Mathematical Modeling Techniques in Food and Bioprocesses

CHAPTER 1

Mathematical Modeling Techniques in Food andBioprocesses: An Overview

Ashim K. Datta and Shyam S. Sablani

CONTENTS

1.1 Mathematical Modeling ............................................................................................................2

1.2 Classification of Mathematical Modeling Techniques.............................................................3

1.3 Scope of the Handbook.............................................................................................................4

1.4 Short Overview of Models Presented in this Handbook..........................................................4

1.4.1 Physics-Based Models (Chapter 2 through Chapter 8) ................................................4

1.4.1.1 Molecular Dynamic Models ...........................................................................5

1.4.1.2 Lattice Boltzmann Models (Chapter 2) ..........................................................5

1.4.1.3 Continuum Models (Chapter 3 through Chapter 6) .......................................6

1.4.1.4 Kinetic Models (Chapter 7) ............................................................................6

1.4.1.5 Stochastic Models (Chapter 8)........................................................................6

1.4.2 Observation-Based Models (Chapter 9 through Chapter 15) .......................................7

1.4.2.1 Response Surface Methodology (Chapter 9)..................................................7

1.4.2.2 Multivariate Analysis (Chapter 10) ................................................................7

1.4.2.3 Data Mining (Chapter 11)...............................................................................9

1.4.2.4 Neural Network (Chapter 12) .........................................................................9

1.4.2.5 Genetic Algorithms (Chapter 13) ...................................................................9

1.4.2.6 Fractal Analysis (Chapter 14).........................................................................9

1.4.2.7 Fuzzy Logic (Chapter 15)...............................................................................9

1.4.3 Some Generic Modeling Techniques (Chapter 16 through Chapter 18)......................9

1.4.3.1 Monte-Carlo Technique (Chapter 16) ..........................................................10

1.4.3.2 Dimensional Analysis (Chapter 17)..............................................................10

1.4.3.3 Linear Programming (Chapter 18) ...............................................................10

1.4.4 Combining Models ......................................................................................................10

1.5 Characteristics of Food and Bioprocesses..............................................................................10

Acknowledgments............................................................................................................................11

References........................................................................................................................................11

1

q 2006 by Taylor & Francis Group, LLC

1.1 MATHEMATICAL MODELING

A model is an analog of a physical reality, typically simpler and idealized. Models can be

physical or mathematical and are created with the goal to gain insight into the reality in a more

convenient way. A physical model can be a miniature, such as a benchtop version of an industrial

scale piece of equipment. A mathematical model is a mathematical analog of the physical reality,

describing the properties and features of a real system in terms of mathematical variables and

operations. The phenomenal growth in the computing power and its associated user-friendliness

Need for understandingthe detailed mechanisms

Availability of time and resources, depending on the state of a-prioriknowledge of the physics

Use fundamental laws to developphysics-based model

Obtain experimental datato develop observation-based model

Validate modelagainst experimental data

Possibly validate against additionalexperimental data

Extract knowledge from the model

Not really necessary

Strong need

Constrained

Available

HANDBOOK OF FOOD AND BIOPROCESS MODELING TECHNIQUES2using sensitivity analysis

Use model in optimization and control Figure 1.1 A simple overview of model development and use.


have allowed models to be more realistic and have fueled rapid growth in the use of models in

product, process, and equipment design and research. Many advantages of a model include (1)

reduction of the number of experiments, thus reducing time and expenses; (2) providing great

insight into the process (in case of a physics-based model) that may not even be possible with

experimentation; (3) process optimization; (4) predictive capability, i.e., ways of performing what

if scenarios; and (5) providing improved process automation and control capabilities.

Mathematical models can be classified somewhat loosely depending on the starting point in

making a model. In observation-based models, the starting point is the experimental data from

which a model is built. It is primarily empirical in nature. In contrast, the starting point for physics-

based models is the universal physical laws that should describe the presumed physical phenomena.

Physics-based models are also validated against experimental data, but in physics-based models the

experimental data do not have to exist before the model. The decision on whether to build an

observation-based or a physics-based model depends on a number of factors, including the need and

available resources, as shown in Figure 1.1. After a model is built, its parameters can be varied to

see their effectsthis process is termed parametric sensitivity analysis. A model can also be used to

control a process. These conceptual steps are also shown in Figure 1.1.

1.2 CLASSIFICATION OF MATHEMATICAL MODELING TECHNIQUES

Classification of mathematical models can be in many different dimensions (Gershenfeld 1999),

as shown in Figure 1.2. As implied in this figure, there is a continuum between the two extremes

for any particular dimension noted in this figure. For example, it can be argued that even a model

that is obviously physics-based, such as a fluid flow in a porous media, has permeability as a

parameter that is experimentally measured and is made up of many different parameters character-

izing the porous matrix and the fluid. It is possible to use a lattice Boltzmann simulation for the

same physical process that will not need most of these matrix and fluid parameters and, therefore,

can be perceived as more fundamental.

The chapters in this text cover much of the range shown in Figure 1.2 for any particular

dimension. Physics-based (first-principle-based) vs. data-driven models is the primary dimension

along which the chapters are grouped. Scale of models is another dimension covered here. The

lattice Boltzmann simulation in Chapter 2, for example, is at a smaller scale than the macroscale

First-principle based

Data-driven

Microscale

Macroscale

Deterministic

Stochastic Analytical

Numerical

Figure 1.2 Various dimensions of a model. This is not an exhaustive list.

MATHEMATICAL MODELING TECHNIQUES IN FOOD AND BIOPROCESSES: AN OVERVIEW 3


continuum models in Chapter 3 through Chapter 6. Another dimension is deterministic vs.

stochastic. For example, the deterministic models in Chapter 3 through Chapter 6 can be made

stochastic by following the discussion in Chapter 7. Analytical vs. numerical method of solution is

another dimension of models. Numerical models have major advantages over analytical solution

techniques in terms of being able to model more realistic situations. Thus, Chapter 2 through

Chapter 5 cover mostly numerical solutions, although some references to analytical solutions are

provided as well.

1.3 SCOPE OF THE HANDBOOK

Each chapter in this book describes a particular modeling technique in the context of food and

bioprocessing applications. Entire books have been written on each of the chapters in this hand-

book. However, these books are frequently not with food and bioprocess as the main focus. Also, no

one book covers the breadth of modeling techniques included here. The motivation behind this

handbook was to bring many different modeling techniques, as varied as physics-based and obser-

vation-based models, under one umbrella with food and bioprocess applications as the focus.

Because the end goal of even very different modeling approaches, such as physics-based and

observation-based models, can be the same (e.g., to understand and optimize the system), any

two modeling techniques can be conceptually thought of as competing alternatives. This is more

so in food and bioprocess applications in which the processes are complex enough that the super-

iority of any one type of modeling technique in an industrial scenario that demands quick answer is

far from obvious. Another reason for discussing various models under one roof is that different

types of models can be pooled to obtain models that combine the respective advantages. Succinct

discussion of each model in the same context of food and bioprocess can help trigger such possi-

bilities. The modeling techniques selected in the handbook are either already being used or have a

great potential in food and bioprocess applications. Emphasis has been placed on how to formulate

food and bioprocess problems using a particular modeling technique, away from the theory behind

the technique. Thus, the chapters are generally structured to have a short introduction to the

modeling technique, followed by the details on how that technique can be used in specific food

and bioprocess applications. Although optimization is often one of the major goals in modeling,

optimization itself is a broad topic that could not be included (with the exception of linear program-

ming) in this text because of its extensive coverage of modeling, and the reader is referred to the

excellent article by Banga et al. (2003).

1.4 SHORT OVERVIEW OF MODELS PRESENTED IN THIS HANDBOOK

A short description of each type of model presented in this handbook is presented in this section.

There is no such thing as the best model because the choice of a model depends on a number of

factors, the most obvious ones being the goal (whether to know the detailed physics), the modelers

background (statistics vs. engineering or physics), and the time available (physics-based models

typically take longer). Some of this is also noted in the schematic in Figure 1.1.

1.4.1 Physics-Based Models (Chapter 2 through Chapter 8)

Physics-based models follow from fundamental physical laws such as conservation of mass and

energy and Newtons laws of motion; however, empirical (but fairly universal) rate laws are needed

to apply the conservation laws at the macroscopic scale. For example, to obtain temperatures using

a physics-based model, combine conservation of energy with Fouriers law (which is empirical)

HANDBOOK OF FOOD AND BIOPROCESS MODELING TECHNIQUES4


of heat conduction. The biggest advantages of physics-based models are that they provide insight

into the physical process in a manner that is more precise and more trustable (because we start from

universal conservation laws), and the parameters in such models are measurable, often using

available techniques.

Physics-based models can be divided into three scales: molecular, macro, and meso (between

molecular and macro). An example of a model at the molecular scale is the molecular dynamic

model discussed later. Models such as the lattice Boltzmann model discussed in this book are in the

mesoscale. Macroscopic models are the most common among physics-based models in food.

Examples of macroscopic models are the commonly used continuum models of fluid flow, heat

transfer, and mass transfer. As we expand food and biological applications at micro- or nanoscale,

such as in detection of microorganisms in a microfluidic biosensor, scales will be approached where

the continuum models in Chapter 2 through Chapter 5 will break down (Gad-el-Hak 2005). Simi-

larly, in very short time scales, continuum assumption breaks down, and mesoscale or molecular

scale models become necessary (Mitra et al. 1995). General discussion of models when continuum

assumption breaks down can be seen in Tien et al. (1998).

Physics-based models today are less common in food and bioprocessing product, process, and

equipment design than in some manufacturing, such as automobile and aerospace. This can be

primarily attributed to variability in biomaterials and the complexities of transformations that

food and biomaterials undergo during processing; however, this scenario is changing as the appro-

priate computational tools are being developed. In fact, the physics-based model (such as

computational fluid dynamics, or CFD) is one of the areas in food process engineering experiencing

rapid growth.

1.4.1.1 Molecular Dynamic Models

Molecular dynamic (MD) models are physics-based models at the smallest scale. In its most

rudimentary version, repelling force between pairs of atoms at close range and attractive force

between them over a range of separations are represented in a potential function (such as Lennard

Jones), for which there are many choices (Rapaport 2004). The spatial derivative of this potential

function provides the corresponding force. Forces between one atom and a number of its neighbors

are then added to obtain the combined force, and Newtons second law of motion is then used to

obtain the acceleration from the force. This acceleration is then numerically integrated to obtain the

trajectory describing the way the molecule would move. Physical properties of the system can be

calculated as the appropriate time average over the trajectory, if it is of sufficient length. Although

applications of molecular dynamics relevant to food processing (such as protein functionality and

solution properties of carbohydrates) have been reported (Schmidt et al. 1994; Ueda et al. 1998),

there appears to be very little ongoing work in applying MD to systems of direct relevance to food

processing. Thus, MD has been excluded from this handbook.

1.4.1.2 Lattice Boltzmann Models (Chapter 2)

The lattice Boltzmann (LB) method is physics-based, but at an intermediate scale (referred to as

mesoscale) between the molecular dynamic model mentioned above that is at the microscale and

continuum models mentioned below that are at the macroscale, where physical quantities are

assumed to be continuous. LB is based on kinetic theory describing the dynamics of a large

system of particles. The continuum assumption breaks down at some point going from the macro-

scale toward the microscale. Examples of such systems can be colloidal suspensions, polymer

solutions, and flow-through porous media. This is where the lattice Boltzmann model is useful

and is currently being pursued in relation to food processes.



Other mesoscale simulations are also being used in food. For example, in Pugnaloni et al.

(2005), large compression and expansion of viscoelastic protein films are studied in relation to

stability of foams and emulsions during formation and storage.

1.4.1.3 Continuum Models (Chapter 3 through Chapter 6)

Continuum models presented in Chapter 3 through Chapter 6 primarily deal with transport

phenomena, i.e., fluid flow, heat transfer, and mass transfer. These physics-based models are

based on fundamental physical laws. Typically, these models consist of a governing equation that

describes the physics of the process along with equations that describe the condition at the boundary

of the system. The conditions at the boundary determine how the system interacts with the surround-

ings. Mathematically, they are needed to obtain particular solutions of the governing equation. The

solution of the combined governing equation-boundary condition system can be made as exact an

analog of the physical system as desired by including as much detail of the physical processes

as necessary.

Physics-based models have several advantages over observation-based models: (1) they can be

exact analogs of the physical process; (2) they allow in-depth understanding of the physical process

as opposed to treating it as a black box; (3) they allow us to see the effect of changing parameters

more easily; and (4) models of two different processes can share the same basic parameter (such

as mass diffusivity and permeability measured for one process can be useful for other processes).

The disadvantages of a physics-based model are as follows: (1) high level of specialized technical

background is required; (2) generally more work is required to apply to real-life problems; and (3)

often longer development time and more resources are needed.

In the past 10 years or so, physics-based continuum models have really picked up because of

the available powerful and user-friendly software. These software programs do have limitations,

however, that apply to food related problems because of complexities in the process and significant

changes in the material due to processing. For example, rapid evaporation, as is true in baking,

frying, and some drying operations, is hard to implement in most of these software. Also, these

continuum models rely heavily on properties data that are only sparsely available for food systems.

There are other physics-based continuum models for which more details could not be included

because of the scope of this handbook. For example, electromagnetic heating of food such as

microwave and radio frequency heating is modeled using the governing Maxwells equations,

some details of which are provided in Chapter 3. Likewise, solid mechanics problems in food,

such as during chewing, puffing, texture development, etc., are governed by the equations of solid

mechanics, which also are not included in the book.

1.4.1.4 Kinetic Models (Chapter 7)

Kinetic models mathematically describe rates of chemical or microbiological reactions. They

generally can be considered to be physics-based. However, in complex chemical and microbiolo-

gical processes, as is true for food and bioprocesses, the mechanisms are generally hard to obtain

and are not always available. The kinetic models for such systems are more data-driven than

fundamental (as could be true for simple systems).

1.4.1.5 Stochastic Models (Chapter 8)

The physics-based continuum models have material properties that are typically measured.

These models are often treated as deterministic ones, i.e., the parameter values are considered

fixed. However, due to biological and other sources of variability, these measured parameters can

have random variations. For example, viscosity of a sample can have random variation because of

HANDBOOK OF FOOD AND BIOPROCESS MODELING TECHNIQUES6


its biological variability. In a fluid flow model that uses viscosity, the final answer of interest, such

as pressure drop, would also have the random fluctuations corresponding to the random variations

in viscosity. Inclusion of such random variations makes the physics-based models more realistic.

Techniques to include such uncertainty are presented in Chapter 6 and Chapter 8.

1.4.2 Observation-Based Models (Chapter 9 through Chapter 15)

The physics-based modeling process described in part I assumes that a model is known, which is

frequently difficult to achieve in complex processes. Although a physics-based model may also be

adjusted based on measured data, observation-based models (see Figure 1.3) are inferred primarily

from measured data. Observational models are black box models to different degrees in relation to

the physics of the process. The classical statistical models can have a model in mind (often based on

some understanding of the process) before obtaining the measured data. This makes them less of a

black box than models such as neural network or genetic algorithm that are frequently completely

data driven; no prior assumption is made about the model and no attempt is made to physically

interpret the model parameters once the model is built. Loosely speaking, though, all observational

models are referred to as data-driven models. For this handbook (Figure 1.3), we separate the

classical statistical models from the rest of the observation-based models and refer to the rest as

data-driven models.

There are many practical situations in which time and resources do not permit a complete

physics-based understanding of a process. Physics-based models often require more specialized

training and/or longer development time. In some applications, detailed understanding provided by

the physics-based model may not be necessary. For example, in process control, detailed physics-

based models often are not needed, and observation-based models can suffice. Observation-based

models can be extremely powerful in providing a practical, useful relationship between input and

output parameters for complex processes. The types of data available and the purpose of modeling

usually influence the kind of observation models to be used. General information on how to choose

a model for a particular situation is hard to locate. An excellent Internet source guiding data-driven

model choice and development can be seen in NIST (2005). Because observation-based models are

built from data without necessarily considering the physics involved, use of such models beyond

the range of data used (extrapolation) is more difficult than in the case of physics-based models.

1.4.2.1 Response Surface Methodology (Chapter 9)

This is a statistical technique that uses regression analysis to develop a relationship between the

input and output parameters by treating it as an optimization problem. The principle of experi-

mental design is used to plan the experiments to obtain information in the most efficient manner.

Using experimental design, the most significant factors are found before doing the response surface

and finding the optimum. This method is quite popular in food applications. It is important to note

that finding the optimum using response surface is not limited to experimental data. Physics-based

models can also be used to generate data that can be optimized using the response surface method-

ology similar to the method for experimental data (Qian and Zhang 2005).

1.4.2.2 Multivariate Analysis (Chapter 10)

Multivariate analysis (MVA) is a collection of statistical procedures that involve observation

and analysis of multiple measurements made on one or several samples of items. MVA techniques

are classified in two categories: dependence and interdependence methods. In a dependence tech-

nique, the dependent variable is predicted or explained by independent variables. Interdependence

methods are not used for prediction purposes and are aimed at interpreting the analysis output to opt



Techniques that can be useful in either model Physics-based Observation-based

Mesoscale (L. Boltzmann)

Macroscale continuum Stochastic Monte-Carlo

Dimensional analysis

Linear programming

Classical statistical

Neural network

Fuzzy logic

Genetic algorithm

Fractal analysis

Modeling of food and bioprocesses

Fluid flow

Heat transfer

Mass transfer

Heat & Mass transfer

Kinetics

Multivariate analysis

Data mining

Response surface meth.

Part I Part III Part II

Datadriven

2

3

4

5

6

7

8

9

10

11 12

13

18 17 16

15

14

MicroscaleMol. Dynamics

Figure 1.3 Various models presented in this handbook and their relationships.

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1.4.2.3 Data Mining (Chapter 11)

Data mining refers to automatic searching of large volumes of data to establish relationships and

identify patterns. To do this, data mining uses statistical techniques and other computing method-

ology such as machine learning and pattern recognition. Data mining techniques can also include

neural network analysis and genetic algorithms. Thus, it can be seen as a meta tool that can combine

a number of modeling tools.

1.4.2.4 Neural Network (Chapter 12)

An artificial neural network model (as opposed to a biological neural network) is an intercon-

nected group of functions (equivalent to neurons or nerve cells in a biological system) that can

represent complex inputoutput relationships. The power of neural networks lies in their ability to

represent both linear and nonlinear relationships and in their ability to learn these relationships

directly from the modeled data. Generally, large amounts of data are needed in the learning process.

1.4.2.5 Genetic Algorithms (Chapter 13)

Genetic algorithms are search algorithms in a combinational optimization problem that mimick

the mechanics of the biological evolution process based on genetic operators. Unlike other optimi-

zation techniques such as linear programming, genetic algorithms require little knowledge of the

process itself.

1.4.2.6 Fractal Analysis (Chapter 14)

Fractal analysis uses the concepts from fractal geometry. It has been primarily used to charac-

terize surface microstructure (such as roughness) in foods and to relate properties such as texture,

oil absorption in frying, or the Darcy permeability of a gel to the microstructure. Although fractal

analysis may use some concepts from physics, the models developed are not first principle-based.

Processes governed by nonlinear dynamics can exhibit a chaotic behavior that can also be modeled

by this procedure. Applications to food have been only sporadic.

1.4.2.7 Fuzzy Logic (Chapter 15)

Fuzzy logic is derived from the fuzzy set theory that permits the gradual assessment of the

membership of elements in relation to a set in contrast to the classical situation where an element

strictly belongs or does not belong to a set. It seems to be most successful for the following: (1)

complex models where understanding is strictly limited or quite judgmental; and (2) processes in

which human reasoning and perception are involved. In food processing, the applications have been

in computer vision to evaluate food quality, in process control, and in equipment selection.

1.4.3 Some Generic Modeling Techniques (Chapter 16 through Chapter 18)

Included in this part of the book are three generic modeling techniques that are somewhatfor the best and most representative model. MVA is likely to be used in situations when one is not

sure of the significant factors and how they interact in a complex process. It is also a popular

modeling process in food.

MATHEMATICAL MODELING TECHNIQUES IN FOOD AND BIOPROCESSES: AN OVERVIEW 9universal and can be used in either physics-based or observation-based model building or for

optimization once a model is built.


1.4.4 Combining Models

1.5 CHARACTERISTICS OF FOOD AND BIOPROCESSES

the future.The industry in this area is characterized by a lower profit level and less room for drastic

changes, than, for instance, automotive and aerospace industries. This translates to lower invest-

ment in research and development, which in turn leads to the generally lower level of technical

sophistication as compared to other industries. Modeling, particularly physics-based modeling,

often requires time and resources that are not available in the food industry. Consequently, withsame material; (3) because the material contains large amounts of water, unless temperatures are

low, there is always evaporation in the food matrix. This evaporation is hard to handle in physics-

based models and increases complexity of the process; and (4) many food processes involve

coupling of different physics (e.g., microwave heating involves heat transfer and electromagnetics),

thus compounding complexities. As novel processing technologies are introduced and combination

technologies such as hurdle technology become more popular, complexities will only increase inSome characteristics of food and bioprocesses are as follows: (1) they often involve drastic

physical, chemical, and biological transformation of the material, during processing. Many of these

transformations have not been characterized, primarily because of the following: (1) such a large

variety of possible materials; (2) their biological origin, variabilities are significant, even in theVarious modeling approaches can be combined to develop models that are even closer to reality

and that have greater predictive power. For example, a physics-based model can be combined

with an observation-based model by treating the output from the physics-based model as analogous

to experimental data. See, for example, Eisenberg (2001) or work in a different application

(Sudharsan and Ng 2000). Such a combined model is useful when only a portion of the system

can be represented using a physics-based model or when the parameters in the physics-based model

are uncertain. Two or more observation-based modeling techniques can also be combined (e.g.,

Panigrahi 1998), which is sometimes referred to as a hybrid model. A challenge, however, is to

combine diverse methods in a seamless manner to provide a model that is easy to use.1.4.3.1 Monte-Carlo Technique (Chapter 16)

Monte Carlo refers to a generic approach whereby a probabilistic analog is set up for a math-

ematical problem, and the analog is solved by stochastic sampling. Chapter 7 shows the application

of this technique to physics-based models.

1.4.3.2 Dimensional Analysis (Chapter 17)

This is typically an intermediate step before developing mostly physics-based (but can be data-

driven) models that is used to reduce the number of variables in a complex problem. This can

reduce the computational or experimental complexity of a problem.

1.4.3.3 Linear Programming (Chapter 18)

This is a well-known technique that is used for the optimization of linear models. It can be used

in the context of a physics-based or a data-driven model.

HANDBOOK OF FOOD AND BIOPROCESS MODELING TECHNIQUES10the exception of a handful of large multinational companies, modeling in general and physics-based

modeling in particular are viewed as less critical and somewhat esoteric. It is expected that as the


computer technology continues to advance, modeling (particularly physics-based modeling) will

become easier and perhaps more of a viable alternative in the industry.

ACKNOWLEDGMENTS

Author Datta greatly acknowledges discussions with Professor James Booth of the Department

of Biological Statistics and Computational Biology, Professor John Brady of the Department of

Food Science, Professor Jean Hunter of the Department of Biological and Environmental Engin-

eering, and Mr. Parthanil Roy of the School of Operations Research and Industrial Engineering, all

Ueda, K., Imamura, A., and Brady, J. W., Molecular dynamics simulation of a double-helical b-Carrageenan

MATHEMATICAL MODELING TECHNIQUES IN FOOD AND BIOPROCESSES: AN OVERVIEW 11hexamer fragment in water, The Journal of Physical Chemistry A, 102(17), 27492758, 1998.of Cornell University.

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Banga, J. R., Balsa-Canto, E., Moles, C. G., and Alonso, A. A., Improving food processing using modern

optimization methods, Trends in Food Science and Technology, 14, 131144, 2003.

Eisenberg, F. G., Virtual experiments using computational fluid dynamics. Proceedings of 7th Conference on

Food Engineering, American Institute of Chemical Engineers, New York, 2001.

Gad-el-Hak, M., Liquids: The holy grail of microfluidic modeling, Physics of Fluids, 17, 113, 2005.

Gershenfeld, N., The Nature of Mathematical Modeling, Cambridge: Cambridge University Press, 1999.

Mitra, K., Kumar, S., Vedavarz, A., and Moallemi, M. K., Experimental evidence of hyperbolic heat conduc-

tion in processed meat, Journal of Heat Transfer, Transactions of the ASME, 117(3), 568573, 1995.

NIST. 2005. NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/

pmd/pmd.htm.

Panigrahi, S., Neuro-fuzzy systems: Applications and potential in biology and agriculture, AI Applications,

12(13), 8395, 1998.

Pugnaloni, L. A., Ettelaie, R., and Dickinson, E., Brownian dynamics simulation of adsorbed layers of inter-

acting particles subjected to large extensional deformation, Journal of Colloid and Interface Science, 287,

401414, 2005.

Qian, F. P. and Zhang, M. Y., Study of the natural vortex length of a cyclone with response surface method-

ology, Computers and Chemical Engineering, 29(10), 21552162, 2005.

Rapaport, D. C., The Art of Molecular Dynamics Simulation, Cambridge: Cambridge University Press, 2004.

Schmidt, R. K., Tasaki, K., and Brady, J. W., Computer modeling studies of the interaction of water with

carbohydrates, Journal of Food Engineering, 22(14), 4357, 1994.

Sudharsan, N. M. and Ng, E. Y. K., Parametric optimization for tumor identification bioheat equation using

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Francis, 1998.q 2006 by Taylor & Francis Group, LLC

Table of ContentsCHAPTER 1: Mathematical Modeling Techniques in Food and Bioprocesses: An Overview1.1 MATHEMATICAL MODELING1.2 CLASSIFICATION OF MATHEMATICAL MODELING TECHNIQUES1.3 SCOPE OF THE HANDBOOK1.4 SHORT OVERVIEW OF MODELS PRESENTED IN THIS HANDBOOK1.4.1 Physics-Based Models (Chapter 2 through Chapter 8)1.4.1.1 Molecular Dynamic Models1.4.1.2 Lattice Boltzmann Models (Chapter 2)1.4.1.3 Continuum Models (Chapter 3 through Chapter 6)1.4.1.4 Kinetic Models (Chapter 7)1.4.1.5 Stochastic Models (Chapter 8)

1.4.2 Observation-Based Models (Chapter 9 through Chapter 15)1.4.2.1 Response Surface Methodology (Chapter 9)1.4.2.2 Multivariate Analysis (Chapter 10)1.4.2.3 Data Mining (Chapter 11)1.4.2.4 Neural Network (Chapter 12)1.4.2.5 Genetic Algorithms (Chapter 13)1.4.2.6 Fractal Analysis (Chapter 14)1.4.2.7 Fuzzy Logic (Chapter 15)

1.4.3 Some Generic Modeling Techniques (Chapter 16 through Chapter 18)1.4.3.1 Monte-Carlo Technique (Chapter 16)1.4.3.2 Dimensional Analysis (Chapter 17)1.4.3.3 Linear Programming (Chapter 18)

1.4.4 Combining Models

1.5 CHARACTERISTICS OF FOOD AND BIOPROCESSESACKNOWLEDGMENTSREFERENCES

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Mathematical Modeling Techniques in Food and Bioprocesses