dublin
(University College Dublin) Dublin, Ireland http://www.ucd.ie/sun/
Optimization in Food Engineering, edited by Ferruh Erdoˇgdu (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 Gülüm ˛Sumnu and Serpil Sahin (2008)
Optimization in
Food Engineering
Edited by
Ferruh Erdoˇ
gdu
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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,
Contents
Series Editor ’s Preface Preface
Series Editor Editor Contributor s
PART I Modeling: Signi fi cance, Fundamentals,
and Methods
Chapter 1 Signi ficance of Mathemat ical Model ing and Simu lation for Optimizat ion
Quang Tri Ho, Hibru Kelemu Mebatsion, Bart Nicolaï, and Pieter Verboven
Chapter 2 Anal ytical So lutions in Conducti on Heat Transfer Proble ms
Ferruh Erdogdu and Mahir Turhan
Chapter 3 Num erical Solutions : Finit e Di fference Meth ods
T. Koray Palazoglu and Ferruh Erdogdu
Chapter 4 Num erical Solutions : Finit e Eleme nt and Finit e Vol ume Meth ods
Rui C. Martins, Vitor V. Lopes, António A. Vicente, and José A. Teixeira
PART II Optimization
Chapter 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 Chen
Chapte r 7 Ran dom-Ce ntroid Optim ization
Shuryo Nakai, Yasumi Horimoto, Jinglie Dou, and Roxana A. Verdini
Chapte r 8 Mult i-Obje ctive Optimizat ion in Food Engineer ing
Cheah Keen Seng and Gade Pandu Rangaiah
Chapte r 9 Appl ications o f the Minimu m Princi ple of Pontryag in for Solvi ng Opt imal Con trol Proble ms
Andrey V. Kuznetsov
Chapte r 10 Neur al Net works and Gene tic Algo rithms
Yang Meng and Hosahalli S. Ramaswamy
Chapte r 11 Com putational Fluid Dynami cs for Optimizat ion in Food Process ing
Ferruh Erdogdu
Chapte r 12 Dyna mic Optim ization
J. Ricardo Pérez-Correa, Claudio A. Gelmi, and Lorenz T. Biegler
Chapte r 13 Tab u Search: Deve lopment, Algori thm, Performance , and Appl ications
Mekapati Srinivas and Gade Pandu Rangaiah
Chapte r 14 Eig envalue Optim ization Techniqu es for Nonline ar Dyna mic Anal ysis and Des ign
Luis G. Matallana, Aníbal M. Blanco, and J. Alberto Bandoni
Chapte r 15 Com plex Meth od Opt imization
Ferruh Erdogdu and Murat O. Balaban
Chapte r 16 Mixe d Integer Linear Program ming Schedul ing in the Fo od Indust ry
Chapter 17 Mixe d Integer Nonline ar Programmi ng: Appl ications to Food Dehydrati on and Deep Chilling
Panagiotis P. Repoussis and Christos T. Kiranoudis
PART III Optimization Studies for Different
Food Processes
Chapter 18 Opt imization and Control Strat egy to Improve the Performance of Batch Reactors
Iqbal M. Mujtaba
Chapter 19 Pu lsed Microw ave Heating of Foods: Tempera ture Measu reme nt and Optim ization
Sundaram Gunasekaran
Chapter 20 Opt imization of Freeze-Dry ing Pr ocess Applied to Fo od and Biolog ical Product s: From Response Surface Meth odolog ies to an Inter active Tool
Michèle Marin, Stéphanie Passot, Fernanda Fonseca, and Ioan Cristian Trelea
Chapter 21 Opt imization of Spray Dry ing of Sugar-Ri ch Foods
Vinh Truong
Chapter 22 Struct ural Optim ization Techniqu es for Deve loping Bev erage Con tainers
Koetsu Yamazaki, Jing Han, and Sadao Nishiyama
Chapter 23 Opt imization for Con tinuous Shorte st Paths in Transpor tation
J. Miguel Díaz-Báñez
Chapter 24 Rea l-Time Nonline ar Optimal Control of Refrige ration Proce sses
Chapte r 25 Opt imization of Apple Juice Extract ion
María Teresa González and Martín Juan Urbicain
Chapte r 26 Opt imization of Canned Food Process ing
Ricardo Simpson and Arthur A. Teixeira
Chapte r 27 Opt imal Design of Contin uous Thermal Process ing wi th Plate Heat Exc hangers
Jorge Andrey Wilhelms Gut and José Maurício Pinto
Chapte r 28 Pro cess Optim ization Strategies to Reduce Var iability in Thermal Process ing of Pac kaged Fo ods
Kevin Cronin and Philippe Baucour
Chapte r 29 Loa ding Opt imization
Reinaldo Morabito and Vitória Pureza
Chapte r 30 Opt imization of the Arrays of Impingi ng Jets
Muhiddin Can and A. Burak Etemoglu
Chapte r 31 Opt imal Operational Planni ng in the Fruit Indust ry Su pply Cha in
Guillermo L. Masini, Aníbal M. Blanco, Noemí C. Petracci, and J. Alberto Bandoni
Chapte r 32 Opt imizin g the Manage ment of Curing Cha mbers
Series Editor’s Preface
CONTEMPORARY FOOD ENGINEERING
Food engineering is the multidisciplinaryfield of applied physical sciences combined with the knowledge of product properties. Food engineers provide the technological knowledge transfer essential to the cost-effective production and commercialization of food products and services. In particular, food engineers develop and design processes and equipment in order to convert raw agricultural materials and ingredients into safe, convenient, and nutritious consumer food products. However, food engineering topics are continuously undergoing changes to meet diverse consumer demands, and the subject is being rapidly developed to reflect market needs.
In the development of food engineering, one of the many challenges is to employ modern tools and knowledge, such as computational materials science and nano-technology, to develop new products and processes. Simultaneously, improving food quality, safety, and security remain critical issues in food engineering study. New packaging materials and techniques are being developed to provide more protection to foods, and novel preservation technologies are emerging to enhance food security and defense. Additionally, process control and automation regularly appear among the top priorities identified in food engineering. Advanced monitoring and control systems are developed to facilitate automation and flexible food manufacturing. Furthermore, energy saving and minimization of environmental problems continue to be an important food engineering issue and significant progress is being made in waste management, efficient utilization of energy, and reduction of effluents and emissions in food production.
The Contemporary Food Engineering series, consisting of edited books, attempts to address some of the recent developments in food engineering. Advances in classical unit operations in engineering applied to food manufacturing are covered as well as such topics as progress in the transport and storage of liquid and solid foods; heating, chilling, and freezing of foods; mass transfer in foods; chemical and 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 are aimed at professional food scientists, academics researching food engineering problems, and graduate level students.
The books’ editors are leading engineers and scientists from many parts of the world. All the editors were asked to present their books to address the market need and pinpoint the cutting-edge technologies in food engineering.
Furthermore, all contributions are written by internationally renowned experts who have both academic and professional credentials. All authors have attempted to
provide critical, comprehensive, and readily accessible information on the art and science 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 and researchers in universities and research institutions.
Preface
Food engineering has gained more and more significance in the last couple of decades. Mathematical models have been used to better understand and improve food processing operations, and in this concept, various optimization approaches have played a significant role. As a result, there has been a dramatic increase in the efficiency and reliability of optimization methods for different problem categories.
Optimization methods can be easily applied in food processing as long as the changes during a process can be predicted mathematically. This case, of course, depends on the presence of mathematical models. Since heat, mass, and momentum transfers are major mechanisms in food processing, mathematical models describing these phenomena are also required for further mathematical-based optimization procedures. Within this context, mathematical optimization plays an important role in optimizing different food processing operations.
Excellent text and reference books are available for educational and research purposes in thefield of optimization and food processing. It will therefore be quite significant to combine the advantages in this field for further optimization strategies to improve the quality and safety of food processes and optimal operating policies in the food industry.
Based on this concept, Optimization in Food Engineering has been divided into the following sections to serve as a reference for professional food scientists, food engineers, academicians, and graduate level students working in the field of food engineering and processing.
This book consists of three parts. In thefirst part, the significance 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 mathematmathemat-ical modeling techniques provides significant information for further research and developments in food processing. In addition, the changes predicted by a model in a given process are required if the given process is described other than by trial-and-error physical experiments.
In the second part, optimization and different optimization techniques are pre-sented. This part begins with statistical optimization techniques and continues with Pontryagin’s method, multi-objective and dynamic optimization techniques, and mixed integer linear and nonlinear programming methodologies. In addition, limi-tations and possibilities of using neural networks and genetic algorithms and com-putational fluid 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 a broad area for different processes starting from the optimization strategies to improve the performance of batch reactors to the optimization of conventional thermal processing, microwave heating, freeze drying, spray drying, and refrigeration systems. Different food processing areas are presented for optimization purposes,
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 the optimization procedures that need to be applied.
Series Editor
Professor Da-Wen Sunwas born in Southern China and is a world authority on food engin-eering research and education. His main research activities include cooling, drying, and refrigeration processes and systems; qual-ity and safety of food products; bioprocess simulation and optimization; and computer vision technology. His innovative studies on vacuum cooling of cooked meats, pizza qual-ity inspection by computer vision, and edible films for shelf-life extension of fruits and vegetables have been widely reported in national and international media. Results of his work have been published in over 180 peer-reviewed journal papers and more than 200 conference papers.
Professor Sun receivedfirst 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 thefirst Chinese national to be permanently employed in an Irish university when he was appointed college lecturer at National University of Ireland, 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 Engineering and director of the Food Refrigeration and Computerized Food Technology Research Group in University College Dublin.
As a leading educator in food engineering, Sun has contributed significantly to the field of food engineering. He has trained many PhD students, who have made their 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-ally and delivered keynote speeches at international conferences. As a recognized authority in food engineering, he has been conferred adjunct=visiting=consulting professorships from 10 top universities in China including Zhejiang University, Shanghai Jiaotong University, Harbin Institute of Technology, China Agricultural University, South China University of Technology, and Jiangnan University. In recognition of his significant contribution to food engineering worldwide and for his outstanding leadership in thefield, the International Commission of Agricultural Engineering (CIGR) awarded him the CIGR Merit Award in 2000 and again in 2006 and the Institution of Mechanical Engineers based in the United Kingdom named him Food Engineer of the Year 2004, in 2008 he was awarded CIGR Recognition Award in recognition of his distinguished achievements as top one percent of Agricultural Engineering scientists around the world.
He is a fellow of the Institution of Agricultural Engineers. He has also received numerous awards for teaching and research excellence, including the President’s Research Fellowship, and has received the President’s Research Award from Uni-versity College Dublin on two occasions. He is a member of the CIGR executive board and honorary vice president of CIGR; editor-in-chief of Food and Bioprocess Technology—An International Journal (Springer); series editor of Contemporary Food Engineering (CRC Press=Taylor & Francis); former editor of Journal of Food Engineering (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 Food Sciences. He is also a chartered engineer registered in the U.K. Engineering Council.
Editor
Dr. Ferruh Erdogdu is an associate professor of food engineering at the University of Mersin, Mersin, Turkey. He was born in Eregli, Turkey, and graduated from the Department of Food Engineering at Hacettepe University in Ankara in 1992 with honors and the highest GPA. In 1994, he succeeded in a nationwide exam by the Ministry of National Education of Turkey to pursue masters and PhD degrees in food engineering in the United States.
Dr. Erdogdu received his master of engineering degree in 1996 and PhD in 2000 at the University of Florida, Gainesville, Florida. While working with Dr. Murat O. Balaban at the University of Florida, he maintained a status of distinguished scholar. He received outstanding academic achievement awards from College of Engineering (1997–2000) and College of Agriculture (1999), and won the student paper competition hosted by the food engineering division of the Institute of Food Technologists (IFT). After receiving his PhD, he conducted his postdoctoral work at the University of California, Davis, California, with Dr. R. Paul Singh.
In 2001, Dr. Erdogdu joined the faculty of food engineering at the University of Mersin where he has been teaching undergraduate- and graduate-level courses on topics in food engineering. In 2007, he was appointed holder of a scholarship within the Swedish–Turkish Programme by the Swedish Institute for studies=research work at Lund University, Lund, Sweden. Ferruh is the author or coauthor of more than 30 research papers published in internationally known peer-reviewed journals, 4 book chapters, and more than 50 presentations. He is the coauthor of the books Virtual Experiments in Food Processing, published in 2004, and Industrial Scale Food Freezing Simulation Software published by the World Food Logistics Organization. Dr. Erdogdu is a professional member of the IFT. He has been serving on the editorial board of the Journal of Food Process Engineering since 2003 and assisting in review processes for Journal of Food Engineering; Journal of Food and Biopro-cess Technologies; Journal of Food Technology and Biotechnology; International Journal of Engineering; Computers and Chemical Engineering; and Chemical Process Engineering.
His current research interests include mathematical modeling and optimization of heat, mass, and momentum transfer operations in food processing.
Contributors
Murat O. BalabanFishery Industrial Technology Center University of Alaska Fairbanks Fairbanks, Alaska
J. Alberto Bandoni
Planta Piloto de Ingeniería Quimica Bahía Blanca, Argentina
Philippe Baucour
Institut FEMTO-ST, Department of CREST
Belfort, France
Lorenz T. Biegler
Department of Chemical Engineering Carnegie Mellon University
Pittsburgh, Pennsylvania
Aníbal M. Blanco
Planta Piloto de Ingeniería Quimica Bahía Blanca, Argentina
Jose Bon
Research Group Analysis and Simulation of Agro-food Processes Food Technology Department Polytechnic University of Valencia Valencia, Spain
Muhiddin Can Uludag University
Faculty of Engineering and Architecture Mechanical Engineering Department Gorukle Campus
Bursa, Turkey
Kun-Nan Chen
Department of Mechanical Engineering Tungnan University
Taipei, Taiwan
Ming-Ju Chen
Department of Animal Science and Technology
National Taiwan University Taipei, Taiwan
Kevin Cronin
Department of Process and Chemical Engineering
University College Cork Cork, Ireland
J. Miguel Díaz-Báñez Universidad de Sevilla
Departament de Matemática Aplicada II Escuela Superior de Ingenieros
Sevilla, Spain
Philip Doganis
National Technical University of Athens,
School of Chemical Engineering, Zografou Campus,
Athens, Greece
Jinglie Dou
University of British Columbia Food, Nutrition and Health
Vancouver, British Columbia, Canada
Ferruh Erdogdu
Department of Food Engineering University of Mersin
Çiftlikköy-Mersin, Turkey
A. Burak Etemoglu
Uludag University, Faculty of Engineering and Architecture Mechanical Engineering Department Gorukle Campus
Fernanda Fonseca INRA
Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France
Claudio A. Gelmi
Department of Chemical and Bioprocess Engineering Pontificia Universidad Católica
de Chile Santiago, Chile
María Teresa González
Planta Piloto de Ingeniería Química Bahía Blanca, Argentina
Sundaram Gunasekaran Biological Systems Engineering
Department
University of Wisconsin-Madison Madison, Wisconsin
Jorge Andrey Wilhelms Gut Department of Chemical
Engineering—Escola Politécnica
University of São Paulo São Paulo, Brazil
Jing Han
Universal Can Corporation Shizuoka, Japan
Quang Tri Ho
BIOSYST-MeBioS, K.U.Leuven Leuven, Belgium
Yasumi Horimoto
University of British Columbia Food, Nutrition and Health
Vancouver, British Columbia, Canada
Christos T. Kiranoudis School of Chemical Engineering Department of Process Control and
Plant Design
National Technical University of Athens
Athens, Greece
Andrey V. Kuznetsov Department of Mechanical and
Aerospace Engineering North Carolina State University Raleigh, North Carolina
Vitor V. Lopes
Institute of Systems and Robotics Technical University of Lisbon Lisbon, Portugal
Michèle Marin AgroParisTech, INRA Joint Research Unit Génie et
Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France
Rui C. Martins
BioInformatics—Molecular and Environmental Research Centre University of Minho
Braga, Portugal
Guillermo L. Masini Facultad de Ingeniería
Departamento de Mecánica Aplicada Universidad Nacional del Comahue Neuquén, Argentina
Luis G. Matallana
Planta Piloto de Ingeniería Química Bahía Blanca, Argentina
Hibru Kelemu Mebatsion BIOSYST-MeBioS, K.U. Leuven Leuven, Belgium
Yang Meng
Department of Food Science
McGill University Macdonald Campus Ste-Anne-de-Bellevue, Quebec, Canada
Reinaldo Morabito
Department of Production Engineering Universidade Federal de São Carlos São Paulo, Brazil
Iqbal M. Mujtaba
School of Engineering, Design and Technology
University of Bradford Bradford, England
Antonio Mulet
Research group Analysis and Simulation of Agro-food Processes Food Technology Department Polytechnic University of Valencia Valencia, Spain
Shuryo Nakai
University of British Columbia Food, Nutrition and Health
Vancouver, British Columbia, Canada
Bart Nicolaï
BIOSYST-MeBioS, K.U. Leuven Leuven, Belgium
Sadao Nishiyama
Universal Can Corporation Tokyo, Japan
T. Koray Palazoglu
Department of Food Engineering University of Mersin
Çiftlikköy-Mersin, Turkey
Stéphanie Passot AgroParisTech
Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France
J. Ricardo Pérez-Correa
Department of Chemical and Bioprocess Engineering
Pontificia Universidad Católica de Chile Santiago, Chile
Noemí C. Petracci
Planta Piloto de Ingeniería Química Bahía Blanca, Argentina
José Maurício Pinto
Advanced Control and Operations Research Technology Group Praxair, Inc.
Danbury, Connecticut
Vitória Pureza
Department of Production Engineering
Universidade Federal de São Carlos São Paulo, Brazil
Hosahalli S. Ramaswamy Department of Food Science McGill University Macdonald
Campus
Ste-Anne-de-Bellevue, Québec, Canada
Gade Pandu Rangaiah Department of Chemical and
Biomolecular Engineering National University of Singapore Singapore, Republic of Singapore
Panagiotis P. Repoussis School of Chemical Engineering Department of Process Control and
Plant Design
National Technical University of Athens
Athens, Greece
Haralambos Sarimveis National Technical University
of Athens
School of Chemical Engineering Athens, Greece
Cheah Keen Seng
Department of Chemical and Biomolecular Engineering National University of Singapore Singapore, Republic of Singapore
Ricardo Simpson
Departamento de Procesos Químicos Biotecnológicos, y Ambientales Universidad Técnica Federico
Santa María Valparaíso, Chile
Mekapati Srinivas
Department of Chemical and Biomolecular Engineering National University of Singapore Singapore, Republic of Singapore
Arthur A. Teixeira
Department of Agricultural and Biological Engineering
Institute of Food and Agricultural Sciences
University of Florida Gainesville, Florida
José A. Teixeira
Institute for Biotechnology and BioEngineering
Centro de Engenharia Biológica, University of Minho
Braga, Portugal
Ioan Cristian Trelea AgroParisTech
Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France
Vinh Truong
Department of Chemical Engineering Nong Lam University
Ho Chi Minh, Vietnam
Mahir Turhan
Department of Food Engineering University of Mersin
Çiftlikköy-Mersin, Turkey
Martín Juan Urbicain (Deceased) Planta Piloto de Ingeniería
Química
Bahía Blanca, Argentina
Pieter Verboven
BIOSYST-MeBioS, K.U. Leuven Leuven, Belgium
Roxana A. Verdini Instituto de Desarrollo
Technológico para la Industria Química
Santa Fe, Argentina
António A. Vicente
Institute for Biotechnology and BioEngineering
Centro de Engenharia Biológica University of Minho
Braga, Portugal
Koetsu Yamazaki
Division of Innovative Technology and Science
Graduate School of Natural Science and Technology Kanazawa University Kanazawa, Ishikawa, Japan
Part I
Modeling: Significance,
1
Significance of
Mathematical Modeling
and Simulation for
Optimization
Quang Tri Ho, Hibru Kelemu Mebatsion,
Bart Nicolaï, and Pieter Verboven
CONTENTS
1.1 Introduction ... 3 1.2 Heat and Mass Transfer Modeling ... 4 1.2.1 General Considerations ... 4 1.2.2 Case Study: Permeation–Diffusion–Reaction Model
of Gas Exchange in Pear Fruit ... 6 1.3 Kinetics Modeling... 7 1.3.1 General Considerations ... 7 1.3.2 Case Study: Model for the Respiration of Fruit ... 9 1.4 Model Parameters ... 11 1.4.1 Thermophysical Properties ... 11 1.4.2 Kinetics Parameters ... 11 1.5 Solution Methods ... 14 1.5.1 General Considerations ... 14 1.5.2 Case Study: Solution of Three-Dimensional Gas Exchange
and Respiration in Pear Fruit ... 14 1.6 Towards Food Process Modeling at Different Scales:
Multiscale Modeling ... 15 1.7 Conclusion ... 16 Acknowledgments... 16 Nomenclature ... 16 Greek Letters ... 17 References ... 17
1.1 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 mical changes , qu ality, and safet y). Math ematical aspects of form ulating and solving models 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 hysical properties and model param eters, such as varia bility and parameter estimat ion, are mentioned , and tool s for food process simulat ion at different scale s are presen ted.
The merits and limitati ons of food proces s simulation are demon strated by means of 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 of pears. Pears are typi cally stored under a contr olled atmospher e with reduced O2 and
increased CO2 level s to extend their commerci al stor age lif e, whic h can be as long as
9 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 the product. Ot her fruit such as apples are considerabl y less sensitiv e to varia tion s in low o xygen condit ions. This is probably related to diff erences in ga s concent ration gradients 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 rnal gas exchange woul d be, nevert heless, very valuab le to guide commerci al storage practices since disorde rs un der controlle d atmospher e related to ferm entation are a prime cause of concern . Opt imal stor age condit ions of new cultivars are generally determin 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 of respiration 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 modeling of heat and mass trans fer is introduced. In Se ction 1.3, model ing of kinet ics is described. Secti on 1.4 dea ls with model param eters. Solution metho ds are mentioned
in Sectio n 1.5 and in Se ction 1.6 a future perspectiv e is given in terms of multisca le modeling of food proces ses. Finally some con clusions are draw n in Sec tion 1.7. The chapter is intended to give the reader a framework and flavor for the following detailed chapters.
1.2 HEAT AND MASS TRANSFER MODELING
1.2.1 G
ENERALC
ONSIDERATIONSIn general, heat and mass transport occurs by diffusion as well as convective mechanisms. In its physical definition, diffusion is due to the spontaneous net movement 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 from a region of higher temperature to a region of lower temperature. However, diffusion is often used as an apparent mechanism encompassing more complex micro- and
nanoscale phenomena such as pressure driven flow, Knudsen flow, and capillary flow. The driving force behind convective transport is a pressure gradient in the case 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 restricted to a single Newtonian system. This means that the materials for which there is a linear relationship between shear stress and velocity gradient, such as water or air will be considered. More complicated fluids such as ketchup, starch solutions etc., are so-called non-Newtonianfluids, and the reader is referred to standard books on rheology for more details. In case of solid materials, convection will not be significant.
Applying the conservation principle to a fixed infinitesimal control volume dx1dx2dx3one obtains the mass continuity, momentum and energy, and mass fraction
equations, written in index notation for Cartesian coordinates xi(i¼ 1, 2, 3 for the
x-, y-, and z-direction, respectively), and whenever an index appears twice in any term, summation over the range of that index is implied (for example,@ruj
@xj becomes @ru1 @x1 þ @ru2 @x2 þ @ru3
@x3 in reality). This model can be applied to any such system
@r @t þ @ruj @xj ¼ 0 (1:1) @rui @t þ @rujui @xj ¼ @ @xjh @ui @xjþ @uj @xi @x@ i pþ2 3h @uj @xj þ fi (1:2) @rH @t þ @rujH @xj ¼ @ @xj k@T @xj þ@p@tþ Q (1:3) @rXa @t þ @ @xj rujXa¼ @ @xi rDa @ @xi Xaþ ra (1:4)
For a full derivation of these equations we refer to any text book onfluid mechanics. The system of at leastfive equations (three equations for the velocity components plus the continuity and the energy equation), and added with a mass fraction equation 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 densityr and pressure p and temperature T. The constitutive equation relates the enthalpy h to the pressure and the temperature by means of the heat capacity c¼ @H@T p. There are no conclu-sive general rules for implementation of boundary conditions for the Navier–Stokes equations to have a well-posed problem because of their complex mathematical nature. For incompressible and weakly compressible flows, it is possible to define Dirichlet boundary conditions (fixed values of the variables mostly for an incoming flow), Neumann boundary conditions (fixed gradients, mostly for an outgoing flow), and wall boundary conditions (a wall function reflecting the system behavior at the solid boundaries at the edge of the system considered). Initial values must be provided for all variables.
1.2.2 C
ASES
TUDY: P
ERMEATION–D
IFFUSION–R
EACTIONM
ODELOF
G
ASE
XCHANGE INP
EARF
RUITThe tissue structure of pear fruit is considered to contain mainly two phases: intracellular liquid phase of the cells and air-filled intercellular space. Assuming local equilibrium at a certain concentration of the gas component i in the gas phase Ci,g (mol m3), concentration of the compound in the liquid phase of fruit tissue
normally follows Henry’s law. If the tissue has a porosity «, the volume-averaged concentration Ci,tissue(mol m3) of species i is then defined as
Ci,tissue¼ « Ci,gþ (1 «) R T Hi Ci,g (1:5)
with Hiis Henry’s constant of component i (i is O2, CO2, or N2). From this definition,
following expression for the gas capacity (ai) of the component i of the tissue is
derived
ai¼ « þ (1 «) R T Hi¼
Ci,tissue
Ci,g
(1:6) A permeation–diffusion–reaction model was constructed describing the diffusion and 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 by
Ho et al. (2008)
ai
@Ci
@t þ r (uCi) ¼ r DirCiþ Ri (1:7)
with boundary conditions at the external surface of the pear:
Ci¼ Ci,1 (1:8)
where
Ri is the production term of the gas component i related to O2consumption
or CO2production
r (m1) is the gradient operator
The index1 refers to the gas concentration of the ambient atmosphere. The first term in Equation 1.7 represents the accumulation of gas i, the second term permeation transport driven by an overall pressure gradient, the third term molecular diffusion due to a partial pressure gradient, and the last term consumption or production of gas i because of respiration or fermentation. If, for example, oxygen is consumed in the fruit center, it creates a local partial pressure gradient, which drives molecular diffu-sion. However, if the rates of transport of different gasses are different, overall pressure gradients may build up and cause permeation transport. Permeation through the barrier of tissue by the pressure gradient is described by Darcy’s law (Geankoplis, 1993), which in fact is an apparent form of the momentum equation outlined above
u ¼ K mrP ¼ K R T m r X Ci (1:9) The relation between gas concentration and pressure was assumed to follow the ideal gas law (P¼ CRT). A typical model simulation result for gas exchange in pear tissue is shown in Figure 1.1. An axi-symmetric geometry model was created for the pear. Simulation showed that, due to the respiration of the tissue, the O2gas
partial pressure decreased from surface to the pear center while CO2decreased in the
opposite direction. An increase of N2from surface to the pear center was also found.
The profiles are strongly dependent on the tissue properties. In the past, it was assumed that gas transfer barriers were restricted to the skin layers of fruit. Here, it was demonstrated that the resistance to gas exchange of thefleshy part of fruit is also significant and could lead to oxygen deficiency.
1.3 KINETICS MODELING
1.3.1 G
ENERALC
ONSIDERATIONSMany food processes are associated with a kinetic aspect, an attribute that changes with time (e.g., microbial activity, active components of disinfectants in cool rooms, color changes), and the resulting kinetic reaction must be solved. Therefore, the reactions rates, property changes, and heat releases must be calculated as a part of the solution. Consider the following reaction
Aþ B ! C (1:10)
where the reaction rate Rc(mol s1) is defined
92 90 88 86 84 82 80 3.5 3 2.5 2 1.5 1 0.5 N2
partial pressure (kPa)
CO
2
partial pressure (kPa)
O2
partial pressure (kPa)
20 18 16 14 12 10 8 6 4 0 0 0 0 0.02 0.02 0.02 0.02 x (m) x (m) x (m) x (m)
FIGURE 1.1 Finite element mesh of pear geometry and simulated gas partial pressure distribution in pear intact fruit using permeation–diffusion–reaction model. Simulation was carried out at 18C, 20 kPa O2, 0 kPa CO2at 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.
Rc¼ d dt[A] ¼ d dt[B] ¼ d dt[C] ¼ kf[A]n[B]m[C]o kb[A]p[B]q[C]r (1:11)
with kf the forward rate constant and kb the backward rate constant. The rate
constants can be modeled by the following Arrhenius-like expression
kf ,b¼ aTbe E R 1 T 1 Tref (1:12) where
a and b are the empirical constants E is the empirical activation energy Trefis the reference temperature
The heat of reaction can be calculated from the heats of formation of the species and depends on temperature. The reaction leads to sources=sinks in the conservation and energy equations.
The most widely studied kinetics is that of enzymes. They are involved in many aspects of food quality, catalyzing the complex underlying biochemical reactions that 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 into a product (P) with the help of enzyme (E):
S!E P (1:13)
The rate of reaction rPcan be expressed in terms of either the change of substrate
concentration, CSor the product concentration, CP
rP ¼
dCS
dt ¼ dCP
dt (1:14)
It is important to know how the reaction rate is influenced by reaction conditions such as substrate, product, and enzyme concentration if you want to understand the effectiveness and characteristics of an enzymatic reaction. If the initial reaction at different levels of substrate and enzyme concentrations are measured, we often obtain a series of characteristic curves, where the reaction rate is proportional to the substrate concentration (first-order reaction) at low values of substrate concentration, and does not depend on the substrate concentration (zero-order reaction) at high values of substrate concentration which means the reaction goes gradually from first- to zero-order as the concentration of the substrate is increased. The maximum reaction rate, Vmaxis proportional to the enzyme
concen-tration. This was what Henri observed in 1902, and he proposed the following rate equation
rP¼ VmaxCS KMþ CS (1:15) where Vmax(mol m3s1)
KM (mol m3) are kinetic parameters which need to be experimentally
determined
KMis the substrate concentration required for an enzyme to reach half of its maximum
velocity. This equation describes many experimental results well. A quantitative theory exists to support the observed enzyme kinetics and is still widely used today under the name Michaelis–Menten kinetics.
1.3.2 C
ASES
TUDY: M
ODEL FOR THER
ESPIRATION OFF
RUITRespiration is one of the most important processes in fruits. Extended Michaelis– Menten kinetics is widely used as a semiempirical model to describe the relationship of the respiration to the O2 and CO2 concentration, and the whole respiration
pathway 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 O2by
respiration as formulated by RO2¼ Vm,O2 PO2 (Km,O2þ PO2) 1 þ PCO2 Kmn,CO2 (1:16) where Vm,O2(mol m
3s1) is the maximum oxygen consumption rate
P (kPa) is the partial pressure for O2and CO2
Km (kPa) is the Michaelis–Menten constant for O2 consumption and
noncompetitive CO2inhibition
RO2(mol m
3s1) is the O
2consumption rate of the sample
The equation for production rate of CO2consists of an oxidative respiration part and
a fermentative part (Peppelenbos et al., 1996)
RCO2 ¼ rq,ox RO2þ Vm, f ,CO2 1þ PO2 Km, f ,O2 (1:17) where Vm,f,CO2(mol m
3s1) is the maximum fermentative CO
2production rate
Km,f,O2(kPa) is the Michaelis–Menten constant of O2inhibition on fermentative
CO2production
rq,oxis the respiration quotient at high O2partial pressure
RCO2(mol m
3s1) is the CO
The effect of temperature was described by Arrhenius’ law (Hertog et al., 1998)
Vm,O2¼ Vm,O2,refexp
Ea,VmO2 R 1 Tref 1 T (1:18) Vm, f ,CO2 ¼ Vm, f ,CO2,refexp
Ea,Vmf CO2 R 1 Tref 1 T (1:19) where
Vm,O2,ref and Vm, f,CO2,ref (mol m
3s1) are the maximal O
2 consumption and
maximal fermentative CO2production rate at Tref¼ 2938K, respectively
Ea,Vm(kJ mol1) is the activation energy for O2consumption and fermentative
CO2production
Typical respiration rates of pear tissue are given in Figure 1.2. The estimated param-eters for Vm,O2and Vm,f,CO2of cortex tissue were (2.39 0.14) 10
4mol m3s1
and (1.61 0.13) 104 mol m3 s1, respectively. Km,O2, a measure for the
saturation of respiration with respect to O2 was relatively small and equal to
(1.00 0.23) kPa. A significant but low inhibition effect of CO2on O2consumption
of pear cortex tissue was found (Kmn,CO2¼ 66.4 21.3 kPa). The respiration quotient
rq,oxwas 0.97 0.04 and showed that the O2consumption was about the same as
the oxidative CO2production. The value Km,f,O2is a measure of the extent to which
fermentation can be inhibited by O2. The estimated value of 0.28 0.14 kPa implies
that fermentation was already inhibited at very low levels of O2concentration.
O2 partial pressure (kPa) O2 partial pressure (kPa)
5 0 10 0 0 1 1 0 2 2 2 4 6 8 10 ⫻10⫺4 ⫻10⫺4 RO 2 (mol m ⫺ 3 s ⫺ 1) RCO 2 (mol m ⫺ 3 s ⫺ 1)
FIGURE 1.2 O2consumption and CO2production rate in pear tissue disks at 208C; Solid
lines (—) and dashed lines (- -) indicate the respiration model at 0 and 10 kPa CO2while the
symbols () and (o) indicate the experiment at 0 and 10 kPa CO2. (Adapted from Ho, Q.T.
1.4 MODEL PARAMETERS
1.4.1 T
HERMOPHYSICALP
ROPERTIESThe therm ophysical properties k, r, and c may be tem perature dependen t (due to insuf ficient probl em dec omposit ion, i.e., the underl ying physi cochemical changes are not modeled expli citly) so that the problem becom es no nlinear. Thermo physical properties 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 been published, whi ch relate the thermophys ical properties of agric ultural product s and food materials to thei r c hemical compo sition. In general, both he at capaci ty and density can be calculated with suf ficient accuracy , but the model s for thermal conductivi ty requi re some assum ptions about the orien tation of d ifferent mai n chemical constitu ents with respect to the direction of heat flow.
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 in diffusivity 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 the case study under consi deration here, one should meas ure gas concent ration as a function of time and space, to whi ch the mass trans fer model is fitted by o ptimizing the apparen t diff usivity value. This is usual ly achiev ed by an iter ative least square s procedu re. Althoug h one c an careful ly desig n experi ments to imp rove the q uality of the fitting, in many cases, variability due to the food material composition and structure must be investigated by, e.g., analysis of variance to reveal significant effects (Ho et al., 2006a,b). Estimated diffusivities of pear tissue are given in
Table 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 in the axial and radial directions. The higher diffusivity in the axial direction compared to that along the radial direction is probably due to the fact that vascular bundles may be not fullyfilled with sap during storage of the fruit and facilitate gas exchange. Higher diffusivity of CO2 compared to O2and N2 is probably due to the larger
solubility of CO2in water than that of O2and N2. In addition, while O2and N2would
be transported mostly through the apoplast, CO2 would also diffuse through the
cytoplasm.
1.4.2 K
INETICSP
ARAMETERSThe kinetics parameters may be determined byfitting the proposed kinetics model to the 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 of respiration of pears, the data on O2 consumption and CO2 production rates are
pooled, and the same weight can be attributed to both gases. Accuracy of the estimated kinetics parameters reflecting the variability and experimental error struc-ture can be expressed by confidence intervals, and asymptotic confidence intervals can be calculated from the asymptotic covariance matrix C of the parameters
where
J is the Jacobi an mat rix with respec t to the estimat ed param eters s 2 is the mean square d error
The asym ptot ic (1 a)% con fide nce interval on the i th parameter estimat e Pi was
calculated from Pi t 1 a 2 , n p ffiffiffiffiffiffiffiCi, i p ( 1:21 ) where
t is the Studen t t -distributi on n is the numbe r of meas urements p is the numbe r of param eters Ci,i is the ith diagon al elem ent of C
Correla tion coeffi cients ri,j of estimat ed model param eters i and j indi cate the
strength and direc tion of the relations hip betw een estimated model param eters i and j . These coef ficient s can be compu ted from
ri, j ¼
Ci , j
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ci , i Cj, j
p ( 1:22 ) The correl ation coeffi cients of kinet ics parameter s of pear tissue respiration from
Figure 1.2 are given in Tab le 1.2. The correl ation coef ficient s are all smaller than 0.71 suggesting that the model is not over-parameterized.
TABLE 1.1
Gas Transport Properties of Pear Tissue Diffusivity
(m2s1) on Tissue MeasurementEstimated Values Based
DO2,skin (1.86 0.70) 10 10 a DCO2,skin (5.06 3.3) 10 10 a DN2,skin (1.06 0.29) 10 10 b DO2,r (2.8 1.59) 10 10 b DCO2,r (2.32 0.41) 10 9 a DN2,r (2.67 1.62) 10 10 b DO2,z (1.10 0.40) 10 9 b DCO2,z (6.97 2.19) 10 9 a DN2,z (1.06 0.66) 10 9 b
Note: 95% confidence limits. Indices skin, r, and z refer
to 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.,
TABLE 1.2
Correlation Coefficient 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 0.01 Km,O2 0.70 1 0.22 0.26 0.58 0.03 0.12 0.01 Kmn,CO2 0.50 0.23 1 0.02 0.04 0.00 0.10 0.00 rq,ox 0.56 0.26 0.02 1 0.43 0.05 0.11 0.02 Km,f,O2 0.38 0.58 0.04 0.43 1 0.30 0.06 0.10 Vm,f ,CO2,tissue 0.02 0.03 0.00 0.05 0.30 1 0.00 0.34 Ea,VmO2 0.20 0.12 0.10 0.11 0.06 0.00 1 0.00 Ea,VmfCO2 0.01 0.01 0.00 0.02 0.10 0.34 0.00 1 ß 2008 by Taylor & Francis Group, LLC.
1.5 SOLUTION METHODS
1.5.1 G
ENERALC
ONSIDERATIONSUnless simplifications are made, the models presented in this chapter cannot be solved by analytical means. In many industrial applications, however, simplifications are very well possible. At afirst simplification level, the geometry can be simplified. For simple shapes such as cylinders, spheres, and blocks one canfind, with certain conditions on the model parameters, (usually they have to be constants!), analytical solutions as a function of time and spatial coordinates. At a second level, overall balances can sometimes be made excluding the spatial dimension. One then typically ends up with ordinary differential equations that can be solved quite efficiently with the latest numerical solvers. At the last simplification level, one is also able to exclude the time dimension and overall balances leading to the algebraic equations. If one does have to rely on numerical means to solve mathematical models, there are a number of efficient methods available. For this purpose the problem is first reduced significantly by requiring a solution only for a discrete number of points (the so-called grid) rather than for each point of the space-time continuum through which the heat and mass transfer proceed. The original governing partial differential equations are accordingly transformed into a system of difference equations and solved 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 longer exact, but only an approximation of the exact solution. Fortunately, the approxima-tion error can be decreased substantially by increasing the number of discretizaapproxima-tion points at the expense of additional computing time.
Various discretization methods have been used in the past for the numerical solution of heat and mass transfer problems arising in food technology. Among the most commonly used arefinite difference method, finite element method, and finite volume method. It must be emphasized that, particularly in the case of nonlinear heat transfer problems, the numerical solution must always be validated. It is very well possible that a plausible, convergent but incorrect solution is obtained. At least a grid dependency study must be carried out to verify whether the solution basically remains the same when the computational grid is refined.
1.5.2 C
ASES
TUDY: S
OLUTION OFT
HREE-D
IMENSIONALG
ASE
XCHANGEAND
R
ESPIRATION INP
EARF
RUITNumerical solution can be applied to solve the governing partial differential equa-tions of heat and mass transfer using thefinite element method. In the axi-symmetric case study, 2719 quadratic finite elements with triangular shape were used and required less than 5 min of CPU time on a desktop PC. For the mass transfer model involving a kinetic reaction term, for example consumption of O2inside the
fruit, mathematical equation of the reaction may not exclude negative concentrations. Numerical problems may then be expected when the concentration approaches zero resulting in nonphysical negative results. Here is an example of two alternative approaches to solve the problem for O2exchange in intact fruit involving respiration:
1. To ensure that the O2 concent ration cann ot becom e negativ e due to O2
consumpti on ( RO2 0), the respi ration term in the perm eation –diff usion
reaction model was modi fied for O2 and CO 2 in the solution .
If CO2,g < 0 then RO 2¼ 0 and RCO2¼ Vm ,f ,CO2 .
If CO2,g 0 then R O2 and RCO2 are descri bed by thei r original equations
above. 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 to be positive . For examp le, exponent ial transform ation of the mai n variab le was used to impose positive values for the O2 concent ration.
CO2 , g ¼ exp ( UO 2 ) (1:23)
Hence, the mass transfer equation for O2 transforms into
(exp ( UO2 ) a O2 ) @
UO2
@ t þ r (u exp ( UO2 ))
¼ r ( DO2 exp ( UO2 )) r UO 2þ RO2 ( 1:24)
At the boundary UO2,r¼ ln (CO2,1). Similarly, exponential transformation
of the O2concentration was applied in the other equations. Both methods
avoided nonrealistic errors in the computations of gas exchange in fruit (Figure 1.1).
1.6 TOWARDS FOOD PROCESS MODELING AT DIFFERENT
SCALES: MULTISCALE MODELING
Many problems related to mathematical modeling of foods and food processes is the poor understanding of the microscopic and nanoscopic mechanisms that affect the macroscopic behavior of the food or process that is being modeled. As a conse-quence, apparent material properties that have to be expressed in complex equations in relation to other variables are used, and sophisticated experiments to find these relationships are required. Hence, more then often the validity range is quite limited, and variability is large.
For the case of biological materials like the pear fruit, the macroscopic properties likely depend on various microscopic histological and cellular features such as tissue types, geometric properties of the cell, presence of an adhesive middle lamella between individual cells, cellular water potential, mechanical properties of the cell wall, presence of intercellular spaces, and many more. These features cover a wide range of spatial scales, from nanoscopic (plasmodesmata, plasma membranes), over microscopic (cell wall–middle lamella complex, cell geometry), to macroscopic
(actual geometry of the material). The material properties of the continuum model, such as diffusion properties incorporate both actual physical material constants such as the diffusivity of water and air but also the microscale geometry of the tissue and intracellular space (Mebatsion et al., 2008).
Multiscale models are basically a hierarchy of submodels, which describe the material behavior at different spatial scales in such a way that the submodels are interconnected. As a result, investigation of the microstructure becomes a prerequi-site to understand transitional theoretical frameworks and modeling techniques to bridge the gap between length scale extremes. Multiscale modeling may involve challenging physical processes such as transport phenomena. Sometimes it is suf fi-cient tofind the solution of the coarser scale by including procedures to construct the equations on the coarser scale that account for the contribution of finer scales. However, this amounts to writing effective equations for the macroscale that account for lower scales, and it is a difficult task. Alternatively, equations for the fine scale itself can be solved. The up-scaling of fine scale solutions to a macroscale solution is known as homogenization. Homogenization has been defined as a collection 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 use simplerfine scale equations that are considerably less expensive to solve, and whose solutions have the same coarse scale properties. While still in its infancy, multiscale 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 transport phenomena and kinetic changes taking place were outlined, and that the main problem for mathematical modeling of food processes is poor understanding of the biochemical and microscopic mechanisms that cause macroscopic changes in appearance and appreciation of foods was demonstrated. As a result, considerable uncertainty in food process simulation and optimization was to be managed.
ACKNOWLEDGMENTS
The authors wish to acknowledge financial support by the Flanders Fund for Scientific 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). Pieter Verboven is Fellow of the Industrial Research Fund at the K.U. Leuven.
NOMENCLATURE
c Specific heat J kg18C1
fi External body forces, including the
gravitational force N m3
H Static enthalpy J kg1
Hi Henry’s constant of component I mol m3kPa1
k Thermal conductivity W m18C1
K Permeation coefficient m2
P Pressure Pa
Q Heat source or sink W m3
ra Source or sink of component a of the
material kg m3s1
R Unversal gas constant 8.314 J mol1K1
Ri Production term of the gas component mol m3s1
T Temperature 8C, K
u Apparent velocity vector m s1
ui(i¼ 1, 2, 3) Cartesian components of the velocity
vector U(u1, u2, u3) m s1
Xa Mass fraction of a component a
of the material t Time s
G
REEKL
ETTERS m, h Viscosity Pa s r Density kg m3REFERENCES
Chang, R., Physical Chemistry with Applications to Biological Systems, MacMillan Publishers, New York, 1981.
Chevillotte, P., Relation between the reaction cytochrome oxidase-oxygen and oxygen uptake in cells in vivo, J. Theor. Biol., 39, 277, 1973.
Geankoplis, J.C., Transport Processes and Unit Operations, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1993.
Hertog, M.L.A. et al., A dynamic and generic model on the gas exchange of respiring produce: the 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., 41,
113, 2006a.
Ho, Q.T. et al., A permeation–diffusion–reaction model of gas transport in cellular tissue of plant materials, J. Exp. Bot., 57, 4215, 2006b.
Ho, Q.T. et al., A continuum model for metabolic gas exchange in pear fruit, PLoS Comput. Biol., 4(3): e1000023, 2008. doi: 10.1371=journal.pcbi.1000023.
Lammertyn, J., Comparative study of the O2, CO2 and temperature effect on respiration
between ‘‘Conference’’ pear cells in suspension and intact pears, J. Exp. Bot., 52, 1769, 2001.
Mebatsion, H.K. et al., Modelling fruit (micro)structures, why and how, Trends Food Sci. Tech., 19, 59, 2008.
Peppelenbos, H.W. et al., Modelling oxidative and fermentative carbon dioxide production of fruit and vegetables, Post. Biol. Tech., 9, 283,1996.
2
Analytical Solutions
in Conduction Heat
Transfer Problems
Ferruh Erdogdu and Mahir Turhan
CONTENTS
2.1 Introduction ... 19 2.2 Analytical Solutions ... 21 2.3 Application and Use of Analytical Solutions ... 26 2.4 Conclusion ... 27 Nomenclature ... 27 Greek Letters ... 28 Subscripts ... 28 References ... 28
2.1 INTRODUCTION
In process optimization studies, the first step generally would be to have a math-ematical model for the given process. Exact solutions of the differential equations describing the process and numerical solutions (finite difference and finite element solutions) are preferred for this objective. Since all the variables affecting the process and the physical–chemical changes occurring in the medium can be defined in a numerical model, applying these solutions in the optimization models would result in longer run-time solutions. Therefore, exact solutions are sometimes preferred for testing the models for convergence of the optimization algorithms. In this chapter, the exact solutions, mostly preferred in the food engineering literature, for the conduction heat transfer problems are reviewed.
A general heat transfer problem encountered in food process engineering area is to determine the steady or unsteady (transient) state temperature distribution in solid food products where the initial temperature distribution and the boundary conditions are specified. An unsteady temperature distribution includes not only the temperature variation from point to point in the medium but also with time (Kakac and Yener, 1993). This problem includes finding exact solution of the governing diffusion equation for different geometries or different coordinate systems. The simplest
case of diffusion equation with constant and isotropic thermophysical properties is given as follows: r2T¼1 a @T @t (2:1)
wherea (thermal diffusivity) is given by a ¼r ck
p (2:2)
The Laplacian (r2) of temperature in various coordinate systems is as follows (Kakac and Yener, 1993):
Rectangular (Cartesian): r2 T ¼@ 2T @x2 þ @2T @y2 þ @2T @z2 (2:3) Cylindrical: r2T ¼1 r @ @r r @T @r þ 1 r2 @2T @f2þ @2T @z2 (2:4) Spherical: r2T ¼ 1 r2 @ @r r2 @T @r þ1 r2 @ @m (1 m2) @T @m þ 1 r2 (1 m2) @2T @f2 (2:5) wherem ¼ cos u.
Physical significance of thermal diffusivity is associated with the speed of heat propagation into the solid product (Ozisik, 1993). The higher the thermal diffusivity the faster the heat transfer rate is the general belief for thermal diffusivity. In a recent publication, Palazoglu (2006) reported that the speed of heat penetration was a function of thermal diffusivity and heat transfer coefficient combination rather than the thermal diffusivity itself.
The exact solution, called analytical solution, of Equation 2.1 for Cartesian (for infinite and finite slab shaped geometries), cylindrical (for infinite and finite cylinder geometries), and spherical (for sphere) coordinate systems are generally used in the literature to especially verify numerical solutions and to develop numerical schemes and grid generation methods (Cai et al., 2006). The first step in obtaining the analytical solution is to choose the orthogonal coordinate system of which the coordinate surfaces will be coinciding with the boundary surfaces of the solid product (Ozisik, 1993). For example, Cartesian coordinate system is used for rectangular bodies while cylindrical coordinate system is used for cylinder shapes and spherical coordinate system is used for sphere shapes.
To simplify the analytical solutions, generally one-dimensional solution repre-senting an infinite slab for Cartesian coordinates, an infinite cylinder for cylindrical coordinates, and sphere for spherical coordinates are applied assuming all the surfaces facing equivalent boundary conditions. An additional solution for the given coordinates requiring the temperature change in only one dimension might be a semi-infinite solution approach. Lumped system solution is another method-ology independent of any coordinate system.
Even though the situations described by analytical solutions represent a small proportion in heat transfer analysis, these solutions can be applied on the basis leading to check the solutions given, for example, by numerical solutions. Therefore, the objective of this chapter is to summarize the usefulness of analytical solutions with 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 significant role in heat transfer simulations for design and optimization purposes 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 the literature to obtain the transient temperature distribution in such shaped foods.
These solutions are obtained using some analytical techniques including Laplace transform and the method of separation of variables. Separation of variables has been widely used in solving the heat conduction problems where the homogeneous equation systems are readily handled (Ozisik, 1993). This method is based on expanding a function in terms of Fourier series. In this method, the dependent variable (temperature, T in the given cases here) is assumed to be the product of independent variables (location, x and time, t). This method is applied when the governing equation and differential equations representing the boundary and initial conditions are homogeneous and linear. For the cases of nonhomogeneous condi-tions where there is more than one nonhomogeneous condition, the superposition techniques are applied to split the problem into simpler problems (Cengel, 2007).
One typical use of analytical solutions is to validate numerical solutions where the constant thermophysical (thermal conductivity, specific heat and density) and constant boundary conditions with uniform initial temperature distribution are used and to develop numerical schemes and grid generation methods (Cai et al., 2006). Analytical solutions give a concise parametric solution in ideal problems of academic interest or to check numerical simulations as explained above. The analytical solutions are generally restricted by the following assumptions:
. Solid object conforming to a regular geometry (i.e., slab, cylinder, or sphere) with the exception of lumped system analysis
. Constant thermal properties and physical dimensions
If the initial temperature distribution and surrounding medium temperatures are not constant and are given by certain functions, an analytical solution can still be obtained, but the results might be complex compared to the constant temperature
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 can be modified for infinite slab (n ¼ 0), infinite cylinder (n ¼ 1), and sphere (n ¼ 2) geometries as follows: 1 a @T @t ¼ 1 xn @ @x x n@T @x (2:6) Equation 2.6 can then be written for a one-dimensional heat transfer in an infinite cylinder and sphere, respectively
1 a @T @t ¼ 1 x @T @xþ @2T @x2 (2:7) 1 a @T @t ¼ 2 x @T @xþ @2T @x2 (2:8)
The case of@T@t ¼ 0 is the simplest approach in a heat transfer analysis leading to the steady 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 assumpintegra-tion is not valid, integration in time with requirement of an initial condition must be used.
Boundary conditions across the surfaces usually encountered in conduction heat transfer are prescribed surface temperature, prescribed heat flux, and convection boundary:
. Prescribed surface temperature: Surface temperature as constant or function of space and time
. Prescribed heatflux: Heat flux across the boundaries specified at constant or as a function of space and time
. Convection boundary condition: Equivalence of convection over the boundary to the conduction towards the solid geometry
k @T@x
s ¼ h (Tjs T1
) (2:9)
The analytical solutions are obtained using convection boundary present across the surface boundary and a symmetry condition at the center (special case of the prescribed heatflux where q00and thereforedT
dxis equal to 0) with a constant uniform
initial temperature distribution by applying separation of variables solution method-ology to Equations 2.3 through 2.5
T(x,t) T1 Ti T1 ¼X1 n¼1 Cn(x) exp m2n a t L2 h i (2:10) wheremnand Cn(x) and are given for regular shaped geometries of slab, cylinder,