Food Processing Operations Modeling Design and Analysis, Second Edition (Food Science and...

FOODPROCESSINGOPERATIONSMODELING

S E C O N D E D I T I O N

Design and Analysis

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© 2009 by Taylor & Francis Group, LLC

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

Boca Raton London New York

FOODPROCESSINGOPERATIONSMODELING

S E C O N D E D I T I O N

Design and Analysis

ED ITED BY

Soojin JunJoseph M. Irudayaraj

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v

Table of ContentsPreface .....................................................................................................................vii

Editors ......................................................................................................................ix

Contributors ............................................................................................................xi

Chapter 1 Introduction to Modeling and Numerical Simulation ..........................1

K.P. Sandeep, Joseph Irudayaraj, and Soojin Jun

Chapter 2 Aseptic Processing of Liquid and Particulate Foods .......................... 13

K.P. Sandeep and Virendra M. Puri

Chapter 3 Modeling Moisture Diffusion in Food Grains during Adsorption ..... 53

Kasiviswanathan Muthukumarappan and S. Gunasekaran

Chapter 4 Computer Simulation of Radio Frequency Heating ........................... 81

Yifen Wang and Jian Wang

Chapter 5 Infrared Radiation for Food Processing ........................................... 113

Kathiravan Krishnamurthy, Harpreet Kaur Khurana, Soojin Jun, Joseph Irudayaraj, and Ali Demirci

Chapter 6 Modeling of Ohmic Heating of Foods .............................................. 143

Soojin Jun and Sudhir Sastry

Chapter 7 Hydrostatic Pressure Processing of Foods ....................................... 173

J. Antonio Torres and Gonzalo Velazquez

Chapter 8 Pulsed Electric Field (PEF) Processing and Modeling .................... 213

Si-Quan Li

Chapter 9 Fouling Models for Heat Exchangers ............................................... 235

Sundar Balsubramanian, Virendra M. Puri, and Soojin Jun

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vi Table of Contents

Chapter 10 Ozone Treatment of Food Materials ............................................... 263

Kasiviswanathan Muthukumarappan, Colm P. O’Donnell, and Patrick J. Cullen

Chapter 11 UV Pasteurization of Food Materials ............................................. 281

Kathiravan Krishnamurthy, Joseph Irudayaraj, Ali Demirci, and Wade Yang

Chapter 12 Stochastic Finite Element Analysis of

Thermal Food Processes.................................................................303

Bart M. Nicolaï, Nico Scheerlinck, Pieter Verboven, and Josse De Baerdemaeker

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vii

PrefaceThe second edition of Food Processing Operations Modeling: Design and Analysis has its unique value far beyond an extension of the previous edition. The key focus of

the second edition is to address novel food processing technologies that are of immense

and infusion of new processes and instrumentation, tomorrow’s consumers will have

access to safe, nutritious, high-quality products via novel food processing technologies.

and pulsed ultraviolet treatments are representative novel techniques to alternate the

traditional food processing methods. The fundamental principles and associated

numerical approaches are some of the key elements addressed in this edition.

Chapter 7 on HPP includes modeling studies to describe microbial kinetics and

Chapter 8, PEF processing is a non-thermal method of food preservation that uses

short bursts of electricity for microbial inactivation with little detrimental effect to

food quality. Along with the fundamentals of the PEF system and operation, novel

food applications and supportive numerical models have been described. Accurate

prediction and analysis of fouling dynamics based on an understanding of chemis-

to respond is discussed in Chapter 9. An introduction to fouling models for heat

coupled with the reaction scheme of milk protein under fouling, is also detailed. The

bactericidal effects of ozone have been documented for a wide variety of organisms,

including Gram positive and Gram negative bacteria as well as spores and vegetative

cells. In Chapter 10, chemical and physical properties of ozone, its generation, and

the antimicrobial power of ozone have been explained as well as many advantages of

ozone use in the food industry.

UV-light used as a bactericidal agent is a portion of electromagnetic spectrum rang-

ing from 100 to 400 nm wavelengths and has the potential to denature the microbial

DNA by forming thymine dimmers, leading to microbial inactivation. Chapter 11 will

inactivation.

In addition, new modeling approaches for infrared heating that include the tem-

perature dependence of spectral distribution and ohmic heating coupled with CFD

tools have been addressed. Modeling of multi-phase food products with various elec-

trical conductivities has been introduced in the chapter on ohmic heating. Distortion

lar domain shapes is one of key interests to food engineers whose effort it is to predict

the accurate thermal performance of ohmic heaters.

We have seen very few books available on modeling the complexities involved

in different food processing operations at this level. This book is unique in the sense

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interest in relation to food safety and quality. With rapid adaptation, modification,

High pressure processing (HPP), pulsed electric field (PEF), ohmic heating, ozone

heat and physical properties of foods can be efficiently interpreted. As described in

of electric field due to several factors such as heterogeneous food materials and irregu-

computational fluid dynamics (CFD) in which the pressure dependence of latent

try and fluid mechanics useful in predicting how real process equipment is likely

exchangers accounting for the hydrodynamics and thermodynamics of fluid flow,

elaborate on various models available and the influence of different factors on microbial


viii Preface

of applying the theories to solve practical problems relevant to food process engi-

neering at a higher level. This book is not intended to be a complete book on model-

ing the numerous food processing operations. In providing the theoretical basis for

selected operations along with case studies, the reader can gain a clear and intuitive

understanding of the concepts and factors involved in modeling food systems. Using

this opportunity, the chapter contributors also wish to engage readers with further

in-depth discussions about challenging subjects.

We would like to thank all the authors for their sincere contribution of time and

effort in making this possible. It has been our pleasure to put together all of their

efforts in one single stage. Many thanks again.

Soojin Jun, PhDJoseph M. Irudayaraj, PhD

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ix

EditorsSoojin Jun was born in 1970 in Seoul, Korea, and received BS (1996) and MS degrees

(1998) in food science and technology from Seoul National University, Korea and a

PhD degree (2002) in agricultural and biological engineering from The Pennsylvania

State University, University Park, Pennsylvania. Currently, he is an assistant profes-

sor in the Human Nutrition, Food and Animal Sciences Department, University of

Hawaii, Honolulu. He is the author or coauthor of over 30 referred journal articles

and papers and his research interests are in novel food processing technologies, nan-

otech and applications, biosensors, food packaging, and food safety engineering.

Dr. Jun is also a member of the Institute of Food Technologists and American Soci-

ety of Agricultural and Biological Engineers.

Joseph M. Irudayaraj received his PhD from Purdue University in food and bio-

process engineering, MS degrees in biosystems engineering and computer sciences

from University of Hawaii, and BS from Tamil Nadu Agricultural University

(India). Presently, he is an associate professor in the Department of Agricultural

and Biological Engineering and co-director of the Physiological Sensing facility

at Purdue University, West Lafayette, Indiana. He has authored more than 125 ref-

ereed journal publications in the areas of food systems simulation, modeling and

design, sensors for quality assessment, and biosensors. His present research thrust

is in the exploration of diffusion and kinetic studies for disease diagnosis using

single molecule imaging and nanotechnology.

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xi

Contributors

Josse De BaerdemaekerDepartment of Biosystems

Katholieke Universiteit Leuven

Leuven, Belgium

Sundar BalsubramanianDepartment of Biological and

Agricultural Engineering

Louisiana State University

Agricultural Center

Baton Rouge, Louisiana

Patrick J. CullenSchool of Food Science and

Environmental Health

Dublin Institute of Technology

Dublin, Ireland

Ali DemirciDepartment of Agricultural and

Biological Engineering

The Pennsylvania State University

University Park, Pennsylvania

S. GunasekaranDepartment of Biological Systems

Engineering

University of Wisconsin

Madison, Wisconsin

Joseph IrudayarajDepartment of Agricultural &


Purdue University

West Lafayette, Indiana

Soojin JunDepartment of Human Nutrition,

Food and Animal Sciences

University of Hawaii

Honolulu, Hawaii

Harpreet Kaur KhuranaDepartment of Human Nutrition

Food and Animal Science

University of Hawaii

Honolulu, Hawaii

Kathiravan KrishnamurthyDepartment of Food and Animal

Sciences

Alabama A&M University

Normal, Alabama

Si-Quan LiDepartment of Research and

Development

Galloway Company

Neenah, Wisconsin

Kasiviswanathan Muthukumarappan Department of Agricultural and

Biosystems Engineering

South Dakota State University

Brookings, South Dakota

Bart M. NicolaïDepartment of Biosystems


Leuven, Belgium

Colm P. O’DonnellUCD School of Agriculture,

Food Science and Veterinary

Medicine

University College Dublin

Dublin, Ireland

Virendra M. PuriDepartment of Agricultural and


The Pennsylvania State University

University Park, Pennsylvania

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xii Contributors

K.P. SandeepDepartment of Food Science

North Carolina State University

Raleigh, North Carolina

Sudhir SastryDepartment of Food, Agricultural and


The Ohio State University

Columbus, Ohio

Nico ScheerlinckDepartment of Biosystems


Leuven, Belgium

J. Antonio TorresDepartment of Food Science &

Technology

Oregon State University

Corvallis, Oregon

Gonzalo VelazquezDepartment of Food Science &

Technology, UAM Reynosa-Aztlán

Universidad Autónoma de Tamaulipas

Tamaulipas, México

Pieter VerbovenDepartment of Biosystems


Leuven, Belgium

Jian WangDepartment of Biological Systems

Engineering

Washington State University

Pullman, Washington

Yifen WangDepartment of Biosystems

Engineering

Auburn University

Auburn, Alabama

Wade YangDepartment of Food and Animal

Sciences

Alabama A&M University

Normal, Alabama

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1

1 Introduction to Modeling and Numerical Simulation

K.P. Sandeep, Joseph Irudayaraj, and Soojin Jun

CONTENTS

1.1 Introduction .......................................................................................................1

1.3 Numerical Formulation .....................................................................................3

1.5 Boundary and Initial Conditions .......................................................................5

1.6 Errors, Consistency, Stability, Compatibility, and Convergence ......................6

1.7 Solution of the Finite Difference Equations .....................................................6

1.7.1 Direct Methods ......................................................................................6

1.7.2 Iterative Methods ...................................................................................7

1.8 Linearization .....................................................................................................8

1.9 Introduction to the FEM ...................................................................................8

1.9.1 How it Works .........................................................................................8

1.9.2 Discretization .......................................................................................9

1.9.3 Interpolating Functions .........................................................................9

1.9.4 Element Matrix Formation to Obtain Global Matrix ............................9

1.9.5 Boundary Conditions .............................................................................9

1.9.6 Solution of the System of Equations.................................................... 10

1.9.7 Summary of the Steps Involved in a Typical Finite Element .............. 10

1.9.8 Future Applications ............................................................................. 10

1.10 CFD Modeling ................................................................................................ 10

1.11 Commercial Codes and Resources Available ................................................. 11

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

1.1 INTRODUCTION

Mathematical modeling is a very useful tool to (relatively) quickly and inexpensively

ascertain the effect of different system and process parameters on the outcome of

a process. It minimizes the number of experiments that need to be conducted to

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1.2 Classification of Partial Differential Equations ................................................3

1.4 Classification and Generation of Grids .............................................................4

determine the influence of various parameters on the safety and quality of a process.


2 Food Processing Operations Modeling: Design and Analysis

Parametric analyses can be conducted to understand the relative effects of different

parameters.

The use of approximate methods to solve problems described by partial differ-

ential equations has been employed for various reasons including, but not limited to,

the lack of availability of analytical solutions or empirical correlations, simplicity of

solution technique, ability to quickly perform parametric analyses, and also because

it serves as a means for quickly honing in on the range of parameters to be used in

experimental studies or for design purposes.

There are three main categories into which mathematical modeling falls—

method falls under the differential method category. Under the integral method

weighted residuals. The method of weighted residuals can be further divided into

four categories—collocation method, subdomain method, Galerkin’s method, and

categorized into two groups—cell-centered schemes and nodal point schemes. The

ment method in that it uses a similar approach but for the surface or boundary under

Monte Carlo method falls under the stochastic method. This is a computationally

intensive and probabilistic method used primarily when the number of independent

variables is large.

niques used to solve problems associated with food processing. Relatively simple

problems can be tackled with ease by commercially available software. Complicated

scratch.

The FDM has been very popular owing to its simplicity in formulation and ease in

in the sections that follow). However, it should be noted that stability, compatibility,

and convergence tests (described later on in this chapter) should be conducted when

developing new methods to ensure that the technique yields a feasible solution.

In addition, the FVM is now the most commonly used technique in development

of its applications [1]. This involves the disretization of the equations over the entire

strategy would give the best results and require the least computing time. However,

in a criterion suggested for the solution of heat and mass transfer problems for food

materials, it was recommended that if the solution region represents a simple rectan-

element methods could be adopted. To capture the behavior of the physics and the

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differential method, integral method, and stochastic method. The finite difference

we have the variational method, finite volume method (FVM), and method of

least squares method while the finite volume (or control volume) method can be

variational method and the method of weighted residuals form the basis of the finite

element method (FEM). The boundary element method is a sub-set of the finite ele-

consideration. It can be used in conjunction with the finite element or FVM. The

The FEM and the finite difference methods (FDM) are the most popular tech-

problems require either modification of commercial codes or writing the code from

modification (especially while introducing different relaxation factors as can be seen

finite volume.

For modeling of food processing, it is often difficult to decide which solution

gular domain then the traditional finite difference methods should be the preferred

discretization strategy [2]. When the boundary conditions are irregular then the finite

conservation laws more rigorously, the finite volume techniques are preferred [3].

of computational fluid dynamics (CFD) codes and has been extensively used in many

solution domain and rigorous conservation of mass and heat flux on each face of the


Introduction to Modeling and Numerical Simulation 3

1.2 CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS

on whether or not there is a product of two terms containing either the dependent

variable or its derivatives. If a PDE is linear in its highest order derivative, but in one

or more of the lower order derivatives, it is called a quasi-linear PDE. The order of a

PDE is the highest power of the derivative in the equation.

Consider the following second order PDE:

A B C D E2 2 2

2

∂ Φ∂

∂ Φ∂ ∂

∂ Φ∂

∂Φ∂2x x y y x

+ + + +∂∂Φ∂

Φy

0+ + =F G

ents A, B, C, D, E, F, and G can be functions of x, y, or Φ.

The above PDE is said to be elliptic if B2 − 4AC < 0, parabolic if B2 − 4AC = 0,

and hyperbolic if B2 − 4AC > 0 at all points in the domain.

Auxiliary variables are usually introduced to convert the second order PDEs to

then be used for solving the system of equations too.

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

equation are either constant or if variable, their derivatives do not appear anywhere in

the equation. The schemes that maintain the discretized version of the conservation

statement exactly (except for round-off errors) for any grid size over any region in the

domain for any number of grid points is said to have the conservative property.

The non-conservative form of the continuity equation is as follows:

ρ ∂∂

ρ ∂∂

∂ρ∂

∂ρ∂

ux

vy

ux

vy

0+ + + =

The conservative form of the same equation is as follows:

∂∂

ρ ∂∂

ρ Δ ρx

uy

v( )+ ( ) 0 (= ⋅ =or V) 0�

Equilibrium problems (or jury problems) are problems for which the solution of the

PDE is required in a closed domain for a given set of boundary conditions. Equilibrium

problems are boundary value problems and are governed by elliptic PDEs.

Marching (or propagation) problems are transient or appear to be transient problems

and the solution of the PDE is required in an open domain for a given set of initial and

boundary conditions. Problems in this category are either initial value or initial boundary

value problems. Marching problems are governed by hyperbolic or parabolic PDEs.

1.3 NUMERICAL FORMULATION

When dealing with the unsteady state heat equation or the scalar (linear or non-linear)

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Partial differential equations (PDEs) are classified as linear or non-linear depending

The coeffici

the first order PDEs at least for the purpose of classification. This formulation may

law form or divergence form) if the coefficients of all the derivative terms in the

Numerical formulations are based on the classification of the governing equation.



Burger’s equation, formulations applicable to parabolic equations are used. When deal-

ing with the wave equation, formulations for hyperbolic equations are used, and when

dealing with Laplace’s equation, formulations for elliptic equations are used. Formu-

lations for all types of equations can be explicit or implicit. Explicit formulations are

simple, but the number of computations and the stability of the formulation (addressed

in the next section) are some of its drawbacks.

The Navier–Stokes equations are hyperbolic in the inviscid region and parabolic

in the viscous region. For steady state conditions, they are hyperbolic in the inviscid

region and elliptic in the viscous region. The scalar equations which are similar to

the Navier–Stokes equations are the Burgers equations (linear and non-linear). Thus,

the starting point for solving the Navier–Stokes equations involves understanding the

methods employed to solve the Burgers equations.

Some of the commonly used explicit formulations for parabolic equations are the

forward time central space (FTCS) method, Richardson’s method, and the DuFort–

Frankel method; while some of the commonly used implicit methods are the Laasonen

nine-point methods are the commonly used methods to address elliptic problems.

ing, LAX method, midpoint leapfrog method, Lax–Wendroff method, Rusanov or

Burstein–Mirin method, and Warming–Kutler–Lomax (WKL) method are some of

the commonly used explicit methods for hyperbolic equations. Euler’s backward time

central space (BTCS) and the Crank–Nicolson methods are two of the commonly

used implicit methods for hyperbolic equations.

Multi-step (or splitting) methods are usually used for non-linear problems and

prediction) of the variable at an intermediate time step and the second step involves

correcting it and hence multi-step methods are also called predictor-corrector meth-

ods too. The Richtmyer formulation, Lax–Wendroff multi-step method, MacCormack

method, and the Warming and Beam (upwind) method are some of the commonly

used multi-step methods with hyperbolic equations.

1.4 CLASSIFICATION AND GENERATION OF GRIDS

In order to solve the partial differential equations that represent the physical problem,

the domain of interest has to be divided into grid lines and the points of intersection

of these gridlines are called nodes. The accuracy of the solution depends on many

solution process can proceed in an ordered sequential manner from one node to the

next. The advantages of using the complicated unstructured grid system are that they

2-D geometries, the most common method of unstructured grid generation involves

boundaries). Advancing front method and the Delaunay method are two of the com-

monly used techniques for triangulation of the domain.

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method, Crank–Nicolson method, and the Beta formulation. The five-point and

Euler’s forward time forward space (FTFS), Euler’s FTCS, first upwind differenc-

sometimes with linear problems too. In this method, the finite difference equations

are written out at two or more time steps. The first step involves determination (or

factors including grid spacing. Grids are classified as structured or unstructured

depending on whether or not there is a set pattern of identification of nodes and if the

can be used to fit irregular, singly-connected and multiply connected domains. For

discretizing the domain into triangles (the most flexible shape to fit various kinds of



The grid system used could be orthogonal—Cartesian, cylindrical, spherical

angular. Due to the complex geometries of the domain of interest and the possibility

or necessity of having more grids close to boundaries, the physical domain is trans-

formed into a computational domain (by twisting or stretching), where the grids are

rectangular.

Grid generation can be divided into three main categories—algebraic (simple

and fast, using one of many algebraic equations or interpolation techniques), partial

differential equation (elliptic, hyperbolic or parabolic), and conformal mapping using

and is generated before solving the problem) or adaptive (grids move toward regions

of steep gradients as the solution process proceeds).

Some of the desirable features of a grid system are: (1) A mapping that ensures

one-to-one correspondence with grid lines of the same family not intersecting; (2)

grid point distribution is smooth; (3) grid lines are orthogonal or close to orthogonal;

and (4) option for grid point clustering exists.

Grid point clustering (or grid embedment) is a technique used to increase the

formed by appropriate choice of functions used in the transformation of coordinates.

Two of the common ways of handling grid embedment is by the meshing of the grid

and the separate regions method (in which there are two types of grids—interface

grid is performed to obtain values of the variable).

One of the easiest ways of obtaining staggered grids is by shifting the grid verti-

cally or horizontally by half a grid space. This technique is used to improve the stabil-

ity criterion by coupling of variables when the governing system of equations can be

solved sequentially. Thus, there is a primary and secondary set of grids with different

of the incompressible Navier–Stokes equations and consider a grid point in the system

and DuFort–Frankel methods are two of the commonly used methods with staggered

grids. Another technique used for coupling of equations is the multilevel (multigrid)

method and has been used for the diffusion, Poisson, and Navier–Stokes equations.

1.5 BOUNDARY AND INITIAL CONDITIONS

A boundary condition (BC) is said to be of the Dirichlet kind if the value of the

dependent variable is given along the boundary. If the derivative of the dependent

variable is given along the boundary, it is said to be a Neumann BC. If the BC at the

boundary is given as a linear combination of Dirichlet and Neumann BCs, it is said

to be a Robin BC. If the BC along a part of the boundary is of the Dirichlet type, and

another part is of the Neumann type, the overall BC is said to be a Mixed BC.

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(depending on the boundary configuration of system) or non-orthogonal such as tri-

complex variables. Grid systems are also classified as fixed (independent of solution

number of grid points around a specific grid point or around a grid line. It is per-

of each point near the interface of the coarse and fine grid on the solution variable)

and non-interface; at the fine grid boundary, interpolation of the values at the coarse

variables being specified on the primary and secondary grids. Consider the example

where the pressure is specified. Immediately to the right and left of this point, the

x-component of the velocities is specified and immediately to the top and bottom of

this point, the y-component of the velocities is specified. The Marker and cell method

method (where weighting factors are introduced to determine the relative influence



1.6 ERRORS, CONSISTENCY, STABILITY, COMPATIBILITY, AND CONVERGENCE

errors (tend to decrease the amplitude of the wave) and that of second order accurate

methods are known as dispersion errors (tend to cause oscillation of the solution).

formulation.

Discretization error. It is the error in the solution of a PDE due to transformation

of the continuous problem to a discrete problem, and it is the difference between the

difference formulation (without round-off error). It is thus the error in the solution

due to truncation and any errors due to the BCs.

Round-off error. It is the error associated with rounding off numbers in math-

ematical operations.

approximates the PDE. A formulation is said to be consistent if the truncation error

tends to zero as the mesh size tends to zero. Methods which are of the order Δt or Δx

are consistent as error tends to zero as the mesh size tends to zero. However, schemes

that are of the order Δt/Δx may potentially be inconsistent unless it is ensured that

Δt/Δx tends to zero.

Stability. A scheme is said to be stable if errors (round-off, truncation etc.) do not

grow as the scheme proceeds (or marches) from one step to another and is hence strictly

types of errors exist—discretization or round-off (computational). It is important to

control the growth of these errors so that the solution is stable. Two standard methods

exist for stability analysis—discrete perturbation stability analysis, and von Neumann

(Fourier) stability analysis. The latter method is simpler and more commonly used.

Convergence. Usually, a consistent and stable scheme is convergent. Convergence

for convergence”. Although this theorem has not been proven for non-linear PDEs,

it is also used for them.

1.7 SOLUTION OF THE FINITE DIFFERENCE EQUATIONS

met, the set of equations have to be solved. Several direct and iterative methods exist

for solving them, and they are discussed in the following sections.

1.7.1 DIRECT METHODS

Cramer’s rule. Simple, but extremely time consuming. Number of operations = (N + 1)!, for N unknowns.

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The errors associated with first-order accurate methods are known as dissipation

Truncation error. It is the error introduced by truncating terms in the finite dif-

ference formulation. It is the difference between the PDE and the finite difference

exact solution of the PDE (without round-off error) and the exact solution of the finite

Consistency. It relates to the extent to which the finite difference formulation

applicable to marching problems only. In the solution of finite difference equations two

relates to the solution of the finite difference formulation approaching the solution to

the PDE as the mesh size is refined. According to Lax’s equivalence theorem, “Given

a properly posed initial value problem and a finite difference approximation to it that

satisfies the consistency condition, stability is the necessary and sufficient condition

Once the finite difference equations have been formulated and the stability criteria



especially tridiagonal system of equations. Approximately N3 multiplications are

required for solving N equations. To improve accuracy, equations are rearranged such

Some of the other direct methods include the LU decomposition method, error

vector propagation (EVP) for the Poisson equation [4], odd–even reduction method

[5], and the fast Fourier transform method [6,7].

Direct methods require an exorbitant number of arithmetic operations and they

are usually restricted by one or more of the following: Type of coordinate system

type of BCs.

1.7.2 ITERATIVE METHODS

Usually an initial solution is guessed, new values computed, and the process contin-

ued until convergence is obtained. If a formulation results in only one unknown, it is

called a point iterative method and if the formulation involves more than unknown

a line iterative method. Some of the commonly employed iterative techniques are

listed below.

Alternating direction implicit (ADI) method for parabolic equations. The ADI

method is a sub-set of the approximate factorization method (replacement of original

two- or three-dimensional cases.

Fractional step method for parabolic equations. This technique involves split-

ting of a multidimensional problem into a series of 1-D problems and solving them

sequentially.

Alternating direction explicit (ADE) methods for parabolic equations. They do

not require tridiagonal matrices to be inverted and can be used for 1-D equations

also.

Jacobi method. Initial values of the variable are either prescribed or guessed (at

iteration step are used) to solve for the variable at the grid point (i, j) at the new itera-

tion step.

Point Gauss–Siedel method. This is an improvement of the Jacobi method. In

this method, the values of the variable computed at the new iteration step are imme-

diately used in the computation of the variable at all grid points at the new time step

(as soon as they become available). It has a much higher convergence rate than the

Jacobi method.

Line Gauss–Siedel method. This method is applied when there are three

as the point Gauss–Siedel method results in a system of linear equations with a tridi-

Gauss–Siedel method.

Successive over-relaxation (SOR). This is a technique used to accelerate any

iterative procedure based on guessing the trend of a solution and modifying the

solution appropriately. A parameter, ω (0 < ω < 2), is used to multiply a set of terms

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Gaussian elimination. It is an efficient means of solving algebraic equations,

that the largest coefficients occupy the diagonal (this process is called pivoting).

(e.g. Cartesian); type of domain (e.g. rectangular); size of coefficient matrix, and

(usually three unknowns that result in a tridiagonal coefficient matrix), it is called

finite difference formulation by tridiagonal formulation). This method applies to

the first iteration step) and the value of the variable at all grid points (at the previous

unknowns. The finite difference equation, when processed under the same guidelines

agonal coefficient matrix. This method has a faster convergence rate than the point



in the equation used for a method such as the Gauss–Siedel method. If 0 < ω < 1, it

is called under relaxation, and if 1 <ω < 2, it is called over-relaxation. Over relax-

ation is similar to linear extrapolation (and is used usually for Laplace’s equation

with Dirichlet BCs), while under-relaxation is used when the solution is oscillating

(usually used for non-linear elliptic equations). Determining the optimum relax-

ation factor (ωopt) for various types of equations and BCs can greatly accelerate the

convergence.

1.8 LINEARIZATION

Consider a non-linear term such as u(∂u/∂x). All the values at time j are known for a

given location i and the values at i + 1 are to be determined. Three of the commonly

used linearization techniques are listed below:

u(∂u/∂x) becomes:

uu u

i ji j i j

,, ,+ −1

Δx

There is only one unknown (ui + 1,j

lation is linear.

Iterative. This method involves updating the lagged value till convergence is

reached. The formulation for this method is:

uu u

i jk i j

k ki j

,

, ,++ −1

1

Δx

i + 1,j is the value at the previous location, ui,j. Once ui + 1,j is ki + 1,j is updated and a new solution is obtained

and this process is continued until the convergence criterion has been met.

Newton’s iterative linearization. This method uses the technique of evaluating

the change in a variable between two iterations and dropping second order terms to

arrive at the following expression for the non-linear term:

2 ( )1, 1,1 2

1,1

1, ,u u u u ui j

ki jk k

i jk

i j i j+ ++

++− −

+

k

ΔΔx

1.9 INTRODUCTION TO THE FEM

1.9.1 HOW IT WORKS

The elements are connected to each other at points called nodes. The nodes typically

lie on the element boundary where adjacent elements are connected. In addition to

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Lagging. The coefficient is used at the known value, i. Thus the formulation for

) in this expression and the finite difference formu-

For the first iteration, udetermined at k + 1, the coefficient u

The finite element discretization procedure reduces the given region into a finite

number of elements. A collection of the elements is called the finite element mesh.



boundary nodes, an element may also consist of a few interior nodes. The nodal

able within the elements. The nature of the solution and the degree of approximation

depend not only on the size of the elements but also on the interpolating functions

which should satisfy compatibility and continuity conditions.

tional or weighted residual method. The variational approach has its foundations in

variational calculus and requires the use of a functional while the weighted residual

following steps.

1.9.2 DISCRETIZATION

This involves dividing the problem domain into subdomains. Generally for a one-

dimensional problem this is very simple. However the degree of complexity increases

with the number of dimensions and the non-uniformity of the object in question. Dis-

cretization or division of the domain into smaller components can be accomplished

by choosing a variety of different element shapes and nodes. The choice of the type

of element and the number of nodes in an element are left to the discretion of the

engineer/scientist and are based on experience.

1.9.3 INTERPOLATING FUNCTIONS

variable over the element. Interpolating functions generally are polynomials that can

be easily integrated and differentiated subject to certain continuity requirements

imposed at the element boundaries.

1.9.4 ELEMENT MATRIX FORMATION TO OBTAIN GLOBAL MATRIX

Depending upon the choice of the procedure (variational or weighted residual method)

element matrices are calculated by transforming the elements from the global to

a local coordinate system where integration and differentiation are performed and

then back transformed into the global matrix. Depending upon the element connec-

tivity or the nodes in the element the elements matrix is incorporated into the global

matrix. Similar calculations are performed for each element and the global matrix is

assembled using the element matrix.

1.9.5 BOUNDARY CONDITIONS

Before solving for the unknown variables. boundary conditions are imposed to the glo-

bal matrix. The two types of boundary conditions are natural and essential boundary

conditions. Natural boundary conditions are convective boundary conditions while

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points depict the field variable or the unknown, defined in terms of approximating

or interpolating functions within each element. The nodal values of the field variable

and the interpolating functions for the elements define the behavior of the field vari-

Solution using the finite element technique is obtained predominantly by varia-

approach used the governing equations. The finite element procedure consists of the

Once the elements are defined the next step is to assign nodes to each element to

choose the appropriate interpolating function to represent the variation of the field

essential boundary conditions are constant or specified boundary conditions [7].



1.9.6 SOLUTION OF THE SYSTEM OF EQUATIONS

The assembled equations consist of a set of simultaneous equations that can be

solved using the matrix solvers. For time-dependant problems the unknown nodal

scheme is generally chosen.

1.9.7 SUMMARY OF THE STEPS INVOLVED IN A TYPICAL FINITE ELEMENT

(b) Derive element matrices for the system.

(c) Evaluate element equations and assemble element matrix to form the global

matrix.

(d) Impose boundary conditions.

(e) Solve the system of equations using an appropriate solver.

(f) Postprocessing—graphics, calculation of gradients etc.

1.9.8 FUTURE APPLICATIONS

Most of the future growth expected will be in the application and validation of

for solving problems with nonlinear and random material properties and boundary

conditions will increase. Interest in the application of the Finite element method

in biological systems and more direct integration of the technique with the actual

design will also be given priority. Another crucial area that will demand attention is

in solving micro-structural problems in engineering and biological sciences. Other

parallel processing.

1.10 CFD MODELING

solutions requires a large amount of insight into the problem that has to be solved,

and the appropriate implementation of both physical models and numerical schemes,

mainly using FDM, FEM, and FVM. However, it is well known that the FVM approach

can form the governing equations to better account for changes in mass, momentum,

domain. Though the overall solution will be conservative in nature, the FVM method

can be sensitive to skewed elements which can prevent convergence if such elements

turbulence model, multiphase, and meshing features such as unstructured or sliding

meshes have been addressed and successfully resolved by using the commercialized

CFD codes.

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values are a function of time and hence an appropriate finite difference time-s tepping

(a) Discretize the problem domain and construct the finite element mesh.

the finite element results by experimental data. Further refinement of the existing

finite element procedures will also increase [8]. Appropriate solution procedures

areas that will demand attention are adaptive finite elements and the application of

either at the user interface or through user-defined codes within the software [9]. The

The CFD codes provide understanding of the physics of a flow system through non-

intrusive flow, thermal, and concentration field predictions. Obtaining accurate CFD

CFD codes required to discretize modeled fluid continuum are numerically obtained

and energy because fluid crosses the boundaries of discrete spatial volumes within the

are in critical flow regions. The outstanding issues associated with convection scheme,



1.11 COMMERCIAL CODES AND RESOURCES AVAILABLE

Commercial codes available these days are often also packaged with pre- and post-

processing modules. Pre-processing involves transformation of the physical problem

into computational domain and generating a grid mesh in the computational domain.

Post-processing involves presenting the data obtained by the code in graphical form.

There are many commercial codes available to solve most standard problems

involving standard governing equations, boundary conditions, and relatively simple

geometries. They are available on different platforms—PC, Unix, SGI etc. There are

many universities and research groups that offer codes, services or perhaps would be

interested in collaborative efforts. Some of the resources for pre-processing, process-

ing, and post-processing are listed below.

Adina R&D (http://www.adina.com)

Phoenics (http://www.cham.co.uk)

NEKTON, IcePak and MixSim software

Innovative Research, Inc. (http://www.inres.com)

CFD Research Corporation (http://www.cfdrc.com)

Amtec (http://www.amtec.com)

CFX ANSYS (http://www.ansys.com/)

Femlab Comsol (http://www.femlab.com)

I-deas NX Siemens Product Lifecycle Management (PLM) Software

(http://www.plm.automation.siemens.com)

REFERENCES

1. Talukdar, P., Steven, M., Issendorff, F.V., and Trimis, D. 2005. Finite volume method

in 3-D curvilinear coordinates with multiblocking procedure for radiative transport

problems. International Journal of Heat and Mass Transfer 48: 4657–66.

of techniques used for modeling and numerically simulating the drying process. In

Mathematical Modeling and Numerical Techniques in Drying Technology 1–82. NY:

Marcel Dekker.

3. Ranjan, R., Irudayaraj, J., and Jun, S. 2001. A three-dimensional control volume

approach to modeling heat and mass transfer in foods materials. Transactions of the ASAE 44(6): 1975–82

5. Buneman, O. 1969. A compact non-iterative Poisson solver. Institute for Plasma

Research SUIPR Report 294. CA: Stanford University.

6. Hockney, R.W. 1965. A fast direct solution of Poisson’s equation using Fourier analysis.

Journal of the Association for Computing Machinery 12: 95–113.

7. Hockney, R.W. 1970. The potential calculation and some applications. Methods in Computational Physics 9: 135–211.

9. Norton, T., and Sun, D. 2007. An overview of CFD applications in the food industry. In

Computational Fluid and Dynamics in Food Processing 1–41. NY: CRC Press.

55534_C001.indd 1155534_C001.indd 11 10/22/08 8:19:41 AM10/22/08 8:19:41 AM

2. Turner, I.W., and Perre, P. 1996. A synopsis of the strategies and efficient resolution

8. Reddy, J.N. 1993. An introduction to the finite element method. NY: McGraw-Hill.

Fluent (http://www.fluent.com), Gambit, FLUENT, FIDAP, POLYFLOW,

4. Roache, P.J. 1972. Computational fluid dynamics. NM: Hermosa.


http://www.adina.com

http://www.cham.co.uk

http://www.fluent.com

http://www.inres.com

http://www.cfdrc.com

http://www.amtec.com

http://www.ansys.com

http://www.femlab.com

http://www.plm.automation.siemens.com

13

2 Aseptic Processing of Liquid and Particulate Foods

K.P. Sandeep and Virendra M. Puri

CONTENTS

2.1 Introduction ..................................................................................................... 14

2.2 Type of Processing .......................................................................................... 16

2.2.1 Critical Factors and Problems Associated with Processing ................ 16

2.2.2 Relevant Historical Background .......................................................... 17

2.3 Fluid Mechanics Aspects of Processing ......................................................... 17

2.3.1 Types of Fluids .................................................................................... 17

2.3.2 Dimensionless Numbers Governing Flow ........................................... 18

2.3.3 Friction Factor .....................................................................................20

2.3.4 Pumps and Pumping Requirements .................................................... 21

2.3.5 Residence Time Distribution of Fluid Elements and Particles ............22

2.3.6 Forces Acting on Fluid Elements and Particles During Flow .............24

2.3.6.1 Equations of Motion of the Fluid ..........................................24

2.3.6.2 Linear Dynamic Equations for Particles ...............................25

2.3.6.2.1 Magnus Lift Force ...............................................25

2.3.6.2.2 Saffman Lift Force ..............................................25

2.3.6.2.3 Drag Force ...........................................................27

2.3.6.2.4 Buoyancy Force (acting in the

y-direction only) ..................................................27

2.3.6.3 Angular Dynamic Equations for Particles ............................28

2.3.7 Techniques to Determine Fluid and Particle Velocity ........................29

2.4 Heat Transfer Aspects of Processing ..............................................................29

2.4.1

2.4.2 Steam Quality ......................................................................................30

2.4.3 Dimensionless Numbers Governing Heat Transfer .............................30

2.4.4

2.4.5 Transient Heat Transfer within Particles ............................................. 32

2.4.6 Hydrodynamic and Thermal Entrance Lengths .................................. 33

2.4.7

2.4.8

2.4.9 Heating Media and Equipment ............................................................ 37

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Convective Heat Transfer Coefficient ..................................................29

Heat Transfer Coefficient in Straight Tubes ........................................ 33

Natural (free) and Forced Convection ................................................. 32

Heat Transfer Coefficient in Helical Tubes .........................................36



2.4.10 Co- and Counter-current Heat Exchangers ......................................... 37

2.4.11 Governing Heat Transfer Equations and Energy Balance................... 38

2.4.11.1 Energy Balance in the Heat Exchanger ................................ 39

2.4.11.2 Energy Balance in the Holding Tube .................................... 39

2.4.11.3 Energy Balance in the Cooling Section ................................40

2.4.12 Fouling and Enhancement of Heat Transfer ........................................40

2.4.13 Techniques to Estimate the Temperature History of a Product ..........40

2.5 Microbiological and Quality Considerations .................................................. 41

2.5.1 Federal Regulations and HACCP ........................................................ 41

2.5.2 Kinetics of Microbial Destruction, Enzyme Inactivation,

and Nutrient Retention ........................................................................ 41

2.5.2.1 Process Lethality and Cook Values ...................................... 43

2.5.2.2 Commercial Sterility of the Product .....................................44

2.6 From an Idea to Commercialization ...............................................................44

2.7 Concluding Remarks ....................................................................................... 47

Nomenclature ........................................................................................................... 47

References ................................................................................................................50

2.1 INTRODUCTION

Aseptic processing involves sterilization of a food product (in a direct or indirect

high temperature for a short period of time (in comparison with conventional can-

ning) in aseptic processing yields a high quality product. The demand for high qual-

ity shelf-stable products has been the driving force for commercialization of aseptic

processing. Deaeration (prior to sterilization) is usually an integral part of aseptic

processing as removal of air enhances product quality and increases the shelf-life

of a product. It also stabilizes the product prior to processing. Care should be taken

not based on the initial raw product. Another important part of an aseptic processing

ing of the product at processing temperatures which can be as high as 125–130°C. An

aseptic surge tank provides the means for product to be continuously processed even

if the packaging system is not operational due to any malfunction. It can also be used

to package the sterilized product while the processing section is being resterilized.

prior to processing is of utmost importance. This is what is referred to as presteriliza-

tion. The recommended heating effect for presterilization (using hot water) of the pro-

cessing equipment for low-acid foods is the equivalent of 121.1°C for 30 minutes. The

used for sterilization. Presterilization of an aseptic surge tank is usually done by satu-

rated steam and not hot water due to the large volume associated with the surge tank.

eliminating the need for refrigeration, easy adaptability to automation, use of any

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contact heat exchanger), followed by holding it for a specified period of time (in a

to ensure that all process calculations are performed after the deaeration stage and

holding tube), cooling it, and finally packaging it in a sterile container. The use of

system is the back pressure valve which provides sufficient pressure to prevent boil-

corresponding combination for acid or acidified products is 104.4°C for 30 minutes.

This often involves acidification of the water (to below a pH of 3.5 for acid products)

Better product quality (nutrients, flavor, color, texture), less energy consumption,

Sterilization of the processing system, packaging system, and the air flow system


Aseptic Processing of Liquid and Particulate Foods 15

advantages of aseptic processing over the conventional canning process. Some of

the reasons for the relatively low number of aseptically processed products include

quality control of raw products, better trained personnel, and better control of process

variables and equipments. Some of the disadvantages of aseptic processing include

increased shear rates, degradation of some vitamins (some vitamins are stable at pas-

teurization temperatures but not at sterilization temperatures), separation of solids

Thus it can be seen that not all products can be aseptically processed to yield a high

quality product.

Due to some of the stringent regulatory requirements of aseptic processing, many

processors adopt an aseptic process, but package it in non-aseptic containers. This

results in products that are called ‘extend shelf-life products’. Such processes are

easier to adopt, require less monitoring (since the resulting product–package com-

One such process involves ultra-pasteurization of milk wherein extended shelf-life

can be obtained.

Notwithstanding the problems associated in producing aseptically processed

foods, several companies have adopted this technology. Some of the products that are

aseptically processed include fruit juices, milk, condensed milk, coffee creamers, pud-

dings, soups, butter, gravies, and jelly. Some of the companies that deal with aseptic

processing and packaging equipment are International Paper, Tetra Pak, Combibloc,

Elopak, Cherry Burrell, Alfa Laval, ASTEC, VRC, APV, FranRica, Benco, Scholle,

Bosch, and Metal Box.

The pH of a food product is a critical factor in determining the type of process-

ing to be adopted and the class of viable microorganisms of concern. Foods are

usually divided into three pH groups while designing a thermal process: high-acid

foods which have pH values less than 3.7, foods with pH values between 3.7 and

4.6 and the low-acid foods with pH values greater than 4.6. For low-acid foods, the

anaerobic conditions that prevail in aseptic processing are ideal for growth of some

toxin-producing microorganisms such as Clostridium botulinum. To obtain a com-

mercially sterile product, all pathogenic microorganisms must be destroyed during

aseptic processing.

Bacteria are the primary organisms of concern in food processing. They multiply

teria is generally divided into seven stages—lag phase (no growth or even a decrease

in numbers), accelerated growth phase (rate of growth is increasing), logarithmic

phase (most rapid and constant increase in numbers), deceleration phase (rate of

growth is decreasing), stationary phase (numbers remain constant), accelerated

a constant rate). In order to extend the shelf-life of products, one of the techniques is

bacteria. Once bacteria reach the third stage (logarithmic phase), spoilage will occur

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slower filler speeds and higher overall cost. Aseptic processing also requires better

bination does not need to be sterile), and are easier to file with regulatory agencies.

by the process called fission wherein one cell splits into two cells. The growth of bac-

death phase (rate of death is increasing), and final death phase (numbers decrease at

to prolong the first two phases (lag and accelerated growth phase) of the growth of

and fats, precipitation of salts, and change in flavor or texture of the product relative

to what consumers are accustomed to. Minimization of the off-flavors produced can

be accomplished by steam injection (short heating time) followed by flash-cooling.

size package, use of flexible packages, and cheaper packaging costs are some of the



freezing, drying, reduction in available oxygen, and reduction in initial number of

bacteria. These techniques must be accompanied by other practices such as the use

of appropriate packaging and storage conditions.

2.2 TYPE OF PROCESSING

Techniques to process and preserve foods range from retorting (canning) to frozen

refrigeration, and drying of foods. Not all products can be processed or preserved using

the same technique. Feasibility of processing and the quality of the end-product deter-

mine the type of processing and preservation technique employed for various foods.

The quality of canned foods is not very high since products are subjected to heat

treatment for an extended period of time. On the other hand, the short processing

times involved during aseptic processing leads to the production of a high quality

product. Recovery of heat from the heat exchangers used in aseptic processing also

do not require further control like refrigeration of frozen foods. Refrigerated foods

(after pasteurization) require careful monitoring of the storage and distribution tem-

perature. They also have a shorter shelf-life than aseptically processed products and

hence their range of distribution is limited. The quality of frozen foods is generally

high, but they need to be thawed and then cooked. The thawing process can result in

uneven heating zones especially if a microwave oven is used. In addition, depending

on the storage period, the energy requirements for freezing can be a major portion

of the total cost involved.

2.2.1 CRITICAL FACTORS AND PROBLEMS ASSOCIATED WITH PROCESSING

Some of the factors that affect the choice of the type of process include the viscos-

ity of the product and presence of large particles and/or low thermal conductivity

particles. The simplest type of food product is a homogeneous low viscosity liquid

product. Direct heating by steam injection or steam infusion is a commonly employed

method for heating such products. For higher viscosity products, plate and tubular

heat exchangers are employed. For extremely viscous products, a scraped surface

heat exchanger is usually used. When relatively high viscosity products containing

large particles and/or low thermal conductivity particulates are involved, dielectric

(microwave) and ohmic heating are two commonly employed methods. The density

of the particles is also an important issue to be considered and will be addressed in

layer surrounding the particle, which in turn is a function of the thermo-physical

presents a problem of some particles being subjected to less thermal treatment than

others. If the heating time is based on mean velocity, the faster moving particles

will be under-sterilized while the slower moving particles will be over-sterilized.

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rapidly. Some of the techniques to prolong the first two phases are refrigeration,

storage. Some of the other techniques of processing and preservation include hot-fill,

makes it more energy efficient. In addition, the products are shelf-stable and hence

and rheological properties of the fluid and the relative velocity between the particle

and the fluid. This boundary layer governs the convective heat transfer coefficient at

the section that deals with residence time distribution.

the particle-fluid interface. In addition, the existence of a residence time distribution

Heat transfer from the carrier fluid to the particle is a function of the boundary



essential in determining the thermal treatment that any product has received.

2.2.2 RELEVANT HISTORICAL BACKGROUND

The work of Olin Ball and the American Can Research Department laid the founda-

process was developed [1]. This was followed by the Avoset process in 1942 (steam

injection of the product coupled with retort or hot air sterilization of packages such as

in a tubular heat exchanger, metal container sterilization using superheated steam

at temperatures as high as 450°F since dry heat requires higher temperature than

package—tetrahedron package. The late 1960s saw the advent of the Tetra Brick

aseptic processing machine and the late 1970s saw the advent of the Combibloc

were established. One of the major landmarks in the history of aseptic processing is

the approval of use of hydrogen peroxide for the sterilization of packaging surfaces

by the FDA in 1981. In recent years, a major break-through for the aseptic processing

industry was in 1997 when Tetra Pak received a no-objection letter from the FDA for

aseptic processing of low-acid foods containing large particulates.

2.3 FLUID MECHANICS ASPECTS OF PROCESSING

mechanics aspects that are important in designing an aseptic process. These param-

and more importantly that of the particles are the factors that eventually are used in

designing holding tubes.

2.3.1 TYPES OF FLUIDS

dependent or time-independent depending on whether the shear stress experienced

law of viscosity—shear stress (σ) and shear rate (γ.) are linearly related, while non-

The Herschel–Bulkley model, given below, is the most commonly used model to

σ σ γ= +0 K n(.

)

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tion of aseptic processing in the US as early as 1927 when the HCF (heat, cool, fill)

wet heat, followed by aseptic filling and sealing of cooled product in a superheated

cans and bottles) and the Dole-Martin aseptic process in 1948 (product sterilization

steam environment). The early 1960s was marked with the advent of a form-fill-seal

eters and the system configuration, in turn are important factors that determine the

Knowledge regarding the spread of residence times for the fluid and particles is

choice of the pump to be used. The residence time distribution of the fluid elements,

that affect the viscosity of a fluid or suspension. Fluids are characterized as time-

by a fluid under a constant shear rate varies as a function of time. If the shear stress

Time, shear rate, temperature, and particle concentration are some of the factors

increases with time, it is called a rheopectic fluid, and if the shear stress decreases

with time, it is called a thixotropic fluid. Time-independent fluids are divided into

Newtonian fluids do not have a linear relationship of shear stress versus shear rate.

describe the flow behavior of most liquid food products:

(blank carton) aseptic system. Soon, aseptic filling in drums and bag-in-box fillers

two categories, Newtonian and non-Newtonian. Newtonian fluids obey Newton’s


The type of fluid, flow characteristics, and fluid properties are some of the fluid


0

0

0 0

the ratio of shear stress to shear rate is not a constant. Apparent viscosity is the

ratio of the shear stress to shear rate and is always expressed along with the shear

e

into picture. One of the equations used to determine the effective viscosity of a

suspension is:

μ μ Φ Φe 1 2.5 14.1= + +( )2

Temperature is a major factor that affects the viscosity of Newtonian and non-

equation is the most commonly used equation to determine the effect of temperature

μ = Be E R Ta g− /

Thus, to determine the Arrhenius parameters B and Ea, a graph of ln(μ) versus 1/T is

a g

2.3.2 DIMENSIONLESS NUMBERS GOVERNING FLOW

Nu d

n n

n n

n nGRe

2

33 1=

+

−

−

ρ⟨ ⟩K[( )/ ] 2

NRe =ρ

μud

if the Reynolds number is greater than 10,000. In the intermediate Reynolds number

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In the above equation, σ is the yield stress, K is the consistency coefficient, and n is the

flow behavior index. For a Newtonian fluid, σ = 0, K = μ, n = 1. A pseudoplastic fluid

v iscosity increases with an increase in shear rate. When small particles of low con-

For a non-Newtonian fluid, the concept of apparent viscosity is introduced since

rate since it varies with shear rate. For a pseudoplastic fluid, the apparent viscosity

is one for which σ = 0, n < 1 while a dilatant fluid is one for which σ = 0, n > 1.

decreases with an increase in shear rate, while for a dilatant fluid, the apparent

centration (Φ) are suspended in a f luid, the concept of effective viscosity (μ ) comes

Newtonian fluids. For a Newtonian fluid, the Arrhenius model, given by the following

on the viscosity of a fluid:

made, the slope of which is –E /R and the intercept is ln(B). Thus, the flow behavior

of the fluid as a function of temperature can be modeled.

steady streamline f low and the flow is referred to as laminar flow. At higher flow

rates, the flow becomes erratic and is referred to as turbulent flow. Reynolds number

is the non-dimensional number that is used to characterize the type of flow and the

When a fluid flows through a tube at low velocities, the flow is characterized by

generalized Reynolds number (valid for power-law fluids, in addition to Newtonian

fluids) is defined as:

The above expression reduces to the following form for a Newtonian fluid:

For flow in a straight tube of circular cross-section, laminar flow conditions are said

to exist if the Reynolds number is less than 2,100 and the flow is said to be turbulent



secondary (radial) direction. This is due to the radial pressure gradient that develops

rotating vortices in the cross-section of the tube. The strength of these vortices

tube is the Dean number (NDe

N ND

De Re=d

The use of helical holding tubes as a means of narrowing the RTD of particles has

been suggested by several researchers in the past. The narrowing of the RTD was

vature (λc >> 1) and low Dean numbers (NDe = NRe/√λc << 17). Dean [3] solved the

Navier–Stokes equation and obtained an approximate expression for the velocity of

tion of the Navier–Stokes equations which are valid over a wide range of curvature

and Reynolds numbers.

providing transition Reynolds number as high as 6,000 to 8,000 in a curved tube as

compared to 2,100 in a straight tube. Koutsky and Adler [6] pointed out that the pres-

sure drop in a tube formed into a helix can be up to four times as great as that in an

in helices at Reynolds numbers up to 8,000 or more. Both these facts imply the

the momentum, mass, and heat transfer and an increase in Reynolds number is also

known to decrease the axial dispersion.

The results of some of the studies that have been conducted to determine the

mentioned here. White [7] conducted experiments with oil and water for different

curvatures of helical tubes. For NRe > 100 and d/D = 1/50, the curved pipe had a

greater resistance than a straight pipe of same diameter and length. The resistance

Re = 6,000 (∼ NRe

Re ∼ 9,000 and

when d/D = 1/2050, turbulence was seen at NRe ∼ 2,250 to 3,200. Many equations

have been developed for predicting the critical Reynolds number that separates

55534_C002.indd 1955534_C002.indd 19 10/22/08 8:24:41 AM10/22/08 8:24:41 AM

coefficient without going into the turbulent regime is by using coiled tubes. Flow in

), which is defined as follows:

obtained analytical expressions for the velocity profile valid for large radii of cur-

region, the flow is said to be in transition. Reynolds number is thus a convenient

non-dimensional quantitative measure of the type of flow in different flow systems

(different pipe diameters, f low rates etc.).

Laminar flow conditions offer the advantage of simplicity in computations

involving flow and heat transfer equations. However, the major drawback of laminar

flow is the relatively low heat transfer coefficient. One way to enhance heat transfer

coiled tubes is characterized by flow in the primary (axial) direction and also in the

due to the centrifugal force. The secondary flow is characterized by two counter-

and pitch of the coil. The non-dimensional number that characterizes flow in a coiled

attributed to the development of secondary flow. Dean [2] was the first to analyze

depends on many factors such as the tube to coil diameter ratio, flow rate, viscosity,

mathematically the phenomenon of secondary flow in helically-coiled tubes. Dean

the fluid as a function of position. Truesdell and Adler [4] obtained a numerical solu-

Taylor and Yarrow [5] found that secondary flow could stabilize laminar flow,

identical straight tube. They also found that stable laminar flow can be maintained

existence of strong secondary flow in helices. Secondary flow is known to increase

critical value of Reynolds number that separates laminar and turbulent flow are

to flow became 2.9 times that in a straight pipe at N when flow

becomes turbulent). When d/D = 1/15, turbulent flow was seen at N



[8] is:

NdD

Re

1/2

c2100 1 12= +

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

It is important to note that most equations similar to the one developed above have a

r

to coil diameter ratio, pitch, or other factors.

lower than that in a straight tube. Several correlations have been developed to predict

form below [9]:

V

R u

Nc

2

De2

2

1 0.0306288

0.012

i

π=

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

+−NNDe

24

288

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

where u– is the average velocity in a straight pipe of the same radius under the same

axial pressure gradient.

Thus, it is important to make comparisons of Dean numbers while dealing with

tubes. It can also be seen that decreasing the coils diameter enhances the extent of

since decreasing the coil diameter results not only in enhanced mixing and heat

transfer, but also in an increase in the pressure drop.

2.3.3 FRICTION FACTOR

drop between the inlet and outlet of the tube. The pressure drop depends on the type

f f

tion and they are determined as follows:

EP

fu Ld

ff

2

= =Δρ

2

In the above equation, f is t

pipe is given by fs = 16/NRe

Moody [10] diagram. An alternative way to determine friction factor is to use the

following equation by Colebrook [11] and perform an iterative analysis:

1 1

3 7

1 255

f d N f= − +

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

4Re

ln.

.ε

55534_C002.indd 2055534_C002.indd 20 10/22/08 8:24:42 AM10/22/08 8:24:42 AM

laminar flow from turbulent flow. One such equation developed by Srinivasan et al.

It is known that for a given pressure gradient, the flow rate in a coiled tube is

ange of applicability. The limitations may be to the range of Reynolds number, tube

the flow rate in a helical tube. One such correlation is presented in a non-dimensional

flow in helical tubes, just like comparisons of Reynolds numbers are made in straight

secondary flow. Optimization is performed to choose the appropriate coil diameter

As a fluid flows through a pipe, friction impedes axial flow and creates a pressure

of flow (laminar, transition or turbulent), type of fluid, and the type of pipe. As the

fluid flows through a tube, there is loss in energy (E ) and pressure (ΔP ) due to fric-

he friction factor and it varies with the type of pipe, flow

conditions and system geometry. The friction factor for laminar flow in a straight

and for turbulent flow it is usually determined from the



These are standard methods for determining friction factors in straight pipes. How-

ever, for curved pipes, there are many additional factors such as coil diameter, pitch,

and Dean number that come into play, and hence there is no standard formula or

procedure available. Manlapaz and Churchill [12] have presented a list of studies

conducted for determination of friction factors in curved pipes.

Choice of the pump for any food processing operation, including aseptic processing

depends on many factors including whether the product has particulates in it, the extent

any food processing operation involves the use of the Bernoulli’s equation:

EP

Ep fPE KE= + + +Δ Δ Δρ

where ΔPE and ΔKE are the changes in potential and kinetic energies, respectively.

Ep is the energy supplied by pump and Ef is total loss in energy due to friction.

The above equation can also be written as:

gZU P

E gZU P

112

1p 2

22

2f

2+ + + = + + +

2Ψ ρ Ψ ρE

In the above equation, Ψ

hence have the units J/kg which is also the same as m2/s2. The subscripts 1 and 2 in

the above equation refer to the intake port and delivery port, respectively. Once Ep is

determined, the power rating of the pump is determined by the following equation:

Power p= m Ei

Once the power of the pump is determined, the next step is to determine the type of

positive displacement. In a centrifugal pump, product enters the center of an impeller

and due to centrifugal force, moves to the periphery. At this point, the liquid experi-

ences maximum pressure and is forced out into the pipeline. For a centrifugal pump,

varies as the square of the pump speed; and the power required varies as the cube of

tors involved in pump selection are as follows:

2. Net positive suction head required (NPSHR)—depends on impeller design.

It is required to maintain stable operation of pump including avoiding

cavitation.

55534_C002.indd 2155534_C002.indd 21 10/22/08 8:24:43 AM10/22/08 8:24:43 AM

pump to be used. Pumps are broadly classified into two categories, centrifugal and

pumps), direct force is applied to a confined liquid to make it move. Some of the fac-

of slippage (if applicable), piping arrangement, fittings present, and the flow behavior

2.3.4 PUMPS AND PUMPING REQUIREMENTS

of the fluid. The first step in determining the type and rating of a pump required for

is a constant and is equal to 0.5 for laminar flow and 1.0

for turbulent flow. All the terms in Bernoulli’s equation are energy per unit mass and

the volumetric flow rate is directly proportional to the pump speed; the total head

the pump speed. In a positive displacement pump (rotary, reciprocating, axial flow

1. Flow rate of fluid.



3. Net positive suction head available (NPSHA)—depends on absolute pres-

sure, vapor pressure of liquid, static head of liquid above center line of

pump, friction loss in the suction system.

2.3.5 RESIDENCE TIME DISTRIBUTION OF FLUID ELEMENTS AND PARTICLES

The FDA only credits heat treatment experienced in the holding tube, which makes

its design critical. An important factor to be taken into account in designing the hold-

ing tube is the fact that the residence time in the holding tube should be based on the

tube of circular cross-section, the maximum velocity occurs at the center of the hold-

imum velocity occurs at the axis of the tube which means that the minimum residence

time corresponds to the residence time of particles located along the center-line of the

tube. Consequently, these particles receive the least amount of heat treatment. Thus,

the holding tube length required to achieve the required F0 value (time-temperature

this will result in an over-processed product. This is where the residence time distribu-

tion (RTD) of the particles comes into picture. To understand RTD, we begin with the

under laminar conditions in a pipe of circular cross-section:

u urR

= −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥2 1

2

2

the tube, they spend different times in the tube, and the distribution of these times

is the RTD of the particles. The RTD of the particles depends a great deal on the

and also the size, density, and concentration of particles. Analysis of particle RTD is

relatively simple when there is only one type of particle in a system. However, when

55534_C002.indd 2255534_C002.indd 22 10/22/08 8:24:43 AM10/22/08 8:24:43 AM

5. Characteristic pump curves (graph of head, power consumption, and effi-

ment of the pump (since the pump is operating at a different temperature and specific

effect) can be calculated based on the knowledge of this minimum residence time, but

4. Properties of fluid (such as density, viscosity).

ciency versus volumetric flow rate).

is affected by the degree of its deviation from the behavior of a Newtonian fluid. The

degree of deviation is characterized by the flow behavior index, n, for Ostwall-de-

Waale fluids. For a Newtonian fluid flowing under laminar conditions in a straight

specific volume of the product at the hold tube temperature and not on the displace-

ing tube and its magnitude is twice the average velocity of the fluid. For pseudoplastic

fluids (n < 1), differences between the maximum and average velocity becomes smaller

volume varies with temperature). The velocity profile of the fluid in the holding tube

as n decreases. In other words, the velocity profile becomes flatter. For the extreme

case (n = 0), the plug flow profile is attained. However, for most cases (n > 0), the max-

following equation which describes the velocity profile for flow of a Newtonian fluid

Thus, it can be seen that different fluid elements (at different radial locations) spend

bution of times spent by various fluid elements within the tube is referred to as the

different amounts of time in the tube. For instance, a fluid element traveling at the

RTD of the fluid elements. Similarly, when different particles are flowing through

RTD of the fluid. It also depends on flow rate and viscosity of the carrier medium,

center of the tube will travel twice as fast as the average fluid element. The distri-



different types of particles (especially particles of different densities) are present in

present as the only particle type in suspension. For instance, in a mixture of two

types of particles, denser particles (which traveled slowly at the bottom of the tube

when present alone) could be sped up by rarer particles due to collisions, and in turn,

the rarer particles could get slowed down. Thus, an analysis has to be performed for

each combination of particle types present in a system and direct inferences cannot

be made from RTD of each particle type separately.

The existence of an RTD for the particles results in some particles receiving more

heat treatment than others in the holding tube. From a safety standpoint, the fastest

cle residence time. Thus, it can be seen that if the particle RTD is narrow, the quality

of the product would be high since the difference between the fastest and slowest

particle residence time is not very high. The wider the RTD of the particles in the

holding tube section, the more non-uniform the process. One of the techniques that

unn

urR

n n

=++

−⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪⎪

+3 1

11

1( )/

⎩⎩⎪⎪

⎫⎬⎪⎪

⎭⎪⎪

unn

umax =++

3 1

1

i

have been conducted by several researchers in the past. Some of these studies include

those of Dutta and Sastry [13], Palmieri et al. [14], Sancho and Rao [15], Sandeep and

Zuritz [16], and Baptista et al. [17]. Several studies have been conducted to determine

the RTD of particles in helical holding tubes too because the RTD in helical holding

those of Chen and Jan [18], Tucker and Withers [19], Ahmad et al. [20], and Sandeep

ticle size, or particle concentration results in a decrease in the RTD of particles while

an increase in viscosity results in an increase in RTD. However, it should be noted

that in determining the fastest particle (to compute process lethality based on this

particle), RTD studies should be conducted for that particular combination of particle

sizes and concentrations involved since experiments have shown that the fastest par-

ticle in a single particle situation is the neutrally buoyant particle that travels through

the center of the tube, while in mixed particle type situations, it is usually a particle

55534_C002.indd 2355534_C002.indd 23 10/22/08 8:24:44 AM10/22/08 8:24:44 AM

a product, the flow behavior is quite different from the situation when they are each

can be used to narrow the RTD of the fluid and particles is the use of helical tubes.

particle is what is of concern and the holding tube length is based on the fastest parti-

When a non-Newtonian (power-law) fluid flows through a straight tube under

Thus, for a pseudoplastic fluid (n < 1), the maximum velocity is given by:

less than twice the average fluid velocity. Thus, the RTD of the fluid is narrower for

a pseudoplastic fluid in comparison with that for a Newtonian fluid. Hence, the RTD

tubes is narrower than that in conventional holding tubes. Some of the studies include

of particles is also narrower when the carrier medium is a pseudoplastic fluid.

Hence it can be seen that the maximum velocity in the case of a pseudoplastic flu d is

Studies on RTD of fluid elements and particles in conventional holding tubes

et al. [21]. The general trend from these studies is that an increase in flow rate, par-

laminar flow conditions, the velocity profile is given by the following equation:



and hence lethality). Slower moving particles of lower thermal diffusivities could

very likely receive less heat treatment than faster moving particles of higher thermal

2.3.6 FORCES ACTING ON FLUID ELEMENTS AND PARTICLES DURING FLOW

system. The density, size, and shape of the particle are important particle characteris-

2.3.6.1 Equations of Motion of the Fluid

equations. The continuity equation is as follows:

∂ρ∂

∇ ρfft

+ ⋅ =( )u 0

This reduces to:

∇ ⋅ =u 0

as follows:

ρ ρ ∇ τf f

DuDt

g= + :

u

τ Δ Δ Δ= −⎡

⎣⎢⎢

⎤

⎦⎥⎥

−

mn

1

2( : )

( )/1 2

where, the second invariant of the strain rate tensor, 1/2(Δ : Δ) is given by:

1

2( : )Δ Δ

∂∂

∂∂

=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

+⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟

2f

2

fu

x

v

y⎟⎟⎟⎟⎟

+⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+ +

2

f

2

f∂∂

∂∂

w

z

v

x

∂∂∂

∂∂

∂∂

u

y

w

y

v

z

f

2

f f

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

+ +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟⎟

+ +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

2

f f

2

∂∂

∂∂

w

x

u

z

55534_C002.indd 2455534_C002.indd 24 10/22/08 8:24:45 AM10/22/08 8:24:45 AM

ticle is not always the critical particle (the particle that receives least heat treatment

In order to solve for the trajectory and velocity of various particles during flow in a

conductivities. This factor further complicates determination of process lethality.

tube, we need to know the fluid flow characteristics. The equations that govern the

of slightly higher or slightly lower density. It should also be noted that the fastest par-

a tube along with a carrier fluid, they experience various forces. These forces are

responsible for the translation and rotation of the particles as they flow through the

tics that affect the motion of the particles while the viscosity, flow rate, and density of

the fluid are the important fluid characteristics that affect the motion of the particles.

The motion of the fluid is described by the continuity equation and three momentum

for an incompressible fluid.

The three momentum equations for the fluid phase are given in vector notation

ids, the Ostwald-de Waele model is used [22]:For non-Newtonian fl

flow of a fluid are the continuity and momentum equations. As particles flow through



This takes into account the spatial variation of viscosity. Thus, the apparent viscosity

determined for the corresponding temperature. The effect of temperature and concen-

tration of particles on the effective viscosity of suspensions have been modeled by sev-

2.3.6.2 Linear Dynamic Equations for Particles

The three linear dynamics equations for the particles (in vector notation) are as follows:

mtppkd

d

VF

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

=∑ k

where mp is the mass of a single particle, Vpk and Fk (k = x, y, z) are the velocities of the

particles and forces acting on the particle in the x, y, and z directions, respectively.

2.3.6.2.1 Magnus Lift ForceThe Magnus lift force acts in a direction perpendicular to the direction of motion

of the particle and it is this force that causes the curving of a spinning sphere. The

expression to compute the Magnus lift force (Frk) is given by:

Frk f3

p f= × −πρ a ΩΩ ( )V V

where Ω is the angular velocity of the particle. In the above expression, the differ-

ence in velocities Vp − Vf is called the slip velocity or relative velocity (Vr). The vector

product Ω × Vr is determined as follows:

ΩΩ ΩΩ× = = − + − +Vr ijk j rk y r z r z r x rε V i j k( ) ( ) (Ω Ω Ω Ωw w w w ΩΩ Ωx r y rw w− )

Substituting the above equation in the equation for computing the force results in

the following three expressions for the Magnus lift force in the x, y, and z directions,

respectively:

F a w w v vx3

f y p f p f= − − −π ρ Ω Ω[ ( ) ( )]

F a u u w wy3

f z p f p f= − − −π ρ [ ( ) ( )]Ω Ω

F a v v u uz3

f x p f p f= − − −π ρ Ω Ω[ ( ) ( )]

The experimental works of researchers [27,28] indicated that particles migrated radi-

ally even in the absence of rotation. Thus, there is some other force that contributes

toward the lift forces experienced by particles.

2.3.6.2.2 Saffman Lift ForceSaffman [29] developed an expression for the lift force acting on a particle during

55534_C002.indd 2555534_C002.indd 25 10/22/08 8:24:46 AM10/22/08 8:24:46 AM

eral researchers [23,24,25]. Thus, the above equations can be used to describe the flow

of the fluid at any particular shear rate (at any particular location in the tube) can be

behavior of a power-law fluid containing particles under non-isothermal conditions.

Particles suspended in a viscous fluid are subjected to the following forces [26]:

an unbounded shear flow. The shear lift force is independent of the particle rotation



unless the rotation speed is much greater than the rate of shear. For a freely rotating

particle, Ω = ½ |K|, where Ω is the angular velocity of the particle and |K| is the mag-

nitude of the vorticity vector. Oliver [27] and Theodore [28] found that if the particle

away from the axis of the tube, thereby moving the particle away from the axis. In

vector notation, the Saffman lift force on a particle is given by:

Fs f2

1/2

p f=⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟× −6 46. ( )ρ a

v

KK V V

p f

The expression for K in the above equation can be obtained as follows:

K = × =

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

∇ ε ∂∂

∂∂

∂∂

V ijk

i j k

x y z

u v w

= −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+ −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟i j

∂∂

∂∂

∂∂

∂∂

wy

vz

uz

wx

⎟⎟⎟⎟+ −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟k

∂∂

∂∂

vx

uy

r

Vp − Vf = Vr = iur + jvr + kwr

Evaluating the above expressions results in the scalar forms of the Saffman lift force

F av u

zwx

xs f2

1/2

=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−⎛⎝⎜⎜6 46. ρ ∂

∂∂∂K ⎜⎜

⎞⎠⎟⎟⎟ − − −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ −( ) ( )w w

vx

uy

v vp f p f

∂∂

∂∂

⎡⎡

⎣⎢⎢

⎤

⎦⎥⎥

F av v

xuy

ys f2

1/2

=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−⎛

⎝⎜⎜6 46. ρ ∂

∂∂∂K ⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ − − −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ −( ) (u u

wy

vz

w wp f p

∂∂

∂∂ ff )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

F av w

yvz

sz f2

1/2

=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−⎛

⎝⎜⎜6 46. ρ ∂

∂∂∂K ⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ − − −

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ −( ) ( )v v

uz

wx

u up f p f

∂∂

∂∂

⎡⎡

⎣⎢⎢

⎤

⎦⎥⎥

The above expressions are valid if the tube Reynolds number is much greater than

unity and the particle is not very close to the axis of the tube. However, it should be

55534_C002.indd 2655534_C002.indd 26 10/22/08 8:24:47 AM10/22/08 8:24:47 AM

in the x, y, and z direction, respectively as follows:

ne the relative or slip velocity, V , as follows:We now defi

velocity was smaller than the fluid velocity, the lift force acted toward the axis of the

tube and if the particle velocity was greater than the fluid velocity, the lift force acted

where K is the curl of the fluid velocity, v is the kinematic viscosity, and a is the

radius of the particle. V and V are the velocities of particle and fluid, respectively.



noted that the above requirements may not be met in many situations and hence the

expression for Saffman force must be used with caution.

2.3.6.2.3 Drag Force

F C ad d f f p f p= − −1

2

2ρ π V V ( )V V

d

Cd

p

p= +24

1 0 15 0 687

Re( . Re ). for 1 < Rep < 1000

Re( )( )

p

f p=−ρ

μ2a V V

2.3.6.2.4 Buoyancy Force (acting in the y-direction only)The expression to compute the buoyancy force exerted on the particle (acting only in

the y-direction) is given by:

F gb f p= −( / ) ( )4 3 3π ρ ρa

Substituting the above four equations into the linear dynamic equation for the par-

ticle results in:

mt

av

pp

f p

d

d

VK V V=

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟× −6 46 2

1 2

. (

/

ρK

ff f p f

d f f p f p

) ( )

( )

+ × −

+ − − +

π ρ

ρ π

a

C a

3

21

2

4

3

ΩΩ V V

V VV V ππ ρ ρa3( )f f− g

The above equation can be rewritten in the following manner to obtain the expres-

sions for the linear dynamic equations for the particles in the x, y, and z directions,

respectively (with the gravity force acting in the y-direction):

x-direction:

mV

ta

v uz

wx

pp

f

d

d=

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−6 46 2

1 2

.

/

ρ ∂∂

∂∂K

⎛⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ − − −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟( ) (w w

vx

uy

vp f p

∂∂

∂∂

−−⎡

⎣⎢⎢

⎤

⎦⎥⎥vf )

+ − − −πρf y p f z p fa w w v v3[ ( ) ( )]Ω Ω

+ − −1

2

2C a u ud f f p f pρ π V V ( )

55534_C002.indd 2755534_C002.indd 27 10/22/08 8:24:48 AM10/22/08 8:24:48 AM

where the drag coefficient, C is obtained from the following equation [30]:

and the particle Reynolds number is defined by:

The expression for the drag on a particle in a viscous fluid is given by:



y-direction:

mV

ta

v vx

uy

pp

f

d

d=

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−6 46 2

1 2

.

/

ρ ∂∂

∂∂K

⎛⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ − − −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟( ) (u u

wy

uz

p f

∂∂

∂∂

uu up f−⎡

⎣⎢⎢

⎤

⎦⎥⎥)

+ − − −πρ Ωf z p f x p fa u u w w3[ ( ) ( )]Ω

+ − − + −1

2

4

3

2 3C a v v a gd f f p f p f fρ π π ρ ρV V ( ) ( )

z-direction:

mV

ta

v wy

vz

pp

f

d

d=

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−6 46 2

1 2

.

/

ρ ∂∂

∂∂K

⎛⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ − − −

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟( ) (v v

uz

wx

vp f p

∂∂

∂∂

−−⎡

⎣⎢⎢

⎤

⎦⎥⎥vf )

+ − − −πρf x p f y p fa v v u u3[ ( ) ( )]Ω Ω

+ − −1

2

2C a w wd f f p f pρ π V V ( )

This accounts for the description of the translation of the spheres. However, the par-

ticles undergo rotation too, and to account for the rotational motion of the sphere, the

angular momentum equations of the particle phase must be solved.

2.3.6.3 Angular Dynamic Equations for Particles

The three angular dynamics equations for the particles are as follows:

It

kk

d

d

ΩΩ⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟=∑T

where I is the moment of inertia (I = 2/5 mp a2 for sphere) and Tk (k = x, y, z) is the

Substitution of the expression for the torque into the angular momentum equation

results in the following sets of equations for the x, y and z directions, respectively:

d

d

x

p

x

Ω Ωt a

vz

wy

= +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−

⎡

⎣⎢⎢

15

82

μρ

π ∂∂

∂∂

⎤⎤

⎦⎥⎥

d

d

y

p

y

Ω Ωt a

uz

wx

= +⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟−

⎡

⎣⎢⎢

⎤

⎦

15

82

μρ

π ∂∂

∂∂

⎥⎥⎥

d

d p

z

Ω Ωz

t auy

vx

= +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−

⎡

⎣⎢⎢

15

82

μρ

π ∂∂

∂∂

⎤⎤

⎦⎥⎥

55534_C002.indd 2855534_C002.indd 28 10/22/08 8:24:49 AM10/22/08 8:24:49 AM

local torque exerted by the viscous fluid on the surface of the particles.



The linear and angular dynamics equations for the particles have to be solved simul-

2.3.7 TECHNIQUES TO DETERMINE FLUID AND PARTICLE VELOCITY

important, are usually not the target, since the species of concern are the slow-heating

particles. Particle residence times, residence time distributions, and velocities can be

determined by using a stop-watch, digital image analysis, LASER-Doppler velocim-

etry, and also with the aid of magnetically tagged particles.

2.4 HEAT TRANSFER ASPECTS OF PROCESSING

Some of the heat transfer aspects that are of importance in designing an aseptic pro-

and thermal properties of the product, mode of heat transfer (conduction or con-

vection), design of the heat exchanger, and the heat resistance of microorganisms,

enzymes, and nutrients.

During heating or sterilization of a solid–liquid mixture by heat treatment using

fers heat to the surface of the particles by convection. Further heating of the interior

temperature and the center temperature of particles can be due to the low convective

to enhance the thermal diffusivity of particles (other than by reformulation), efforts

have been geared towards measuring and enhancing the convective heat transfer

2.4.1 CONVECTIVE HEAT TRANSFER COEFFICIENT

Q = hfp Ap (Tps − Tf)

fp) include the shape of the particle, surface roughness, the posi-

in a coiled tube is fully three-dimensional and the rate of heat transfer can be much

55534_C002.indd 2955534_C002.indd 29 10/22/08 8:24:50 AM10/22/08 8:24:50 AM

dye tracers, and fine particles are used. Magnetic resonance imaging can also be used

cess are the convective heat transfer coefficient, effect of temperature on the physical

conventional means, the liquid part of the mixture gets heated first and then it trans-

Convective heat transfer coefficient has been described [31] as the thermal lag

Some of the factors that affect the convective heat transfer coefficient between a

taneously along with the four equations of motion for the fluid phase in order to

completely describe the flow dynamics of the suspension.

The average fluid velocity can be calculated once the volumetric flow rate of the prod-

uct is known. To determine the distribution of fluid residence times, salt injections,

under certain circumstances to obtain a fluid flow profile. Fluid flow profiles, though

of the particles take place by conduction. The difference between the bulk fluid

heat transfer coefficient between the fluid and the surface of the particle or due to

coefficient between the fluid phase and particles.

between the particle surface temperature and the fluid temperature for a particle

the low thermal diffusivity of the particles. Since there is not much that can be done

being heated in a fluid. This is mathematically written as follows:

fluid and particle (h

properties of the fluid.

Heat transfer for laminar flow in a straight tube (heat exchanger or holding tube)

tion of the particle in the tube, particle concentration, type of flow, and the thermo-

is usually low since there is very little mixing in the radial direction. However, flow

higher than that in a straight tube. This is due to the development of secondary flow



imposed as a result of the centrifugal forces present in the curved section. The sec-

2.4.2 STEAM QUALITY

Steam quality refers to the amount of steam that is in vapor phase in saturated steam,

with superheated steam having a quality of 1. The amount of energy given out by

steam is given by the following equation:

Q m H H= −st s c

i( )

with

Hs = (X) Hv + (1 − X) Hc

In the above equations, Hs, Hv, and Hc are the enthalpies of steam, pure vapor, and

pure condensate respectively and ‘X’ is the steam quality. Thus, when steam quality

is 1 (or 100%), all the steam is in vapor state and the enthalpy of steam is the same as

the enthalpy of pure vapor and when steam quality is 0, all the steam is in condensate

form and the enthalpy of steam is the same as the enthalpy of pure condensate.

The enthalpy of pure vapor and pure condensate can be determined from saturated

steam tables at the corresponding saturation temperature and pressure. The enthalpy

of saturated steam can then be determined as a weighted mean of these enthalpies,

once the steam quality is known. Thus, it can be seen that higher the quality of

steam, higher the amount of energy transferred from steam to the product.

2.4.3 DIMENSIONLESS NUMBERS GOVERNING HEAT TRANSFER

dimensionless quantity called Nusselt number. Nusselt number is a dimensionless

temperature gradient at the surface and is given by:

NhDk

T yT T D

Nuc

f

surface

surface c

d d

/

Te

= = −−

=

( / )

( )

mmperature gradient at surface

Average temperrature gradient throughout the system

Nusselt number also represents the ratio of the diameter of the tube to the equivalent

thickness of the laminar boundary layer. Empirical correlations have been developed

to determine Nusselt number as a function of various other dimensionless quantities

ing forced and free convection.

55534_C002.indd 3055534_C002.indd 30 10/22/08 8:24:50 AM10/22/08 8:24:50 AM

called Reynolds number, convective heat transfer coefficient is represented by a

ondary flow serves as a means of redistributing fluid elements in the radial direction,

(in the directions normal to the main direction of flow) due to the pressure gradient

and thereby transferring heat more efficiently between the bulk of the fluid and the

fluid elements near the tube wall.

Just like fluid flow characterization is done by the use of a dimensionless quantity

including Reynolds number under different flow and heat transfer conditions includ-



Ng T T D v Dn

Grf f surface vertical ver=

− −β ρ ∞2 3 14( ) ( ttical

2

n

n nK n n

−

−+

1 2

13 1 2

)

{ [( ) / ] ( )}

The quantity β is t

following expression:

β⟨ ⟩

∂⟨ ⟩∂

=⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟⎟

1

V

V

TP

For the case of free convection, Nusselt number is a function of Grashof number.

number reduces to the following form:

Ng T T D

Grf f surface vertical=

−β ρμ

∞2 3

2

( )

Prandtl number, which is the ratio of momentum diffusivity (μ/ρ) and thermal dif-

fusivity (k/ρcp), comes into picture in the determination of Nusselt number for both

forced and free convection. The generalized Prandtl number (valid for power-law

Nc K n n

v d k

n n

n nGpf

f

/Pr

[( ) ] ( )=

+ −

− −

3 1 2

4

1

1 1

Nc

kPr = pf f

f

μ

The Biot number is the ratio of the internal (conductive) and external (convective)

resistance offered to heat transfer in an object. It comes into picture for calculations

NhDk

D kh

Bic

s

c s/

/

Internal resistance

Exter= = =

1 nnal resistance

Unsteady state heat transfer is characterized by another dimensionless quantity,

Nt

Dk D D

c D tFo

c

c c

p c

/

/

Rate of heat c= = =

αρ2

2

3

1( ) oonduction

Rate of heat storage

55534_C002.indd 3155534_C002.indd 31 10/22/08 8:24:51 AM10/22/08 8:24:51 AM

in free convection only (where buoyancy effects are significant). The expression for

he coefficient of volumetric thermal expansion and is given by the

involving unsteady state heat transfer and is defined as follows:

which is the Fourier number and is defined as follows:

the generalized Grashof number (valid for power-law fluids, in addition to Newtonian

Grashof number is the ratio of buoyancy to viscous forces and is of importance

fluids) is given by:

When dealing with Newtonian fluids, the expression for the generalized Grashof

fluids, in addition to Newtonian fluids) is given by:

For Newtonian fluids, the above expression reduces to the following form:



2.4.4 NATURAL (FREE) AND FORCED CONVECTION

Convective heat transfer can take place by natural or forced means. In natural con-

contact with a hot (or cold) surface, thereby resulting in buoyancy forces. The Nus-

selt number in this case is a function of the Grashof number (NGr) and the Prandtl

number and it takes the following form:

NhLk

c N NNuc

Grc2= = 1( )Pr

c

teristic length is given by:

Lc

Cross-sectional area

Wetted perimeter=

⎛

⎝⎜4⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

In some situations, a combination of natural and forced convection takes place.

Thus, the relative importance of the two has to determined. If (NGr) (NPr) < 8 ×105,

the effect of natural convection can be neglected and forced convection governs

the heat transfer. Another method of determining the relative importance of nat-

ural and forced convection is by determining the ratio of the Grashof number

(measure of the buoyancy force) and the square of the Reynolds number (measure

of the inertial force). If the ratio is close to unity, the effects of forced and free

convection have to be taken into account. The magnitude of the Froude num-

ber also can be used to determine the relative importance of natural and forced

convection.

2.4.5 TRANSIENT HEAT TRANSFER WITHIN PARTICLES

Two basic approaches exist to solve the problem of heat transfer involving par-

entire particle is at the same temperature and the second method takes into account

the temperature gradient within particles. The lumped capacitance method is valid

only if the Biot number (NBi) is less than 0.1. If the lumped capacitance method is

applicable, the following equation can be used to determine the temperature (T)

within an object at any given time (t):

T TT T

hAc V

ti

−−

= −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

∞

∞ ρexp

p

It should be noted that when the lumped parameter method is valid, the resistance to

heat transfer due to conduction is negligible in comparison to that due to convection.

However, if the Biot number is greater than 0.1, the Heisler chart is used to determine

the temperature at the center of an object. When the Biot number is greater than 40,

the resistance to heat transfer due to convection is negligible in comparison with that

due to conduction.

55534_C002.indd 3255534_C002.indd 32 10/22/08 8:24:52 AM10/22/08 8:24:52 AM

ticles—the first method, called the lumped capacitance method, assumes that the

vection flow occurs due to the differences in the density of the fluid as it comes into

In the above equation, L is the characteristic length. For internal flows, the charac-



2.4.6 HYDRODYNAMIC AND THERMAL ENTRANCE LENGTHS

the cooling section and hence care has to be taken in designing these components of

the aseptic processing system depending on the product.

and fully developed region. The entrance region is the region where the velocity and

to become fully developed (hydraulic length or hydrodynamic entry length, lh) is

given by the following equation [32]:

l N dh = 0 05. ( )( )Re

The above equation is also referred to as the Langhaar equation. In the transient

(thermal length, lt) is given by the following equation:

l N dt Pe= 0 036. ( )( )

where the Peclet number (NPe) is given by:

Nu d

Pef

f

=α

The Peclet number, given by the above equation (for Newtonian and non-Newtonian

2.4.7 HEAT TRANSFER COEFFICIENT IN STRAIGHT TUBES

While considering transfer of heat between a heating medium (such as water or steam)

commonly are listed below.

between steam and the wall of the heat exchanger [33]:

hk g

T T do hx

st st

st w hx o hx

( )

. .

( ) (

.( )

=−

0 7253 0 2 0ρ λ

))

/

μ

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

1 4

55534_C002.indd 3355534_C002.indd 33 10/22/08 8:24:53 AM10/22/08 8:24:53 AM

region, the velocity profile is fully developed, while the temperature profile is still

developing. The length required for the temperature profile to become fully developed

medium and the outer wall of the inner tube (outside heat transfer coefficient) and

the convective heat transfer coefficient between the product and the inner wall of the

inner tube (inside heat transfer coefficient). Different techniques exist to determine

ficients come into picture—convective heat transfer coefficient between the heating

the heat transfer coefficient based on the state of the heating medium (liquid or gas)

The following equation is used to compute the (outside) heat transfer coefficient

The flow behavior of fluids in a pipe depends a great deal on the temperature. As most

products get heated, they become less viscous and flow much easier than when it is

at room temperature. Thus, it is possible to have turbulent flow in the heat exchanger

and holding tube (where the viscosity of the product is very low) and laminar flow in

As a fluid flows through a straight pipe and convective heat transfer is taking

place, the flow can be divided into three regions—entrance region, transient region

temperature profiles are still developing. The length required for the flow (laminar)

fluids) is a product of the Reynolds number and the Prandtl number.

and a product for flow in a tubular heat exchanger, two convective heat transfer coef-

and whether the flow is in a tube or in an annulus. Some of the correlations used



product is computed using the following equation [33]:

Nn

n

mc

k LNu

p f

f hx

=+⎛

⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎛

⎝

⎜⎜⎜2 03 1

4

1 3

.

/( )

i

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

1 30 14

/.

mmw

fi lm), given by the

following expression:

T T Tfilm wall fluid= +1

2( )

In the holding tube, the following equation is used to determine the (inside) heat

tr

Nn n nn n n

Nu =+ + ++ + +

⎛

⎝⎜⎜⎜⎜

⎞8 0

15 23 9 1

31 43 13 1

3 2

3 2.

⎠⎠⎟⎟⎟⎟

In the cooling section, the following equations are used to determine the inside and

t

Nn mc

kLNu

p=+⎛

⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟2 03 1

4

1 3

.

/ i

⎟⎟⎟⎟⎟

1/3

N N NNu G G= 0 023 0 8 1 3. Re.

Pr/

Reynolds number is less than 2,100).

puted using the following correlation [35]:

N N Nd

dNu G Gp= +

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟

2 0 28 37 0 233 0 143. . Re.

Pr. ⎟⎟⎟⎟⎟

1 787.

sented [36] as a function of Peclet number and the local Graetz number, where the

Graetz number (NGz) is given by:

55534_C002.indd 3455534_C002.indd 34 10/22/08 8:24:54 AM10/22/08 8:24:54 AM

The (inside) heat transfer coefficient between the wall of the heat exchanger and the

ansfer coefficient between the product and the wall of the holding tube [34]:

ou side heat transfer coefficients [33]:

determining the outside heat transfer coefficient, the outside diameter of the tube

The properties of the fluid are determined at the film temperature (T

Laminar flow:

Turbulent flow:

tube and the properties of the fluid undergoing processing have to be used. While

tion (both internal and external flow) is considered to be laminar if the generalized

While determining the inside heat transfer coefficient, the inside diameter of the

and the properties of the cooling water have to be used. The flow in the cooling sec-

The surface heat transfer coefficient between the fluid and the particle is com-

Nusselt number expressions for heat transfer from a power-law fluid under laminar

flow conditions under an uniform wall heat flux boundary condition have been pre-



Nmc

kxN N

x dGz

p= =i

Re Pr

/

The most commonly used equations for determining the Nusselt number for laminar

for NReNPr (d/L) < 100

NN N d L

N NNu = +

+3 66

0 085

1 0 0 045.

. [ ( / )]

. . [

Re Pr

Re Pr (( / )] .

.

d L 0 66

0 14

μμ

b

w

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

for NReNPr (d/L) > 100

N N NdL

Nub

w

= ×⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎛

⎝⎜⎜⎜⎜

⎞

⎠1 86

0 33

. Re Pr

. μμ

⎟⎟⎟⎟⎟

0 14.

by the following equation [37]:

Nn

nNNu Gz

1/3 b

w

=+⎛

⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟2 0

3 1

4.

μμ ⎟⎟⎟

0 14.

H

ber is provided by Perry and Chilton [38] to determine the convective heat transfer

friction is given by the following equation:

j f NH = = −1

20 023 0 2. Re

.

number:

N N NLd

Nu G Gc= +

⎛⎝⎜⎜⎜

⎞⎠

2 0 8 4703 0 553 0 2716. . Re.

Pr. ⎟⎟⎟⎟

−⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

−0 6272 0 1142. .R rR

When dealing with a scraped surface heat exchanger, the convective heat transfer

N N N NNu Re Pr b=1 2 0 5 0 33 0 26. . . .

55534_C002.indd 3555534_C002.indd 35 10/22/08 8:24:55 AM10/22/08 8:24:55 AM

Thus, it can be seen from the above equation that the heat transfer coefficient will be

Researchers [39] determined the convective heat transfer coefficients between a

They found that the convective heat transfer coefficient increased with decreasing

heat transfer coefficient was higher for a particle near the wall.

coefficient is determined using an equation such as the one presented below [40]:

flow in horizontal pipes are given as follows:

Another commonly used correlation for laminar flow heat transfer in pipes is given

higher for pseudoplastic fluids (n <1) as compared to that for Newtonian fluids.

For transition flow, a graph of the Colburn j-factor ( j ) versus the Reynolds num-

coefficient. The statement of the Colburn analogy between heat transfer and fluid

fluid and a particle and developed the following correlation to determine the Nusselt

particle size or viscosity and increasing flow rate. It was also found that convective



A more general equation has been developed [41] which takes into account the speed

N cu D D

ND

uC

Nu

s=−( )⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

( )1

22ρ

μπ

Pr

/Ω⎛⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

0 62 0 55. .

( )DD

Nsb

00 53.

For viscous liquids, c1 = 0.014, c2 = 0.96.

For non-viscous liquids, c1 = 0.039, c2 = 0.70.

Thus, depending on the situation presented, the appropriate equation to determine

each of the equations have a range of applicability and also assumptions involved.

2.4.8 HEAT TRANSFER COEFFICIENT IN HELICAL TUBES

tubes, there is no simple correlation that can be applied to a wide range of process

conditions. Nevertheless, different correlations are available for different situations

and some of them have been presented below.

heat transfer in curved tubes:

N N N NNu De Pr Defor and 0= ≤ ≤3 31 20 12000 115 0 0108. . . ..005 ≤ ≤NPr .0 05

N N N NNu De Pr Defor and 0.7= ≤ ≤0 913 80 12000 476 0 2. . . ≤≤ ≤NPr 5

For constant wall temperature heat transfer in curved tubes, the following

correlation was developed [43]:

N N N N NNu De Pr Defor and 0.7= ≤ ≤ ≤0 836 80 12000 5 0 1. . .PPr ≤ 5

Other researchers [44] developed the following Nusselt number correlations for dif-

ferent ranges of Dean numbers:

N N N N N NNu De De Defor and (= ≤1 7 202 1 6 2 0 5. ( ) )Pr/

Pr. >>100

N N N NNu Defor= < <0 9 20 1002 1 6. ( )Re Pr/

N N NdD

Nu pr for=⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ <0 7 1000 43 1 6

0 07

. Re. /

.

NNDe <8300

They concluded that the effect of d/D can be neglected for Dean numbers less than

100 in the fully developed thermal region. They also found that for all cases with

(N 2

De NPr) > 100, the Nusselt number in the fully developed thermal region was

55534_C002.indd 3655534_C002.indd 36 10/22/08 8:24:55 AM10/22/08 8:24:55 AM

the convective heat transfer coefficient should be used. It should also be noted that

of rotation and the type of fluid:

Researchers have conducted several studies on flow and heat transfer in helical tubes.

These studies have included various flow and fluid types, tube diameters, and coil

radii. Due to the complexities that are involved in flow and heat transfer in helical

The following correlations have been developed [42] for constant wall heat flux



proportional to NPr1 6/ and that for the thermal entry region, the Nusselt number was

proportional to NPr1 3/ .

2.4.9 HEATING MEDIA AND EQUIPMENT

Heating the product can be accomplished by direct contact of the hot medium and the

product or by indirect contact. Direct contact heating is accomplished by steam injec-

tion (injecting steam into a product) or steam infusion (passing a product in a thin

layer through a chamber of steam). Both of these methods are rapid means of steril-

izing the product. The product is then cooled by evaporation in a vacuum chamber.

There are several types of indirect contact heat exchangers. Some of them are

the plate, tubular, shell and tube, scraped surface, microwave, and ohmic. Some of

the non-thermal techniques include the use of high pressure, irradiation, and pulsed

liquid foods with small particles. Tubular and shell and tube heat exchangers can

be used with relatively low viscosity products with the limiting factor being the

size of the particulates. Tubular heat exchangers can be double tube, triple tube

exchanger (SSHE) provide mixing and also prevent burning of products onto the

wall of the heat exchanger. A SSHE is also suitable for products with large par-

ticulates. Microwave (dielectric heating) and ohmic (electrical resistance heating)

heating results in rapid and simultaneous heating of the liquid and particulate

phases of the product. However, they can also result in non-uniform and runaway

heating. Some of the other techniques such as the use of radio frequency, pulsed

under investigation for commercialization on a large scale. On a smaller scale, the

use of pulsed light, ultrasound, and ultraviolet radiation have been attempted with

limited success.

Some of the methods that handle liquids and particulates separately are the

Jupiter system (particles are sterilized by steam in a double cone pressure vessel),

rotaholder (a tubular sterilizer in which particles are held back for extra time using

ized bed by steam and cooled by sterile Nitrogen) and have been described in more

detail elsewhere [45]. The disadvantages of these methods are the added costs for

separating and recombining the liquid and particulate phases and the complexities

introduced in the overall process.

2.4.10 CO- AND COUNTER-CURRENT HEAT EXCHANGERS

Co- and counter-current heat exchangers are used abundantly in the food processing

industry. Thus, it is important to be able to determine the overall heat transfer coef-

can easily be determined by the following equation:

Q mc T=•

p( )Δ

55534_C002.indd 3755534_C002.indd 37 10/22/08 8:24:56 AM10/22/08 8:24:56 AM

electric field. Plate heat exchangers have a large surface area and hence rapid heat

electric fields, irradiation, membrane separation, and high pressure are currently

ficient in these types of heat exchangers. The total energy transferred to the product

transfer can take place (due to the turbulent flow conditions) to liquid foods and

or corrugated tube heat exchangers with flow in the co-current, counter-c urrent

or cross-flow mode. With viscous products, the blades of the scraped surface heat

forks), and the fluidized bed system (particles are separately sterilized in a fluid-



where ΔT is the rise in temperature of the product. This information is then used to

following equation:

Q UA T= lm lmΔ

where the logarithmic mean temperature difference (ΔTlm) is given by:

Δ Δ ΔΔ Δ

TT T

T Tlm =

−1 2

1 2ln( / )

with ΔT1 and ΔT2 being the difference between the temperatures of the hot and cold

lm is the

logarithmic mean area and is given by:

AA A

A Almo i

o i/=

−ln( )

with Ao and Ai being the outside and inside surface areas, respectively.

It is thus possible to compute the effectiveness of various heat exchangers based

the same inlet conditions, it can be shown that the amount of energy transferred from

2.4.11 GOVERNING HEAT TRANSFER EQUATIONS AND ENERGY BALANCE

The energy equation in spherical coordinates is as follows:

ρ ∂∂

∂∂

∂∂ θ

∂∂

cTt r r

krTr r

p =⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+

1 12

2

2 sin θθθ ∂

∂θ θ∂

∂ϕ∂∂ϕ

kT

rk

Tsin

sin

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+

⎛

⎝⎜1

2 2⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+ S

By symmetry,

∂∂θ

∂∂ϕ

T T= = 0

Also, S = 0 for the case where there is no heat source term.

Thus, the energy balance equation reduces to:

ρ ∂∂

∂∂

∂∂

cTt r r

krTr

p =⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

12

2

For constant properties, the above equation reduces to the following form:

α ∂

∂∂∂

∂∂r r

rTr

Tt2

2 0⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟− =

55534_C002.indd 3855534_C002.indd 38 10/22/08 8:24:57 AM10/22/08 8:24:57 AM

calculate the overall heat transfer coefficient in the heat exchanger making use of the

on the above outlined procedure to compute the overall heat transfer coefficient. For

fluids at the inlet and exit of the heat exchanger, respectively. The quantity A

the hot to the cold fluid is higher in the case of a counter-current heat exchanger.



This is the equation that has to be then solved to determine the temperature distribu-

tion within spherical particles.

2.4.11.1 Energy Balance in the Heat Exchanger

U A T T m c T T N h A T TEhx hx st f f pf f f p f f pp( ) ( ) (− = − + −

i ′ss )

In the above equation, T f and T ps

temperatures, respectively and are given by:

T T T T T Tf f f ps ps psand= + = +1

2

1

2( ) ( )′ ′

The terms Tf’ and Tps

’

step (or the new spatial location). Also, the surface area of the heat exchange surface,

Ahx, is computed as follows:

A dhx hx= π Δx

the heat exchanger (Uhx):

1 1 1

U A h A h Ax

hx lm hx o hx o hx i hx i hx

hx

( ) ( ) ( ) ( ) ( )

= + +kk Ahx lm(hx)

where the logarithmic mean area (Alm) is given by:

AL R R

R Rlm

o i

o i

=−2π ( )

ln( / )

The terms 1/hoAo, Δr/kAlm, 1/hiAi represent the resistances to heat transfer from the

steam to the outside wall of the heat exchanger, from the outside wall of the heat

exchanger to the inside wall of the heat exchanger, and from the inside wall of the

heat exchanger to the product, respectively.

Thus, the two unknowns, namely, Tf′ and Tps

′ can be solved for.

2.4.11.2 Energy Balance in the Holding Tube

The overall energy balance equation in the holding tube is as follows:

m c T T U A T T N h A T Ti

f pf f f ht ht f air p f p fp( ) ( ) (′ − = − + − pps)

55534_C002.indd 3955534_C002.indd 39 10/22/08 8:24:58 AM10/22/08 8:24:58 AM

The following equation is used to determine the overall heat transfer coefficient in

In the heat exchanger, the steam loses heat to the fluid and the particles suspended in

the fluid. The overall energy balance equation in the heat exchanger is as follows:

are the mean fluid and mean particle-surface

are the fluid and particle-surface temperatures at the new time

In the holding tube, the fluid loses heat to the particles and also to the surroundings.



the holding tube (Uht):

1 1 1

U A h A h Ax

ht lm(ht) o hx o hx i hx i hx

ht= + +( ) ( ) ( ) ( ) kk A

xk Aht lm ht

ins

ins lm ht( ) ( )

+

peratures in the heat exchanger was then used to determine the temperatures of the

2.4.11.3 Energy Balance in the Cooling Section

overall energy balance equation in the cooling section is as follows:

m c T Ti

f pf f f cs cs f cw p f p f pp( )− =′

ss

cs

the heat exchanger with all parameters for the heat exchanger being replaced by the

corresponding parameters for the cooling section.

exchanger. This is referred to as fouling and can greatly impede the rate of transfer

of heat. After a certain time, when the heat transfer rate becomes unacceptably low,

decreased productivity and increased costs. This is more of a problem with viscous,

eliminate it totally. The use of scraped surface and helical heat exchanger also mini-

mizes fouling. Another detrimental factor associated with fouling is the fact that

or holding tube). This in turn could result in a decrease in the actual accumulated

lethality which could potentially result in an unsafe product.

2.4.13 TECHNIQUES TO ESTIMATE THE TEMPERATURE HISTORY OF A PRODUCT

TM). Another tech-

data to a computer. Infrared imaging is a technique that can be used to obtain surface

temperature information. In order to use infrared imaging for particles, the particles

55534_C002.indd 4055534_C002.indd 40 10/22/08 8:24:59 AM10/22/08 8:24:59 AM

The following equation is used to determine the overall heat transfer coefficient in

The overall heat transfer coefficient in the cooling section (U ) is computed using

U A (T −T )+ N h A (T −T )

the same equation that was used to determine the overall heat transfer coefficient in

with the aid of clamps or compression fittings (such as Swagelok

A similar approach to that used in the determination of the fluid and particle tem-

fluid and particles in the holding tube.

In the cooling section, the fluid and particles lose heat to the cooling water. The

As a product flows through a system, it tends to stick to the hot surface of the heat

proteinaceous, and starchy foods and is predominant under laminar flow conditions.

2.4.12 FOULING AND ENHANCEMENT OF HEAT TRANSFER

The use of appropriate flow rate and temperature can minimize fouling, but not

with increased fouling, the cross-sectional area available for flow decreases, thereby

the system has to be shut down and cleaned using a CIP solution. This translates to

increasing flow velocity and decreasing the residence time (in the heat exchanger

the use of thermocouples or RTDs in the flow regime which may be installed in-line

nique to determine temperature within a fluid is the introduction of tracer capsules

The most commonly used method to determine the temperature within a product is

(or data tracers) in the flow, retrieving it at the exit, and downloading the temperature



are retrieved at the exit, sliced, and imaged to determine variation of temperature at

any cross-section of the particle. Thermochromic dyes that change color with time

also within particles. Thermoluminescent markers (that emit a certain wavelength of

light depending on the temperature) can be used to determine the temperature within

2.5 MICROBIOLOGICAL AND QUALITY CONSIDERATIONS

During processing, there are several factors that the processor takes into account. First

and foremost comes the safety of the process and compliance with regulatory require-

ments. Other factors that come into picture are the extent of enzymatic inactivation

and nutrient retention. Thus, the process is designed such that it is safe and results in

maximum nutrient retention and the appropriate level of enzymatic inactivation.

2.5.1 FEDERAL REGULATIONS AND HACCP

Unlike in European countries, where regulations are based on spoilage tests, the

latitude for variability in process conditions. In the U.S., different regulatory agencies

and rules apply to different products. For example, UHT milk processing is covered

under title 21 (parts 108, 113, 114) of the code of federal regulations (CFR). The

process should also adhere to the pasteurized milk ordinance (PMO). When meat is

involved, the regulations are imposed by the USDA. In addition to these regulations,

certain states have state regulations imposed on certain processes. During the past

few years, HACCP has gained tremendous importance and its implementation has

been extended by the FDA to various products after its initial application to certain

2.5.2 KINETICS OF MICROBIAL DESTRUCTION, ENZYME INACTIVATION, AND NUTRIENT RETENTION

There are several techniques to estimate the extent of heat treatment received by vari-

ous components in a food. The most direct technique to do this is by measuring the

situation. Thus, indirect mechanisms, such as the change in color of a dye or the extent

of sucrose inversion are used. As far as microorganisms go, one of the techniques

to ascertain the extent of microbial destruction is by using an alginate particle with

spores of Bacillus stearothermophillus embedded in it. The gel ensures that the micro-

organisms do not leak out and result in inaccurate degree of microbial destruction.

When vegetative cells of bacteria are subjected to harsh conditions (high heat or

lack of nutrients), they form a hard proteinaceous coating outside the cell that can with-

stand the harsh conditions, and go into a passive stage and the organisms in this state

are called spores. Inactivating vegetative cells of bacteria can be achieved relatively

easily (few minutes at ∼ 80°C) while inactivating the spores requires relatively high

heat treatment (few minutes at ∼ 120°C). The heat resistance of bacteria (vegetative cells

55534_C002.indd 4155534_C002.indd 41 10/22/08 8:25:00 AM10/22/08 8:25:00 AM

and melting point indicators (that melt at a specific temperature) are some of the

FDA requires microbiological tests to prove the safety of a process with sufficient

acidified and low-acid canned foods.

other techniques that can be used to determine the temperature within fluids and

clear fluids on-line.

temperature at the desired locations. However, this is not easy in a continuous flow



and spores) is affected by previous events such as incubation temperature (resistance

increases as incubation temperature is raised closer and closer to the optimum growth

temperature), age (least resistant in logarithmic growth phase and most resistant in the

last part of the lag phase and also in the stationary phase), growth medium (more nutri-

tious the growth medium, more resistant the spore), and drying (some spores become

more heat resistant after drying). Other factors affecting heat resistance are presence of

ionic species, oxygen content, water activity (moist heat is generally more effective than

dry heat), pH (acid medium is usually more effective than alkaline medium which is

usually more effective than neutral medium), salts and sugars (high concentrations are

effective in reducing their resistance), and proteins and fats (the presence of these mate-

rials increases the heat resistance). Thus, it is important to determine the heat resistance

of the organisms of concern in the substrate of interest and under the appropriate pro-

cessing condition. It should also be noted that bacteria which tend to clump together are

generally more resistant to heat and care has to be exercised when dealing with them.

Most chemical and microbiological reactions encountered in thermal processing

lncc

k t0

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟= − T

where c0 is the initial concentration of the species under consideration, c is the con-

centration after time t, and kT is the rate of the reaction. For microbial destruction,

the concentration in the above equation is replaced by the number of viable micro-

organisms, and the rate of reaction is replaced by the decimal reduction time (DT) to

yield the following equation (expressed in base 10):

logNN

tD0

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟= −

T

with the decimal reduction time (in minutes) being related to the rate of the reaction

(in seconds) by the following expression:

Dk

T

T

=2 303

60

.

When a semi-logarithmic plot is made between the number of viable microorgan-

isms (on the y-axis on a logarithmic scale) and time in minutes (on the x-axis), a

straight line is obtained. The slope of this line is equal to the negative reciprocal of

the decimal reduction time. This graph is also referred to as the survivor curve, ther-

mal death curve, inactivation curve or the thermal death time (TDT) curve.

The dependence of the rate of the reaction, kT, on temperature, is given by the

following equation by Arrhenius:

k Be E R Ta gT = _ /

with B being a constant which is referred to as the collision number or frequency

factor and Ea being the activation energy. In the above equation, both B and Ea are

55534_C002.indd 4255534_C002.indd 42 10/22/08 8:25:01 AM10/22/08 8:25:01 AM

are first order equations and are given by the following equation:



assumed to be independent of temperature. However, there are other models that do

not make this assumption. For Clostridium botulinum, researchers [46] determined

the appropriate value of B to be 2 × 1060 s−1 and Ea to be 310.11 kJ/mol−K.

Another commonly used technique to express the dependence of the rate of reac-

tor as the ratio of the reaction rates at two temperatures. When these temperatures

are 10°C apart, the quotient indicator is then referred to as Q10 and is given by:

Qk

k

D

DT

T

T

T10

10

10

10= = =+

+

10/z

The Q10 value for Clostridium botulinum is 10.

2.5.2.1 Process Lethality and Cook Values

The decimal reduction time of bacteria depend strongly on temperature and is given

by the following expression:

logD

DT T

zref

ref⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟= −

−

where Dref is the decimal reduction time at a reference temperature of Tref and z is the

temperature change required for an order of magnitude change in the decimal reduc-

tion time. The z value for Clostridium botulinum is 18°F (or 10°C).

Lethal rate (LR), which is a measure of the rate of inactivation of the microor-

ganisms at any given temperature, is given by:

LR ref ref

T

= =−10( )/T T z DD

For a constant temperature process, the above approach can be used. However, when

the process temperature changes, F value is used to calculate the total lethal rate as

follows:

F dt dt

t

T T z

t

ref= =∫ ∫ −( )

( )/LR

0 0

10

The F value at a reference temperature of 250°F (or 121.1°C) and a z value of 18°F

(10°C) is referred to commonly as the F0 value and is thus evaluated as shown

below:

F dtT

t

0250 18

0

10= −∫ ( )/

Another term that is commonly encountered in aseptic processing is lethality. Lethal-

ity is the ratio of the F0 value of the process to the F0 value required for commercial

sterility. Thus, process lethality must be at least unity for commercial sterility.

55534_C002.indd 4355534_C002.indd 43 10/22/08 8:25:02 AM10/22/08 8:25:02 AM

tion on temperature is the quotient indicator method which defines a quotient indica-



An F0 value of 5 minutes indicates that the process is equivalent to a heat treat-

ment of 5 minutes at 250°F. It can be thus seen that many combinations of time

and temperature can yield an F0 value of 5 minutes. The appropriate combination

of time and temperature that is used for processing is based on other factors such

as nutrient retention and enzyme destruction. This is where the cook value (C) of a

process comes into picture. The cook value is a measure of the extent of destruction

of enzymes or nutrients and is given by:

C dtT T z

t

ref c= −∫ 10

0

( )/

with zc being analogous to z for microorganisms. Similar to F0, C0 is the reference

cook value based commonly on a reference temperature of 100°C and the zc value

is much higher than that for microorganisms (e.g. 33°C for thiamine destruction).

Graphical methods or optimization models are then used to determine the optimum

time-temperature combination that renders the product safe and also retains the

maximum possible amount of nutrients.

2.5.2.2 Commercial Sterility of the Product

vative method is generally employed. The conservative approach involves the assump-

tion that particles neither receive lethality nor any heat treatment in the heat exchanger.

This approach is referred to as the ‘hold only’ approach. The other two commonly

used approaches are the ‘F0 hold’ and the ‘total system’ approach. In the ‘F0 hold’

approach, it is assumed that particles gain heat treatment in the heat exchanger, but

not lethality. In the ‘total system’ approach, it is assumed that particles gain heat treat-

ment and also accumulate lethality in the heat exchanger. The reason for not including

lethality accumulated in the cooling section is that it is possible for particulates to

break in the cooling section and thereby get cooled rapidly and hence not receive the

assumed heat treatment and hence lethality.

2.6 FROM AN IDEA TO COMMERCIALIZATION

In order to commercially produce aseptically processed low-acid foods (pH > 4.6)

containing large particulates (diameter > 4.6 mm), there are several hurdles to over-

come. It all begins with an idea for the product. Let us for example consider a product

such as 1/2" carrot cubes (10% w/w) suspended in a 1% CMC solution. This is the

product that we would like to aseptically process, package, and market as a high

quality shelf-stable product.

cubes (mildly blanched) and a supplier who can provide easy to dissolve, shear-stable

CMC powder of satisfactory initial microbiological count. The next step is to deter-

mine the tentative process layout. This includes the choice of pump, heat exchanger,

holding tube, cooling unit, and packaging equipment.

Let us begin by determining the appropriate pump to be used. Since we are

dealing with large particulates, a piston pump such as the 20 hp Marlen twin-piston

55534_C002.indd 4455534_C002.indd 44 10/22/08 8:25:03 AM10/22/08 8:25:03 AM

When insufficient data are available regarding cold spots in the product, a very conser-

The first step is to identify a supplier who can provide high quality 1/2" carrot



pump (Model 629A, Marlen Research Corp., KS) will be an appropriate choice. This

damage to the particles. The product is batched in a 200 gallon tank using an Admix

Rotosilver submersible High-shear Mixing unit (Admix Inc., Londonderry, NH).

The next step is to identify the appropriate heating system. In situations where

there are large particles, a scraped surface heat exchanger or a volumetric heating

system is usually employed. Here, we select a scraped surface heat exchanger

(SSHE) equipped with steam seals for aseptic processing applications (Model 6×9,

Alfa Laval, Newburyport, MA) as the pre-heater (the speed of rotation of the rotor is

set at 175 rpm and measured using an optical tachometer) and a 30 kW, 40.68 MHz

to room temperature and a certain elevated temperature and the RF heater is the

penetration of the electromagnetic waves.

The holding tube is one of the most important parts of the aseptic processing

system as this is where the product receives its heat treatment from a commercial

sterility standpoint. A stainless steel helical holding tube assembly (coil diameter 1")

uniform heat treatment of the product.

The product is then cooled in a hydrocoil cooling unit (ASTEC, IA). A hydrocoil

cooling unit will result not only in rapid cooling of the product, but will also be gentle

enough to the product so as to not cause disintegration of the particulates. The cool-

continuously circulating through the system).

The packaging unit used in this case is a single lane, two-head Metal Box cup

Filler (model SL1-15, Metal Box, Reading, UK) with a peroxidase spray system and

Louisville, KY). The package used was a 12-oz cup with an aluminum foil used to

heat-seal the top.

The data acquisition system consisted of a datalogger (Model CR10, Campbell

at the entrance and exit of the pump, SSHE, RF heater, holding tube, and cooling

section using type T thermocouples (Omega Engineering, Stamford, CT) installed

Back pressure is provided to the system by means of a lobe pump (Model 45U2,

Waukesha-Cherry Burrell, Delavan, WI) along with a T-junction with a manual con-

trol back pressure valve in the vertical section and a wire mesh screen that ensured

passed through to the lobe pump.

The next step is to use mathematical modeling to design the length of the holding

tube based on the properties of the product, process parameters, and other system

55534_C002.indd 4555534_C002.indd 45 10/22/08 8:25:04 AM10/22/08 8:25:04 AM

MA) as the final heater. The SSHE serves to bring up the temperature of the product

finisher which has the effect of minimizing the difference in temperature between

an agitated raque tank (100 gallon PF-2-5-1A agitated filler, Food Systems Inc.,

Scientific, Logan, UT) and multiplexer capable of handling 32 channels which were

with sanitary fittings.

type of pump will not only result in uniform product flow rate, but also minimal

continuous flow radio frequency heater (Model 464, Radio Frequency Co., Millis,

the fluid and particle since a low frequency (40.68 MHz) translates to a high depth of

is used. A helical holding tube results in the development of secondary flow and

ing medium is chilled water flowing through the system at a high flow rate (20 gpm

hence causes mixing of the solid–liquid mixture and hence translates to a relatively

of chilled water constantly flowing in and out of the system and 200 gpm of water

run using the software PC208W 3.0. The temperatures of the fluid are monitored

that only the fluid portion passed through the back pressure valve and the particles



parameters. For the heat exchanger and the holding tube sections, the theory of the

mathematical modeling has been presented in previous sections of this chapter.

However, for modeling heat transfer in the RF heater, another model has to be used.

Based on the modeling studies, the length of the holding tube is appropriately chosen

(conservative estimate). During the modeling, care should be taken to account for

studies should be conducted to determine the rheological behavior of CMC as a func-

tion of shear rate, time, and the high temperatures encountered during processing.

This can be done using a controlled stress rheometer (Stresstech, ATS Rheo Systems,

Bordentown, NJ). During the modeling, conservative estimates of convective heat

calorimeter.

ent magnetic strengths) into several cube-shaped tracer particles in order to deter-

mine the residence times of the particles in various sections of the aseptic processing

system (especially the holding tube section). Care is taken to compensate for the

higher density of the magnets than the tracer particle since particle density is a major

factor affecting the fastest particle residence time. Magnetic coils situated outside

the tubes of the processing line picked up the signals produced by the motion of

the magnets throughout the system, and this enables us to determine the residence

times and hence the residence time distribution and also the fastest particle residence

created by the RF system. Based on statistics, it has been shown that the residence

of collecting the fastest 1% of the particles.

The next step is to perform biological validation tests. These tests are performed

at various stages of the process—just after start-up, during the middle of the run,

and just before shut-down. These tests account for variations during the process

and also for factors such as fouling. The validation tests are conducted at different

temperatures to document a positive/negative result at the end of the process. This

will aid in determining the minimum allowable process temperature that will result

in a safe process. Microbiological validation tests are done using PA 3679 inoculated

within alginate particles. Care should be taken to ensure that the spores do not leach

5

decimal reduction time of the organisms used is determined by means of thermal

the predicted temperatures and lethalities in order to ensure that the model results in

a conservative prediction of process lethality.

down and care should be taken to ensure that these parameters remain within an

acceptable range. Some of the parameters include hydration time, mixing/batching

55534_C002.indd 4655534_C002.indd 46 10/22/08 8:25:04 AM10/22/08 8:25:04 AM

transfer coefficients and thermal diffusivities are used. Thermal conductivity is mea-

change in viscosity of the suspension as a function of time. To aid this, benchtop

sured using the line heat source probe and specific heat is measured using a mixing

The first phase in experimental studies is to incorporate tiny magnets (of differ-

time. The magnets are of low enough strength to not affect the electromagnetic field

times of at least 299 particles must be determined in order to have a 95% confidence

10 spores per particle is used, a final count of <1 would indicate a safe process. The

death time studies. Based on all of these tests, a process is designed and finally

verification of the established process has to be conducted. During this process of

verification, comparisons are made between actual temperatures and lethalities to

Once verification is successful, all the process and system parameters are noted

out into the fluid. If the target for the process was a 5D process, and an initial load of

time, temperatures at various locations, product flow rate, back pressure, and product



with the FDA using form 2541C. A comprehensive overview of the procedures and

has been given in a report elsewhere [47]. This is based on the workshops organized

by the Center for Advanced Processing and Packaging Studies (CAPPS) and the

National Center for Food Safety and Technology (NCFST).

2.7 CONCLUDING REMARKS

Aseptic processing has undergone a variety of changes since its inception as far as

equipment, operating procedures, and critical points of concern. With the advent

of new technologies to inactivate microorganisms, some of the existing problems,

such as slow heating of particles, can be overcome. Nevertheless, new technologies,

since the target microorganism, extent of enzymatic inactivation and other factors

might change. Despite the hurdles posed by aseptic processing, the high quality of

the end-product will make this technology more prevalent in the US market as con-

sumers are becoming more conscious about the nutritive value of foods and leading

a healthy lifestyle.

NOMENCLATURE

a Radius of particle, m

A Surface area, m2

B Arrhenius parameter, Pa−s

c Final concentration of species

co Initial concentration of species

p−K

c1, c2 Constants

C Cook value, min

d

d Diameter of tube, m

D Diameter of helical coil, m

Dc Characteristic dimension, m

Ds Diameter of shaft of SSHE, m

DT Decimal reduction time at temperature T, min

Dvertical Vertical dimension, m

Ea Activation energy, J/kg−mol

Ef Loss in energy due to friction, J/kg

Ep Energy to be supplied by pump, J/kg

f Friction factor

F Force, N

F0 F-value when reference temperature is 250°F and z-value is 18°F, min

Fb Buoyancy force, N

Fd Drag force, N

Frk Magnus lift force, N

55534_C002.indd 4755534_C002.indd 47 10/22/08 8:25:05 AM10/22/08 8:25:05 AM

properties. The final step in commercialization of the product involves process filing

processes involved in process filing for a product such as the one discussed above

such as the use of high pressure or pulsed electric field, have to be carefully studied,

c Specific heat, J/kg

C Drag coefficient



Fs Saffman lift force, N

g Acceleration due to gravity, m/s2

h 2−K

Hc Enthalpy of condensate, J/kg

Hs Enthalpy of steam, J/kg

Hv Enthalpy of vapor, J/kg

I Moment of inertia, kg-m2

jH Colburn j-factor

k Thermal conductivity, W/m−K

kT Reaction rate constant at temperature T, s−1

n

K Curl of velocity

lh Hydrodynamic entry length, m

lt Thermal entry length, m

L Length of tube, m

Lc Characteristic length, m.

mp Mass of particle, kg

n Flow behavior index

N Final bacterial count

Nb Number of blades

NBi Biot number

NDe Dean number

NFo Fourier number

NGGr Generalized Grashof number

NGPr Generalized Prandtl number

NGRe Generalized Reynolds number

NGr Grashof number

NGz Graetz number

NNu Nusselt number

No Initial bacterial count

NP Number of particles

NPe Peclet number

NPr Prandtl number

NRe Reynolds number

NRec Critical Reynolds number

P Pressure, Pa

Q Energy transferred, W

Q10 Quotient indicator

r Radial location, m

R Radius of tube, m

Rep Particle Reynolds number

Rg Universal gas constant, J/kg-mol-K

S Source term, W/m3

t Time, s

T Temperature, K

55534_C002.indd 4855534_C002.indd 48 10/22/08 8:25:06 AM10/22/08 8:25:06 AM

Convective heat transfer coefficient, W/m

K Consistency coefficient, Pa-s

m Mass flow rate, kg/s



Ti Initial temperature, K

Tk

T∞ Free stream temperature, K

u Velocity, m/s–u Average velocity, m/s

u, v, w x, y, and z components of velocity respectively, m/s

U 2

.3

V Velocity, m/s

<V> Volume, m3

X Quality of steam

z Temperature change required for an order of magnitude change in decimal

reduction time, °C

Z Height, m

GREEK LETTERS

α Thermal diffusivity, m2/s2

−1

ρ Density, kg/m3

ε Roughness of pipe, m

ϕ, θ Spherical coordinates

σ Shear stress, Pa

σ0 Yield stress, Pa

γ. Shear rate, s−1

λ Latent heat of vaporization, J/kg

λc Curvature

μ Viscosity, Pa−s

μe Effective viscosity, Pa−s

ΔPf Pressure loss due to friction, Pa

Δr, Δx Thickness, m

ΔT Temperature difference, K

ψ Constant

Ω Angular velocity, rad/s

Φ Particle concentration

SUBSCRIPTS

c Coiled tube

cs Cooling section

f Fluid

fp Fluid–particle interface

ht Holding tube

hx Heat exchanger

i Inside

ins Insulation

55534_C002.indd 4955534_C002.indd 49 10/22/08 8:25:06 AM10/22/08 8:25:06 AM

Overall heat transfer coefficient, W/m -K

β Coefficient of volumetric thermal expansion, K

Local torque exerted by fluid, N-m

V Volumetric flow rate, m /s

b Bulk fluid



lm Logarithmic mean

max Maximum

o Outside

p Particle

ps Particle surface

ref Reference temperature

s Straight tube

st Steam

w Wall

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sterilization during laminar flow. Journal of Food Science 39: 1047–54.


53

3 Modeling Moisture Diffusion in Food Grains during Adsorption

Kasiviswanathan Muthukumarappan and S. Gunasekaran

CONTENTS

3.1 Introduction .....................................................................................................54

3.2 Moisture Diffusion in Food Grains ................................................................ 55

3.2.1 Various Moisture Transport Mechanisms ........................................... 55

3.2.1.1 Knudsen Diffusion ................................................................ 55

3.2.1.2 Stefan Diffusion .................................................................... 55

3.2.1.3 Mutual Diffusion ................................................................... 55

3.2.1.4 Poiseuille Flow ...................................................................... 55

3.2.1.5 Condensation–Evaporation Theory ...................................... 57

3.2.1.6 Capillary Flow ...................................................................... 57

3.2.1.7 Liquid Diffusion .................................................................... 57

3.2.1.8 Surface Diffusion .................................................................. 57

3.2.2 Coupled Heat and Moisture Transport ................................................ 57

3.2.3 Characterization of Shape for Modeling Moisture

Diffusion in Grains .............................................................................. 58

3.3 Modeling Moisture Diffusion in Food Grains ................................................ 59

3.3.1 Theoretical Considerations .................................................................. 59

3.3.1.1 Boundary Condition .............................................................. 59

3.3.2 Numerical Formulation ....................................................................... 61

3.4 Moisture Diffusion in a Corn Kernel During Adsorption .............................. 61

3.4.1 Corn Structure ..................................................................................... 61

3.4.2 Moisture Diffusivity Determination ................................................... 62

3.4.2.1 Germ ..................................................................................... 62

3.4.2.2 Pericarp .................................................................................66

3.4.2.3 Soft and Hard Endosperms ................................................... 70

3.4.3 Finite Element Simulation of Corn Moisture Adsorption ................... 74

3.5 Recommendations ........................................................................................... 76

Nomenclature ...........................................................................................................77

References ................................................................................................................ 78

55534_C003.indd 5355534_C003.indd 53 10/22/08 8:27:50 AM10/22/08 8:27:50 AM



3.1 INTRODUCTION

Food grains are hygroscopic and hence adsorbs or desorbs moisture depending upon

the environment. Moisture diffusivity is a physical property of measurement, which

aids in studying the moisture diffusion mechanism. Moisture gradients prevalent

within a food grain due to the moisture adsorption/desorption phenomenon may lead

to the development of internal stresses [1,2]. The internal as well as external stresses

because they are quite susceptible to breakage during handling and cause problems

in storage, shipping, and processing [3]. If the stresses developed within the kernels

development. However, such an estimation requires accurate determination of mois-

ture diffusivity of the grain components and a description of moisture adsorption

and/or desorption mechanisms.

Adsorption and desorption are different mechanisms and there exists a hyster-

esis between them. That is, the equilibrium moisture content attained by grains via

desorption is higher than that via adsorption for a particular temperature and relative

humidity condition. There have been many theories to explain this hysteresis. Chung

and Pfost [4] postulated that more sorption sites or polar sites are available to water

vapor for the desorption process than for the adsorption process. That is, the mois-

ture transport mechanisms of desorption and adsorption are different. Desorption

and adsorption processes are subjected to the same physical laws and thus can be

treated analogously. Variations in the rate of desorption and adsorption (diffusion)

occur due to the boundary conditions at the medium interface and may cause the

apparent hysteresis in the sorption isotherms. It should be emphasized that during

low temperature deep bed drying, while some parts of a large mass lose moisture

(desorption), others are simultaneously gain moisture (adsorption). Thus, models

that cover both desorption and adsorption processes are needed.

Extensive research work has been done on drying of different grains with a pri-

mary focus on modeling diffusion of moisture [5,6] and determining moisture dif-

fusivities of major grain components. However, only limited information is available

on diffusion of moisture in grains during adsorption [7,8]. In general, the moisture

diffusivity of grains during adsorption is lesser (at least one order of magnitude) than

during desorption. For example, rice kernels had a moisture diffusivity of 1.3 ×10−7

during desorption and 1.2 ×10−8 m2/h during adsorption. Experimental methods of

diffusivity determination, collecting moisture content data at various points inside

the corn kernel over a time period, require sophisticated sensors and are cumber-

some. Mathematical models, based on physical principles, can potentially predict

with reasonable accuracy the moisture distribution inside the kernel during adsorp-

tion. However, for improved accuracy the mathematical models require the moisture

diffusivities of kernel components during adsorption. Moisture diffusivity of grain

components during adsorption is also needed to better understand the moisture trans-

port in grain conditioning, storage, deep-bed drying and aeration processes.

uct quality are needed. In this chapter, different mathematical models to determine

the moisture diffusivity of individual components of any heterogeneous food grain

55534_C003.indd 5455534_C003.indd 54 10/22/08 8:27:51 AM10/22/08 8:27:51 AM

cause grain kernels to fissure. Fissured or stress-cracked kernels are objectionable

can be calculated accurately, better processes can be designed to reduce fissure

For efficient processing operations quantitative and predictive models relating to

the physical properties of food to transient time-moisture profiles that determine prod-


Modeling Moisture Diffusion in Food Grains during Adsorption 55

are described. As an example, the developed models were validated using the mois-

ture adsorption data in a corn kernel. Moisture diffusivity of individual components

of a corn kernel namely pericarp, germ, soft and hard endosperms were determined

inside the corn kernel.

3.2 MOISTURE DIFFUSION IN FOOD GRAINS

3.2.1 VARIOUS MOISTURE TRANSPORT MECHANISMS

The mechanisms of moisture movement within a product can be primarily summa-

rized as water–vapor transport mechanisms and liquid–water transport mechanisms

(Figure 3.1). The water–vapor transport mechanism consists of Knudsen diffusion,

liquid diffusion, and surface diffusion [9]. Among these, diffusion is the dominant

mechanism.

3.2.1.1 Knudsen Diffusion

One of the water–vapor transport mechanisms within a product may be explained

in terms of the Knudsen diffusion mechanism as outlined in Figure 3.1. This type

the mean free-path of molecules is more than the pore size and the molecules col-

density and the Knudsen vapor diffusivity within the product. The size and amount

as represented in the equation column.

3.2.1.2 Stefan Diffusion

pressure, and the Stefan vapor diffusivity within the product. A constant diffusivity

is assumed.

3.2.1.3 Mutual Diffusion

Mutual diffusion is predominant in solids with large pores, whose size is much more

than the free-path of the diffusing vapor molecules. The roles Knudsen and mutual

diffusions perform are commensurable within a certain range of pore sizes and gas

pressures.

3.2.1.4 Poiseuille Flow

55534_C003.indd 5555534_C003.indd 55 10/22/08 8:27:51 AM10/22/08 8:27:51 AM

using the finite difference, analytical, and finite element methods, respectively. The

developed finite element model was also used to predict the moisture distribution

of diffusion occurs in gas-filled solids with small pores, or under low pressure when

Stefan diffusion, Mutual diffusion, poiseuille flow, and condensation–evaporation.

On the other hand the liquid–water transport mechanisms consists of capillary flow,

lide with the walls more often than between themselves. Molecule reflection from

the walls is normally diffuse. In this case, the water flux is a function of the vapor

of pores, tortuosity, and the geometry of the solid matrix affect the water vapor flux

In the case of Stefan diffusion, the water flux is a function of the vapor pressure, total

Poiseuille flow is pressure-induced flow in a long duct. It is also called channel flow.

In this case, it is assumed that there is laminar flow of an incompressible Newtonian



Type Picture Equations

2 2Π3

Knudsendiffusion

Stefandiffusion

Mutual

diffusion

Poiseuille

Condensation–

evaporation

Capillary

flow

Liquid

diffusion

Surface

diffusion

d

h

d

nd

w

nP

RTkP – P

w

Mwd

w

1w

= –ετβDkw

∇ρ

nd

w1w

= –ετDwg

∇ρ

–ετDwg

∇Pw

∇P

Dkw

Dwg

d ( )RTk

Mw

=

=

= 2.5 × 10–5 (m2/s)

1/2

nd2ρ32µ

dw

n nw (∇T

k,·····)

–ρ1 χ ∇θ

–ρ1 Dwg ∇ω

w

–ρ1 Dsu ∇ω

w

dw

n 1w

–ετ

=

=

n1

w =

n1

w =

=

FIGURE 3.1 Various mechanisms of moisture transport in porous materials. (From S

Bruin, KChAM Luyben. 1980. Drying of food materials: A review of recent developments.

Advances in Drying. Vol. 1, 155–215. New York: Hemisphere Publishing Corporation.)

55534_C003.indd 5655534_C003.indd 56 10/22/08 8:27:52 AM10/22/08 8:27:52 AM

flow



drop ΔP in a pipe of length L and diameter d << L.

3.2.1.5 Condensation–Evaporation Theory

Water vapor within the solid is condensed near the surface. This assumes that the

rate of condensation is equal to the rate of evaporation at the surface of the solid, and

allows no accumulation of water in the pores near the surface. This theory takes into

account the simultaneous diffusion of heat and mass, which assumes that the pores

are a continuous network of spaces in the solid.

3.2.1.6 Capillary Flow

Moisture which is held in the interstices of solids, as liquid on the surface, or as free

moisture in cell cavities, moves by gravity and capillarity, provided that passageways

point, as in textiles, paper, and leather, and to all moisture above the equilibrium

such as soil, sand, and clays.

3.2.1.7 Liquid Diffusion

The movement of liquids by diffusion in solids is restricted to the equilibrium

moisture content below the point of atmospheric saturation and to systems in which

moisture and solid are mutually soluble. This applies to the drying of clays, wood,

soaps, and pastes.

3.2.1.8 Surface Diffusion

Surface diffusion is observed during adsorption of a diffusing substance by a solid.

Since the equilibrium surface gas concentration increases with an increase in partial

pressure of the adsorbed species, a surface concentration gradient of a diffusing

substance appears in the surface layer of a pore. Under certain conditions like high

The mechanisms described above refer only to a single-component diffusion.

Multicomponent diffusion in porous solids is very complex. Because of its complex

nature, it has been inadequately investigated.

3.2.2 COUPLED HEAT AND MOISTURE TRANSPORT

The behavior of any food material during drying and rewetting depends on the heat

and mass transfer characteristics of the product being dried or wetted. Knowledge

of the temperature and moisture distribution in these products is vital for equipment

and process design, quality control and choice of appropriate storage practices.

Wang and Hall [10] stated that if temperature distribution within the medium is

uniform, the assumption of moisture concentration as the driving force is adequate.

55534_C003.indd 5755534_C003.indd 57 10/22/08 8:27:53 AM10/22/08 8:27:53 AM

applies to liquids not held in solution and to all moisture above the fiber-saturation

moisture content at atmospheric saturation, as in fine powders and granular solids,

fluid of viscosity μ induced by a constant positive pressure difference or pressure

for continuous flow are present. In drying, liquid flow resulting from capillarity

temperature, this may enhance the total flow of a diffusing component.



This assumption is reasonable since grains respond to temperature differences more

rapidly than to moisture differences [11]. Further, Sharaf-Eldeen et al. [12] reported

that the body temperature of grains approached the drying air temperature in a small

fraction of the total drying time and thus temperature could be omitted in the model.

Young [13] described a mathematical model for drying of a porous sphere using the

diffusion equation for both moisture and heat transfer, assuming that moisture diffu-

number is greater than 60 (negligible temperature gradient).

About a decade ago, coupled heat and mass transfer equations have been solved

for an isotropic sphere with constant material properties [14]. In 1992, Irudayaraj et al.

[15] developed a comprehensive model that describes the heat and mass transfer in

a wide range of food grains (soybean, barley, and corn kernels) with varying mate-

rial properties. Their simulated results from the heat and mass transfer models agreed

well with the experimental results. Recently, Irudayaraj and Wu [16] developed models

incorporating heat, mass, and pressure transfer equations to describe the moisture dif-

fusion process in a barley kernel during soaking. The results obtained from the heat,

mass, and pressure transfer show a marked difference from the results obtained from

the heat and mass transfer model. This indicated that a pressure gradient exists during

Coupling the effect of moisture and temperature may be important for accurately mod-

eling the drying process. But for adsorption, the coupling effect may not be important

because the adsorption process takes much longer (48–50 h) than the desorption pro-

cess (6–10 h).

3.2.3 CHARACTERIZATION OF SHAPE FOR MODELING MOISTURE DIFFUSION IN GRAINS

Thin-layer models can be divided into three groups: (1) empirical models, (2) semi-

empirical models, and (3) theoretical models. Among these, theoretical models provide

the most information about the moisture transport inside grains.

Young and Whitaker [17] and Whitaker and Young [18] evaluated different

an empirical model during drying of peanuts. They found that the diffusion equa-

tion better represented the drying than did the empirical model. Moreover, they

and Okos [19] qualitatively explored the diffusion path of gaseous sulfur dioxide

(SO2) into yellow dent corn. They showed that SO2 enters at the tip cap, moves

up through the area between the pericarp and seed coat, and then diffuses into

the endosperm. Their observations clearly indicate that the pericarp of corn acts

as a diffusion barrier even to gaseous SO2. Modeling diffusion in cereal grains

using the spherical diffusion model is thus inappropriate because the model does

Recently, Eckhoff and Okos [20] modeled the gaseous SO2 sorption by corn as an

insulated cylinder with one end open for diffusion. Walton et al. [21] cautioned that

55534_C003.indd 5855534_C003.indd 58 10/22/08 8:27:53 AM10/22/08 8:27:53 AM

sivity is a linear function of moisture content. He defined a modified Lewis number

and suggested that the moisture diffusion equation alone is sufficient if the Lewis

the soaking process, causing additional moisture movement due to filtration effect.

diffusion equations for a plane sheet, finite and infinite cylinders, and a sphere and

concluded that the finite-cylinder model best fit the experimental data. Eckhoff

the diffusion coefficient that is determined with one geometric shape couldn’t be

not adequately reflect the true nature of the diffusional processes via the tip cap.



used with another geometric shape. Muthukumarappan and Gunasekaran [22–25]

evaluated the effect of different shapes in determining the moisture diffusivity of

tion data.

3.3 MODELING MOISTURE DIFFUSION IN FOOD GRAINS

3.3.1 THEORETICAL CONSIDERATIONS

of Fick’s law of diffusion developed by Crank [26] were used. The differential equa-

tion with initial and boundary conditions to describe the system are:

∂∂

∂∂

∂∂

M

t xD

M

xm nm=

⎡

⎣⎢⎢

⎤

⎦⎥⎥ =1, (3.1)

∂∂M

Xx= = ≥0 0 0; ; t

(3.2)

M M x l l t= = − >e 2 2; , ; 0 (3.3)

M M l x l t= − < < =0 2 2; ; 0 (3.4)

The following assumptions were made in solving Equation 3.1:

1. Initial moisture content is uniform throughout the kernel.

2. Grain is isothermal during adsorption; i.e. the heat transfer equations may

be neglected.

3. Moisture diffusivity is constant throughout the adsorption process.

4. Grain components are homogeneous and isotropic.

5. Expansion of the kernel during adsorption is negligible.

3.3.1.1 Boundary Condition

The boundary condition given in Equation 3.3 implies that the moisture content at

the kernel surface reaches equilibrium with the environment instantaneously. New-

man [27] disputed this assumption during drying. Shivhare et al. [28] assumed that

the surface moisture reaches equilibrium exponentially during microwave drying

of corn. Muthukumarappan [8] found, during adsorption, that the numerical model

adsorption curve better than the model with assumption of instantaneous equilib-

rium surface moisture. Walton et al. [21] assumed a boundary condition, which is

55534_C003.indd 5955534_C003.indd 59 10/22/08 8:27:54 AM10/22/08 8:27:54 AM

corn samples and found that the infinite slab model fitted the experimental adsorp-

A typical kernel is irregular in shape. Therefore, three geometries, namely, an infinite

slab, an infinite cylinder, and a sphere were considered. The corresponding solutions

with assumption of exponentially varying surface moisture fitted the experimental

a function of the convective mass transfer coefficient in developing a drying model

for corn. However, the experimental convective mass transfer coefficient values at



different temperature and relative humidity (RH) conditions during adsorption are

a function of adsorption time. Following these investigations, the surface moisture

content was assumed to vary exponentially with adsorption time. The boundary con-

dition describing the moisture content at the kernel surface (Ms, %) and the surface

moisture ratio (MRs) can be written as:

M kt M M Ms e 0 0= − − × − +[ exp( )]1 [ ] (3.5)

MRss 0

e 0

=−−

= − −M MM M

kt[ exp( )]1 (3.6)

Moisture ratio (MR) of a multicomponent system can be modeled as:

MR MRi ( ) ( )t X ti,j

ij j=∑ (3.7)

where i = grain type (1 = soft and 2 = hard) and j = grain components

Instead of geometrically modeling the whole kernel, individual components

(namely pericarp, soft and hard endosperms and germ) can be modeled independently

using Equation 3.7. The number of components are those components that have

differing properties namely pericarp, soft and hard endosperms and germ for a corn

kernel. Similarly other grains namely rice, wheat and soybean can be modeled using

the above approach.

The Cartesian coordinates were used to represent the grain as a two-dimensional

body. The general diffusion equation, which describes the moisture transport, has

the form:

∂∂

Δ ΔM

tD M= ( ) (3.8)

In two-dimensions it becomes,

∂∂

∂∂

∂∂

∂∂

∂∂

M

t xD

M

x yD

M

y=

⎡

⎣⎢⎢

⎤

⎦⎥⎥ +

⎡

⎣⎢⎢

⎤

⎦⎥⎥ (3.9)

The initial and boundary conditions are:

M M t= =0 , 0 (3.10)

and

M Kt M M M , ts e 0 0 0 on= × >[ exp( )]1− − − +[ ] Ω (3.11)

where, Ω constitutes the complete boundary surface for the body.

55534_C003.indd 6055534_C003.indd 60 10/22/08 8:27:55 AM10/22/08 8:27:55 AM

not currently available. This makes it difficult to predict surface moisture content as



3.3.2 NUMERICAL FORMULATION

The element equations were developed by transforming the governing differential

equations using the Galerkin’s weighted residual approach. After the formulation,

� �M C M Kj ij

j

n

j

j

n

ij[ ] [ ]

= =∑ ∑+ =

1 1

0 (3.12)

where the element moisture capacitance matrix,

[ ]C N N x yij i j

A

= ∫ d d (3.13)

and the element moisture conductance matrix,

[ ]KN

x

N

x

N

y

N

yD x yij

i j i j

A

= +⎡

⎣⎢⎢

⎤

⎦⎥⎥∫ ∂

∂∂∂

∂∂

∂∂

d d (3.14)

Assembling the element matrices in Equation 3.12 using Equation 3.13 and Equation

3.14, the global matrix equation can be written as:

[ ]{ } [ ]{ }C M K M� + = 0 (3.15)

where [C] and [K] are the global moisture capacitance and conductance matrices.

The solution of Equation 3.15 will result in moisture values at every time step in the

domain of interest.

For the transient case under consideration, an implicit technique (backward dif-

tions incorporating the known boundary conditions, had the following form:

([ ] [ ]) [ ]C t K M C M tFt t t t t+ = ++ +Δ ΔΔ Δ (3.16)

3.4 MOISTURE DIFFUSION IN A CORN KERNEL DURING ADSORPTION

3.4.1 CORN STRUCTURE

broad at the apex of its attachment to the cob. The kernel is composed of germ,

kernel and is strongly adherent to the seed coat. The hard endosperm is found on the

55534_C003.indd 6155534_C003.indd 61 10/22/08 8:27:56 AM10/22/08 8:27:56 AM

the element equations can be written in a simplified form as:

ference scheme), which is unconditionally stable, was used. The final system of equa-

Corn is a complex cereal grain. The kernel is flattened, wedge-shaped, and relatively

soft (floury) and hard (horny) endosperm, and pericarp. The pericarp surrounds the



sides and back of the kernel and bulges in toward the center at the sides. The soft

the germ. The pericarp, the outermost part of the kernel and a major part of what

the millers know as hull, is composed of several layers. Most important is the outer

layer, the epidermis, which is more or less cutinized on its outer surface. Cutin is

relatively impervious to moisture, so the cutinized surface of the epidermis acts as a

barrier to moisture movement during adsorption. Typically a soft corn kernel is com-

posed of 5% pericarp, 10% germ, 48% soft endosperm and 37% hard endosperm.

And a hard kernel is composed of 4% pericarp, 9% germ, 21% soft endosperm and

66% hard endosperm [8].

3.4.2 MOISTURE DIFFUSIVITY DETERMINATION

Chittenden and Hustrulid [29] reported that mean diffusivity of shelled corn varied

linearly with the initial moisture content; they concluded that actual diffusivity should

depend also on moisture content at any point within the kernel. Steffe and Singh

moisture content. But Hsu et al. [30] demonstrated that during soaking of soybeans,

the moisture diffusivity is strongly dependent upon moisture content of the seeds

and that the diffusion equation with constant diffusivity is inadequate in describing

the water absorption curve. More recently Lu and Siebenmorgen [31] modeled the

moisture diffusion in rough, brown, and milled rice during adsorption with constant

diffusivity. Their predictions agreed well with the experimental adsorption data.

temperature. This has been demonstrated during drying of rough rice [32,33,34];

brown and milled rice [35]; peanut [18]; corn [21]; and wheat [36]. Chu and Hustrulid

medium and moisture content of corn during drying. Recently, Lu and Siebenmorgen

[31] described the dependency of diffusivity of rough, brown and milled rice on

temperature by an Arrhenius-type function during adsorption. Further they agreed

diffusivity may depend on moisture content of a material and follow an exponential

relative humidity of the environment is not clearly understood.

3.4.2.1 Germ

Three types of samples, namely, corn germ, soft corn (FR27 × MO17), and hard

corn (P3576) were tested. Corn germ obtained from Archer Daniels Midland (ADM)

Company was used. ADM used steam tube dryers to remove moisture from the corn

germ. The germ had oil content of 44–48% and initial moisture content of 3–4%

[38]. Two types of corn, namely soft and hard, of different densities 1229 and 1327

kg/m3 respectively, were used in this study. The soft corn was grown on the Agri-

cultural Research Station Farm at the Purdue University, West Lafayette, IN and

combine-harvested at about 27% moisture content during Fall 1990. The hard corn

was obtained from Frito-Lay Inc., Sidney, IL at about 15% moisture content during

55534_C003.indd 6255534_C003.indd 62 10/22/08 8:27:56 AM10/22/08 8:27:56 AM

endosperm fills the crown (upper part) of the kernel, extends downward to surround

[5] verified that liquid diffusivity of rice components did not vary with the initial

The fluid parameters affecting moisture diffusion are temperature and RH.

Diffusion of moisture is generally enhanced by the temperature of the fluid medium

(air) and has an exponential relationship (Arrhenius-type) with the inverse of the fluid

[37] reported the diffusion coefficient as a function of temperature of the fluid

that the relative humidity had some influence on moisture diffusivity. Therefore, the

relationship with fluid temperature. But the dependence of diffusivity of grains on



Spring 1991. The corn samples were dried using natural air at a temperature of 23°C

and RH of 55%.

The dried corn samples were hand-cleaned to remove the broken kernels. The

moisture content of the samples was determined, by the oven method [38], to be

about 9–10%. The samples were stored in a refrigerator maintained at 5°C and

58% RH until the experiments. The adsorption tests were conducted in a controlled

environment chamber (2.21 × 0.74 ×1.95 m) available in the Biotron at the Univer-

sity of Wisconsin-Madison. The environment for this experiment consisted of four

air temperatures of 25, 30, 35, and 40°C with each at two RH values of 75 and 90%.

Temperature and RH of the air in the chamber were maintained within 0.1°C and

1.0%, respectively. Air was circulated constantly at 0.5 m/s during the tests.

Two 50g samples of each variety of corn and 25g samples of germ were placed

in individual perforated wire-meshed containers. The individual sample depth was 10

mm. This depth was chosen in order to acquire thin-layer adsorption data. The con-

tainers were individually supported with a load cell (Omega Model No. LCL-227G;

rated capacity of 227 g). The output signals from the load cells were transmitted to a

high-speed analog I/O board (DAS16G1, National Instruments) and a personal com-

digital data from the output signal were acquired through a DAS16G1 interface card.

The data were saved on the computer using the software EASYEST LX. The volt-

age values from the load cell were converted to the corresponding mass values after

calibrating each load cell with a set of known masses. The weight measurements were

taken at 15-min interval. The tests were conducted for 48–72 h during which the sam-

ple moisture content reached near-equilibrium with the chamber.

Fick’s law of diffusion model considers the geometry of corn germ and corn kernel

multivariate secant method [39]. The moisture diffusivity was estimated by minimiz-

ing the sum of square deviations (SSD) between the experimental and theoretical corn

adsorption data. The characteristic dimensions of the germ and corn kernel were deter-

mined at the initial moisture content. Further details of the dimension measurement

can be found in Muthukumarappan and Gunasekaran [22]. As a preliminary analysis,

the diffusivities of corn germ and corn kernels exposed to air at 25°C and 90% RH

tion behavior of corn germ and corn kernels. Therefore, the moisture diffusivity of

corn germ and corn kernels at other humid conditions was determined using only the

0.99, 2.08 and 2.40 mm for corn germ, kernels of FR27 × MO17 and P3576 variety

corn, respectively. A multifactor analysis of variance [40] was performed to study the

effects of air temperature, relative humidity, and type of corn on moisture diffusivity.

moisture at the kernel surface is approaching equilibrium with the environment.

In selecting the K values, it was assumed that the moisture content at the surface

reaches equilibrium halfway through the adsorption process, i.e. MR = 0.5. This

assumption is reasonable because the diffusivity is traditionally determined by cal-

culating the time taken to reach MR = 0.5 [26]. Moreover, the effect of varying MR

55534_C003.indd 6355534_C003.indd 63 10/22/08 8:27:57 AM10/22/08 8:27:57 AM

puter (Diversified Systems 486) through an Expansion Multiplexer (EXP-16). The

as an infinite slab, an infinite cylinder, and a sphere used for diffusivity determination.

The first ten terms of each model were considered using the non-linear, least square

were determined using infinite slab, infinite cylinder, and sphere models. Based on the

SSD values it was found that the infinite slab model best predicts the moisture adsorp-

infinite slab model. The characteristic dimensions used in the infinite slab model were

The surface adsorption coefficient (K) in Equation 3.6 indicates how fast the



Therefore, the time taken to reach the moisture ratio of 0.5 was determined from

the experimental data. This time value was substituted in Equation 3.6 to obtain

the K value. At this time the surface moisture ratio should be, theoretically, unity.

However, a value of MRs = 0.998 was assumed since MRs = 1 only at t = ∞. Values

of MRs ranging from 0.99 to 0.999 were evaluated and it was found that MRs = 0.998

conditions are presented in Table 3.1. Using the K values in Table 3.1, the surface

moisture values were calculated as a function of time. These surface moisture val-

diffusivity of corn samples with varying surface moisture content was estimated

moisture ratio) and the theoretical data using the non-linear, least square multivari-

ate secant method [39].

The moisture ratio for corn germ and corn samples (FR27 × MO17) exposed to

air at 35°C and 75% RH are presented in Figure 3.2. Due to the smaller size of corn

germ compared to the whole kernel, the germ approached the equilibrium moisture

the germ than for the corn samples. Compared to the experimental values for both

corn germ and corn kernels the model initially overpredicted and then underpre-

dicted the moisture ratio.

The moisture diffusivity values obtained for corn germ and composite corn kernel

at different humid air conditions are presented in Table 3.2. In general, the moisture

diffusivity increased with increasing temperature. In classical theory [41], increased

temperature is interpreted to mean an increase in the average energy for each mode

of motion of vapor (translational, rotational and vibrational motions). Therefore, an

increase of temperature must mean an increase in the probability that the energy

of the mode of motion required for interaction will attain a high value more fre-

quently. The moisture diffusivity of germ decreased with increasing RH. During the

adsorption tests, the concentration gradient is higher at higher RH. This higher con-

centration gradient should lead to higher moisture diffusivity values. But the opposite

trend was observed. At this time there is no conclusive explanation for this trend. The

effective moisture diffusivity of the composite corn kernels. This could be due to

TABLE 3.1−1) Values Used in Equation 3.6

Corn without Pericarp Corn with Pericarp

Temperature (oC) Germ FR27×MO17 P3576 FR27×MO17 P3576

25 0.776 0.672 0.388 0.487 0.371

30 1.381 1.036 0.829 0.606 0.592

35 1.776 1.130 0.888 0.710 0.777

40 2.072 1.243 1.036 1.036 0.921

55534_C003.indd 6455534_C003.indd 64 10/22/08 8:27:58 AM10/22/08 8:27:58 AM

(MR = 0.6, 0.7, 0.8 and 0.9) on the goodness of fit was also investigated. From the

preliminary investigation it was found that MR = 0.5 assumption best fitted the data.

gave the best fit to the adsorption data. The values of K for all the environmental

ues were then used to calculate the modified moisture ratio with time. The moisture

by minimizing the sum of square deviations between the experimental (modified

moisture diffusivity values of corn germ were about two to five times lower than the

Surface Adsorption Coefficient (K, h

content more rapidly than the corn. This is reflected by the higher moisture ratio for



presence of oil in the germ. The difference was higher at low temperatures (25 and

30°C) and lower at high temperatures (35 and 40°C). From the multifactor analysis of

variance tests, it was found that the differences in moisture diffusivity values obtained

0.05 level. The mean moisture diffusivity value of corn germ varied from 0.15 ×10−7

to 1.18 ×10−7 m2/h for different air temperature and RH conditions.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 10 20 30 40

Time, h

Corn germ

Corn

Model predicted

Modif

ied m

ois

ture

rat

io

FIGURE 3.2

moisture content assumption.

TABLE 3.2Moisture Diffusivities (m2/h) of Pericarp, Germ, Soft and Hard Endosperms of a Corn

Adsorption Condition Moisture Diffusivity

Pericarp (×10−8)

Germ (×10−7)

Soft Endosperm

(×10−7)

Hard Endosperm

(×10−7)Composite

(×10−7)Temperature (oC) RH (%)

25 75 0.41 0.27 0.825 0.450 0.97

80 0.34 0.17 0.733 0.320 0.68

90 0.30 0.15 0.546 0.420 0.60

30 75 0.45 0.54 1.014 0.652 1.01

80 0.42 0.20 0.997 0.566 0.90

90 0.42 0.18 0.923 0.549 0.78

35 75 0.52 0.66 1.245 0.733 1.24

80 0.49 0.33 1.142 0.680 1.20

90 0.47 0.24 1.056 0.639 0.88

40 75 0.57 1.18 1.460 0.919 1.40

80 0.53 0.40 1.221 0.687 1.32

55534_C003.indd 6555534_C003.indd 65 10/22/08 8:27:58 AM10/22/08 8:27:58 AM

Modified moisture ratio for corn germ and corn kernels exposed to air at 35°C

and 75% RH. The Fick’s analytical model was used for an infinite slab with varying surface

at different air temperatures, RH, and type of corn were statistically significant at the



The moisture diffusivity values of corn germ obtained during the adsorption

study are lower than the published values during drying (desorption). For example,

within the 10–24% moisture content range, reported diffusivity values of germ

ranged from 5.974 ×10−7 to 34.731×10−7 m2/h [6]. In general, the Fick’s diffusion

model better predicted the adsorption of corn germ than that of the composite corn

kernel. One of the possible reasons is that the germ is more homogeneous than the

composite corn kernel.

Temperature dependency of the moisture diffusivity of corn germ was modeled

as an Arrhenius-type function used by Lu and Siebenmorgen [31]:

D ABT

= exp −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

a

(3.17)

2

tion) values are summarized in Table 3.3. This model was satisfactory as evidenced

by high R2 values.

3.4.2.2 Pericarp

Two types of corn were used namely soft (FR27 × MO17) and hard (P3576) with den-

sities of 1229 and 1327 kg/m3, respectively. To remove the corn pericarp, preliminary

TABLE 3.3

a

for Temperature Dependency of Moisture Diffusivity of Corn Germ, Pericarp, Soft and Hard Endosperms

Component RH (%) A B R2

75 1.122 × 105 8645.3 0.964

Germ 80 3.568 5727.4 0.956

90 0.028 4313.9 0.981

75 4.917 × 10−7 2116.1 0.991

Pericarp 80 3.961 × 10−6 2783.3 0.967

90 3.293 × 10−4 4137.1 0.928

75 9.338 × 10−3 3463.7 0.996

Soft endosperm 80 2.795 × 10−3 3127.4 0.911

90 0.418 × 102 6081.5 0.901

75 3.282 × 10−5 1884.8 0.991

Hard endosperm 80 0.2243 4657.7 0.903

90 1.803×10−2 3862.3 0.978

* D = moisture diffusivity (m2/h); Ta = absolute temperature (K).

55534_C003.indd 6655534_C003.indd 66 10/22/08 8:27:59 AM10/22/08 8:27:59 AM

The model coefficients A and B and the corresponding R (coefficient of determina-

Coefficients of Arrhenius-type Model (D =A exp(−B/T )*)

Coefficients



experiments were conducted by soaking corn kernels in 25°C water for 15, 30, 60, and

120 s. The pericarp was carefully removed using a razor blade. Corn kernels soaked

for 30 s absorbed less than 1% moisture and facilitated easy removal of the pericarp.

The adsorption tests were conducted as explained in Section 3.4.2.1.

The corn kernel was modeled as a slab core (corn without pericarp composed of

soft and hard endosperms and germ) surrounded by a slab shell (pericarp) as shown

in Figure 3.3. First, adsorption of corn without pericarp (the slab core) was pre-

dicted numerically by solving the diffusion equation (in Cartesian coordinates) and

the diffusivity of corn without pericarp was determined. Next, adsorption of corn

with pericarp (the slab core and shell) was predicted numerically. The diffusivity of

corn pericarp was determined using the diffusivity of corn without pericarp and the

experimental adsorption data for corn with pericarp.

The following assumptions were made in developing the adsorption models:

1. Mechanism of moisture transport is diffusion.

2. Corn is isothermal during adsorption; i.e. the heat transfer equations may

be neglected.

Material 1

Corn without pericarp

Material 2

pericarp

D1

l1

l1

Δ Δ l2

D2

j + 1jj – 1

l2

FIGURE 3.3 Schematic of a corn kernel modeled as a slab core (corn without pericarp) and

slab shell (pericarp).

55534_C003.indd 6755534_C003.indd 67 10/22/08 8:28:00 AM10/22/08 8:28:00 AM



dimensional and end effects may be neglected.

4. Moisture diffusivity is constant throughout the adsorption process.

5. Germ, soft and hard endosperms, and pericarp are homogeneous and

isotropic.

6. Expansion of the corn kernel during adsorption is negligible.

The differential equation with initial and boundary conditions as shown in Equa-

equations proposed by Crank and Nicolson [42] and summarized by Strikwerda [43]

were used to solve the differential equations. Due to symmetry, only one-half of the

whole system was considered for the analysis (Figure 3.3). Equation 3.1 should satisfy

Muthukumarappan [8] the nodal moisture contents were predicted.

The average moisture content of the corn can be obtained by volume averag-

ing the nodal moisture content values. Therefore, the moisture ratio of corn during

adsorption can be stated as a function of time (t):

vM(x)dv M

M Mf (t)

∫ −−

0

e 0

= (3.18)

the Thomas algorithm [43]. One program was used to determine the diffusivity of

corn without pericarp and the other program was used to determine the diffusivity of

corn pericarp. The listing of these programs can be found in Muthukumarappan [8].

Space intervals of 0.01 mm and 0.001 mm were used for corn without pericarp and

corn pericarp, respectively. An interval of 0.25 h was used for the time marching.

The thicknesses of corn with and without pericarp were determined for 50 kernels

each using a micrometer. They were 4.16 and 4.04 mm for the soft corn with and

without pericarp and 4.80 and 4.66 mm for the hard corn with and without peri-

carp, respectively. Moisture ratio was calculated from the average moisture content.

The moisture diffusivity was estimated by minimizing the sum of squares deviation

between the experimental and predicted moisture ratio data. A multifactor analysis

of variance [40] was performed to study the effects of air temperature, RH, and peri-

carp on moisture diffusivity.

which the kernels surface moisture approaches the equilibrium moisture as described

in Section 3.4.2.1. The values of K for all the environmental conditions (Table 3.1)

were used to calculate the surface moisture contents as a function of time. These

surface moisture values were then used in the numerical model as the time-varying

boundary condition.

The variation of moisture ratio with time for corn samples (FR27 × MO17)

exposed to air at 35°C and 75% RH, is presented in Figure 3.4. The corn samples

without pericarp attained higher moisture ratios than the samples with pericarp.

Compared with the experimental data for corn samples without pericarp the model

55534_C003.indd 6855534_C003.indd 68 10/22/08 8:28:01 AM10/22/08 8:28:01 AM

3. Geometrically, the corn kernel is an infinite slab, i.e. the diffusion is one-

tions 3.1 through 3.4 were used with n = 2. The Crank–Nicolson finite difference

every point inside the system. Using the finite difference formulations explained in

Two FORTRAN programs were written to solve the finite difference equations using

The surface adsorption coefficient (K) was determined to represent the rate at

overpredicted during the first 10 h of adsorption and then underpredicted. But the



opposite trend was obtained for the samples with pericarp. This might be related to

the resistance of the pericarp to moisture movement interacting with the boundary

condition.

The moisture diffusivity values of pericarp at different humid air conditions are

presented in Table 3.2. In general, the moisture diffusivity decreased with increasing

RH and increased with increasing temperature. Similar trends were observed for corn

germ and composite corn kernels during adsorption tests and possible explanations

for these trends can be found in Section 3.4.2.1. The moisture diffusivities of compos-

ite corn samples were much higher (about two orders of magnitude) than the pericarp.

This shows that the pericarp offers substantial resistance to moisture migration into

corn kernels.

From the multifactor analysis of variance test, it was found that the differences

in moisture diffusivity values between air temperatures, RH, and corn without

ture diffusivity of pericarp varied from 0.30 ×10−9 to 0.57 ×10−9 m2/h for different

air temperature and RH conditions. Diffusivity values of the pericarp obtained in

this adsorption study are lower than the published values during drying (desorp-

tion). For example, within 10–24% moisture content, reported diffusivity values of

pericarp varied from 0.568 ×10−7 to 3.299 ×10−7 m2/h [6]. The moisture diffusivi-

ties of pericarp were much lower (about two orders of magnitude) than the germ

(Table 3.2). The mean moisture diffusivity of corn germ varied from 0.15 ×10−7 to

1.18 ×10−7 m2/h for different air temperature and RH conditions. This shows that

the pericarp offers higher resistance to moisture movement into corn kernels than

the germ.

2

tions are summarized in Table 3.3. In view of the high R2 values, the Arrhenius-type

model satisfactorily described the temperature dependency of diffusivity.

Without pericarp

ExperimentalPredicted

With pericarp

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Mois

ture

rat

io

0.00 10 20 30 40

Time, h

Experimental

Predicted

FIGURE 3.4 Experimental moisture ratio for corn samples (FR27 × MO17) exposed at 35°C

55534_C003.indd 6955534_C003.indd 69 10/22/08 8:28:01 AM10/22/08 8:28:01 AM

and 75% RH and predicted moisture ratio of corn samples using finite difference method.

pericarp and pericarp were statistically significant at 0.05 level. The mean mois-

Temperature dependency of the diffusivity of corn pericarp was fitted to the

Arrhenius-type model shown in Equation 3.17. The model coefficients and the corre-

sponding R (coefficient of determination) values for corn pericarp at all RH condi-



3.4.2.3 Soft and Hard Endosperms

The diffusivities of the soft and hard endosperm were determined using Equation

3.10 along with the adsorption data for soft (FR27 × MO17) and hard (P3576) corn.

The details of sample preparation and adsorption tests are presented in Section

3.4.2.1. The mass of individual components was determined by carefully breaking

of moisture in the individual components was estimated from the total amount of

moisture in each corn type (Si) and the component mass fraction (Xij). The soft corn

was composed of 5% pericarp, 10% germ, 48% soft endosperm, and 37% hard endo-

sperm. The hard corn was composed of 4% pericarp, 9% germ, 21% soft endosperm,

and 66% hard endosperm [8]. It was assumed that the differences in moisture diffu-

sion between the two types of corn were due to different amounts of soft and hard

endosperms in both types of corn. The amount of moisture in each component (Mj)

was determined by normalizing the total mass of both types of corn to the compo-

nent’s total mass as:

M S X

Xji ij

ji

=∑∑

(3.19)

The diffusivities of the soft and hard endosperm were determined using Equation

3.19 along with the adsorption data for soft (FR27 × MO17) and hard (P3576) corn.

The mass of individual components was determined by carefully breaking and

of moisture in the individual components was estimated based on the component

mass fraction (X11, X12, X13 and X21, X22, X23 for germ, soft and hard endosperms of

each type of corn). Then, the amount of moisture in each component was determined

by normalizing the total mass of both types of corn to the component’s total mass

as explained below:

If C1 = total amount of moisture in corn type 1;

C2 = total amount of moisture in corn type 2;

X11 = mass fraction of germ in corn type 1;

X12 = mass fraction of soft endosperm in corn type 1;

X13 = mass fraction of hard endosperm in corn type 1;

X21 = mass fraction of germ in corn type 2;

X22 = mass fraction of soft endosperm in corn type 2;

X23 = mass fraction of hard endosperm in corn type 2;

then the

amount of moisture in germ =++

C X C X

X X1 11 2 21

11 21

(3.20)

amount of moisture in soft endosperm =++

C X C X

X X1 12 2 22

12 22

(3.21)

55534_C003.indd 7055534_C003.indd 70 10/22/08 8:28:02 AM10/22/08 8:28:02 AM

and weighing the component fragments from five kernels for each corn type. Amount

weighing the component fragments from five kernels from each corn type. Amount



amount of moisture in hard endosperm =++

C X C X

X X1 13 2 23

13 23

(3.22)

This procedure can be used for other grains and food materials. Recently, Kang and

Delwiche [44] used this approach for modeling the moisture diffusion in wheat ker-

nels during soaking. In determining the soft and hard endosperms diffusivity values,

element solutions developed by Muthukumarappan [8] the nodal moisture content of

a corn kernel during adsorption was predicted. The average kernel moisture content

(as distinguished from the nodal moisture values) is estimated to be the mass aver-

age value. Assuming constant density, the mass average moisture of a body (M) is

MM x y dm

dmtV

V

= ∫∫

( , )for every Δ (3.23)

mine the diffusivity of individual components, and (2) simulate the moisture adsorption

of grains. The listing of the program is presented in Muthukumarappan [8].

element predicted moisture content data of the individual components. A corn kernel

without pericarp was considered for the analysis. The cross-section of the corn ker-

nel with four distinct regions of germ, pericarp, soft and hard endosperm is shown

in Figure 3.5. The cross-section presented is through the narrowest dimension of the

kernel. The two-dimensional cross section was selected based on average dimension

of two hard corn kernels. Two hard corn kernels were cut through the narrowest

dimension of the kernel. Then the cut kernels were mounted on 10 ×10-mm alu-

minum cylindrical stubs using double-sided sticky tape. Further, silver paint was

applied around the sides of the kernel. The mounted samples were sputtered with

gold to a thickness of about 270°A using a Bio-Rad Polaron Division Gold Coater

(Model E5000M SEM Coater). The samples were examined in a scanning electron

microscope (Model Hitachi S-570) at an accelerating potential of 10 kV and corre-

is shown in Figure 3.6. The two-dimensional model in Cartesian co-ordinates con-

sists of 53 elements.

The diffusivity values of the germ determined previously (Section 3.4.2.1)

using the analytical model was used in the three component FEM. Since we

have two variables to optimize (the diffusivity of soft and hard endosperm), two

sperm model’, an initial diffusivity value of the soft endosperm was assumed

and the hard endosperm diffusivity value was predicted. And for the second

model, called ‘soft endosperm model’, the previously estimated hard endosperm

55534_C003.indd 7155534_C003.indd 71 10/22/08 8:28:03 AM10/22/08 8:28:03 AM

the finite element model described in the previous Section 3.3.2 was used. Four-noded

quadratic finite elements were used for discretization of the domain. Using the finite

defined by Haghighi and Segerlind [14] as

A computer program for two-dimensional steady-state field problems written by

Segerlind [45] was modified to solve the time-dependent diffusion problem. The modi-

fied computer program was written in Fortran77. This program can be used to (1) deter-

The diffusivity values were estimated by optimizing the experimental and finite-

sponding dimensions were determined. A finite-element discretization of the kernel

models were considered simultaneously. For the first model, called ‘hard endo-



diffusivity was used and the new soft endosperm diffusivity was predicted. This

procedure was repeated until the sum of square deviations (SSD) between the

experimental and predicted moisture data was minimized. A subroutine based

on the Gold Section search method [46] was used to optimize the diffusivity

evaluation process.

The experimental and FEM predicted moisture ratios of soft and hard endo-

sperms, exposed to air at 35°C and 90% RH, are presented in Figure 3.7. The moisture

ratio of soft endosperm was higher than that of hard endosperm at intermediate

times. The difference in moisture diffusion rates between the soft and hard endo-

sperms might be due to the packing of starch granules in both types of endosperm.

The starch granules within the hard endosperm cells are small and tightly packed

compared to large and loosely organized granules in the soft endosperm cells. The

soft endosperm might have more sorptive sites for water vapor compared to the hard

endosperm. The moisture diffusivity values of soft and hard endosperm for all the

humid air conditions are presented in Table 3.2.

The moisture diffusivity of hard endosperm was lower than the soft endosperm.

The average diffusivity of soft and hard endosperms increased with increasing air

temperature and decreased with increasing air relative humidity. These trends are

similar to those for corn germ and corn pericarp (Section 3.4.2.1 and Section 3.4.2.2)

Soft endosperm

Hard

endosperm

Germ

0 20

1

2

3

4

5

6

7

8

9

10

11

Len

gth

, m

m

Thickness, mm

4

Hard

endosperm

FIGURE 3.5 Cross-section of a corn kernel through its narrowest dimension.

55534_C003.indd 7255534_C003.indd 72 10/22/08 8:28:04 AM10/22/08 8:28:04 AM



Soft endosperm

Hard endosperm

Predicted

0 100.0

0.1

0.2

Mois

ture

rat

io

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

20 30 40

Time, h

FIGURE 3.7 Moisture ratio of soft and hard endosperms exposed to air at 35°C and 75%

11

10

9

8

7

6

5

4

3

2

1

00 2 4

Thickness, mm

Len

gth

, m

m

FIGURE 3.6 Finite element discretization of the corn kernel.

55534_C003.indd 7355534_C003.indd 73 10/22/08 8:28:04 AM10/22/08 8:28:04 AM

RH using finite element method.



and have been further explained in Section 3.4.2.1. From the multifactor analysis of

variance [40], it was found that the differences in moisture diffusivity between soft

and hard endosperms for different air temperatures and relative humidity were sta-

The mean moisture diffusivity of the soft endosperm exposed to air at 35°C and

80% RH (1.14 ×10−7 m2/h) is the largest, followed by hard endosperm (0.68 ×10−7

m2/h), germ (0.33 ×10−7 m2/h) and pericarp (0.49 ×10−9 m2/h). From these results,

it is evident that the pericarp offers the most resistance to moisture diffusion fol-

lowed by germ, hard endosperm, and soft endosperm. Based on these values, the

diffusivity of composite corn is expected to be less than 1.14 ×10−7 m2/h. However,

the mean moisture diffusivity of composite corn kernels exposed to air at 35°C and

80% RH is 1.20 ×10−7. This value is higher than expected because it was obtained via

and Gunasekaran [22] compared the diffusivities of corn kernels for three differ-

ent geometry representations. They found that the diffusivity values of corn kernels

allow for a better comparison.

the Arrhenius-type model shown in Equation 3.17 for corn germ and pericarp. The 2

corn endosperm at all RH conditions are summarized in Table 3.3. In view of the

high R2 values, the Arrhenius-type model satisfactorily described the temperature

dependency of diffusivity.

3.4.3 FINITE ELEMENT SIMULATION OF CORN MOISTURE ADSORPTION

A corn variety of FR27 × MO17 was used. The FEM described in the previous

Section 3.4.2.3 was used to simulate the moisture diffusion into a corn kernel. A

sional model in Cartesian co-ordinates consists of 85 elements. The diffusivity

values obtained from the experimental data (Section 3.4.2) were used along with

The corn moisture adsorption was simulated using the FEM and analytical

model. The composite moisture diffusivity values reported in Table 3.2 were used

in the analytical model to simulate the corn moisture adsorption. The experimen-

tal, analytical and FEM simulated moisture ratio of FR27 × MO17 corn samples

exposed to air at 35°C and 75% RH are presented in Figure 3.8. In general, FEM

simulated the experimental moisture ratio very well. The analytical model poorly

tion (MSSD) was used as an indicator to determine the prediction accuracy of the

models studied. Based on the MSSD values, the FEM predictions were clearly

better than the corresponding analytical solutions. This may be because individual

55534_C003.indd 7455534_C003.indd 74 10/22/08 8:28:05 AM10/22/08 8:28:05 AM

tistically significant at the 0.05 level.

one-dimensional analytical models rather than the finite element analysis. Moreover,

the analytical model assumed a regular infinite slab geometry for a corn kernel and

the finite element analysis assumed an actual irregular shape. Muthukumarappan

using the infinite slab geometry were about 1 to 2.5 times the diffusivity values using

the infinite cylinder geometry. Thus the use of more nearly identical models would

Temperature dependency of the diffusivity of corn endosperms were fitted to

model coefficients and the corresponding R (coefficient of determination) values for

finite-element discretization of the kernel is shown in Figure 3.6. The two dimen-

the necessary initial and boundary conditions for the finite element simulation.

over-predicted the finite element model in the early stage of adsorption and under-

predicted the model in the final stage of adsorption. Mean sum of squares devia-



component moisture diffusivities of corn were considered for the FEM while com-

posite moisture diffusivity was considered for the analytical model. Ruan et al.

process using a magnetic resonance imaging technique. From the images they

between the germ and endosperm, and through the cross and tube cells of the

pericarp layers. Then it quickly diffused into the germ, and slowly diffused into

the endosperm. From these observations it is clear that the moisture diffusion in a

corn kernel is a complex phenomena and more work is needed to better understand

this behavior.

The FEM predicted nodal moisture contents were transformed to contour plots

sure to air at 35°C and 90% RH is presented in Figure 3.9. The moisture gradient

between the center and surface of corn kernels during simulated moisture adsorption

at 25, 30, and 35°C each at 90% RH is presented in Figure 3.10. In general, the mois-

ture gradient inside a corn kernel during adsorption at 25°C was lower than at 35°C.

of adsorption (up to 5 h). This is because the different moisture diffusivity values

and varying boundary condition were used in the simulation model. The moisture

gradient between the center and boundary of a corn kernel exposed to air at 25, 30,

and 35°C each at 90% RH was about 4% after 1 h, reaching a maximum of about 9%

after 7.5 h, and declined during subsequent adsorption. These times compare well

with Sarwar and Kunze’s [1] experimental observations; the corn samples took about

8 h of adsorption. This shows that the difference in moisture gradient may cause the

was maximum.

1.0

0.8

0.6

0.4

0.2

0.00 10 20 30 40

Time, h

Experimental

Finite element

Analytical

Mois

ture

ratio

FIGURE 3.8 Moisture ratio of corn kernels exposed to air at 35°C and 75% RH.

55534_C003.indd 7555534_C003.indd 75 10/22/08 8:28:06 AM10/22/08 8:28:06 AM

[47] presented 3-D transient moisture profiles of corn kernels during a steeping

reported that the steepwater moved first into the corn kernel through the space

using Surfer software [48]. The moisture profiles for corn samples after 1 h of expo-

The temperature effect on moisture gradient was significant during the early stage

1 h of exposure for fissures to start developing when exposed to 92% RH at 21°C.

Further they reported that all the kernels exposed to 92% RH at 21°C fissured within

kernels to fissure. In addition, all the kernels may fissure when the moisture gradient



3.5 RECOMMENDATIONS

Future research could be conducted in developing a model to predict possible fail-

A simultaneous heat and mass transfer model could also be developed to predict

9.00

8.00

7.00

6.00

5.00

4.00

3.00

Kern

el le

ngth

, m

m

2.00

1.00

0.000.00 1.00 2.00 3.00 4.00

Kernel thickness, mm

10.2

9.2

9.2

9.2

11.2

FIGURE 3.9 35°C and 90% RH during adsorption.

55534_C003.indd 7655534_C003.indd 76 10/22/08 8:28:06 AM10/22/08 8:28:06 AM

Moisture profile (% wb) within a corn kernel after 1 h of exposure to air at

ures in grains during adsorption and needs to be verified with experimental results.



temperature and humidity conditions. A storage model may be developed using the

thin-layer moisture adsorption models presented in this chapter.

NOMENCLATURE

[C] Global moisture capacitance matrix

[Cij] Element moisture capacitance matrix

Dm Mass of an element

D Diffusivity, m2/h−1

[K] Global moisture conductance matrix

[Kij] Element moisture conductance matrix

M Mass average moisture content of a body

Mt Moisture content at time t (h), % wb

M0 Average initial moisture content of the kernel, % wb

Me Equilibrium moisture content of the kernel, % wb

Ms Surface moisture content of the kernel, % wb

t Adsorption time, h

x,y Cartesian co-ordinates

Δt Time step, h

Ω Boundary surface of the body

D1 Diffusivity of corn without pericarp, m2/h

10

8

6

4

2

0

0 10 20 30 40 50 60 70

Time, h

25°C

30°C

35°C

Mois

ture

gra

die

nt, %

wb

FIGURE 3.10 Moisture gradient (% wb) within a corn kernel with time when exposed to 25,

30, and 35°C each at 90% RH air condition.

55534_C003.indd 7755534_C003.indd 77 10/22/08 8:28:07 AM10/22/08 8:28:07 AM

the temperature and moisture profiles inside a grain during adsorption at different

K Surface adsorption coefficient, h



D2 Diffusivity of pericarp, m2/h

Dm Diffusivity of material m, m2/h

j−1,j,j + 1

l1 Half-thickness of corn without pericarp, m

l2 Half-thickness of corn, m

m Number of components included in the adsorption model

RH Relative humidity, %

Δl1,Δl2 Space interval of corn without pericarp and pericarp, mm

F Varying boundary condition

{M} Nodal moisture values at time t, %{M} Nodal moisture values at time t + Δt, %Mj Amount of moisture in each component jMR(t) Moisture ratio as a function of time, (M−M0)/(Me−M0)

MRi(t) Moisture ratio of corn type iMRj(t) Moisture ratio of the jth component

Si Total amount of moisture in each corn type iTa Absolute temperature, K

Xij Ratio of the mass of the jth component to total mass for each corn type i, fraction

REFERENCES

1. G Sarwar, and OR Kunze. 1989. Relative humidity increases that cause stress cracks in

corn. Transactions of the ASAE 32(5): 1737–43.

enced by drying and rehydration processes. Transactions of the ASAE 25(3): 768–72.

3. S Gunasekaran, and MR Paulsen. 1985. Breakage resistance of corn as a function of

drying rates. Transactions of the ASAE 28(6): 2071–76.

4. DS Chung, and HB Pfost. 1967. Adsorption and desorption of water vapor by cereal

grains and their products. Part I, III. Transactions of the ASAE 10(4): 549–51, 555–57.

5. JF Steffe, and RP Singh. 1980. Liquid diffusivity of rough rice components. Transactions of the ASAE 23(3): 767–74, 782.

yellow-dent corn components. Transactions of the ASAE 30(2): 522–28.

7. MK Misra. 1978. Thin-layer drying and rewetting equations for shelled yellow corn.

PhD thesis, University of Missouri-Columbia, MO.

8. K Muthukumarappan. 1993. Analysis of Moisture diffusion in corn kernels during

adsorption. PhD thesis, University of Wisconsin-Madison, WI.

9. S Bruin, KChAM Luyben. 1980. Drying of food materials: A review of recent devel-

opments. Advances in Drying. Vol. 1, 155–215. New York: Hemisphere Publishing

Corporation.

10. JK Wang, and CW Hall. 1961. Moisture movement in hygroscopic materials: A math-

ematical analysis. Transactions of the ASAE 4(1): 33–36.

11. TB Whitaker, HJ Barre, and MY Hamdy. 1969. Theoretical and experimental studies

the ASAE 12(5): 668–72.

ear corn-I: Mathematical description of the moisture history of fully-exposed ears of

corn. ASAE Paper No. 78–6005. St. Joseph, MI: ASAE.

55534_C003.indd 7855534_C003.indd 78 10/22/08 8:28:08 AM10/22/08 8:28:08 AM

Spatial nodes defined in Figure 3.1

6. MA Syarief, RJ Gustafson, and RV Morey. 1987. Moisture diffusion coefficients for

of diffusion in spherical bodies with a variable diffusion coefficient. Transactions of

2. GM White, IJ Ross, and CG Poneleit. 1982. Stress crack development in popcorn as influ-

12. YI Sharaf-Eldeen, JL Blaisdell, and MY Hamdy. 1978. Factors influencing drying of



13. JH Young. 1969. Simultaneous heat and mass transfer in a porous, hygroscopic solid.

Transactions of the ASAE 12(5): 720–25.

14. K Haghighi, and LJ Segerlind. 1988. Modeling simultaneous heat and mass transfer

629–37.

15. J Irudayaraj, K Haghighi, and RL Stroshine. 1992. Finite element analysis of drying

with application to cereal grains. Journal of Agriculture Engineering Research 53:

209–29.

16. J Irudayaraj, and Y Wu. 1995. Effect of pressure on moisture transfer during moisture

adsorption. Drying Technology 13: 1603–17.

17. JH Young, and TB Whitaker. 1971. Evaluation of the diffusion equation for describing

thin-layer drying of peanuts in the hull. Transactions of the ASAE 14(2): 309–12.

18. TB Whitaker, and JH Young. 1972. Simulation of moisture movement in peanut ker-

nels: Evaluation of the diffusion equation. Transactions of the ASAE 15(1): 163–66.

19. SR Eckhoff, and MR Okos. 1989. Diffusion of gaseous sulfur dioxide into corn kernels.

Cereal Chemistry 66(1): 30–33.

20. SR Eckhoff, and MR Okos. 1990. Sorption kinetics of sulfur dioxide on yellow dent


21. LR Walton, GM White, and IJ Ross. 1988. A cellular diffusion-based drying model for


22. K Muthukumarappan, and S Gunasekaran. 1990. Vapor diffusivity and hygro-

scopic expansion of corn kernels during adsorption. Transactions of the ASAE 33(5):

1637–41.

23. K Muthukumarappan, and S Gunasekaran. 1994. Moisture diffusivity of corn kernel

components during adsorption Part I: Germ. Transactions of the ASAE 37(4): 1263–68.

24. K Muthukumarappan, and S Gunasekaran. 1994. Moisture diffusivity of corn kernel com-

ponents during adsorption Part II: Pericarp. Transactions of the ASAE 37(4): 1269–74.

25. K Muthukumarappan, and S Gunasekaran. 1994. Moisture diffusivity of corn kernel

components during adsorption Part III: Soft and Hard Endosperms. Transactions of the ASAE 37(4): 1275–80.

26. J Crank. 1975. The mathematics of diffusion. 2nd ed. London: Oxford University

Press.

27. AB Newman. 1931. The drying of porous solids: Diffusion and surface emission equa-

tions. Transactions of the AICHE 27: 203–20.

28. US Shivhare, GSV Raghavan, and RG Bosisio. 1991. Modeling of microwave-drying of

corn through diffusion phenomena. ASAE Paper No. 91–3520. St. Joseph, MI.

29. DH Chittenden, and A Hustrulid. 1966. Determining drying constants for shelled corn.


30. KH Hsu, CJ Kim, and LA Wilson. 1983. Factors affecting water uptake of soybeans

during soaking. Cereal Chemistry 60(3): 208–11.

31. R Lu, and TJ Siebenmorgen. 1992. Moisture diffusivity of long-grain rice components.


32. CY Wang, and RP Singh. 1978. A single layer drying equation for rough rice. ASAE

Paper No. 78–3001. St. Joseph, MI.

33. WJ Chancellor. 1968. Characteristics of conducted-heat drying and their comparison

with those of other drying methods. Transactions of the ASAE 11(6): 863–67.

Journal of Agriculture Engineering Research 27(6): 489–93.

35. JF Steffe, and RP Singh. 1980. Diffusivity of starchy endosperm and bran of fresh and

rewetted rice. Journal of Food Science 45(2): 356–61.

36. HA Becker. 1960. On the absorption of liquid water by the wheat kernel. Cereal Chemistry 37(3): 309–23.

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in an isotropic sphere – A finite element approach. Transactions of the ASAE 31(2):

34. JF Steffe, and RP Singh. 1982. Diffusion coefficients for predicting rice drying behavior.



37. ST Chu, and A Hustrulid. 1968. Numerical solution of diffusion equations. Transactions of the ASAE 11(5): 705–8.

38. ASAE Standards. 1990. ASAE S352.2 moisture measurement – unground grain and seeds. 35th ed. 353. St. Joseph, MI: ASAE.

39. SAS Institute Inc. 1987. SAS/STAT guide for personal computers Ver. 6 ed. Cary, NC:

SAS Institute Inc.

40. STSC. 1991. STATGRAPHICS user’s guide. Ver.5 ed. Rockville, MD: STSC, Inc.

41. WM Clark. 1952. The laws of mass-section: Rates and reaction. In Topics in physical chemistry. Baltimore, MD: The Willies & Wilkins Company.

42. J Crank, and P Nicolson. 1947. A practical method for numerical evaluation of solutions

of partial differential equations of the heat conduction type. Proceedings of Cambridge Philosophical Society 43: 50–67.

43. JC Strikwerda. 1989. Finite difference schemes and partial differential equations. 1st

ed. CA: Wadsworth, Inc.

44. S Kang, and SR Delwiche. 1999. Moisture diffusion modeling of wheat kernels during

soaking. Transactions of the ASAE 42(5): 1359–1365.

Sons, Inc.

46. SLS Jacoby, JS Kowalik, and T. Pizzo. 1972. Iterative methods for nonlinear optimiza-tion problems. NJ: Prentice-Hall, Inc.

ing steeping using microscopic NMR imaging. Presented at the 1991 International

Summer Meeting, Paper No. 913055. 2950 Niles Road, St. Joseph, MI.

48. Surfer. 1990. Surfer reference manual. Ver.4. Golden, Colorado: Golden Software, Inc.

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45. LJ Segerlind. 1984. Applied finite element analysis. 2nd ed. New York: John Wiley and

47. R Ruan, JB Litchfield, and SR. Eckhoff. 1991. Simultaneous and nondestructive

measurement of transient moisture profiles and structural changes in corn kernels dur-


81

4 Computer Simulation of Radio Frequency Heating

Yifen Wang and Jian Wang

CONTENTS

4.1 Introduction ..................................................................................................... 82

4.2 Radio Frequency Heating Systems ................................................................. 82

4.2.1 Radio Frequency Power Generators .................................................... 82

4.2.2 Radio Frequency Applicators ..............................................................84

4.3 Dielectric Properties ......................................................................................85

4.3.2 Transmission Properties ......................................................................87

4.3.3 Measurement of Dielectric Properties ................................................. 89

4.3.3.1 Open-ended Coaxial Probe Methods .................................... 89

4.3.3.2 Transmission Line Method ...................................................90

4.3.3.3 Resonance Cavity Method ....................................................90

4.4 Computer Simulation ...................................................................................... 91

4.4.1 Techniques for Solving Electromagnetic Problem .............................. 91

4.4.2 Finite-Difference Time Domain Method ............................................92

4.4.3 Finite Element Method ........................................................................95

4.4.4 Coupling Problem ................................................................................95

4.4.5 Previous Simulation Works .................................................................96

4.4.6 Commercial Electromagnetic Software ..............................................96

4.4.7 Examples of Computer Simulation ......................................................96

4.4.7.1 Simulation on Homogeneous Food .......................................96

4.4.7.1.1 Assumptions ..........................................................99

4.4.7.1.2 Governing Equations........................................... 100

4.4.7.1.3 Model .................................................................. 100

4.4.7.1.4 Simulation Results .............................................. 101

4.4.7.2 Simulation on Heterogeneous Food .................................... 101

4.5 Conclusions ................................................................................................... 102

Nomenclature ......................................................................................................... 108

References .............................................................................................................. 109

55534_C004.indd 8155534_C004.indd 81 10/22/08 8:30:36 AM10/22/08 8:30:36 AM

4.3.1 Definition of Dielectric Properties ......................................................85



4.1 INTRODUCTION

Heating is one of the most essential methods for food preservation. Conventional

heating methods use external heat sources including hot water and steam. Heat is

transferred by conduction, convection, and radiation. The poor thermal conducting

Dielectric heating, which includes radio frequency (RF) and microwave heating,

offers the possibility of fast heating in solid and semi-solid foods. Over the past 60

years, numerous studies have been reported on microwave (300–30,000 MHz) and

RF (10–300 MHz) heating. An advantage of dielectric heating over the conventional

thermal processing is the rapid heating by direct interaction between electromagnetic

ization of individual molecules or causes migration of ions within the material as it

alternates at high frequency [6]. The main difference between RF and microwaves

is wavelength. The wavelength at designated RF heating frequencies (6.78, 13.65,

27.12, and 40.68 MHz) is 22–360 times as great as that of the two commonly used

microwave frequencies (915 and 2450 MHz). This allows RF energy to penetrate

dielectric materials more deeply than microwaves. Therefore, RF heating may be

particularly useful when applied to large size packaged food products including the

6 lb army ration because of its deep penetration. RF heating offers the possibility of

fast heating in solid and semi-solid foods that can overcome the limits of uneven and

slow heating inherent in conventional retorting. To gain a better understanding of the

heating process, to predict heating patterns within the heated region, and to develop

new formulas and appropriate processes for treating them are real challenges due

to the complicity of the physical properties of foods and the interaction mechanism

With the rapid development of computer technology and software over recent years,

computer simulations based on mathematical electromagnetic models may help to face

these challenges. However, the large variety of food compositions, geometric shapes,

and processing requirements make the simulation of RF heating complicated.

4.2 RADIO FREQUENCY HEATING SYSTEMS

Before introducing simulation of RF dielectric heating in food processing, RF heat-

4.2.1 RADIO FREQUENCY POWER GENERATORS

There are two fundamental approaches of equipment design according to different

generators. They are the free running oscillator system (conventional RF heating

equipment) and the Crystal Oscillator Source Matched Impedance Generator system

(COSMIG) [11].

The conventional power oscillator RF heating system consists of a main power, a

high voltage transformer, a self-excited oscillator with one or more triodes, a high volt-

The RF applicator and foods are part of the power generator circuit; a change in the

55534_C004.indd 8255534_C004.indd 82 10/22/08 8:30:37 AM10/22/08 8:30:37 AM

ability of food, especially solid food, results in non-uniform and inefficient heating.

fields, and foods that are hermetically sealed in microwavable packages [1–5]. Heat is

generated within certain materials when the electromagnetic field reverses the polar-

between the electromagnetic field and foods [7–10].

age rectifier, a tank circuit, and a work circuit, as shown schematically in Figure 4.1.

ing systems (including generators and applicators) are briefly introduced.


Co

mp

uter Sim

ulatio

n o

f Rad

io Freq

uen

cy Heatin

g 83

Main power

+

–

Rectifier Oscillator High-voltage

transformer

Tank

circuit

Work circuit

Variable

inductance

tuning

Load

Variable

capacitance

applicator

FIGURE 4.1 Schematic diagram of a conventional RF heating equipment. (From Wig, T.D. 2001. Computer simulation of dielectric heating. Ch. 7 in

Sterilization and Pasteurization of Foods using Radio Frequency Heating. PhD thesis, Pullman, WA: Washington State University. Based on Orfeuil, M.

1987. Electric Process Heating. Columbus, OH: Battelle Memorial Institute.)

55534_C004.indd 8355534_C004.indd 83

10/22/08 8:30:37 AM

10/22/08 8:30:37 AM



capacitance or inductance of the work circuit affects the power coupled from the tank

circuit to the load [12]. The RF power is typically coupled from tank circuit to work

circuit by changing the space interval between the electrodes and/or by adjusting the

length of variable inductor in the work circuit. The power oscillator design is able to

tional design is also simple to construct and the system is relatively inexpensive.

However, during a heating process, variations in applicator separation, food prod-

uct dielectric properties, and other factors may change the capacitance and quality

factor of the applicator in the circuit, in turn shifting the intrinsic frequency of the

applicator. It is intolerable in regions where strict operating frequency limitations

are enforced. The Federal Communications Commission (FCC) assigned 6.78, 13.56,

50 Ω RF heating equipment, and uses a different approach for the generation of RF

power. The system, demonstrated in Figure 4.2, consists of an oscillator, a power

output of the power is transferred to a load through the transmission line and imped-

ing frequency that meets the requirement of international electromagnetic compat-

generator. However, the much higher cost and limitation of power output obstructs

4.2.2 RADIO FREQUENCY APPLICATORS

ing systems are used, the RF applicator should be designed to meet production

Applicator

Load

50

Line

OscillatorPower Impedance

matching

Generator: 50 Source

FIGURE 4.2 Schematic diagram of a conventional RF heating equipment. (From Wig, T.D.

2001. Computer simulation of dielectric heating. Ch. 7 in Sterilization and Pasteurization of Foods using Radio Frequency Heating. PhD thesis, Pullman, WA: Washington State Univer-

sity. Based on Orfeuil, M. 1987. Electric Process Heating. Columbus, OH: Battelle Memorial

Institute.)

55534_C004.indd 8455534_C004.indd 84 10/22/08 8:30:38 AM10/22/08 8:30:38 AM

reach high overall efficiency because the load is part of the circuit [13]. The conven-

27.12 and 40.68 at radio frequencies for industrial, scientific, and medical (ISM) usage

[14]. Therefore, power amplifier generators were introduced to solve the problem.

The output of the amplifier in COSMIG (power amplifier) system is designed at

a fixed output impedance, normally 50 Ω. So the COSMIG system is also known as

amplifier, a 50 Ω transmission line, an impedance matching circuit, and a work

circuit [10,15]. In this type of system, a stable, fixed frequency oscillator supplies a

radio frequency signal to a power amplifier, which supplies power to the load. The

ance matching circuit, which is used to match the impedance of the amplifier and the

The power amplifier system has the advantage of a stringently controlled operat-

ibility. It also physically separates the matching circuit from the generator. Finally

it improves the working circuit, the process control system, and the efficiency of the

the wide use of the power amplifier system.

Whether free running oscillator systems or power amplifier dielectric heat-

amplifier

load to avoid the power reflection [12].


Computer Simulation of Radio Frequency Heating 85

requirements such as the nature and shape of the material being heated. The size

and shape of the applicator may vary tremendously, but they are sorted into three

main commercially available types [6,13,15]:

is the most common and has the simplest design. The object to be heated

is placed between two electrodes which form a parallel plate capacity

(Figure 4.3a). It is mostly used for a block of material or a relatively thick

material.

electrodes lie on the same level and are parallel to the plane of the product

to be heated. The polarity of each pair of electrodes is opposite (Figure

4.3b). It is often used in drying applications and for relatively thin layers

(approximately 10 mm) of products.

gered on either side (Figure 4.3c). It is often used for products of intermedi-

ate thickness.

4.3 DIELECTRIC PROPERTIES

In order to properly simulate dielectric heating on biomaterials including foods, it is

desirable to determine the factors that affect the rate of heating throughout the prod-

uct. The dielectric properties of foods are the principal parameters that determine

the coupling and distribution of electromagnetic energy during dielectric heating

[17]. The dielectric properties of foods are often temperature dependent [18], and

therefore must be known over the full range of temperatures experienced by the

product to allow simulation of heating behavior.

(ISM) applications according to international agreement [18]. The Federal Communi-

cations Commission (FCC), the responsible regulatory agency in the US, has adopted

this frequency allocation scheme [14] with a few additional requirements. Dielectric

heating applications typically operate within the ISM frequency bands. Those heaters

that operate at the frequencies 6.78, 13.56, 27.12, and 40.68 MHz are considered to

microwave apparatus.

Most biological materials behave as lossy insulators, tending to both store and dis-

electric properties, which are normally described in terms of the complex relative

permittivity, εr:

ε ε εr r r= −’ ’’j (4.1)

55534_C004.indd 8555534_C004.indd 85 10/22/08 8:30:38 AM10/22/08 8:30:38 AM

2. Fringe-field applicator (Strayfield electrodes). In this case, several pairs of

3. Staggered through-field (Garland electrodes) applicator. Conceptually, this

configuration is a modified through-field applicator with electrodes stag-

Several frequency bands are reserved for use in industrial, scientific, and medical

be RF heaters, and those that operate at 915 and 2450 MHz are usually identified as

4.3.1 DEFINITION OF DIELECTRIC PROPERTIES

sipate electrical energy in response to an imposed electromagnetic field, in the

same fashion as capacitors and resistors [19]. These abilities are defined by di-

1. Through-field applicator (flat electrodes). A through-field RF applicator



where j = −1 . The real part of the relative complex permittivity, ε’r, known as

the relative dielectric constant, describes the ability of a material to store energy

permittivity, ε’’r , known as the relative loss factor, describes the ability of a material

in heat generation [16,20–21].

(a)

(b)

(c)

VRF

Product

VRF

Product

ProductElectrodes VRF

FIGURE 4.3

Dielectric drying. Drying Technology 14(5): 1063–98.)

55534_C004.indd 8655534_C004.indd 86 10/22/08 8:30:39 AM10/22/08 8:30:39 AM

RF heating applicator. (a) Simple through-field RF applicator. (b) Fringe-field

applicator. (c) Staggered through-field applicator. (From Jones, P.L., and Rowley A.T. 1996.

in response to an applied electric field. The imaginary part of the relative complex

to dissipate energy in response to an applied electric field, which typically results



The corresponding properties of materials that describe their interaction with

expressed in terms of the complex relative permeability μr:

μ μ μr r r= −’ ’’j (4.2)

Together, εr and μr describe the behavior of a material in an electromagnetic

and permeability are given by: εo = 8.854 × 10−12 Farads/meter and μo = 4π × 10−7

Henrys/meter, respectively. Most natural biological materials do not interact with

[7,22]. Therefore, a relative permeability of μr = 1 − j0 is often assumed, and it

is common that no attempt is made to characterize the magnetic properties of

foods.

Heat generation during dielectric heating can take place by several mechanisms,

including ionic conduction and dipolar relaxation. At lower frequencies (below 200

MHz, depending on the material) gross electron conductivity plays a major role

relaxation often dominates, whereby molecules (typically water molecules) absorb

energy due to the repeated reorientation of their polarization in response to the elec-

in dielectric heating. The precise contribution of each mechanism by which energy

is dissipated is not always easily determined, and is often irrelevant. When relaxa-

tion effects are discounted, the loss factor of a material is related to its direct current

(DC) conductivity σ by the following Equation:

σ ωε ε= =0 r 0 rε π ε’’ ’’2 f (4.3)

where f is the temporal frequency and ω is the radian frequency, which are related

by ω = 2πf. It is often convenient to express the sum of the conductive and dielec-

tric loss in terms of an equivalent loss factor. This is adequate for use in dielectric

heating studies. The loss factor and frequency data can be used in Equation 4.3 to

compute an equivalent conductivity.

4.3.2 TRANSMISSION PROPERTIES

by von Hippel [19]:

γ = α + jβ (4.4)

The real part of the complex propagation factor, known as attenuation factor,

describes the diminution of the electric portion of an imposed electromagnetic

55534_C004.indd 8755534_C004.indd 87 10/22/08 8:30:39 AM10/22/08 8:30:39 AM

an applied magnetic field, functioning in the same manner as an inductor, can be

field. They are relative values, since they describe a material’s interaction with

electromagnetic field with respect to free space (vacuum), whose permittivity

the magnetic portion of the electromagnetic field to generate heat. Instead, virtually

all of the energy absorption in a material is due to interaction of the electric field

in dissipating electromagnetic fields. At microwave frequencies, however, dipolar

tric field. At even higher frequencies, relaxation of individual atoms can play a role

eling electromagnetic field, is described in terms of a complex propagation factor (γ)

The transmission properties, which determine energy flow in a material within trav-



ing to:

α ε εε

= +⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ −

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

21 1

π2f

c’

’’’

⎡⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1

2

(4.5)

where c is the speed of light in vacuum, 2.998 × 108 m/s. The imaginary part of the

complex propagation factor β, known as phase constant, describes the phase shift of

a plane wave propagating through a dielectric material.

p

depth or attenuation distance, is a parameter that describes the distance an incident

electromagnetic wave can penetrate beneath the surface of a material before its elec-

amplitude at the surface [23]. According to Lambert’s law:

E E ez 0z= −α (4.6)

0 z

traveling within a dielectric material, and α is the attenuation constant. When

E d E e( ) /p = 0 , with Equation 4.5 and Equation 4.6, a relation can be derived for dp:

dc

f

p

rr

r

=

+⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2 1 1

2

π ε εε

’’’’

⎧⎧⎨⎪⎪⎪

⎩⎪⎪⎪

⎫⎬⎪⎪⎪

⎭⎪⎪⎪

1

2

(4.7)

normally used in electrical engineering and is not commonly applied in food engi-

neering [24].

The average power dissipation per unit volume in a dielectric subjected to an

P f Eav 02π= 2 ε ε’’ Watts/meter3 (4.8)

or, using the equivalent conductivity:

Pav = σE2 Watts/meter3 (4.9)

where f is the frequency in Hertz, σ is the equivalent conductivity in Siemens/meter,

distance an incident electromagnetic wave can penetrate beneath the surface of a

55534_C004.indd 8855534_C004.indd 88 10/22/08 8:30:40 AM10/22/08 8:30:40 AM

field as it penetrates a material. The attenuation factor α can be calculated accord-

The electric field penetration depth (d ) of a material, also known as the skin

tric field intensity is diminished by a factor of 1/e (e, Naprian base, 2.71828) of its

where, E is incident electric field intensity at the surface of a material, E is electric

field intensity at distance z from the surface in the direction of the electric wave

This fundamental Equation 4.7, which yields an electric field penetration depth, is

electromagnetic field can be obtained using the expression:

and E is the electric field intensity in Volts/meter. The most important penetration

concept in food engineering, known as power penetration depth [24], defines the



material as the power decreases to 1/e of its power at the surface. Since the power is

P P ez 02 z= − α

(4.10)

where P0 z

within a dielectric material, and α is the attenuation constant. When P d P e( ) /p = 0 ,

with Equation 4.7 and Equation 4.10, the power penetration depth is given by:

dc

f

p

rr

r

2

=

+⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥2 2 1 1π ε ε

ε’

’’’ ⎥⎥

⎧⎨⎪⎪⎪

⎩⎪⎪⎪

⎫⎬⎪⎪⎪

⎭⎪⎪⎪

1

2

(4.11)

which is precisely half of the amplitude penetration depth. Penetration depth in this

proportional to frequency. Therefore, it would be expected that deeper penetration

would be obtained to lower frequencies, and that higher frequencies would result

in greater surface heating. Increased penetration can reduce the overall variation in

properties themselves vary with frequency, penetration depth does not vary exactly

etrate deeply into most moist food products [25], where dielectric constants and loss

factors are relatively low. Electromagnetic waves in the radio frequency range are

generally regarded as having deep penetration into most foods.

4.3.3 MEASUREMENT OF DIELECTRIC PROPERTIES

Both the determination of heating rate and penetration depth depend on the value

of dielectric properties of food. Dielectric constant and loss factor are the most

important factors for dielectric heating. However, the lack of knowledge about the

dielectric properties of various foods as functions of food composition, food tem-

perature, and frequency of processing electromagnetic wave restricted our ability

to design optimum dielectric heating systems and properly simulate heating [26].

Measurement of dielectric properties is necessary for the successful simulation and

application of dielectric heating.

Several methods can be used to measure the dielectric properties, including

open-end coaxial probe methods, transmission line methods, and resonance cavity

methods [27].

4.3.3.1 Open-ended Coaxial Probe Methods

The open-ended coaxial probe method uses a coaxial probe with the open end in con-

tact with the material-under-test. During the measurement, a signal is sent by a vec-

55534_C004.indd 8955534_C004.indd 89 10/22/08 8:30:41 AM10/22/08 8:30:41 AM

proportional to square of the electric field intensity:

is the incident electric field power density at the surface of a material, P

is electric field power density at distance z in the direction of electric wave traveling

study is defined as power penetration depth and will be calculated using Equation 4.11.

Given fixed dielectric properties, the penetration depth of a material is inversely

electric field, which can, in turn, improve heating uniformity. Since the dielectric

as 1/f. In general, an electromagnetic field having a short wavelength does not pen-

tor network analyzer or an impedance analyzer, and reflected back by the material.



dielectric properties of the tested material [28–30].

The open-ended coaxial probe method has the advantage of ease to use and is

suitable for all kinds of material, especially for liquid and semi-solid food material.

It has a large frequency range for measurement (10–20 GHz). It also needs little

sample preparation. However, it has some restrictions. The method has limited accu-

racy in dielectric constant and low loss factor resolution. Three major error sources

for measurement are the cable stability, air gaps, and sample thickness.

network analyzer or impedance analyzer in both amplitude and phase, inaccuracy

measurement. The sample surface, which contacts the coaxial probe, should be made

racy of measurement. Therefore, the thickness of the sample should be greater than

the recommended minimum sample thickness, tmin [31], which is expressed as:

tmin =20

ε r

(mm).

Normally, the sample thickness should typically be greater than 1cm. The solid

Although the open-ended coaxial probe method has some limitations, it is an

ideal method for measuring the dielectric properties of liquids or semisolids, and it is

one of the most widely and commonly used methods in the food research community

[32–33]. Most of the time, the accuracy of measured dielectric properties is adequate

for dielectric heating research [34].

4.3.3.2 Transmission Line Method

ple. The transmission line may be either rectangular or coaxial [27]. During the mea-

surement, a vector network analyzer is used to detect the change of the impedance

the measuring result, the dielectric properties of the tasted material are calculated

by software.

Both the accuracy and sensitivity of transmission line method are higher than those

of open-ended coaxial probe method. However, the measuring range is narrower than

the open-ended coaxial probe method and the samples need to be carefully prepared to

4.3.3.3 Resonance Cavity Method

For resonance cavity method, a sample will be put in a cavity with high quality

resonance. Due to the insertion of the sample, both the center resonance frequency

55534_C004.indd 9055534_C004.indd 90 10/22/08 8:30:42 AM10/22/08 8:30:42 AM

This method will need the cross-section of a transmission line to be filled by a sam-

fit for the cross-section of the transmission line. The way of measurement makes it dif-

ficult for the transmission line method to test liquid and semisolid sample material.

The magnitude and phase change of the reflected wave are used to calculate the

As the instability and flexing of cable may distort the detective signal sent by

during measurement is introduced. It is better to minimize the cable flexing because

the existence of air gaps can distort the detective signal and influence accuracy of

as flat and smooth as possible. If the sample is too thin, the detective signal can pen-

etrate the sample and introduce insufficient reflection, thereby influencing the accu-

samples must have a flat surface.

and propagation characteristics due to the fllling of dielectric material. According to



fc and quality factor Q change [35]. Two changes of the parameter are measured by

the vector network analyzer, and use special software to determine the dielectric

properties of the sample material. Resonance cavity method can be very accurate.

It is sensitive to very low values of loss factor. However, the method provides the

4.4 COMPUTER SIMULATION

Historically, the prediction of the electromagnetic wave distribution within dielectric

heating equipment mainly relied on an experimental approach. Time consuming and

high cost are always problems for researchers. Computer modeling recently shows a

great potential to predict the wave distribution due to the rapid evolution of computer

calculating ability.

4.4.1 TECHNIQUES FOR SOLVING ELECTROMAGNETIC PROBLEM

Several techniques can be used to provide insight into electromagnetic heating

numerical. Experimental techniques are expensive, time consuming, and usually

techniques are used for problems associated with complicated constructions. The

among the most commonly used in electromagnetics [36].

Analytical and numerical methods solve the dominant equations, Maxwell

equations, for all the electromagnetic problems. Maxwell equations that describe

∇ ∂∂

× = +H JDt

(4.12)

∇ ∂∂

× = −EBt

(4.13)

∇ ρ⋅ =D e (4.14)

∇ ⋅ =B 0 (4.15)

where E (Vm−1 −1 −2

density, B (Wb m−2 −2) is electric current density,

and ρe is electric charge density.

spatial position and time. However, in many practical systems, the time variations

55534_C004.indd 9155534_C004.indd 91 10/22/08 8:30:43 AM10/22/08 8:30:43 AM

dielectric properties of only one frequency for a specific resonance cavity and the

cavities are difficult to design and use.

phenomena. Those techniques can be classified as experimental, analytical, and

provide exact electromagnetic field distribution. However, the technique can only

solve the problem with very limited and extremely simple configurations. Numerical

finite difference time domain method (FDTD) and finite element method (FEM) are

electromagnetic fields are [37]:

) is electric field, H (A m ) is magnetic field, D (C m

) is magnetic field density, J (A m

E, H, D, B, and J represent the instantaneous field vectors as the function of

do not allow adequate flexibility for parameter variation. Analytical techniques can

) is electric flux



are of cosinusoidal form and are referred to as time-harmonic. Under time-harmonic

∇ ω ωε× = + = +H J D J Ej j (4.16)

∇ ω ωμ× = − = −E B Hj j (4.17)

∇ ∇ ε ρ⋅ = ⋅ =D E e (4.18)

∇ ∇ μ⋅ = ⋅ =B H 0 (4.19)

where E, H, D, B, and J represent the corresponding complex spatial forms which

are only a function of position.

used to solve the electromagnetic equations [39]. The two most common sub-methods

ment frequency domain (FEFD). Dibben and Metaxas [40] reported that time domain

simulations of microwave heating are much faster than frequency domain simulations

since time domain overcomes the problems of ill conditioning. Another advantage of

the time domain method is when solutions for several frequencies are needed, it can

release several results from a single solution.

is now used in place of FETD to some degree. The FDTD method increases the num-

ber of cells, consequently, increases the computational demands at a small matrix. But

4.4.2 FINITE-DIFFERENCE TIME DOMAIN METHOD

time-dependent curl equations.

Several key attributes combine to make the FDTD method a useful and powerful

discretized in space and time in a straightforward manner. Second, since the method

radiation, scattering, and coupling problems [36].

equations is the numerical approximation of the derivative of a function f(x). The

central-difference formula is [36]:

′ =+ − −

+f xf x x f x x

xO x( )

( / ) ( / )( )0

0 0 22 2Δ ΔΔ

Δ (4.20)

55534_C004.indd 9255534_C004.indd 92 10/22/08 8:30:43 AM10/22/08 8:30:43 AM

conditions, the Maxwell’s equation can be modified to [38]:

Among a lot of numerical methods the finite element method (FEM) is mainly

of the finite element method are finite element time domain (FETD) and finite ele-

Recently finite difference time domain (FDTD) method has gained acceptance and

FEM has superiority over FDTD in handling complicated product configuration.

The finite-difference time-domain (FDTD) method is a convenient, easy-to-use, and

efficient method for solving electromagnetic scattering problems. It was first intro-

tool. First is the method’s simplicity; Maxwell’s equations in differential form are

tracks the time-varying fields throughout a volume of space, FDTD results lend

themselves well to scientific visualization methods. These in turn provide the user

with excellent physical insights into the behavior of electromagnetic fields. Finally,

The fundamental scheme for finding the finite difference solution of Maxwell’s

where Δx is a sufficiently small interval.

duced in 1966 by Yee [41] and was developed by Taflove [42–44] to solve Maxwell’s

the geometric flexibility of the method permits the solution of a wide variety of



( , , ) ( , , )i j k i x j y k z≡ Δ Δ Δ (4.21)

and any function of space and time in the solution region can be presented as:

F i j k F i j k n t( , , ) ( , , , )= δ δ δ Δ (4.22)

where δ = Δx = Δy = Δz is the space increment, Δt is the time increment, and i, j, k, n

are integers.

In applying Equation 4.20 and using notation in Equation 4.21 and Equation

4.22, Equation 4.12 and Equation 4.13 can be approximated to [36]:

Hz

Ey

Ex Ez

Hy

Y

X

Z

Ez

Ey

Ey

Ex

Ex

Ez

Hx

(i, j ,k)

FIGURE 4.4 Luedecke, L., and Feng, H. 1998. Dielectric properties of cottage cheese and surface treat-

ment using microwaves. Journal of Food Engineering 37(4): 389–410.)

55534_C004.indd 9355534_C004.indd 93 10/22/08 8:30:44 AM10/22/08 8:30:44 AM

A grid point in a solution region as shown in Figure 4.4 can be defined as [36]:

Positions of field components in a unit cell. (From Herve, A.G., Tang, J.,



H i j k H i j kxn

xn+ −+ + = + +1 2 1 21 2 1 2 1 2 1/ /( , / , / ) ( , / , /22

1 2 1 2

1 2 1

)( , / , / )

( , / , )

++ +

+ + −

δμ δ

ti j k

E i j k Eyn

y

×nn

zn

zn

i j k

E i j k E i j k

( , / , )

( , , / ) ( , , /

+

+ + − + +

1 2

1 2 1 1 22)

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(4.23a)

H i j k H i j kyn

yn+ −+ + = + +1 2 1 21 2 1 2 1 2 1/ /( / , , / ) ( / , , /22

1 2 1 2

1 1 2

)( / , , / )

( , , / )

++ +

+ + −

δμ δ

ti j k

E i j k Ezn

z

×nn

xn

xn

i j k

E i j k E i j k

( , , / )

( / , , ) ( / , ,

+

+ + − + +

1 2

1 2 1 2 11)

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(4.23b)

H i j k H i jzn

zn+ −+ + = + +1 2 1 21 2 1 2 1 2 1 2/ /( / , / , ) ( / , / ,kk

ti j k

E i j k Exn

x

)( / , / , )

( / , , )

++ +

+ + −

δμ δ1 2 1 2

1 2 1×

nn

yn

yn

i j k

E i j k E i j

( / , , )

( , / , ) ( , / ,

+

+ + − + +

1 2

1 2 1 1 2 kk)

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(4.23c)

E i j ki j k t

i jxn+ + = −

++

1 1 2 11 2

1 2( / , , )

( / , , )

( / , ,

σ δε kk

E i j kt

i jxn

)( / , , )

( / , ,

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

+ ++

1 21 2

δε kk

H i j k H i jzn

zn

)

( / , / , ) ( / ,/ /

δ

×+ ++ + − +1 2 1 21 2 1 2 1 2 −−

+ + − − ++ +

1 2

1 2 1 2 11 2 1 2

/ , )

( / , , / ) (/ /

k

H i j k H iyn

yn // , , / )2 1 2j k +

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ (4.23d)

E i j ki j k t

i jyn+ + = −

++

1 1 2 11 2

1 2( , / , )

( , / , )

( , / ,

σ δε kk

E i j kt

i jyn

)( , / , )

( , / ,

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

+ ++

1 21 2

δε kk

H i j k H i jxn

xn

)

( , / , / ) ( , // /

δ

×+ ++ + − +1 2 1 21 2 1 2 1 2,, / )

( / , / , ) (/ /

k

H i j k H izn

zn

−

+ − + − ++ +

1 2

1 2 1 2 11 2 1 2 // , / , )2 1 2j k+

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ (4.23e)

E i j ki j k t

i j kzn+ + = −

++

1 1 2 11 2

1( , , / )

( , , / )

( , , /

σ δε 22

1 21)

( , , / )( , , /

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

+ ++

E i j kt

i j kzn δ

ε 22

1 2 1 2 1 21 2 1 2

)

( / , , / ) ( / ,/ /

δ

×H i j k H i jy

nyn+ ++ + − − ,, / )

( , / , / ) ( ,/ /

k

H i j k H i jxn

xn

+

+ − + −+ +

1 2

1 2 1 21 2 1 2 ++ +

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥1 2 1 2/ , / )k (4.23f)

55534_C004.indd 9455534_C004.indd 94 10/22/08 8:30:45 AM10/22/08 8:30:45 AM



To ensure the accuracy of the computed results, the spatial increment must be small

compared to the wavelength (usually ≤ λ/10) or minimum dimension of the scatterer.

Δt must satisfy the following stability condition [36,45]:

u tx y z

maxΔΔ Δ Δ

≤ + +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−1 1 1

2 2 2

1

2

(4.24)

where umax is the maximum wave phase velocity within the model.

dynamic electromagnetic solutions when the size of the structure is comparable

with the wavelength, so the method is normally applied to the structures of a size

between 0.1 and 20 times of wavelengths [46]. For the structures whose physical size

4.4.3 FINITE ELEMENT METHOD

structural analysis [47]. In 1968, the method was applied to electromagnetic prob-

lems. Although the concept and programming of FEM is not as simple and easy as

numerical technique for handling problems involving complex geometries and inho-

mogeneous media [36].

Basically a four-step scheme is applied to solve problems by FEM [48]:

elements.

2. Deriving governing equations for a typical element.

3. Assembling of all elements in the solution region.

4. Solving the system of equations obtained.

4.4.4 COUPLING PROBLEM

The traditional electromagnetic simulation software concentrates on the applications

such as microwave circuits and communication equipment design, where thermal

effects are normally neglected. However, to simulate a RF heating process, thermal

of the simulation process. During the RF heating two physical factors, temperature

dielectric properties change with temperature. The changes in dielectric properties

phenomena is one of the most critical factors for the successful simulation of RF heat-

ing processes. Therefore, electromagnetic modeling and thermal modeling must be

55534_C004.indd 9555534_C004.indd 95 10/22/08 8:30:46 AM10/22/08 8:30:46 AM

To ensure the stability of the finite difference scheme of equations, the time increment

The principle of the FDTD method makes it more effective in finding the

is smaller than 0.1 of wavelength, the field distribution is close to quasi-static and in

general the methods of quasi-static field solutions are advised to be used.

Originally finite element method (FEM) was developed and applied in the field of

finite difference method and method of moment, it is a more powerful and versatile

1. Discretizing the solution region into a finite number of sub-regions or

effect analysis, besides electromagnetic field investigation, becomes an essential part

and electromagnetic field intensity, will interrelate with each other because the dis-

sipated energy produced by electromagnetic field heats the materials while their

integrated to provide real-time modification of the dielectric properties of the product

in order to modulate the electromagnetic and thermal field through a feedback loop.

in turn influence the electromagnetic field distribution. The coupling of two physical



4.4.5 PREVIOUS SIMULATION WORKS

Several previous works have been conducted to numerically simulate the RF drying

and heating processes. Neophytou and Metaxas [39,49] demonstrated the capabil-

distribution. Baginski, Broughton and Christman [51] and Marshall and Metaxas

[52] showed the potential of computer simulation to model RF drying processes,

and to couple the electromagnetic and thermodynamic phenomenon during the

simulation.

4.4.6 COMMERCIAL ELECTROMAGNETIC SOFTWARE

Computer simulation can be used to model complicated geometries, simulate many

electromagnetic conditions, and analyze a lot of problems.

A lot of electromagnetic simulation softwares are is commercially available

which can be operated on Windows, UNIX, and LINUX platforms [53]. Yakovlev

[54] and Kopyt and Gwarek [55] reported more than 17 different software packages

in their reviews of EM modeling software, and compared the license price, computer

operating system, and status in microwave power engineering (Table 4.1). Among

all the software packages, Ansoft HFSSTM (Ansoft, Corp, Pittsburgh, PA), ANSYS

Multiphysics (ANSYS, Inc, Canonsburg, PA), COMSOL Multiphysics® (COMSOL,

Inc., Los Angeles, CA), MAFIA and CST Microwave Studio® (CST GmbH, Welles-

ley Hills, MA), MARC® (MSC.Software Corporation, Palo Alto, CA), XFdtd®

(Remcom, Inc, State College, PA), and QuickWave-3D (QWED, Warsaw, POLAND)

are the most commonly used.

Several software packages, such as Ansoft HFSS®, EMAS, and COMSOL

Multiphysics®, have the ability to couple the electromagnetic and thermal aspects,

allowing users to simulate both thermal conditions and electromagnetic problems.

Datta [56] used EMAS coupled with NASTRAN (heat transfer modeling) in trying to

solve the electromagnetic problems coupled with thermal changes in dielectric heating

processes. They transferred data from one modeling to another and updated physical

properties and heat generation in each round. There are other software packages, like

QuickWave-3D, allowing users to customize and connect software interface with their

own programming code, in order to enhance the ability of original software to couple

electromagnetic aspect with thermal aspect.

4.4.7 EXAMPLES OF COMPUTER SIMULATION

Two examples of computer simulation on homogenous and heterogeneous foods,

respectively will follow. Details on computer simulation procedures, governing

equations, assumptions, models and results are illustrated.

4.4.7.1 Simulation on Homogeneous Food

Figure 4.5 illustrates computer simulation procedures. An appropriate simulation

module was chosen based on governing equations that revealed the physical basis of

55534_C004.indd 9655534_C004.indd 96 10/22/08 8:30:46 AM10/22/08 8:30:46 AM

ity of finite element method (FEM) to model the RF heating system. Chan, Tang,

and Younce [50] studied RF heating patterns in foods due to electromagnetic field

simulated physical phenomena. The constants and variables were then predefined.



TABLE 4.1Summary of Some Commercially Available Software in 2004

Company Code PurposeOperating System and

requirements

ADINA, Inc.

http://www.adina.com

Adina-T Heat transfer analysis

problems

Windows, UNIX

Adina-F Analysis of

compressible and

Adaptive Research Corp.

http://www.adaptive

research.com

CFD2000

transfer in electronic

systems. Radiative and

conjugate heat transfer

models

Windows

ALGOR, Inc.

http://www.algor.com

Professional

Multiphysics

Multiphsics Windows 98/2000/NT/

Me/XP

Professional CFD Heat transfer analysis Windows 98/2000/NT/

Me/XP

Ansoft, Corp.

http://www.ansoft.com

HFSS

simulation

Windows, Linux Solaris

ANSYS, Inc.

http://www.ansys.com

CFX

dynamics (CFD)

package

UNIX Compaq/HP/

SUN/SGI/IBM

Windows NT/2000/XP

Linux

Multiphysics Multiphysics Windows 2000/XP Linux

Compaq/HP/SUN/

SGI/IBM

Design Space Static structural and

thermal, dynamic,

weight optimization,

vibration mode, and

safety factor

simulations

Windows 2000/XP/

NT 4.0 Linux Compaq/

HP/SUN/SGI/IBM

Professional Structural/thermal

analysis

Windows 2000/XP/NT

4.0 Linux Compaq/

HP/SUN/SGI/IBM

CD adapco Group

http://www.cd-adapco.com

Star-CD Heat transfer, reacting

physics and others

Windows; UNIX

HP/SGI/IBM; Linux

COMSOL

http://www.comsol.com

COMSOL

Multiphysics

Multiphysics Windows 2000 or

later, Linux 2.4.x kernel,

glibc-2.2.5 or later,

Solaris 8, 9, 10

Flomerics Group PLC FLO/EMC

analysis

Windows, Sun Solaris

(continued)

55534_C004.indd 9755534_C004.indd 97 10/22/08 8:30:47 AM10/22/08 8:30:47 AM

of solids and field

Electromagnetic field


incompressible flow

Air flow and heat

and fluid flow analysis

Computational fluid

flows, multiphase

http://www.flomerics.com



After the geometry of the model was built, the sub-domain properties and boundary

conditions were assigned. A convergence study was conducted to investigate con-

vergence of the simulation and to determine the optimized meshing that comprises

the calculation time and accuracy of simulation. Finally, the computing solution was

calculated and the simulation results were analyzed.

TABLE 4.1 (Continued)

Company Code PurposeOperating System and

Requirements

Flow Science, Inc. FLOW-3D Fluid modeling,

thermal modeling,

dielectric phenomena

and others

Windows NT/XP/

2000; Linux

UNIX DEC/HP/

IBM/Sun/SGI

Fluent, Inc. FLUENT Multiphysics Windows NT/2000/

XP; UNIX

SGI/HP/IBM/SUN

FIDAP Multiphysics Windows NT/2000/XP;

UNIX SGI/HP/IBM/SUN

Infolytica, Corp.

http://www.infolytica.com

ThermNet Standalone or coupled

thermal simulation

Windows

FullWave High frequency

electromagnetic

simulation

MSC Software Corp

http://www.msc

software.com

Marc Coupled thermal-

structural interactions

and others

Windows NT/2000;

Linux

UNIX Sun/HP/IBM/

SGI/Compaq

MSC Nastran Heat transfer and others Windows NT/2000;

Linux; UNIX

Sun/HP/IBM/SGI/

Compaq/Cray/

Fujitsu/Nec

QWED

http://www.qwed.com.pl

QuickWave-3D

analysis

Windows

Remcom, Inc.

http://www.remcom.com

XFdtd Wave electromagnetic

solver, temperature

rise calculation and

others

Windows 2000/XP, Red

Hat, Silicon Graphics,

Sun Solaris, Mac OSX,

HPUX

Vector Fields Ltd.

com

Opera 3D

and thermal analysis

Windows 98/Me/

NT/2000/XP; UNIX

SUN/HP/SGI;

Zeland Software, Inc

http://www.zeland.com

FIDELITY

analysis

Windows

Source: Yakovlev, V.V. 2002. Review of commercial EM modeling software suitable for modeling of

microwave heating-update. Presented at the 4th IMMG workshop ‘Computer modeling and microwave power industry’, January 7, 2002, Seattle, WA.

55534_C004.indd 9855534_C004.indd 98 10/22/08 8:30:47 AM10/22/08 8:30:47 AM



http://www.vectorfields.


http://www.flow3d.com

http://www.fluent.com



4.4.7.1.1 AssumptionsThe accuracy of the computer simulation directly relies on the exactness of the

numerical model, which affect the demand for the CPU and memory capacities. To

achieve a compromise between the simulation accuracy and computer capability,

assumptions have to be made to simplify the heating system.

Firstly, quasi-static condition was assumed, since the wavelength of the electro-

magnetic wave in RF heating is much larger than the dimension of equipment. The

RF heating is mainly caused by dipolar and interfacial polarization [58], which is only

netic, which is mainly affected by the structure of heating equipment, was overlooked.

sel and the heat convection between circulating water and food were neglected. The

assumption was fair since we concentrated on the heating distribution at the center

be ignored.

Problem analysis

Governing equation selection

Geometry modeling

Sub-domain and boundary condition setting

Mesh generation

Obtain convergentresult?

NO

Computing solution

Results analysis

YES

Convergencestudy

FIGURE 4.5 Flow chart of the computer simulation procedure. (From Wang, J. 2007.

of Philosophy Dissertation. Department of Biological System Engineering. WA, USA:

Washington State University, Pullman.)

55534_C004.indd 9955534_C004.indd 99 10/22/08 8:30:48 AM10/22/08 8:30:48 AM

Constant and variable definition

Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor

of dielectric properties of material on electric field, the propagation of electromag-

Without considering the magnetic field, the quasi-static analysis in the current study

caused by the electric field. As our research mainly concentrated on the influence

was acceptable. Secondly, the flow of circulating water inside the pressure-proof ves-

layer of the sample trays; the influence from circulating water was small and could



4.4.7.1.2 Governing EquationsThe general governing equations for solving typical RF heating problems can be

described by Maxwell’s equations [38] as shown in Equations 4.16 through 4.19.

According to the assumptions in section 4.7.1.1 the governing equations for this case

can be deduced as [59]:

∇2 0V = (4.25)

were obtained. The time-averaged power density, P, was generated by transferring

the electric energy to heat the food material as shown by [58]:

P = ω ε εE20 ’’r (4.26)

where P is the dissipated power density.

The power density in Equation 4.26 was then treated as the heat source in the

the governing equation for heating transfer by conduction is expressed as:

ρ ∂∂

∇ ∇CTt

k T Pp − ⋅ =( ) (4.27)

where ρ is the density, Cp is heat capacity, k is the thermal conductivity, and T is the

temperature.

4.4.7.1.3 ModelBecause of its advantage in solving multiphysical problems, the COMSOL

Multiphysics was used to simulate the models. COMSOL Multiphysics is a modeling

package for the simulation of the physical process that can be described with partial

modeling of the RF heating system is shown in Figure 4.6 and Figure 4.7.

The dielectric properties of food were measured and the linear regression

analysis was used to investigate the relation between temperature and dielectric

properties. The dielectric properties of circulating water were measured during

the experiment. In the heat transfer analysis, heat effects were only considered

in the food sample domain to simplify the model. The thermal conductivity and

were assigned to the numerical model. Dell Precision 870 workstation with 2

Dual-Core 2.80GHz Intel® Xeon™ Processors, and 12 GB memory was utilized

thousands, which was adequate for obtaining convergent solution, in the model it

took about 1 hour for one simulation run. The procedures for computing the solu-

55534_C004.indd 10055534_C004.indd 100 10/22/08 8:30:48 AM10/22/08 8:30:48 AM

By solving Equation 4.25, the electric potential (V) and electric field intensity (E)

heat transfer phenomena to couple the electromagnetic field with thermal field, and

differential equations. The software uses the finite element method to model systems

of coupled physical phenomena through predefined templates [60]. The geometry

specific heat of foods were measured. All of the measured physical properties

to perform the simulation. The predefined finer mesh sizes were used to auto-

matically mesh the model, and the total number of finite elements was 100 to 120

tion are shown in the flow chart in Figure 4.8.



4.4.7.1.4 Simulation ResultsThe hot spot and cold spot were the most critical positions to evaluate the unifor-

mity of the heating process. Therefore, temperatures at cold and hot spots at the

horizontal central layer of the sample tray were compared between experiment and

simulation result to verify the simulation results. The thermal image provided the

of the sample tray. As shown in Figure 4.9, the hot spots were located at the places

near the corner of the sample, and the cold spots were at the center of the sample.

The temperature values at the same positions, as shown in Figure 4.10, were drawn

from the numerical solution.

4.4.7.2 Simulation on Heterogeneous Food

Differences in loss factors among the ingredients of heterogeneous food may result

in non-uniform heating in an RF heating system. Macaroni and cheese is one of few

heterogeneous foods in the literature on RF sterilization [61] even though heteroge-

neous foods are more common in the food market. Computer simulation can help

to obtain more insight and understanding of RF heating patterns on heterogeneous

food.

Cavity0.7

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

0.7

0.6

0.5

0.4

0.3

0.2

0.10.079

0

0.5

0.4

0.2

0

–0.2

–0.4

–0.5

yz

x

Upper

electrodeInductor

FIGURE 4.6 Model of RF heating system. (From Wang, J. 2007. Study of electromagnetic

Department of Biological System Engineering. WA, USA: Washington State University,

Pullman.)

55534_C004.indd 10155534_C004.indd 101 10/22/08 8:30:49 AM10/22/08 8:30:49 AM

field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation.

final heating pattern and temperatures at hot spot and cold spots at the central layer



Assumptions governing equations are similar to those in homogeneous food. The

geometry modeling for heterogeneous food is also similar to that of homogeneous

except for the packaged food as shown in Figure 4.11

to obtain convergent results. The same computer system was used for heterogeneous

foods as for homogeneous foods. Due to the complexity of numerical model to simu-

late heterogeneous food, it took 4–5 hours for one simulation run.

The relatively uniform heating results from both the experiment and computer

simulation suggest potential for attaining safety along with high quality when heat-

ing heterogeneous food with RF as long as the different components are in close

2 (Figure 4.13) indicated a concentration at the tray corners and inside the noodles.

Accordingly, the power density at these locations was higher than the rest of the

food. However, there was no apparently severe overheating at the noodle and corner

intensity concentration.

4.5 CONCLUSIONS

but affected the simulation accuracy. Therefore, a comprise has to be achieved to

Circulatingwater0.2

0.1

0

–0.1

–0.2

0.1

0.079

0.2

0.1

0

–0.1

–0.2yz

x

Aluminum

Food

Aluminum bottom cover of

pressure-proof vessel

FIGURE 4.7 in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of

Biological System Engineering. WA, USA: Washington State University, Pullman.)

55534_C004.indd 10255534_C004.indd 102 10/22/08 8:30:51 AM10/22/08 8:30:51 AM

film

Model of vessel. (From Wang, J. 2007. Study of electromagnetic field uniformity

The predefined finer mesh sizes were used to automatically mesh the model, and

the total number of finite elements was 100 to 120 thousands, which were sufficient

proximity to each other and have sufficient heat transfer. The simulation results were

checked at two planes (Figure 4.12). The electric field distribution in planes 1 and

of tray (Figure 4.14). Sufficient heat transfer among the sauce, noodles, cheese and

beef mitigated the power density concentration that was caused by the electric field

Assumptions simplified the governing equation and saved computer resources,



and heating rate with limited computer resources.

The results also indicated that computer simulation has the potential to further

systems. Computer simulation can also be a useful tool to further improve the RF

heating systems.

Assign initial condition

Start time step

FEM EM simulationto obtain electric potential V

and time averaged power density P

FEM heat transfer simulationto obtain temperature T

step?

NO

Solution output

YES

Update dielectric properties based on T

FIGURE 4.8 Flow chart of the procedure in solution calculation. (From Wang, J.

Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA:


55534_C004.indd 10355534_C004.indd 103 10/22/08 8:30:52 AM10/22/08 8:30:52 AM

Calculate electric field E

Reach final time

2007. Study of electromagnetic field uniformity in radio frequency heating applicator.

provide a reasonable indication of electromagnetic field distribution, heating pattern

improve similar simulations and aid the construction and modification of RF heating



Position of

cold spot

Position ofhot spot

80.0°C

40.0°C

80

70

60

50

40

FIGURE 4.9 Typical thermal image of the central layer of RF processed food sample.

applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering.

WA, USA: Washington State University, Pullman.)

FIGURE 4.10 Typical numerical solution of the central layer of processed food sample.



Position of

cold spot

Y

X

Position of

hot spot

8080

75

70

65

60

55

50

45

4040

55534_C004.indd 10455534_C004.indd 104 10/22/08 8:30:52 AM10/22/08 8:30:52 AM

(From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating




Circulatingwater

0.2

0.1

0.217

0

0.079

0.2

0.1

0

yz

x

Aluminumfilm

Cavity

Meat ballsSauce and

Cheese

FIGURE 4.11 Model of quarter of packaged heterogeneous food and circulating water.



Plane 1

0.2

0.1

0

0.217

0.079

0

0.1yx

z

Plane 2

FIGURE 4.12 Two planes for checking the numerical solutions. (From Wang, J. 2007. Study

Dissertation. Department of Biological System Engineering. WA, USA: Washington State

University, Pullman.)

55534_C004.indd 10555534_C004.indd 105 10/22/08 8:30:54 AM10/22/08 8:30:54 AM


of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy



Noodles

Meat balls

220

200

280

(V/m)

270

260

250

240

230

210

(a)

Meat balls

Noodles

380

(V/m)

150

300

250

200

350

130

(b)

xyz

xyz

FIGURE 4.13



55534_C004.indd 10655534_C004.indd 106 10/22/08 8:30:57 AM10/22/08 8:30:57 AM

Electric field distribution at (a) plane 1 and (b) plane 2. (From Wang, J. 2007.




Noodles

Meat balls

119

123

(°C)

121

120

122

118

(a)

118

122.5

(°C)

120

119.5

119

118.5

122

121

120.5

Meat balls

Noodles

121.5

(b)

xz

y

xz

y

FIGURE 4.14 Temperature distribution at (a) plane 1 and (b) plane 2. (From Wang, J. 2007.



55534_C004.indd 10755534_C004.indd 107 10/22/08 8:30:59 AM10/22/08 8:30:59 AM




RF heating systems with the conventional power oscillator design are mostly

adopted in the US market. For the conventional power oscillator design, the RF

applicators and foods are part of the power generator circuit. The variations in appli-

cator separation, food product dielectric properties, and other factors may change

the capacitance and quality factor of the applicator in the circuit. Therefore, further

studies are thus preferred to include the whole circuit of the RF heating system when

modeling to obtain more accurate simulation results.

NOMENCLATURE

c Speed of light in vacuum (m s−1)

p

dp Power penetration depth (m)

E0 −1)

−1)

f Frequency (Hz)

k Thermal conductivity (W m−1 K−1)

P Dissipated power density (W)

P0

Pav Average power dissipation per unit volume (W m−3)

tmin minimum sample thickness (m)

V Scalar electric potential (V)

A Magnetic vector potential−2)

−2)−1)

−1)

J Electric current density (A m−2)

Δt Time increment (s)

α Attenuation factor (Np m−1)

β Phase constant (Rad m−1)

γ Complex propagation factor

δ, Δx, Δy, Δz Space increment (m)

εc Complex permittivity

εr Relative complex permittivity

εo Permittivity of free space (F m−1)

εr

‚ Relative dielectric constant

εr

‚‚ Relative loss factor

μr Relative complex permeability

μo Permeability of free space (H m−1)μr

‚ Real part of relative complex permeability

μr

‚‚ Imaginary part of relative complex permeability

ρe Electric charge density (C m−3)

σ Electric conductivity (S m−1)

ω Angular frequency (Rad s−1)

55534_C004.indd 10855534_C004.indd 108 10/22/08 8:31:01 AM10/22/08 8:31:01 AM

d Electric field penetration depth (m)

Incident electric field intensity (V m

E Electric field intensity (V m

Incident electric field power density (W)

B Magnetic field density (Wb m

E Electric field intensity (Vm

H Magnetic field intensity (A m

D Electric flux density (C m



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113

5 Infrared Radiation for Food Processing

Kathiravan Krishnamurthy, Harpreet Kaur Khurana, Soojin Jun, Joseph Irudayaraj, and Ali Demirci

CONTENTS

5.1 Introduction ................................................................................................... 113

5.2 Basic Laws of Infrared Radiation ................................................................. 115

5.3 Interaction of IR Radiation with Food Components ..................................... 116

5.4 Applications of IR Heating in Food Processing Operations ......................... 118

5.4.1 Drying and Dehydration ................................................................... 118

5.4.2 Integrated Drying Technologies: IR and Convective Drying ............ 119

5.4.3 Pathogen Inactivation ........................................................................ 120

5.4.3.1 Effect of Power and Sample Temperature ........................... 120

5.4.3.2 Effect of Peak Wavelength and Bandwidth ......................... 120

5.4.3.3 Effect of Sample Depth ....................................................... 121

5.4.3.4 Types of Microorganisms.................................................... 121

5.4.3.5 Inactivation Mechanism ...................................................... 122

5.4.3.6 Types of Food Materials ..................................................... 123

5.4.4 IR Heating in Other Miscellaneous Food Processing Operations .... 123

5.5 Sources of IR Heating ................................................................................... 123

5.6 Quality and Sensory Changes by IR Heating ............................................... 126

5.7 IR Heat Transfer Modeling ........................................................................... 128

5.8 Selective Heating by Infrared Radiation ....................................................... 131

5.9 Thermal Death Kinetics Model .................................................................... 135

5.10 Conclusion and Future Research Potential ................................................... 137

References .............................................................................................................. 138

5.1 INTRODUCTION

cess of any unit operation. Heat transfer occurs through one of three methods,

conduction, convection, and radiation. Foods and biological materials are heated

primarily to extend their shelf life or to enhance taste. In conventional heat-

ing, which is achieved by combustion of fuels or by an electric resistive heater,

heat is generated outside of the object to be heated and is conveyed to the mate-

rial by convection of hot air or by thermal conduction. By exposing an object to

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Energy conservation is one of the key factors determining profitability and suc-



infrared (IR) radiation (wavelength of 0.78–1000 μm), the heat energy generated

can be directly absorbed by food materials. Along with microwave, radiofrequency

(RF), and induction, IR radiation transfers thermal energy in the form of elec-

tromagnetic (EM) waves and encompasses that portion of the EM spectrum that

borders on visible light and microwaves (Figure 5.1). Certain characteristics of

make it more effective for some applications than others. IR heating is also gaining

time in comparison to conventional heating. Recently, IR radiation has been widely

applied to various thermal processing operations in the food industry such as dehy-

dration, frying, and pasteurization [1].

Food systems are complex mixtures of different biochemical molecules, biological

polymers, inorganic salts, and water. The infrared spectra of such mixtures originate

with the mechanical vibrations of molecules or particular molecular aggregates

within a very complex phenomenon of reciprocal overlapping [2]. Amino acids,

polypeptides, and proteins reveal two strong absorption bands localized at 3–4 μm

and 6–9 μm. On the other hand, lipids show strong absorption phenomena over the

entire infrared radiation spectrum with three stronger absorption bands situated at

3–4 μm, 6 μm, and 9–10 μm, whereas carbohydrates yield two strong absorption

bands centered at 3 μm and 7–10 μm [3,4].

mid-infrared (MIR), and far-infrared (FIR), corresponding to the spectral ranges of

0.75–1.4 μm, 1.4–3 μm, and 3–1000 μm, respectively [1]. In general, FIR radiation

is advantageous for food processing because most food components absorb radiative

energy in the FIR region [3].

Over the past several years, IR heating has been predominantly applied in the

made in the area of IR heating of foods. This chapter is in line with the current

developments in the area of IR heating and serves as a base for its widespread

upcoming practical applications in food processing. Therefore, the aim of this

chapter is to evaluate the existing knowledge in the area of IR heating, provide

insight for the relation between product properties, engineering processes, and

present an up-to-date view on further research. Along with the sound theoretical

FIGURE 5.1 Electromagnetic wave spectrum.

1019

10–5 10–4 10–3 10–2 10–1 101 102 103 1041

1018 1017 1016 1015 1014 1013 1012 1011

Microwave

Wavelength, µm

Frequency, Hz

Infrared

Visible

UltravioletX rays

Gamma rays

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popularity because of its higher thermal efficiency and fast heating rate/response

IR radiation can be classified into three regions, namely, near-infrared (NIR),

electronics and allied fields with little practical application in the food process-

ing industry. However, in the last few years significant research efforts have been

IR heating such as efficiency, wavelength, and reflectivity set it apart from and


Infrared Radiation for Food Processing 115

background on IR heating, the chapter also encompasses application of IR heat-

ing in food processing operations such as drying, dehydration, blanching, thaw-

ing, pasteurization, sterilization, and other miscellaneous food applications

such as roasting, frying, broiling, and cooking, as well as in-depth assessment

of pathogen inactivation. The effect of IR heating on sensory, physicochemical,

nutritional, and microstructural quality of foods and its comparison with other

existing common methods of heating such as convection and microwave heating

are discussed as well.

5.2 BASIC LAWS OF INFRARED RADIATION

The amount of the IR radiation that is incident on any surface has a spectral

dependence because energy coming out of an emitter is composed of different

wavelengths and the fraction of the radiation in each band, dependent upon the

temperature and emissivity of the emitter. The wavelength at which the maximum

radiation occurs is determined by the temperature of the IR heating elements.

This relationship is described by the basic laws for blackbody radiation such as

Planck’s law, Wien’s displacement law, and Stefan-Boltzman’s law, as summa-

rized in Table 5.1 [1,5].

TABLE 5.1Basic Laws Pertaining to Infrared Radiation

Basic Laws Aspects Addressed/Explanation

Planck’s law

E Thc

n ehc n kTbλ λλ πλ

( , )[ ]/

=−

2

1

02

2 5 0

Gives spectral blackbody emissive power distribution E Tbλ λ( , ).

Wien’s displacement law

λmax =2898

T

Gives the peak wavelength (λmax), where spectral distribution of

radiation emitted by a blackbody reaches maximum emissive power.

Stefan-Boltzmann’s law

E T n Tb ( ) = 2 4σb

an infrared source.

H H uλ λ λσ= −0 exp( )*

Gives the transmitted spectral irradiance (Hλ W/m2 · μm) in non-

homogeneous systems.

ρ τ+ + =α 1

total incoming radiation, absorptivity (α): ratio of absorbed part of

incoming radiation to the total incoming radiation, and

transmissivity (τ): ratio of transmitted part of incoming radiation to

the total incoming radiation (Figure 5.2)

k: Boltzmann’s constant (1.3806 × 10 − 23 J/K); n: refractive index of the medium (n for vacuum is 1 and,

for most gases, n is very close to unity); λ: the wavelength (μm); T: source temperature (K); c0: speed of

light (km/s); h: Planck’s constant (6.626 × 10 − 34 J·s); σ: Stefan-Boltzmann constant (5.670 × 10−8

W/m2K4); λmax: peak wavelength; Hλ0: incident spectral irradiance (W/m2 ⋅ μm); u: mass of absorbing 2

λ* 2

Source: From Krishnamurthy, Khurana, Jun, Irudayaraj, and Demirci. Infrared heating in food process-

ing: an overview. Comp Reviews in Food Science and Food Safety. Blackwell, Jan 2008, v.7. With

permission.

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Gives the total power radiated (E (T)) at a specific temperature from

Modified Beer’s law

medium per unit area (kg/m ); σ spectral extinction coefficient (m /kg).

Reflectivity (ρ): ratio of reflected part of incoming radiation to the



5.3 INTERACTION OF IR RADIATION WITH FOOD COMPONENTS

The effect of IR radiation on optical and physical properties of food materials is

crucial for the design of an infrared heating system and optimization of a thermal

process of food components. The infrared spectra of such mixtures originate with

the mechanical vibrations of molecules or particular molecular aggregates within a

When radiant electromagnetic energy impinges upon a food surface, it may

induce changes in the electronic, vibrational, and rotational states of atoms and mol-

ties at different wavelengths by food components differ. The type of mechanisms

for energy absorption determined by the wavelength range of the incident radiative

energy can be categorized as: (1) Changes in the electronic state corresponding to the

wavelength range 0.2–0.7 μm (ultraviolet and visible rays); (2) changes in the vibra-

tional state corresponding to wavelength range 2.5–1000 μm (FIR); and (3) changes

in the rotational state corresponding to wavelengths above 1000 μm (microwaves)

mechanism of changes in the molecular vibrational state, which can lead to radiative

heating. Water and organic compounds such as proteins and starches, which are the

main components of food, absorb FIR energy at wavelengths greater than 2.5 μm [1].

Sandu [3] reported that most foods have high transmissivities (low absorptivities) at

wavelengths smaller than 2.5 μm.

Due to a lack of information, data on absorption of infrared radiation by the

principal food constituents can be regarded as approximate values. The key absorp-

tion ranges of food components are as visualized in Figure 5.3 [3]. It depicts the

principal absorption bands of the major food components compared to the absorp-

tion spectrum of water, indicating that the absorption spectra of food components

overlap with one another in the spectral regions considered. The effect of water on

absorption of incident radiation is predominant over all the wavelengths, suggesting

that selective heating based on distinct absorptivities for a target food material can

be more effective when predominant energy absorption of water is eliminated. The

Irradiation

Absorbed radiation

Transmitted radiation

FIGURE 5.2

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[6]. In general, the food substances absorb FIR energy most efficiently through the

ecules. As food is exposed to infrared radiation, it is absorbed, reflected, or scattered

very complex overlapping phenomenon [2].

(a blackbody does not reflect or scatter), as shown in Figure 5.2. Absorption intensi-

Reflected radiation

Extinction of radiation (absorption, transmission and reflection).



summarized in Table 5.2 [4].

some absorption; and the remaining light leaves the material close to where it enters.

tion obtained visually. For materials with a rough surface, both regular and body

TABLE 5.2The Infrared Absorption Bands for Chemical Groups and Relevant Food Components

Chemical Group Absorption Wavelength (μm) Relevant Food Component

Hydroxyl group (O-H) 2.7–3.3 Water, sugars

Aliphatic carbon-hydrogen bond 3.25–3.7 Lipids, sugars, proteins

Carbonyl group (C=O) (ester) 5.71–5.76 Lipids

Carbonyl group (C=O) (amide) 5.92 Proteins

Nitrogen-hydrogen group (-NH-) 2.83–3.33 Proteins

Carbon-carbon double bond (C=C) 4.44–4.76 Unsaturated lipids

Source: Rosenthal I. 1992. Electromagnetic radiations in food science. Berlin, Heidelberg: Springer-Verlag.

(From Krishnamurthy, Khurana, Jun, Irudayaraj, and Demirci. Infrared heating in food processing:

an overview. Comp Reviews in Food Science and Food Safety. Blackwell, Jan 2008, v.7. With

permission.)

1.0

0.5

02 3 4 5 6 7 8 9 10 11 12 13 14 15

Wavelength (µm)

Tra

nsm

issiv

ity

W

P L

S

LP L

S

P = Proteins

L = Lipids

S = Sugars

W = Water

FIGURE 5.3 Principal absorption bands of the main food components compared with water.

55534_C005.indd 11755534_C005.indd 117 10/22/08 12:06:02 PM10/22/08 12:06:02 PM

infrared absorption bands for chemical groups and relevant food components are

[7]. Regular reflection takes place at the surface of a material. For body reflection,

Regular reflection produces only the gloss or shine of polished surfaces, whereas

Interactions of light with food material and the crucial optical principles such

as regular reflection, body reflection, and light scattering were discussed by Birth

body reflection produces the colors and patterns that constitute most of the informa-

reflection can be observed. For instance, at NIR wavelength region (λ < 1.25 μm),

approximately 50% of the radiation is reflected back, while less than 10% radiation

the light enters the material, becomes diffuse due to light scattering, and undergoes

is reflected back at the FIR wavelength region [8]. Most organic materials reflect



different colors and patterns [5].

The infrared optical characteristics of different media are also theoretically

discussed by demonstrating the necessity of the scattered radiation during measure-

ments [9]. It was experimentally observed that as the thickness of the layer increases,

no theoretical explanation of this phenomenon was presented.

5.4 APPLICATIONS OF IR HEATING IN FOOD PROCESSING OPERATIONS

The application of infrared radiation to food processing has gained momentum due

to its inherent advantages over the conventional heating systems. Infrared heating

has been applied in drying, baking, roasting, blanching, pasteurization, and steriliza-

tion of food products.

5.4.1 DRYING AND DEHYDRATION

Infrared heating has an imperative place in drying technology and extensive research

work has been conducted in this area. Most dried vegetable products are prepared

conventionally using a hot-air dryer. However, this method is inappropriate when

dried vegetables are used as ingredients of instant foods because of the low rehy-

dration rate of the vegetables. Freeze-drying technique is a competitive alternative;

however it is comparatively expensive.

new process for the production of high-quality dried foods at low cost [1]. The

use of IR radiation technology for dehydrating foods has numerous advantages

ess control parameters, and space saving along with clean working environment

[1,10,11,12].

Therefore, FIR drying operations have been successfully applied in recent years

for drying of fruit and vegetable products such as potatoes [13,14], sweet potatoes

[15], onions [12,16], kiwifruit [17], and apples [18,19]. Drying of seaweed, vegeta-

also found its application in food analysis to measure water content in food products

[20,21].

Generally, solid materials absorb infrared radiation in a thin surface layer. How-

ever, moist porous materials are penetrated by radiation to some depth and their

transmissivity depends on the moisture content [22]. Energy and mass balance devel-

oped by Ratti and Mujumdar [23] accounts for the shrinkage of the heated particle

and absorption of infrared energy. Theoretical calculations showed that intermittent

infrared drying with energy input of 10 W/m2 becomes equivalent to convective dry-2

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including reduction in drying time, alternate energy source, increased energy effi-

Application of FIR drying in the food industry is expected to represent a

ciency, uniform temperature in the product while drying, better-quality finished

ing in which the heat transfer coefficient would be as high as 200 W/m K.

4% of the total reflection producing a shine of polished surfaces. The rest of the

reflection occurs where radiation enters the food material and scatters, producing

a simultaneous decrease in transmittance and increase in reflection occurs. However,

products, a reduced necessity for air flow across the product, high degree of proc-

bles, fish flakes, and pasta is also done in tunnel infrared dryers. Infrared drying has



Factors affecting IR drying kinetics have been studied by several researchers.

surface temperature of the radiator. Optimization of the FIR heating process for

shrimp dehydration suggested that the effect of plate distance on the drying rate was

in the plate and air temperature [24]. Nowak and Levicki [18] reported that infrared

drying of apple slices was an effective and much faster method of water removal

than convective drying under equivalent parameters. Exploring the IR convective

drying of onion slices, Sharma et al. [16] observed that the drying time increased

with the increase in air velocity at all infrared powers applied; however, it reduced

with an increase in infrared power and the drying took place in the falling drying

rate period.

5.4.2 INTEGRATED DRYING TECHNOLOGIES: IR AND CONVECTIVE DRYING

Even though IR drying is a promising novel method, it is not a panacea for all drying

processes. It appeals, because it is fast and produces heating inside the material being

dried, but its penetrating powers are limited [25,26]. Prolonged exposure of a bio-

logical material to IR heat results in swelling and ultimately fracturing of the mate-

rial [27]. Fasina et al. [28] showed that IR heating changes the physical, mechanical,

chemical, and functional properties of barley grains. IR heating of legume seeds to

140°C caused cracking on the surface [29]. However, a combination of intermittent

infrared heating and continuous convection drying of thick porous material resulted

sidered as surface treatment similar to other radiation technologies.

Application of combined electromagnetic radiation and conventional convective

as it gives a synergistic effect. Afzal et al. [30] reported that during the combined

convective and IR heating process of barley, the total energy required was reduced

by about 156, 238, and 245% as compared with convection drying alone at 40, 55 or

70°C, respectively. Datta and Ni [31] discussed the application of combined infrared,

microwave, and hot air heating food materials. Mongpreneet et al. [12] evaluated

the dehydrating synergy generated when using ceramic-coated radiators and a high-

vacuum environment to study drying of welsh onion.

Development of a continuous drying apparatus equipped with FIR heaters, NIR

heaters, and hot air blast can reduce the economic costs, drying time, and operating

temperature. However, vegetable size should be restricted to no more than 5 mm in

ous combined infrared and convective dryer for vegetables. The synergistic effect of

infrared and hot air led to rapid heating of the materials, resulting in a higher rate

of mass transfer. The evaporation of water took 48% less time and 63% less energy

consumption in combined mode drying as compared to convective drying.

Recently, the concept of FIR heating immediately after convective drying

(approximately 40°C) for drying of paddy has been utilized in the paddy industry in

Japan [33,34]. Gabel et al. [35] compared the drying and quality characteristics of

sliced high-solids onions dried with catalytic infrared (CIR) heating and forced air

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Masamura et al. [14] confirmed increased drying rates of potatoes with increasing

not significant, whereas the drying rate increased monotonically with an increase

in better product quality and energy efficiency [10]. Thus, IR radiation can be con-

heating is considered to be more efficient over radiation or convective heating alone,

thickness to improve drying efficiency [1]. Hebbar et al. [32] developed a continu-



convection (FAC) heating. CIR both with and without air recirculation had higher

maximum drying rates, shorter drying times, and greater drying constants than FAC

at moisture contents greater than 50% (d.b.).

A combination of IR heating with freeze-drying in sweet potatoes could reduce

the processing time by less than a half [36]. The effect of NIR on reduction of freeze-

drying time of beef was investigated by Burgheimer et al. [37]. The authors con-

cluded that shorter wavelength resulted in rapid drying and thus reduced drying

time. Drying time with infrared heating was reduced to 7 h, as opposed to 11 h with

convectional drying.

5.4.3 PATHOGEN INACTIVATION

IR heating can be used to inactivate bacteria, spores, yeast, and mold in both liq-

on the following parameters: Infrared power level, temperature of food sample,

peak wavelength, and bandwidth of infrared heating source, sample depth, types

of microorganisms, physiological phase of microorganisms (exponential or station-

ary phase) and types of food materials. Therefore, several researchers have investi-

gated the effects of these parameters on inactivation of pathogenic microorganisms

as follows.

5.4.3.1 Effect of Power and Sample Temperature

Increase in the power of infrared heating source produces more energy and thus total

energy absorbed by microorganisms (M/Os) increased, leading to increased levels of

microbial inactivation. Sterilization of wheat surface was investigated by Hamanaka

et al. [38]. Surface temperature increased rapidly as infrared rays directly heated the

surface without any need for conductors. Therefore, irradiating powers of 0.5, 1.0,

1.5, and 2.0 kW resulted in 60, 80, 125, and 195°C inside the experimental device,

and 45, 65, 95, and 120°C on the surface of wheat stack, resulting in reductions of

0.83, 1.14, 1.18, and 1.90 log10 CFU/g total bacteria after a 60 s treatment, respec-

tively. Dry heat inactivation of B. subtilis spores by infrared radiation was investi-

gated by Molin and Ostlund [39]. D values of B. subtilis at 120, 140, 160, and 180°C

were 26 min, 66 s, 9.3 s, and 3.2 s, respectively. Shorter treatment time was enough

to inactivate pathogens at higher temperatures and the estimated Z value was 23°C.

E. coli population was reduced by 0.76, 0.90, and 0.98 log10 after 2 min exposure to

IR radiation when the temperature of the bacterial suspension was maintained at 56,

58, and 61°C, correspondingly [40].

5.4.3.2 Effect of Peak Wavelength and Bandwidth

As indicated earlier, food and microbial components absorb certain wavelengths of

key components in order to ensure pathogen inactivation and minimize changes in

food quality. It would be feasible to selectively heat the M/Os present in food prod-

ucts without adversely increasing the temperature of sensitive food components. Jun

and Irudayaraj [41] utilized selective infrared heating in the wavelength range of

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uid and solid foods. Efficacy of microbial inactivation by infrared heating depends

infrared radiation. Therefore, it is beneficial to investigate the absorption pattern of



Fusarium proliferatum in corn meal. The selected wavelength denatures the pro-

tein in microorganisms, leading to a 40% increase in inactivation of A. niger and

F. proliferatum, compared to normal IR heating. For instance, a 5-min treatment

with nonselective and selective heating resulted in approximately 1.8 and 2.3 log10

CFU/g reduction of A. niger. Similarly, reductions of 1.4 and 1.95 log10 CFU/g of

F. proliferatum were obtained with 5 min of nonselective and selective heating,

respectively. Although the sample temperatures after selective or non-selective

infrared heating were identical, absorption of energy by fungal spores was higher in

selective heating, leading to a higher lethal rate [41].

Total energy decreases as the peak wavelength increases. Therefore, NIR radia-

tion with short wavelength has relatively higher energy level than FIR radiation with

subtilis treated with three infrared heaters (A, B, and C) having different peak wave-

lengths (950, 1100, and 1150 nm) and radiant energies (4.2, 3.7, and 3.2 μW/cm2/nm),

respectively. Air-dried Bacillus subtilis solution placed on a stainless steel Petri dish

was treated with infrared heating after adjusting the water activity using a desiccator.

Surface temperature of Petri dish was 100°C after a 2 min exposure for all the heat-

ers. Pathogen inactivation was higher with heater A than those with heaters B and C,

although temperature was the same for all the heaters. For example, at water activity

of 0.7, decimal reduction times of heaters A, B, and C were approximately 4, 12, and

with the radiation spectrum [42].

5.4.3.3 Effect of Sample Depth

The penetration depth of IR radiation is very low. An increase in the sample depth

slows down the bulk temperature increase of foods [43]. A 90% reduction in IR

power was observed within a thin layer of 40 μm in bacterial suspension [44]. There-

fore, the effect of IR radiation on the microbial inactivation diminishes as the sample

thickness increases. Decreasing the sample depth also accelerates the inactivation

of spores [45] and E. coli and S. aureus [46]. The ratio of number of injured cells to

the number of survivors increased as the depth decreased. For example, S. aureus population was reduced by approximately 2 and 5 log10 CFU/mL at 321°K, when the

sample depths were 2.9 and 0.9 mm, respectively. Similarly, E. coli population in the

samples with 1.3 and 2.2 mm in depth resulted in approximately 1.33 and 1.66 log10

CFU/mL reductions, respectively at 321°K.

5.4.3.4 Types of Microorganisms

Resistances of bacteria, yeasts, and molds to infrared heating might be differ-

ent due to their structural and compositional differences. In general, spores are

more resistant than vegetative cells. When Bacillus subtilis spores in physiologi-

cal saline were exposed to infrared heating, a spore population increased up to

ing in shoulder and tailing effects. Upon infrared heat treatment, vegetative cells

were inactivated followed by activation of spores. An initial increase in B. subtilis

55534_C005.indd 12155534_C005.indd 121 10/22/08 12:06:05 PM10/22/08 12:06:05 PM

5.88–6.66 μm using optical bandpass filters for inactivation of Aspergillus niger and

longer wavelength. Hamanaka et al. [42] studied the inactivation efficacy of Bacillus

22 min, respectively. Therefore, it is obvious that inactivation efficiency is associated

five times in the first 2 min, followed by subsequent exponential reduction, result-



population was caused by heat shock germination of spores. A 10-min treatment

with infrared heating resulted in more than 90% reduction in B. subtilis population

[47]. Hamanaka et al. [42] also reported a shoulder effect where B. subtilis spores

were germinated.

Cereal surface is often contaminated with spore formers like Bacillus, Aspergillus, and Penicillium. Wheat was treated with infrared heating at 2.0 kW

for 30 s, followed by cooling for 4 h, and again treated for 30 s with infrared heat-

ing to obtain a 1.56 log10 CFU/g reduction. The irradiation helped in activation

of spores into vegetative cells and the second irradiation effectively inactivated

spore formers. Furthermore, intermittent treatment can minimize the quality

changes, as continuous treatment longer than 50 s resulted in discoloration of

wheat surface [38].

Naturally occurring yeasts in honey were completely inactivated with an 8-min

infrared heat treatment [48]. The temperature of the honey was raised to 110°C after

the treatment, resulting in microbial reduction of 3.85 log10 CFU/mL.

5.4.3.5 Inactivation Mechanism

Inactivation of M/Os by IR heating may include inactivation mechanism similar

to that of ultraviolet light (DNA damage) and microwave heating (induction heat-

ing) in addition to thermal effect, as infrared is located between ultraviolet and

microwave in the electromagnetic spectrum [38]. Thermal inactivation can damage

DNA, RNA, ribosome, cell envelope, and proteins in microbial cell. Sawai et al. [43]

investigated the inactivation mechanism of E. coli treated with infrared radiation in

phosphate buffer saline. They proposed that sub-lethally injured cells will become

more sensitive to an inhibitory agent which has an inhibitory action on the damaged

portion of the cell. Four inhibitory agents, namely, penicillin (PCG; inhibits cell

wall synthesis), chloramphenicol (CP; inhibits protein synthesis), rifampicin (RFP;

inhibits RNA synthesis), and nalidixic acid (NA; inhibits DNA synthesis) were used

for the enumeration of pathogens. An 8-min infrared radiation at a wattage of 3.22

kW/m2 resulted in approximately 1.8, 1.9, 2.7, and 3.2 log10 reduction of E. coli, when

NA, PCG, RFP, and CP enriched agars were used for enumeration, respectively.

When no inhibitory agents were present, a 1.8 log reduction was obtained. This

observation implies that approximately 0.1, 0.9, and 1.4 log reductions were caused

by inhibitory actions of PCG, RFP, and CP, respectively. With conductive heat-

ing, similar damages were observed; however, RNA, protein, and cell wall showed

more vulnerability to IR heating than conductive heating. The order of magnitude

of infrared damages was as follows: Protein>RNA>cell wall>DNA. RFP inhibits

RNA polymerase in E. coli and CP binds ribosomal subunits and inhibits peptidyl-

transferase reactions [43].

Sawai et al. [45] reported that for both stationary and exponential phase cells,

sensitivity to NA increased as the sample temperature increased. However, there

was only a small increase, indicating that minimal damage occurred in the DNA. In

particular, exponential phase cells had more cell wall and membrane damage than

stationary phase cells. However, more serious injuries to RNA polymerase occurred

for stationary phase cells, compared to exponential phase cells [45]. Transmission

electron microscopic observation and infrared spectroscopy of IR-treated S. aureus

55534_C005.indd 12255534_C005.indd 122 10/22/08 12:06:06 PM10/22/08 12:06:06 PM



content leakage, and mesosome disintegration [49].

5.4.3.6 Types of Food Materials

As described earlier, IR radiation has a poor penetration capacity. However, the sur-

face temperature of food materials increases rapidly and heat is transferred inside

by thermal conduction. Typical thermal conductivities of solid foods are much lower

than liquid foods. Convective heat transfer occuring inside the liquid foods under IR

heating can contribute to an increase in the lethality of microbes. A summary of the

study pertinent to pathogen inactivation in different types of food materials such as

solid, liquid, and nonfood materials is given in Table 5.3.

5.4.4 IR HEATING IN OTHER MISCELLANEOUS FOOD PROCESSING OPERATIONS

The usefulness of IR heating has also been demonstrated in various other food

processing applications such as roasting, frying, broiling, heating, and cooking meat

and meat products, soy beans, cereal grains, cocoa beans, and nuts.

orized IR broiling is a unique and innovative method. Khan and Vandermey [50] pre-

pared ground beef patties by IR broiling in a conveyorized broiler. Results showed

that due to high temperatures and short cooking times, the infrared broiler could

produce more servings per hour, compared to conventional gas heating. In addition,

it was found that ground beef patties broiled by tube broiler did not have any adverse

effects on the cooking quality (number of samples cooked/min, percentage shrink-

and overall acceptability), as compared to conventional gas broiling method. Sakai

and Hanzawa [1] reported on the performance of infrared-based systems with con-

tive study indicated energy savings of 45–70% with infrared heating. Abdul-Kadir

et al. [51] conducted imbibition studies and cooking tests to evaluate the effect of IR

heating on pinto beans (Phaseolus vulgaris) heated to 99 and 107°C. IR-heating was

found to improve rehydration rate and degree of swelling of pinto beans; however,

Studies on color development during IR roasting of hazelnuts were reported

by Ozdemir and Devres [52]. Olsson et al. [53] found that infrared radiation and jet

impingement, as compared with heating in a conventional household oven, increased

the rate of color development of the crust, and shortened the heating time of parbaked

baguettes during post-baking. Furthermore, the fastest color development was obtained

by combining infrared and impingement heating. The rate of water loss increased due

to a higher heat transfer rate, but the total water loss was reduced because of a shorter

heating time. In general, the formed crust was thinner for IR-treated baguettes.

5.5 SOURCES OF IR HEATING

Two conventional types of infrared radiators used for process heating are electric or

55534_C005.indd 12355534_C005.indd 123 10/22/08 12:06:06 PM10/22/08 12:06:06 PM

cells clearly verified cell wall damage, cytoplasmic membrane shrinkage, cellular

ventional ovens for baking rice crackers and for roasting fish pastes. The compara-

cooking time of pinto beans was significantly increased.

gas-fired heaters. These two types of IR heaters generally fit into three temperature

With the growing interest in flame-broiling and rapid cooking methods, convey-

age, number of servings/h) or sensory quality (appearance, flavor, texture, juiciness


124 Fo

od

Processin

g Op

eration

s Mo

delin

g: Design

and

An

alysis

TABLE 5.3Inactivation of Pathogenic Microorganisms by Infrared Heating

Pathogen Food/non-food Material Temperature/energy Time Log Reduction* Reference

Solid FoodsMonilia fructigena Strawberry ∼50°C++ 10 s 2.5–5.2 log (estimated) [60]

Total bacterial count Wheat or soybean surface 1.5 kW 10 s ∼3.0 [47]

Total aerobic plate count Onion 80°C (average 2226 W/m2) ∼24 min 1.72 ± 0.45 log10 CFU/10g [35]

Coliform counts Onion 80°C (average 2226 W/m2) ∼24 min 4.04 ± 0.47 log10 CFU/10g

Yeast and mold Onion 80°C (average 2226 W/m2) ∼24 min 1.26 ± 0.14 log10 CFU/10g

Wheat surface 2.0 kW 60 s ∼1.9 log10 CFU/g [38]

Listeria monocytogenes Turkey frankfurters 70°C++ 82.1 s 3.5 ± 0.4 log10 CFU/cm2 [65]

75°C++ 92.1 s 4.3 ± 0.4 log10 CFU/cm2

80°C++ 103.2 s 4.5 ± 0.2 log10 CFU/cm2

Aspergillus niger spores Corn meal 72°C+++ 6 min 1.8 [41]

Aspergillus niger spores Corn meal 68°C+++ 6 min 2.3

Fusarium proliferatum spores Corn meal 72°C+++ 6 min 1.4

Fusarium proliferatum spores Corn meal 68°C+++ 6 min 1.95

Listeria monocytogenes Oil-browned deli turkey 399°C around product surface 75 s 3.7 log10 CFU/mL [69]

Liquid FoodsYeast Honey 0.2 W/cm2 8 min ∼3.85 log10 CFU/mL+ [48]

55534_C005.indd 12455534_C005.indd 124

10/22/08 12:06:07 PM10/22/08 12:06:07 PM

(with an optical filter: 5.45–12.23 μm)

(with an optical filter: 5.45–12.23 μm)

Natural bacterial microflora


Infrared

Rad

iation

for Fo

od

Processin

g 125

Nonfood MaterialsBacillus subtilis Stainless steel plate at water

activity of 0.7

4.2 μW/cm2/nm (peak wavelength: 950 nm)



4 min**

22 min**

12 min**

[42]

E. coli Nutrient agar (depth = 0) 4.36 × 103 6 min ∼2.30–2.48 log10 CFU/plate+ [87]

(depth = 1 mm from surface) 4.36 × 103 6 min ∼ 0.70 log10 CFU/plate

(depth = 2 mm from surface) 4.36 × 103 6 min ∼ 0.66 log10 CFU/plate

Bacillus subtilis spores Steel plate 180°C 3.2 s** [39]

E. coli Phosphate buffer saline 3.22 kW/m2 8 min 1.8 log10 CFU/mL [43]

E. coli Phosphate buffer 61°C 2 min 0.98 log10 CFU/mL [40]

Aspergillus niger spores Physiological suspension 1.0 kW 40 s 4.0 to 5.0 [47]

Bacillus subtilis spores Physiological suspension 1.0 kW 10 s ∼1.0

* In (log10

** D value.+ No growth observed after treatment.++ Surface temperature.+++ Temperature of corn meal.

Source: From Krishnamurthy, Khurana, Jun, Irudayaraj, and Demirci. Infrared heating in food processing: an overview. Comp Reviews in Food Science and Food Safety.

Blackwell, Jan 2008, v.7. With permission.

55534_C005.indd 12555534_C005.indd 125

10/22/08 12:06:08 PM10/22/08 12:06:08 PM

CFU/mL), unless specified.



ranges [54]: 343–1100°C for gas and electric IR, and 1100–2200°C for electric IR

only. IR temperatures are typically used in the range of 650–1200°C to prevent char-

ring of products. The capital cost of gas heaters is higher, while the operating cost is

cheaper than that of electric infrared systems. Electrical infrared heaters are popular

because of installation controllability, ability to produce prompt heating rate, and

ing the desired wavelength for a particular application. In general, the operating

heaters ranges from 30 to 50% [54]. The spectral region suitable for industrial proc-

ess heating ranges from 1.17 to 5.4 μm, which corresponds to 260–2200°C [55].

Infrared radiation is transmitted through water at short wavelength, whereas at

longer wavelengths it is absorbed at the surface [1]. Hence, drying of thin layers

give better results at the NIR region. Studies to investigate the superiority of FIR to

NIR radiation have also been found in the literature. Sakai and Hanzawa [1] have

discussed the effects of the radiant characteristics of heaters on the crust formation

Radiant heating with an NIR heater led to a greater heat sink into food samples,

resulting in formation of relatively wet crust layers, compared to dry layers formed

by FIR heaters. However, the rate of color development by FIR heaters was greater

with NIR heaters, primarily due to a more rapid heating rate on the surface.

using infrared sources at λmax of 2.7 μm (MIR) and at λmax of 4.0 μm (FIR). With

a higher energy source (MIR), change in core temperature followed closely the

change in surface temperature with a shorter cooking time. Fat content of the food

was found to be independent of core temperature. However, with the lower energy

source (FIR), the increasing rate of core temperature was dependent on the fat con-

tent, showing that targeted core temperature was achieved more quickly as the fat

content increased.

FIR energy penetration into the food has gained ceaseless concern. Hashimoto et

al. [56,57] studied the penetration of FIR energy into sweet potato and found that FIR

radiation absorbed by the vegetable model was damped to 1% of the initial values at

a depth of 0.26–0.36 mm below the surface, whereas NIR showed a similar reduction

at a depth of 0.38–2.54 mm. Sakai and Hanzawa [1] reported the penetration depth

of the FIR energy did not affect the temperature distribution inside the food. Fur-

ther, they indicated that FIR energy penetrates very little, almost all the energy being

converted to heat at the surface of the food, which was consistent with the study of

Hashimoto et al. [25], evaluating FIR heating technique as a surface heating method.

Table 5.4 shows the penetration depth of NIR energy into various food products [58].

5.6 QUALITY AND SENSORY CHANGES BY IR HEATING

during IR heat treatment for ensuring commercial success. Several researchers have

studied the quality and sensory changes of food materials during IR heating.

Application of infrared radiation in a step wise manner by slowly increasing the

power, with short cooling between power levels, resulted in less color degradation

55534_C005.indd 12655534_C005.indd 126 10/22/08 12:06:08 PM10/22/08 12:06:08 PM

efficiency of an electric IR heater ranges from 40 to 70%, while that of gas-fired IR

seems to be more efficient at the FIR region, while drying of thicker bodies should

Sheridan and Shilton [55] evaluated the efficacy of cooking hamburger patties

It is crucial and beneficial to investigate the quality and sensory changes occurring

cleaner form of heat. Electric infrared emitters also provide flexibility in produc-

and color development at the surfaces of foods, such as white bread and wheat flour.



than with intermittent infrared heating [59]. Reductions in overall color change of

37.6 and 18.1% were obtained for potato and carrot, respectively. The quality of beef

produced by infrared dehydration was similar to that of conventional heating as indi-

cated by the surface appearance and taste tests [37]. Longer infrared heat treatments

may darken the color of onion due to browning [35].

Hebbar et al. [48] suggested that a 3–4-min infrared heat treatment was adequate

for producing commercially acceptable products, with reduction in yeast cells and

acceptable changes in hydroxymethylfurfural and diastase activity. Infrared heating

raised the internal temperature of the strawberries not above 50°C, while the surface

temperature was high enough to effectively inactivate microorganisms. Therefore,

infrared heating can be used for surface pasteurization of pathogens without deterio-

rating the food quality [60].

in rancidity development [61]. Compared to regular freeze-drying, IR-assisted

freeze-drying of yam brought about lower color differences as well as faster dehy-

dration. Furthermore, infrared heating leading to a higher dehydration ratio implies

that infrared heating reduces serious product shrinkage [62].

IR heat-treated lentils were found to be darker than raw lentil, though there was

infrared heat treatment, in addition to having a more open microstructure; thus,

enhancing the rehydration characteristics [63].

Sensory evaluation of ground beef patties treated by infrared heating and gas

TABLE 5.4Penetration Depth of NIR (0.75–1.4 μm) into Food Products

Product Spectral Peak (μm) Depth of Penetration (mm)

Dough, wheat 1.0 4–6

Bread, wheat 1.0 11–12

Bread, biscuit, dried 1.0 4

0.88 12

Grain, wheat 1.0 2

Carrots 1.0 1.5

Tomato paste,

70–85% water 1.0 1

Raw potatoes 1.0 6

Dry potatoes 0.88 15–18

Raw apples 1.16 4.1

1.65 5.9

2.36 7.4



permission.

55534_C005.indd 12755534_C005.indd 127 10/22/08 12:06:09 PM10/22/08 12:06:09 PM

significant difference between two treatments [50]. However, the appearance of

The evaluation of full-fat flour made from IR-heat treated soybeans maintained

no visible indication [63]. Cell walls of lentils were less susceptible to fracture after

freshness similar to fresh flour for 1 year. However, untreated samples resulted

broiling in terms of flavor, texture, juiciness, and overall acceptability showed no



gas-broiled patties were rated higher than infrared heating, as seen by the scores of

10.94 and 9.62 for gas broiling and IR heating, respectively. Pungency of onions fol-

lowing infrared radiation decreased with reduction in moisture [35]. Infrared heating

of carrots provided less damage to the tissue than blanching, as observed by lower

relative electrolyte leakage values and microscopic observations [64]. Furthermore,

infrared-treated carrots had higher tissue strength while effectively inactivating the

enzymes on the carrot surface.

Although infrared heat-treated turkey samples were slightly darker than the

difference in color values as measured by L*, a*, and b* values [65]. When menu

servings of peas were held at 50–60°C for 2 h by IR lamps, the quality of peas dete-

riorated and resulted in unacceptable products [66]. Bitterness and protein solubility

of peas were reduced after IR heat treatment [67]. Furthermore, canola seeds had

higher dehulling capacity after infrared heating [67]. Head rice yield was improved

by infrared heating while the whiteness of the rice was maintained [68].

Chlorophyll content of dehydrated onions treated by infrared increased with an

increase in irradiation power [12]. Infrared heating provided a more appealing brown

color and roasted appearance to deli turkey, in addition to effectively pasteurizing

the surface [69]. Infrared heating and jet impingement of bread resulted in rapid

drying and enhanced color development, compared to conventional heat treatment

[53]. Though the thickness of bread crust increased faster, a short IR treatment time

enabled the formation of thinner crust.

various food products. As the literature review substantiates, IR heating does not

antioxidant activities.

5.7 IR HEAT TRANSFER MODELING

Modeling of infrared heat transfer inside food has been a research-intensive area

because of the complexity of optical characteristics, radiative energy extinction, and

combined conductive and/or convective heat transfer phenomena.

Diffusion characteristics in relation to radiation intensity and thickness of slab were

inside food systems under FIR radiation. The radiation energy driving internal mois-

ture movement during FIR drying of a potato produced the activation energy for diffu-

sion inversely proportional to the slab thickness [13]. Sakai and Hanzawa [1] assumed

that most FIR radiation energy would be absorbed at the surface of a food system due

to the predominant energy absorption of water. Energy would thereafter be transported

by heat conduction in the food. Based on this assumption, a governing equation and

boundary conditions to explain heat transfer derived from energy balance in a food

distribution in samples was in good agreement with model predictions, permitting con-

trol of the surface temperature to retain food properties without overtreatment.

Abe and Afzal [76] investigated four mathematical drying models, namely, an

exponential model, a Page model, a diffusion model based on spherical grain shape,

55534_C005.indd 12855534_C005.indd 128 10/22/08 12:06:10 PM10/22/08 12:06:10 PM

controls after treatments, refrigerated storage for an hour resulted in no significant

change the quality attributes of foods significantly, such as vitamins, protein, and

explored using the finite element method to explain the phenomenon of heat transfer

system were solved using Galerkin’s finite element method. The measured temperature

Table 5.5 briefly summarizes the effect of IR treatment on nutritional quality of



and an approximation of the diffusion model to address the thin-layer infrared dry-

ing characteristics of rough rice. They found the Page model as most satisfactory for

describing thin-layer infrared radiation drying of rough rice. Similarly, Das et al.

while studying the drying characteristics of high-moisture paddy.

TABLE 5.5 Effect of Infrared Treatment on Nutritional Quality of Food Products

Food ProductParameters Effecting Nutritional Quality Effect of Treatment Reference

Barley Germination

rate at 55ºC

25% increase by combination of IR

heating and convectional heating

[30]

Wheat Germination rates

(heat treatment

for 63 s each)

Convectional heating: 90–97%

Intermittent IR heating: 80–86%

Continuous IR heating: 78–85%

[70]

Lentils Phytic acid content Untreated: 2.34%

High density IR heating (170ºC): 1.06%

[63]

Full fat

soybeans

Protein solubility Infrared heating: 84%

Spouted bed drying: 82%

Extrusion: 73%

[71]

Lentils Protein solubility Untreated: 74.7%

High density IR heating (170ºC): 50.9%

[63]

Soymilk Protein digestibility Untreated: 83.2%

IR heat treated (110–115ºC): 86.5%

[72]

Crude canola

oil

Phosphorus contents Untreated canola seeds: 46 ppm

IR heat treated (123ºC): 273 ppm

[67]

Sulfur contents Untreated canola seeds: 1.4 ppm

IR heat treated (123ºC): 4.4 ppm

[67]

Soymilk Available lysine content Untreated: 4.64 g/16g N

IR heat treated (110–115ºC): 6.14 g/16g N

[72]

Fried chicken Thiamine retention Reheated by IR heating: 81–84%

Convection heating: 86–96%

[73]

Orange juice D-values for vitamin C

degradation at 75°C

Convectional heating: 27.02 min

Ohmic heating: 23.72 min

Infrared heating: 23.76 min

[74]

Full fat

soybeans

Reduction in urease

activity at 140°C and

28% moisture (d.b.)

Infrared heating: 53%

Spouted bed drying: 30%

[71]

Peanut hulls Antioxidant activities

(total phenolic

compounds in water

extract, after 60 min)

FIR irradiation: 141.6 μM

FIR heating: 90.3 μM

[75]

Radical scavenging

activities

FIR irradiation: 48.83%

FIR heating: 23.69%



permission.

55534_C005.indd 12955534_C005.indd 129 10/22/08 12:06:10 PM10/22/08 12:06:10 PM

[77] also reported that Page model adequately fitted the experimental drying data



which would require the least computing time [78]. However, in a proposal sug-

gested for the solution of heat transfer problems for food materials, it was recom-

mended that if the solution region represents a simple rectangular domain, then

strategy [79].

Tsai and Nixon [80] investigated the transient temperature distribution in a

multilayer composite, semi-transparent or transparent, absorbing and emitting

the initial and boundary conditions in consideration of the effects of both thermal

radiation and conduction within each layer and convection on both exterior sur-

faces were solved by a hybrid numerical algorithm, using a fourth-order explicit

space variable.

IR frying was successfully predicted by the model developed based on combined

infrared radiation and convection heating [81]. Heat conduction equation was solved

was determined by the step size of these differences.

A control volume formulation for the solution of a set of three-way coupled heat,

moisture transfer, and pressure equations with an IR source term was presented in

three dimensions. The solution procedure uses a fully implicit time-stepping scheme

to simulate the drying of potato during infrared heating in three-dimensional Car-

tesian coordinates. Simulation indicated that the three-way coupled model predicted

the temperature and moisture contents better than the two-way coupled heat and

mass transfer model. The overall predictions agreed well with the available experi-

mental data and demonstrated a good potential for application in grain and food

drying [82].

Togrul [19] investigated infrared drying of apple to create new suitable models

including combined effects of drying time and temperature. In order to explain the

Page, Wang and Singh, Henderson and Pabis, logarithmic, diffusion approach,

Midilli equation) were developed and validated. The variation of moisture ratio

with time could be well described by the model developed by Midilli et al. [83].

of infrared drying of apple were derived wherein the model derived from modi-

Moreover, a single equation was derived to predict the moisture ratio change dur-

ing infrared drying (0–240 min) of apple in the temperature range of 50–80°C.

The developed model is expected to predict drying behaviors of other vegetables

and fruit.

55534_C005.indd 13055534_C005.indd 130 10/22/08 12:06:11 PM10/22/08 12:06:11 PM

In general, numerical methods applied to solve the set of equations are finite

elements, finite difference, and finite volume or the control volume method. It is

often difficult to decide which solution strategy would give the best results and

the traditional finite difference methods should be the preferred discretization

Runge-Kutta method for the time variable and a finite difference method for the

numerically using the finite difference method. The infinitesimal differentials were

The experimentally measured temperature distribution of slices of beef during

replaced by differences of finite size and the degree of accuracy of the representation

drying behavior of apple, ten different drying models (Newton, Page, modified

fied Page II had lowest root mean squared error (RMSE), mean bias error (MBE),

and chi-square along with highest modeling efficiency and regression coefficient.

medium exposed to a thermal radiative heat flux. The governing conditions with

Sixty-six different model equations relating the temperature and time dependence

simplified Ficks diffusion (SFFD) equation, modified Page Equation-II, and



5.8 SELECTIVE HEATING BY INFRARED RADIATION

Very few attempts have been made to study selective heating in the food industry as

well as in nonfood research areas. Certain studies have been found in the literature

applied to electronics [84,85]. These studies on electronics showed the accessibility

and the spectral distribution of the radiative source. However, the studies did not

elaborate on the details or its implementation.

and control of the accurate wavelength should be considered for optimization of the

process. In practice, the IR source emits radiation covering a very wide range. Hence, it

In the context of food processing, wavelengths above 4.2 μm are most desirable

for an optimal IR process of food system due to predominant energy absorption of

water in the wavelengths below 4.2 μm [86]. Lentz et al. [87] discussed the impor-

tance of IR-emitting wavelength for thermal processing of dough. Excessive heating

of the dough surface and poor heating of the interior was observed when the IR

spectral emission was not consistent with the wavelengths best absorbed for dough.

Excessive surface heating, in the absence of corresponding heat removal to the inte-

rior, gave rise to crust formation thus inhibiting heat transfer.

From the earliest, Shuman and Staley [88] discussed that orange juice has a

minimum absorption at the range between 3 and 4 μm, whereas dried orange sol-

ids have a maximum absorption at the same region. When using an IR source with

the maximum peak at wavelength of 4 μm, the radiation energy was not properly

absorbed by orange juice; however, dried orange solids could absorb IR energy pre-

dominantly. Hence, the IR source was controlled to emit the spectral ranges between

5 and 7 μm to obtain desirable absorption of orange juice. Their work clearly shows

the importance of spectral control of the IR source to manipulate the delivery of heat

A study by Bolshakov et al. [89] suggested that a maximum transmission of IR

radiation should cover the spectral wavelength of 1.2 μm obtained by analysis of

the transmittance spectrograms of lean pork for deep heating of pork. A two-stage

max of 3.5–3.8 μm (FIR) and the second stage for greater penetra-

max of 1.04 μm (NIR). Higher moisture

content and sensory quality of the products were obtained using combined FIR and

NIR heaters, compared to the conventional method. A similar study explored by

Dagerskog [81] used two alternative types of infrared radiators for frying equipment,

ature was 2340°C at 220 V rating, corresponding to λmax of 1.24 μm as NIR region,

and tubular metallic electric heaters (Backer 500W, type 9N5.5) at a temperature of

680°C at 220 V, corresponding to λmax of 3.0 μm as FIR region. It was observed from

of the radiation decreased, indicating that although the short-wave radiation (NIR)

had a higher penetrating capability than the long-wave radiation (FIR), the heating

55534_C005.indd 13155534_C005.indd 131 10/22/08 12:06:11 PM10/22/08 12:06:11 PM

Most infrared heaters consist of lamps emitting the spectrum with one specific peak

wavelength corresponding to a fixed surface temperature. The type of infrared emitter

of selective heating based on the relation between the optical properties of objects

is a challenge to cut off the entire spectral distribution to obtain a specific bandwidth.

amounts to specific food materials.

frying process designed consisted of the first stage to aim surface heat transfer by

which were quartz tube heaters (Philips 1 kW, type 13195X) whose filament temper-

radiant flux with λtion of heat transfer by radiant flux with a λ

the study that both penetration capacity and reflection increased as the wavelength

effects were almost the same due to body reflection.



There seems to be a lack of consistent methods to explore the intrinsic selective

heating process in the area of food engineering. Note that Dagerskog and Österström

in their frying experiment of pork to transmit only the wavelength above 1.507 μm,

which turned out be a good example for design of selective IR heating systems to

emit the spectral regions of interest.

Recently, Jun [90] developed a novel selective FIR heating system, demonstrating

the importance of optical properties besides thermal properties when electromag-

netic radiation is used for processing. The system had the capability to selectively

ters that can emit radiation in the spectral ranges as needed. Simulation of the heat

transfer phenomena in the food domain was done in one dimension (Figure 5.4)

because only the top surface of food sample was exposed to the incident IR radiation.

The governing differential equation can be described by:

ρ ∂∂

∂∂

∂∂

CTt

kT

z

qzpr= −

2

2 (5.1)

r2 3

p

t is the time, and z is the distance (m).

If the initial temperature (T0) is assumed to be uniform, the initial boundary

condition is given by

T z t T z D t( , ) = ≤ ≤ =0 for 0 0; (5.2)

where T0 is the initial sample temperature (°C) and D is the sample thickness (m)

Soy protein Glucose

Glass plate

n = N (Boundary 2)

n = 1 (Boundary 1)

n = 2

n = 3

25.4 mm

Infrared radiation

qconv

qrad,out

qabs

qcond

denotes a thermocouple

0dz

dq

7 mm

2 mm

Cylindrical vessel

Rotated at 3 rpm by an AC motor (3.8 W)

FIGURE 5.4 Schematic of the discretized food domain with food holder. (From Jun, S.

2002. Selective far infrared heating of food systems. Ph.D. Dissertation, The Pennsylvenia

State University, USA.)

55534_C005.indd 13255534_C005.indd 132 10/22/08 12:06:12 PM10/22/08 12:06:12 PM

[5] first used a bandpass filter (Optical Coating Laboratory, Inc., type no. L-01436-7)

heat higher absorbing components to a greater extent using optical band pass fil-

where, q is the heat flux (W/m ), T is the temperature (°C), ρ is the density (kg/m ),

C is the specific heat of food sample (J/kg⋅°C), k is thermal conductivity (W/m⋅°C),



Considering convection (qconv) and radiation losses (qrad,out) at boundary 1 and

conduction loss (qcond) at boundary 2, the boundary conditions can be given by

kTz

h T T q E T Ez

∂∂

ε λ=

= − − + −0

1 11conv r b b( ) ( ) ( ) ( ( ) (∞ TT t∞⎡⎣ ⎤⎦ >) ,at the top 0

and

kTz

q Nk

dT T

z D

N∂∂ ∞

=

= − −rglass

glass

at the bott( ) ( ) oom, t > 0 (5.3)

Here, Eb(T) is the total emissive power at a given source temperature obtained using

Planck’s integral, as presented by

E T E

hc

n eb b

0

5( , )

[ –1]0( ) = =λ

∞

∫ T d hc n kTλ λ πλ λ

2 02

2

00

∞

∫ dλ (5.4)

The IR radiation is not only absorbed by the surface but also penetrates into the food

q n q S z nr absexp(– ( – 1))( ) = ⋅ ⋅Δ (5.5)

where, qabs−1), and Δz is the grid size (m) [81].

boundary 1 denotes an assembly of FIR source lamps, boundary 2 denotes a cone-

shaped waveguide to keep IR radiation from dispersing out into the air, boundary

3 denotes an opening outlet, and boundary 4 denotes a sample surface. Under the

IR source (boundary 1)

Waveguide (boundary 2)

Sample (boundary 4)

Opening (boundary 3)

T2, ε

2, q

2 =

0 (insulation)

T4, ε

4, q

4

T3, ε

3, q

3

T1, ε

1, q

1

FIGURE 5.5 2002. Selective far infrared heating of food systems. Ph.D. Dissertation, The Pennsylvenia

State University, USA.)

55534_C005.indd 13355534_C005.indd 133 10/22/08 12:06:13 PM10/22/08 12:06:13 PM

is the extinction coefficient (m

Figure 5.5 shows a simplified schematic of the FIR heating system where

Simplified gray and diffuse enclosure of the heating chamber. (From Jun, S.

system, causing a local radiative heat flux of

is the initial radiant heat flux absorbed by food sample on the surface, S



condition that the IR lamps (boundary 1) and the waveguide (boundary 2) are gray

using the energy balance equation for each surface [91], given as

qF q E T Fi

i jj

i j j i iε ε− −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ = ( )−

=

−∑ 11

1

3

b ii j j i

j

E T i

q A q q

−

=∑

+ = =

b

1

1, 2, 3

A fo

( ),

( ,

1

3

1 3 3 20 0

=

rr insulation)

(5.6)

2

F is the view factor, and ε is the emissivity of each boundary. Equation 5.6 for each

boundary can be solved simultaneously.

abs, is dependent upon the

tion and the view factor as obtained from the opening (boundary 3) to the food

surface (boundary 4). This relationship can be expressed as

q F qabs = ⋅ ⋅ ⋅−3 4 3α(λ) τ(λ) (5.7)

3

integral form of q3 with respect to the wavelength. F3–4 can be calculated using the

equation set formulated for disk to a parallel coaxial disk with the same radius [91],

as given by

F X x Xa r

r3 4

22 2

2

1

24 1− = − −{ } = +

+, (5.8)

where r is the radius (m) of boundary 3 and boundary 4, and a is the distance (m)

between boundary 3 and boundary 4.

Equation 5.1 through Equation 5.3 because this technique is relatively simple and

very accurate for highly transient problems. It is seen from Figure 5.2 that the food

of accuracy of the representation is determined by the step size of these differences.

Finite difference formulas obtained from Taylor series expansions such as forward

difference and centered difference are used to approximate time and space deriva-

change with respect to time at different locations in the food domain are derived in

a discretized form (for n = 2 to N − 1) as,

T Tk t

C zT T T

tnt

nt

nt

n nt+

+ −= + − + −1

2 1 122

ΔΔ

Δρ p( )

( )t

ρρC zq n q n

p

r rΔ( ( ) ( ))+ − −1 1 (5.9)

55534_C005.indd 13455534_C005.indd 134 10/22/08 12:06:14 PM10/22/08 12:06:14 PM

spectral absorptivity (α) of food, the spectral distribution of filtered infrared radia-

where, τ is the filter transmissivity which is a function of the wavelength (λ). The

hence, food absorptivity (α) and filter transmissivity (τ) can be combined into an

An explicit (forward in time) finite difference method is applied to solve

domain is subdivided as indicated by the grid points for each layer. The infinitesimal

differentials are replaced by differences of finite size (time and space) and the degree

tives in the partial differential equation [80]. The first derivatives of temperature

and diffuse reflectors, the total heat flux absorbed by food sample can be calculated

where, q is the heat flux (W/m ) absorbed or emitted by each boundary (Figure 5.3),

The amount of heat flux absorbed by food surface, q

incoming IR heat fl ux transmitting the boundary 3, q is spectral dependent and



Hence for n = 1

T Tk t

C zT

k tC z

ht t t1

11 2 2 2

2 2 2+ = + − +ΔΔ

ΔΔρ ρp p

co

( ) ( )

nnv

p

p

r

ΔΔ

ΔΔ

tC z

T

tC z

q h

t

ρ

ρ

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

− − −

1

21( ( ) TT E T E T

tC z

q q∞ ∞ε(λ)( )ρ

+ − − −b b

p

r r( ) ( ) ) ( ( ) (1 2 1Δ

Δ)))

(5.10)

and n = N

T Tk t

C zT

k tC z

kNt

Nt

Nt+−= + − +1

2 1 2

2 2 2ΔΔ

ΔΔρ ρp p( ) ( )

gglass

glass p

ΔΔt

d C zTN

t

ρ

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

+22Δ

ΔΔt

C zq N

k

dT

ρ p

rglass

glass

( )+⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−∞

ttC z

q N q Nρ p

r rΔ( ( ) ( ))− −1

(5.11)

The temperature at time, t + 1 is explicitly expressed as a function of neighboring

be restricted by the stability criterion, which otherwise may be diverging and

for the explicit solution of Equations 5.9 through Equations 5.11 to be stable is

given by

ΔΔ

Δt

z C

k h z≤

+( )

( )

2

2

ρ p (5.12)

Based on Equation 5.9 through Equation 5.11, N simultaneous equations for N

nodal points can be formed and the unknown temperatures determined using

Matlab (v. 5.2, Natick, MA). Simulated temperature distributions at the upper

and mid section of the food domain were obtained and compared with the experi-

mental results. Applicability of this technique was demonstrated by selective

heating of soy protein and glucose. Soy protein was heated about 6°C higher than

glucose after 5 min of heating, exhibiting a reverse phenomenon when heating

with experimental data, thus supporting the mechanism of selective IR heating

(Figure 5.6) [92].

5.9 THERMAL DEATH KINETICS MODEL

Hashimoto et al. [93] developed a simple integrated model to predict the survivors

of E. coli under predicted temperature distribution during FIR pasteurization. Ana-

lytical and numerical models of bacterial spores have been developed to predict

55534_C005.indd 13555534_C005.indd 135 10/22/08 12:06:15 PM10/22/08 12:06:15 PM

temperatures at an earlier time t. The explicit finite difference solution should

never reach the final solution [20]. The common stability criterion to be satisfied

without the filter. Simulation results from the developed models were consistent

microbial spore growth during sterilization. Stumbo [94] first validated a model with



Time (sec)

200 50 100 150 200 250 300

30

40

50

60

70

80

Time (sec)

0 20 40 60 80 100 120 140

Tem

pera

ture

(oC

)T

em

pera

ture

(oC

)

20

30

40

50

60

70

80

90

Exp. Sim.

Soy protein

Glucose

Exp. Sim.

Soy protein

Glucose

(a)

(b)

FIGURE 5.6 Comparison between the simulated temperatures (denoted by ‘Sim.’) of soy

protein and glucose, and the measured data (denoted by ‘Exp.’) at the top surface during IR

infrared heating system–design and evaluation (Part I). J Drying Technol 21(1): 51–67.)

55534_C005.indd 13655534_C005.indd 136 10/22/08 12:06:16 PM10/22/08 12:06:16 PM

heating (a) without filter and (b) with filter. (From Jun S, and Irudayaraj J. 2003, Selective far



To overcome limitations of traditional models to predict spore populations during

treatment, especially under ultra-high temperatures, new models including spore

activation have been proposed [95]. The populations in a suspension of bacterial

spores subjected to lethal heat treatment were simulated using a composite model

involving simultaneous, independent activation and inactivation of dormant but via-

ble spores, and inactivation of activated spores.

Jun [90] developed an integrated model that combined the thermal death kinet-

ics with the IR heat transfer model and could predict the survivors of fungal spores

based on temperature prediction. Selective IR heating was found to differentially

deliver a higher degree of lethality to individual fungal spores. The denaturation of

the protein band as a target spectral region of selective heating might also partially

contribute to an increase in the lethality of fungal spores.

Recently, Tanaka et al. [60] combined Monte Carlo FIR radiation simulations

suitability of the method for surface decontamination in strawberries. The model

include radiation, convection, and conduction in a fast and comprehensive way.

Computations were validated against measurements with a thermographic camera.

FIR heating showed more uniform surface heating than air convection heating, with

a maximum temperature well below the critical limit of about 50°C. To improve the

system functionality in terms of heating rate and temperature uniformity, several

factors can be considered, i.e. system rotation, optimized heating cycles, and dif-

ferent heater geometries. The projected modeling approach can be used to achieve

such goals in a comprehensive manner and the model should be extended to consider

mass transfer and volumetric dissipation of the radiation power.

5.10 CONCLUSION AND FUTURE RESEARCH POTENTIAL

IR heating is a unique process; however, presently, the application and understanding

of IR heating in food processing are still in its infancy unlike the electronics and

allied sector where IR heating is a mature industrial technology. It is further evident

from this chapter that IR heating offers many advantages over convection heating

time-saving as well as increased production line speed. IR heating is attractive pri-

marily for surface heating applications. In order to achieve energy optimum and

nation of IR heating with microwave and other common conductive and convective

modes of heating holds great potential. It is quite likely that the utilization of IR

heating in the food processing sector will augment in the near future, especially in

the area of drying and minimal processing.

Over the last three decades several studies have been conducted to address vari-

ous technological aspects of IR heating for food processing. However, research needs

for the upcoming years may include the following:

55534_C005.indd 13755534_C005.indd 137 10/22/08 12:06:17 PM10/22/08 12:06:17 PM

first-order inactivation of uniformly activated spores during a sterilization process.

was a powerful tool to evaluate and address complex heating configurations that

efficient practical applicability of IR heating in the food processing industry, combi-

with convection-diffusion air flow and heat transfer simulations to investigate the

including greater energy efficiency, heat transfer rate, and heat flux that results in



maximum optical response of the target object when the emission band of

infrared and the peak absorbance band of the target object are identical.

Such manipulations of IR radiation for selective heating of foods could be

very useful.

2. Detailed insight into the theoretical explanation of IR effects especially

with regards to its interaction with food components, changes in taste and

3. Application of catalytic infrared (CIR) heating: CIR heating uses natural

gas or propane, which is passed over a mesh catalyst pad to produce ther-

mal radiant energy through a catalytic reaction. This reaction occurs below

magnetic radiant energy from CIR has peak wavelengths in the range of

medium- to far-infrared. The peak wavelengths match reasonably well with

the three absorption peaks of liquid water, which could result in rapid mois-

ture removal. Since CIR directly converts natural gas to radiant energy, it is

4. 3D modeling of food products: Studies on IR heating have generally been

applied to foods with a simple 1D or 2D geometry. There is a paucity of

information in the area of advanced 3D radiation modeling. Most cru-

cially, integrating microbial death kinetics with chemical kinetics due to

IR heating will provide a holistic approach to understand the of complex

microbial and chemical process kinetics and interactions as well as the

system design.

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143

6 Modeling of Ohmic Heating of Foods

Soojin Jun and Sudhir Sastry

CONTENTS

6.1 Introduction ................................................................................................... 143

6.2 Basic Principles ............................................................................................. 145

6.3 Case Study I: 2D Modeling ........................................................................... 146

6.3.1 Background ........................................................................................ 146

6.3.2 Packaging .......................................................................................... 147

6.3.3 Model Development........................................................................... 148

6.3.4 Model Validation ............................................................................... 152

6.3.5 Deliverables ....................................................................................... 158

6.4 Case Study II: 3D Modeling ......................................................................... 159

6.5 Case Study III: Multi-Phase Ohmic Heating ................................................ 164

6.6 Conclusion ..................................................................................................... 169

Nomenclature ......................................................................................................... 169

References .............................................................................................................. 169

6.1 INTRODUCTION

Ohmic heating technology has been investigated for heating various materials for a

long time. The basic principle of ohmic heating is that electrical energy is converted

to thermal energy within a conductor. Typically, an alternating current is applied

across the material (Figure 6.1). Because heating occurs by internal energy gen-

eration within the conductor, the method results in a remarkably even distribution

of temperatures within the material. Since the energy is almost entirely dissipated

within the heated material, there is no need for heat intervening heat exchange walls,

while in ohmic heating, ionic motion results in heat generation [2]. Ohmic heating

requires electrodes that make good contact with the food; however, microwave heat-

ing needs no physical contact [3].

A large number of potential applications exist for ohmic heating, including

blanching, evaporation, dehydration, fermentation, and extraction. In the use of

the application as a heat treatment for microbial control, ohmic heating provides

55534_C006.indd 14355534_C006.indd 143 10/22/08 10:04:24 AM10/22/08 10:04:24 AM

6.4.1 Model Verification ............................................................................. 159

thus the process has close to 100% energy transfer efficiency [1]. Various techniques

are available which use electric fields. In microwave or radio-frequency (RF) heat-

ing, a high frequency electric field excites the water molecules within the materials



rapid and uniform heating, resulting in less thermal damage to the product. A high

quality product with minimal structural, nutritional, or organoleptic changes can be

manufactured in a short operating time [4]. Ohmic heating is currently being used

for the processing of whole fruits, syruped fruit-salad and fruit juices in Japan and

the United Kingdom. Ohmic heating has shown to enhance drying rates [5–7] and

extraction yields [5,8] in certain fruits and vegetables.

critical factors that distinguish ohmic heating from conventional heating processes.

The rate of heat generation in ohmic heating depends strongly on the electrical con-

ductivity of the food. However, heterogeneous foods (e.g. liquid and particulate food

ent heat generation rates at distinct localized regions. The consequent heat transfer

within a phase or between phases of liquid–particulate mixtures further complicates

the determination of temperature distribution in ohmically heated systems. Addi-

tional factors contributing to the complexity of the ohmic heating process include

possible heat channeling causing hot spots and cold spots, complex coupling between

ohmic heating process for sterilizing food products is essential for process valida-

tion—an actual demonstration of the accurate reliability—and safety of the process

[11]. Mathematical modeling provides insight into the heating behavior of the ohmic

processes. The temporal and spatial temperature distribution obtained from a reli-

able mathematical model can provide information for the calculation of sterility and

cook value that will save time and money for validation experiments, and process

and product design.

The performance of a mathematical model for ohmic heating relies in part on

accurate inputs of material properties and parameter values. In most of the existing

models, the electrical conductivity values of both solid and liquid phases were con-

sidered as constants or as linear functions of temperature; however, often times these

values are inconsistent under varying ohmic heating conditions. There is always a

need to investigate these properties under a consistent condition as the ohmic heating

processes [12].

I

V R

Electrical analogOhmic heating

Alternating

current power

supply

Electrodes

Food

FIGURE 6.1 The concept of ohmic heating.

55534_C006.indd 14455534_C006.indd 144 10/22/08 10:04:25 AM10/22/08 10:04:25 AM

Modeling ohmic heating is a difficult task due to the unique characteristics and

materials may have significantly different electrical conductivity) may exhibit differ-

the temperature and electrical fields, and process parameters such as particle size,

shape and orientation to the electrical field [9,10]. Understanding the behavior of the


Modeling of Ohmic Heating of Foods 145

6.2 BASIC PRINCIPLES

equation [8]:

∇[σ(T) ∇V] = 0 (6.1)

where V is the voltage and σ is the electrical conductivity which varies as a

function of temperature (T). The most important parameter in the applicability

of ohmic heating is the electrical conductivity of the material. Most foodstuffs,

which contain water in excess of 30% and dissolved ionic salts have been found to

such as fats, oils, sugar and syrups are not suitable as their conductivities are too

low [13].

The temperature distribution is determined according to the following

Equation:

ρ ∂∂

∇ ∇CTt

k T Sp ( )= + (6.2)

p

conductivity. The symbol S is the internal energy source term which is generated by

S = σ(T) |∇V|2 (6.3)

For most aqueous based materials, the electrical conductivity increases linearly with

temperature [14].

σ(T) = σref (1 + mT) (6.4)

where σref is the electrical conductivity at a reference temperature and m is a

ity of liquid fruit products like juices and purees [14–16]. Mitchell and de Alwis

[17] measured electrical conductivity of pear and apple at 25°C. Castro et al. [18]

reported electrical conductivity of fresh strawberry over 25–100°C temperature

range.

Electrical properties of meat have also been investigated in recent years [19].

Conductivities of chicken [14,17], beef [14,20] and pork [21] have been measured,

of chicken breast over the sterilization temperature range. Shirsat et al. [23] reported

conductivities of different pork cuts and observed that lean is highly conductive

compared to fat, however, conductivity measurements were performed only at 20°C.

Recently, Sarang [24] summarized electrical conductivities and temperature model

parameters for various foodstuffs as shown in Table 6.1.

55534_C006.indd 14555534_C006.indd 145 10/22/08 10:04:26 AM10/22/08 10:04:26 AM

Electric field distribution within an ohmic heater is calculated by solving Laplace’s

conduct sufficiently well for ohmic heating to be applied. Non-ionized materials

where ρ is the density, C is the specific heat, t is the time, and k is the thermal

electric field.

temperature coefficient. Much research has been done on the electrical conductiv-

but the type of meat cut was not specified. Tulsiyan et al. [22] measured conductivity



6.3 CASE STUDY I: 2D MODELING

6.3.1 BACKGROUND

The primary goal of food systems in long-duration space missions is to provide the

crew with a palatable, nutritious, and safe food system and minimize volume, mass,

and waste [25]. The relevant food processing technologies must satisfy mission

constraints, including maximizing safety and acceptability of the food and minimiz-

ing crew time, storage volume, power, water usage, and the maintenance schedule. At

present, the galleys of the US space shuttle and International Space Station (ISS) have

been equipped with a rehydration station and convection oven, permitting addition

of hot or cold water and providing the ability to heat food to serving temperatures.

Even refrigerators and freezers have not yet been installed. However, long duration

methods to increase the shelf life. Food sterilization accomplished by heat would be

one of the most likely process requirements. Regardless of mission types, the need

would still exist for reheating meals to serving temperature and heating water for

personal use.

This work has involved the development of a package with a pair of electrodes,

to permit ohmic heating. The package is generally considered to pose a disposal

TABLE 6.1Electrical Conductivity–Temperature Model Parameters

σref (S/m) m (oC)–1

Fruits Apple-golden 0.089 0.049

Apple-red 0.079 0.057

Peach 0.179 0.056

Pear 0.124 0.041

Pineapple 0.076 0.060

Strawberry 0.234 0.041

Chicken Breast 0.663 0.020

Tender 0.567 0.021

Thigh 0.329 0.026

Drumstick 0.428 0.024

Separable fat 0.035 0.049

Pork Top loin 0.564 0.018

Shoulder 0.527 0.020

Tenderloin 0.551 0.021

Beef Bottom round 0.504 0.019

Chuck shoulder 0.456 0.023

Flank loin 0.318 0.038

Top round 0.472 0.024

Source: Sarang, S. 2008. Ohmic heating for thermal processing of low-acid foods

containing solid particulates. Ph.D. Thesis, The Ohio State University.

55534_C006.indd 14655534_C006.indd 146 10/22/08 10:04:27 AM10/22/08 10:04:27 AM

manned space flights beyond low earth orbit requires advanced food preservation



problem after use. Solid waste treatment in space for Advanced Life Support (ALS)

environment [26]. The overall intention of the proposed project is to investigate if

food in packaging could be sterilized by ohmic heating to yield a superior quality

product with long shelf life; to enable ohmic reheat in transit, thereby minimizing

the ESM; and potentially reusing the container post-food consumption, to contain

and sterilize waste.

ohmic heating with the package being reusable. The only related work found in the

similarity in packaging structure [27]. However, no detailed investigation of this

subject or its implementation is available.

In ohmic heating, electrochemical processes at the electrode/solution interfaces

must be avoided or minimized. Clearly, safety would be the primary consideration

throughout the duration of the mission. High frequency alternating currents allowing

electrochemical reaction [28,29]. Samaranayake [30] observed that pulsed ohmic

compared to conventional 60 Hz sinusoidal ohmic heating. It is, therefore, expected

that pulsed ohmic heating with high frequency and long delay time between pulses

would effectively avoid the worst scenario such as electrolytic gas production.

6.3.2 PACKAGING

trodes to permit ohmic heating of food materials (Figure 6.2a). Flexible packaging,

such as multilayered laminates, provides an alternative to the rigid container.

and nylon) on one side for external scratch resistance and a heat sealable polymer

is made. The electrode assembly made of aluminum foil with 2 mm in thickness is

placed between a folded laminate, with the electrodes extending out and heat sealed

on the edges.

The pouch inside the ohmic cell (Figure 6.2b) was powered with high

frequency pulsed alternating current generated by an Integrated-Gate-Bipolar-

Transistor (IGBT)-based power supply [31]. The power supply was developed to

generate the square waveforms with frequency of 10 kHz and duty cycle (pulse

widths/period) of 0.2, which were the parameters optimized to minimize electro-

chemical reactions on the electrodes [32]. Voltage and current data were collected

in a data acquisition unit (DAQ). During the sterilization process, the external air

assist line and pressure regulator were used to counterbalance internal pressures

and to suppress boiling. In addition, cold water circulation was installed and used

for post-process cooling.

55534_C006.indd 14755534_C006.indd 147 10/22/08 10:04:27 AM10/22/08 10:04:27 AM

None of the past work has dealt with heating of foods in a flexible package using

literature involved flexible batteries using food packaging materials, in terms of the

A package developed using flexible pouch materials incorporates a pair of foil elec-

applications requires that the material be safely processed and stored in a confined

only minimal charging of electrical double layers were found to significantly inhibit

heating could significantly reduce electrochemical reaction on electrode surfaces,

This study was aimed to optimize electrode configurations in a pouch, to yield

the most uniform, yet rapid heating thermal profiles in static systems.

Multilayered laminates (Smurfit-Stone Container, Co., Milwaukee, WI) consist of a

thin metal film (7 mm aluminum) that has a protective polymer film (4 mm polyester

(4 mm polyethylene) film on the other side, which becomes the interior after the pouch



To investigate the uniformity of heating of food materials within the package, three

cm in width) on the bottom right of the package, while Pouch B has V-shaped elec-

trodes (3 cm in width) at each end. Pouch C has one electrode (2 cm in width) on the

top middle and the other two electrodes (1 cm in width) at each end on the bottom.

6.3.3 MODEL DEVELOPMENT

Understanding that a reliable mathematical model can provide information for the

calculation of sterility and cook value which will save time and money for process

DAQ Data acquisition unit

PC Personal computer

REG Pressure regulator

V Voltage measurement

A Current measurement

Foil electrodes

Flexible package

Ohmic cell

Power supplyV

A

DAQ

PC

REG

REG

Air assist

Cooling water

Water drain

(a)

(b)

FIGURE 6.2 heating system (b). (From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic

36, 2005. With permission.)

55534_C006.indd 14855534_C006.indd 148 10/22/08 10:04:28 AM10/22/08 10:04:28 AM

A schematic of flexible package with foil electrodes (a) and pulsed ohmic

heating of foods inside the flexible package. Journal of Food Process Engineering, 28, 417–

different electrode configurations (pouches A, B, and C) were designed, as shown in

Figure 6.3. Pouch A has one electrode (2 cm in width) on the top left and the other (2



and product design, an ohmic heating model was developed to optimize the electrode

To develop the 2D transient model for foods under ohmic heating process, the

following assumptions are made for simplicity:

permitting the use of a 2D model. This assumption is useful since our main

intention is not process evaluation but rather design optimization.

2. Chicken noodle soup and black beans are considered to be in a single phase

wherein the electrical conductivity values of both solid and liquid materials

are considered as an identity with linear functions of temperature.

of temperature and pressure.

4. Heat losses to the environment are negligible.

This approach assumes no convection is implemented in Equation 6.2. This is a

reasonable assumption since our primary application involves a microgravity

environment, where no natural convection can occur. When there is little convective

heating, the temperature differences between different regions of a food system will

be more pronounced.

Electrode

(a)

(b)

(c)

FIGURE 6.3 C. (From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods

permission.)

55534_C006.indd 14955534_C006.indd 149 10/22/08 10:04:29 AM10/22/08 10:04:29 AM

inside the flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With

3. The thermophysical properties of the fluids are considered to be independent

Different electrode configurations: (a) Pouch A, (b) Pouch B, and (c) Pouch

configuration for temperature uniformity.

1. The temperature profiles are uniform along with the package length,



The corresponding initial conditions and boundary conditions are given as

follows.

T(x, y, t) = T0, ∀x, ∀y, t = 0, (6.5)

Electrical boundary conditions:

V (x, y, t)|wall = 0 V (ground) or U0, ∀t, x and y ∈ [electrodes], (6.6)

∇V x y t n( , , ) ⋅ =→

wall

0 , ∀t, x and y ∈ [pouch surfaces except electrodes], (6.7)

Thermal boundary condition:

k T x y t n∇ ( , , ) ⋅ =→

wall

0 , ∀t, x and y ∈ [pouch surfaces], (6.8)

where T0 is the initial temperature of food, U0 is the electric potential, and n→

is the

normal vector.

The governing Equations 6.1 through 6.4 are solved using the commercially

available CFD software Fluent (v. 6.1). The software is customized using user

the original platform. Basic C++ codes used for UDFs are provided as follows.

DEFINE_SOURCE (cell_source, cell, thread, dS, eqn) /* energy source term */

{

long double source; /* source term */

mag=NV_MAG(C_UDSI_G(cell,thread,0));

source = (A*C_T(cell,thread)+B)*mag*mag; /* Equations 6.3 and 6.4 */

dS[eqn] = A*mag*mag;

return source;

}

DEFINE_DIFFUSIVITY (cell_elect_conduct, cell, thread, i) /* electrical conductivity */

{

double theta;

return (theta = A* C_T(cell, thread)+B); /* Equation 6.4 */

}

/* This UDF is used to store the values of voltage gradient in a UDS (UDS-1) for

postprocessing. Note that UDS-0 is used to calculate voltage values for each cell and

this UDS (UDS-1) is used to store values of voltage gradients in each cell. To activate

hook up this UDS by choosing v_grad from the list in the UDF function hook up.

55534_C006.indd 15055534_C006.indd 150 10/22/08 10:04:30 AM10/22/08 10:04:30 AM

defined functions (UDFs) to solve the electric field model, which is not employed in

long double mag; /* electric field */

this UDS, first increase the number of UDS to 2, compile this group of UDF, then

Define -->User-defined -->Function --> Function hook. At Adjust function, choose



v_grad. Since UDS-1 is used only to store the values of the voltage gradient, don't

enum

{

V,

MAG_GRAD_V,

N_REQUIRED_UDS

};

DEFINE_ADJUST (v_grad, domain)

{

Thread *t;

int nt;

cell_t c;

face_t f;

int ns;

/* Fill the UDS with magnitude of voltage gradient. */

domain = Get_Domain(1);

thread_loop_c(t,domain)

{

if(NULL ! = THREAD_STORAGE(t,SV_UDS_I(V)) &&

NULL ! = T_STORAGE_R_NV(t,SV_UDSI_G(V)))

{

begin_c_loop(c,t)

{

C_UDSI(c,t,MAG_GRAD_V) = NV_MAG(C_UDSI_G(c,t,0));

}

end_c_loop_all(c,t)

}

}

thread_loop_f(t,domain)

{

if(NULL ! = THREAD_STORAGE(t,SV_UDS_I(V)) &&

NULL ! = T_STORAGE_R_NV(t->t0,SV_UDSI_G(V)))

{

begin_f_loop(f,t)

{

F_UDSI(f,t,MAG_GRAD_V) = C_UDSI(F_C0(f,t),t->t0,MAG_GRAD_V);

}

end_f_loop(f,t)

}

}

}

Paved triangular meshes of the geometry for packages with the three different

55534_C006.indd 15155534_C006.indd 151 10/22/08 10:04:31 AM10/22/08 10:04:31 AM

solve this UDS. Solve only the flow, energy, and uds-0 Equations. */

electrode configurations were constructed using the GAMBIT 2.0 pre-processor.



A, B, and C were 468, 1048, and 924, respectively (Figure 6.3).

6.3.4 MODEL VALIDATION

the package with one unsealed end. Temperature values of thermocouple probes (T-

type, Omega, TMQSS-062U-6) installed at seven different locations (Figure 6.4)

The sensor locations are coordinated using X1 to X5 in the x direction (horizontal

plane), and Y1 to Y3 in the y direction (vertical plane). The ends of thermocouple

probes were placed to be located at an 8.5-cm depth of a package. The sensors were

electrodes. A critical issue was to install and maintain the thermocouple probes

ungrounded thermocouple probes used in this study offer basic electrical isolation

since the thermocouple junction is detached from the probe wall. However, they

have slower response time than grounded or exposed thermocouples, permitting

only either 1D or 2D measurement by nature.

1 2 3 4 5

3

2

1

X

Y

1/2 L

FIGURE 6.4 Temperature sensor locations inside the package (L = package length). (From

Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods inside the

55534_C006.indd 15255534_C006.indd 152 10/22/08 10:04:31 AM10/22/08 10:04:31 AM

flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With permission.)

Through mesh refinement study, the optimum numbers of mesh elements for Pouches

For model verification, temperature values were measured at different locations inside

were isolated to eliminate the signal interference with the electric field before being

transmitted to the data acquisition unit (Campbell Scientific, 21X Micrologger).

allocated to determine the likely cold and hot zones for different configurations of

at fixed locations during heating since the packaging material is flexible. The



Among thermostabilized ISS menu items supplied from NASA’s Johnson

Space Center, the select chicken noodle soup and black beans were used for the

reheating experiment. The formulation and thermal properties of chicken noodle

soup and black beans are listed in Table 6.2. Thermal properties of each sample

were calculated from the values reported for the material using their mass or volume

fractions [33,34].

obtained for chicken noodle soup and black beans (Figure 6.5). For both samples,

the electrical conductivity increases with temperature with a linear correlation, as

V/m, Pouch B between 1 and 2457 V/m, and Pouch C between 24 and 4985 V/m.

TABLE 6.2Formulations and Thermal Properties for Chicken Noodle Soup and Black Beans

Chicken Noodle Soup Black Beans

Ingredient Percentage (%) Ingredient Percentage (%)

Fettuccine 7.68 Black beans 74.66

1.77 Crushed tomatoes 11.35

0.58 Cumin 0.25

0.35 Oregano 0.15

Salt 0.25 Black pepper 0.1

Black pepper 0.06 Salt 0.09

Parsley 0.03 Green chillies 3.55

Poultry seasoning 0.03 Onions 9

Water (slurry) 10.25 Garlic 0.85

Water 13.62

Chicken broth 22.13

Half and half 3.41

Chicken 22.13

Carrots 8.85

Celery 4.43

Onions 4.43

Total 100 100

Thermal PropertiesDensity (kg/m3) 1033 1157

3388

0.486

170

3977

Thermal conductivity

(W/m°C)

0.553

Net weight (g)

55534_C006.indd 15355534_C006.indd 153 10/22/08 10:04:32 AM10/22/08 10:04:32 AM

Medium toasted flour

Filling aid starch

Modified food starch

Specific heat (J/kg°C)

Changes in electrical conductivity values with field strength and temperature were

expected. The coefficients corresponding to Equation 6.4 are presented in Table 6.3.

The trends of electrical conductivity values with respect to field strength are not

obvious from this figure, which rather seem to be identical.

Figure 6.6 shows the simulated electric field distributions for different electrode

configurations. For even comparison, the potential difference between two electrodes

was set to be 100 V. Pouch A has electric field strength ranging between 243 and 2421



Ele

ctri

cal

conduct

ivit

y (

S/c

m)

0.009

0.012

0.015

0.018

0.021

0.024

0.027

9.18 V/cm

15.8 V/cm

28.4 V/cm

Temperature (oC)

10 20 30 40 50 60 70 80 900.010

0.015

0.020

0.025

0.030

8.4 V/cm

14.2 V/cm

(a) Chicken noodle soup

(b) Black beans

FIGURE 6.5 ture for (a) chicken noodle soup and (b) black beans. (From Jun, S. and Sastry, S. Modeling

Food Process Engineering, 28, 417–36, 2005. With permission.)

TABLE 6.3

Soup and Black Beans

Field Strength (V/cm) σ0 m

Chicken noodle soup 9.18 0.0054 0.044

15.8 0.0067 0.030

28.4 0.0092 0.015

Black beans 8.4 0.0068 0.041

14.2 0.0058 0.049

Source: From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heat-

417–36, 2005. With permission.

55534_C006.indd 15455534_C006.indd 154 10/22/08 10:04:32 AM10/22/08 10:04:32 AM

and optimizing of pulsed ohmic heating of foods inside the flexible package. Journal of

ing of foods inside the flexible package. Journal of Food Process Engineering, 28,

Changes of electrical conductivity values with field strength and tempera-

Coefficients of Electrical Conductivities for Chicken Noodle



is more desirable than the others. However, it should be noted that the electric

Understanding the pouch geometry, the area adjacent to the electrode edge, which is

key factor dominating the thermal performance inside the package.

under ohmic heating, Figure 6.7 compares the 2D model-predicted temperature

values with the experimental data of chicken noodle soup in Pouches A and B, and C

after 300 s of heating. The RMS voltage supplied for Pouch A and B was 48.6 V, and

Pouch C had 27 V, leading the reheating temperature to increase to 80°C during the

same heating period. Pouch C was powered by a lower voltage since the geometrical

distance between the bipolar electrodes is shorter than the others. The standard

deviation for experimental results (n = 3) was 3.3°C.

The temperature values measured at coordinates (X2, Y3) and (X4, Y1) in

Figure 6.7a, and (X2, Y1), (X2, Y3), (X4, Y1), and (X4, Y3) in Figure 6.7b were

higher than those at other coordinates by a minimum of 15°C. It clearly shows

FIGURE 6.6

Electrode

(a)

(b)

(c)

V/m 5.00e+ 03

4.75e+ 03

4.50e+ 03

4.25e+ 03

4.00e+ 03

3.75e+ 03

3.50e+ 03

3.25e+ 03

3.00e+ 03

2.75e+ 03

2.50e+ 03

2.25e+ 03

2.00e+ 03

1.75e+ 03

1.50e+ 03

1.25e+ 03

1.00e+ 03

7.50e+ 02

5.00e+ 02

2.50e+ 02

0.00e+ 00

55534_C006.indd 15555534_C006.indd 155 10/22/08 10:04:33 AM10/22/08 10:04:33 AM

Simulated field strength in (a) Pouch A, (b) Pouch B, and (c) Pouch C.

Apparently, Pouch A has a relatively uniform distribution in field strength, which

field strength near the edges of electrodes goes close to the maxima. Obviously,

the distance between the two electrodes is crucial to determine the field strength.

closer to the opposite electrode, would have higher values of field strength, eventually

causing the field overshoot at the insulative boundary. It would rarely occur between

electrodes in parallel [35]. The existence and strength of field overshoot might be a

As verification of the thermal behavior of chicken noodle soup in a package

the existence of the overshoot of electric field strength near the electrode edges,



Tem

per

ature

(oC

)T

emper

ature

(oC

)T

emper

ature

(oC

)

(a)

(b)

(c)

X

80

60

40

20

0

80

60

40

20

0

80

60

40

20

0

12

34

5 1

2

3

12

34

5 1

2

3

12

34

5 1

2

3Y

X

Y

X

Y

: Experimental data

: Simulation data

FIGURE 6.7 and measurement data in (a) Pouch A, (b) Pouch B, and (c) Pouch C after 300 s of heating.

(From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods

permission.)

55534_C006.indd 15655534_C006.indd 156 10/22/08 10:04:34 AM10/22/08 10:04:34 AM

inside the flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With

Comparison of temperature profiles of chicken noodle soup between simulation



permitting correspondingly more heating. The overall model-predicted data are

close to the actual measurements with minimum R2 of 0.88 (maximum error of

5°C); however, the prediction errors at both corners of packages, i.e. (X1, Y2) and

observed under-prediction error may be attributed to local non-homogeneities of

Figure 6.7c shows the closeness between the prediction and measurement data

with R2 of 0.83 and maximum prediction error of 10.6°C. The cold zone dominating

Pouch C occurs at the middle coordinating at (X3, Y2), rather than each corner of the

package, (X1, Y2) and (X5, Y2).

chicken noodle soup and black beans, perhaps due to similar electrical conductivities.

Black beans in different types of packages show thermal behavior analogous to

chicken noodle soups (Figure 6.8 on the left). After 300 s of heating, Pouch A had a

temperature distribution between 23 and 80°C and Pouch B between 37 and 80°C.

On the other hand, Pouch C had a greater temperature variation between 12 and

80°C. The 2D dynamic model for black beans predicts thermal results that are in

good agreement with the experimental data with a minimum R2 of 0.80, except those

at the coordinates, (X1, Y2) and (X5, Y2). The average percentage deviation between

simulated and measured temperature values was 10.7%.

12 and 40°C in the case of the reheating scheme, the corresponding areas were clipped

and compared for quantative analysis (Figure 6.8 on the right). Calculations show that

Pouches A, B, and C have the ratios of cold zone area to the entire package area as 0.67,

(a)

(b)

(c)

80 12

oC

40

FIGURE 6.8 Simulated temperature distribution (left side) and cold zone (right side) of

black beans in (a) Pouch A, (b) Pouch B, and (c) Pouch C after 300 s of heating.

55534_C006.indd 15755534_C006.indd 157 10/22/08 10:04:35 AM10/22/08 10:04:35 AM

(X5, Y2), increased significantly with the maximum prediction error of 14°C. The

chicken noodle soup, filled with solid chunky particulates.

There was little discrepancy of simulated electric field distributions between

Defining the cold zone as the area covering the temperature distribution between



0.02, and 0.42, respectively. Pouch B with V-shaped electrodes is, therefore, expected

to be more likely to perform uniform heating of black beans within the package.

overshoots near each electrode edge in symmetry might have compensating effects to

help increase the temperature at the adjacent areas such as (X1, Y2) and (X5, Y2).

The ratios of cold zone, average temperature, power consumption, temperature

variation (ΔT) of black beans in the Pouch B under ohmic heating are enumerated

and presented in Figure 6.9. The power consumption was calculated by integration

heat, and temperature increment. The calculated values were validated by comparing

with a set of measurements which had a RMS voltage of 48.6 V, and RMS current of

3.1 A, producing 150.7 W as power consumption. A package with narrow electrodes,

i.e. 0.063 in dimensionless width, would have low power consumption of 81 W,

whereas developing predominant cold zones of 91.1%, low average temperature

of 27.4°C, and huge temperature variation of 59.3°C. The widest electrode might

require higher electric power of 245 W with an increased temperature deviation

food heating in the package.

6.3.5 DELIVERABLES

temperature during reheating process. Unlike electrodes in parallel, the pouch

Dimensionless electrode width

0.05 0.10 0.15 0.20 0.25 0.30

Cold

zone a

rea r

atio

0.0

0.2

0.4

0.6

0.8

1.0

Avera

ge te

mpera

ture

(oC

)

25

30

35

40

45

50

55P

ow

er

(W)

50

100

150

200

250

300

Tem

pera

ture

variation (

oC

)

35

40

45

50

55

60

65

70

Cold zone area ratioAverage temperature (oC)Power (W)Temperature variation (oC)

FIGURE 6.9 Optimization of electrode width with respect to cold zone ratio, power con-

sumption, average temperature, and temperature variation.

55534_C006.indd 15855534_C006.indd 158 10/22/08 10:04:36 AM10/22/08 10:04:36 AM

Obviously, the field strength at the corners of Pouch B, coordinating at (X1, Y2) and

(X5, Y2), is weaker than that of Pouch A, as aforementioned; however, the two field

of internal heat contents with respect to the area, being determined by mass, specific

between the coldest and hottest, 68.1°C. Consequently, the electrode configuration

with dimensionless width of 0.147 would be close to the best fit for uniformity in

The electrode configuration could be optimized to ensure uniformity in food



electrodes were found to induce the overshoot of voltage gradient on the edges. The

conductivities are needed for further study to improve the accuracy of the model

prediction for a multiphase food system.

6.4 CASE STUDY II: 3D MODELING

A 2D ohmic model that accounts for electro-thermal performance within a packaging

system was capable of predicting the heating patterns of ISS menu items under ohmic

reheating process [31]. The 2D dynamic model could identify hot and cold spots,

However, the 2D simulation would be unable to provide a complete thermal picture of

foods over the entire pouch, since end effects would not be considered. Such effects

are critical in sterilization calculations. For example, the current density at the seal-

ended corner of the pouch may be lower than expected, resulting in localized under-

treatment of food materials. We have noted the presence of a so-called ‘shadow area’

Computational Fluid Dynamics (CFD) software Fluent (v. 6.1) was used to solve

the governing equations in the 3D environment. UDFs for electric potential and

TGrid meshing schemes in the GAMBIT pre-processor (v. 2.0) were applied to a

volume in which tetrahedral mesh elements were dominant but where there may

be elements that possess other shapes such as hexahedral, pyramidal, and wedge.

automatically added or deleted where needed for better resolution. In doing so, the

overshoot of voltage gradient at the interface between the two adjacent boundaries

could be resolved without completely regenerating the mesh.

6.4.1 MODEL VERIFICATION

Temperatures were measured within a tomato soup sample (details below) using T-

type thermocouple wires (Omega Engineering, Inc., Stamford, CT) at seven different

locations inside the package during ohmic heating (Figure 6.10a). Thermocouple

signals were transmitted to the data acquisition unit (Agilent 39704A, Agilent Tech-

nologies, Inc., Palo Alto, CA) via a signal conditioning module, to eliminate signal

locations covered half the area of the pouch. The locations are as named in Figure

6.2a, with two locations [middle (M) and bottom (B)] in the horizontal plane (z

direction), and four locations [left (L), center (C), right (R), and electrode (E)] in

the vertical plane (x direction). The selection of location E was intended to identify

the potential cold spot inside the V-shaped electrodes. Thermocouple wires were

and aligned perpendicular to the pouch surface so as to be centrally located (y = 0;

Figure 6.10b). Temperature measurements were triplicated. Comparisons between

simulated and measured temperature values at different locations were conducted by

using ANOVA in MINITAB® (v.13, Minitab Inc., State College, PA).

55534_C006.indd 15955534_C006.indd 159 10/22/08 10:04:37 AM10/22/08 10:04:37 AM

field overshoots and non-homogeneities of food characterized by various electrical

permitting optimization of the electrode configuration to minimize cold zones.

of electric field in the cross-sectional domain of 2D model [36].

potential gradient were coded and coupled to solve the electric field model. The

By using solution-adaptive refinement of meshes in Fluent, grid points could be

interference with the electric field. Due to geometric symmetry of the pouch, sensor

installed through a Swagelok® fitting which was located at the center of one of pouch

surfaces and then firmly taped onto the inner pouch layer. The sensor tips were bent



Condensed tomato soup (11 oz, Campbell Soup Company, Camden, NJ) was

purchased from a grocery store, and was mixed with distilled water in a volumetric

ratio of 1:1. Thermal properties of the sample were calculated based on food com-

position data [33,34], and were: density of 1020 kg/m3, viscosity of 3000 × 10–6 Pa·s,

trical conductivity was characterized as a function of temperature, increasing with a

linear correlation, σ (S/m) = 0.032*T (°C) + 0.98.

M

B

L C R E

Symmetry

x

z

(a)

(b)

y

x

Thermocouple wiresTaping

Sensor tip

y = 0RCL

FIGURE 6.10 Temperature sensor locations at the xz-plane (a) and the xy-plane (b) inside the

package (M, middle; B, bottom; L, left; C, center; R, right; and E, edge). (From Jun, S. and Sastry,

55534_C006.indd 16055534_C006.indd 160 10/22/08 10:04:38 AM10/22/08 10:04:38 AM

S. Reusable pouch development for long term space mission: 3D ohmic model for verification of

sterility efficacy. Journal of Food Engineering, 80(4), 1199–1205, 2007. With permission.)

specific heat of 4020 J/kg°C, and thermal conductivity of 0.6 W/m°C. Sample elec-



Figure 6.11 shows the computational domain and coordinate system of the pouch

which has electrodes on opposite sides. The dimensions of the pouch used in mod-

eling were: width 0.1 m, height 0.02 m and length 0.08 m. Because of symmetry,

only half of the pouch geometry needs to be modeled. A regular, structured grid of

tetrahedral mesh elements was created to discretize the domain using 79,041 ele-

ments. As shown in circled areas A, special attention needs to be paid to the seal

ended portion of the pouch since the gap between V-shaped electrodes narrows in

this region. Also there is no electrode within 1.4 cm from the bottom end of the

pouch due to the presence of reseal lines. This could result in low current density,

and may be a potential cold zone.

0.0002 and 2050 V/m. The potential patterns appeared to be perpendicular to the x

getting close to the maximum value, as observed in our previous work [25]. On the

overwhelmingly occur at these locations.

The simulated temperature patterns of tomato soup for selected heating times

(100, 400, and 800 s) are presented in Figure 6.13. It may be noted that one set

A

A

B

M

L

C

R

E

X

Y

Z

FIGURE 6.11 Pouch geometry and grid mesh with coordinated thermocouple locations (M,

middle; B, bottom; L, left; C, center; R, right; and E, edge). (From Jun, S. and Sastry, S. Reus-

55534_C006.indd 16155534_C006.indd 161 10/22/08 10:04:38 AM10/22/08 10:04:38 AM

Figure 6.12 shows the simulated electric potential and field distributions inside a

package when the supplied RMS voltage was 42 V. The field strength ranged between

direction; however, the electric field strength near the edges of electrodes overshoot

other hand, the food-filled space bounded by the V-shaped electrodes experienced

relatively weak field strength below 440 V/m. Since the electro-heat generation is

proportional to the square of field strength (Equation 6.3), cold spots are expected to

efficacy. Journal of Food Engineering, 80(4), 1199–1205, 2007. With permission.)

able pouch development for long term space mission: 3D ohmic model for verification of sterility



(a)4.20e+014.03e+013.91e+013.78e+013.65e+013.53e+013.40e+013.28e+013.15e+013.02e+012.90e+012.77e+012.65e+012.52e+012.39e+012.27e+012.14e+012.02e+011.89e+011.76e+011.64e+011.51e+011.39e+011.26e+011.13e+011.01e+018.82e+007.56e+006.30e+005.04e+003.78e+002.52e+00

YX Z

Y

ZX

1.26e+000.00e+00

2.05e+031.97e+031.91e+031.85e+031.78e+031.72e+031.66e+031.60e+031.54e+031.48e+031.41e+031.35e+031.29e+031.23e+031.17e+031.11e+031.05e+039.84e+029.23e+028.61e+028.00e+027.38e+026.77e+026.15e+025.54e+024.92e+024.31e+023.69e+023.08e+022.46e+021.85e+021.23e+026.15e+012.55e–04

(b)

FIGURE 6.12 (From Jun, S. and Sastry, S. Reusable pouch development for long term space mission: 3D ohmic

With permission.)

55534_C006.indd 16255534_C006.indd 162 10/22/08 10:04:40 AM10/22/08 10:04:40 AM

Simulated 3D (a) electric potential and (b) field distributions inside the pouch.

model for verification of sterility efficacy. Journal of Food Engineering, 80(4), 1199–1205, 2007.



of cold zones were always located inside the V-shaped electrodes, which showed a

temperature of 95°C when the maximum pouch temperature was 139°C. The ther-

mal distribution in the middle of the pouch appeared to be a valley with elongated

noted previously. Secondly, even colder zones were noted at the corners of the pouch

(a)

(b)

(c)

150

150

150

A

A –0.05

0

0.050.1

0.05

0

Length

Width

100

50

100

50

–0.05

Width0

0.050.1

0.05

0

Length

–0.05

0

0.050.1

0.05

0

Width

Length

100

50 Tem

per

ature

(°C

)

Tem

per

ature

(°C

)

Tem

per

ature

(°C

)

FIGURE 6.13 Simulated 3D temperature distribution in the pouch at selected heating times;

(a) 100 s, (b) 400 s, and (c) 800 s. (From Jun, S. and Sastry, S. Reusable pouch development for

Engineering, 80(4), 1199–1205, 2007. With permission.)

55534_C006.indd 16355534_C006.indd 163 10/22/08 10:04:42 AM10/22/08 10:04:42 AM

long term space mission: 3D ohmic model for verification of sterility efficacy. Journal of Food

depression of surfaces between hills. This thermal profile reflected the extra heating

at the interface between electrodes and insulating materials due to the field overshoot



(Circle A in Figure 6.13c). Note that during simulation, the lowest temperature values

of 53.3°C after 800 s of heating were found in these zones, even when the maximum

temperature in the pouch was at 139°C. Location A is a region to which electrodes

do not extend, thus resulting in relatively limited heating at this zone. Elimination of

such hot and cold zones represents a major goal in design optimization. For example,

electrodes are arranged parallel to one another, and electrodes and insulating layers

cross at right angles. However, redesigning also involves numerous other considera-

tions including practicality, mechanical strength and economics.

Figure 6.14 compares model-predicted temperature values with experimen-

tal data. The standard deviation for experimental results (n = 3) was 4.5°C. The

3D ohmic model produces thermal results that are in good agreement with the

experimental data (P > 0.53). The average percentage deviation between simulated

and measured temperature values was 6.2%. As expected in Figure 6.6, the tempera-

ture values measured at ML, MR, BL, and BR were higher than those at MC and BC

with a minimum of 3°C at 800 s of heating. Also location ME shows the existence of

undertreated spots which could be below 95°C, as discussed previously. While this

temperature was higher than the edge location represented Circle A of Figure 6.13c,

however, it still poses a challenge for producing high-quality sterilized products. The

location BC showed lower temperatures than MC, which might be due to conductive

heat loss to the cold bottom edge where the ohmic current density is low.

Sample temperatures measured at location ME (edge) were lower than model

predictions at the earlier stages of simulations, but were higher at the later time peri-

ods of the simulation. The greatest error (around 6°C) was associated with this loca-

tion, and represented an underprediction. This error implies that the expected value

of the pouch at this point does not lend itself to easy determination of convective

heat transfer. It is possible to improve predictions by improved assumptions regard-

ing conditions at this location. A second implication is that the model represents an

underprediction; thus its predictions are conservative, and may result in over- rather

than under-processing. Under a microgravity environment, convective heat transfer

would be minimal, resulting in lower heat losses than expected and higher tempera-

tures within the pouch. There is a need for further research on minimizing cold spots

in a pouch via modeling and redesign.

A three-dimensional model for heat transfer within an ohmic heating pouch

equipped with electrodes shows generally good agreement with experimental data,

with the exception being at the edges at the later stages of simulation, when under-

cold zones, suggesting the need to further optimize pouch design for more uniform

heating. In particular, the zones within the V formed by metal foil electrodes, and

concern, and will need further design optimization.

6.5 CASE STUDY III: MULTI-PHASE OHMIC HEATING

Modeling of multi-phase food products which have various electrical conduc-

55534_C006.indd 16455534_C006.indd 164 10/22/08 10:04:43 AM10/22/08 10:04:43 AM

the field overshoot is unlikely to occur in a rectangular shaped domain in which two

of heat transfer coefficient was not valid at the edge locations; indeed, the V-shape

prediction was observed. Simulations suggest the presence of significant hot and

the edge of the pouch, where current densities are lowered, are identified as points of

tivities has been reported. Distortion of electric field due to several factors such



Tem

pera

ture

(oC

)

0

20

40

60

80

100

120

140

Time (s)

0 200 400 600 8000

20

40

60

80

100

120

Simulated L & R

Simulated C

Simulated E

L

C

R

E

M

B

FIGURE 6.14 tomato soup in the pouch. (From Jun, S. and Sastry, S. Reusable pouch development for long

neering, 80(4), 1199–1205, 2007. With permission.)

as heterogeneous food materials and irregular shapes of domain is one of key

interests to food engineers whose efforts is to predict the accurate thermal per-

formance of ohmic heaters. Sterilization of solid–liquid mixtures by ohmic heat-

ing requires the assurance that all parts of the food or biomaterial in question

are treated adequately to ensure inactivation of pathogenic spore formers. The

55534_C006.indd 16555534_C006.indd 165 10/22/08 10:04:44 AM10/22/08 10:04:44 AM

Comparison of predicted and measured transient temperature profiles of

term space mission: 3D ohmic model for verification of sterility efficacy. Journal of Food Engi-



is our lack of knowledge of (and inability to measure) temperature at the slow-

est-heating location within the entire system. Ohmic heating poses even greater

challenges in measurement than conventional heat exchange processes due to the

to predict cold-spot temperatures; indeed modeling is a prerequisite to success

case scenarios as being associated with a single solid piece (inclusion particle) of

substantially different electrical conductivity than its surroundings. Two poten-

tially hazardous situations were modeled, both involving an inclusion particle,

but one involving a static medium surrounding the solid; and the other involving

coding in a FORTRAN 90 program, and compared and evaluated for predictions

of particle cold-spot and average medium temperatures under conditions which

would likely lead to a worst case heating scenario as demonstrated in the experi-

mental part of this study.

able slightly underpredicting temperature) prediction of mixture cold-spot tempera-

tures than the static model when the cold-spot occurs within the particle; typically

occurring when the medium is more conductive than the solid (Figure 6.15). How-

solid is more conductive than the medium (Figure 6.16) [39].

Notably, the cold spot is within the solid when the medium is more conductive,

of the current. Under this condition, the cold zone is within the medium at shadow

zones immediately in front/back of the particle. When the particle is more conduc-

tive, the coldest zone is within the medium when the particle size is small; however,

at large particle sizes, the particle cold spot approaches (and, in case of the mixed

dition the static model is still conservative.

One of the most critical factors to be taken into consideration is likely to be

fat content. If a fat globule is present within a highly electrical conductive region,

where currents can bypass the globule, it may heat slower than its surroundings due

to its lack of electrical conductivity. Under such conditions, any pathogens poten-

tially present within the fat phase may receive less treatment than the rest of the

product. Heating of the fat phase may then depend on the rate at which energy is

transferred from the surroundings. Based on the foregoing discussion, a high heat

tends to moderate heating in such situations. If a fat-rich (low conductivity) phase

the process. Due to the complexity and great number of possible situations in food

55534_C006.indd 16655534_C006.indd 166 10/22/08 10:04:44 AM10/22/08 10:04:44 AM

fundamental problem in continuous flow sterilization of solid–liquid mixtures

ing equations for the mixed fluid and static models were solved iteratively using

Results indicate that the mixed fluid model provides a more conservative (prefer-

ever, the static fluid model provides more conservative prediction of the mixture

cold-spot temperatures when the cold-spot is within the fluid; typically when the

fluid model, eventually becomes lower than) the medium cold spot. Under such con-

faster than the surrounding fluid. In any case, care must be taken in establishing

presence of an electric field. This necessitates the use of mathematical modeling

in this process. The work done by Sastry and Salengke [37] has identified worst-

a mixed fluid, with a circuit theory analysis for the electric field [38]. The govern-

the Galerkin–Crank–Nicholson algorithm (Galerkin three-dimensional finite

element method in space; Crank–Nicholson finite difference scheme in time) by

except when the solid size becomes sufficiently large to intercept a large fraction

transfer coefficient may not necessarily relate to the worst case, since fluid motion

is aligned to significantly intercept the current, it is possible for such a zone to heat



processing, it may be prudent to investigate all potentially likely scenarios in proc-

ess evaluation. The models compared in this work each have their respective merits,

enabling large-system, high solids content simulation and low computation cost in

Laplace model.

(a) Y

X

Z

67.87 80.81 93.75 106.68 119.62 132.56

Temperature (°C)

Y

X

Z

67.87 80.81 93.75 106.68 119.62 132.56

Temperature (°C)

(b)

FIGURE 6.15 Color mapping of modeled temperature distributions within an ohmic heater

after 150 s of ohmic heating, for a single solid cylindrical ‘inclusion’ particle 1/3 as conduc-

55534_C006.indd 16755534_C006.indd 167 10/22/08 10:04:45 AM10/22/08 10:04:45 AM

tive as the fluid; (a) well-mixed fluid (b) static fluid.

the case of the circuit model, and rigorous, fine-detail predictions in the case of the



(a) Y

X

Z

111.34 120.17 129.00 137.83 146.66 155.49

Temperature (°C)

111.34

Z

X

120.17 129.00 137.83 146.66 155.49

Temperature (°C)

(b) Y

FIGURE 6.16 Color mapping of modeled temperature distributions within an ohmic heater

after 150 s of ohmic heating, for a single solid cylindrical ‘inclusion’ particle twice as conduc-

55534_C006.indd 16855534_C006.indd 168 10/22/08 10:04:47 AM10/22/08 10:04:47 AM

tive as the fluid; (a) well-mixed fluid (b) static fluid.



6.6 CONCLUSION

Ohmic heating is one of the successful alternative food processing methods because

patterns of foods under ohmic heating are uniquely dependent upon their electrical

conductivities, which are also a function of temperature. Modeling of ohmic heating

provides a valuable insight in determining the relevant process parameters to prevent

any food components from being under-processed, which is required for food safety

purpose. These days, the power of CFD to model complex food processes is predomi-

nant and it seems that its adoption is inevitable and progressive and aimed for com-

and mass transfer, phase change, solid and liquid interactions and such. The case

studies provided in this chapter show how CFD models can work for the ohmic heat-

ing process. Although limited to a special process environment such as microgravity,

the developed CFD model with little convective heat transfer taken into consideration

could not be a hurdle for its application to complete ohmic heating processes.

The challenge of ohmic heating modeling exists in interactive visualization of a

continuous ohmic heating system, in which the transient orientation of all the parti-

to be incorporated.

NOMENCLATURE

p

k Thermal conductivity (W/m°C)

k0 Pre-exponential factor

m T Temperature (°C)

t Time (s)

S Internal energy source (W/m2)

V Voltage (V)

σ Electrical conductivity (S/cm)

σ0 Referencing electrical conductivity (S/cm)

ρ Density (kg/m3)

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173

7 Hydrostatic Pressure Processing of Foods

J. Antonio Torres and Gonzalo Velazquez

CONTENTS

7.1 Introduction ................................................................................................... 173

7.2 Hydrostatic Pressure Processing (HPP) of Foods ......................................... 174

7.2.1 Principles of High Pressure Processing ............................................ 175

7.2.1.1 Niche Opportunities for HPP Foods ................................... 176

7.2.1.1.1 Consumer Demand for Fresh Foods .................. 176

7.2.1.1.2 Pressure Processing Effect is Unique ................ 178

7.2.1.1.3 Product is a High Microbial Risk to

Producer ............................................................. 180

7.2.1.1.4 Product has a High Added Value and

Thermal Lability ............................................... 185

7.2.1.2 Mechanisms of Microbial Inactivation by Pressure ........... 185

7.2.1.2.1 Vegetative Bacteria ............................................ 185

7.2.1.2.2 Bacterial Spores ................................................. 188

7.3 Pressure Assisted Thermal Processing (PATP) of Foods ............................. 189

7.3.1 Reaction Kinetics Analysis ............................................................... 189

7.4 Low Hydrostatic Pressure (LHP) Disinfestation of

Dry Fruits and Vegetables .............................................................................202

7.5 Conclusions ...................................................................................................203

Nomenclature .........................................................................................................205

References ..............................................................................................................205

7.1 INTRODUCTION

New food processing technologies meeting consumer expectations for increased food

safety, extending shelf life and improving product quality are needed today. Consum-

ers are demanding fresh foods and products minimally affected by processing so as to

preserve desirable compositional and sensory properties. For example, in the United

States raw milk may be purchased directly from farms in 28 states, and in four states

it may be purchased in retail stores [1]. This consumer interest in untreated milk is

troublesome information for agencies monitoring food safety since its potential con-

tent of microbial pathogens poses a serious health risk to consumers. An analysis of

the incidence of pathogenic bacteria in raw milk from 70 farms showed that 4.9 and

3.4% were positive for Listeria monocytogenes and L. innocua, respectively [2,3].

55534_C007.indd 17355534_C007.indd 173 10/22/08 10:52:12 AM10/22/08 10:52:12 AM



Milk from 36 transport tankers tested for the presence of L. monocytogenes showed

a 2.8–11% contamination incidence [2].

High hydrostatic pressure processing (HPP), a relatively new technology to the

nutritional changes to foods [5–7]. On the other hand, the effectiveness of thermal

processing technologies explains why it remains as the prevailing method to achieve

microbial safety and the inactivation of enzymes and microorganisms responsible

for food spoilage. However, the high temperatures used in these processes cause sig-

losses. For example, high-temperature short-time (HTST) pasteurization treatments

(72°C for 15 s) impart a slight cooked, sulphurous note that has become acceptable

to milk consumers but its refrigerated shelf life is only approximately 20 days. Ultra

pasteurization (UP), a process similar to HTST pasteurization using more severe

ality

and causes more nutrient damage but yields milk with a refrigerated shelf life of

approximately 30 days [8]. Pressure treatments of 400 MPa for 15 min or 500 MPa

for 3 min at room temperature achieves microbiological reductions similar to thermal

pasteurization [9] but it is not used commercially because long pressure processing

ate temperature (55°C) extend the refrigerated shelf life of milk to over 45 days [10]

HTST treatments [11]. Finally, ultra high temperature (UHT) processing (135–150°C

for 3–5 s) yields milk that is stable at room temperature for 6 months; however, this

acceptance in important markets [14].

Future advances are expected from the synergistic effects of using high pressure

and high temperature combinations in the rapidly evolving pressure-assisted thermal

processing technology (PATP). PATP is not yet a commercial application and will

require more complex safety validation procedures than HPP, particularly for the case

the inactivation of bacterial spores and recent studies suggest that pressure can lower

the degradation rate of product quality caused by high temperature treatments. The

lowering by pressure of the rate of thermal degradation reactions could preserve qual-

inhibit formation reactions for potential toxicants [15].

This chapter reviews the current use of HPP technology for pasteurization and

other applications, and the promising future of PATP technology for the production

of shelf stable foods. Novel applications such as relatively low hydrostatic pressure

(LHP) disinfestation of dehydrated fruits and vegetables currently under develop-

ment are also presented.

7.2 HYDROSTATIC PRESSURE PROCESSING (HPP) OF FOODS

High pressure processing at refrigeration, ambient or moderate heating temperature

allows inactivation of pathogenic and spoilage microorganisms in foods with fewer

55534_C007.indd 17455534_C007.indd 174 10/22/08 10:52:15 AM10/22/08 10:52:15 AM

nificant chemical changes in foods. Particularly important are thermal degradation

times are not financially viable. HPP treatments (586 MPa for 3 and 5 min) at moder-

while retaining milk volatile profiles similar to those observed after conventional

of low-acid foods (pH under 4.5). PATP conditions are sufficiently severe to achieve

ity factors and constituents with important health benefits to consumers. It may also

food industry [4] inactivates microorganisms without causing significant flavour and

reactions leading to off-flavours, destruction of nutrients and other product quality

treatments (e.g. 1 s at 89ºC, 0.1 s at 96ºC or 0.01 s at 100ºC) lowers flavour qu

process induces strong ‘cooked’ off-flavour notes [12,13] thus limiting its consumer


Hydrostatic Pressure Processing of Foods 175

changes in food quality as compared to conventional technologies [6,7,16]. Pressure

acts by disrupting mainly hydrogen bonds without affecting covalent bonds. There-

fore, high pressure processing (HPP) treatments at low (approximately 0–30°C) and

moderate (approximately 30–50°C) temperature cause minimum losses in quality

factors associated with small molecules such as vitamins, pigments and volatile

products make them often indistinguishable from untreated controls [19]. Five decimal

reductions in pathogens including Salmonella typhimurium, S. enteritidis, Listeria monocytogenes, Staphylococcus aureus and Vibrio parahemolyticus can be achieved

by HPP [20–25].

7.2.1 PRINCIPLES OF HIGH PRESSURE PROCESSING

Unlike thermal processing and other preservation technologies, HPP effects are uni-

form and nearly instantaneous throughout the food and thus independent of food

ings to full-scale production. The key HPP equipment technologies are the pressure

Oil at ∼20 MPa is fed on the high-pressure oil side of the main pump piston which

∼600 MPa (Figure 7.1). When the main piston reaches the end of its displacement,

the system is reversed and high-pressure oil is then fed to the other side of the main

limitation of pressure vessel construction from a single block limits them to ∼25 l for

operating pressures in excess of 400 MPa. Prestressing by wire-winding and other

technologies is used for safe and reliable commercial-size vessels operating at higher

pressures. Typically, the same technology is used for the yoke holding the top and

bottom seals (Figure 7.2). Wire winding increases equipment costs leading to the

MPa separated by a technology barrier at ∼400 MPa from higher cost operations

such as guacamole salsa production at ∼600 MPa (Figure 7.3). A second technol-

ogy barrier exists at ∼650 MPa and above this pressure level there are no vessels

available for commercial applications. However, the next generation of equipment

High pressure seal

Inlet

Main piston High pressure fluid piston

Outlet

High pressure fluidLow pressure oilHigh pressure oil

FIGURE 7.1 High pressure pump technology. (Adapted from Torres, J.A. and Velazquez, G.

Commercial opportunities and research challenges in the high pressure processing of foods.

Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 17555534_C007.indd 175 10/22/08 10:52:15 AM10/22/08 10:52:15 AM

geometry and equipment size. This has facilitated the scale-up of laboratory find-

vessels and the high hydrostatic pressure generating pumps or pressure intensifiers.

current definition of low cost operations such as oyster shucking requiring 200–400

flavours [4,17,18]. Research has confirmed that the sensory characteristics of HPP

has an area ratio of 30:1 with respect to the high-pressure fluid piston displacing

into the high pressure vessel a food-grade contact fluid, typically purified water at

pump piston and the high-pressure fluid exits on the other pump side. The casting



is expected to reach ∼700 MPa and operate at temperatures higher than 100°C to

inactivate bacterial spores [26]. The possibility exists also that equipment suppliers

may offer in the future large vessels for low pressure applications such as disinfesta-

tion (Figure 7.3).

7.2.1.1 Niche Opportunities for HPP Foods

HPP is an alternative processing technology that has reached consumers with a vari-

ety of new products. The successful introduction of a new technology demands the

HPP, an additional constraint is the large capital investment which is overcome by

operating HPP plants at full capacity. Therefore, the processing of seasonal com-

modities requires identifying a product mix achieving maximum utilization of the

equipment investment. The following sections provide examples of the many oppor-

tunities in which HPP has a clear competitive advantage.

7.2.1.1.1 Consumer Demand for Fresh FoodsThe classical example of satisfying a consumer demand for a fresh product is the

pressure processing of avocado. The treatment required is under 650 MPa for ∼1

consumer demand for products with acceptable shelf-life, convenient to use and free

from chemical additives. Refrigerated fresh-cut fruit salads that consumers demand

Monoblock

Wire wound

No pressure Pressure

Yoke

Vessel plug

Wire wound

vessel

Vessel plug

Concentric cylinders

FIGURE 7.2 High pressure vessel technologies. (Adapted from Torres, J.A. and Velazquez, G.

Commercial opportunities and research challenges in the high pressure processing of foods.

Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 17655534_C007.indd 176 10/22/08 10:52:17 AM10/22/08 10:52:17 AM

identification of specific competitive advantages over existing ones. In the case of

min. HPP avocado is now in national distribution because there was an unsatisfied


Hyd

rostatic Pressu

re Processin

g of Fo

od

s 177

Moderate pressure High pressure Research

0 1000 2000 3000 4000 5000 6000 7000 8000

0 1 2 3 4 5 6 7 8

0 1000 2000 3000 4000 5000 6000 7000 8000

0 100 200 300 400 500 600 700 800

kPsi

Atm

MPa

Kg/cm2

KBar

Low pressure

0 10 20 30 40 50 60 70 80 90 100 110 120

Pasteurization

Low coste.g., oysters

Technology limitTechnology step

High coste.g., avocado

Sterilization

Potential:Disinfestation

FIGURE 7.3 High pressure technology barriers. (Adapted from Torres, J.A. and Velazquez, G. Commercial opportunities and research challenges in

the high pressure processing of foods. Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 177

55534_C007.indd 177

10/22/08 10:52:18 AM

10/22/08 10:52:18 AM



for healthiness, convenience and labor-savings reasons are another example [27].

During cutting and packaging, these products may be contaminated with Salmonella,

Escherichia coli O157:H7 and other pathogens of concern. Early research focusing

limitations of laboratory units available at the time [28] and not current equipment

technology. Current vessels operate at higher pressure allowing processing times in

the 1–3 min range reducing processing costs [29–31].

In 1996, non-pasteurized apple juice was traced to an E. coli O15:H7 outbreak

affecting seven western USA states and British Columbia, Canada. Although this

outbreak affected only an estimated 60 people, public attention was high because

the cases included a 2-year-old girl who suffered permanent renal damage and a 16-

month-old infant who died from cardiac and respiratory arrest [32]. E. coli O15:H7

found in an unopened container was used as evidence to support new juice regula-

tions requiring a 5-log decimal reduction in pathogens coming primarily from animal

fecal contamination [33]. Inactivation of enzymes and spoilage microorganisms

in HPP-treated juice have been extensively studied [34,35]. No viable counts were

observed during storage of apple juice inoculated with a pathogenic cocktail includ-

ing O15:H7 treated at 545 MPa for 1 min under refrigeration temperature and kept 1

month at room temperature or 2 months under refrigeration [4]. CO2(g)-assisted HPP

inactivation of pectinmethylesterase (PME) in Valencia orange juice has been inves-

tigated [36]. Non-carbonated and carbonated juices subjected to conditions ranging

temperature has shown that CO2(g)-assisted HPP increases the rate but not the extent

of PME inactivation.

The demand for juices with no thermal treatment remains strong in important

markets and high pressure processors seek to satisfy it by products that meet the

new pasteurization requirement and that in the future could be labelled ‘fresh’ if

undistinguishable from fresh-squeezed juice. Color, vitamin C content, and anti-

oxidant levels of pressure-pasteurized apple and pulp-free orange juice show no

[19]. Triangular test sensory evaluations using 101 apple and 221 orange juice con-

sumers show that fresh-squeezed and HPP-treated juice are indistinguishable. These

studies show that pressure treated juices are safe and similar to fresh-squeezed sam-

ples [19,37–44].

7.2.1.1.2 Pressure Processing Effect is UniqueThe classical example of unique pressure effect is oyster shucking by HPP. In 2002,

California banned the sale of untreated Gulf of Mexico oysters harvested between

April and October costing local producers an estimated $20 million/year loss and

is a good example of the need for new processing technologies for raw products.

The process discovered in 1997 places live oysters under pressures of 240–350 MPa

for 3 min. These moderate pressure treatments denature the abductor muscle and

oysters can be opened without knife damage. Replacing the laborious and costly

100 years [45]. Most importantly, the HPP treatment eliminates a high safety risk

to production workers, extends refrigerated shelf-life to three weeks and reduces the

55534_C007.indd 17855534_C007.indd 178 10/22/08 10:52:18 AM10/22/08 10:52:18 AM

from 200 to 600 MPa pressure, 30–300 s dwell time and 15–50°C final processing

significant differences between pressure-treated and control samples (Figure 7.4)

hand-shucking is the most significant development in the oyster industry in the past

on long moderate-pressure processes, typically 5–15 min at ∼400 MPa, reflected the



V. cholerae, V. cholerae non-O:1, V. hollisae and V. mimicus [5].

Another example of unique pressure effect is the moderate hydrostatic pressure

(MHP) treatment proposed to improve the early shreddability of Cheddar cheese

[46–48]. MHP (345 MPa, 483 MPa for 3 and 7 min) applied to fresh Cheddar cheese

curd induces immediately a microstructure resembling that of ripened cheese. Scan-

ning electron microscopy (SEM) shows major changes in the microstructure of

Cheddar cheese immediately after HPP treatment (Figure 7.5). Transmission elec-

tron microscopy (TEM) shows similar microstructure changes (Figure 7.6) in Ched-

dar cheese treated at 275 MPa for 100 s [49]. Sensory evaluations of pressurized

Cheddar cheese [47,48] show that MHP treatments improves the visual and tactile

sensory properties of shredded Cheddar cheese (Figure 7.7). By reducing the pres-

ence of crumbles, increasing the mean shred particle length, improving its length

uniformity and enhancing surface smoothness, it is possible to obtain shreds from

unripened cheese with high visual acceptability and improved tactile handling. The

increase in these and other desirable physical properties suggest that when pressure is

applied, proteins are partially denatured and form a more continuous cheese matrix.

Cheese processors could use MHP to eliminate ripening as a preliminary step for

shredding and still obtain products with equal tactile and visual attributes to those

produced from Cheddar cheese ripened for about 30 days. The advantages would be

Apple juice

Ascorbic acid ORAC FRAP 0

2

4

6

Control HPP

FRAP ORAC

0

10

20

30

L a b

Orange juice

0

20

40

60

80

L a b

0

2

4

6

mM

Tro

lox e

quiv

./L

mM

Tro

lox e

quiv

./L

FIGURE 7.4 Characterization of pressure treated juices by colour, vitamin C, ORAC and

FRAP determinations. (Adapted from Torres, J.A. and Velazquez, G. Commercial oppor-

tunities and research challenges in the high pressure processing of foods. Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 17955534_C007.indd 179 10/22/08 10:52:19 AM10/22/08 10:52:19 AM

microbial risk to consumers by inactivating Vibrio parahemolyticus, V. vulnificus,

refrigerated storage savings of over $30/1000 kg cheese and a simplified handling of



cheese for shredding. MHP has been shown to be effective both in Cheddar cheese

manufactured by stirred and milled curd technology [47,48], an encouraging obser-

vation suggesting that pressure treatments could improve the early shreddability of

other natural cheeses of commercial interest. Shredded cheese is the most common

ingredient cheese sold through retail, food-processing and foodservice marketing

channels.

7.2.1.1.3 Product is a High Microbial Risk to ProducerThis is the case when best production practices do not yield pathogen-free products;

however, the product is on the market because of a strong consumer demand. For

example, seafood processors cannot guarantee absence of Listeria monocytogenes

dence [50]. Good manufacturing and handling practices for cold smoked salmon

yield at best <1 L. monocytogenes cfu/g which explains why products do not pass

the detection procedures used by regulatory agencies with a higher sensitivity of 0.04

cfu/g. No outbreak cases have been associated with cold-smoked salmon; however,

product examinations by regulatory agencies have led to frequent product recalls

losses. HPP could reduce this risk with a process and product formulation developed

to minimize protein damage and thus changes in product texture.

345 MPa/3 min

Control

(b) Stirred

Control

345 MPa/3 min

(a) Milled

10 µm

10 µm10 µm

10 µm

FIGURE 7.5 Scanning electron microscopy (SEM) analysis to visualize changes in the

microstructure of Cheddar cheese immediately after moderate pressure treatments. (Adapted

from Torres, J.A. and Velazquez, G. Commercial opportunities and research challenges in the

high pressure processing of foods. Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 18055534_C007.indd 180 10/22/08 10:52:20 AM10/22/08 10:52:20 AM

in cold-smoked products and they are aware that FDA surveys find this pathogen

with a 17% frequency. Even hot-smoked fish and shellfish processors are concerned

because the same surveys find them to contain L. monocytogenes with a 4% inci-

(Table 7.1). These recalls are extremely costly in terms of financial and reputation



during post-capture manipulation and processing, particularly during deboning (or

(a)

(b)

1 µm

10 µm

1 µm

(c)

FIGURE 7.6 Transmission electron microscopy (TEM) analysis to visualize changes in the

microstructure of Cheddar cheese immediately after moderate pressure treatments. (Adapted

from Torres, J.A. and Velazquez, G. Commercial opportunities and research challenges in the

high pressure processing of foods. Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 18155534_C007.indd 181 10/22/08 10:52:22 AM10/22/08 10:52:22 AM

A related example is the production of restructured fish products from under-

raw fish has the risk of food borne diseases associated with microbial contamination

utilized species such as arrowtooth flounder (Atheresthes stomias). Using minced



ing. HPP can reduce this microbial load and could be an alternative to induce gel

paste treated at 400–600 MPa for 5 min yields products with appropriate mechani-

cal and functional properties [51]. These high pressure conditions are appropriate to

inactivate parasites and most spoilage microorganisms.

Milled

345Mpa-3min 345Mpa-7min 483Mpa-3min 483Mpa-7minControl

VO = Visual oiliness

SS = Smooth surface

ML = Mean length

UL = Uniformity of length

PC = Presence of crumbles

TO = Tactile oiliness

A = Adhesiveness

0

2

4

6

8VO

SS

ML

ULPC

TO

A

Day 1

0

2

4

6

8VO

SS

ML

ULPC

TO

A

Day 27

0

2

4

6

8VO

SS

ML

ULPC

TO

A

Day 1

0

2

4

6

8VO

SS

ML

ULPC

TO

A

Day 27

Stirred

FIGURE 7.7 Visual and tactile sensory properties of shredded Cheddar cheese obtained

from control and pressure treated curd. (Adapted from Serrano, J. Efecto de altas presiones

en la microestructura de quesos, Aplicación en el rallado de queso Cheddar para uso comer-

cial. MSc dissertation, Oregon State University, Querétaro, Qro. México, 2003; Serrano, J.,

Velazquez, G., Lopetcharat, K., Ramirez, J.A., and Torres, J.A. Effect of moderate pressure

treatments on microstructure, texture, and sensory properties of stirred-curd Cheddar shreds.

Journal of Dairy Science 87, 3172–82, 2004. and Serrano, J., Velazquez, G., Lopetcharat, K.,

Ramirez, J.A., and Torres J.A. Moderately high hydrostatic pressure processing to reduce

production costs of shredded cheese: Microstructure, texture, and sensory properties of

shredded milled curd cheddar. Journal of Food Science 70(4), S286–93, 2005.)

55534_C007.indd 18255534_C007.indd 182 10/22/08 10:52:23 AM10/22/08 10:52:23 AM

filleting and mincing), protein solubilizing with salt, product forming and packag-

formation without heat to obtain products closer to raw fish. Arrowtooth flounder



depends on pressure level, holding time [51] and temperature [52]. Gels obtained

freeze-drying or thermal abuse show poor mechanical properties [53]. However,

mechanical properties of heat-induced gels (90°C for 30 min) and actually improved

the ones for the gels obtained by inducing their subsequent setting [54]. Pressure

treatments induced favorable changes in the mechanical properties of heat-induced

Pressure induces a protein denaturation/aggregation different from that resulting

from freezing, drying or heating where denaturation is immediately followed by irre-

versible aggregation affecting negatively the mechanical properties of gels. Differ-

ation of muscle proteins [55]. It appears that pressure induces a protein aggregation

dominated by side-to-side interactions of proteins with a low degree of denaturation

and not by aggregation of proteins with major changes in molecular conformation.

Pressure treatments break intermolecular bonds with reaction rate constant k1 induc-

ing changes in protein conformation favoring a protein aggregation reaction with

TABLE 7.1Examples of Smoked Salmon Recalls, 1999–2001

Date Product

04-Dec-01 Frozen smoked salmon

07-Mar-01 Bear Candy smoked salmon

26-Jul-00 Jensen’s Old Fashioned Smokehouse Inc. smoked king salmon

12-Apr-00 Chef Daniel Boulud Atlantic smoked salmon

12-Apr-00 Scandinavian Smoke House salmon

12-Apr-00 Chef Daniel Boulud smoked Atlantic salmon

27-Mar-00 Craigellachie smoked Scottish salmon

14-Mar-00 Grants traditional oak-smoked salmon

10-Mar-00 Royal Baltic Smoked Captain salmon

10-Mar-00 Imperial European-style smoked salmon

11-Jan-00

10-Jan-00 Imperial European-style smoked salmon

19-Sep-99 Blue Ribbon Smoked Fish Co. smoked salmon

18-Nov-99 Kendall Brook smoked Atlantic salmon

09-Nov-99 Tuv Taam sliced smoked Nova salmon

23-Dec-99 Royal Baltic smoked salmon

06-Apr-99 Perona Farms smoked salmon

01-Apr-99 Perona Farms smoked salmon

Source: http://www.foodsafetynetwork.ca/, adapted from [4]

55534_C007.indd 18355534_C007.indd 183 10/22/08 10:52:24 AM10/22/08 10:52:24 AM

Highland Crest finest smoked Scottish salmon

Denaturation/aggregation of myofibrillar proteins induced by high pressure

from myofibrillar proteins previously denatured/aggregated by freezing, dehydration,

pressure treating fish paste at 300 MPa for 30 min at 5°C did not deteriorate the

gels when compared with gels obtained by heating untreated fish paste [51].

ential scanning calorimetry (DSC) thermograms of fish paste subjected to 200 MPa

for 10 min at 7°C differ from those for a fish paste control indicating partial denatur-


http://www.foodsafetynetwork.ca


reaction rate constant k2 (Figure 7.8). The reaction rate constants expected would be

k1< k2 and as a consequence, an almost immediate aggregation of these extensively

denatured proteins would take place. However, protein aggregation with reaction

rate constant k3 appear to occur preferentially by side-to-side interactions of proteins

with low degree of denaturation and the overall aggregation reaction can be charac-

terized by k3 >>(k1 + k2). Evidence of this type of aggregation has been reported in

tion improves the mechanical properties of heat-induced gels, a situation similar to

H

H

HPP

HH2O

Native protein

Denaturated protein

Preferred agregation:

side-to-side interaction

of near-native proteins

Crosslinking of denatured proteins

HPP k1

HPP

k2

HH2O

H2O

H2O

Protein structure showing intramolecular interactions

Denaturated protein structure with broken intramolecular interactions

Pressure induced crosslinking protein interactions

Pressure effect on the protein system

k3

FIGURE 7.8 Suggested mechanism for pressure-induced changes in proteins where k1, k2

and k3 are reaction constants such that k1 < k2 and k3 >>(k1 +k2). (Adapted from Uresti, R.M.,

Velazquez., G., Ramírez, J.A., Vázquez, M., and Torres, J.A. Effect of high pressure treatments

(Atheresthes stomias). Journal of the Science of Food and Agriculture 84(13), 1741–49, 2004.)

55534_C007.indd 18455534_C007.indd 184 10/22/08 10:52:24 AM10/22/08 10:52:24 AM

pressure-induced fish proteins gels [54]. Because of this, pressure-induced aggrega-

on mechanical and functional properties of restructured products from arrowtooth flounder



the favorable protein aggregation induced at low temperature by transglutaminase in

the so-called setting or suwari phenomenon. Setting induces protein aggregation only

brillar proteins aggregate immediately inhibiting the transglutaminase effect [56].

Sugars and polyols at concentrations below 8% are used to inhibit protein inter-

treated samples while sucrose and trehalose are unable to inhibit this aggregation in

and maltodextrin [57]. The mechanism involved in the protection by sugar and polyol

stabilizers against pressure damage on protein functionality has been associated with

inhibition of the side to side protein interactions as illustrated in Figure 7.9. Sugars

and polyols stabilize proteins by the mechanism of solute exclusion from the hydration

ratio of proteins. Solutes are excluded from the surface of the proteins and do not react

with them [57,58–60]. However, this hypothesis is not fully accepted yet. The possibil-

ity of interactions between sugars and proteins or lipids has been reported by using

nuclear magnetic resonance (NMR) and quantum chemical methodology [61].

7.2.1.1.4 Product has a High Added Value and Thermal LabilityBiologically active compounds are an important market as the consumer interest

in functional foods continues to expand. Sales of specialty supplements, functional

foods, nutraceuticals and natural personal care products are soaring creating a world-

wide opportunity. A major trend that ensures continuing growth in the demand for

to 46% in 2000. Concurrently, those who believe they need added nutrients increased

from 54% in 1994 to 70% in 2000 (Multi-Sponsor Surveys 2001, Princeton, NJ)

[4,123]. HPP can help meet the challenge of producing from natural sources and

without damaging biologically active compounds ingredients with low microbial

spoilage counts and free of pathogens.

7.2.1.2 Mechanisms of Microbial Inactivation by Pressure

7.2.1.2.1 Vegetative BacteriaMechanistic studies now emerging in the literature show that HPP inactivates micro-

organisms by interrupting cellular functions responsible for reproduction and sur-

vival (Figure 7.10). HPP can damage microbial membranes and thus affect transport

onstrating that leaks occur while cells are held under pressure [62]. Membrane damage

occurs later than cell death and this suggests that dye exclusion measurements assess-

ing this pressure effect can be used to characterize microbial pressure inactivation [63].

Knowledge of cell damage and repair mechanisms could lead to new HPP applica-

tions [64,65]. For example, lysis of starter bacteria induced by HPP treatments could

promote the release of intracellular proteases important in cheese ripening. Viability,

morphology, lysis and cell wall hydrolase activity measurements suggest that high

pressure can cause inactivation, physical damage, and lysis in Lactobacillus lactis [66].

55534_C007.indd 18555534_C007.indd 185 10/22/08 10:52:25 AM10/22/08 10:52:25 AM

for partially denatured myofibrillar proteins (mostly myosin). Fully denatured myofi-

actions. Sorbitol inhibits the aggregation of myofibrillar proteins of raw pressure-

higher stabilizing effect on the ATPase activity of fish myosin than lactitol, sucrose

these products is a diminishing confidence that our diet satisfies our nutritional needs.

In 1994, 70% of women believed their diet met their nutritional needs, a figure down

low-temperature, pressure-treated arrowtooth flounder mince [51]. Sorbitol shows a

phenomena involved in nutrient uptake and disposal of cell waste. Intracellular fluid

compounds have been found in the cell suspending fluid after pressure treatment dem-



Treatments at 300 MPa have shown by TEM intracellular and cell envelope damage of

the cheesemaking strains Lactococcus lactis subsp. cremoris MG1363 and SK11. Cell

whereas cells treated at >400 MPa had decreased cell wall hydrolase activity. How-

samples indicating that this pressure activates cell wall hydrolase activity or increases

cell wall accessibility to the enzyme.

HPP

Stabilizing agent

H2O

H OH2O

HPP

Native proteins

Pressure induced

side-to-side aggregation

Native proteins

Stabilized proteins against

side-to-side aggregation

H2O

HH2O

Protein structure showing intramolecular interactions

Stabilizing agent molecule

Pressure induced crosslinking protein interactions

Pressure effect on the protein system

FIGURE 7.9 against pressure induced aggregation. (Adapted from Uresti, R.M., Velazquez, G., Ramírez,

J.A., Vázquez, M., and Torres, J.A., Effect of high pressure treatments on mechanical and

mias). Journal of the Science of Food and Agriculture 84(13), 1741–49, 2004.)

55534_C007.indd 18655534_C007.indd 186 10/22/08 10:52:26 AM10/22/08 10:52:26 AM

Mechanism of stabilization of sugars and polyols on myofibrillar proteins

suspensions treated at 200 or 300 MPa did not differ significantly from the control,

ever, cells treated at 100 MPa released significantly more reducing sugar than all other

functional properties of restructured products from arrowtooth flounder (Atheresthes sto-



Increasing our knowledge of the behaviour of bacterial membrane proteins sub-

jected to pressure under different conditions (e.g. pH or aw) will lead to effective

membranes of untreated Salmonella typhimurium reveal three major and 12 minor

protein bands but only two major bands after pressure treatments [67]. One band is

more pressure resistant in acidic pH media suggesting a different protein conformation

at this condition. HPP treatments, 345 MPa for 5 min at 25°C, alter the cell walls of

Leuconostoc mesenteroides and make cell membranes permeable [68]. This damage

reduces the potential gradient across membranes, preventing cells from synthesizing

ATP, which activates the autolytic enzyme degradation of cell walls. Cells treated at

400 MPa for 10 min in pH 5.6 citrate buffer show no growth after 48 h of culture on

plate count agar. Cells can be examined by SEM, membrane integrity by propidium

even though membrane potential decreases from −86 to −5 mV [69,70]. Membrane

damage in S. typhimurium can also be measured by pH differential (pHin–pHout). Mor-

phological changes increasing with pressure correlate with a progressive decrease of

the pH differential, intracellular potassium, and ATP concentration [71,72].

Denaturation

Activeenzyme

Inactiveenzyme

Renaturation

Enzymes

Nutrients

Waste

Leakage

Membranes

(a)

(b)

Hydrostatic pressure effects on cellular functions. (Adapted from Torres, J.A.

and Velazquez, G. Commercial opportunities and research challenges in the high pressure

processing of foods. Journal of Food Engineering 67, 95–112, 2005.)

55534_C007.indd 18755534_C007.indd 187 10/22/08 10:52:27 AM10/22/08 10:52:27 AM

FIGURE 7.10

hurdle preservation technologies. For example, electrophoretic profiles of the outer

ies reveal no significant changes in cellular morphology while PI staining followed

iodide (PI) staining and changes in membrane potential by flow cytometry. SEM stud-

by flow cytometry shows a small population proportion with membrane integrity loss



The outer membrane (OM) providing a protective barrier to Gram-negative bac-

teria is susceptible to pressure-mediated permeabilization. The kinetics of OM and

cytoplasmic membrane permeabilization induced by pressure treatments can be deter-

only slightly even after pressure treatments resulting in a >6 log decrease in viable

observed prior to cell death. Reversible OM damage occurs rapidly and is in thermo-

dynamic equilibrium with pressure conditions while irreversible OM damage is time

dependent. Pressure (200 or 400 MPa) resistance of exponential-phase E. coli NCTC

8164 cells is highest for cells grown at 10°C and decreases with growth temperature

up to 45°C [74]. By contrast, pressure resistance of stationary-phase cells is lowest

in cells grown at 10°C and increases with growth temperature reaching a maximum

at 30–37°C before decreasing at 45°C. This pressure effect can be correlated to the

proportion of unsaturated fatty acids in the membrane lipids which decreases with

growth temperature in both exponential- and stationary-phase cells. In exponential-

sure resistance has been observed.

7.2.1.2.2 Bacterial SporesAlthough the application of 400–800 MPa inactivates pathogenic and spoilage bac-

teria [4,75–77], the inactivation of bacterial spores has been a major challenge to HPP

process developers as these spores are extremely resistant to pressure. Therefore,

current HPP products on the market rely on refrigeration, reduced water activity or

low pH to prevent bacterial spore outgrowth.

surized at 980 MPa for 40 min at room temperature [78]; however, combining tem-

peratures higher than 50°C with pressures above 400 MPa can be effective. For

example, treating Bacillus subtilis spores at 404 MPa and 70°C for 15 min can

spores of Clostridium sporogenes, considered a non-toxigenic equivalent to pro-

teolytic C. botulinum and an important food spoilage bacteria, to 400 MPa at 60°C

for 30 min at neutral pH yields only 1 DR [80]. The effect of 15 min hydrostatic

pressure treatments (550 and 650 MPa) at 55 and 75°C in citric acid buffer (4.75

the gene that encodes the C. perfringens enterotoxin (cpe) on the chromosome (C-

cpe), four isolates carrying the cpe gene on a plasmid (P-cpe), and two strains of

C. sporogenes was studied to develop an effective spore inactivation strategy [81].

Treatments at 650 MPa, 75°C and pH 6.5 were found to be moderately effective

against spores of P-cpe (approximately 3.7 DR) and C. sporogenes (approximately

2.1 DR) but not for C-cpe (approximately 1.0 DR) spores. Treatments at pH 4.75

were moderately effective against spores of P-cpe (approximately 3.2 DR) and C. sporogenes (approximately 2.5 DR) but not of C-cpe (approximately 1.2 DR) when

combined with 550 MPa at 75°C. However, when pressure was raised to 650 MPa

under the same conditions, high inactivation of P-cpe (approximately 5.1 DR) and

55534_C007.indd 18855534_C007.indd 188 10/22/08 10:52:28 AM10/22/08 10:52:28 AM

Spores of six Bacillus species showed no significant inactivation when pres-

achieved five decimal reductions (DRs) at neutral pH [79]. However, subjecting

and 6.5 pH) on spores of five isolates of Clostridium perfringens type A carrying

mined by staining pressure-treated cells with the fluorescent dyes propidium iodide

(PI) and 1-N-phenylnaphtylamine (NPN), respectively [73]. PI fluorescence increases

cell counts while increased NPN fluorescence, indicating OM permeabilization, is

phase cells, pressure resistance increased with greater membrane fluidity, whereas in

stationary-phase cells, no simple relationship between membrane fluidity and pres-



C. sporogenes (approximately 5.8 DR) spores, and moderate inactivation of C-cpe

(approximately 2.8 DR) spores were observed. These studies show the important

need for further advances in high-pressure treatment strategies to inactivate bacte-

modeling of spore germination appears promising.

Many models have been developed to predict the growth of C. perfringens in

meat products during the cooling stage [e.g. 82–84]. However, modeling of ger-

mination has not been fully studied. Models for B. cereus spore germination in

the presence of L-alanine based on the Weibull function have been proposed [85].

Germinants, that is, compounds that promote spore germination, were recently

in the presence of free amino acids (e.g. L-asparagine) but fast in the presence

of potassium chloride [81]. The Weibull function was used to model C-cpe spore

germination as affected by pH, germinant concentration, and spore germination

temperature [81]. An empirical predictive germination rate model for the germina-

tion of any C. perfringens type A food poisoning isolate as a function of spore ger-

mination temperature was also constructed. These advances in spore inactivation

will further enhance the opportunities to develop shelf-stable food products based

on PATP technology.

7.3 PRESSURE ASSISTED THERMAL PROCESSING (PATP) OF FOODS

The extent to which the severity of pressure-assisted thermal processing (PATP) con-

ditions is increased to enhance microbial inactivation and shelf life must be carefully

approached. Unfortunately very few reports have been published on PATP effects on

chemical changes in foods. A brief published summary reports that in a sugar-amino

derived compounds [86]. Another study reported that reported that although HPP

does not lower the consumer acceptability of orange juice it could change its volatile

milk [11]. Moreover, at the same temperature, a principal component analysis of the

tion different from that observed for UHT milk with sensory properties rejected by

consumers [14].

7.3.1 REACTION KINETICS ANALYSIS

The analysis of PATP effects can be investigated from a reaction kinetic analysis

point of view to illustrate the advantages of this technology. During a reaction, the

change in concentration (c) of a given compound with respect to time (t) can be

expressed as [88]:

d

d

nct

kc= (7.1)

55534_C007.indd 18955534_C007.indd 189 10/22/08 10:52:28 AM10/22/08 10:52:28 AM

rial spores more efficiently. A focus on a mechanistic understanding and process

identified for C-cpe spores. At 30–50°C, spores of C-cpe isolates germinate slowly

profile [87]. Milk subjected to HPP treatments equal or less severe than 586 MPa

and 60°C for up to 5 min had a volatile profile similar to that of HTST-pasteurized

volatile profile in milk under higher hydrostatic pressure was displaced in a direc-

acid model solution pressure influences the thermally induced formation of Maillard-



where k is the reaction rate constant at the experimental pressure and temperature

while n is the reaction order. Integrating Equation 7.1 yields the following linearized

kinetic expressions:

Zero order: c – c0 = kt (7.2)

First order: log(c) – log(c0) = kt (7.3)

Second order: 1 1

0c ckt− = (7.4)

The regression curve obtained by simple linear regression of the concentration com-

cient (R2) for one of the three kinetic models is then used to calculate the rate constant

k. Experimental conditions imply that the c0 value in these equations is a ‘pseudo-

initial concentration’. Multiple linear regression analysis is used to test the difference

between the linearized intercepts at the same temperature for each pressure level

tion c0 which is then reported as an average value for each temperature level. The

difference between this concentration and the one found in untreated samples repre-

sent the effect of time when the food has not yet reached the vessel temperature and

pressure including time for pressure come-up and come-down.

Calculated pseudo-initial concentration c0 values for raw milk samples subjected

to pressure (482, 586, 620, and 655 MPa), temperature (45, 55, 60, and 75°C before

compression) and time (1, 3, 5, and 10 min) treatments in a 2.2-l high-pressure ves-

sel (Engineered Pressure Systems Inc., Haverhill, MA) equipped with a temperature

controller and a high-pressure pump (Model P100-10FC, Hydro-Pac Inc., Fairview,

PA) were recently reported [15]. All milk samples were previously equilibrated to

25°C and processed immediately with vessel loading (1 min) and unloading times

(1.5 min) kept constant for all runs. The average pressure come up time was 40 s.

After treatment, samples were placed immediately in a saturated salt slurry and

ice bath before storage at −38°C until analysis by headspace solid-phase micro-

19 other volatiles were analyzed using a headspace solid-phase microextraction and

[90]. It should be noted that the volatile formation reported in Table 7.2 and Table

7.3 includes the effect of the temperature increase due to adiabatic heating during

sample compression [91]. In the 400–1000 MPa range, milk temperature increases

approximately 3°C for every 100 MPa [92]. In addition, even though milk is a well-

buffered food system, reaction rates may be affected by the temporary pH shift

an alignment of water molecules resulting in a more compact arrangement around

charged species [94–98]. This temporary pH shift cannot be measured experimen-

tally as pH probes are currently unavailable for measurements at high pressure lev-

els. The strategy of a pressure-independent buffer system used in studies of reaction

55534_C007.indd 19055534_C007.indd 190 10/22/08 10:52:29 AM10/22/08 10:52:29 AM

pared with time at constant pressure and temperature with the best correlation coeffi-

tested to confirm that all regression lines start at the same pseudo-initial concentra-

SPME/GC-PFPD) for eight volatile sulphur compounds [89]. Dimethyl sulfide and

induced by pressure [93]. As pressure increases, the columbic field of ions produces

extraction and gas chromatography with pulsed-flame photometric detection (HS-

gas chromatography with flame ionization detection (HS-SPME-GC/FID) technique



TABLE 7.2Effect of Hydrostatic Pressure and Temperature on the First-Order Reaction Rate Constant for the Formation of Various Straight-Chain Off-Flavour Aldehydes in Milk*

T† (°C) P‡ (MPa)

Hexanal Heptanal Octanal Nonanal Decanal

k R2 k R2 K R2 k R2 k R2

45 482 0.010a 0.87 0.006a 0.81 0.001a 0.92 0.003a 0.82 0.002a 0.85

586 0.046b 0.99 0.049b 0.94 0.038b 0.92 0.023b 0.92 0.004ab 0.80

620 0.084c 0.99 0.085c 0.99 0.035b 0.91 0.034c 0.93 0.008b 0.91

655 0.105d 0.99 0.127d 0.99 0.049c 0.86 0.045d 0.98 0.017c 0.91

c0§ 1.58 0.34 0.25 2.69 3.69

55 482 0.012a 0.89 0.011a 0.92 0.005a 0.80 0.007a 0.84 0.006a 0.84

586 0.049b 0.97 0.054b 0.96 0.043b 0.98 0.037b 0.93 0.009ab 0.95

620 0.091c 0.97 0.102c 0.98 0.062c 0.98 0.048c 0.98 0.013b 0.94

655 0.106d 0.97 0.130d 0.99 0.079d 0.90 0.065d 0.98 0.019c 0.97

c0 2.88 0.37 0.44 2.91 5.77

60 482 0.021a 0.91 0.022a 0.91 0.009a 0.81 0.010a 0.89 0.009a 0.83

586 0.064b 0.97 0.074b 0.98 0.065b 0.96 0.045b 0.96 0.012a 0.94

620 0.092c 0.98 0.114c 0.96 0.088c 0.99 0.056c 0.98 0.016b 0.87

655 0.108d 0.97 0.134d 0.99 0.098d 0.97 0.067d 0.97 0.023c 0.90

c0 4.63 0.56 0.77 4.22 5.88

75 482 0.029a 0.96 0.033a 0.93 0.019a 0.92 0.014a 0.96 0.016a 0.91

586 0.084b 0.99 0.088b 0.96 0.119c 0.96 0.057b 0.98 0.021b 0.92

620 0.099c 0.98 0.125c 0.98 0.107b 0.93 0.071c 0.99 0.025c 0.93

655 0.109d 0.97 0.138d 0.98 0.128d 0.95 0.082d 0.98 0.030d 0.90

c0 12.41 2.10 1.15 8.03 7.46

craw|| 1.51 0.26 0.24 2.54 3.42

c75°C/655 MPa/ 10 min# 144.5 36.9 18.3 49.2 16.1

a, b, c, d

pressure values at constant temperature.* k = Rate constant (min−1 2

† T = Temperature.‡ P = Pressure.§ c0 = Pseudo-initial concentration (μg/kg), i.e. concentration in raw milk plus changes due to sample

handling before and after pressure treatment.|| Concentration (μg/kg) measured in raw milk.# Maximum concentration (μg/kg) measured in milk samples treated at 75°C and 655 MPa for 10 min.

Source: Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of vola-

tile formation in milk subjected to pressure-assisted thermal treatments. Journal of Food Science 72(7),

E389–98, 2007.

55534_C007.indd 19155534_C007.indd 191 10/22/08 10:52:30 AM10/22/08 10:52:30 AM

Different letters mean significant difference (α = 0.05) of the rate constant for different hydrostatic

) with R = correlation coefficient.



TABLE 7.3Effect of Hydrostatic Pressure and Temperature on the Zero-Order Rate Constants for the Formation of Various Off-Flavour Compounds in Milk*

T† (°C) P‡ (MPa)

2-Methylpropanal 2,3-Butanedione

k R2 k R2 k R2

45 482 0.016a 0.88 0.009a 0.83 0.112a 0.90

586 0.018a 0.86 0.010a 0.84 0.165b 0.94

620 0.017a 0.81 0.010a 0.85 0.205c 0.96

655 0.016a 0.91 0.013b 0.93 0.208c 0.97

c0§ 0.73 0.44 1.62

55 482 0.018a 0.85 0.009a 0.90 0.119a 0.90

586 0.023b 0.83 0.014a 0.94 0.227b 0.89

620 0.021b 0.91 0.022b 0.94 0.221b 0.85

655 0.022b 0.87 0.024b 0.86 0.242b 0.88

c0 0.54 0.37 8.57

60 482 0.019a 0.83 0.014a 0.94 0.140a 0.87

586 0.024b 0.83 0.020b 0.89 0.236b 0.94

620 0.023b 0.934 0.040c 0.92 0.275c 0.91

655 0.021a 0.93 0.053d 0.98 0.280c 0.94

c0 0.48 0.41 14.56

75 482 0.023a 0.82 0.029a 0.88 0.217a 0.84

586 0.030a 0.90 0.045b 0.97 0.314b 0.82

620 0.027a 0.92 0.055c 0.96 0.371c 0.80

655 0.030a 0.92 0.082d 0.99 0.394c 0.86

c0 0.38 0.55 16.65

craw|| 0.75 0.36 1.03

c75°C/655 MPa/ 10 min# 0.64 1.24 23.2

a, b, c, d

pressure values at constant temperature.* k = Rate constant (min−1 2

† T = Temperature.‡ P = Pressure.§ c0 = Pseudo-initial concentration (μg/kg), i.e. concentration in raw milk plus changes due to sample han-

dling before and after pressure treatment.|| Concentration (μg/kg) in raw milk.# Maximum concentration (μg/kg) in milk treated at 75°C and 655 MPa for 10 min.

Source: Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of volatile

formation in milk subjected to pressure-assisted thermal treatments. Journal of Food Science 72(7), E389–

98, 2007.

55534_C007.indd 19255534_C007.indd 192 10/22/08 10:52:31 AM10/22/08 10:52:31 AM

Hydrogen Sulfide

Different letters mean significant difference (α = 0.05) of the rate constant for different hydrostatic




kinetics in model systems or the inactivation of bacterial spores [81] cannot be used

in studies with milk.

All pseudo-initial concentration c0 values shown in Table 7.2 and Table 7.3 for

compounds following 1st and zero order kinetics, respectively, except those for 2-

above the amount present in raw milk as a result of the sample heating during the

handling steps for each HPP run. Values of c0 did not change with pressure (multiple

treatment applied, sample heating before and after each HPP run, sample loading,

vessel closing, and sample unloading steps were responsible for the increase in the

amount of volatiles above the amount present in raw milk.

Activation energies (Ea) for volatile formation in milk at constant high hydro-

static pressure can be calculated using the Arrhenius Equation 7.5 in its linearized

form (Equation 7.6). The slope of this curve (−Ea/R with R = universal gas constant,

8.314 × 10−3 kJ mol−1 K−1) and the intercept (ln k0 with k0 = pre-exponential rate con-

stant) are calculated as follows.

k k eE

RT=−

0

a

(7.5)

ln( ) ln( )k kERT

= −0a

(7.6)

A quantity derived from the pressure dependence of the rate constant k (Equation

partial molar volumes of the transition state and the sums of the partial volumes of

the reactants at the same temperature and pressure [99]. When pressure is applied,

ΔV* < 0 leads to an increase in reaction rate while ΔV* > 0 has the opposite effect.

The greater the magnitude of ΔV* (positive or negative) the higher the sensitivity of a

chemical reaction to pressure while reactions with ΔV* = 0 are pressure independent

[100]. Equation 7.7 can be integrated to obtain Equation 7.8 where lnA is the inte-

gration constant. Values for ΔV* as a function of temperature are then calculated by

linear regression of lnk versus pressure p.

ΔV RTk

p*

ln= −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

∂∂

T

(7.7)

ln ln( )*

k AV pRT

= −Δ

(7.8)

The 1st-order kinetic constants observed for straight chain aldehydes in milk pro-2

a

Hexanal formation had the lowest Ea value decreasing from 35.2 to 0.9 kJ mol−1 at the

maximum pressure tested (Table 7.4). Compounds with formation reaction following

55534_C007.indd 19355534_C007.indd 193 10/22/08 10:52:32 AM10/22/08 10:52:32 AM

methylpropanal and 2,3-butanedione, increased significantly with test temperature

linear regression with 95% confidence) confirming that regardless of the pressure

7.7) is the partial activation volume (ΔV*), is defined as the difference between the

cessed under PATP fitted well (R > 0.9) the Arrhenius model with activation energy

(E ) values decreasing significantly with pressure (Table 7.4 and Table 7.5) [15].

zero order kinetic models (Table 7.3) showed also a good fit to the Arrhenius model


194 Fo

od

Processin

g Op

eration

s Mo

delin

g: Design

and

An

alysis

TABLE 7.4Effect of Hydrostatic Pressure on the Activation Energy for the Formation in Milk of Various Off-Flavour Compounds*

P† (MPa)

Straight Chain Aldehydes (First Order Reaction Kinetics) Other Compounds (Zero Order Reaction Kinetics)

Hexanal Heptanal Octanal Nonanal Decanal 2-Methyl-propanal 2,3-Butanedione

Ea R2 Ea R2 Ea R2 Ea R2 Ea R2 Ea R2 Ea R2 Ea R2

482 35.2 0.91 53.8 0.93 88.3 0.92 48.5 0.90 66.9 0.93 11.1 0.99 39.0 0.91 21.1 0.92

586 19.5 0.93 19.2 0.91 36.9 0.94 28.0 0.93 51.7 0.96 14.6 0.96 47.1 0.98 19.2 0.97

620 4.8 0.98 11.6 0.93 34.5 0.91 22.3 0.97 36.3 0.99 15.0 0.96 52.2 0.90 19.0 0.94

655 0.9 0.93 2.4 0.96 29.1 0.94 17.8 0.91 18.4 0.95 18.7 0.97 57.8 0.92 20.0 0.98

* Ea = Activation energy (kJ mol−1 2

† P = Pressure.

of Agricultural and Food Chemistry 54(24), 9184–92, 2006a.

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Hydrogen Sulfide


Source: Adapted from Vazquez-Landaverde, P.A., Torres, J.A., and Qian, M.C. Effect of high pressure-moderate temperature processing on the volatile profile of milk. Journal



(R2 > 0.9); however, Ea values were affected differently by pressure, increasing for

2-methylpropanal and 2,3-butanedione and remaining practically unchanged for

hydrogen sulphide regardless of the pressure level (Table 7.4 and Table 7.5). Pressure

increased the formation of straight-chain aldehydes, decreased that of 2-methylpro-

panal and 2,3 butanedione, and did not affect that for hydrogen sulphide (Table 7.4).

A remarkable observation was that the concentration of the other 18 volatiles ana-

lyzed in this study on PATP-milk did not increase during pressurization time (slope

P-value > 0.05, R2 < 0.60) for all pressure and temperature levels tested. Predicted

pseudo-initial concentration c0 values for some of these compounds increased with

treatment temperature (Table 7.6), but the concentration of these volatiles remained

stable during pressurization time, indicating that only the heating time at atmospheric

pressure before and after each HPP treatment was responsible for the increase in

these volatiles above the level found in raw milk.

The lack of contribution to the concentration of most milk volatiles by heating time

formation of these volatiles even though initial vessel temperature varied from 45 to

75°C, and compression to 482–655 MPa for 1–10 min further increased milk tempera-

ture due to adiabatic heating. Past research on the formation of these compounds had

shown a concentration increase with processing temperature [89,90,101–105]. This

suggested that the formation of these compounds was inhibited by pressure. Firm

evidence of the inhibition by pressure could have been obtained by PATP milk experi-

ments in the range 0.1–482 MPa. However, this range was outside the scope of the

reported work. The pressure range covered in this PATP-milk study, 482–655 MPa,

TABLE 7.5Relationship Between the Activation Energy and the Hydrostatic Pressure for the Formation of Various Off-Flavour Compounds in Milk*

Compound Relationship† R2

Hexanal Ea = −0.20x +134.84 0.96

Heptanal Ea = −0.29x +197.25 0.99

Octanal Ea = 0.002x2 − 2.65x +894.08 0.99

Nonanal Ea = −0.18x +135.16 0.99

Decanal Ea = −0.001x2 +1.86x − 394.38 0.99

2-Methylpropanal Ea = 0.03x − 8.28 0.90

2,3-Butanedione Ea = 0.10x − 12.44 0.96

* Ea = Activation energy (kJ mol−1).† x = Hydrostatic pressure (MPa).

Source: Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and

Torres, J.A. Kinetic analysis of volatile formation in milk subjected

to pressure-assisted thermal treatments. Journal of Food Science

72(7), E389–98, 2007.

55534_C007.indd 19555534_C007.indd 195 10/22/08 10:52:34 AM10/22/08 10:52:34 AM

under pressure was a most interesting finding. High hydrostatic pressure inhibited the



was chosen to approach conditions achieving microbial inactivation including bacte-

rial spores [15].

reactions in PATP milk can be obtained by comparing the effect of thermal treat-

ments at conventional and PATP conditions.

Volatiles in milk subjected to conventional and PATP treatments are shown in

Table 7.7. The concentration of volatiles in milk subjected to 620 MPa for 5 min at 75°C

was compared to a heat treatment at atmospheric pressure that simulated the temperature

TABLE 7.6Effect of Temperature During Sample Loading (3.5 min) into the Pressure Vessel on the Average Pseudo-Initial Concentration* (c0) of Various Off-Flavour Compounds in Milk

Compound

Temperature (°C)Relationship Intercept (I)

vs. Temperature (T) R245 55 60 75

Ethyl acetate (μg/kg) 0.19a 0.21a 0.27b 0.29b I = 0.003T − 0.03 0.854

2-Methylbutanal (μg/kg) 0.14a 0.24b 0.24bc 0.28c I = 0.004T − 0.02 0.817

2-Pentanone (μg/kg) 0.14a 0.15a 0.15a 0.18b I = 0.001T − 0.07 0.925

3-Methyl-1-butanol (μg/kg) 0.09b 0.07a 0.07a 0.08ab NR NR

2-Hexanone (μg/kg) 0.07a 0.13b 0.15b 0.28c I = 0.007T–0.25 0.982

2-Furaldehyde (μg/kg) 0.29a 0.43b 0.44b 0.47b NR NR

2-Heptanone (μg/kg) 1.14a 1.44b 2.05c 4.01d I = 0.002T 2 − 0.22T +5.91 0.996

2-Octanone (μg/kg) 1.49c 1.26b 1.17b 0.97a I = −0.017T +2.22 0.975

2-Nonanone (μg/kg) 0.47a 0.48a 0.57ab 0.64b I = 0.006T–0.18 0.893

2-Decanone (μg/kg) 0.62ab 0.69b 0.58ab 0.55a NR NR

2-Undecanone (μg/kg) 0.30ab 0.52c 0.22a 0.38b NR NR

Methanethiol (μg/kg) 5.20a 5.21a 5.59a 7.76b I = 0.004T 2 − 0.40T +15.04 0.999

21.87c 19.16b 20.01bc 14.42a NR NR

3.97a 3.95a 3.91a 3.89a NR NR

24.84a 33.39b 22.08a 34.72b NR NR

30.83a 43.96b 36.82ab 122.83c I = 0.141T 2 − 14.05T +378.32 0.968

Dimethyl sulfoxide (mg/kg) 0.66ab 0.71b 0.57a 1.29c NR NR

Dimethyl sulfone (mg/kg) 0.87b 0.91b 0.79a 0.99c NR NR

a, b, c, d Different letter for each compound indicates statistical difference between the pseudo-initial con-

centration (Tukey HSD 95%).* c0 = Pseudo-initial concentration (μg/kg), i.e. concentration in raw milk plus changes due to sample

handling before and after pressure treatment.

NR = No relationship between intercept and temperature (R2 < 0.800).

Source: Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of vola-

tile formation in milk subjected to pressure-assisted thermal treatments. Journal of Food Science 72(7),

E389–98, 2007.

55534_C007.indd 19655534_C007.indd 196 10/22/08 10:52:35 AM10/22/08 10:52:35 AM

Carbon disulfide (ng/kg)

Dimethyl sulfide (μg/kg)

Dimethyl disulfide (ng/kg)

Dimethyl trisulfide (ng/kg)

Further confirmation of the inhibitory effects of pressure on several chemical


Hyd

rostatic Pressu

re Processin

g of Fo

od

s 197

TABLE 7.7Comparison between the Effect of HPP and Heat Treatment at Atmospheric Pressure on the Formation of Off-Flavour Compounds in Milk*

Treatment Hexanal Heptanal Octanal 2-Heptanone 2-Octanone 2-Nonanone MeSH DMS DMDS

620 MPa, 75°C, 5 min† 44.7 7.33 7.02 4.01 0.97 0.64 7.7 3.89 0.03

Simulated heat

treatment‡ 16.9 2.6 5.1 5.2 4.8 8.6 24.8 8.44 0.06

* Concentration in μg/kg.† This study.‡ Heat treatment under atmospheric pressure, equivalent to the temperature values and times a high pressurized sample of milk treated at 620 MPa and 75°C for 5 min would

be subjected to, and is equal to 3.5 min at 75°C to account for sample handling before pressurization, and 5 min at 93.6°C to account for adiabatic heating.

Source: Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of volatile formation in milk subjected to pressure-assisted thermal treat-

ments. Journal of Food Science 72(7), E389–98, 2007.

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10/22/08 10:52:36 AM



and time during the HPP treatment, that is, 3.5 min at 75°C to account for sample

handling, plus 5 min at 93.6°C, a temperature chosen to account for milk adiabatic heat-

ing during pressurization of milk initially at 75°C. Straight-chain aldehydes were pres-

ent at higher concentrations in HPP-treated samples while methyl ketones and some

sulphur compounds were present at lower concentrations than in the heat-only samples.

treatment at atmospheric pressure as had been suggested by a principal component

analysis of experimental PATP and commercial HTST and UHT milk [11].

The inhibitory effect of pressure on the formation of several sulphur compounds

offers a promising improvement in milk processing technology. The increase in the

[106] because of their low sensory thresholds [107] and could affect the consumer

acceptance of milk processed with pressure-assisted thermal treatments. However,

consumer sensory studies on the impact of these aldehydes in PATP-treated milk

duced during conventional thermal treatments [11]. It has been reported that vola-

tile sulphur compounds are mainly responsible for the development of the cooked

powerful sulphur-containing aroma compound in heated milk [11] with a low sen-

sory threshold and an unpleasant rotten cabbage aroma [110]. Dimethyl sulphide is

also an important compound commonly present in milk at concentrations above its

sensory threshold [90] and has a sulphury aroma [107]. Hydrogen sulphide also has

cation techniques [11] indicated that hydrogen sulphide may not be as important to

the aroma of heated milk. Inhibition of methyl ketones formation by pressure is also

of importance since their concentration increase has been associated to the develop-

olds suggest that they could be less important than previously thought [107], some

researchers have indicated that methyl ketones could act in a synergistic manner to

The interpretation of the formation of volatiles in milk based on the analysis of

temperature and pressure effects on their kinetic constant (k) using the energy of

activation and activation volume models assume that the conversion of a reactant

into a volatile passes through an intermediate or active state. The activation energy

(Ea) needed to reach the activated state is always a positive value (Figure 7.11a),

of the activated state and the partial volumes of the reactants [99] can be negative,

positive or zero (Figure 7.11b). Therefore, reactions with Ea decreasing with pressure

will be consistent with an increased volatile formation in PATP-milk as compared to

conventionally treated milk (Figure 7.11c). Moreover, these reactions will have nega-

tive ΔV* values and thus k-values at constant temperature will increase with pressure

(Figure 7.11d). The opposite behaviour will be observed for reactions with Ea values

increasing with pressure (Figure 7.11c and Figure 7.11d), while no pressure effects on

Ea values (Figure 7.11c) will correspond to reactions with ΔV* = 0 (Figure 7.11d). For

reactions with ΔV*>>0, values for reaction kinetic constants at high pressure would

be so low that volatile formation would not be observed (Figure 7.11d). Table 7.8

55534_C007.indd 19855534_C007.indd 198 10/22/08 10:52:36 AM10/22/08 10:52:36 AM

These findings support previous conclusions that the effect of pressure-assisted ther-

mal treatments on the volatile profile of milk is different to that of an equivalent heat

are needed because the aroma profiles obtained are very different from those pro-

an unpleasant eggy, sulphury aroma, but a recent analysis using improved quantifi-

whereas ΔV* values defined as the difference between the partial molar volumes

concentration of saturated aldehydes is thought to cause the stale off-flavour in milk

flavour defect in heated milk [14,102,108,109]. Methanethiol is probably the most

ment of stale-heated flavour in UHT milk [111]. Although their high sensory thresh-

impart a perceptible flavour [112].


Hyd

rostatic Pressu

re Processin

g of Fo

od

s 199

Pressure

ΔV = 0

Absolute temperature–1

ln k

ln k

Ea increases

650 MPa

Ea decreases650 MPa

0.1 and 650 MPa

No change in Ea

V*

Reagents

[Activated state]* w/ΔV*< 0

ΔV*

[Products]

[Activated state]* w/ΔV* > 0

Reaction pathway

E

[Products]

Reagents

[Activated state]*

Ea>0

Reaction pathway

ΔV*< 0

ΔV*> 0ΔV*> 0

482–655 MPa

(a)

(b)

(c)

(d)

Reaction w/ΔV*= 0

ΔV*

If ΔV*= 0

If ΔV*< 0

If ΔV*> 0

FIGURE 7.11 a

volume ΔV*. (c) Effect of temperature on the reaction kinetic constant as a function of ΔV*. (d) Effect of pressure on the reaction kinetic constant as a

function of ΔV*. (Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of volatile formation in milk subjected to pres-

sure-assisted thermal treatments. Journal of Food Science 72(7), E389–98, 2007.)

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10/22/08 10:52:37 AM

Analysis of pressure and temperature effects on reaction kinetics. (a) Definition of energy of activation E . (b) Definition of activation


200 Fo

od

Processin

g Op

eration

s Mo

delin

g: Design

and

An

alysis

TABLE 7.8Effect of Temperature on the Activation Volumes for the Formation of Various Off-Flavor Compounds in Milk*

T (°C)

Hexanal Heptanal Octanal Nonanal Decanal 2,3-Butanedione

ΔV* R2 ΔV* R2 ΔV* R2 ΔV* R2 ΔV* R2 ΔV* R2 ΔV* R2

45 −3.72 0.99 −4.78 0.99 −6.18 0.89 −4.29 0.97 −3.09 0.91 −3.51 0.87 −1.01 0.97

55 −3.60 0.98 −4.03 0.99 −4.52 0.96 −3.60 0.98 −1.74 0.92 −1.60 0.92 −1.15 0.90

60 −2.71 0.99 −3.01 0.99 −4.04 0.94 −3.16 0.96 −1.41 0.89 −2.13 0.87 −1.19 0.96

75 −2.31 0.96 −2.51 0.98 −3.32 0.88 −3.10 0.96 −1.01 0.96 −1.62 0.94 −1.04 0.99

∗ ΔV* = Activation volume (×10−5 m3 mol−1 2

Source: Adapted from Vazquez-Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of volatile formation in milk subjected to pressure-assisted thermal treat-

ments. Journal of Food Science 72(7), E389–98, 2007.

55534_C007.indd 200

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10/22/08 10:52:38 AM

Hydrogen Sulfide




shows the activation volume change ΔV* for volatile formation in HPP treated milk.

ΔV* values for straight-chain aldehydes in PATP milk are negative; indicating that an

increase in pressure would lead to an increase in reaction rate constants. This is con-

sistent with the observations of k values for straight-chain aldehydes increasing with

pressure (Table 7.2). In addition, ΔV* values for straight-chain aldehydes decrease

in absolute value with temperature, meaning that at higher temperatures, formation

of aldehydes is less sensitive to pressure changes. ΔV* values for hydrogen sulphide

remained fairly stable regardless of HPP temperature (Table 7.8), an observation also

consistent with the lack of pressure effect on its Ea value (Table 7.4). ΔV* values for

2,3-butanedione are negative and appear to be affected by temperature, but with an

unclear trend (Table 7.8). ΔV* values for 2-methylpropanal are not shown because 2

of methyl ketones and some sulphur compounds are observed under pressure. Since

these compounds are formed at atmospheric pressure, ΔV* values for their formation

reactions must be positive, and very large, because these reactions are completely

inhibited by pressure.

The increase, decrease or lack of change caused by pressure and temperature on

the formation of volatiles in PATP milk can be explained with no need to assume

alternative reaction pathways. This suggests that there are no reaction pathways for

the 27 volatiles measured with a small formation rate at low pressure that would

increase with pressure because of negative ΔV* values (Figure 7.12) [15]. Findings

in this study on PATP milk represent a dramatic improvement in the understanding

of the effect of temperature and pressure on reaction rates in foods and support the

need for further research on pressure-assisted thermal processes to develop products

Pressure

ln k

ΔV*< 0

482–655 MPa

Minimum rate for detectable formationwithin chosen experimental time

FIGURE 7.12 Hypothetical pathway for volatile formation at constant temperature and high

pressure with negligible rate at conventional pressure (e.g. 0.1 MPa). (Adapted from Vazquez-

Landaverde, P.A., Qian, M.C., and Torres, J.A. Kinetic analysis of volatile formation in milk

subjected to pressure-assisted thermal treatments. Journal of Food Science 72(7), E389–98,

2007.)

55534_C007.indd 20155534_C007.indd 201 10/22/08 10:52:38 AM10/22/08 10:52:38 AM

they did not fit the Arrhenius model (R < 0.648). No changes in the concentration



meeting current consumer demand for foods with minimal effects of processing.

in pressure treatments as differences in chemical changes between milk subjected

to conventional and pressure-assisted thermal treatments can be interpreted on the

basis of the kinetic analysis here presented and reactions already well-described

in the literature. This is very important to food processors that use high pressure

processing, as it will enhance the acceptance of pressure processing as a novel tech-

nology alternative for improved-quality foods. The likelihood that pressure-assisted

thermal processes produce new compounds of unknown safety appears to be low.

PATP is a promising alternative to preserve not only quality factors and desirable

formation of potential toxicants.

7.4 LOW HYDROSTATIC PRESSURE (LHP) DISINFESTATION OF DRY FRUITS AND VEGETABLES

Consumers have become increasingly concerned about the quality and safety of the

food reaching their table, including dry and fresh fruits and vegetables. Fumigation

of these products with ethylene bromide has been a standard disinfestation method

in the global trade of these products to meet quarantine restrictions issued to prevent

the introduction of pests [113,114]; however, the use of this chemical is becoming

more and more restricted because of its impact on the ozone layer [115,116]. In addi-

tion, chemical treatments are not well accepted by consumers because of potential

health risks of chemical residues in their diet. An effective alternative to chemical

disinfestation is irradiation but this technology is considered undesirable by some

consumers and disallowed by regulations in many countries [117]. Hot water dipping

used for quarantine treatments of tropical and subtropical fruits [118,119] has been

reported to have heat transfer limitations (fruit shape/size and maturity), increase

respiratory metabolism, induce skin damage and requires large use of freshwater of

good sanitary quality for fruit cooling. In these hydrothermal treatments long heat

application times are necessary to reach larvae that can penetrate deep into the fruit.

heat-conducting differences among fruits of different maturity, fruits containing lar-

vae far from fruit surfaces, failure to account for fruit temperature differences prior

to treatment, and failure to account for lot-size dependent times before the hydro-

thermal treatment tank can reach the required lethal temperature.

Pests of worldwide commercial importance include Anastrepha ludens a native

of Northeastern Mexico, particularly the states of Nuevo León and Tamaulipas, and

[121]. When detected in Florida and California, each infestation has required inten-

sive and massive eradication and detection procedures. Its larvae feed and develop

on many deciduous, subtropical, and tropical fruits and some vegetables. The Carib-

eggs under the skin of fruit just as they begin to ripen, often where some break in

55534_C007.indd 20255534_C007.indd 202 10/22/08 10:52:39 AM10/22/08 10:52:39 AM

Most significantly, alternative reaction formation mechanisms are not likely involved

constituents with important health benefits to consumers, but may inhibit also the

Reports of larvae survival to these hydrothermal treatments [120] appear to reflect

the Mediterranean fruit fly (Ceratitis capitata), one of the most destructive fruit pests

bean fruit fly (Anastrepha suspense), a near relative of the Mexican fruit fly (A. ludens) is one of several species of fruit flies indigenous to the West Indies and its

larvae attack tropical and subtropical fruits. As other harmful flies, females deposit



the fruit skin has already occurred. Infested fruit often drop and those staying on the

plant may have no outward signs of infestation [122].

Preliminary studies conducted at Oregon State University on coddling moth,

have shown that egg and larvae inactivation in dry fruits and vegetables is possible

with relatively low hydrostatic pressure (LHP) treatments. Disinfestation conditions

would be independent from fruit size and geometry because pressure transmission

to any location within the product is essentially instantaneous [4]. LHP treatments at

pressures as low as 125 MPa have been shown to inactivate the Mediterranean fruit

ing time and treatment temperature. Unfortunately, this is the only inactivation data

reported to assist in the design of LHP treatments. At present, the pressure inactiva-

tion conditions for the eggs of insect of commercial importance in California and the

cooperation with the California Kearney Agricultural Center. Preliminary observa-

encouraging.

7.5 CONCLUSIONS

reported over a century ago when a 5–6 log-cycle total count reduction was achieved

TABLE 7.9Inactivation of Mediterranean Fruit Fly (Ceratitis capitata) Eggs by Relative Low Hydrostatic Pressure (LHP) Treatments

Pressure (Pa)

Temperature (oC)

0 12.5 25 32.5 40

Time (mn)

5 10 20 5 10 20 5 10 20 5 10 20 5 10 20

0 - o o - - o - - o - - o - - o

5 - - - - - - - - - - - - o o o

15 - o - - - - - - - - - - o o o

30 - o - - - - - - - - - - o o o

50 - o - o o o o o o o o o u u u

75 - o - - - - o o o - - - - - -

100 u u u o o u o o o o o o u u u

125 - - - u u u u u u u u u - - -

150 - - - u u u u u u u u u u u u

o= eggs hatching; u= eggs inactivated; - = condition not tested.

Journal of Food Processing and Preservation 19, 161–64, 1995.

55534_C007.indd 20355534_C007.indd 203 10/22/08 10:52:40 AM10/22/08 10:52:40 AM

Pacific Northwest (Table 7.10) are being determined at Oregon State University in

tions of the quality of LHP-treated dry fruits, including raisins, figs and apricots are

The application of high hydrostatic pressure for the processing of foods was first

Source: Adapted from Butz, P, and Tauscher, B. Inactivation of fruit fly eggs by high pressure treatment.

and at least one peer-reviewed published report on the Mediterranean fruit fly [115],

fly. As shown in Table 7.9, inactivation was almost independent of pressure hold-



without using heat by treating milk at 670 MPa for 10 min. HPP technology is nowa-

days a well-established food technology with commercial success stories all over the

world [124, 125]. Unlike thermal processing and most other preservation technolo-

gies, HPP effects are uniform and nearly instantaneous throughout the food and thus

TABLE 7.10

Common Name Primary Crop(s) Status in Oregon*

Lepidopterans

Oriental fruit moth Grapholita molesta Stone fruit Limited distribution in

western Oregon

Peach twig borer Anarsia lineatella Stone fruit, almonds Statewide

Codling moth Cydia pomonella Pome fruit, walnuts Statewide

Navel orange worm Amyelois trasitella Almonds, pistachio,

walnuts

Not known to occur

Oblique banded leaf

roller

Choristoneura rosaceana

Pome fruit, pistachio Statewide

Coleopterans

Dried fruit beetle Carpophilus hemipterus

Figs, raisins Infests stored products

at least on NW Oregon

Ten-lined June beetle Polyphylla decemlineata

Almonds, walnuts Statewide

Hemipterans

San Jose scale Diaspidiotus perniciosus

Stone fruit, almonds Statewide

Citricola scale Coccis pseudomagnolarum

Figs, citrus Not known to occur

Green peach aphid Myzus persicae Stone fruit, pome fruit Statewide

Diptera

Bactorcerus oleae Olives Not known to occur

Rhagoletis complete Walnuts Statewide

Acari

European red mite Panonychus ulmi Almonds, stone &

pome fruit

Statewide

Almonds, stone &

pome fruit

Statewide

Two spotted spider mite Tetranychus urticae Almonds, stone &

pome fruit

Statewide

* Provided by Rick Westcott, Rich Worth, and Kathleen Johnson at the Oregon Dept. of Agriculture, July

10, 2007.

55534_C007.indd 20455534_C007.indd 204 10/22/08 10:52:41 AM10/22/08 10:52:41 AM

Infestation of Commercial Concern in California and the Pacific Northwest

Scientific Name

Pacific spider mite Tetranychus pacificus

Olive fruit fly

Walnut husk fly



independent of food and equipment geometry and size. This has facilitated the scale-

rapid commercialization of HPP processing technology.

The next generation of high hydrostatic pressure units will allow the combina-

tion of pressure and thermal treatments. It is interesting to note that it is again experi-

mental research on milk that has demonstrated the promising advantages of PATP

technology. Only thermal degradation reactions leading to the formation of alde-

hydes were accelerated by high pressure. The formation of most volatiles reported

inhibited by pressure. Most importantly, new reaction formation mechanisms were

not likely involved in volatile formation in PATP-milk and this will be particularly

important in the European market where novel technologies regulations could limit

the commercialization of PATP applications for the development of shelf-stable

foods. Also, the application of the Le Chatelier principle frequently used to explain

the high quality of pressure-treated foods, often with no supporting experimental

evidence, is not necessary to demonstrate that PATP promises to cause much less

chemical damage to foods than conventional thermal treatments. This will allow

meeting the current consumer demand for foods minimally affected by processing

so as to preserve desirable compositional and sensory properties while meeting also

a demand for enhanced food safety.

NOMENCLATURE

c Concentration, g L−1

t Time, min

k Reaction rate constant, units vary with reaction order

n Reaction order

c0 Pseudo-initial concentration

Ea Activation energy, kJ mol−1

R Universal gas constant, 8.314×10-3 kJ mol−1 K−1

k0 Pre-exponential rate constant

ΔV* Partial activation volume

A Integration constant

p Pressure, MPa

R2

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213

8 Pulsed Electric Field (PEF) Processing and Modeling

Si-Quan Li

CONTENTS

8.1 Introduction ................................................................................................... 213

8.2 Pulsed Electric Field (PEF) Processing Technology and Mechanisms

for Microbial Inactivation ............................................................................. 216

8.2.1 PEF Technology and Hardware Development................................... 216

8.2.1.1 Basics of PEF Technology .................................................. 216

8.2.1.1.1 Critical Components in a Typical PEF

System................................................................ 218

8.2.1.1.2 Critical Parameters Determining the

8.2.2 Mechanisms for PEF Inactivation of Microorganisms and

Enzymes ............................................................................................223

8.2.3 Modeling of PEF Inactivation of Microorganisms ............................227

8.2.4 Trends in PEF Research .................................................................... 229

8.2.4.1 Hardware Development has a Long Journey Ahead ........... 229

8.2.4.2 Application Studies of PEF Technology in the

Near Future ......................................................................... 229

8.2.4.2.1 Pasteurization of High Acid or

8.2.4.2.2 PEF Combined with Mild Heat for Shelf

Stable High Acid Food Products .......................230

8.2.4.2.3 Research and Applications in PEF Assisted

Food Processing ................................................230

8.3 Conclusions ...................................................................................................230

Nomenclature ......................................................................................................... 231

References .............................................................................................................. 231

8.1 INTRODUCTION

55534_C008.indd 21355534_C008.indd 213 10/22/08 10:12:16 AM10/22/08 10:12:16 AM

Efficacy of PEF Processing ............................... 221

Acidified Food ................................................... 229

Pulsed electric field (PEF) processing involves subjecting food products to certain

controllable pulsed electric fields to inactivate microorganisms and enzymes and to



PEF technology in food processing practices is improvement in product quality due

to the non-thermal characteristics of PEF. Due to the fact that only minimum amount

of heat is generated during PEF processing, food products pasteurized using PEF

compounds, such as vitamin C, vitamin B, and immunoglobulins. The non-thermal

characteristics of PEF processing also make it possible to avoid the generation of heat-

purchasing intent and higher willingness to pay more for PEF processed orange juice

than heat pasteurized orange juice, as demonstrated in an auction experiment [1]. Suc-

cessful commercial application of PEF technology in the food industry can be tracked

back to 2005 by Genesis, a juice processing company based in Oregon, USA [2].

at Oregan County Fair. As reported in the Salem Stateman Journal (August 18, 2005),

a co-owner of Genesis claimed that “I was told people were crying, though I did not

pulser) and Genesis were nominated for 2007 IFT Industrial Achievement Award to

recognize their contribution in introducing PEF technology into food industry prac-

tices. The PEF processed juices helped the company win their customers back and

In fact, the advantages of applying PEF technology in food processing practices have

attracted increasing attention since the 1960s, particularly after the mid-1990s when

demand for fresher and healthier food products with minimum processing dramati-

cally increased due to consumer’s increased awareness of the impact of diet on human

health. It has been believed that, along with increase in consumer awareness of the

role that diet plays in determining people’s health, PEF and other non-thermal food

processing technologies, such as high pressure processing will be the major trends in

food processing innovation. More and more food researchers and manufacturers will

be involved in the global effort to apply PEF in food processing practices to meet the

increasing demands of fresher and healthier food products by consumers.

PEF technology has been developed and evaluated by different walks of research-

ers since the 1960s. In 1960, a patent was issued to Doevenspeck [3] who creatively

tivation effects. In 1967, Sale and Hamilton [4,5] reported that the inactivation effects

of PEF on the selected microorganisms are highly positive and, as a result, stimulated

studies on inactivation of microorganisms by PEF, a novel non-thermal food pasteuriza-

inactivate Escherichia coli, Staphylococcus aureus, Micrococcus lysodeikticus, Sarcina lutea, Bacillus subtillus, B. cereus, B. megatherium, Clostridium welchii, Saccharo-

−1 were applied

as a series of direct current pulses from 2 to 20 μs to the suspensions of microorgan-

at 19.5 kV was 10°C. The contribution of thermal inactivation to the total inactivation

the two most important parameters responsible for the inactivation of microorganisms.

55534_C008.indd 21455534_C008.indd 214 10/22/08 10:12:17 AM10/22/08 10:12:17 AM

modify the properties of the food components. The number one benefit of applying

induced toxic compounds in food products. Consumers showed significantly higher

Genesis became the first company to introduce a PEF product in an emotional debut

see that myself”. Diversified Technologies, Inc. (DTI, the builder of the high voltage

applied uniform electric fields for inactivating microorganisms. Doevenspeck found that

the intensity of the electric fields applied onto the microbial cells have different inac-

tion. The authors reported that electric pulses of high voltage electric fields up to 25 kV

myces cerevisiae, and Candida utilis. Electrical fields up to 25 kV × cm

isms. Temperature increase during pulsed electric field treatment for ten pulses of 20 μs

observed was literally negligible. Electric field strength and the total treatment time are

technology are fresher in flavor, more nutritious and richer in heat-labile bioactive

attracted many new customers due to the significantly fresher flavor of the products.


215

However, different cell structural properties, such as size of the cell, properties of the

tion effects by PEF. Generally cells with larger size, such as yeast and molds, are more

negative microorganisms are more sensitive to PEF treatment than Gram positive micro-

organisms. The authors argued that the inactivation is caused by the lysis of protoplasts,

and leakage of intracellular contents. Loss of β-galactosidase activity and plasmolyzing

ability in a permease-negative mutant of E. coli were observed during their experiments.

Dunn and Pearlman [6] successfully developed the required PEF apparatus and estab-

lished the protocol methodology for food PEF treatments.

In the 1990s, more researchers were attracted to studies on PEF by its potential

application as a non-thermal pasteurization alternate in food industry. Castro and his

co-workers [7] argued that PEF treated food retains ‘fresh’ physical, chemical, and

nutritional characteristics and possesses a satisfactory ambient shelf life. The intro-

duction of this new technology has the potential to provide consumers with micro-

biologically safe, minimally processed, nutritious, and fresh-like foods. The authors

suggested that high-intensity PEF is potentially the most important non-thermal

pasteurization/sterilization technology available to replace or complement thermal

Lactobacillus delbrueckii [9,10]. PEF treatment at 16 kV × cm−1 with 60 pulses at a

pulse duration ranging from 200 to 300 μs inactivates 4–5-log microbial population

ture was maintained below the lethal temperature. The results suggest that the micro-

bial inactivation effect of PEF treatment is not due to thermal effect. Stepwise PEF −1 with

pulse duration time of 2 μs was reported to be able to achieve a 9 log-cycle reduction

in microbial population [11]. Cell suspension of the samples were treated in a static

chamber and maintained at 7, 20, or 30°C and the maximum temperature change by

many other microbial inactivation experimental results have been reported [12–20].

microorganism growth stage and processing temperature on inactivation effects of

E. coli by PEF. The authors reported that at 3–40°C processing temperature, square-

wave showed higher inactivation effects on E. coli suspended in simulated milk

strength of 36 kV × cm−1. Logarithmic-phase cells were more sensitive than station-

ary and lag-phase cells. This result implies that when we treat food samples not only

effects but also the microorganisms themselves.

the introduction of the new generation of continuous, automatically controlled PEF

55534_C008.indd 21555534_C008.indd 215 10/22/08 10:12:18 AM10/22/08 10:12:18 AM

Pulsed Electric Field (PEF) Processing and Modeling

sensitive to the electric fields than their smaller peers like bacterial cells and virus. Gram

processes. Destruction of bacterial cells under high intensity electric fields is due

primarily to the field-induced rupture of cell membrane and not to ohmic heating

[8]. The premium effect of electric field strength on microbial inactivation was also

confirmed by applying PEF to cultures of E. coli, S. aureus, Bacillus subtillis, and

in model foods such as simulated milk ultrafiltrate [9]. The cell suspension tempera-

treatment on E. coli at electric field strength ranged from 35 to 75 kV × cm

each pulse, measured with a fiberoptic temperature probe, was only 0.3°C. Since then

ultrafiltrate (SMUF) than exponential decaying wave pulses at same electric field

Significant breakthrough in PEF hardware development was reported [21] with

cell membranes, growth phases of the microorganisms and influences from media in

which the cells grow are critical factors and show significant influences on the inactiva-

Pothakamury and co-workers [10] reported their studies about influences of

the PEF parameter should be counted as the influential factor for the inactivation

processing systems, OSU-4, using co-field flow chamber systems. The innovative



PEF system using H-bridge electronic pulse generating circuit and solid state high

speed switch devices allows researchers to treat samples continuously with accurate −1 with a treat-

ment time up to 400 μs. System adjustability in pulse duration time and delay time as

mercial scale PEF processing system were assembled at The Ohio State University,

Ohio, USA in 2001 and 2003, respectively. The up-scaled PEF processing facilities

cacy on food preservation and effects on food quality and functionalities at a level

close to food industrial practices [22,23]. The OSU-5 PEF system had a capacity of

200–500 L/h for food processing and up to 2000 L/h for waste water processing.

Increased consumer awareness of food impacts on public health has provoked

a dramatic increase in demands of functional foods and better understanding of the

functional components, particularly heat labile bioactive compounds such as vitamin

C, B and immunoglobulin G. In responding to this increased demand, studies on PEF

effects on heat labile bioactive compounds have been carried out. Compared to conven-

tional thermal pasteurization on single strength orange juice, PEF processing showed

soymilk product [25], while thermal process at the same pasteurization power caused

noglobulin G activity during a thermal process is because of the loss of the beta-sheet

secondary structure, which is needed for an IgG molecule to function normally, and

PEF does not cause any detectable change in IgG secondary structures [26]. Li and his

Although other non-thermal technologies, such as high pressure processing (HPP),

may also have some saving effects on IgG immunoactivity, PEF effects on IgG clearly

showed different mechanisms [27]. So far, PEF has been illustrated as most effective in

saving heat labile bioactive compounds and effective in microbial inactivation.

In this chapter, the development and current status of PEF technology will be

discussed and the mechanisms and modeling of the processing effects will also be

summarized. Hardware and software issues will be discussed to facilitate the under-

standing of this non-thermal technology. Potential applications in food industry and

chapter.

8.2 PULSED ELECTRIC FIELD (PEF) PROCESSING TECHNOLOGY AND MECHANISMS FOR MICROBIAL INACTIVATION

8.2.1 PEF TECHNOLOGY AND HARDWARE DEVELOPMENT

8.2.1.1 Basics of PEF Technology

PEF processing involves the application of pulses of high voltage (typically 20–80

kV × cm−1

55534_C008.indd 21655534_C008.indd 216 10/22/08 10:12:19 AM10/22/08 10:12:19 AM

temperature control. Electric field strength can be up to 50 kV × cm

evaluate the efficacy of PEF technology. Based on the design concepts employed in

OSU-4 systems and pilot scale PEF processing systems (OSU-5), world’s first com-

at The Ohio State University provided the necessary tools for evaluation of PEF effi-

significant saving on vitamin C content both right after and 6 months after processing

[24]. PEF caused no significant loss in bovine milk immunoglobulin G in an enriched

over 80% immunoglobulin G activity loss. Later studies confirmed that loss of immu-

co-workers proposed the shape factor concept to define the changes in IgG activity.

) to foods between two electrodes, in a finely controlled manner to inactivate

well as options in pulse polarity provides researchers with tremendous flexibility to

possible trends of this technology will be briefly discussed at the later part of this


Pulsed Electric Field (PEF) Processing and Modeling 217

microorganisms and enzymes existing in food products, and/or to modify certain prop-

erties of selected food components. A typical continuous laboratory scale PEF system

is illustrated in Figure 8.1a and Figure 8.1b. PEF technology is an emerging technology

which may be used as a non-thermal alternate for conventional food pasteurization.

PEF treatment is conducted at ambient, sub-ambient or slightly above ambient tem-

perature for a very short time (typically between 30 and 200 μs). PEF processing can

be a batch process using static treatment chambers or a continuous practice with some

specially designed chamber systems (Figure 8.2). Today, pilot and commercial scale

PEF processing are continuous practices while laboratory investigation can be either a

batch or a continuous process, depending on the sample types and the purposes of the

test. Only a minimal amount of heat is generated during PEF processing, thus mini-

mizing the heat impairment on food quality [28]. Food products processed using PEF

technology may be fresher, more nutritious, with more heat sensitive physiologically

approaches [24,25,28].

PLC

Oscilloscope

Marked area A

Thermocouple

readers

Syringes

Step motors

Pulse

generator

Water bath

AA B C D

DCBA

(a)

(b)

FIGURE 8.1 OSU-4J PEF system (a) and schematic diagram of a typical lab scale OSU-4

PEF system (b).

55534_C008.indd 21755534_C008.indd 217 10/22/08 10:12:20 AM10/22/08 10:12:20 AM

bioactive components and better flavor compared to those processed by conventional



PEF has been under development since the 1960s but is now close to ready for

industrial practices in the food and water industry. This is due to recent develop-

ments in electronic technology and innovation in treatment chambers, such as the

mercial scale PEF food processing system was set up at The Ohio State University

in the early 21st century. Since 2005, PEF technology has found its position in fresh

juice processing companies, such as Genesis Company. PEF processed fruit and veg-

etable juices are preferred based on market research and consumer surveys as well as

realistic sales by the companies who apply PEF in their juice making practices.

8.2.1.1.1 Critical Components in a Typical PEF SystemIn a typical PEF processing system, the following subunits are critical to the overall

systematic functionality:

a. Power supplierPEF processing requires stable power supplies, particularly stable voltage supply. An

effective power supplier is always needed to provide primary power needs. However,

Chamber 1 Chamber 2 Chamber 3 Chamber 4

Gap distance

PEF zone

Sample Out

Sample In

GRD

HV

Cooling coils

Thermocouple

readers

T4

T3

T2

T1

FIGURE 8.2 pled with OSU-4 PEF system.

HV GNDGND

PEF Zones

FIGURE 8.3 ment zones.

55534_C008.indd 21855534_C008.indd 218 10/22/08 10:12:22 AM10/22/08 10:12:22 AM

safety and improving PEF efficacy. PEF technology is also seeking more applica-

tions in medicine, breeding and weed control industries. The world’s very first com-

Schematic diagram of a typical co-field flow treatment chamber system cou-

Modified co-field flow PEF chamber system with two compactly paired treat-

compactly paired co-field flow chamber system (Figure 8.3) ensuring operational



on the biological targets, a set of capacitors are needed to ensure the output of electric

potential cross the paired electrodes is high enough to meet the needs of expected

b. Pulse generatorThe pulse generator includes a signal generator, pulse generating electric circuits

and solid phase high speed switching devices. A signal generator is the command-

ing device that controls the width of electric pulses, pulse replication rate, pulse

delay time and the polarity of the electric pulses. A signal generator is actually the

interface that allows operators to interfere to pulse generating process with designed

parameters. A very popular signal generator, also called a pulse generator or trig-

ger generator, is model 9310 trigger generator (Quantum Composer Inc., Bozeman,

Montana, USA), which provides lots of controlling options, internal or external, even

remote control. Although single shot mode can be used for kinetic studies using

static treatment chamber, continuous pulse mode is almost always used when PEF

A commonly used circuit is the so called H-bridge circuit (Figure 8.4) which

can produce square shaped electric pulses at high accuracy in pulse width. Currently

H-bridge circuit is widely used in PEF processing systems from lab scale to pilot and

commercial scale, such as in OSU PEF systems. The H-bridge circuit provides the

conveniences of accurate control and monitoring of pulse width, with the aids from

the solid phase high speed switching devices, such as IGBT and IGCT from Semi-

cron. IGCT devices have been successfully used in many OSU-4 PEF systems. Never-

theless, in order to build a high effective pulse generating system with high accuracy

in pulse width and intensity control, close attention is also needed to the quality of the

communication between the individual switching devices and the noise level at bus

area. Interruption of communication between the subunits and the switching devices

by the electromagnetic noise will cause the system malfunction, even crash due to

tems. Maintaining system temperature stable and close to room temperature is as

Rc

C1

U1

U3

U4U

2

Transformer

PEF chambers

V+

–

FIGURE 8.4 H-bridge electric circuits for generating square waveform pulses (Circuit dia-

gram of a laboratory-scale high voltage pulse generator (OSU-4A). Rc refers to the charging

resistor, V refers to a power supplier, and C1 refers to the capacitor. U1, U2, U3 and U4 refer to

switch 1, switch 2, switch 3 and switch 4, respectively).

55534_C008.indd 21955534_C008.indd 219 10/22/08 10:12:23 AM10/22/08 10:12:23 AM

due to the nature of high intensity of the electric fields needed for the expected effects

electric field strength.

is critical to ensure high efficiency and stable performance of pulse generating sys-

processing is in continuous flow mode.

over flow of system memory. Cooling of switching devices and the circuits connected



critical as the fundamental design of the system, because the switching devices will

teurization purposes. Currently both water cooling and forced air cooling are in use

in the PEF processing systems with built-in pulse generators. Interruption of com-

munication between subunits and temperature control issues are the most commonly

technical challenges encountered by users in practice.

c. Control and monitoring systemControl and monitoring subunit is critical to the overall function of the whole PEF

system. The subunit assigns each designed parameter to be carried out in an accurate

and timely manner. Both (Programmable Logistic Control) PLC and microprocessor

unit are used to accurately control the system to function normally and monitor the

performance in a timely manner. The key for a PEF control and monitoring system to

function normally is to ensure the accuracy of parameter measurement and smooth/

effective communication between the commanding system and executive units.

System noise and electromagnetic signal interruption must be minimized by all means.

One easily overlooked problem is to monitor the peak voltage as the mean to calculate

voltage at different locations, such as chamber inlet, paired electrodes, transformer

output, bus, capacitor discharge, etc., for their own convenience. However, the most

reliable location for measuring the peak output voltage is at the two paired electrodes.

The measurement error can be minimized by doing so since the system energy diffu-

sion and voltage drop from transformer to treatment chamber is avoided.

impact the normal function of the controlling and monitoring system. An effective

cooling system is needed to provide optimal temperature condition for the controlling

cooling are commonly used in many different versions of PEF equipment, depending

on the heat load generated during operation.

d. Fluid handling system

units, delivering units, receiving units and monitoring systems such as temperature

the delicately generated electric pulses meet with food products, and is the center

other subsystems serve the purpose of either generating delicately controlled electric

rate or making sure the food product passes through the treatment chamber at a uni-

the PEF treatment zone, ensuring the uniformity through every single portion of the

chamber volume.

application, and now are still in wide use for kinetic studies. The advantages of using

55534_C008.indd 22055534_C008.indd 220 10/22/08 10:12:24 AM10/22/08 10:12:24 AM

drop their efficiency and accuracy under an elevated temperature. It is particularly

important when the system is operated under higher electric field strength for pas-

the observed electric field strength. Different designers may choose to measure the

pulses at designed field strength, accurate pulse width and stable pulse replication

PEF treatment chambers were first designed in static mode used for laboratory

static chambers include the uniform distribution of electric fields in most cases, and

Attention is also needed to closely watch the temperature fluctuation which will

and monitoring system to sustain stable performance. Forced air cooling and fluid

A PEF fluid handling system includes PEF chamber system, pump(s), pressure control

monitors and flow meter (Figure 8.1b). PEF chamber system (Figure 8.2) is where

of a fluid handling system and the heart of the whole PEF processing system. All

form and stable flow mode. A well designed PEF chamber is the essence of uniform

distribution of the electric field strength and delicately controlled flow profiles in



the more accurate control over pulse numbers and total PEF treatment time. Single

shot test types are also feasible using static treatment. However, with static chamber,

increasing sample temperature due to accumulated heat within the chamber is of

great concern when justifying the righteousness of a reported result, if the pulse

tices due to their limited production capacity.

potential industrial applications, such as liquid food pasteurization. There have been

many different designs in pursuit of a successful application in the food industry,

currently allows the food industry to operate PEF processing at a high production

capacity up to 2000 L⋅h−1. Further scale-up of production capacity has been on-going

since the early 2000s.

e. Aseptic packagingAseptic packaging system may not be needed in lab scale kinetic research, but is

necessary for many pilot and commercial practices. It allows the PEF processed food

products to be packaged in an aseptic mode avoiding/minimizing the post-process

PEF processing system. A Banco® aseptic packaging system, with in-line thermo-

forming of multilayer cups, adjustable in cup size, was hooked together with the

The assembly had been in successful operation for research purposes and for indus-

try plant tests.

8.2.1.1.2

on PEF inactivation effects against microorganisms. Primary parameters that deter-

trodes divided by their gap distance, L.

E V L= × –1 (8.1)

strength’ and widely accepted by different groups of researchers. However, readers

the whole treatment zone due to the nature of many different chamber designs. For

be close to ideally uniform across the whole chamber, although the distortion of

electric lines may happen at the edge of the cylinder and at the interface between

55534_C008.indd 22155534_C008.indd 221 10/22/08 10:12:25 AM10/22/08 10:12:25 AM

replication rate is too fast, a significant amount of pulses were applied during experi-

ments. Static chambers have also found themselves of difficult use in industrial prac-

ferent models of aseptic filling/packaging systems can be easily hooked up with a

world first commercial scale PEF processing system at The Ohio State University.

Critical Parameters Determining the Efficacy of PEF ProcessingTable 8.1 illustrates the primary and secondary factors that have significant impacts

mine PEF processing effects are electric field strength (E) and total (PEF) treatment

time (t). There are several different definitions regarding electric field strength, E.

The most commonly used one is so called ‘nominal average electric field strength’,

E, which is defined as the electric potential or voltage, V, cross the two paired elec-

The nominal average electric field strength is generally simplified as ‘electric field

should notice that, in many cases, electric field strength is not uniform throughout

instance, in the case of disc type static chamber design electric field strength could

However, on the other hand, continuous flow PEF chambers have found lots of

from early designs such as co-axis, co-field and co-flow types to the later co-field

flow design [29] to the recently improved compact co-field flow chamber module with

multiple chambers in a tightly packed chamber set. The co-field flow chamber system

recontamination. Thanks to its sanitary continuous flow characteristics, many dif-



happens at the central axis.

electric pulses. In the case multiple treatment chambers are used, total PEF treatment

time (t) during the whole process is used to anticipate the treatment effects. Total

PEF treatment time is determined by the geometry (inner diameter, d, and the gap

distance between the two paired electrodes, L) of the chamber, number of chambers

rr

food matrix. The total PEF treatment time can be calculated using Equation 8.2:

t = × × × × × ×0.25 fr2 –1rrπ τd L p n (8.2)

When monopolar pulses are used, pulse replication rate is the same as pulse fre-

quency, which is the reciprocal of toper time set in the trigger generator. However,

when bipolar pulses are used, prr is two times higher than pulse frequency, that is

p frr p2= × (8.3)

TABLE 8.1

Primary Parameters Secondary Factors

Processing temperature, T

Total PEF treatment time, t Medium pH

Ionic strength

Presence of antimicrobial agents

Pulse types: square waveform or exponential decaying waveform

Polarity of pulses: single or bipolar

Number of pulses per chamber

Pulse width, τ

Geometry characteristics of PEF chambers

Types of chambers: static or continuous

Electrode erosion status

Flow types: laminar or turbulent

Cooling application between chamber pairs

Accuracy of monitoring and controlling

Growth phase of microorganism

Species and strains of microorganisms

etc.

55534_C008.indd 22255534_C008.indd 222 10/22/08 10:12:26 AM10/22/08 10:12:26 AM

Some Important Factors Determining the Efficacy of a PEF Processing

Electric field strength, E

design, electric field strength can be significantly decreased along the radius in the

direction from edge to geometric central point. The minimal electric field strength

PEF treatment time is defined as the time when food is actually exposed to the

different phases within sample matrix. On the other hand, in the case of co-field flow

used (n), pulse width (τ), pulse replication rate (p ) and the volumetric flow rate (fr) of



In general, the higher the E and t, the higher microbial and enzymatic inactivation

effects, provided the other conditions are same. It is also worth to notice that the

number of pulses per chamber, treatment temperature, chamber dimensions and

on the effects of a PEF treatment. Uniformity of PEF treatment has been one of the

biggest concerns about the application of PEF in commercial food processing. Size

or geometry of the PEF treatment chamber is another big concern in the case of PEF

system scale-up.

It is not surprising if one observes that PEF effects reported by one group are

different with that demonstrated by others due to the fact that there are so many other

factors than those well described in the report that may also affect the overall result.

cantly to the overall turned-out and requires that one pays close attention to.

Secondary factors (Table 8.1) are those that are relatively less determining com-

overall PEF effects. The importance of keeping close attention to the contribution of

secondary factors to overall PEF microbial inactivation effect, is never overstressed.

Processing temperature, medium pH, growth phase of targeted microorganisms,

availability of nutrients, presence of protecting or inhibiting components, types of

pulses (square waveform pulses—Figure 8.5a or exponential decaying waveform

pulses—Figure 8.5b), pulse number per chamber, pulse width, pulse polarity, elec-

trode erosion and geometry characteristics are commonly in the list of important

monitor the secondary factors can result in misleading conclusions.

8.2.2 MECHANISMS FOR PEF INACTIVATION OF MICROORGANISMS AND ENZYMES

The mechanisms involved in inactivation of microorganisms and enzymes have been

Although several different mechanisms have been proposed, the most well accepted

theory or mechanism is the electrical breakdown theory proposed by Zimmermann

[30]. The electrical breakdown theory, as illustrated in Figure 8.6, entails electric

is in its normal thickness. The transmembrane potential is the potential between

the electric charges distributed on the outer and inner surfaces of the membrane.

electric charges are induced on membrane surfaces along the direction of the applied

are distributed on the membrane surfaces. As a result, more compression force is

generated across the membrane due to the increased attraction between the positive

55534_C008.indd 22355534_C008.indd 223 10/22/08 10:12:27 AM10/22/08 10:12:27 AM

Other than primary parameters, there are also a group of factors contributing signifi-

pared with the primary factors, E and t, but may also show significant impact on

secondary factors and significantly affect the overall effect of PEF. Failure to closely

investigated by many researchers since PEF was first suggested for cell destruction.

deformable film with positive charges on the outer surface and negative charges on

the inner surface. When there is no external electric field applied, the cell membrane

When there is no external electric field, most microbial cells have a transmembrane

potential of approximately 10 mV. However, once an external electric field is applied,

electric field. The amount or density of induced electric charges on the surfaces is

proportional to the strength or intensity of the applied electric field. The stronger

the applied electric field, the higher the density of the induced electric charges that

pulse duration time are important parameters that may have significant influences

breakdown of cell membrane. The membrane itself is a dielectric, flexible and



force, the increased compression results in a thinner membrane than it used to be.

Nevertheless, although it gets much thinner, the membrane still functions normally

to such a point that the transmembrane potential reaches 1 V, at which cellular mem-

brane gets so thin that it can not hold its complete structure at the weakest textural

positions (such as ion channels, penetrating proteins, interface of different phases,

etc.), the membrane will start to form small pores or holes along the direction of the

dramatically increased due to reduction in membrane thickness and deformation in

when transmembrane potential exceeds a critical value of 1 V, membrane destruction

or irreversible membrane changes can be observed and the pores or holes grow bigger

–12000

–10000

–8000

–6000

–4000

–2000

0

2000

4000

6000

8000

10000

0 5 10 15 20

Time (µs)

(b)

(a)

Voltage (V)

–60

–40

–20

0

20

40

60

Current (A)

Voltage

Current

3.54 v–5.44 v

Ch2 period6.714 µs

Low signalamplitude

SampleTek Run: 50MS/s

1

Ch1 1 v 1 v M 1 µs Ext –440 mV 18 Jun 199909:48:59

2 *

Ch2

FIGURE 8.5 Two types of typical electric pulses: (a) square and (b) exponential decaying.

55534_C008.indd 22455534_C008.indd 224 10/22/08 10:12:27 AM10/22/08 10:12:27 AM

and the cell survives the electric stress well. Further increasing electric field strength

external electric field. At this point, membrane conductivity and permeability can be

membrane structure. Along with further increasing external electric field strength

and negative charges. Since the membrane is flexible and deformable to external



and bigger. As a result, cytoplasma of the cell leak out via the induced pores, and thus

with cell suspension at 10 kV × cm−1

in microorganisms, the structural changes due to a transmembrane potential higher

than 1 V gives rise to irreversible loss of membrane function as the semi-permeable

barrier between the cell and its environment. Membrane damage is believed to be the

primary mechanism responding to the inactivation effects on cells by PEF. The cor-

relation between the effects of the pulse treatment on cell inactivity and membrane

damage, which was measured by little to no spheroblast formation (Table 8.2), sug-

gested that cell inactivation and membrane damage have a very high positive associa-

tion. This implies that membrane damage is a direct cause of cell death.

Other researchers [31–33] also suggested that when viable cells are subjected to

PEF, certain levels of transmembrane potential, depending on the cell size and the

tivation is approximately 1 V or higher, depending heavily on the different species

of microorganisms and their growth conditions. This is because different microor-

ganisms have different membrane composition and structure. Some cell membranes

membrane potential or slightly higher is the critical voltage, higher than which pores

will form inside the membrane and cause so called dielectric breakdown [30].

Cell shape is a very important factor determining the transmembrane potential

ucting

membrane, the induced potential is given by the equation:

V f r Em c= × × (8.4)

m

strength Ec, r refers to cell radius while f refers to form factor for spherical shape.

Zimmermann et al. [34] derived a mathematical equation to calculate the membrane

+++++++++

–

–

–

–

–

–

–

–

+

+

+

+

+

–

–

–

–

–

MediumCytoplasm

Cytoplasm

medium

Stages: a b c d

Big pores

allowing

cytoplasm

escape+++++++++

–––––––––

+++++++++

–––––––––

FIGURE 8.6 Illustration of cell membrane reversible and irreversible dielectric breakdown.

m

ical transmembrane potential, reversible breakdown; (d) large pores are formed, irreversible

dielectric breakdown. (Redrawn from U Zimmerman. 1986. Electric breakdown, electroper-

meabilization, and electrofusion. Review of Physiology, Biochemistry, and Pharmacology

105: 175–256.)

55534_C008.indd 22555534_C008.indd 225 10/22/08 10:12:28 AM10/22/08 10:12:28 AM

(a) Cell membrane with a potential V when there is no external electric field; (b) membrane

compression when cell is exposed to an external electric field; (c) small pore formation at crit-

resulting in death of the cells. Sale and Hamilton [5] first reported this phenomenon

electric field strength. The authors reported that

electric field strength, will be induced. The critical electric field strength for cell inac-

structure and not sensitive to electric field treatment. But for most cells, 1 V of trans-

induced under certain electric field. For spherical cells surrounded by non-cond

where V refers to the transmembrane potential induced by the external electric field

fluxes into

need higher transmembrane voltage to form pores since they may be less flexible in



potential Vm for nonspherical cells. The equation is based on the assumption that the

cell shape consists of a cylinder with two hemispheres at each end. In this case the

form factor f for rod-shaped microorganism is given by

f L= ×(1 0.33 )− d (8.5)

where L refers to the length of particle and d refers to the diameter of the cylinder.

So for rod-shaped cells the induced potential is given by:

V L d r Em1

c[ (1 0.33 ) ]= × × ×− − (8.6)

c

the cells.

So we know the critical voltage for cell inactivation is the induced transmembrane

to predict the inactivation effects after a given treatment time.

PEF inactivation effect on Lactobacillus brevis cells suspended in phosphate

buffer of pH 7.1 was consistent with the aforementioned theory [8]. The destruc-

tion of the cell membrane was primarily due to the pore formation and the increase

in the permeability of the membranes. When treated in the range of 24–80°C and

5–30 kV × cm−1, particularly when treated at 60°C and 25 kV × cm−1, an increase

in the chloride ion (Cl−) concentration was observed after PEF treatment. Most

importantly, the increase happened only in the case when L. brevis cells presented

in the test media and PEF was applied. The results suggest that the increased con-

centration of chloride ion is primarily due to the lysis of the cells, which leads to the

release of cellular matrix media—particularly those with small particle size—into

the test media.

TABLE 8.2Staphylococcus aureus Activity After PEF Treatment

Electric Field Strength (kV × cm−1) Survivors (%) Protoplasts Not Lysed (%)

0.00 100.0 100.0

9.25 100.0 100.0

14.25 35.0 43.0

19.50 0.9 16.0

24.00 0.3 3.0

27.50 0.6 1.5

on microorganisms, I. Killing of bacteria and yeast. Biochemistry and Biophysics Acta 148: 781–88.

55534_C008.indd 22655534_C008.indd 226 10/22/08 10:12:30 AM10/22/08 10:12:30 AM

Source: AJH Sale, and WA Hamilton. 1967. Effect of high electric fields

where E is external electric field strength applied, and r refers to the diameter of

potential, not the external electric field itself. However, if the processing targeted at

a same or similar shaped microorganism, external electric field strength can be used



Another well-known theory is the electroporation mechanism which was pro-

posed by Vega-Mercado and co-workers [35] in 1996 (as illustrated in Figure 8.7).

The authors reported that the plasma membranes of cells exposed to external elec-

then swelling and eventually rupture of the cell membrane—as a result, the cell

died. This theory adds valuable information to better understand the lethal effect

of PEF against microorganisms. However, it was derived from the dielectric break-

down theory, which was proposed in 1986 by Zimmermann [30] and also focuses

on the formation of pores crossing cell membrane and leaching of function plasma

contents.

Readers have to be aware of the fact that, although lots of reports mention the

observation of pores formation across membranes when comparing the PEF treated

dead cells with their untreated peers, no direct evidence during PEF operation shows

that pores are actually induced and how the pores are formed by exposing microbial

mation of the pores is most likely to happen at the two polar areas of the cell along

8.2.3 MODELING OF PEF INACTIVATION OF MICROORGANISMS

Sensoy and co-workers [36] reported their kinetic studies about the inactivation

kV × cm−1

tested during the experiment. These authors raised an inactivation kinetic model to

describe the inactivation behaviors of the microbial cells upon the changes of PEF

properties. Four models were established for the calculation of survival fractions of

Pore initiation Water influx Membrane rupture

Water

Swelling Cell lysis Inactive cell

FIGURE 8.7 Electroporation mechanism proposed for PEF microbial inactivation. (From

Vega-Mercado, H., Pothakamury, U.R., Chang, F.J., Barbosa-Canovas, G.V., and Swanson,

B.G. Inactivation of Escherichia coli by combining pH, ionic strength and pulsed electric

55534_C008.indd 22755534_C008.indd 227 10/22/08 10:12:31 AM10/22/08 10:12:31 AM

Electric field

fields hurdles. Food Research International 29(2), 117–21, 1996. With permission.)

tric field became permeable to small molecules such as water. The elevated per-

cells to an external electric field. It is very challenging to pursue the direct evidence

during a PEF operation, primarily due to the high electric field strength and the for-

the electric field direction—if the pore formation process indeed happens.

of Salmonella dublin by PEF. The authors used an electric field strength of 15–40

sity pulsed electric field treatment system. The medium temperature of 10–50°C were

missibility of cellular membranes to small molecules caused the influx of water

and treatment time of 12–127 microseconds in a co-field flow high inten-



time (model 3 and 4).

Model 1. s E E K= − −( )× −e c c 1

(8.7)

–1), Ec is –1), kc is constant factor (kV × cm–1).

Model 2 is Peleg’s [37] model in the form of:

Model 2. s E E k= + −( )1 1/ { }

[ / ]e c e (8.8)

where

k t k kt

e e( ) = e01 (8.9)

E t E kt

c c0e( ) = − 2 (8.10)

–1), Ec –1), kc, ke0, Ec0, k1(μs–1), k2 (μs–1) are constant

factors.

Model 3. s t t k= e c( )/− t (8.11)

where s is survival fraction, t is total treatment time (μs), tc is critical treatment time

(μs), kt is a constant factor (μs).

Model 4 was suggested by Hulsheger and co-workers [32] as shown in the

following model

Model 4. s t tE E k= × − −( ) ′

{ }/

cc1

(8.12)

where s is the survival fraction, t is the total treatment time, tc is critical treatment –1), Ec is the critical electric

–1) for the targeted microorganisms, k′ is a constant factor

(kV × cm–1).

All these models were developed to explain the relationship between cell sur-

two critical PEF processing parameters. On the other hand, however, cell shape is

also a very important factor needs to be concerned when we think about microbial

inactivation by PEF technology.

The above-mentioned microbial inactivation models were also applied to practical

research activities in enzyme inactivation. Min, Jin and Zhang [22] reported that all

the four models can be used for prediction of inactivation of lipoxygenase in tomato

juice by PEF with reasonable accuracy. Similarly, Yang and his colleagues [38] applied

these mathematic models to describe the characteristics of pepsin inactivation by PEF

the mechanisms for microbial inactivation and those for enzyme inactivation, the

55534_C008.indd 22855534_C008.indd 228 10/22/08 10:12:33 AM10/22/08 10:12:33 AM

the treated cells based on the electric field strength (model 1 and 2) and total treatment

where s is survival fraction, E is applied electric field strength (kV × cm

critical electric field strength (kV × cm

where s is the survival fraction, E is the applied electric field strength (kV × cm

is critical electric field strength (kV × cm

time, E is the applied electric field strength (kV × cm

field strength (kV × cm

vival fraction or ratio and electric field strength or the treatment time, which are the

using a lab scale continuous system. Although significant differences exist between



authors demonstrated that these mathematic models are useful tools for predictions

of enzymes PEF inactivation. However, the authors didn’t discuss the phenomena in

details and failed to infer it to a more general tool for broader application.

8.2.4 TRENDS IN PEF RESEARCH

8.2.4.1 Hardware Development has a Long Journey Ahead

It is always challenging, exciting and a must-do to continue further hardware devel-

opment to improve the production capacity and system reliability for PEF facili-

ties. It can be foreseen that scientists and engineers will strive to further modify

the designs of PEF systems, particularly the high voltage pulse generator (pulser),

chamber systems and monitoring and controlling subunits. Along with innovation

capacitors and more capable high speed switching devices will be available to allow

electric current in switching devices also sustains the desired performance of the

system and avoids the necessity for over-current shut-down, which can jeopardize

the smoothness of process and compromise the safety of end food products. The

reduced chance of system protective shut-down from higher tolerance to electric cur-

rent also greatly helps improve PEF processing reliability, which is still questionable

for most current PEF systems.

Another major hardware development effort will be in the area of improvement

of current PEF treatment chamber systems. Although there are numerous patents

more uniform treatment and more reliable system behavior. One of the major limita-

tions preventing PEF technology from industrial practice is the current small inner

diameter in treatment chambers, limited by the overall system electric current toler-

ance, particularly with products with large particles or high viscosity. Efforts will

also be invested to redesign current chamber geometry, although it will be extremely

and expected to be under better control by using electrically inert materials or with

innovative compensation technologies.

Monitoring of actual temperature inside a PEF treatment zone is very challeng-

sample temperature before and after PEF treatment zone with a distance for safety

ment in temperature monitoring is also needed for better understanding of PEF inac-

tivation of microorganisms and enzymes in food matrix.

8.2.4.2 Application Studies of PEF Technology in the Near Future

8.2.4.2.1

PEF will continue to focus on applications in high acid food processing. Fruit juices,

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and new development in electronic and semiconductor technology, more efficient

engineers to design PEF systems for higher electric field strength with high toler-

and different designs in efforts for efficient and reliable treatment chamber systems,

it still remains as one of the most difficult challenges for higher production capacity,

difficult, for uniform electric field distribution. Electrode erosion will be investigated

ing due to the high intensity of applied electric fields. The current method measures

purpose and to avoid possible interruption from operating PEF. Significant improve-

Pasteurization of High Acid or Acidified FoodPEF first found its commercial application in juice processing. In the near future,

ance in electric current flowing through the treatment chambers. High tolerance of



and carbonated ones are suitable for PEF pasteurization. PEF can be combined

together with other treatments to use as part of an innovative hurdle technology to

applications in medical and other biological industries, such as enzyme manufactur-

8.2.4.2.2 PEF Combined with Mild Heat for Shelf Stable High Acid Food Products

PEF is not effective in inactivating bacterial spores. PEF alone is not able to pro-

cess for shelf stable food products, even for high acidic foods. However, when com-

bined with a mild heat treatment, PEF processing can successfully result in a model

salad dressing shelf stable at room temperature for over a year without growth of

requires dramatically higher temperature and much longer holding time to achieve

shelf stable samples. The same phenomenon was observed in freshly made ranch

salad dressing. The promising future of this combined process of PEF paired with

a mild heat treatment will attract more researchers and manufacturers to explore in

a broader scope of application for shelf stable high food products, because of the

8.2.4.2.3 Research and Applications in PEF Assisted Food ProcessingAnother big area in the near future would be PEF assisted food processing. Preliminary

rate of sugar from sugar beets, and juice yield rate from diced fruit. PEF treatment also

helps remove unwanted components from food materials. It was also reported that PEF

treatment also improved the texture of gluten in bread dough. The research provides

ening processing time, and to increase yield rate. The selective inactivation of PEF pro-

cessing to microorganisms and proteins provides tremendous opportunities to apply it

into many high value or value-added applications. It is reasonable to believe this will

be an intensively investigated area in the near future.

8.3 CONCLUSIONS

PEF technology is a non-thermal or minimal heat alternate for conventional pasteur-

willing to pay more than the conventionally made products. This means a bigger

market share for manufacturers who apply PEF technology in their practices. On

the other hand, with more heat-labile bioactive compounds such as vitamin C and

tion, particularly pumpable high acid foods, via a mechanism named cell membrane

dielectric breakdown. The proposed mathematic models, Hulsheger, Fermi and other

55534_C008.indd 23055534_C008.indd 230 10/22/08 10:12:35 AM10/22/08 10:12:35 AM

vegetable juices with acidification and high acid beverages including low alcoholic

improve food safety with enhanced quality attributes. PEF may also find successful

ing and purification.

microorganisms and no significant quality changes [23]. Using heat treatment alone

studies show that low intensity PEF treatment can significantly increase the production

solid foundation for one to consider the benefits of PEF pre-treatment to facilitate a

chemical process, to improve production efficacy, to reduce processing cost by short-

of the processed food products attracts more customers with significant numbers

IgG, the more health benefits consumers can get from consumption of such products.

More importantly, PEF is an effective and efficient technology for food pasteuriza-

advantages of significantly improved flavors.

ization to improve the quality attributes, including fresher flavor profile and more

heat-labile functional compounds, of processed food products. The fresher flavor



tion of microbial pasteurization, even enzyme inactivation by PEF. However, PEF is

a complicated system consisting of many subsystems including power supplier, pulse

tem, and aseptic packaging system. To make things more complicated, while electric

are lots of secondary parameters, such as pH, ion strength, processing temperature,

to the overall microbial inactivation effect by PEF. There are still a long journey

ahead for PEF research and development, particularly in hardware development, to

improve the production capacity and system reliability, before this technology can be

widely commercialized in the food industry. The bright side of PEF also includes the

strong potential that it can be used for many other processing purposes such as PEF

assisted processing. Its bright future will stimulate the research activities in the area

and attract more and more capable researchers around the world.

NOMENCLATURE

E . −1

t Total PEF treatment time, μs3.s−1

prr Pulse replication rate, s−1

f Form factor, m2

Vm Transmembrane potential, volts

Ec . −1

kc Constant factor, kV.cm−1

ke0, Ec0 Constant factors, kV.cm−1

k1 Constant factor, μs−1

k2 Constant factor, μs−1

tc Critical treatment time, μs

kt Constant factor, μs

k′ Constant factor, kV × cm−1

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235

9 Fouling Models for Heat Exchangers

Sundar Balsubramanian, Virendra M. Puri, and Soojin Jun

CONTENTS

9.1 Introduction ................................................................................................... 235

9.2 Fouling Mechanism ...................................................................................... 238

9.2.1 Hydrodynamic and Thermodynamic Models ................................... 239

9.2.1.1 One-Dimensional Models ................................................... 239

9.2.1.2 Two-Dimensional Models ................................................... 243

9.2.1.3 Three-Dimensional Models ................................................245

9.2.2 Dynamic Fouling Model Incorporating

Physio-Chemical Changes ................................................................. 247

9.2.2.1 One Phase Approach ........................................................... 247

9.2.2.2 Two Phase Approach ...........................................................248

9.2.2.3 Three and Four Phase Approaches .....................................250

9.2.3 Cleaning and Economic Models .......................................................254

9.3 Concluding Remarks ..................................................................................... 257

References .............................................................................................................. 258

9.1 INTRODUCTION

One of the most critical and widely used unit operations in the food processing

of heat utilization through heat recycling and better heat transfer. During heat treat-

ment the food products undergo structural and chemical changes. Owing to changes

occurring in the food product some of the constituents like proteins and minerals

ment surface. These deposits are generally referred to as foulants and this phenom-

enon of deposition of food constituents on the equipment surface is termed fouling.

It has been documented in the literature that deposition of fouling layers on the

surface of the food processing equipment results in:

1. Increase in electrical and thermal energy usages due to the decrease in heat

2. Increase in pressure drop across the heat exchanger unit thereby lowering

the overall system performance.

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industry is thermal treatment. Heat exchangers offer an effective and efficient means

precipitate resulting in film-like deposits which adhere to the food processing equip-

transfer coefficient.



3. Additional increase in the use of electrical and thermal energies and water

usage due to the increase in the frequency and duration of cleaning opera-

tions to remove foulants.

Economically, fouling is a burden to the food industry. In the USA, the total costs

of fouling have been estimated to be $7 billion [1]. This includes the costs incurred

due to cleaning of the equipment, loss of production, additional energy consump-

tion, and over-sizing of the heat exchanger unit. The impact of fouling is so severe

that it is estimated that the total fouling costs equates to about 0.25% of the gross

national product of a developed country such as the USA [2]. In the dairy industry

alone fouling accounts for about 80% of the total operating costs involved [3]. Spe-

or redundant equipment, additional downtime for maintenance and repair, loss of

production, cleaning equipment and waste of energy [4,5]. With such a high impact

on the total operating costs, there is a need to minimize or delay the process of foul-

ing of the equipment surface thus prolonging the operation of the equipment and

conserve energy. In dairy industries it is a common practice to shut down the plate

heat exchanger (PHE) operation at least once a day in order to clean the equipment

[6]. The frequent interruptions during processing due to fouling causes extended

plant operation while lowering the desired output. Cleaning the foulants is also a

time consuming and energy intensive process that consumes a substantial amount

of water and chemicals. A typical dairy processing plant handling 75,000 gallons of

milk per day could use up to 110 million gallons of water per year [7]. Rebello et al.

[8] estimates that water (23.9%) and cleaning agents (7.5%) were the top contributors

towards the cleaning costs incurred during removal of foulants. Once the cleaning

ner further adds to the cost of production. Hence, prolonging the operation of the

equipment down-time, thus translating into increased production.

The exact mechanisms and underlying chemical reactions that result in fouling

is still not well understood. However, it has been widely believed that the denatura-

tion of the protein β-lactoglobulin plays a critical role in the fouling process during

dairy processing. The temperature and pH of the product aid in the unfolding of

the protein chain. Once the protein chains unfold, they form aggregates and get

adsorbed on the walls of the contact surface. Subsequently, calcium and phosphate

ions precipitate and add to the layers of adsorbed protein aggregates. This results

in a solid layer spread over the food equipment surface resulting in fouling. The

food processing equipment surface also plays an important role in fouling. Visser

and Jeurnink [9] have listed some of the factors pertaining to the food processing

following conditions that relate to fouling in stainless steel surfaces:

1. Presence of an additional covering layer like chromium oxide that inhibits

corrosion and oxidation of the stainless steel.

during manufacture dictate the nature of the surface charge.

55534_C009.indd 23655534_C009.indd 236 10/22/08 10:13:55 AM10/22/08 10:13:55 AM

cifically, the costs incurred due to fouling includes increased cost due to oversized

heat exchanger unit by reducing the rate of fouling could be beneficial in reducing

2. Surface charge; the cleaning process and the industrial finishing conditions

process is completed, disposal of the effluents in an environmentally friendly man-

equipment surface that have an influence on the fouling process. They observed the


Fouling Models for Heat Exchangers 237

3. Surface energy or in other words, the degree of hydrophobicity.

4. Micro structure of the surface like surface roughness.

5. Presence of residual proteins, microbes, and other contaminants which

were left behind during the earlier processing operations.

6. Type of steel used.

position, location of fouling, operating condition of the heat exchanger and the type

and characteristics of the heat exchanger. While processing milk, factors such as

pH, age of the milk, protein composition, calcium ions present, pre-chilling of milk

24 hours prior to thermal processing, whether the milk is reconstituted or not, and

fouling observed [3].

Two types of fouling deposits have been documented depending on the process

temperatures during dairy processing. At lower processing temperatures (between

75°C and 105°C) the foulants are predominantly proteins [9]. These deposits, i.e.

type A, are soft and bulky [10] containing about 50–60% protein (mostly β-lac-

toglobulin in milk), 30–50% minerals (like calcium and phosphate), and about 4–8%

fat [9]. The type B fouling occurs at temperatures above 100°C and has a hard,

granular structure. These deposits comprise mostly of minerals (about 70–80%),

followed by proteins (15–20%) and fat (4–8%). Table 9.1 summarizes the fouling

deposit characteristics obtained during type A and B fouling [11]. Thus, essentially

during fouling two processes take place: calcium phosphate deposition (mineral

fouling or crystallization fouling) and protein fouling (or chemical reaction fouling).

Both these processes follow different kinetics. Fouling deposits from a range of food

products, including milk [12,13], orange juice [14] and tomato juice [15], have been

studied. In particular, fouling during milk processing has been extensively studied

TABLE 9.1Characteristics of Type A and Type B Fouling Deposit Formed During Heating of MilkDeposit Content Type A Type B

Mineral content (%) 30–40 70–80

Protein content (%) 50–70 10–20

Fat content (%) 4–8 4–8

Temperature of occurrence (°C) 75–110 > 110

Color of deposit White/cream Grey/brown

Characteristics of the deposit Soft, curd like and voluminous Hard, brittle and granular

Type of protein and minerals present β-LG, calcium, phosphate β-casein, α-S1 casein, calcium,

phosphate

Source: From Prakash, S., Datta, N., and Deeth, H.C. Methods of detecting fouling caused by heating of

milk. Food Reviews International 21, 267–93, 2005. With permission.

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Other factors that have an influence on the fouling process are the product com-

seasonal variations are some of the factors that have an influence on the extent of



owing to the importance of thermal processing in the dairy industry. There are vari-

ous factors that contribute to the fouling process during processing; similar to these

mentioned previously for milk. These include, the product composition, tempera-

ture, pH, surface geometry of the processing equipment, presence of air bubbles,

plate corrugation are some of the known critical factors that impact the fouling rate.

To develop fouling models in order to simulate and predict the fouling mechanism

useful in building reliable models that can shed light on the fouling occurring in

surface can be taken into account thus developing models that could closely analyze

model that can closely predict the fouling behaviour under various operating con-

processing conditions. Operating under the best processing conditions will ensure a

balance between safely processed foods and prolonged equipment operation (due to

less fouling). Therefore, this chapter reviews the work to date by various researchers

to develop an optimum fouling model for the fouling mechanisms that occur in heat

exchangers with particular emphasis on plate heat exchangers.

9.2 FOULING MECHANISM

ing mechanism that leads to the fouling formation on the equipment surface. There

are two main schools of thought regarding the fouling process. One thought is that

the fouling process is a bulk-controlled homogenous reaction process independent

of mass transfer or a surface reaction process [9]. The other line of thought is that

thermal boundary layer. The aggregated proteins formed then adhere to the equip-

ment surface and the deposition of the protein is proportional to the concentration

of the aggregated protein in the thermal boundary layer. Fouling models have been

derived based on these assumptions. Once the fouling takes place, the deposit layer

removal of the deposits [3].

There has been extensive research conducted on the fouling in milk process-

ing equipment. However, the exact mechanism of fouling is not fully understood.

It has been agreed upon that when milk is heated, the native protein β-Lg (β-Lac-

toglobulin) chain denatures and exposes the protein molecules containing reactive

sulphhydryl (-SH) groups. These reactive groups from the unfolded (or denatured)

protein react with similar or other milk proteins like casein and α-La (α-lactal-

bumin) to form aggregates. It is here that the fouling mechanism becomes debat-

layer and others believe that it is the aggregated proteins that are involved in form-

ing the fouling layer. Hence, researchers modeling the fouling mechanism follow

55534_C009.indd 23855534_C009.indd 238 10/22/08 10:13:56 AM10/22/08 10:13:56 AM

different food equipment surfaces. With the advent of improvements in the field of

In order to model the fouling process it is imperative to first understand the underly-

able by researchers. Some believe that the denatured proteins form the first fouling

and the mixing intensity which is dependent on both the fluid flow rate and the

accurately is a challenge owing to the various factors that influence the process.

However, a thorough understanding of the chemistry and fluid mechanics are very

computational fluid dynamics (CFD) the detailed geometry of the heat exchanger

the interactions between processing surface geometry and fluid flow. An optimum

ditions like temperature, residence time, and flow rate will help in optimizing the

mass transfer takes place between the bulk fluid containing the proteins and the

is subjected to hydrodynamic forces from the moving fluids resulting in possible



different assumptions. There is literature dealing with modeling assuming that only

the aggregated proteins resulted in fouling [16] and others based on the assumption

that the aggregated proteins are not involved in the fouling process [17]. Yet another

group of researchers believe that the fouling process is due to both the denatured

and aggregated proteins [18,19]. Hence, there are various fouling models proposed

in literature pertaining to different starting assumptions. This results in the ambi-

guity of the actual fouling mechanism during thermal processing of food products.

There also have been attempts made to model the protein adsorption onto stainless

steel surfaces in conjunction with Langmuir-type adsorption isotherms. However,

this approach is disputed by some researchers who believe that the Langmuir-type

9.2.1 HYDRODYNAMIC AND THERMODYNAMIC MODELS

To completely understand the problem of fouling that occurs in heat exchangers, it is

essential to understand the relationship between fouling and the hydrodynamic and

are hence very keen in developing fouling models that can predict the performance

of heat exchangers. It has to be mentioned that most of the research activities related

to heat exchangers have been performed by engineers in auto, aerospace and chemi-

cal industries. Comparatively, fewer studies have been found in the food-processing

area. Hence, we have included some key studies in the non-food industry areas with

our current study on fouling modeling.

9.2.1.1 One-Dimensional Models

exchangers was experimentally and theoretically discussed by Rene et al. [21]. An

experimental model was developed by them that could predict the temperature pro-

This is of particular importance owing to the change in rheological and physical

properties of foods due to changes in temperature. Most studies have been limited

to numerical analysis or analytical formulation of the steady state behaviour in heat

exchangers including multi-stream or multi-channel heat exchangers. However, in

real life conditions heat exchanger systems always undergo transients resulting from

external load variations and regulations. Including the effect of transients in the pro-

The transient response of a multi-pass PHE was studied and a model based on

from shell-and-tube heat exchangers because of the phase lag at the entry and the

studies have been extended to include the phase lag effect in multi pass PHE units.

55534_C009.indd 23955534_C009.indd 239 10/22/08 10:13:56 AM10/22/08 10:13:56 AM

adsorption isotherms will not be an ideal fit for biopolymers [20].

files inside each channel of the PHE. The developed model defined the calorific fac-

posed models is expected to lead to significant enhancements in the food industry.

thermodynamic flow patterns occurring within the heat exchanger. Food engineers

The thermal processing of non-Newtonian food fluids in continuous plate heat

tor which was used to estimate the calorific temperatures of the cold and hot fluids.

the axial heat dispersion in the fluid was developed [22]. This model took into con-

sideration the deviation from ideal plug flow. The fluid flows in PHEs are different

successive channels. The phase lag increases with increase in number of flow chan-

nels because of decrease in fluid velocity in the port. In multi-pass PHE this delay

further increases due to fluid mixing. Studies have analyzed single-pass PHEs with

axial dispersion in fluid taking care of the phase lag effect at the port [23]. The



passes.

Certain assumptions need to be made prior to developing a one-dimensional

model for fouling in PHEs. The following assumptions have been made during the

developments of a one dimensional fouling model [22,24,25].

plate width.

ii. Heat transfer only takes place between channels and not between channels

and ports or through the seals and gaskets.

sure and temperature.

v. The loss of heat to the environment is negligible. Negligible radiation heat

losses are encountered.

of solid plate Figure 9.1, the energy balance over these control volumes taking into

tions related to the channel and plate [26]:

g CpTt

vTx

U Ti i ii

ii

i iρ ∂∂

∂∂c c

c cp+

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟= −( ( 11 2) )+ −T Ti ip c (9.1)

Channel

i–1

Channel

iChannel

i+1

vi

Tci

Plate

iPlate

i–1T

pi

Fluid control

volume

x

Δx

FIGURE 9.1

55534_C009.indd 24055534_C009.indd 240 10/22/08 10:13:57 AM10/22/08 10:13:57 AM

The developed algorithms also took into consideration the mixing of fluids between

i. The flow rate and temperature profiles are uniform across the channel and

iii. The thermal and physical properties of the fluids are not dependent on pres-

iv. Each fluid is split equally between all related channels. In other words, the

same volume of fluid flows across each channel meant for that particular

fluid type.

vi. The flow cross-sectional area of each channel is the same.

vii. There is uniform flow distribution within each channel giving a ‘plug flow’

of fluid inside each channel.

Considering a small control volume of fluid inside the channel and a control volume

account the above mentioned assumptions gives rise to the following fluid flow equa-

One dimensional view of control volume of fluid inside the PHE channel.



δ ρ ∂∂p p p

pc c pi i i

ii i iCp

T

tU T T T

⎡

⎣⎢⎢

⎤

⎦⎥⎥= + −+( ( )1 2 ii ) (9.2)

cith channel formed

between plate i and i + 1; Tpi = temperature of ithci

th channel; Cppi

in ithci

thpi

th plate; gi = gap

between plates i and i + 1; δpi = thickness of ith plate; Ui = overall heat transfer coef-th

The above Equations (9.1) and (9.2) were derived based on the fundamental

energy conservation law and describe the energy transfer between a channel and its

Neighboring plates.

less numbers such as Nusselt number (Nu), Reynolds number (Re) and the Prandtl

number (Pr). The Re and Pr numbers are related to the Nu number by the following

Equation [27,28];

Nu 0.214 (Re 3.2)Pr0.662 0.4= − (9.3)

where the Re and Pr numbers can be derived from the following relationships;

Re , ,= = =ρ

μvD p

kgi

eePr

CD

μ2 (9.4)

e

lated using the relation,

Nue=

hDk

(9.5)

T

1 1 1

U h h k= + +

hot cold

p

p

δ (9.6)

where hhot cold = con-

p

vity of the plate.

Another dimensionless quantity of major importance during fouling modeling

is the Biot number. Due to the deposition of foulants on the heat exchanger surface,

the heat transfer rate changes. The rate of deposition of the foulants is related to the

concentration of the aggregated proteins present in the thermal boundary layer (CA* ).

The Biot number is used to express the rate of change of heat transfer due to fouling

55534_C009.indd 24155534_C009.indd 241 10/22/08 10:13:58 AM10/22/08 10:13:58 AM

plate; Cp = specific heat at con-

ficient in the i

The overall heat transfer coefficient (U) can be calculated using the dimension-

From the Nu number, the convective heat transfer coefficient (h) can be calcu-

he overall heat transfer coefficient (U) can now be calculated by;

where t = time; x = axial position; T = temperature of fluid in i

stant pressure of fluid in i = specific heat at constant pressure of fluid

plate; ρ = fluid density in i channel; ρ = fluid density in i

channel; v = average fluid velocity which can be positive or negative

depending upon the direction of flow.

where k = thermal conductivity of the fluid; h = convective heat transfer coefficient;

μ = viscosity of the fluid; and D = equivalent diameter.

= convective heat transfer coefficient of the hot fluid stream; hvective heat transfer coefficient of the cold fluid stream; and k = thermal conducti-



and is related to the rate of deposition of the aggregated proteins by the following

expression [26]:

∂∂

β ∗Bim A

tk C= (9.7)

e m A∗

centration of aggregated protein in the thermal boundary layer.

under fouling condition, Uf, are related by the Equation

UU

fBi

=+

0

1 (9.8)

e 0

are considered to be important in the design of heat exchangers by engineers. A foul-

ing model which is able to predict the fouling thickness, Biot number and bulk milk

temperature in relation to time and position within a triple tube heat exchanger has

been proposed and demonstrated to be effective [29]. This model could be extended

for other heat exchangers.

and to the terminal block. Thus, the equations for these two channels for a PHE

having N plates can be written as follows [30]:

g CpTt

vTx

U T T1 1 11 1

1 1 1ρ ∂∂

∂∂c

c cp c+

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟= −( )) ( )+ −∞ ∞U T Tc1 (9.9)

g CpTt

vTx

U TN N cNN N

N Nρ ∂∂

∂∂

c cp+

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟= −( ( )1 −− + −∞ ∞T U T TN Nc c) ( ) (9.10)

1 1 1

U h h k∞= + +

hot cold

b

b

δ (9.11)

where kb = thermal conductivity of the front and back terminal blocks of the PHE;

δb = thickness of the front and back terminal blocks of the PHE; and T∞ = ambient

temperature.

One of the useful measurements used for modeling of fouling which helps to

drop. The drawback of using one-dimensional hydrodynamic model for modeling

the performance of PHE during fouling is that this model cannot be used to estimate

vide an estimate of the milk temperature along the plate height. The estimated tem-

perature distributions of the product at various locations along the height of the plate

provided curvatures in the isotherms. The results obtained from this model fueled

interest in the development of two-dimensional and three-dimensional models for

55534_C009.indd 24255534_C009.indd 242 10/22/08 10:13:59 AM10/22/08 10:13:59 AM

wh re Bi = Biot number; β = constant; k = mass transfer coefficient; and C = con-

It can be shown that the Biot number and the overall heat transfer coefficient

wh re, U is the heat transfer coefficient under no fouling conditions. Biot numbers

characterize the geometrical changes in the corrugated plate profiles is the pressure

A quadratic temperature profile model has been developed [31] that could pro-

The first and last channel in a PHE transfers heat to one adjacent fluid channel

the pressure drop varying across the PHE because of over simplified flow streams.



studying the hydrodynamic performance of heat exchangers. Models have also been

developed to estimate the temperature in each channel in the PHE [28]. These empir-

ical models were based on the steady state simulation of PHEs. Simulation results

from these models show an average deviation of about 4.9% from actual experimen-

tal results when the outlet temperatures were compared.

9.2.1.2 Two-Dimensional Models

The equations obtained for one-dimensional models can be expanded to form

directions. In order to compute the two-dimensional models incorporating the veloc-

be solved. The assumptions made in the case of two-dimensional modeling are that

by Kays [32] include the continuity and momentum equations as given below:

Continuity equation: ∂∂

∂∂

ux

vy

+ = 0 (9.12)

x-momentum: ∂∂

∂∂

∂∂ ρ

∂∂

ϑ ∂∂

∂∂

ut

uux

vuy

Px

ux

uy

+ + = − + +⎡

⎣⎢⎢

1 2

2

2

2

⎤⎤

⎦⎥⎥ (9.13)

y-momentum: ∂∂

∂∂

∂∂ ρ

∂∂

ϑ ∂∂

∂∂

vt

uvx

vvy

Py

vx

vy

+ + = − + +⎡

⎣⎢⎢

1 2

2

2

2

⎤⎤

⎦⎥⎥ (9.14)

where ϑ = kinematic viscosity; ρ = density; P = pressure; t = time; u = velocity

component in the x direction; and v = velocity component in the y direction.

noted by Ozisik [33] which is essentially an extension of Equation 9.1 and Equation

9.2 are given as follows:

g CpTt

uTx

vTy

i i ii

ii

iiρ ∂

∂∂∂

∂∂c c

c c c+ +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟= + −−U T T Ti i i i( )( )p p c1 2 (9.15)

δ ρ ∂∂p p p

pc c pi i i

ii i iCp

T

tU T T T

⎡

⎣⎢⎢

⎤

⎦⎥⎥= + −+( ( )1 2 ii ) (9.16)

involved than solving one-dimensional model equations. To simplify the proc-

the momentum equations. A vorticity-stream function approach applicable to

55534_C009.indd 24355534_C009.indd 243 10/22/08 10:14:00 AM10/22/08 10:14:00 AM

and plate, respectively, and have been defined previously.

two-dimensional models by considering the velocity vectors of the flow in x and y

ity and pressure distribution of the flow, the Navier–Stokes (N–S) equations have to

the plate surface is flat and smooth. The two-dimensional N–S flow equation given

The transient energy equation for a two dimensional incompressible flow as

where subscripts i, i + 1, i − 1, c and p refer to the plate i, plate i + 1, plate i − 1, fluid

The solving of the flow equations for the two-dimensional model is much more

ess of solving two-dimensional model flow equations, it is essential to transform

the flow equations into a simpler form by eliminating the pressure terms between



ity vector and streamline functions for a two-dimensional Cartesian coordinate

simpler form.

The vorticity vector is given by:

ω ∂∂

∂∂

= −vx

uy

(9.17)

and the streamline function (ψ) for the velocity vectors, u and v is given by:

∂ψ∂

∂ψ∂y

ux

v= = −, (9.18)

From Equation 9.17 and Equation 9.18, the vorticity vector can be transformed into

the following relationship

∂ ψ∂

∂ ψ∂

ω2

2

2

2x y+ = − (9.19)

From t

to the continuity equation (Equation 9.12). Transformation of the dependent vari-

ables from ‘u, v’ to ‘ω, ψ’ is applied to Equation 9.13 and Equation 9.14 to obtain a

relationship for the vorticity (ω) upon elimination of the pressure term. This trans-

formation leads to the following relationship:

∂ω∂

∂ω∂

∂ω∂

ϑ ∂ ω∂

∂ ω∂t

ux

vy x y

+ + = +⎡

⎣⎢⎢

⎤

⎦⎥⎥

2

2

2

2 (9.20)

One can obtain a differential equation for the pressure term from the momentum

equations. The pressure term can be shown to be a function of the velocity vectors

and the density by the following relation:

∂∂

∂∂

ρ ∂∂

∂∂

∂∂

∂∂

2

2

2

22

Px

Py

ux

vy

uy

vx

+ = −⎡

⎣⎢⎢

⎤

⎦⎥⎥ (9.21)

Reducing this equation by including the streamline function (ψ) the differential

equation for the pressure term can be given by:

∂∂

∂∂

ρ ∂∂

∂∂

2

2

2

2

2

2

2

22

Px

Py x y

+ =⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎛

⎝⎜ψ ψ⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

∂∂ ∂

2 2

ψx y

(9.22)

s

mined if the stream line function is known. Jun et al. [30] have noted that by using

the one-dimensional model for predicting the temperatures at various zones on the

55534_C009.indd 24455534_C009.indd 244 10/22/08 10:14:03 AM10/22/08 10:14:03 AM

two-dimensional modeling is usually used [33]. This approach defines the vortic-

he streamline function definition it is obvious that the relationship is identical

U ing finite-difference approximation and Gauss-Seidel iterative solver, the pres-

system and use these terms to transform the two-dimensional flow equations to a

sure distribution at various locations on a grid for the entire flow field can be deter-



plate large prediction errors (up to 43.2%) were observed. With the use of the two-

dimensional model they observed good agreement of the predicted temperatures

with the experimental temperatures. The average percentage deviation between the

predicted and measured temperature values observed by them was about 6.2%. Since

etry the error observed during prediction could be due to the fact that the actual plate

two-dimensional model was superior in predicting the temperatures across the plate

surface when compared to the one-dimensional model, an interesting fact observed

by the authors [30] was that both the models performed identical while estimat-

tures at various locations on the plate surface fairly accurately the two-dimensional

model could be used as an effective tool to gain information on possible milk fouling

sites. The one dimensional model lacks this potential when compared with the two-

dimensional model. Hence, the two-dimensional model is better suited for studies on

fouling behavior and control than the one-dimensional model.

Regarding the pressure drop, there are various components which contribute

toward the drop in pressure observed in a heat exchanger. Drop in pressure due

are the major contributing components for the pressure drop observed in a heat

exchanger [34]. Out of these factors the pressure drop due to friction is the largest

contributor. This term can be calculated using the relation:

ΔPfm L

D=

4

2

2

ρ h

(9.23)

area; L = plate length; and Dh = hydraulic diameter, which is usually twice the plate

spacing.

The value for the friction factor for a Chevron plate which is the common type

of plate design used in plate heat exchangers in the food processing area can be

obtained from the correlations given by Shah and Focke [35]:

f x y= −Re (9.24)

The values for x and y can be obtained from the published literature [35].

9.2.1.3 Three-Dimensional Models

The use of three-dimensional models to study the fouling behavior in PHE was aided

with the advent of advanced computer software packages. Numerical simulations of

Patankar [37] called the Semi-Implicit Method for Pressure-Linked Equation Revised

(SIMPLER) algorithm was used to numerically solve the governing equations for

continuity, momentum and energy iteratively. The results of the study showed the

55534_C009.indd 24555534_C009.indd 245 10/22/08 10:14:05 AM10/22/08 10:14:05 AM

Chevron plates was first reported in the late 1990s [36]. An algorithm proposed by

the two-dimensional model was based on the assumption of a flat plate surface geom-

geometry was not flat and had a corrugated surface geometry. The plate corrugations

guide the fluid flow to distribute evenly across the whole plate area. Though the

ing the average energy balance of mass flow. By being able to predict the tempera-

to friction, changes in velocity, changes in direction of flow and changes in height

where ΔP = pressure drop; f = friction factor; m = mass flow rate per cross sectional

the turbulent, three-dimensional fluid and heat transfer flow between two parallel



only one channel to study the fouling behaviour will not give a true picture of the

actual fouling process that occurs in multi-channel and multi-pass PHE systems.

of milk between two corrugated plates was studied by Grijspeerdt et al. [38] using

CFD (Computational Fluid Dynamics). The three-dimensional calculations showed

culations. This clearly showed the limitations of two-dimensional calculations for

temperature regions. These regions of elevated temperatures are potential fouling

locations on the plate surface. Eliminating such occurrence is essential to minimiz-

models have immense potential in optimizing the design of plates for heat exchang-

ers. So far the CFD based studies on fouling behavior have been concentrated towards

cal and chemical aspects into three-dimensional model studies of fouling behavior

for various food products will go a long way in better understanding the process of

fouling, and thus, help in better design of control strategies to minimize fouling. For

this, the denaturation of β-LG and its relation to wall adhesion needs to be critically

examined and incorporated in the three-dimensional model. This is in fact not an

easy task and adds to the complexity of three-dimensional model calculations.

the continuity and momentum equations by extending the two-dimensional model

which was described in detail earlier.

∂∂

∂∂

∂∂

ux

vy

wz

+ + = 0 (9.25)

The equations for the momentum in the x, y and z directions are now described by

the following Equations:

x-momentum: ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

ut

uux

vuy

wuz

Px

ux

u+ + + = − + +

1 2

2

2

ρϑ

yyu

z2

2

2+

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂∂

(9.26)

y-momentum: ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

vt

uvx

vvy

wvz

Py

vx

v+ + + = − + +

1 2

2

2

ρϑ

yyv

z2

2

2+

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂∂

(9.27)

z-momentum: ∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

wt

uwx

vwy

wwz

Pz

wx

w+ + + = − + +

1 2

2

2

ρϑ

yyw

z2

2

2+

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂∂

(9.28)

wh

55534_C009.indd 24655534_C009.indd 246 10/22/08 10:14:06 AM10/22/08 10:14:06 AM

potential of using finite element analysis to have a better understanding of the foul-

designing new plate configurations that could reduce fouling. The three-dimensional

ing fouling and that could be done by better plate configuration design. Thus CFD

The continuity equation in this case is defined by

ing phenomenon occurring between a flow channel. However, the results from using

A detailed two-dimensional and three-dimensional study on the flow pattern

the virtual flow velocity fields which were not possible using two-dimensional cal-

calculations help to identify regions of turbulent backflows that could cause elevated

the thermodynamic and hydrodynamic aspects of fluid flow. Incorporating the physi-

The flow equations describing the three-dimensional model can be derived from

ere w is the velocity component of the flow in the z direction.



be written as [39]:

∂∂

∂∂

∂∂

∂∂ ρ

∂∂

∂∂

∂Tt

uTx

vTy

wTz

kC

Tx

Ty

+ + + = + +p

2

2

2

2

2TTz∂ 2

⎡

⎣⎢⎢

⎤

⎦⎥⎥ (9.29)

stituting the thermal diffusivity term α = k/ρCp. The thermal diffusivity is the ratio

of the thermal conductivity to the volumetric heat capacity of a substance. Hence, the

Equation can now be written as

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Tt

uTx

vTy

wTz

Tx

Ty

Tz

+ + + = + +α2

2

2

2

2

22

⎡

⎣⎢⎢

⎤

⎦⎥⎥ (9.30)

Using the above three-dimensional model and coupling it with a CFD software

package FLUENT (Fluent Inc., NH, USA), Jun and Puri [39] studied the fouling

behavior in a PHE system. The three-dimensional model incorporated the surface

This gives the three-dimensional model an advantage over the two-dimensional

model in which a detailed study of the fouling behavior on the PHE surface can be

carried out. Results from the three-dimensional study can be utilized for designing

with a two-dimensional model.

9.2.2 DYNAMIC FOULING MODEL INCORPORATING PHYSIO-CHEMICAL CHANGES

The dynamic fouling model was developed based on the fact that fouling is essen-

tially a transient process. In the beginning the heat exchanger starts clean and slowly

with change in time the foulants build-up in the equipment. With this in mind the

dynamic fouling model was approached under various phases of fouling. Some of

the models just took into account the protein denaturation occurring, while others

took into account the induction period also.

9.2.2.1 One Phase Approach

As discussed earlier, to obtain a comprehensive model that includes the hydrody-

namic and thermodynamic factors of fouling with the physical and chemical con-

tributing factors of fouling it is imperative to understand the protein denaturation

process. The dynamic fouling models study the denaturation of β-LG and its rela-

tionship to the fouling observed.

of 13 plates. They developed a model that can predict the amount of native β-LG at

the outlet of the PHE. The model was tested by simulating the amount of denatured

proteins which was determined based on the steady state conditions and the numeri-

denatured β-LG was then compared with the actual amount obtained from experi-

ments through measurements using immunodiffusion methods. From the developed

55534_C009.indd 24755534_C009.indd 247 10/22/08 10:14:07 AM10/22/08 10:14:07 AM

configuration of the PHE which was not possible with the two-dimensional model.

new surface configurations that can help in minimizing fouling. This is not possible

cal determination of temperature profile for each channel. The simulated quantity of

For a three-dimensional incompressible flow the transient energy equation can

where T is the temperature of the fluid. The above Equation can be simplified by sub-

Delplace et al. [40] studied the complex flow arrangements in a PHE consisting



model they could predict the native β-LG at the outlet of the PHE with an experi-

mental error of less than 10%.

The model for predicting the β-LG was given by

C tCkC t

( ) =+

0

01 (9.31)

For T ≤ 363.15 K, log k = 37.95 – 14.51 (103/T).

For T ≥ 363.15 K, log k = 5.98 – 2.86 (103/T).

where C is the β-LG concentration, C0 is the initial β-LG concentration, k is the sec-

ond order rate constant, t is the time, and T is the temperature. Similar models have

been developed to predict the amount of dry mass deposited based on the steady

of β-LG protein [6,40]. Some of these models were found to be suitable for online

applications.

9.2.2.2 Two Phase Approach

The two phase approach was based on the observation that there may be an induction

phase prior to the actual fouling phase. During the induction period the conditions

the fouling period begins resulting in increased pressure drop and decreased heat

ing period consists of a deposition and removal process. The difference between the

rates of deposition and removal of deposits constitutes a simple model that explains

the rate of build up of deposit on a surface.

d

dD R

mt

= −θ θ (9.32)

e D R

area for the deposition and removal periods. This simple model forms the basis of the

local fouling factor model. Fryer and Slater [41] have suggested a generalized equa-

tion for the fouling deposition process based on the above Equation 9.32:

dBi

de Bid r

fi

tk k

E

R T= −−⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

(9.33)

where kd and kr are, respectively, the rate constants for the deposition and removal

expressed in s − 1. Tfi is the temperature (oC) at the interface of the fouling deposit and

priately to include various factors relevant to fouling such as chemical reaction,

relation result in linear (constant rate), falling rate or asymptotic fouling growth

55534_C009.indd 24855534_C009.indd 248 10/22/08 10:14:08 AM10/22/08 10:14:08 AM

state conditions, predicted temperature profiles and the amount of heat denaturation

of pressure and temperature do not change significantly. These later change when

transfer coefficients. Most models developed deal with the fouling period. The foul-

The general equation described above (Equation 9.32) can be modified appro-

wh re m is the mass deposited, θ and θ are the mass flow rates per unit surface

the process fluid.

mass transfer, fluid shear, and bond resistance. The models developed using this



models (Figure 9.2). The falling rate and constant rate fouling phenomenon is mostly

observed in food processing applications [42]. Constant rate fouling leads to rapid

of calcium sulfate scale deposition on heat transfer surfaces a falling rate of fouling

locations of the heated surface which was determined numerically, the CaSO4 scale

formation rate was not uniform. The assumption made in this study was that there

was no removal of scaling occurring. However, models for CaSO4 fouling including

the removal term have been developed earlier [46]. This model took into account

the rate of particulate fouling, rate of crystallization and the rate of removal of the

deposits. The rates of crystallization and particulate fouling together constitute the

rate of deposition of CaSO4 on the heated surface. The particulate fouling term was

determined taking into account the physical mechanism for particle transport and

adherence. The crystallization term was estimated based upon the ionic diffusion

deposit properties. This model also took into account both linear and asymptotic

fouling conditions.́

In general, two phenomena occur during fouling; heterogeneous nucleation and

crystal growth. Heterogeneous nucleation refers to the nuclei formation on any for-

eign body, just as in the case of heat exchanger surface. The heterogeneous nuclea-

tion can be calculated using the term [44]:

H BNV

R T SN = −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

’exp(ln )

16

3

2 3

3 3 2

π σm

i

(9.34)

where B′ is the pre-exponential factor, N is the Avagadro’s number, Vm is the molar

i

solid–liquid interface temperature, and S is the supersaturation. A low energy sur-

face (having high contact angle) will require higher supersaturation for nucleation to

cation techniques like those that use coatings to alter the surface roughness can help

Induction

period

Fouling

period

Linear fouling

Falling rate fouling

Asymptotic fouling

Time

Fouling

resistance

FIGURE 9.2 Fouling curves.

55534_C009.indd 24955534_C009.indd 249 10/22/08 10:14:09 AM10/22/08 10:14:09 AM

decrease in the heat transfer coefficient. This leads to rapid increase in pressure

volume, σ is the specific interfacial energy, R is the universal gas constant, T is the

occur than for a surface with high energy (having low contact angle). Surface modifi-

drop and blockage of the passage of fluid flow due to the foulants [43]. In the study

growth model was observed [44,45]. Owing to the non-uniform heat flux at various

and the removal term was estimated based upon the hydrodynamics of flow and



in delaying the nucleation occurrence by altering the surface energy [47]. The super-

saturation can be expressed as the ratio of the bulk concentration (cb) to the saturation

concentration (cs). The cs value is calculated from the solubility curve for the particular

the Ti value.

Once the nucleation occurs, the crystallization begins and the fouling layers

begin to form. There are various models that explain crystal growth. For example in

the absence of any removal term, the rate of deposit growth on a heat transfer surface

due to crystallization can be expressed as [48]:

d

d r

b s

r

mt k

c ck

=⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+ − −

⎛

⎝⎜⎜⎜

⎞

⎠β β β1

2

1

4( ) ⎟⎟⎟⎟⎟ +

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ −

⎧⎨⎪⎪

⎩⎪⎪

⎫⎬⎪⎪

⎭⎪⎪

⎡ 2βk

c cr

b s( )

⎣⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥ (9.35)

T r

from the Sherwood number,

Sh Sch= +

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟0 023 1

60 8 0 33. Re . . dx

(9.36)

where Sc is the Schmidt number and dh is the hydraulic diameter at position x. The

Sherwood number is given by:

Shh=

βαd

where α is the thermal diffusivity of the ions.

9.2.2.3 Three and Four Phase Approaches

The above discussed models deal with bulk-controlled homogeneous reaction proc-

esses. In contrast, the three and four phase model deals with surface reaction proc-

ess. This model deals with the varying protein characteristics during denaturation;

native, unfolded, aggregated, and deposited [9]. Usually the three and four phase

approaches go hand-in-hand, because once the protein aggregates are formed, the

fourth phase, i.e. the attachment of the aggregated protein to the contact surface

occurs. A mathematical fouling model where both the surface and bulk reactions

are considered has been proposed in the 1990s [49]. The foundation of this model

was the consideration of the denaturation of β-LG protein as a series of consecutive

reaction kinetics involving unfolding and aggregation. The model can be stated as

follows:

N ↔ U → A (9.37)

The terms N, U and A stand for the native, unfolded and aggregated β-LG protein,

respectively. It can be seen that from the model there is some unfolded protein being

converted back into the native state.

55534_C009.indd 25055534_C009.indd 250 10/22/08 10:14:10 AM10/22/08 10:14:10 AM

salt followed by curve-fitting techniques to obtain a standard equation in relation to

he term k is the rate constant, and the mass transfer coefficient β can be obtained



The rate of disappearance and formation of these protein phases are given by the

relation [18,19]:

− = −d

d

NU N

aN U

bCt

k C k C (9.38)

d

d

UU N

aN U

bA U

cCt

k C k C k C= − − (9.39)

d

d

AA U

cCt

k C= (9.40)

e

reaction. The orders of the reaction vary according to the assumptions made and

the process condition like temperature of denaturation, and also the product being

processed [19,50]. Hence, the values of the orders of the reaction (a, b, c) are not

always the same. For example; Chen et al. [19] considered the orders of the reaction

values of a, b, and c to be respectively, 1, 0, and 2.

If the β-LG protein denaturation process is modeled considering the entire proc-

ess to be irreversible, i.e. no unfolded protein is converted back into its native state,

then Equation 9.36 can be written as:

N → U → A (9.41)

This approach means that all of the native proteins get unfolded and immediately

get converted to its aggregated state. The rate of disappearance and formation of the

different protein states can then be written as:

− =d

d

NU N

aCt

k C (9.42)

d

d

UU N

aA U

cCt

k C k C= − (9.43)

d

d

AA U

cCt

k C= (9.44)

The main mechanism in the fouling process of skim milk is a reaction-controlled

adsorption of the unfolded β-LG protein [39]:

F k CR D U= 1 2. (9.45)

where FR is the fouling rate and D is the deposited β-LG protein. The rate constants

are denoted by kU, kN, kA and kD, and are dependent on the temperature, T as given

by the Arrhenius law:

k An n

E

RTn

=−

e (9.46)

55534_C009.indd 25155534_C009.indd 251 10/22/08 10:14:11 AM10/22/08 10:14:11 AM

wh re C is the protein concentration. The suffix a, b, c, pertain to the orders of the

to be a = 1, b = 1, and c = 2. Jun and Puri [39] have described a simplified model with



where An is the Arrhenius constant, and E is the activation energy for n = U, N, A,

and D and R is the universal gas constant.

Attempts have been made to use the developed models for protein denatur-

ation in conjunction with a process model and cost predictive model to optimize

the process of PHE operation in relation to the desired product quality and safety

[51,52]. It is interesting to note that Grijspeerdt et al. [53] mention that the aggregated

reacted with the milk constituents (M) to form aggregated milk constituents (D).

These aggregated milk constituents were later adsorbed to the heat exchanger wall

(D*) causing fouling. Their reaction scheme was as follows:

N → U (9.47)

2U → A (9.48)

U D DM+⎯→⎯ → ∗ (9.49)

It should be mentioned that the models studied by de Jong [10,51,52] mainly dealt

with the fouling caused by β-LG. Fouling can also be caused by the precipitation

of minerals. The mechanics and nature of mineral fouling is different from that of

protein fouling. The underlying mechanism of mineral fouling is more complex than

protein fouling and this could be the reason of why this phenomenon has been least

studied in detail. Other probable reasons for this phenomenon of fouling getting

lesser attention than protein fouling is that the mineral foulants being less volumi-

nous in occurrence than protein fouling could have a lesser impact on the pressure

layer is known to be proteinaceous in nature. In milk calcium phosphate is the major

mineral component which constitutes the mineral deposits in the fouling process.

Calcium phosphate fouling predominantly occurs at higher temperatures (tempera-

tures greater than 100oC) than protein fouling and the reason of this occurrence

is because calcium phosphate is less soluble at higher temperatures resulting in its

precipitation. The fouling caused by calcium phosphate involves the competition

between different types of reactions involving calcium phosphate, the contact sur-

actual mechanism of calcium phosphate fouling is complex to understand and has

not been fully understood, Visser and Jeurnink [9] have postulated a possible path-

way for the fouling mechanism. According to them, as a pre-cursor to the fouling

phate complex. This complex is transformed to amorphous calcium phosphate (ACP)

and is subsequently converted to hydroxyapatite (HAP) after going through another

phase change, which involves the formation of octa-calcium phosphate (OCP). The

phosphate fouling and is accompanied by an increase in solution turbidity, indicating

that this process might be taking place in the bulk liquid. Due to the formation of

insoluble calcium phosphate this process is generally accompanied by a decrease in

pH [56]. This explains the complex nature of the mineral fouling mechanism.

55534_C009.indd 25255534_C009.indd 252 10/22/08 10:14:12 AM10/22/08 10:14:12 AM

β-LG played a less significant role in the fouling process, while the unfolded β-LG

drop and thermal resistance encountered during the fouling process. Also, the first

phenomenon, the calcium and phosphate ions first form a colloidal calcium phos-

final product formed, namely HAP is the thermodynamically stable form of calcium

face, the solvent and any other solute present in the fluid system [54,55]. Though the



The model proposed by Petermeier et al. [57] follows a slightly different pathway

than the model proposed by de Jong (Equations 9.46 through 9.48) [51,52]. Accord-

ing to their model the pathway for β-LG denaturation is given by:

− =d

d

NU N

Ct

k C (9.50)

d

d

UU N A U D U

Ct

k C k C k C= − −2 (9.51)

d

d

AA U

Ct

k C= 2 (9.52)

d

d

DD U

Ct

k C= (9.53)

The above pathway for protein denaturation and deposit formation indicates that

some of the unfolded β-LG protein gets lost due to the deposition process (Equa-

tion 9.50).

A more comprehensive model has been developed that takes into account the

assumption that for each protein present, mass transfer takes place between the bulk

and thermal boundary layer [58]. But it is the aggregated proteins that can adhere to

the wall in such a way that the amount of deposit is proportional to the concentra-

tion of aggregated protein in the thermal boundary layer. Figure 9.3 and Figure 9.4

this model is an extension of the models proposed by de Jong et al. [51]. The key

assumptions followed while developing this model are as follows:

are then transformed by a second order reaction to form aggregated proteins.

ii. For each protein present whether it is N, U, or A, mass transfer takes place

between the bulk and thermal boundary layer.

iii. Only the aggregated protein is deposited on the wall.

iv. The thickness of the deposit dictates the magnitude of the fouling resis-

tance to heat transfer.

A major difference between the models proposed by de Jong [51] and Toyoda and

Fryer [58] is that in the former case the main mechanism of fouling was believed to

be due to the reaction-controlled adsorption of unfolded β-LG; while in the latter

case the fouling deposit on the walls was assumed to be only due to the aggregated

proteins. Most of the models proposed have been validated with experimental data.

tions and relationships between minerals and denaturation of milk proteins. Such

a model will be more suitable for real world conditions. However, to date due to

the complexity of forming a model that can relate to all the fundamental reactions

55534_C009.indd 25355534_C009.indd 253 10/22/08 10:14:13 AM10/22/08 10:14:13 AM

Unfolded proteins are formed by a first order reaction from native proteins and

It would be beneficial to have a comprehensive model that can encompass the reac-

schematically represent the flow and reaction model dynamics proposed. Basically

i. Proteins react in both the bulk and thermal boundary layer in fluid milk.



that give rise to fouling, it is not possible to point out which fouling model is more

suitable for real world conditions from the impressive array of proposed models.

Also, it is imperative to channel the collective knowledge and wealth of information

available from past research experiences to dwell upon a threshold value of mineral

amount that will accelerate the fouling process and how controlling the bulk tem-

perature can control this mineral precipitation.

9.2.3 CLEANING AND ECONOMIC MODELS

Cleaning of the fouled deposits has been a major concern for food processors as

Channel i

Plate

i–1

Plate

i

y

Δx

Height

Width

y

x

FIGURE 9.3

N* U*

UN A

A*

Reaction

Mass transfer

Adhesion

Thermal

boundary

layer

Wall

FIGURE 9.4 Protein reaction scheme for milk fouling. (From Georgiadis, M.C., and

Macchietto, S. Dynamic modelling and simulation of plate heat exchangers under milk

fouling. Chemical Engineering Science 55, 1605–19, 2000. With permission.)

55534_C009.indd 25455534_C009.indd 254 10/22/08 10:14:14 AM10/22/08 10:14:14 AM

Two-dimensional view of control volume of fluid inside the PHE channel.

Bulk fluid

it dictates the amount of resources and time spent, not to mention the influence



on product quality and safety. Hence, it is natural that the cleaning process has

been studied in detail and various models been proposed. During fouling both

organic and mineral deposits are formed. The deposits formed depend on the prod-

uct processed and the processing conditions. Altering the composition of the liquid

observed. The aim of an appropriate cleaning model developed is to optimize the

cleaning agent and the other cleaning parameters. An important parameter that

needs to be studied to develop a cleaning model is the strength of adhesion of

the foulants with the food processing surface or in other terms the force required

to dislodge the foulants from the food processing surface. This parameter is not

known directly and needs to be determined by other methods. For instance it has

been shown that altering the surface energy of a surface alters the adhesive force

between the foulants and the surface. This is one of the key components in the

design of frictionless coating materials to reduce fouling. Some researchers have

focussed on the sticking probability [64,65] to ascertain the force required to

remove the deposits. Though the sticking probability helps in providing informa-

tion about the probability of a surface being fouled, it can also provide information

regarding the amount of force required to dislodge a particle from the surface. A

study conducted on the adhesive strength while baking tomato paste in an oven at

100oC and the baking time [15]. The study revealed that the adhesive strength of

the tomato starches increased with baking time, but the increase became less sig-

hydration time. It was found that the adhesive strength of tomato starch decreased

by a factor of three initially, and then became constant. The results of the study

show that a larger amount of the tomato foulants can be removed at the initial

chemical concentration. Similar studies on the removal of milk proteins revealed

that the deposits could not be removed completely with water alone. Chemical

proteins decreased with time. Also, the order of use of the acid and base chemicals

ies clearly show that the cleaning models need to be developed keeping in mind

the type of food product processed and the other cleaning parameters involved.

The foulants adhering to the food processing surface are attached by cohesive and

adhesive forces. Studies on the adhesive and cohesive forces will reveal the appro-

priate cleaning model or cleaning protocol to be developed. Various studies have

been conducted on the adhesive and cohesive forces encountered during fouling

to develop appropriate cleaning models [67,68]. The adhesive forces are related to

the foulant and surface interaction and the cohesive strength relate to the particle

and particle interaction. Deposits of tomato paste, bread dough and egg albumin

have less adhesive strengths than cohesive strengths causing them to be removed

in larger chunks from the attached surface. On the other hand, deposits like milk

proteins have more adhesive strengths than cohesive, resulting in their removal in

smaller chunks [67].

55534_C009.indd 25555534_C009.indd 255 10/22/08 10:14:15 AM10/22/08 10:14:15 AM

nificant after 3 h. The same study also focussed on the adhesive strength versus the

food material being processed has a significant influence in the fouling profiles

time involved with reference to the flow velocity, chemical concentration of the

stages of cleaning. This could be the time to decrease the flow rates and cleaning

concentration, flow rates of the cleaning solutions and temperature had a major

influence on the removal of the milk proteins. The force needed to remove the milk

during cleaning had an influence on the milk foulants removal [66]. These stud-



The minimum adhesive energy between the surface and the deposits can be

expressed in terms of the surface energy by the relationship [69]:

γ γ γsurfaceLW

foulantLW

fluidLW=

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+⎡⎣⎢

1

2

⎤⎤⎦⎥ (9.54)

where γ γ γsurfaceLW

foulantLW

fluidLW, , are the Lifshitz van der Waals (LW) surface free energies

the surface free energy of the food contact surface can be reduced, this in turn will

reduce the adhesive forces between the surface and the foulant, thus making it easier

to remove the foulant attached to the surface.

A simple model for the removal of the calcium sulfhate deposits [46] is given by:

Wx

Sremovaldeposit

d

∝∇

(9.55)

where Wremoval = rate of removal of deposits; ∇ = shear stress in N/m2; Sd = strength

of deposit and xdeposit = deposit thickness (m).

The rate of removal of the deposit is time-dependent as the thickness of the

deposit formed and the strength of the deposit vary over time. This relationship does

not take into consideration the cleaning chemicals. A relationship has been proposed

for the adhesive strength per unit area, σadhesive and the deposit thickness [70]. This

relationship indicates that σadhesive increases with deposit thickness.

σ ω ψadhesive s v depositx= + (9.56)

where ωs = work needed to overcome surface bonds; and ψv = force required per unit

volume to overcome the deposit–deposit bonds.

This simple model however, is suited for low surface energies (about 28 mNm–1).

At higher surface energies this model is not as effective indicating that at higher sur-

face energies a different method of breakdown of the deposits could be possible.

A model incorporating the concentration of the chemicals for cleaning was pro-

posed as early as 1957 [71,72]. It is prudent to develop models incorporating the

concentration of the chemicals and the force required to remove the deposits. Such

models that involve the mechanical and chemical aspects of cleaning will be more

comprehensive in studying the cleaning process. The modeling of cleaning process

in heat exchangers is still in its infancy stage. But a lot of emphasis is laid upon the

CIP modeling in recent years. CIP is an energy intensive process. For example the

CIP process accounts for 9.5% of primary energy demand (energy consumption) in

the Dutch dairy industry and accounts for 0.14–0.30 MJ/cycle of thermal energy

requirement for milk pasteurization [73]. To add to this high energy requirement

the incidence of fouling causes an increase of about 8% in energy consumption and

about 21% increase in the total energy consumption related to the operation and

cleaning of milk pasteurization units [74].

With the advent of various food-grade frictionless coatings and emphasis on

energy conservation in the food industry models optimizing the cleaning process

is essential. Attempts have been made to model the optimum cleaning schedules for

55534_C009.indd 25655534_C009.indd 256 10/22/08 10:14:15 AM10/22/08 10:14:15 AM

of the surface, foulants, and the fluid, respectively. This relationship indicates that if



plate heat exchangers [75,76] and also the cost economies involved during cleaning

[77,78] in the petroleum industry. An accurate model predicting the correct direc-

tional change (CDC) values of more than 92% has been developed using neural net-

works for the petroleum industry [79]. CDC is a measure of the prediction capability

of a model to predict the correct direction of change in a variable. Using this model

it would be able to schedule optimum cleaning schedules. Using similar models in

the food industry it would be possible to develop cleaning strategies that will result

in optimum plate heat exchanger performance; saving costs and minimizing energy

usage. Attempts have been made to use neural networks to model the optimum clean-

ing schedule in heat exchangers for the dairy industry by monitoring the overall heat

transfer, deposit thickness and the critical time (time when cleaning is required). This

model worked irrespective of the type of milk (for either goat or cow milk) used since

network model updated the error continuously [80]. The results from the study show

that fouling was highest at low pH and high temperature. Using such models will

revolutionize the food industry and cut costs. A cost model has been proposed to

optimize the performance of a Stirling engine which encounters fouling in the heat

exchangers [81]. Taking into account the various costs encountered due to fouling the

proposed model for the total costs of fouling can be summarized as follows:

CT T

C C Cf

f c

p c u=+

+ +1

( ) (9.57)

where Cf = total costs due to fouling; Cp = costs due to changes in engine perform-

ance; Cc = costs due to cleaning; Cu = costs due to unavailability of the engine;

Tf = time for fouling to develop; and Tc = time required for cleaning.

The optimum time for fouling to develop was derived by the following rela-

tionship after taking into consideration the power requirements, and the engine

performance:

T TC C

a e b eTf c

u c

fu e

c= +++

−2 2( )

. . (9.58)

where efu = energy price for the fuel used as input for the Stirling engine; and

ee = average price of purchased and sold electricity. This value is weighted by the

change in purchased and sold energy due to fouling at the fouling period Tf.

Since, the value of ee depends on the fouling period, Tf, and Tf in turn depends

on the value ee , Equation 9.58 becomes an optimization problem where the values

of ee and Tf are determined by iteration. Georgiadis et al. [82] have modeled in detail

the cost economics involved during fouling in a dairy plant. Their comprehensive

model was derived after solving a set of integral, partial differential and algebraic

equations. The results of their model indicate that the cost factor due to interruption

of the dairy operation due to fouling (because of cleaning) is predominant, but the

sis of energy conservation and reports on the impact of fouling on energy loss any

modeling efforts in the future needs to address the issue of energy loss and their

resulting cost economics to optimize dairy operation.

55534_C009.indd 25755534_C009.indd 257 10/22/08 10:14:16 AM10/22/08 10:14:16 AM

increase in energy consumption due to fouling is not significant. With high empha-

it measured the heat flux directly from the plate heat exchanger unit and the neural



9.3 CONCLUDING REMARKS

The impact of fouling in the food processing industry is an issue of major concern.

With recent efforts towards energy conservation and energy utilization, controlling

savings. There has been a dearth in comprehensive models that can explain the foul-

ing mechanism in detail. This is because fouling occurs due to various processing

and physico-chemical changes. Development of a comprehensive model starts with

studying the existing models that have been attempted to integrate them. Under-

standing the fouling phenomena will help in optimizing the process conditions, and

timely scheduling of cleaning operations that will cut down costs, and increase per-

advent of frictionless coatings and new surfaces, the issue of controlling fouling has

gained momentum. Though the study of fouling in the food processing industry is

motive, aerospace, chemical, petroleum, and marine industries, where the issue of

fouling has been addressed for a long time.

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263

10 Ozone Treatment of Food Materials

Kasiviswanathan Muthukumarappan, Colm P. O’Donnell, and Patrick J. Cullen

CONTENTS

10.1 Introduction ................................................................................................. 263

10.2 What is Ozone? ...........................................................................................264

10.2.1 Production of Ozone ......................................................................266

10.2.1.1 Electrical (Corona) Discharge Method ..........................266

10.2.1.2 Electrochemical (Cold Plasma) Method ........................ 267

10.2.1.3 Ultraviolet (UV) Method ............................................... 267

10.3 Modeling Ozone in Food Materials ............................................................268

10.3.1 Modeling Ozone Diffusion in Liquid Food ...................................268

10.3.2 Analyzing the Ozone Bubbles in the Column ............................... 270

10.4 Microbial Inactivation of Food Materials ................................................... 270

10.4.1 Application of Ozone in Solid Food Materials ............................. 270

10.4.2 Application of Ozone in Liquid Food Materials ........................... 273

10.4.3 Effects of Ozone on Product Quality ............................................ 273

10.5 Safety Requirements ................................................................................... 275

10.6 Disinfection of Food Processing Equipment and Environment.................. 275

10.7 Limitations of Using Ozone ........................................................................ 276

References .............................................................................................................. 276

10.1 INTRODUCTION

Foodborne illness remains the greatest of all food safety threats, with rapidly increasing

population density throughout the world accompanied by the evolution of new micro-

biological strains including Listeria monocytogenes and virulent strains of Escherchia coli. Consumer preference for minimally processed foods free of chemical preserva-

and the passage of new legislation such as the Food Quality Protection Act in the US

have created demand for novel food processing and preservation systems. Bacterial

pathogens in food cause an estimated 80 million cases of human illness, 325,000 cases

of hospitalization and up to 5,000 deaths annually in the US alone, coupled with sig-

the yearly cost of foodborne diseases in the US is $7–8 billion [2].

55534_C010.indd 26355534_C010.indd 263 10/22/08 10:17:24 AM10/22/08 10:17:24 AM

tives, recent outbreaks of foodborne pathogens, identification of new food pathogens,

nificant economic losses [1]. The Center for Disease Control and Prevention estimates



Concerns over food contamination span the spectrum of foodstuffs from liquid to

solids. Solid foods such as muscle origin exposed to microbial contamination during

slaughter and handling are responsible for causing microbial spoilage and foodborne

illness. Hence, the need for better control of foodborne pathogens has been para-

mount in recent years. It has become obvious that current systems of food produc-

tion contain inadequate bacterial interventions (kill or reduction steps). Today, the

only bacterial interventions for meat and poultry are antibacterial rinses during the

styles more and more households partially rely on ready-to-eat (RTE) or so called

fast foods. In the last decade, food companies in the US have introduced processed

meat products that do not require extensive preparations. Ground meat products con-

stitute a major share of this category of food products. With this change in cooking

food safety is becoming a major concern for such products. These products are at

times given little or no thermal treatments before consumption. Newly emerging

microbiological strains such as L. monocytogenes, virulent strains of E. coli, and

assorted viruses and their involvement in causing human illnesses has prompted a

need to improve the microbiological status of RTE meat products. In the US recent

regulations by the FDA governing fruit juice pasteurization has led to a search for

novel nonthermal processes that could ensure product safety and maintain desired

sensory characteristics. Conversly a lack of such regulation in the European Union

has raised concerns over possible pathogenic outbreaks due to recent trends towards

unpasteurized fruit juice consumption.

From the bacterial group, E. coli O157:H7 and L. monocytogenes are emerg-

ing pathogens whereas long-time recognized, Salmonella is still on the number one

position in terms of bacterial agents causing foodborne illness. Table 10.1 provides

estimates of the annual foodborne illness, hospitalization, and deaths for some of the

most common foodborne pathogens in the US. Methods for inactivating these patho-

gens in food would reduce the likelihood of future foodborne disease outbreaks.

There are several processing methods available for inactivation of microorganisms

safety in solid and liquid food and mathematical modeling in liquid food are empha-

sized in this chapter.

10.2 WHAT IS OZONE?

The passage of new legislation such as the Food Quality Protection Act in the US

has created renewed demand for novel food processing and preservation systems.

Also, the accumulation of toxic chemicals in our environment has increased the

focus on the safe use of sanitizers, bleaching agents, pesticides and other chemicals

in industrial processing [3]. Hence, there is a demand for safe and judicious usage

of these chemicals and preservatives in food processing. Ozone is generally recog-

nized as safe status (GRAS) in the US for use in treatment of bottled water and as a

sanitizer for process trains in bottled water plants [4]. In June 1997, ozone received

the GRAS status as a disinfectant for foods by an independent panel of experts,

55534_C010.indd 26455534_C010.indd 264 10/22/08 10:17:26 AM10/22/08 10:17:26 AM

slaughter process, and the final cooking stage. However, with rapidly changing life-

habits, the RTE foods may not be attaining sufficiently high temperatures and hence,

in foods namely thermal, high pressure, pulsed electric field, oscillating magnetic

field, irradiation, and ozonation. Ozonation treatment of food materials for microbial


Ozone Treatment of Food Materials 265

sponsored by the Electric Power Research Institute. In 2001, the Food and Drug

Administration (FDA) allowed the use of ozone as a direct-contact food-sanitizing

agent [5]. This action eventually cleared the way for the use of ozone in the $430

billion food processing industry [5,6]. The use of ozone in the processing of foods

has recently come to the forefront as a result of the FDA approval to use ozone as an

antimicrobial agent for food treatment, storage and processing. The approval serves

to provide the basis for extended use of ozone in food and agricultural industries

with applications ranging from produce washing, recycling of poultry wash water to

seafood sterilization. Ozone has recently gained the attention of food and agricul-

tural industries though it has been used effectively as a primary disinfectant for the

treatment of municipal and bottled drinking waters for 100 years at scales from a

few gallons per minute to millions of gallons per day. Currently, there are more than

3,000 ozone-based water treatment installations all over the world and more than

300 potable water treatment plants in the US [7].

Ozone is a naturally occurring substance found in our atmosphere and it can

also be produced synthetically. The characteristic fresh, clean smell of air follow-

ing a thunderstorm represents freshly generated ozone in nature. Ozone is a form

of oxygen that contains three atoms (O3) compared to the standard two (O2) in a

molecule of oxygen. Structurally, the three atoms of oxygen are in the form of an

isoscales triangle with an angle of 116.8 degree between the two O–O bonds. The

distance between the bond oxygen atoms is 1.27 Å. The name ‘ozone’ is derived from

the Greek word ‘Ozein’ which means ‘to smell’. Ozone as a gas is blue; both liquid

(−111.9°C at 1 atm) and solid ozone (−192.7°C) are an opaque blue-black color [8].

TABLE 10.1Estimated Annual Food Borne Illnesses, Hospitalization, and Deaths Due to Selected Pathogens, US, 2005

Disease/Agent Illness Hospitalization Deaths

BacterialCampylobacter spp. 1,963,141 10,539 99

Clostridium perfringens 248,520 41 7

Escherichia coli O157:H7 62,458 1,843 52

Listeria monocytogenes 2,493 2,298 499

Salmonella, nontyphoidal 1,341,873 15,608 553

Staphylococcus 185,060 1,753 2

Vibrio cholerae, toxigenic 49 17 0

47 43 18

ParasiticToxoplasma gondii 112,500 2,500 375

and RV Tauxe. 2005. Food-related illness and death in the United States. Emerging Infec-tious Diseases 11 (5): 607–25.

55534_C010.indd 26555534_C010.indd 265 10/22/08 10:17:27 AM10/22/08 10:17:27 AM

Vibrio vulnificus

Source: PS Mead, L Slutsker, V Dietz, LF McCaig, JS Bresee, C Shapiro, PM Griffin,



It is a relatively unstable gas at normal temperatures and pressures, is partially sol-

uble in water, has a characteristic pungent odor, and is the strongest disinfectant

currently available for contact with foods [9–11]. The relatively high ( + 2.075 V)

electrochemical potential (E0, V) indicates that ozone is a very favorable oxidizing

agent (Equation 10.1). The various physical properties of ozone are summarized in

Table 10.2

O (g) 2H 2e O (g) H O{E 2.075V}3 2 20+ + + =+ − ⇔ (10.1)

10.2.1 PRODUCTION OF OZONE

Ozone is generated by the exposure of air or another gas containing normal oxy-

gen to a high-energy source. High-energy sources such as a high voltage electri-

cal discharge or ultraviolet radiation convert molecules of oxygen to molecules of

ozone. Ozone must be manufactured on site for immediate use since it is unstable

and quickly decomposes to normal oxygen. The half-life of ozone in distilled water

at 20°C is about 20–30 min [12]. Ozone production is predominantly achieved by

one of three methods: Electrical discharge methods, electrochemical methods, and

ultraviolet (UV) radiation methods. Electrical discharge methods, which are the

consume large amounts of electricity. The other two methods (electrochemical and

UV) are less cost effective.

10.2.1.1 Electrical (Corona) Discharge Method

In this method, adequately dried air or O2 is passed between two high-voltage elec-

trodes separated by a dielectric material, which is usually glass. Air or concentrated

O2 passing through an ozonator must be free from particulate matter and dried to

a dew point of at least −60°C to properly protect the corona discharge device. The

ozone/gas mixture discharged from the ozonator normally contains from 1 to 3%

ozone when using dry air, and 3–6% ozone when using high purity oxygen as the

feed gas [10,11].

TABLE 10.2Physical Properties of Ozone

Physical Properties Value

Boiling point, °C −111.9

Density, kg/m3 2.14

Heat of formation, kJ/mole 144.7

Melting point, °C −192.7

Molecular weight, g/mole 47.9982

Oxidation strength, V 2.075

Solubility in water, ppm (at 20°C) 3

1.658

55534_C010.indd 26655534_C010.indd 266 10/22/08 10:17:28 AM10/22/08 10:17:28 AM

most widely used commercial methods, have relatively low efficiencies (2–10%) and

Specific gravity



electrodes. When a voltage is supplied to the electrodes, a corona discharge forms

between the two electrodes, and the O2 in the discharge gap is converted to ozone

(Figure 10.1). A corona discharge is a physical phenomenon characterized by a

low-current electrical discharge across a gas-containing gap at a voltage gradient,

which exceeds a certain critical value [13]. First, oxygen molecules (O2) are split into

oxygen atoms (O), and then the individual oxygen atoms combine with remaining

oxygen molecules to form ozone (O3).

Considerable electrical energy (5000 V) is required for the ozone producing

verted to heat that, if not rapidly removed, causes the O3 to decompose into oxygen

atoms and molecules, particularly above 35°C. In order to prevent this decomposi-

tion, ozone generators utilizing the corona discharge method, must be equipped with

a means of cooling the electrodes. The temperature of the gas inside the discharge

chamber must be maintained at a temperature between the temperature necessary

for formation of O3 to occur and the temperature at which spontaneous decomposi-

tion of O3 occurs [14]. The cooling is usually accomplished by circulating a coolant

such as water or air over one surface of the electrodes so that the heat given off by

the discharge is absorbed by the coolant.

10.2.1.2 Electrochemical (Cold Plasma) Method

Usually, in the electrochemical method of ozone production, an electrical current

is applied between an anode and cathode in electrolytic solution containing water

and a solution of highly electronegative anions. A mixture of oxygen and ozone

is produced at the anode. The advantages associated with this method are use of

low-voltage DC current, no feed gas preparation, reduced equipment size, possible

generation of ozone at high concentration, and generation in water.

10.2.1.3 Ultraviolet (UV) Method

In the ultraviolet method of O3 generation, the ozone is formed when O2 is exposed

to UV light of 140–190 mm wavelength, which splits the oxygen molecules into

H

He

Electrode (high tension)

Electrode (low tension)

Dielectric

O2

Discharge gap O3

FIGURE 10.1 Ozone generation by corona discharge method.

55534_C010.indd 26755534_C010.indd 267 10/22/08 10:17:28 AM10/22/08 10:17:28 AM

electrical discharge field to be formed. In excess of 80% of the applied energy is con-

The electrodes are typically either concentric metallic tubes or flat, plate-like



oxygen atoms, which then combine with other oxygen molecules to form O3 [10,11].

The method has been reviewed thoroughly by Langlais, Reckhow and Brink [15].

However, due to poor yields, this method has limited uses.

10.3 MODELING OZONE IN FOOD MATERIALS

10.3.1 MODELING OZONE DIFFUSION IN LIQUID FOOD

for determination of the log reduction and bromate formation in any ozone applica-

tion in liquid such as water treatment, microbial inactivation in fruit juices, etc. For

improvement of operational management of ozonation by model control, the model

and water/fruit juice quality parameters. For this purpose an ozone model can be

developed with gas transfer, slow decay and rapid decay, on the basis of some type

the column.

Assuming a liquid and gas transport without dispersion, neglecting decay of

ozone in the gas phase and using the relations of equilibrium, transfer and decay, the

equations for ozone concentration in liquid and the ozone concentration in gas in a

bubble column (Figure 10.2) are given by:

∂∂

∂∂

αc

tu

c

xk RQ

uu d

k c c k c= − + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − − −Lg b

D g O

63

( ) kk cr

∂∂

∂∂ α

αc

tu

c

xk

dk c cg

gg L

b

D g= − + ⋅ ⋅ ⋅ ⋅ −6

( )

where c = concentration of ozone in liquid (g/m3); t = time (s); x = height of the

bubble column (m); u = velocity of liquid through a reactor (m/s); kL = transfer coef-

g w g

through reactor (m/s); db = bubble diameter (m); α = pressure and temperature cor-

rection factor α = (Pg/P0)·(T0 /Tg); P0, Pg = standard and actual pressure (Pa); T0 ,

g D

ing on nature of gas and temperature; cg = concentration of ozone in gas under stan-

dard pressure and temperature (g/Nm3); kO3

(s−1r

−1).

hand side is for transport of ozone, the second term is for transfer of ozone from the

gas phase to the water phase and vice versa, the third term is for slow decay and the

fourth term is for rapid decay of ozone. The equation for the gas phase consists of

both transport and transfer of ozone.

From experience it is known that direct consumption of ozone is larger when

a model system is considered such as the UV254 absorbance of humic substances

is higher and that during the ozonation process a strong degradation of UV254

55534_C010.indd 26855534_C010.indd 268 10/22/08 10:17:29 AM10/22/08 10:17:29 AM

Predicting the ozone profile in a bubble column and contact chambers is important

must be able to predict the ozone profile for changes in different control parameters

T = standard and actual temperature (K); k = distribution coefficient (−), depend-

= first order constant for slow decay

); and k = first order constant for rapid decay (s

In the equation for the ozone concentration in liquid, the first term on the right

of quality change such as UV254 nm absorbance in the influent and the effluent of

ficient (m/s); RQ = gas to liquid flow ratio (Q /Q ) (−); u = velocity of gas phase



occurs [15]. Therefore one can describe the rapid decay as function of the degrada-

tion of UV254, resulting in the following Equation:

k ckY

rUV

sUV UV= ⋅ −( )

∂

∂∂

∂( ) ( )UV UV

rt

ux

k Y c= − − ⋅ ⋅

where kUV

between UV254 and ozone consumption (l/(mg·m)); UV = UV254 in water (m−1);

and UVs = stable UV254 after completion of the ozonation process (m−1).

It can be assumed that not all UV254 is degraded, but, after ozonation, a stable

UV254 exists. This stable UV254 is incorporated in the equation. As the ozonation

progresses, the rate of ozone decay is shifted from rapid and slow to just slow decay

[16]. An expression given by Hughmark [17] could be used for the gas transfer coef-

L L

density of the bubbles in the column varying from a kL-value for a single bubble to

FIGURE 10.2 Schematic of ozone analysis in a bubble column.

ug

db

h

55534_C010.indd 26955534_C010.indd 269 10/22/08 10:17:30 AM10/22/08 10:17:30 AM

= decay coefficient of UV254 (l/(mg·s)); Y = yield factor, gives the relation

ficient, k -value. This expression gives a range for the k -value depending on the



a kL-value for a swarm of bubbles. The partial differential equations of the ozone

model can be numerically integrated where variations in time and space are fol-

lowed. For solving the equations the number of Complete Stirred Tank Reactors

(CSTRs) can be determined based on standard tracer experiments.

10.3.2 ANALYZING THE OZONE BUBBLES IN THE COLUMN

As with any pasteurization technology employed by the food industry it is essential

to identify, monitor and control critical operational parameters. Given the recent

uptake in the direct ozonation of liquid foods it is paramount to understand how

value in terms of product control and development. Hepworth [18] has developed a

novel application of computer vision for measuring bubble size distributions. The

technique incorporates a computer controlled charge coupled device camera to cap-

ture and save bubble images. The technique is also designed to be simple to use and

relatively portable. The technique allows for the measurement of both bubble diam-

eter and velocity. Data from the experiments have been analyzed to predict rates

of bubble nucleation, growth, and motion for a variety of experimental conditions.

Such understanding would facilitate optimization of control parameters including;

10.4 MICROBIAL INACTIVATION OF FOOD MATERIALS

When a cell becomes stressed by viral, bacterial or fungal attack, its energy level is

the third atom of oxygen which is electrophilic, i.e. ozone has a small free radical

electrical charge in the third atoms of oxygen which seeks to balance itself electrically

with other material with a corresponding unbalanced charge. Diseased cells, viruses,

harmful bacteria and other pathogens carry such a charge and so attract ozone and its

by-products. Normal healthy cells cannot react with ozone or its by-products, as they

possess a balanced electrical charge and a strong enzyme system.

Because of its very high oxidation reduction potential, ozone acts as an oxidant

of the constituent elements of cell walls before penetrating inside microorganisms

and oxidizing certain essential components e.g. unsaturated lipids, enzymes, pro-

teins, nucleic acids, etc. When a large part of the membrane barrier is destroyed

causing leakage of cell contents, the bacterial or protozoan cells lyse (unbind) result-

ing in gradual or immediate destruction of the cell. Most pathogenic and foodborne

microbes are susceptible to this oxidizing effect.

10.4.1 APPLICATION OF OZONE IN SOLID FOOD MATERIALS

Ozone is one of the most potent sanitizers known and is effective against a wide

spectrum of microorganisms at relatively low concentrations [12]. Sensitivity of

microorganisms to ozone depends largely on the medium, the method of appli-

cation, and the species. Susceptibility varies with the physiological state of the

culture, pH of the medium, temperature, humidity and presence of additives, such

55534_C010.indd 27055534_C010.indd 270 10/22/08 10:17:31 AM10/22/08 10:17:31 AM

size distributions for the ozone processing of liquid foods would be of significant

ozone behaves when introduced to fluids. Experimentally determined ozone bubble

diffuser characteristics, ozone flowrates, static mixing, etc.

reduced by the outflow of electrons, and becomes electropositive. Ozone possesses



as, acids, surfactants, and sugars [19]. The antimicrobial spectrum and sanitary

applications of ozone in food industry are summarized in Table 10.3.

poultry carcasses. The microbial counts of ozone treated carcasses stored at 4°C

Gorman, Sofos, Morgan, Schmidt and Smith [21] evaluated the effect of various

sanitizing agents (5% hydrogen peroxide, 0.5% ozone, 12% trisodium phosphate,

2% acetic acid, and 0.3% commercial sanitizer), and water (16–74°C) spray- washing

interventions for their ability to reduce bacterial contamination of beef samples in a

model spray-washing cabinet. Hydrogen peroxide and ozonated water were found to

be more effective than the other sanitizing agents. In another study, the effect of dif-

ferent treatments (74°C hot-water washing, 5% hydrogen peroxide, and 0.5% ozone)

in reducing bacterial populations on beef carcasses was studied and the researchers

have found that water at 74°C caused higher bacterial reduction than those achieved

by the other sanitizing agents [22]. Ozone and hydrogen peroxide treatments had

minor effects and were equivalent to conventional washing in reducing bacterial

populations on beef. Silva da, Gibbs and Kirby [23] investigated the bacterial activity

TABLE 10.3Antimicrobial Spectrum and Sanitary Applications of Ozone in Food Industry

Sanitation Dosage Susceptible Microorganisms

Animal >100 ppm HVJ/TME/Reo type 3/murine hepatitis virus

Black berries 0.3 ppm Botrytis cinerea

Cabbage 7–13 mg/m3 Shelf life extension

Carrot 5–15 mg/m3 Shelf life extension

60 μl/L Botrytis cinerea/Scerotinia sclerotiorum

Dairy 5 ppm

Fish 0.27 mg/L P. putida/B. thermospacta/L. plantarum/Shewanella putrefaciens/Enterobacter sp.

0.111 mg/L Enterococcus seriolicida

0.064 mg/L Pasteurella piscicida/Vibrio anguillarum

Media 3–18 ppm E. coli O157:H7

Peppercorn 6.7 mg/L 3–6 log reduction of microbial load

Potatoes 20–25 mg/m3 Shelf life extension

Poultry 0.2–0.4 ppm Salmonella sp./Enterobacteriaceae

Shrimp 1.4 ml/L E. coli/Salmonella typhimurium

Water 0.35 mg/L A. hydrophila/B. subtilis/E. coli/V. cholerae/P. aeruginosa/L. monocytogenes/Salm. typhi/Staph. aureus/Y. enterocolitica

Source: K Muthukumarappan, F Halaweish, and AS Naidu. 2000. Ozone. In: AS Naidu, ed. Natural food anti-microbial systems. Boca Raton, FL: CRC Press, 783–800.

55534_C010.indd 27155534_C010.indd 271 10/22/08 10:17:32 AM10/22/08 10:17:32 AM

Sheldon and Brown [20] investigated the efficacy of ozone as a disinfectant for

were significantly lower than carcasses chilled under non-ozonated conditions.

of gaseous ozone on five species of fish bacteria and reported that ozone in relatively

Alcaligens faecalus/P. fluorescens



low concentration (< 0.27 × 10 −3 g/l) was an effective bactericide of vegetative cells.

Kaothien, Jhala, Henning, Julson and Muthukumarappan [24] evaluated the effec-

tiveness of ozone in controlling Listeria monocytogenes in cured ham. There was a

centration in range of 0.5−1.0 ppm, with exposure time of 1–15 min at an exposure

temperature of 20°C.

Within the food industry, ozone has been used routinely for washing and storage

of fruits and vegetables [25,26]. Ozone can be used during the washing of produce

before it is packaged and shipped to supermarkets, grocery stores and restaurants.

With a 99.9% kill rate, it’s far more effective than current sanitizing methods, such

as commercial fruit and vegetable washes. Also, processors who chill fruits or veg-

etables after harvest using water held at approximately 1°C can ozonate the water to

prevent product contamination. Cooling fruits and vegetables helps slow product res-

piration, and preserve freshness and quality. Fruit and vegetable processing systems

that incorporate ozone-generating technology will be able to produce cleaner food

while using substantially less water. It will destroy bacteria that can cause premature

spoilage of fruits and vegetables while also ensuring a safer product for consumers

without any toxic residues. The ozone dissipates within minutes following the wash-

ing process. Ozone can also be injected or dissolved in process waters of all kinds to

reduce microbial contamination.

Recent investigations involving the use of ozone for dried foods have shown

that gaseous ozone reduced Bacillus spp. and Micrococcus counts in cereal grains,

peas, beans and whole spices were reduced by up to 3 log units, depending on ozone

concentration, temperature and relative humidity conditions [27,28]. Zhao and Cran-

ston [29] used gaseous ozone as a disinfectant in reducing microbial populations in

ground black pepper, observing a 3–6 log reduction depending on the moisture con-

tent with samples ozonized at 6.7 ppm for 6 h. Furthermore, ozonated water has been

applied to fresh-cut vegetables for sanitation purposes reducing microbial popula-

tions and extending the shelf-life [30,31]. Treatment of apples with ozone resulted in

lower weight loss and spoilage. An increase in the shelf-life of apples and oranges by

ozone has been attributed to the oxidation of ethylene. Fungal deterioration of black-

berries and grapes was decreased by ozonation processing [32]. Ozonated water was

found to reduce bacterial content in shredded lettuce, blackberries, grapes, black

pepper, shrimp, beef, broccoli, carrots, tomatoes and milk [19,29,33–35].

to sterilize bacon, beef, bananas, eggs, mushrooms, cheese and fruit [42,43], to pre-

serve lettuce [19], strawberries [44], green peppers [45] and sprouts [46]. Reduction

in mold and bacterial counts could be achieved without any detrimental change in

chemical composition and sensory quality [47]. Microbial studies typically show

pathogenic species most commonly associated with fruit and vegetable products.

Bubbling of ozone in stored apples inoculated with E. coli O157:H7 was found

to be more effective than dipping apples in ozonated water. Bubbling and dipping

55534_C010.indd 27255534_C010.indd 272 10/22/08 10:17:33 AM10/22/08 10:17:33 AM

significant (p > 0.05) reduction (about 90%) in bacterial population, with ozone con-

Ozone has been used in several studies to decontaminate freshly caught fish [36],

poultry products [20,37], meat and milk products [38,39], to purify and artificially

2 log reduction of total counts and significant reduction of spoilage and potentially

provide chilling, fluming, rinsing or washing of meat, poultry, seafood, and eggs to

age wine and spirits [40], to reduce aflatoxin in peanut and cottonseed meals [41],



resulted in 3.7 log and 2.6 log reductions in counts of E. coli, respectively [48]. About

1.3–3.8 log reduction was reported for inactivation of E. coli at ozone concentration

of 0.3–1.0 ppm (O3 demand-free water) at pH of 5.9 and 1.3 to ∼7 log reduction for

Leuconostoc mesenteroides at similar treatment conditions, whereas 0.2–1.8 ppm of

ozone concentration yielded 0.7 to ∼7 log reduction in L. monocytogenes [49].

10.4.2 APPLICATION OF OZONE IN LIQUID FOOD MATERIALS

Most contemporary applications of ozone include treatment of drinking water [50]

and municipal wastewater [7,51]. Effectiveness of ozone against microorganisms

depends not only on the amount applied, but also on the residual ozone in the medium

and various environmental factors such as medium pH, temperature, humidity, addi-

tives (surfactants, sugars, etc.), and the amount of organic matter surrounding the

that can be detected in the medium after application to the target medium. Both the

instability of ozone under certain conditions and the presence of ozone-consuming

materials affect the level of residual ozone present in the medium. Therefore, it is

important, to distinguish between the concentration of applied ozone and residual

ozone necessary for effective disinfection. It is advisable to monitor ozone avail-

targeted microorganisms are suspended and treated in pure water or simple buffers

how ozone will react in the presence of organic matter [54]. Food components are

reported to interfere with bactericidal properties of ozone against microbes [55].

In apple cider, Dock [56] determined that the mandatory 5-log reduction could be

achieved without harming essential quality attributes. Inactivation of E. coli O157:H7 and Salmonella in apple cider and orange juice treated with ozone in combina-

tion with antimicrobials such as dimethyl dicarbonate (DMDC; 250 or 500 ppm)

or hydrogen peroxide (300 or 600 ppm) was evaluated by Williams, Summer and

in a 5-log colony-forming units (CFU)/mL reduction of either pathogen. However,

in their second study they found that all combinations of antimicrobials plus ozone

treatments, followed by refrigerated storage, caused greater than a 5-log CFU/mL

reduction, except ozone/DMDC (250 ppm) treatment in orange juice. They have con-

cluded that the ozone treatment in combination with DMDC or hydrogen peroxide

followed by refrigerated storage may provide an alternative to thermal pasteurization

to meet the 5-log reduction standard in cider and orange juice. Recently a number

of commercial fruit juice processors in US began employing this ozone process for

pasteurization resulting in industry guidelines being issued by the FDA [58].

10.4.3 EFFECTS OF OZONE ON PRODUCT QUALITY

Applying ozone at doses that are large enough for effective decontamination may

change the sensory qualities of food and food products. The effect of ozone treat-

ment on quality and physiology of various kinds of food products have been evalu-

55534_C010.indd 27355534_C010.indd 273 10/22/08 10:17:34 AM10/22/08 10:17:34 AM

cells [19,52]. It is difficult to predict ozone behavior under such conditions and in

the presence of specific compounds. Residual ozone is the concentration of ozone

ability during treatment [53]. Efficacy of ozone is demonstrated more readily when

(with low ozone demand) than in complex food systems where it is difficult to predict

Golden [57]. In their first study they found no combination of treatments resulted

ated by various researchers. Ozone is not universally beneficial and in some cases



may promote oxidative spoilage in foods [59]. Surface oxidation, discoloration or

development of undesirable odors may occur in substrates such as meat, from exces-

sive use of ozone [12,60].

disinfecting wastewater. Dock [56] reported no detrimental change in quality attributes

of apple cider when it was treated with ozone. However, much research still needs to

be conducted before it can effectively be applied to fruit juice. No change in chemical

composition and sensory quality of onion was reported by Song, Fan, Hilderbrand

total sugar content of celery and strawberries [62] during storage periods. Ozone is

expected to cause the loss of antioxidant constituents, because of its strong oxidizing

phenolic content of fresh-cut iceberg lettuce [31]. Contradictory reports are found in

the literature regarding ascorbic acid, with decomposition of ascorbic acid in broc-

ascorbic acid contents for treated and non-treated celery samples. Moreover, increases

in ascorbic acid levels in spinach [64], pumpkin leaves [65] and strawberries [66] were

reported in response to ozone exposure.

Slight decreases in vitamin C contents were reported in lettuce [31]. Ozone treat-

ments were reported to have minor effects on anthocyanin contents in strawberries

[66] and blackberries [34]. The most notable effect of ozone on sensory quality of

fruits was the loss of aroma. Ozone enriched cold storage of strawberries resulted in

reversible losses of fruit aroma [66,67]. This behavior is probably due to oxidation of

vegetative and spore forms, ozone is unlikely to be used directly in foods containing

high-ozone-demand materials, such as meat products [68]. Applying ozone at doses

that are large enough for effective decontamination may change the sensory quali-

ties of these products. Due to increased concern about the safety of fruit, vegetable

and juice products, the FDA has mandated that these must undergo a 5 log reduction

in pathogens. The effect of ozone treatment on apple cider quality and consumer

acceptability was studied over 21 days. Ozone-treated cider had greater sedimenta-

tion, lower sucrose content and a decrease in soluble solids by day 21 [69].

Recently researchers in Spain evaluated the effects of continuous and intermit-

tent applications of ozone gas treatments, applied during cold storage to maintain

postharvest quality during subsequent shelf life, on the bioactive phenolic composi-

tion of ‘Autumn Seedless’ table grapes after long-term storage and simulated retail

display conditions [70]. They found that the sensory quality was preserved with

both ozone treatments. Although ozone treatment did not completely inhibit fungal

time. Continuous 0.1 μL L−1 O3 application also preserved the total amount of

tent sampled at harvest. Total phenolics increased after the retail period in ozone

treated berries. Therefore the improved techniques tested for retaining the quality

of ‘Autumn Seedless’ table grapes during long-term storage seem to maintain or

even enhance the antioxidant compound content.

55534_C010.indd 27455534_C010.indd 274 10/22/08 10:17:35 AM10/22/08 10:17:35 AM

and Forney [47]. Ozonated water treatment resulted in no significant difference in

activity. However, ozone washing treatment was reported to have no effect on the final

Conversely Zhang, Lu, Yu and Gao [62] reported no significant difference between

the volatile compounds. In spite of its efficacy against microorganisms both in the

Richardson [61] reported that ozone helps to control odor, flavor and color while

coli florets reported after ozone treatment by Lewis, Zhuang, Payne and Barth [63].

development, its application increased the total flavan-3-ol content at any sampling

hydroxycinnamates, while both treatments assayed maintained the flavonol con-



10.5 SAFETY REQUIREMENTS

Ozone is a toxic gas and can cause severe illness, and even death if inhaled in high

quantity. It is one of the high active oxidants with strong toxicity to animals and

plants. Toxicity symptoms such as sharp irritation to the nose and throat could result

instantly at 0.1 ppm dose. Loss of vision could arise from 0.1 to 0.5 ppm after expo-

sure for 3–6 h. Ozone toxicity of 1–2 ppm could cause distinct irritation on the upper

part of throat, headache, pain in the chest, cough and drying of the throat. Higher

levels of ozone (5–10 ppm) could cause increase in pulse, and edema of lungs. Ozone

level of 50 ppm or more is potentially fatal [11]. The ozone exposure levels as recom-

mended by the Occupational Safety and Health Administration (OSHA) of the US

are shown in Table 10.4

10.6 DISINFECTION OF FOOD PROCESSING EQUIPMENT AND ENVIRONMENT

Within the food industry much attention is given to the cleaning and sanitizing

operations of food-processing equipment both in preventing contamination of

products and in maintaining the functionality of equipment [71]. Since ozone is

a strong oxidant, it can be used for the disinfection of processing equipment and

10

units when tested in wineries for barrel cleaning, tank sanitation, and clean-in-

place processes [72].

Water containing low concentrations of ozone can be sprayed onto processing

that may be present. Ozone has been shown to be more effective than chlorine, the

most commonly used disinfectant, in killing bacteria, fungi and viruses, and it does

this at one tenth of the concentration. Ozone can react up to 3000 times faster than

chlorine with organic materials and does not leave any residual toxic by-products.

Currently, ozone is the most likely alternative to chlorine in food applications.

TABLE 10.4Approved Levels of Ozone Application

Exposure Ozone Level, ppm

Detectable odor 0.01–0.05

OSHA 8 h limit 0.1

OSHA 1.5 min limit 0.3

Lethal in few minutes >1700

Source: K Muthukumarappan, JL Julson, and AK Mahapatra.

2002. Ozone applications in food processing. In: SK Nanda,

ed. Souvenir 2002 – Proceedings of College of Agricultural Engineering Technology alumni meeting, Bhubaneswar, India,

published by CAET Alumni Association: 32–35.

55534_C010.indd 27555534_C010.indd 275 10/22/08 10:17:36 AM10/22/08 10:17:36 AM

environments. It has been reported that ozone decreased surface flora by 3 log

equipment, walls or floors to both remove and kill bacteria or other organic matter



10.7 LIMITATIONS OF USING OZONE

As discussed earlier applying ozone at doses that are large enough for effective

decontamination may result in changes in the sensory or nutritional qualities of some

food products including; surface oxidation, discoloration and the development of

undesirable odors. Additionally, microorganisms embedded in product surfaces are

more resistant to ozone than those readily exposed to the sanitizer. Hence, suitable

application methods have to be used to assure direct contact of ozone with target

microorganisms. The rapid reaction and degradation of ozone diminish the residuals

of this sanitizer during processing. The lack of residuals may limit the processor’s

Also, there are existing restrictions relating to human exposure to ozone, which

must be addressed. Plant operators seeking to employ ozone will be faced with system

design and process operation challenges. However, ozone monitors and destructors

may be employed to overcome such challenges. The initial cost of ozone generators

may be of concern to small-scale food processors but as the technology improves the

cost of the generators are coming down.

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281

11 UV Pasteurization of Food Materials

Kathiravan Krishnamurthy, Joseph Irudayaraj, Ali Demirci, and Wade Yang

CONTENTS

11.1 Introduction ................................................................................................. 281

11.2 UV and Pulsed UV Light Processing .........................................................282

11.2.1 Interaction of Light and Matter .....................................................282

11.2.2 Pulsed UV Light ............................................................................284

11.2.3 UV Light and Pulsed UV Light Inactivation Mechanisms ...........285

11.2.4 Selected Inactivation Studies by UV Light and

Pulsed UV Light ............................................................................289

11.2.5 Inactivation Modeling ................................................................... 291

11.2.6 Other Applications of UV Light and Pulsed UV Light ................ 293

11.2.7 Effect of UV Light on Food Components and Quality ................. 293

11.2.8 Economics of UV Light Disinfection System ............................... 295

11.2.9 Challenges in the Application of UV Light and Pulsed UV

Light and Future Research Needs ................................................. 295

11.3 Conclusions ................................................................................................. 298

References ..............................................................................................................299

11.1 INTRODUCTION

Consumption of food contaminated with pathogenic microorganisms cause illnesses

and deaths resulting in several billion dollars losses. Because of rigorous governmen-

tal regulations and potential risk of costly recalls, the food industry has been forced

to ensure that their food products are free from pathogenic microorganisms. Further-

more, increased consumer awareness about minimally processed foods and industries’

thirst to reduce the total cost of food processing, propels researchers to investigate the

material while preserving the quality. Owing to the increased consumer demand for

wholesome and fresh-like products, application of novel food processing technologies

UV light have been investigated. UV light and pulsed UV light are two such methods,

which are already approved by the Food and Drug Administration (FDA), for reduction

of pathogens in different food products [1]. Pulsed UV light encompasses energy from

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efficacy of alternative food processing technologies to effectively pasteurize the food

such as pulsed electric field, high pressure processing, ultraviolet (UV) light, pulsed



Ultraviolet, visible and infrared light regions. However, majority of the energy comes

from the UV region. Therefore in this chapter, pasteurization by both UV light and

pulsed UV light will be presented.

UV light has been used as a bactericidal agent from 1928 [2]. UV light is divided

into the following four regions according to their wavelength: vacuum UV (100–

200 nm), UV-C (200–280 nm), UV-B (280–315 nm), and UV-A (315–400 nm) [3].

UV light disinfection is rather one of the widely studied applications. In addition to

inactivating the microorganisms in food, UV light can also increase the Vitamin D

content of the food. UV light had been used for enriching the vitamin D content of

milk a few decades ago.

Pulsed UV light is an emerging technology, wherein the energy is multiplied

several folds by storing the energy in a capacitor and releasing it as a short duration

production of a broadband spectrum ranging from UV to infrared. For the same total

energy, pulsed UV light is four to six times more effective than the conventional UV

light in terms of pathogen inactivation, as suggested by several researchers [28].

Though the total energy is the same, instantaneous energy of pulsed UV light is

several thousand times higher than the conventional UV light due to the very short

pulse duration (in terms of few nanoseconds to microseconds). Pulsed UV light has

been proven to be very effective in inactivating various microorganisms present in

different food products.

11.2 UV AND PULSED UV LIGHT PROCESSING

11.2.1 INTERACTION OF LIGHT AND MATTER

Light consists of discrete fundamental packets of energy known as photons, which

contains energy based on the wavelength of light (Equation 11.1).

E hhc

= =υλ

(11.1)

where, E is the energy of photon, h is the Planck’s constant (6.626 × 10−34 J s), υ the

frequency of light, c is the speed of light in vacuum, and λ is the wavelength of light.

The typical quantum energy of photons is given for the region of pulsed UV light

in Table 11.1. The photons in the UV region have higher energy than visible light

followed by infrared region (Table 11.1). Therefore, photons in the UV region may

account for the predominant inactivation of pathogens. The temperature increase

due to the infrared region is much higher than visible and UV light regions and thus

temperature build-up in pulsed UV light processing can be attributed to the contri-

bution from infrared region.

A major limiting factor for both UV light and pulsed UV light is poor penetration

capacity. This limits the application of these technologies to i) surface sterilization,

ii) clear liquid foods, and iii) thin layers of food materials, to be effective. When

light of initial intensity (I0) falls on a food surface, only a portion of the light is actu-

55534_C011.indd 28255534_C011.indd 282 10/22/08 10:18:55 AM10/22/08 10:18:55 AM

intermittent pulses in a lamp filled with inert gases such as xenon. This leads to the

ally absorbed by the food material, whereas the rest of the energy is reflected back,


UV Pasteurization of Food Materials 283

transmitted, and/or scattered. Intensity of the light decays as it penetrates through

the food material, along a distance of x beneath the food surface as follows [5]

I TI e x= −0 (11.2)

light at a distance x from the surface, I0 is the initial intensity of the light, and x is

the distance below the food surface. During absorption, some amount of light is dis-

sipated as heat and transferred to the inner layers through conduction [5], whereas

the rest can be absorbed by food molecules and microorganisms which may in turn

cause some chemical/physical changes. As the intensity of UV light exponentially

decays within the food material, it is more effective for surface sterilization and

sterilization of highly transparent liquids such as water. However, pulsed UV light,

owing to its high energy and wavelength make-up, can have increased penetration

capacity.

Less transparent foods have to be treated in a thin layer to overcome the penetra-

tion limitation. Futhermore, good mixing can aid in uniform exposure. UV light is

absorbed and penetrates into the microorganism depending upon the chemical com-

position, size of the microorganism, wavelength of interest, and medium of introduc-

tion etc. (Table 11.2). For instance, increase in the size of the microorganism results

in decreased transmission at lower UV wavelengths (Table 11.2). As indicated earlier

(Table 11.1), photons from UV range have high energy. Table 11.3 lists the chemical

bond energy of some common chemical bonds, which corresponds to the energy of

break most of these chemical bonds. Hence, UV light can cause cleavage in organic

TABLE 11.1Characteristics of UV, Visible, and Infrared Regions of Electromagnetic Spectrum

RegionWavelength

(nm)Frequency

(Hz)Photon Energy

(eV)Molar Photon

Energy (kJ/mol)

Vacuum UV 100–200 3.00 × 1016–3.00 × 1015 124–12.4 11975–1197

UV-C 200–280 3.00 × 1015–1.07 × 1015 12.40–4.43 1197–427

UV-B 280–315 1.07 × 1015–9.52 × 1014 4.43–3.94 427–380

UV-A 315–400 9.52 × 1014–7.49 × 1014 3.94–3.10 380–299

Visible light 400–700 7.49 × 1014–4.28 × 1014 3.10–1.77 299–171

Near infrared 700–1400 4.28 × 1014–2.14 × 1014 1.77–0.89 171–85.5

Mid infrared 1400–3000 2.14 × 1014–9.99 × 1013 0.89–0.41 85.5–39.9

Far infrared 3000–10000 9.99 × 1013–3.00 × 1013 0.41–0.12 39.9–12.0

Source: K Krishnamurthy. 2006. Decontamination of milk and water by pulsed UV light and infrared heat-

ing. PhD dissertation, Pennsylvania State University, University Park, PA.

55534_C011.indd 28355534_C011.indd 283 10/22/08 10:18:56 AM10/22/08 10:18:56 AM

where T is the transparency coefficient of the food material, I is the intensity of the

photons in the UV range. Therefore, it is clear that UV light has sufficient energy to



compounds. Furthermore, as UV light has the energy in the magnitude of covalent

bond energy, it mainly breaks the covalent bonds of the target material [4].

As the energy of photons in UV light range is high, they can even cause ioniza-

tion of molecules whereas visible light and infrared causes vibration and rotation of

molecules, respectively. When the molecules absorb the energy, they are elevated

to an excited state. These excited molecules can (i) relax back to the ground state

by releasing the energy as heat; (ii) relax back to the ground state by releasing the

energy as photons, or (iii) can induce some chemical changes [4]. Because of these

mechanisms, pulsed UV light can cause chemical changes and changes due to tem-

perature build-up in the microorganism to varying degrees.

11.2.2 PULSED UV LIGHT

Pulsed UV light is also referred as pulsed light, high intensity light, UV light, broad-

spectrum white light, pulsed white light, and near infrared light [8]. For pulsed UV

light generation, the electrical energy is stored in a capacitor over a short period of

time (few milliseconds) and released as very short period pulses (several nanosec-

krypton), causing ionization of gas and the production of a broad spectrum of light

in the wavelength region of UV to near infrared. Typically the pulse rate is 1–20

pulses per second and the pulse width is 300 ns to 1 ms. Pulsed UV light has very

high energy as evident by the fact that the intensity of pulsed light is 20,000 times

more than that of sunlight [9]. Though the total energy of pulsed UV light can be

comparable to that of continuous UV light, the instantaneous energy is multiplied

several thousand times due to its short pulse width. Due to its increased energy,

pulsed UV light treatment is more effective than continuous UV light treatment for

rapid inactivation of microorganisms [9].

Pulsed light is a broad spectrum radiation from UV light to infrared radiation,

with a typical wavelength range of 100–1100 nm. In a typical pulsed UV light

system, the majority of the energy is produced from the UV light portion. For

instance, a commercial Steripulse-XL® pulsed UV light system produces approxi-

mately 54, 26, and 20% energy from UV, visible, and infrared regions, respectively

[10]. Furthermore, the UV portion of the pulsed UV light has higher energy level

than visible light and infrared region (Table 11.1). Due to its high energy level, UV

TABLE 11.2Percent Transmission to the Center of Selected Cells and Viruses

Biological Sample Diameter (μm)

Percent Transmission at Selected Wavelengths (%)

200 nm 250 nm 300 nm 350 nm

Virus (herpes simplex) 0.15 66 80 100 100

Bacteria 1 33 78 98 100

Yeast 5 1.6 69 97 100

Source: TP Coolhill. 2003. Action spectroscopy: Ultraviolet radiation. In: WM Horspool and F Lenci, eds.

CRC Handbook of Organic Photochemistry and Photobiology. pp. 113.3 Boca Ration: CRC Press.

55534_C011.indd 28455534_C011.indd 284 10/22/08 10:19:00 AM10/22/08 10:19:00 AM

onds to microseconds) and transferred through a lamp filled with inert gas (xenon or



light can ionize the molecules, whereas visible light results in vibration of mol-

ecules and infrared in rotation of molecules.

Previous research shows that pulsed UV light is four to six times more effective

than the continuous UV light [11,28]. Pulsed UV light is gaining attention in recent

sterilization with no toxic by-products [1]. It can be effectively used to inactivate

pathogens on the surface of food or packaging materials. Furthermore, it can also

be used for in-package sterilization if a packaging material can allow UV light to

penetrate [12].

11.2.3 UV LIGHT AND PULSED UV LIGHT INACTIVATION MECHANISMS

UV light exhibits germicidal properties from 100 to 280 nm (UV-C region). The

inactivation occurring between 254 and 264 nm (Figure 11.1). As a typical conven-

tional mercury UV lamp produces UV light at 254 nm, this wavelength is often used

TABLE 11.3Strength of Common Bonds in Biomolecules

Chemical Bond Type WavelengthBond Dissociation Energy (kJ/mole)

N≡N 129 930

C≡C 147 816

C=O 168 712

C=N 195 615

C=C 196 611

P=O 238 502

O–H 259 461

H–H 275 435

P–O 286 419

C–H 289 414

N–H 308 389

C–O 340 352

C–C 344 348

S–H 353 339

C–N 408 293

C–S 460 260

N–O 539 222

S–S 559 214

Source: DL Nelson, and MM Cox. 2001. Lehninger Principles of Biochemistry. pp. 11 New York: Worth Publishers.

55534_C011.indd 28555534_C011.indd 285 10/22/08 10:19:01 AM10/22/08 10:19:01 AM

years because it can provide sufficient antimicrobial inactivation and commercial

inactivation efficiency of UV light follows a bell shaped curve with the maximum

for comparison of the disinfection efficiencies of conventional UV lamps.



As only the absorbed UV light energy induces photophysical, photochemical,

and/or photothermal effects necessary for inactivation of pathogenic microorganisms,

it is crucial to have a proper lamp design to enhance the absorption of energy from

the germicidal range. Among the constituents of the microorganisms, DNA base

pairs readily absorb UV light because of their aromatic ring structure. In general,

pyrimidines (thymine (DNA), cytosine (DNA and RNA) and uracil (RNA)) are

strong absorbers of photons in the UV range, leading to changes in the chemical

structure, resulting in bacterial inactivation [13].

The main inactivation mechanism for UV light is the formation of thymine dimers

in bacterial DNA. Upon formation of the dimers, bacterial DNA cannot be unzipped

for replication and thus cannot reproduce [14,15]. Though cyclobutyl pyrimidine dimer

formation is the main inactivation mechanism, there are other photoproducts formed

during UV light processing including pyrimidine pyrimidinone-[6-4]- photoproduct,

Dewar pyrimidinone, adenine–thymine heterodimer, cytosine photohydrate, thym-

ine photohydrates, single strand break, and DNA-protein crosslink (Table 11.4).

Approximately 77 and 78% of the photoproducts produced by UV-C and UV-B

radiation, respectively are cyclobutyl pyrimidine dimers (Table 11.4). Pyrimidine

pyrimidinone-[6-4]- photoproduct is the next major photoproduct, as they contribute

to approximately 20 and 10% of the total photoproducts formed by UV-C and UV-B,

respectively (Table 11.4). Formation of these photoproducts depends on the wavelength,

DNA sequence, and protein–DNA interactions [16]. The major photochemical changes

that occur in DNA upon UV light exposure includes: DNA chain breakage, cross-link-

ing of strands, hydration of pyrimidines, and formation of dimers between adjacent

residues in the polynucleotide chain [17]. Aromatic amino acids such as phenylalanine

and tryptophane also absorb UV light effectively [18] and thus denature these amino

acids present in the microorganisms.

FIGURE 11.1 Light in Water and Wastewater Sanitation. Boca Raton: W.H. Freeman Publishers.)

% act.

100

80

60

40

20

220 260 300 nm

Rela

tive

bacte

ricid

al effect

254

55534_C011.indd 28655534_C011.indd 286 10/22/08 10:19:01 AM10/22/08 10:19:01 AM

Germicidal efficiency of UV light. (From WJ Masschelein. 2002. Ultraviolet



Though UV light can damage the microorganisms, some of them can repair

themselves by photo-reactivation or dark repair. As the name indicates, photo-

reactivation requires the presence of light, while dark repair is a light-independent

process [19]. Photo-reactivation occurs in the wavelength range of 330–480 nm

because of the activation of DNA photolyase. DNA photolyases splits the thym-

ine dimers which the were formed due to UV light exposure [20] and thus the

microorganism can start replicating again. The photo-reactivated microorganisms

are much more resistant to UV light and thus require higher dose for inactivation

[21–23]. As can be seen from Table 11.5, cells require more energy for inactivation

after being treated once and reactivated. Therefore, higher levels of UV doses are

needed for complete inactivation of pathogenic microorganisms by damaging the

cells beyond repair by photo-reactivation and dark repair.

In addition to the germicidal UV-C portion of the UV light, UV-A (315–400

nm) damages the membrane by the production of active oxygen species and H2O2

[24]. However, UV-A has very little impact on microbial cells unless exogenous

photosensitizers are used with the UV treatment and absorbed by the bacterial

cell [25].

Due to structural differences, spores respond differently to UV light. Riesenman

and Nicholson reported that the resistance in Bacillus subtilis spores was induced by

the thick protein coat [26]. Furthermore, the DNA of a bacterial spore has a different

conformation than the DNA of the vegetative cell. Unlike vegetative cells, Bacillus spores did not produce any detectable amount of thymine containing dimers [27].

The predominant photoproduct produced in spores is 5-thyminyl-5,6 dihydrothym-

ine adduct (also called as “spore photo-product”).

As pulsed UV light has energy from UV, visible, and infrared regions, energy

from all the three regions contribute towards the inactivation. However, inactiva-

tion is expected to be predominately caused by the UV light portion of the broad

TABLE 11.4Photoproducts Produced in DNA because of Absorption of UV Light

Photoproduct

Percentage of Total Photoproducts

UV-C UV-B

Cyclobutyl pyrimidine dimer 77 78

Pyrimidine pyrimidinone[6–4]-photoproduct 20 10

Dewar pyrimidinone 0.8 10

Adenine–thymine heterodimer 0.2 –

Cytosine photohydrate −2.0 −2.0

Thymine photohydrates n/a n/a

Single strand break <0.1 <0.1

DNA-protein cross link <0.1 <0.1

Source: DL Mitchell. 2003. DNA damage and repair. In: WM Horspool, F Lenci, eds. CRC Handbook of Organic Photochemistry and Photobiology. pp. 140.4 Boca Raton: CRC Press.

55534_C011.indd 28755534_C011.indd 287 10/22/08 10:19:02 AM10/22/08 10:19:02 AM



spectrum. Krishnamurthy examined the pulsed UV light treated cells with transmis-

sion electron microscopy and Fourier transform spectroscopy [4]. The author reported

that pulsed UV light induced cell wall breakage, cytoplasm leakage, damage in the

cellular membrane structure, and leakage of the cellular content in Staphylococcus aureus. The temperature increase during the treatment was negligible (increase of

2–3°C) as the cells were treated for only 5 sec with a Xenon-Steripulse-XL® unit.

Therefore, pulsed UV light might have some shocking effect on the cell wall/cyto-

plasmic membrane of bacteria [4] as the effect from the temperature increase was

negligible and photochemical transformation does not lead to physical damage to

the cells. The author proposed that the constant disturbance caused to the bacteria

by exposing it to a repeated cycle of short duration high intensity pulses resulted

in damages to cell wall and cytoplasmic membrane. It is also hypothesized that

pulsed UV light exposure can lead to thermal stress on bacterial cell especially at 2

ria is induced by the differences in the heating and cooling rates of bacteria and the

surrounding matrix [28]. Bacteria can also be overheated due to the differences in

the absorption characteristics of the bacteria and the surrounding medium. Due to

overheating, bacteria become a local vaporization center and may generate a small

Fukunaga, Isobe, Arihara, and Itoh investigated the mechanisms of damage induced

in yeast cells by pulsed light and continuous UV light [29]. The authors reported

that the DNA damage induced by continuous UV light was slightly higher than that

TABLE 11.5UV Light Exposure (at 254 nm) Required for 4-log10 Reduction of Pathogens in Drinking Water

MicroorganismExposure Required without

Reactivation (J/cm2)Exposure Required with

Reactivation (J/cm2)

Citrobacter freundii 80 250

Enterobacter cloacae 100 330

Enterocolitica faecium 170 200

Escherichia coli ATCC 11229 100 280

Escherichia coli ATCC 23958 50 200

Klebsiella pneumoniae 110 310

Mycobacerium smegmatis 200 270

Pseudomonas aeruginosa 110 190

Salmonella typhi 140 190

Salmonella typhimurium 130 250

Serratia marcescens 130 300

Vibrio cholerae wild isolate 50 210

Yersinia enterocolitica 100 320

Source: Hoyer. 1998. Testing performance and monitoring of UV systems for drinking water disinfec-

tion. Water Supply. 16: 424–29.

55534_C011.indd 28855534_C011.indd 288 10/22/08 10:19:03 AM10/22/08 10:19:03 AM

higher flux densities (0.5 J/cm ), leading to cell rupture. Localized heating of bacte-

steam flow causing membrane destruction [29]. Takeshita, Shibato, Sameshima,



of pulsed light. Protein elution because of pulsed UV light was also higher than

that resulting from continuous UV light, suggesting possible leakage of the cellular

contents. Wekhof suggested that the inactivation mechanism of pulsed UV light

includes both the germicidal action of UV-C light and rupture of microorganism

because of thermal stress caused by other UV components [30].

Therefore, the inactivation mechanism of pulsed UV light [4] can be catego-

rized into:

a. Photo-chemical effect: Thymine dimer formation and other photo-

chemical changes in DNA.

b. Photo-thermal effect: Damage caused to the bacterial cell because

of the differences in the heating rates of bacteria and the surrounding

media resulting in localized heating of bacterial cell.

c. Photo-physical effect: Structural damage to bacterial cells caused due

to the disturbances of intermittent high energy pulses.

11.2.4 SELECTED INACTIVATION STUDIES BY UV LIGHT AND PULSED UV LIGHT

Inactivation of microorganisms by UV light had been extensively studied for several dec-

ades, especially for water disinfection. Chang, Ossoff, Lobe, Dorfman, Dumais, Qualls,

S. typhi, Shigella sonnei, Streptococcus fecalis, and S. aureus. The bacterial cells were

resuspended in sterile buffer water and the aggregated groups of bacteria were removed

UV light [31]. E. coli, S. aureus, S. sonnei, and S. tyhi exhibited similar resistance to the

UV light and a 3 log10 reduction was obtained with approximately 7 × 10−3 J/cm2 energy.

However, the resistance exhibited by S. fecalis was higher and required a 1.4-times

higher dose than the above-mentioned microorganisms to obtain a 3 log10 reduction

of inactivation. Stermer, Lasater-Smith, and Brasington also investigated the effect of

UV light on inactivation of bacteria on lamb meat. A 3 log10 reduction of the naturally

cus spp.) was obtained with approximately 4 × 10−3 J/cm2 energy [32].

UV light was also used for inhibition of pathogens on the surface of fresh produce

[33]. The surfaces of red delicious apples, leaf lettuce, and tomatoes were inoculated

with Salmonella spp. or E. coli O157:H7 and treated with UV-C light at a wavelength

of 253.7 nm with different doses ranging from 1.5 to 24 × 10−3 W/cm2. A 3.3 log10

CFU/apple reduction was obtained for E. coli O157:H7 at 24 × 10−3 W/cm2, whereas,

a 2.19 log10 CFU/tomato reduction was obtained for E. coli O157:H7 at 24 × 10−3

W/cm2. Similarly, lettuce inoculated with Salmonella spp. and E. coli O157:H7

resulted in 2.65 and 2.79 log10

of orange juice was treated with UV light at 214.2 W/m2 in order to increase the

shelf life twice [34]. UV light was also successfully used for inactivation of Listeria monocytogenes in goat’s milk. More than 5 log reduction was obtained when the

milk received cumulative energy dose of 15.8 ± 1.6 mJ/cm2 [35]. Wright, Sumner,

Hackney, Pierson, and Zoecklein used UV light for reducing E. coli O157:H7 popu-

lation. Average UV doses of 10.288, 14.713, and 61.005 μW-s/cm2 resulted in average

reductions of 3.1, 3.0, and 5.4 log10 CFU/ml, respectively [36].

55534_C011.indd 28955534_C011.indd 289 10/22/08 10:19:04 AM10/22/08 10:19:04 AM

and Johnson investigated the efficacy of UV light on inactivation of Escherichia coli,

using a 1.0-μm nucleopore polycarbonate membrane and the filtrate was treated with

CFU/lettuce reductions, respectively [33]. A thin film

occurring microfloras of lamb (mostly Pseudomonas, Micrococcus, and Staphylococ-



Due to the differences in the energy and the wavelength range, pulsed UV light

may behave differently than conventional UV light. Therefore, the effectiveness of

a continuous UV light source and a pulsed UV light source for the decontamina-

tion of the surfaces were compared. An almost identical level of inactivation of B. subtillis with 4 × 10−3 J/cm2 of pulsed UV light source and 8 × 10−3 J/cm2 continu-

ous UV light source was reported [11]. For pulsed UV light, the distance from the

lamp source, treatment time, and sample depth are the main factors determining

lamp, lesser volume, and longer treatment times result in increased inactivation.

Sonenshein reported that three pulses (1 sec) of pulsed UV light resulted in more

than 6.5 log10 CFU/ml reduction of Bacillus subtilis spores when the samples were

directly placed at the lamp axis and at the midpoint of the lamp [37]. Complete

inactivation of S. aureus in phosphate buffer was obtained with a 5 sec pulsed UV

light treatment. The reduction corresponds to 7.50 log10 CFU/ml. In case of agar

seeded S. aureus cells, a 5 sec treatment resulted in complete inactivation, yielding

a reduction of approximately 7.50 log10 CFU/ml. The authors also noted that the

temperature increase during this treatment was negligible [38].

for up to 180 sec for various distances from the light source and sample volumes.

The reduction obtained varied from 0.16 to 8.55 log10 CFU/ml, demonstrating the

ability of pulsed UV light to inactivate S. aureus in opaque food product. Complete

inactivation was obtained at (i) 8 cm sample distance from quartz window, 30 ml

sample volume, and 180 sec time combination; and (ii) 10.5 cm sample distance

from quartz window, 12 ml sample volume, and 180 sec treatment time combination

in milk. Milk was treated at 5, 8, or 11 cm distance from UV light strobe at 20,

10 reductions varied from 0.55 to 7.26 log10 CFU/ml. Complete inac-

tivation was obtained in two cases: (i) 8 cm sample distance from quartz window,

30 ml sample volume, and 180 sec treatment time combination; and (ii) 10.5 cm

sample distance from quartz window, 12 ml sample volume, and 180 sec treatment

time combination. Following further enrichment, growth was not observed in most

of the cases [39].

Alfalfa seeds inoculated with E. coli O157:H7 were subjected to pulsed UV light

[40]. The authors obtained reductions of 0.07–4.89 log10 CFU/g at different conditions

(treatment time, distance from the lamp source, and thickness of seed layer). Seeds

treated at different distances from the UV lamp had germination rate over 81% for

up to 60 sec treatment relative to a germination rate of 86% for untreated seeds. This

clearly indicates that pulsed UV light treatment did not reduce the seed viability. The

that about 1 log reduction of E. coli O157:H7 or L. monocytogenes can be achieved

within 60 sec at 8 cm distance from the lamp. This study indicates the potential of

pulsed UV light technology for surface decontamination of muscle foods.

55534_C011.indd 29055534_C011.indd 290 10/22/08 10:19:04 AM10/22/08 10:19:04 AM

the efficacy of the treatment. Therefore, several researchers have studied the effect

of these factors. Though it is product specific, in general shorter distance from the

Milk, artificially inoculated with S. aureus cells were pulsed UV light treated

efficiency. Log

efficacy of pulsed UV light for inactivation of E. coli O157:H7 and Listeria monocy-togenes Scott A on salmon fillets was also investigated [41]. The authors demonstrated

[4]. A continuous flow-through system was developed for inactivation of S. aureus

30, or 40 ml/min flow rate and treated up to three times by recirculation of milk

to determine the effect of number of passes, distance, and flow rate on inactivation



Pulsed UV light can also be used for effective inactivation of spores. A 3-min

pulsed UV light treatment of honey inoculated with Clostridium sporogenes spores

resulted in approximately 89.4% reduction [42]. The authors attributed the low reduc-

tion to poor penetration capacity of pulsed UV light as the honey is really opaque

and viscous. They have also suggested that the heat build-up during pulsed UV

light treatment did not provide any synergistic effect on the inactivation of C. spo-rogenes. Pulsed UV light was also used for inactivation of Aspergillus niger spores

in corn meal [43]. The authors validated the following parameters: processing time

(20–100 sec), voltage input (2000–3800 V), and distance from UV lamp (3–13 cm).

The energy output ranged from 1.8 to 5.7 J/cm2 per pulse at 1.8 cm below the lamp

surface when the voltage was varied from 2000 to 3800 V. The optimal treatment

condition (treatment time: 50 sec, sample distance: 8 cm from the UV lamp, and

input voltage: 3800 V) resulted in a 3.12 log10 reduction of A. niger spores.

11.2.5 INACTIVATION MODELING

a shoulder and/or a tail [47]. Shoulder effect can be attributed to delayed response of a

microorganism to UV light due to injury [44,47]. Photo-reactivation, dark repair, resist-

be attributed to shielding of external particles, clumping of bacteria, and resistant micro-

N N k I t= −0

* *exp ( ) (11.3)

where N = concentration of viable microorganisms after UV light treatment (CFU/ml);

N0 = concentration of viable microorganisms before UV light treatment (CFU/ml); 2

(J/cm2); and t = treatment time (sec). Equation 11.3 can also be represented as [44]:

N N kD

D

D= =−−

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

0 10 10exp( ) (11.4)

210 = UV dose

required to achieve 90% reduction in microbial population.

Therefore,

logNN

DD

0

10

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟= (11.5)

The D10 value of a microorganism evaluates the extent of the resistance to UV light.

Higher D10 values indicate that the microorganism is very resistant to UV light and

requires more energy for inactivation.

From Equation 11.4,

NN

kD

0

= −exp( ) (11.6)

55534_C011.indd 29155534_C011.indd 291 10/22/08 10:19:05 AM10/22/08 10:19:05 AM

can be represented by first-order kinetics [44–46], resulting in a sigmoidal curve with

organisms. The first order inactivation equation [46] can be represented as follows:

k = first order inactivation coefficient (cm /J); I = intensity of UV light energy applied

The dose–response curves for dispersed or free-floating microorganisms by UV light

ance of bacteria are some of the factors influencing the shoulder effect. Tailing effect can

where D = I*t = UV dose delivered or fluence rate (J/cm ); and D



Therefore,

log *ln( ) .

NN

kDk0 1

10 2 303

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟= =

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟⎟* D (11.7)

where log(N0 / N) is the log10 reduction of microbial population and (K / 2.303) is the

10 reduction versus available UV

dosage.

When the microorganism exhibits a shoulder effect, Equation 11.6 can be modi-

N N kD d= − − −0 1 1( ( exp )( ) (11.8)

where d is the intercept of the exponential phase of the dose–response curve with the

effect as follows:

N N NkD k D= +− −0e ep

p( ) ( ) (11.9)

where N0 is the concentration of dispersed microorganisms present; Np is the concen-

tration of particles containing microorganisms; and kp is the inactivation constant for

microorganisms associated with particles [44].

Inactivation models developed for UV light can also be utilized for pulsed UV

light modeling as majority of the energy comes from UV light (typically over 50%

pulsed UV light inactivation modeling. However, due to the contribution of visible

tion in the model. Therefore, researchers have used other models such as the Weibull

equation for describing the inactivation kinetics of pulsed UV light treatment. Bialka,

Demirci, and Puri successfully used the Weibull equation (Equation 11.10) to esti-

mate the microbial inactivation as a function of treatment time, depth of the sample,

and dose during pulsed UV light treatment [48]. The authors obtained R2 values of

0.91, 0.92, 0.98, and 0.96 for estimation of the reduction in the population of E. coli O157:H7 on raspberry, Salmonella on raspberry, E. coli O157:H7 on strawberry,

and Salmonella on strawberry, respectively. The corresponding root mean square

error values were 0.23, 0.06, 0.06, and 0.02, respectively, indicating that the Weibull

model was able to estimate the reduction in the microbial population due to pulsed

UV light.

log.

10

0

1

2 303

NN

t⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

= −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟α

β

(11.10)

where N = number of microorganisms after dose D (CFU/g); N0 = initial number of

microorganisms (CFU/g); t = treatment time (sec or min); α = characteristic time (sec

or min); and β = shape parameter (unitless).

55534_C011.indd 29255534_C011.indd 292 10/22/08 10:19:06 AM10/22/08 10:19:06 AM

slope of the fitted straight line for the plot of log

fied as

y-axis [44]. Similarly, Equation 11.6 can be modified to take into account the tailing

of the total energy). Therefore, the first order kinetics (Equation 11.4) can be used for

light and infrared heating, first order kinetics may not be able to explain all the varia-



The β parameter describes the concavity of the survival curves. A value of β < 1

indicates upward concavity and β > 1 indicates downward concavity.

D value can be estimated by the Weibull model as follows [48]:

D = α( . )2 30 (11.11)

where α = characteristic time (sec or min); and β = shape parameter (unitless).

11.2.6 OTHER APPLICATIONS OF UV LIGHT AND PULSED UV LIGHT

In the wavelength region of 280–310 nm, ergosterol (provitamin) changes into natural

vitamin D3 and hence enriches the food product with vitamin D. UV light exposure

was used as the primary method for vitamin D enrichment in milk several decades

back until vitamin D production became cheaper. UV light enrichment of other prod-

ucts can also be achieved by UV light exposure. Jasinghe and Perera successfully

2

abalone mushrooms were treated with UV-A (315–400 nm), UV-B (290–315 nm),

and UV-C (190–290 nm) for a period of up to 1 h on each side of the mushrooms. The

vitamin D2 content ranged from 22.9 ± 2.7 to 184.0 ± 5.7 μg/g of dry matter, for vari-

ous mushroom and UV light combinations. The authors noted that even treatment of

5 g of shitake mushrooms for 15 min with UV-A or UV-B can provide more than the

recommended allowance of vitamin D for adults (10 μg/day). UV light can also inac-

tivate toxins. Yousef and Marth reported that reductions of 3.6–100% of alfatoxin M1

was achieved in milk with 2–60 min UV light treatment [50].

Pulsed UV light was also used for inactivation of allergens in soybean and

peanuts. Chung, Yang, and Krishnamurthy reported that the allergenicity of the

liquid peanut butter was upto reduced seven folds after pulsed UV light treatment

[51]. They indicated that pulsed UV light treatment effectively inactivated two major

peanut allergens Ara h 1 and Ara h 3. Further optimization of the treatment may

result in potential development of hypoallergenic peanut-based products or bever-

ages such as a smoothie where liquid peanut butter with reduced allergenicity may

be blended with fruit juices. This also opens another avenue for inactivation of other

food allergens.

11.2.7 EFFECT OF UV LIGHT ON FOOD COMPONENTS AND QUALITY

Though, UV light is effective in reducing the microbial population, high dose usage

may result in deterioration of food quality. For instance, depolymerization of starch

can occur under UV light in the presence of air, and sensibilizers (metal oxides, par-

ticularly ZnO) [52]. UV light forms lipid radicals, superoxide radicals, and hydrogen

peroxide [53]. Peroxides produced during UV light exposure may affect the fat soluble

vitamins and colored compounds and may lead to nutritional quality loses and/or dis-

coloration. Super oxide radials can further induce carbohydrate cross-linking, protein

cross-linking, protein fragmentation, peroxidation of unsaturated fatty acids, and loss

OH- and H+ radicals, which in turn interacts with other food components. Further-

more, UV radiation may also denature proteins, enzymes, and amino acids (especially

55534_C011.indd 29355534_C011.indd 293 10/22/08 10:19:07 AM10/22/08 10:19:07 AM

fortified edible mushrooms with vitamin D [49]. The Shitake, oyster, button and

of membrane fluidity function. Water molecules absorb UV photons and produces



amino acids with aromatic compounds) in food material, leading to changes in the

composition. Therefore, UV light treatment can not only change the chemistry of the

food components, but also leads to product quality deterioration when it is applied at

high doses.

treatment, oxygen radicals are formed. These oxygen radicals lead to the forma-

products. UV light can also degrade vitamins, especially vitamins A, B2, and C, by

photo-degradation. Fat soluble vitamins and colored compounds can also be affected

of methionine from methanol which leads to a burnt protein-like, burnt feathers-like,

oration of food materials. Cuvelier and Berset reported that 3 h exposure to UV light

resulted in the fading of paprika gel [55].

Pulsed UV light is also expected to induce some quality changes, when applied in

high doses. Prolonged treatment with pulsed UV light leads to temperature build-up

in color due to non-enzymatic browning, etc. The effect of UV light and pulsed

UV light on sensory quality of different food materials had been studied by several

the sensory attributes, while severe treatments lead to objectionable changes.

cider [56]. A consumer panel sensory evaluation with 40 panelists suggested that

untreated, pasteurized, and UV light treated apple ciders stored for 4 days. Further-

more, the pasteurized apple cider received less acceptable rating for color, cloudi-

ness, and overall product preference than control and UV light treated samples. This

clearly indicates that the quality changes are minimal in UV light treated apple cider

at optimum conditions when compared to thermal pasteurized cider. Consumer pan-

ness than pasteurized cider [56].

A semi-trained panel of four to six people evaluated the quality of minimally

processed white cabbage and iceberg lettuce by pulsed UV light [57]. Off-odors,

which panelists described as “plastic” were present for the pulsed light treated white

cabbage just above the acceptable limit and thus limiting the shelf life of white cab-

bage to a maximum of 7 days. However, the off-odor faded away after a couple of

hours in storage. Therefore, the off-odor can be assumed to disappear before con-

sumption by the consumers. On the other hand, it is interesting to note that pulsed

UV light treated iceberg lettuce received better scores than the control samples

for off-odor, taste, and leaf edge browning, clearly indicating that pulsed UV light

treatment can help in preserving the lettuce quality. Rice reported that the in-pack-

age UV light treatment of white bread slices with the PureBright® system resulted

in bread slices with a fresh like appearance for more than 2 weeks, although, the

55534_C011.indd 29455534_C011.indd 294 10/22/08 10:19:08 AM10/22/08 10:19:08 AM

researchers. In general, moderate treatments did not induce significant changes in

Choi and Nielsen investigated the efficacy of UV light for pasteurization of apple

there was no significant difference (p < 0.05) in the sensory scores for the odor,

elists significantly preferred UV light treated apple cider over thermally pasteurized

UV light may also cause flavor and color changes in food [54]. During UV light

tion of ozone, especially between 185 and 195 nm, causing off-flavors in the food

by the peroxides produced during extended UV light treatment. Light-induced flavor

is caused due to the activation of riboflavin, which is responsible for the conversion

or medicinal-like flavor. Prolonged treatment with UV light can also result in discol-

and thus induce temperature related quality changes such as cooked flavor, change

color, cloudiness, sweetness, acidity, overall flavor, and overall product ratings for

apple cider for color, cloudiness, sweetness, acidity, overall flavor, and overall like-



control slices were dried out and had mold growth [58]. The author also suggested

that the quality of tomatoes treated with pulsed light was acceptable up to 30 days

when stored at refrigerated temperature.

In general, UV light treatment and pulsed UV light treatment of food may not

cause any adverse effect, if applied in moderate amounts. Undesirable changes

in food may occur when food is treated with UV light for an extended period of

time. Foods may need to be treated for shorter less time to achieve the desired

decontamination level, and hence there will not be any adverse change in food

light equipments can also ensure that decontamination is achieved in a short

time.

11.2.8 ECONOMICS OF UV LIGHT DISINFECTION SYSTEM

The cost of UV light disinfection systems and pulsed UV light systems is competi-

tive with other available disinfection technologies. Choi and Nielsen suggested that

it will be cost-effective for the apple cider industry to utilize UV light pasteurization,

because UV light pasteurizers cost about $15,000 [56,59]. It has been reported that

the annual power consumption and lamp replacement costs based on a minimum

dosage of 30,000 MW.s/cm2 for multilamp and single lamp continuous UV light

disinfection sources were $2465 and $3060 for an 8000 h run time [2]. The process-

ing cost for 4 log10 reduction of E. coli in primary waste water by UV light, electron

beam, and gamma irradiation were 0.4 ¢/m3, 1.25 ¢/m3, and 25 ¢/m3, respectively

[60]. This clearly indicates that the UV light treatment is cost-effective for inactiva-

tion of pathogenic microorganisms [4].

Dunn, Bushnell, Ott, and Clark estimated that a 4-J/cm2 pulsed UV light treat-

ment with the PureBright® system will cost 0.1 ¢/ft2 of the treated area, where the

estimated cost includes conservative estimates for electricity, maintenance, and

equipment amortization [61]. Lander also estimated the cost of treatment with the

PureBright® system as 0.1 ¢/ft2, where the estimated cost includes the electricity,

maintenance, and investment in a hooded high intensity lamp and power unit [62].

11.2.9 CHALLENGES IN THE APPLICATION OF UV LIGHT AND PULSED UV LIGHT AND FUTURE RESEARCH NEEDS

One of the major limitations of UV light is the poor penetration capacity. For

instance, UV light can penetrate only up to several millimeters of food material

depending upon the optical properties of the food materials [4]. The penetration

and milk were 0.02–0.1, 10, 10–20, and 300, respectively. This clearly indicates that

type of the food material determines the applicability of UV light for disinfection of

the product. Therefore, UV may be used for disinfection of pathogens in only selec-

tive food products.

55534_C011.indd 29555534_C011.indd 295 10/22/08 10:19:09 AM10/22/08 10:19:09 AM

quality. Further modification and optimization of the UV light and pulsed UV

capacity of UV light decreases as the absorption coefficient increases [21], while

the absorption coefficient of food increases as the color and turbidity of the liquid

increases. The coefficient of absorption for various liquid foods is given in Table 11.6

[63]. For instance, the coefficient of absorption of drinking water, white wine, beer,



UV light can easily penetrate through transparent liquids such as water. How-

ever, foods such as milk have limited penetration due to its opacity. The effective

penetration depth for milk at various wavelengths is given in Table 11.7. It is evident

from the table that the longer wavelength results in increased penetration depth. Due

to its limited penetration capacity, it is essential to apply UV light to a thin layer

of milk. Oppenlander [65] suggested some of the possible arrangements of the UV

TABLE 11.6

Products at 254 nm

Liquid Food Product −1)

Distilled water 0.007–0.01

Drinking water 0.02–0.1

Clear syrup 2–5

White wine 10

Red wine 30

Beer 10–20

Dark syrup 20–50

Milk 300

Source: G Shama. 2000. Ultraviolet light. In: RK Robinson, CA Batt and

PD Patel, eds. Encyclopedia of Food Microbiology. pp. 2212 San Diego:

Academic Press.

TABLE 11.7Wavelength Dependent Effective Penetration Depth of Milk

Wavelength (nm)Effective Penetration

Depth (mm)

250 0.036

275 0.038

300 0.041

400 0.050

500 0.058

600 0.065

700 0.073

800 0.080

Source: H Burton. 1951. Ultraviolet irradiation of milk. Dairy Science Abstracts 13(3): 229–44.

55534_C011.indd 29655534_C011.indd 296 10/22/08 10:19:09 AM10/22/08 10:19:09 AM

Coefficient of Absorption of Various Liquid Food

Coefficient of Absorption (cm



light lamp in a photo-reactor, which can be utilized to enhance the effectiveness of

be essential for effective penetration in milk and other opaque food products, as a

thickness of opaque food products would be crucial for the commercial success of

this technology. The designs proposed by Oppenlander [65] can be extended to other

liquid foods and water disinfection. Addition of some absorption enhancing agents

such as edible colorants can also increase the penetration capacity [5]. Though the

penetration capacity of pulsed UV light is expected to be better than UV light, it

build-up during prolonged treatments. This can also reduce the treatment time and

enhance the quality of the food product. For opaque food materials, photocatalyzers

can also be added during pulsed UV light to enhance the effectiveness.

It will be essential to monitor the actual energy absorbed by the food sample at

different depth levels for both UV light and pulsed UV light treatments, for effective

model development and process validation. This can be challenging for pulsed UV

light as it emits polychromatic radiation ranging from UV to infrared heating. The

contribution of the temperature increase on inactivation of pathogens during pulsed

FIGURE 11.2

Germany: Wiley-VCH.)

RV Q Q R RV

Q

R

RV

Flow of

medium

Flow of

medium

Flow of

medium

Flow of

medium

A B C D

Falling filmflat bed

CFBRTubular flow-through

L L L LL

55534_C011.indd 29755534_C011.indd 297 10/22/08 10:19:10 AM10/22/08 10:19:10 AM

absorption by target material (Figure 11.2). A falling film photo-reactor design might

thin film of food material can be maintained. Other designs which may reduce the

will be essential to treat the food product as a thin film in order to avoid temperature

tube. (From T Oppenlander. 2003. Photochemical Purification of Water and Air. Weinheim,

Arrangements of lamps for flow-through systems. Cross sectional and top-

view of arrangements. A, continuous flow annular photoreactor with coaxial lamp position;

B, external lamp position with reflector (R); C, perpendicular lamp position; D, contact-free

photoreactor types (including falling film, flat bed); L, lamp, RV, reactor vessel; Q, quartz



of pulsed UV light.

One of the drawbacks of pulsed UV light technology is the temperature build-

up during prolonged treatments, as it can lead to quality deterioration in tem-

perature sensitive products. Therefore, the infrared region of the spectrum needs

also minimize the quality deterioration due to temperature increase. Provision of

UV light also emits radiation from 330 to 480 nm, which is responsible for photo-

reactivation (a mechanism to repair the DNA damage caused by UV light). In

spectrum.

A report submitted by Institute of Food Technologists (IFT) to the FDA identi-

1. Effects of individual parameters, such as suspended and dissolved solids

concentration.

4. Development of validation methods to ensure microbiological effectiveness.

5. Development and evaluation of kinetic models

6. Studies to optimize critical process factors.

Similarly, the National Advisory Committee on Microbiological Criteria for Foods

(NACMF) indicated the following future research needs for pulsed light [66]:

and parasites exposed to pulsed light.

activation.

4. Optimization critical process factors and development of protocols to moni-

tor critical factors.

5. Suitability of the technology for solid foods and non-clear liquids.

6. Differences between pulsed light technology and UV (255.4 nm) light treat-

ment, especially with respect to mechanism of inactivation.

11.3 CONCLUSIONS

In general, UV light and pulsed UV light can be potentially used for pasteurization

of several food materials. These technologies are also cost-effective. Furthermore,

quality deterioration during UV light and pulsed UV light processing is minimal

when applied in moderate doses. However, harsh treatments may lead to undesirable

changes in the food quality. Further optimization of these disinfection methods can

open an avenue to a myriad food products. As an emerging technology, pulsed UV

light is still in its primitive stage and thus extensive research has to be done before

the technology can be effectively used on an industrial scale. As penetration capacity

55534_C011.indd 29855534_C011.indd 298 10/22/08 10:19:12 AM10/22/08 10:19:12 AM

UV light must also be clearly identified in order to shed more light on the efficacy

to be filtered out for treatment of temperature sensitive food products. This will

a heat sink may be beneficial in avoiding excessive temperature build-up. Pulsed

order to avoid, photo-reactivation, it may be beneficial to filter out this wavelength

fied the following research needs for UV light [47]:

2. Identification of the pathogens most resistant to UV light.

3. Identification of surrogate microorganisms for pathogens.

1. Data on pulsed light effectiveness for specific commodities.

2. Comparison of resistance of specific pathogens, including bacteria, viruses,

3. Identification of critical process factors and their effect on microbial



is the limiting factor for both pulsed UV light and UV light, food products have to

inactivation mechanisms of pulsed UV light can help us understand the process bet-

ter and may lead to a better and effective equipment design.

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303

12 Stochastic Finite Element Analysis of Thermal Food Processes

Bart M. Nicolaï, Nico Scheerlinck, Pieter Verboven, and Josse De Baerdemaeker

CONTENTS

12.1 Introduction .................................................................................................303

12.2 Numerical Computation of Conduction Heat Transfer ...............................305

12.3 Description of Uncertainty .........................................................................306

12.3.1 Random Variables .........................................................................306

12.3.2 Random Processes .........................................................................308

12.3.3 Random Fields and Random Waves .............................................. 311

12.4 The Monte Carlo Method ............................................................................ 313

12.4.1 Description .................................................................................... 313

12.4.2 Generation of Random Variables and Processes ........................... 314

12.5 The Variance Propagation Algorithm ......................................................... 316

12.5.1 Lumped Heat Capacitance Heat Conduction Problems ................ 316

12.5.2 Heat Conduction Problems ............................................................ 319

12.5.3 Algorithm for Random Variable Parameters................................. 325

12.5.4 Derivatives of C, K and F with Respect to Random Parameters ... 327

12.6 Numerical Solution of Lyapunov and Sylvester Differential Equations ..... 328

12.6.1 Algebraic Lyapunov and Sylvester Equations ............................... 328

12.6.2 Convergence and Stability Analysis .............................................. 330

12.7 Application to Thermal Sterilization Processes ......................................... 336

12.8 Conclusions ................................................................................................. 336

Acknowledgments .................................................................................................. 337

Nomenclature ......................................................................................................... 337

References .............................................................................................................. 338

12.1 INTRODUCTION

For the design of thermal food process operations the temperature in the thermal

center of the food during the process must be known. Whereas traditionally this tem-

perature course is measured using thermocouples, there is a growing interest towards

the use of mathematical models to predict the food temperature during the thermal

treatment [1−4]. The advantages of such an approach include the computation of

55534_C012.indd 30355534_C012.indd 303 10/22/08 12:02:52 PM10/22/08 12:02:52 PM



heat penetration curves corresponding to arbitrary process conditions and container

shapes (e.g., glass jars [2]), the ability to predict overshoot [5], rapid on-line evaluation

of unscheduled process deviations [6], and optimization of thermal processes [7,8].

In the case of conduction heated foods, the heat transfer process is described

through the Fourier equation. As for complicated geometries and time-dependent

boundary conditions usually no analytical solutions are available for the Fourier

equation, and a numerical solution becomes mandatory. Several methods, includ-

by numerous investigators to the numerical solution of the Fourier equation. While

now widely available, they require that the product and process parameters are accu-

rately known. However, in reality these parameters may vary quite extensively, due

to biological variability or unpredictably changing conditions such as the ambient

temperature. Consequently, the temperatures inside the product are stochastic quan-

tities, which must be characterized by statistical means.

Nicolaï et al. [9] reviewed the sources of uncertainty in thermal sterilization

diffusivity, is typically 3–15%, although values as high as 26% have been observed

[10]. Meffert [11] concluded that the possible maximum error in the experimental

determination of the thermal conductivity can be as high as 30–50% at the 95% con-

and non-vacuum-packed potato slabs to establish a temperature increase from 20ºC to

75ºC with an oven setpoint of 80ºC at different positions in commercial steamers and

for different oven types. They found substantial differences in heating time between

packs of the same shelf. According to these authors, the observed heating time vari-

ations were due to the intermittent inputs of the steam used to maintain temperatures

below 100ºC. The standard deviation of the temperature of a well- controlled retort

is typically 1ºC [13]. In a more recent publication [14] it was reported that the retort

temperature variability is normally less than ±0.5ºC. Ramaswamy et al. observed

that the maximum difference between different positions during the holding phase

was between 2.6 and 3.5ºC [15]. The average of the standard deviations of the retort

temperatures at each time was 1.3ºC. From experiments inside a pilot scale water

cascading retort it was found [16] that the average of the standard deviations of the

temperatures at different positions was equal to 0.7ºC during the entire cook-period.

The overall standard deviation during the cook period was 0.9ºC, and the maxi-

mum temperature difference between positions 1 min after the coming-up period

was equal to 3.2ºC. Little information is available on the variability of the surface

25% for his Monte Carlo analyses.

As the thermal inactivation of micro-organisms is highly dependent on the

temperature, it is very well possible to end up in a situation where some foods of the

same batch are microbiologically safe, while others are not. The uncertainty involved

in thermal food process design has therefore been addressed by several authors

[13,17–21] by means of Monte Carlo analyses. Using this method a large number

samples of the random parameters are generated by the computer and for every set the

thermal problem is solved. In the end statistical parameters such as the mean value

and the variance of the temperature at the thermal center can be calculated using

55534_C012.indd 30455534_C012.indd 304 10/22/08 12:02:53 PM10/22/08 12:02:53 PM

ing finite differences [1,3,4], and finite elements [2] have been applied successfully

commercial codes (mostly based on the finite element or finite volume method) are

processes. The coefficient of variation (CV) of the f-value, and, hence, the thermal

fidence level. Sheard and Rodger [12] compared the time required for vacuum-packed

heat transfer coefficient in thermal food processes. Martens [13] used a CV of 10 and


Stochastic Finite Element Analysis of Thermal Food Processess 305

statistical inference. The drawback of the Monte Carlo method is the large amount

of computer time, particularly when the thermal problem is to be solved numerically.

Alternative algorithms have therefore been suggested to calculate the propagation of

described. The main features of the Monte Carlo and variance propagation algo-

rithms will be illustrated by a numerical example.

12.2 NUMERICAL COMPUTATION OF CONDUCTION HEAT TRANSFER

Transient linear heat transfer in solid foods subjected to convection boundary condi-

tions is governed by the Fourier equation

k T Q cTt

∇ ρ ∂∂

2 + = (12.1)

kn

T h T T∂∂

Γ= −( )∞ on (12.2)

T = T0 at t = t0 (12.3)

where T is the temperature (ºC), k the thermal conductivity (W/mºC), ρc the volu-

metric heat capacity (J/m3 ºC), T∞ the (known) process temperature (ºC), n the out-2

surface, Q the heat generation (W/m3), and t the time (s).

For many realistic heat conduction problems no analytical solutions of Equation

12.1 subjected to Equation 12.2 and Equation 12.3 are known. In this case numerical

is subdivided in elements of variable size and shape which are interconnected in a

nod of nodal points. In every element j the unknown temperature is

approximated by a low order interpolating polynomial

u j jTj t t( ) ( )= φφ u (12.4)

where uj(t) is the approximate temperature in element j, u j(t) is the vector contain-

ing the nodal temperatures in element j, and φφ j is the vector of shape functions

corresponding to element j. The application of a suitable spatial discretization tech-

nique such as the Galerkin weighted residual method to Equation 12.1 subjected to

Equation 12.2 and Equation 12.3 results in the following differential system [26]:

C u Ku fddt

+ = (12.5)

55534_C012.indd 30555534_C012.indd 305 10/22/08 12:02:54 PM10/22/08 12:02:54 PM

In this chapter the use of stochastic finite element methods to calculate statisti-

cal characteristics of the temperature field inside conduction heated foods will be

ward normal to the surface, h the convection coefficient (W/m ºC), Γ the boundary

discretization techniques such as the finite difference or finite element method can

be used to obtain an approximate solution. The finite element method in particular is

the Fourier equation. In the framework of the finite element method the continuum

finite number n

parameter fluctuations in space and/or time [22–25].

a very flexible and accurate method for solving partial differential equations such as



u(t = 0) = u0 (12.6)

with u = [ ]u u u1 2 � nnod

I the overall nodal temperature vector, C the capacitance

matrix and K the conductance matrix, both nnod × nnod matrices, and f a nnod × 1

tions of each element (the ‘element matrices’) C j, K j and f j

C j j j= ∫ ρc dV

V j

φφ φφT

(12.7)

K B Bj j j

V

j j

S

k dV h dST

j

T

j

= +∫ ∫ φ φ (12.8)

f j j

S

j

V

hT dS Q dVj j

= +∫ ∫∞φ φ (12.9)

with

Bz

jj

=∂∂φφ

,

where Sj and Vj are the boundary surface and volume of element j, respectively and

z is the position vector. The element matrices are then incorporated in the global

matrices. The matrices K and C are sparse and this property can be exploited advan-

tageously for reducing the CPU time required for the solution of Equation 12.5.

food processing applications such as sterilization of baby food jars [2], cooling of

broccoli stalks [27] and tomatoes [28].

12.3 DESCRIPTION OF UNCERTAINTY

known. However, in reality this is certainly not always the case. In the stochastic

ess parameters is explicitly incorporated in the calculations. It is therefore required

that an appropriate mathematical description of the random parameters is available.

In this section the random variable model, along with its multidimensional exten-

description of these concepts, the reader is referred to the literature [29].

12.3.1 RANDOM VARIABLES

The most simple uncertainty model is that of a random variable. A random variable

X is a real-numbered variable whose value is associated with a random experiment.

55534_C012.indd 30655534_C012.indd 306 10/22/08 12:02:55 PM10/22/08 12:02:55 PM

vector. The system (Equation 12.5) can be solved by finite differences in the time

domain. For the construction of the global finite element matrices C, K and f, it is

most convenient from the programming point of view to first assemble the contribu-

The finite element method has been successfully used in a number of thermal

In the finite element method it is assumed that all parameters are deterministic and

finite element method, knowledge about the uncertainty of the material and proc-

sions such as random process, field and wave, will be introduced. For a more precise



For example, the heat capacity of a potato is a random variable which can vary

between different potatoes. A random variable X can be characterized by its prob-

ability density function f(x) and statistical moments such as the mean value X and

the variance σ2, if existent and known.

X X� ε ( ) (12.10)

� xf x dx( )

−

∫∞

∞

(12.11)

σ ε2 2� ( )X X− (12.12)

� ( ) ( )x X f x dx−−∞

∫ 2

∞

(12.13)

with ε the expectation operator.

Sometimes an experiment will yield values for two or more physical parameters.

Assume, for example, that both the thermal conductivity as well as the volumetric

heat capacity of a material are measured simultaneously. In this case the outcome

of the experiment is called a bivariate (two) or multivariate (more than two) random

variable. The random variables X1 and X2 can then be stacked conveniently in a ran-

and covariance matrix V of the random vector as

X X� ε( ) (12.14)

V X X X X� ε ( )( )− −⎡⎣⎢

⎤⎦⎥

T (12.15)

The i-th diagonal entry of V is the variance σXi2 of random variable Xi; the (i, j)-th

entry of V is the covariance σX Xi j, of random variables Xi and Xj.

As expected, the probability density function f (X1, X2) of a bivariate random

variable is a function of two variables. The bivariate Gaussian density function is

f X XR

RX X

X X X

( )( )

exp(

1 2

1

2

1

2 1

1 2,,=

− −−

2 1/2

1 2

−

πσ σ11 2

2

,X X

X X2

1 1

1)

−⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢⎢

⎧

⎨⎪⎪⎪⎪

⎩⎪ σ⎪⎪⎪⎪

−⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟−⎛

⎝−2

1

RX X X X

X XX X

1 2

2

1 1 2 2, σ σ

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟+

−⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

⎤

⎦

⎥⎥⎥⎥

⎫

⎬

2

2 2

2

X X

Xσ

⎪⎪⎪⎪⎪

⎭⎪⎪⎪⎪

(12.16)

R is ca

55534_C012.indd 30755534_C012.indd 307 10/22/08 12:02:56 PM10/22/08 12:02:56 PM

dom vector X. Similar to the univariate case we can then define the mean value X

often used to describe bivariate random variables. It is defined as

lled the correlation coefficient, and −1 ≤ R ≤ 1.



12.3.2 RANDOM PROCESSES

If a parameter changes in an unpredictable way as a function of the time co-ordinate,

it can be described conveniently by means of a random process. The mean X and

covariance V of a stationary process X with probability density function f(x,t) are

X X= ε( ) (12.17)

� x f x t dx( ),−∞

∫∞

(12.18)

V X t X X t X( ) {[ ( ) ][ ( ) )]}τ ε τ= − + − (12.19)

The covariance function describes how much the current value of the random func-

mean of the process does not change in time and its covariance function is only a

function of the separation time τ. The correlation function R is found by normaliza-

tion of the covariance function:

R (τ) = V (τ)/σ2 (12.20)

A Gaussian stationary white noise process W with covariance

VW W W W, ,= ,( ) ( )τ σ δ τ2 (12.21)

tions. Sample values of W are uncorrelated no matter how close together in time they

which change more smoothly as a function of time. An autoregressive random proc-

ddt

X t addt

X t a X t W tm

m

m

m m( ) ( ) ( ) ( )+ + + =−

−1

1

1� (12.22)

where a1, a2,…, am are constants, m ≥ 1, and W(t) is a stationary Gaussian white

noise process with W a Xm=1 m, and their high frequency content decreases with increas-

ing order m. The (Gaussian) random variable initial condition corresponding to the

ε[ ( )]X t X0 = (12.23)

ε σ[ ( ) ]X t X02 2− = (12.24)

55534_C012.indd 30855534_C012.indd 308 10/22/08 12:02:58 PM10/22/08 12:02:58 PM

defined by

tion will affect its future values. By definition of a stationary random process, the

are. However, white noise does not exist in reality as it has an infinite energy content

ess of order m is defined by the following stochastic differential equation

the coefficients a ,…,a

stochastic differential Equation 12.22 is defined as

where δ is the Dirac delta, can be used to describe very rapid unpredictable fluctua-

and variance. Autoregressive processes provide a tool to incorporate fluctuations

. The time scale of the fluctuations is dependent on



Note that a random variable parameter X can be modeled as a trivial case of an

AR(1) process:

ddt

X = 0 (12.25)

AR(m) processes are a special case of the class of physically realizable stochastic

processes which comprise most of the random processes seen in practice [30].

In order to describe the smoothness of a random process by means of a sin-

θ τ τ=−∞

+∞

∫ R d( )�

It gives an indication of the time beyond which a future value of a random process

will not be affected anymore by its current value. In Table 12.1 the variance, the

processes. For the latter, the characteristic polynomial

ξ ξm mma a+ + + =−

11 0� (12.26)

has two real or two complex conjugate roots, resulting in non-oscillating or oscil-

lating correlation functions, respectively. In Figure 12.1 some correlation functions

tion are compared. If θ → 0 then the process approximates a white noise process.

On the other hand, if θ → +∞ then the values of the realization at arbitrary points

TABLE 12.1Autocovariance Function and Scale of Fluctuation of AR(1) and AR(2) Processes

AR(1) V

a

x x xa

, ( )

/

τ σ

θ

τ=

=

−2

1

1

2

e

AR(2) real

roots

Vx x x, ( ) [( )]τ σ ξ ξ ξ ξ

ξ

ξ τ ξ τ= −( ) − −22 1 2 1

11 2e e

with 1 aand the roots of2ξ

ξ ξ

θ

21 2

1 2

0

2

+ + =

=

a a

a a/

AR(2) complex

roots

V e pap

px x x

a

, ( ) cos( ) sin( )τ σ τ ττ

= +⎡

⎣⎢⎢

⎤

⎦

−2

1

2 11

2⎥⎥⎥

⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

− =with p a1= ( /a a a12 1 2

1 24 2) // θ

55534_C012.indd 30955534_C012.indd 309 10/22/08 12:02:59 PM10/22/08 12:02:59 PM

defined as

gle measure, Vanmarcke [29] introduced the concept scale of fluctuation, which is

autocovariance function and the scale of fluctuation are given for AR(1) and AR(2)

and corresponding realizations of an AR(2)-process with different scales of fluctua-



FIGURE 12.1 Correlation functions (a) and corresponding realizations (b) of a AR(2)-

–1000 –500 0 500

(a)

1000

Rx,

x( t)

t (s)Δ

Δ

0 200

= 100 sθ

θ

θ

= 10 s

= 1 s

400 600 800 1000

Time (s)

x (t

)

(b)

55534_C012.indd 31055534_C012.indd 310 10/22/08 12:03:00 PM10/22/08 12:03:00 PM

process with different scales of fluctuation.



are completely correlated. In this case the random process concept is far too sophis-

ticated to describe the physical quantity since all the meaningful probabilistic fea-

tures of the quantity can be captured by a simple random variable model.

The correlation functions and corresponding realizations of different types of

measure of how frequent the process wiggles around the mean-axis, irrespective of

the order of the process.

It is convenient to write the autoregressive process (Equation 12.22) in the fol-

lowing state space form [31]:

ddt

t t W tX AX B( ) ( ) ( )= + (12.27)

where

X =/

//

− −

− −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

X

dX dt

d X dt

d X dt

m m

m m

�2 2

1 1

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

B

0

0

0

1

�

⎤⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

− − −−

⎡

⎣

⎢⎢⎢⎢

A

0 1 0

0 0 1

1 1

�

� � �

�

�a a am m

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

The vector X is called the state vector, the matrix A the companion matrix and the

vector B is an auxiliary vector.

12.3.3 RANDOM FIELDS AND RANDOM WAVES

Often a physical quantity varies randomly as a function of the time and/or space

coordinates. Examples include the temperature in an oven, the thermophysical prop-

erties of heterogeneous materials such as foods, hydraulic properties of soils, elastic

venient mathematical framework to describe such phenomena [29].

of random waves. The random wave model is a straightforward extension of the

is referred to the literature [29].

55534_C012.indd 31155534_C012.indd 311 10/22/08 12:03:01 PM10/22/08 12:03:01 PM

properties of construction materials, etc. The random field concept provides a con-

random field model combined with the random process model. A full account of

random fields and random waves is beyond the scope of this chapter, and the reader

random processes are shown in Figure 12.2. Clearly the scale of fluctuation is a

A parameter which fluctuates both in space and time can be described by means



FIGURE 12.2 Correlation functions (a) and corresponding realizations (b) of several types

–1000 –500

(a)

0 500 1000

Rx,

x( t)

t (s)Δ

Δ

(b)

AR(2) complex roots

AR(2) real roots

AR(1)

0 200 400 600 800 1000

Time (s)

x (t

)

55534_C012.indd 31255534_C012.indd 312 10/22/08 12:03:02 PM10/22/08 12:03:02 PM

of autoregressive processes with the same scale of fluctuation.



12.4 THE MONTE CARLO METHOD

12.4.1 DESCRIPTION

The Monte Carlo method was introduced by John von Neumann and Ulam during

rial which arose in the development of the atomic bomb [32]. The code name of the

project was Monte Carlo, from where the method inherits its name.

generated by means of a random generator. For every parameter set the heat conduc-

tion problem is solved by analytical or numerical means, and the solution is stored

for future use. This process is repeated a large number of times n, and in the end

the statistical characteristics are be estimated. For the mean and the variance of the

solution T at arbitrary space-time co-ordinates, the following non-biased estimation

formulas can be applied

Tn

Tj

nj� =

=∑1

1

(12.28)

2

1

1

1ˆ ( )σ =

−−

=∑n

T Tj

nj 2 (12.29)

where Tj is the solution in the j-th Monte Carlo run and the symbol “ˆ” means “esti-

mate of”.

If T is a linear function of the random parameters (e.g. ambient temperature) and

if the latter are normally distributed, then T is also normally distributed. In this case

T� and 2σ̂ are given by [33]

T tn

T T tn

i� �− ≤ ≤ +0.975 0.975

ˆ ˆσ σ (12.30)

χ σ

σχ0 025

2 2

2

0 9752

1 1

. .

−≤ ≤

−n nˆ

(12.31)

where the t and χ2 are student t and χ2 distributed and are to be evaluated with n–1

degrees of freedom. For n ≤ 30 they are tabulated in all textbooks on introductory

statistics; for n ≥ 30 it can be shown [33] that 2 2 1χ2 − −n is normally distributed

with zero mean and unit variance. The Student’s t distribution then approximates a

normal distribution.

The formulas in Equation 12.30 and Equation 12.31 are only valid if the tem-

perature T is linearly dependent on the random parameter(s). Even if this is not so,

parameters is not too large.

55534_C012.indd 31355534_C012.indd 313 10/22/08 12:03:02 PM10/22/08 12:03:02 PM

World War II for studying random neutron diffusion problems in fissionable mate-

In the Monte Carlo finite element method, samples of the random parameters are

the confidence intervals for

e.g., in the case of random thermal conductivity, these formulas can be used as a first

order approximation of the real confidence intervals if the variability of the random



For n = 100 and 1000, Equation 12.31 becomes

0 74 1 28 1002

2. < < . =ˆ

σσ

for n (12.32)

0 91 1 09 10002

2. < < . =ˆ

σσ

for n (12.33)

wide. It can therefore be concluded that the large number of repetitive simulations

necessary to obtain an acceptable level of accuracy is a major drawback of the

to generate the parameter samples can outweigh by far the actual CPU time required

the random samples is therefore imperative, as it may considerably reduce the total

CPU time.

A further drawback of the Monte Carlo method is the fact that the stochastic

(joint) probability density functions.

12.4.2 GENERATION OF RANDOM VARIABLES AND PROCESSES

Uniformly distributed random numbers are now most commonly generated by means

of a congruential generator. In this method, a discrete random number xi+1 is derived

from a previous one, xi, based on a fundamental congruence relationship

xi+1 = (axi + c)mod m, i = 0, 1, … (12.34)

where the multiplier a, the increment c and the modulus m are nonnegative integers.

3 is equal to 2. The recursion is started with a starting value x0, the seed. It has

been shown statistically that the xi are uniformly distributed on the interval (0,m).

Uniformly distributed random numbers on the unit interval (0,1) can be obtained by

dividing the xi by m. Obviously, after at most m recursions, the random sequence will

repeat itself. Conditions on a, c and m can be found such that the period after which

the sequence will repeat itself is maximal [32]. The following values give random

numbers of reasonable quality [34]: m = 233 280, c = 49 297, and a = 9 301, and do

It is emphasized here that the implementation of a good random number gen-

erator is by no means a trivial task. Inappropriate constants a, m and c can lead to

numbers which are highly correlated. It is therefore suggested to use random number

generators which are provided with standard mathematical packages such as Nag

(The Numerical Algorithms Group, Oxford, UK) or IMSL (USA). The generators

which are included in compilers must be used with special care—for example, those

55534_C012.indd 31455534_C012.indd 314 10/22/08 12:03:03 PM10/22/08 12:03:03 PM

This means that, even for n = 1000, the relative confidence band is almost 20%

Monte Carlo method, particularly when in each run a finite element problem must

be solved. If the random parameters are of the random field type, the time required

to solve the finite element problem. A careful choice of the algorithms to generate

parameter set must be completely specified in the probabilistic sense, including

The modulo (‘mod’) is defined as the remainder of the integer division, e.g., 5 mod

not cause integer overflow on most systems.



implemented in some commercial C compilers generate random numbers of very

poor quality. For a discussion, see Press et al. [34].

Random variates with non-uniform probability density function f(x) can be

obtained from uniformly distributed random numbers on (0,1) by several methods,

including the transformation method and the acceptance–rejectance method. For

example, consider the following transformation due to [35]

Z1 = (−21n U1)1/2 cos 2π U2 (12.35)

Z2 = (−21n U1)1/2 sin 2π U2 (12.36)

It can be shown that if U1 and U2 are two uniformly random numbers in (0,1), then Z1

and Z2 are two uncorrelated standard (μ = 0, σ2 = 1) Gaussian random numbers. The

histogram in Figure 12.3 was produced from 1000 Gaussian numbers according to

Equation 12.35, Equation 12.36 and Equation 12.34 with the above given numerical

values of a, c and m.

Algorithms for other probability density functions are described elsewhere [32].

Samples (or realizations) of an AR(m) random process can be generated recur-

sively by time discretization of the corresponding differential equation. For example,

for an AR(1) process we have that

ddt

X a X W+ =1 (12.37)

By applying an implicit Euler discretization we obtain the following time series:

(1 + a1Δt)X(t + Δt) = X(t) + ΔtZ (t + Δt) (12.38)

or

X(t + Δt) = (1 + a1Δt)−1 (X(t) + ΔtZ (t + Δt)) (12.39)

FIGURE 12.3 Histogram of a measurement experiment of a random variable U. The full

line is the limiting probability density function.

–3.6 –2.8 –2.0

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

f(u)

–1.2 –0.4 0.4 1.2 2.0 2.8

U

55534_C012.indd 31555534_C012.indd 315 10/22/08 12:03:04 PM10/22/08 12:03:04 PM



with Z discrete time white noise (a sequence of Gaussian random numbers). It can be

shown [36] that the variance σZ2 of Z is equal σ ΔZ t2 / . The algorithm is bootstrapped

ance of the generated sample will be smaller than the target variance. Methods to gen-

erate samples of AR(m) processes of arbitrary order are compared by Nicolaï [37].

12.5 THE VARIANCE PROPAGATION ALGORITHM

The major drawback of the Monte Carlo method is the considerable amount of CPU

time required to obtain accurate estimates of the stochastic characteristics of the

Carlo method. For a full account of the algorithm, the reader is referred to Melsa

and Sage [36].

12.5.1 LUMPED HEAT CAPACITANCE HEAT CONDUCTION PROBLEMS

In order to explain the variance propagation algorithm, we will consider the fol-

lowing simple lumped capacitance heat transfer problem [38]. Consider a sphere of

radius r0 with thermal capacity c and density ρ. The sphere is initially at a uniform

temperature T0. At time t = 0 the sphere is immersed in a water bath at temperature

T∞. The temperature of the sphere will approach T∞ with a rate which depends on

capacitance method it is assumed that, because of the high thermal conductivity of

the solid medium, the temperature inside the solid is uniform at any instant during

the transient heat transfer process. This hypothesis holds if the Biot number, Bi, sat-

Bi = < .hLk

0 1 (12.40)

where L is the characteristic length of the solid which in the case of a sphere is usu-

0

leads to the following differential equation:

ρcddt

Th

rT T= −

3

0

( )∞ (12.41)

After integration the following formula for the temperature course is found

T T T Th

crt= + − −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟∞ ∞( )exp0

0

3

ρ (12.42)

For simplicity we will assume that T∞ is an AR(1) random process described by

means of the following differential equation:

ddt

T a T W∞ ∞+ =1 (12.43)

55534_C012.indd 31655534_C012.indd 316 10/22/08 12:03:05 PM10/22/08 12:03:05 PM

with a random value x of the process as specified in Equation 12.23 and Equation

12.24. A sufficiently small time step should be selected because otherwise the vari-

temperature field. The variance propagation algorithm is an alternative to the Monte

the surface heat transfer coefficient h at the solid–liquid interface. In the lumped

isfies the following constraint

ally defined as L = 3r [38]. It is easy to show that applying an overall energy balance



with W a white noise process with mean W a T= ∞1 . We can combine Equation

12.43 and Equation 12.41 into the following global system:

ddt

W W tx g x h= + −( ) ( )( ) (12.44)

with

x =⎡

⎣⎢⎢

⎤

⎦⎥⎥

T

T∞

(12.45)

g =−

−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

3 3

0

1

hcr

Th

crT

W a T

ρ ρ∞

∞

0

⎥⎥

(12.46)

h =⎡

⎣⎢⎢

⎤

⎦⎥⎥

0

1 (12.47)

the covariance matrix of the solution of Equation 12.44 are given by

ddt

x g x= [ ] (12.48)

ddt

T

WTV V V h hx x x x x x, , ,= +

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ +

∂∂

∂∂

σgx

gx

2 (12.49)

with

∂∂

∂∂

gx

g xx

�x t

t tt

( )

[ ( ) ]

( )

,

Equation 12.48 and Equation 12.49 are called the variance propagation algorithm.

Equation 12.49 is a matrix differential equation of the Lyapunov type.

If we combine Equation 12.44 through Equation 12.52 we obtain the following

system:

ddt

Th

crT T= −

3

0ρ ∞( ) (12.50)

ddt

T∞ = 0 (12.51)

ddt

hcr

hcr

a

Vx x, =−

−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

3 3

0

0 0

1

ρ ρ⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

+−

−V Vx x x x, ,

30

3

0

0

1

hcr

hcr

a

ρ

ρ⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 0

0 2σW

(12.52)

55534_C012.indd 31755534_C012.indd 317 10/22/08 12:03:06 PM10/22/08 12:03:06 PM

It can be shown [36] that first order approximate expressions for the mean vector and



with

Vx xT T T

T T T

, =⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

σ σσ σ

2

2

,

,

∞

∞ ∞

(12.53)

and σT T, ∞ the covariance of T and T∞. The initial conditions are given by

T t T( )= =0 0 (12.54)

T t T∞ ∞( )= =0 (12.55)

Vx xT

T

t, ( )= =⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

00

0

0

2

2

σ

σ ∞

(12.56)

Equation 12.50 expresses that the mean solution can be found by solving the orig-

inal differential equation for the mean value of the random parameter. Equation

expected since an autoregressive process is stationary).

Equation 12.52 can be elaborated further to yield

ddt

hcr

hcrT T T Tσ

ρ ρσ2

0

2

0

6 6= − +σ , ∞

(12.57)

ddt

hcr

hcr

aT T Tσρ

σρ, ∞ ∞

= − +⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟3 3

0

2

0

1

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟σT T, ∞

(12.58)

ddt

aT T Wσ σ σ∞ ∞

21

2 22= − + (12.59)

As T∞ is stationary, σT∞

2 not a function of time so that Equation 12.59 reduces to

σ σT W a∞

2 212= / (12.60)

The solution of Equation 12.58 can readily be found through direct integration

σ ρρ

σρT T

h crh cr a

hcr

a2 0

0 1

2

0

1

3

31

3=

// +

− − +⎛

⎝

⎜

∞exp

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎧⎨⎪⎪⎪

⎩⎪⎪⎪

⎫⎬t⎪⎪⎪⎪

⎭⎪⎪⎪

(12.61)

After substitution of Equation 12.61 in Equation 12.57 and subsequent integration

we can derive the following expression for σT2

55534_C012.indd 31855534_C012.indd 318 10/22/08 12:03:07 PM10/22/08 12:03:07 PM

12.51 confirms that the mean value of the random parameter is constant (which we



σ ρρ

σ ρρT T

h crh cr a

h crh cr a

2 0

0 1

2 0

0 1

3

3

3

3=

// +

+/

/ −∞σσ

ρ

ρ

Th

crt

h c r

∞

2

0

2 2 202

6

18

3

exp

(

−⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

−/

hh cr a h cr ah

craT/ + / −

− +⎛

⎝

⎜

ρ ρσ

ρ0 1 0 1

2

0

13

3

)( )exp

∞

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

t (12.62)

In the special case of a random variable, we can simplify the above expression by

putting a1 = 0, so that we have

σ σ σρ

σT T T Th

crt2 2 2

0

262= + −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−

∞ ∞ ∞exp exxp −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

3

0

hcr

tρ

(12.63)

A sample of the random process ambient temperature and the corresponding tem-

perature course in the sphere are shown in Figure 12.4. The parameter values were

as follows: ρ = 1000 kg/m3, c = 4180 J/kg°C, r0 = 0.01 m, T0 = 20°C, h = 10 W/m2°C,

T ∞ = 80°C, σT∞ = 5°C, and θ = 600 s.

thermal inertia of the sphere. There was a very good agreement between the mean

temperature of the sphere calculated by means of the Monte Carlo and the variance

propagation algorithm (not shown). In Figure 12.5 the time course of the variance of

the temperature of the sphere is shown. The results obtained by means of the vari-

ance propagation algorithm and the Monte Carlo method with 1000 or 5000 runs

were comparable. However, the variances obtained by means of the Monte Carlo

method with 100 runs are scattered.

the sphere are shown in Figure 12.6.

12.5.2 HEAT CONDUCTION PROBLEMS

For the extension of the variance propagation to conduction limited problems we will

start from the spatially discretized system (Equation 12.5). We will further assume

that T∞, h and Q are autoregressive processes of order mT∞, mh, and mQ, respectively,

dt

t td

W tx A x BT T T T T∞ ∞ ∞ ∞ ∞= +( ) ( ) ( ) (12.64)

ddt

t th h hx A x Bh ht W( ) ( ) ( )= + (12.65)

ddt

t t tQ Q Q Qx A x B( ) ( ) ( )= + WQ (12.66)

55534_C012.indd 31955534_C012.indd 319 10/22/08 12:03:08 PM10/22/08 12:03:08 PM

A Crank–Nicolson finite difference scheme in the time domain was used to

The mean value and 95% confidence interval for the temperature prediction in

as defined by the following state space equations

solve Equation 12.41. The high frequency fluctuations are smoothed because of the



with WT∞, Wh, WQ white noise processes of, in general, different covariance. As the

thermophysical properties k and ρc usually do not change as a function of time, they are

modeled as random variables by means of the following trivial differential equations

ddt

kddt

c= =ρ 0 (12.67)

FIGURE 12.5 Temperature variance in sphere subjected to random process ambient tem-

perature. —: variance propagation; ∗: Monte Carlo (nMC=1000); +: Monte Carlo (nMC=100);

O: Monte Carlo (nMC=5000).

6

5

4

3

2

Var

iance

of

tem

per

ature

(°C

2)

1

0

–10 20 40 60 80 100 120

Time (min)

FIGURE 12.4 Realization of AR(1) ambient temperature and corresponding temperature

in sphere.

90

80

70

60

50

Tem

per

ature

(°C

)

40

30

200 20 40 60

Time (min)80 100 120

55534_C012.indd 32055534_C012.indd 320 10/22/08 12:03:10 PM10/22/08 12:03:10 PM



with appropriate initial conditions. Obviously, T0 is modeled as a random variable

as well.

heat conduction in the form of Equation 12.44. It is easy to see that this can be

accomplished through the following choice of x, g and h

x

u

x

x

x=

∞T

h

q

k

cρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(12.68)

g

Ku f

A B

A B

A B=

− +

∞

C−

∞ ∞ ∞+

+

+

1( )

T Tx W

x W

x W

T T

h h h h

Q Q Q Q

000

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(12.69)

FIGURE 12.6 in sphere subjected to random process ambient temperature.

90

80

70

60

50

40

30

200 20 40 60 80 100 120

Time (min)

Tem

per

ature

(°C

)

55534_C012.indd 32155534_C012.indd 321 10/22/08 12:03:11 PM10/22/08 12:03:11 PM

Mean value (−) and 95% confidence interval (∗) for temperature prediction

As with the lumped capacitance problem, a first step is to write the stochastic



h

0 0 0

0 0

0 0

0 0=

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

∞B

B

B

T

h

Q

0 0 0

0 0 0

⎥⎥⎥⎥⎥⎥⎥⎥

(12.70)

w =−−−

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

∞W

W

W W

T T

h h

Q Q

W

W

∞

(12.71)

with 0 null vectors of appropriate dimension, and

V W Ww,w

,

,( ) [ ( ) ( )] ( )τ ε τ δ τ

σ

σ� t t

W T

W hT + =

∞

2

2

0 0

0 0

0 00 2σW Q,

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(12.72)

After substitution of Equation 12.68 through Equation 12.71 in Equation 12.48 and

Equation 12.49 and subsequent rearrangement, the following system is obtained

ddt

u C Ku f= − +−1( ) (12.73)

ddt k c

V C KVK

uVC u

Vu,u1

u,u u, u,= − − −− ∂∂

∂∂ρ ρk

T d

dtcc

T

TT

T

⎡

⎣

⎢⎢⎢

+ + −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟∞

∂∂

∂∂

∂∂∞

fV

f Kuu, h h ⎟⎟ +

+ − − −

Vf

V

KVK

uVC

u, u,

u,u u,

hT

QT

kT

∂∂

∂∂

∂∂ρ

Q

k cc

T h h

cd

dt

uV

fV

f Ku

u,

u,

ρ

∂∂

∂∂

∂∂∞

T

TT

⎡

⎣

⎢⎢⎢

+ + −⎛

⎝∞

⎜⎜⎜⎜⎜⎞

⎠⎟⎟⎟⎟ +

⎤

⎦⎥⎥

−Vf

V Cu, u,h QT T

T

T

Q∂∂

(12.74)

ddt TT TTV C V Vu,x u,x xK

T∞ ∞ ∞ ∞∞

= − +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟−1 ∂

∂f

,⎟⎟⎟⎟+ V Au T

T,xT∞ ∞

(12.75)

ddt h hh h

V C KVK

u Vu,x u,x=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

−1 − + −∂∂

∂∂

fhh u hh h, ,x x

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥+ V AT

(12.76)

55534_C012.indd 32255534_C012.indd 322 10/22/08 12:03:12 PM10/22/08 12:03:12 PM



ddt QQ Q QQV C KV V Vu,x u x x= − +

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+−1

, ,

∂∂

fuu A,xQ Q

T (12.77)

ddt kk kV C KV

Kuu, uk = − −

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

−1 2, σ ∂

∂ (12.78)

ddt

V KVC

uu, c u, cρ ρ∂∂ρ

= − −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟−C 1 2σρc c

ddt ⎟⎟⎟

(12.79)

where the notation C−T denotes the transpose of the inverse of C . C, K and f are

assembled using the mean values of ρc, k, T∞, h, and Q. The initial condition for

Equation 12.73 is given by

u u( 0) 0t = = (12.80)

Vu u I, = σT0

2 (12.81)

where I is an nnod × nnod unity matrix. Further, since the initial temperature is uncor-

related with k, ρc, h, T∞, and Q, the other initial conditions are equal to null matri-

ces of appropriate dimension. Equation 12.73 through Equation 12.81 constitute the

variance propagation algorithm for stochastic heat conduction problems.

Observe that the above algorithm can be extended to take into account nonlin-

ear heat conduction with temperature dependent thermal properties since Equation

12.48 and Equation 12.49 are applicable to general nonlinear systems. The corre-

sponding algorithm has a similar overall structure as the above algorithm and has

been described in detail by Nicolaï [37]. As it is essentially based on a linearization

tion problems only. The applicability of this algorithm for heat conduction problems

with phase changes is currently being investigated by the authors. A more extended

variance propagation algorithm for heat conduction problems with random process

and random wave parameters was described recently [25].

Equation 12.74 is of the general form

ddt

t t t tV AV V A B( ) ( ) ( ) ( )= + +T

with V, A, and B square matrices of equal dimension, and is called a Lyapunov

matrix differential equation. Equation 12.75 through Equation 12.77 are of the

general form

ddt

t t t tV AV V B C( ) ( ) ( ) ( )= + + (12.82)

55534_C012.indd 32355534_C012.indd 323 10/22/08 12:03:13 PM10/22/08 12:03:13 PM

of the finite element formulation of the (nonlinear) heat conduction equation, it can

however be expected to be sufficiently accurate for smooth nonlinear heat conduc-



with A, B square matrices, and V and C matrices which are in general not square.

Equation 12.82 is called a Sylvester matrix differential equation. Equation 12.78 and

Equation 12.79 are of the form

C V KV hddt

t t( ) ( ) ( )+ =t

wi− −

nod.

This structure is similar to that of Equation 12.73, and further on it will be outlined

that this fact can be exploited advantageously.

The matrices V Vx x xT T∞ ∞,x , ,h h, and Vx xQ Q, in Equation 12.75 through Equation 12.77

can be computed by straightforward application of the variance propagation algo-

rithm to the Equation 12.64 through Equation 12.66, respectively, which yields for

example for T∞

ddt T T T T T T T T

TT TV A V V A Bx x x x x ,x∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞, ,= + + σ22 BT

T∞ (12.83)

It can be proven that AR(m) processes driven by stationary white noise are stationary

[36]. This implies that the mean and the covariance of the AR(m) process does not

change in time. Consequently, the time derivatives in the left hand sides of Equation

12.83 vanish and the following algebraic matrix Lyapunov equation is obtained

A V V A B BT T T T T T

TT T T

T

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞x x x ,x, + + =σ2 0 (12.84)

Similarly,

A V Vh hT

T hT

h h h Tx ,x x ,x h+ + =∞ ∞

A B B 0σ2 (12.85)

A V VQ QT

T QT

Q Q Q Tx ,x x ,x+ + =∞ ∞

A B B 0Qσ2 (12.86)

The above equations can now be combined conveniently in the algorithm outlined

in Table 12.2.

Step 2 is calculated in advance, as well as the partial derivatives of K, C and f with respect to the random parameters (see later). The other steps are merged into

a time stepping scheme in which the mean temperature vector and all covariance

matrices are updated each time step. The linear differential systems Equation 12.73,

Equation 12.77 through Equation 12.79 can be solved using a similar time stepping

ference algorithm the following recursive relationship can be used

C

K uC

uΔ

ΔΔ Δt

ttt t t t+

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

+ − = +f (12.87)

55534_C012.indd 32455534_C012.indd 324 10/22/08 12:03:14 PM10/22/08 12:03:14 PM

th C and Κ the finite element matrices, and V and h vectors of dimension n

algorithm as in the case of a deterministic problem. For an implicit Euler finite dif-



Observe that the matrix C / +Δt K is to be triangularized only once.

12.5.3 ALGORITHM FOR RANDOM VARIABLE PARAMETERS

If all parameters are random variables, the mean value and the covariance matrix

derived by Nicolaï and Baerdemaeker [22]. The variance propagation algorithm

is then equivalent with the perturbation algorithm. Without loss of generality, this

equivalence will be proven below for the simple case of a random variable ini-

tial condition and thermal conductivity. This fact can be exploited advantageously,

as the perturbation algorithm involves only the solution of vector differential

equations.

Proof

variable parameters starts with a system of differential equations which describe the

sensitivity of the nodal temperature vector with respect to the random parameters.

For the case of a random variable initial condition and thermal conductivity, the

following system is obtained [22]

C K fddt

u u+ = (12.88)

Cddt k k k

∂∂

∂∂

∂∂

uK

u K⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+ = − (12.89)

Cddt T T

∂∂

∂∂

uK

u

0 0

0⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

+ = (12.90)

subject to the initial conditions

∂∂

∂∂

u

u

k

T

=

=

0

11 10

[ ]� T

TABLE 12.2Variance Propagation Algorithm

Step 1 Compute u from Equation 12.73 with initial condition (Equation 12.80)

Step 2 Solve the Lyapunov matrix Equation 12.84 through Equation 12.86

Step 3 Compute V V Vu x u x u x u u, , , ,, , ,T h Q V k∞V and ,ρc

from Equation 12.75 through Equation 12.79

Step 4 Compute Vu,u (t) by solving the Lyapunov matrix differential Equation 12.74

55534_C012.indd 32555534_C012.indd 325 10/22/08 12:03:15 PM10/22/08 12:03:15 PM

can be calculated by means of the first order perturbation algorithm which was

The first order perturbation algorithm for heat conduction problems with random



The covariance matrix is then calculated from

Vu u u u

uu, =⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ +

⎛

⎝⎜⎜

∂∂

∂∂

σ ∂∂

∂∂k k T

T

k2

0 0T ⎜⎜⎞

⎠⎟⎟⎟⎟

T

σT0

2 (12.91)

Differentiation of Equation 12.91 with respect to time yields

ddt

ddt k k

kVu u

u u, =⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

∂∂

∂∂

σT

22 +⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ +

∂∂

∂∂

σ ∂u uk

ddt k

ddt

T

k2 uu u

u

∂∂∂

∂∂

T0

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

+

T

T

T

T0

2

0

0σ

⎛⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

ddt T

T

T∂∂

σu

0

20

(12.92)

Further, Equation 12.89 and Equation 12.90 can be rearranged as

ddt k k k

∂∂

∂∂

∂∂

∂uC K

uC K

uC

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟= − − −− − −1 1 1 KK

u∂k

(12.93)

ddt T T

∂∂

∂∂

uC K

u

0

1

0

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

= − − (12.94)

Substitution of Equation 12.93 and Equation 12.94 in Equation 12.92 results in

ddt k k k

Vu K

uu

u u,1= − +⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟⎛⎝⎜⎜⎜

−C K∂∂

∂∂

∂∂

⎞⎞⎠⎟⎟⎟ − +

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

−

T

k kk k k

σ ∂∂

∂∂

∂∂

σ2 u u KuK C−T 2

CC K− ⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ −

⎛1 ∂

∂∂∂

σ ∂∂

∂∂

u u u uT T T T

T

T0 0

2

0 00 ⎝⎝

⎜⎜⎜⎞

⎠⎟⎟⎟⎟

−T

TKT TC σ0

2 (12.95)

u( )k,T0 around u

u uu u

≅ + +∂∂

Δ ∂∂

Δk

TkT0

0 (12.96)

55534_C012.indd 32655534_C012.indd 326 10/22/08 12:03:16 PM10/22/08 12:03:16 PM

The perturbation algorithm is based on the following first order Taylor expansion of



where

Δ = −

Δ = −

k k k

T T T0 0 0

From Equation 12.96 it follows that

V u u

uu k kk k

k, [( )( )]= − − =ε ∂∂

σ2 (12.97)

V u uu

u,T TTT0 00 0

0

2= − − =ε[( )( )] ∂∂

σT (12.98)

After substitution of Equation 12.91, Equation 12.97 and Equation 12.98 in Equation

12.95 the following Equation is obtained

ddt k

kTV C V

KuV Vu,u u,u u,= − +

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟−−1 K K

∂∂ u, uu

KV C+

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

−∂∂k

Tu u,k

TT (12.99)

Further, right multiplication of Equation 12.93 by σk2 and using Equation 12.97

gives

ddt k

k k kV C V CK

u, u= − −− −1 1 2K ,

∂∂

σu (12.100)

Equation 12.99 and Equation 12.100 are equivalent to Equation 12.74 and Equation

12.5.4 DERIVATIVES OF C, K AND F WITH RESPECT TO RANDOM PARAMETERS

The derivatives of C, K and f with respect to the random parameters can be com-

puted by differentiation of the element matrices and subsequent incorporation in the

global derivative matrices. The following expressions are easily derived from Equa-

tion 12.7 through Equation 12.9.

55534_C012.indd 32755534_C012.indd 327 10/22/08 12:03:17 PM10/22/08 12:03:17 PM

12.78 for the given stochastic specifications. This concludes the proof.



∂∂

=

∂∂

=

∂∂

=

∫

∫

CC

jj j

V

jj j

V

jj j

T

j

T

j

dV

kdV

h

ρφφ φφ

φφ φφ

KB B

K TT

j

j

j

dS

hT dS

Th dS

Q

S

jj

S

jj

S

j

∫

∫

∫

∂∂

=

∂∂

=

∂∂

=

f

f

f

∞

∞

φφ

φφ

φφφφ j

V

dVj

∫

12.6 NUMERICAL SOLUTION OF LYAPUNOV AND SYLVESTER DIFFERENTIAL EQUATIONS

12.6.1 ALGEBRAIC LYAPUNOV AND SYLVESTER EQUATIONS

The variance propagation algorithm requires the numerical solution of Sylvester

matrix differential equations of the form

d

dtt t t t t tV( ) A( )V( ) V( )B( ) C( )= + + (12.101)

where A, B and C are real matrices of dimensions r × r, s × s, and r × s, respectively,

so that V is of dimension r × s. If B = AT and r × s, the Sylvester differential Equation

12.101 reduces to a Lyapunov equation.

As with vector differential equations, implicit as well as explicit methods can

explicit algorithm is obtained by substitution of the differential operator in Equation

V(t + Δt) = [I + ΔtA(t)]V(t) + ΔtV(t) B(t) + ΔtC(t) (12.102)

The algorithm involves matrix multiplication and addition and is particularly simple

to implement. However, as in the case of ordinary differential equations, it will be

shown later that the algorithm is only conditionally stable provided that a suitable

time step has been chosen.

55534_C012.indd 32855534_C012.indd 328 10/22/08 12:03:18 PM10/22/08 12:03:18 PM

be applied for the numerical solution of matrix differential equations. A first order

12.101 by a first order forward difference operator:



An unconditionally stable implicit algorithm is obtained by substitution of the dif-

1

2

1

2I A V( ) V( ) I− +

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ + + −Δ Δ Δ Δt t t t t t t t( ) ΔtBB( )

V( ) C( )

t t

t t t t

+⎡

⎣⎢⎢

⎤

⎦⎥⎥

= + +

Δ

Δ Δ (12.103)

where the equality V V V= +12

12

is used. Equation 12.103 can be written as

DX + XE = F (12.104)

with

D I A( )

E I B( )

= − +⎡

⎣⎢⎢

⎤

⎦⎥⎥

= − +⎡

⎣⎢⎢

⎤

1

2

1

2

Δ Δ

Δ Δ

t t t

t t t⎦⎦⎥⎥

= + +

= +

F V( ) C ( )

X V( )

t t t t

t t

Δ Δ

Δ

An equation of the form Equation 12.104 is called an algebraic Sylvester equation

and is solved as follows [39].

First, D is reduced to lower real Schur form D′ by an orthogonal similarity trans-

formation U:

′ = =

′′ ′

′ ′ ′

D U DU

D

D D

D D

T

r r

1 1

2 1 2 2

1 2

0 0

0

,

, ,

, ,

�

�

� � �

� DDr r,

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

where the diagonal submatrices ′Di i, are of order at most two and UUT = I.

Similarly, E is reduced to upper real Schur form E′ by an orthogonal similarity

transformation U′:

′ = ′ ′ =

′ ′ ′′ ′

E E

E E E

E EU UT

s

s

1,1 1,1 1,

2,2 2,

�

�

� � �

0

0 0 �� ′

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥Es s,

55534_C012.indd 32955534_C012.indd 329 10/22/08 12:03:18 PM10/22/08 12:03:18 PM

ferential operator in Equation 12.101 by a first order backward difference operator:



where again ′Ei i, is of order at most two. Substitution of D and E in Equation 12.104

by UD′UT and, respectively, premultiplication by UT and postmultiplication by U′ yields the following system:

D′X′ + X′E′ = F′ (12.105)

with

F′ = UT FU′ (12.106)

X′ = UT XU′ (12.107)

The advantage of the transformation of Equation 12.104 to Equation 12.105, is that

the latter equation can be written as a system of mutual uncoupled algebraic matrix

equations of order at most two. These are equivalent to an ordinary algebraic system

of at most four equations which can be solved using appropriate techniques (e.g.,

variants of the method of Gauss). The solution must then of course be backtrans-

formed using Equation 12.107.

Other algorithms of the Runge Kutta and BDF type of order 1–6 for the solution

of Lyapunov equations are described by Scheerlinck et al. [40].

12.6.2 CONVERGENCE AND STABILITY ANALYSIS

A convergence and stability analysis of the explicit and implicit Euler methods for

the solution of Lyapunov matrix differential equations is now presented for a sto-

chastic heat conduction problem with random variable initial temperature and ran-

dom process ambient temperature. Although this is a very simple case, it allows the

investigation of some interesting features of the algorithms. A more comprehensive

stability and convergence analysis for stochastic heat conduction problems with ran-

dom process ambient and initial temperature is described elsewhere [41].

From Equation 12.74 it follows that the variance propagation algorithm for linear

lowing Lyapunov equation:

ddt

V AV V A( ) ( ) ( )t t t= + T (12.108)

V(t) = V0 at t = 0 (12.109)

with

A = −C K−1

A and V are square matrices of dimension nnod × nnod. Note that A is constant if

subject to the initial condition (Equation 12.109), can be solved numerically using

the explicit or implicit Euler method as outlined above. These methods are subcases

of the more general class of linear multistep methods which are among the most

55534_C012.indd 33055534_C012.indd 330 10/22/08 12:03:19 PM10/22/08 12:03:19 PM

heat conduction problems with random field initial temperature is given by the fol-

the surface heat transfer coefficient does not change over time. Equation 12.108,



popular methods for the solution of differential equations. The convergence and

stability theory of linear multistep methods is well established for scalar and vec-

tor differential equations [42,43] and can readily be extended to matrix differential

equations.

For future use a general expression for the exact solution of Equation 12.108 is

now derived. Assume that A has nnod distinct eigenvalues. A can than be written as

A = H Λ H−1 (12.110)

where H is the matrix of eigenvectors and Λ the diagonal matrix of eigenvalues. Note

that if there are eigenvalues with multiplicity larger than one, A can no longer be

diagonalized. The analysis can then be based on the Jordan canonical form, but this

is not elaborated further here. Substitution of Equation 12.110 in Equation 12.108,

premultiplication with H−1, and postmultiplication with H−T yields the following

matrix differential equation

d

dtW W WA= +Λ (12.111)

where

W H VH� − −1 T (12.112)

Since Equation 12.111 is completely uncoupled, it can be derived that its solution is

given by

Wi, j = ci, j exp [(λi + λj)t] (12.113)

where the ci, j are the integration constants which can be determined by imposing the

initial condition

W0 = H−1V0H−1,

with λi, λj the eigenvalues of A, i, j = 1, … , nnod

The general multistep matrix method is now introduced. For this purpose,

assume the following matrix differential equation:

d

dtt tV F V( ) ( , )=

with an appropriate initial value. A general linear multistep matrix method of order

α βl

l

k

n l l

l

k

n lt t=

+

=

+∑ ∑′ = ′0 0

V F( ) ( )Δ t (12.114)

55534_C012.indd 33155534_C012.indd 331 10/22/08 12:03:20 PM10/22/08 12:03:20 PM

k is defined as



where tn + l = (n + l)Δ t; V’(tn + l ) is the approximation of V(tn + l ); F’(tn + l ) = F(tn + l, V’tn + l );

l l

1 1

initial sequence V’(tj), j = 0, k − 1 must be provided for the algorithm to start.

Application of Equation 12.114 to the Lyapunov Equation 12.108 gives

α βl l

l

k

n l n l n lt t t=

+ + +∑ ′ = ′ + ′⎡⎣⎢

0

V AV V A( ) ( ) ( )Δ t T ⎤⎤⎦⎥

=∑

l 0

κ

(12.115)

An obvious property to be met by the general linear matrix method is that, in the

limit Δt → 0, the approximate solution V’(tn), n = 1,…, N = tf/Δt, converges to the

f f

same time n → ∞. This can be stated more precisely as follows:

be convergent if

tntfixed

lim ( ) ( )Δ →

′0V Vt tn n= (12.116)

holds for all n and for all starting values V′(tl) for which

lim ( ) (t ), 0, –1Δt→

= =0

′V Vt kl l l (12.117)

The conditions for convergence of the linear matrix multistep method are summa-

rized in the following theorem.

it is consistent

α αl

l

k

l

l

k

l

l

k

l= == = =

∑ ∑ ∑0

0 0 0

; β (12.118)

and zero-stable, which means that no root ξi of the characteristic polynomial

α ξll

k

il

=∑ =

0

0 (12.119)

is larger than one in modulus, and every root with modulus 1 is simple

Proof Let V′i be the i-th column of V′. Equation 12.115 can be rearranged as

α Δl

l

k

n l l

l

k

n lt t=

+

=

+∑ ∑′ = ′0 0

V A* *( ) ( )β t (12.120)

55534_C012.indd 33255534_C012.indd 332 10/22/08 12:03:21 PM10/22/08 12:03:21 PM

and α and β are the coefficients of the method. For both the explicit and the implicit

Euler method k = 1. The values of the coefficients α and β are given in Table 12.3. An

exact solution V(t), t ∈[0,t ]. The time final t is hereby kept constant, so that at the

Definition 1 The linear multistep method defined by Equation 12.115 is said to

Theorem 1 The method defined by Equation 12.115 is convergent if and only if



where

′ =

′

′

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

V

V

V

*

nnod

1

� (12.121)

′ =

′+ ′

′ + ′

=∑A

AV V A

AV V A

*

1 11

i ii

n

n i n i

,

nod

nod nod,

�

ii

n

=∑

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

1

nod

(12.122)

Equation 12.120 is a linear multistep vector algorithm. From Lambert [43], it follows

that under the conditions in Equation 12.118 and Equation 12.119

t

n n

n

tt t

fixed

* *lim ( ) ( )Δ →

′ =0V V (12.123)

for all tn, n = 1,2,…, provided that the initial sequence is chosen such that

lim ( ) ( )Δt

l lt t l … k→

∗ ∗= ′ , = , , −0

0 1V V (12.124)

Equation 12.123 and Equation 12.124 are obviously equivalent to Equation 12.116

and Equation 12.117 so that the theorem is proven.

Where Theorem 143 deals with the behavior of the approximate solution V′ if

local errors are accumulating in an adverse fashion. This is the subject of the linear

stability theory [43]. Before proceeding further, the following lemma is proven.

Lemma 1 The eigenvalues of A = − −C K1 are real and negative

Ax x x= − =−C K1 λ

Left multiplication of both sides by xTC yields

−x x xT Kx = TC λ

x (real or imaginary) and thus also if x is an eigenvector

x xT K > 0 (12.125)

55534_C012.indd 33355534_C012.indd 333 10/22/08 12:03:22 PM10/22/08 12:03:22 PM

Δt tends to zero, it is also interesting to investigate whether for a fixed time step the

Proof Let λ and x be an eigenvalue–eigenvector pair of A. Then, by definition,

Since K and C are both positive definite, the following relations hold for any vector



x xTC > 0 (12.126)

and both expressions are real scalars. As a consequence, λ must be real and negative.

By repeating the above derivation for each eigenvalue–eigenvector pair of A, the

proof is completed.

Using Lemma 1 it follows from Equation 12.112 and Equation 12.113 that, for

t → ∞, all solutions V(t) of Equation 12.108 satisfy

||V(t)|| → 0

be absolutely stable if, for a given Δt, the approximate solution V′(tn) of Equation

||V ′(tn)|| → 0 (12.127)

if n → ∞

The conditions for absolute stability of the linear multistep matrix method (Equation

12.115) are given by the following theorem

Theorem 2 Let A have nnod distinct eigenvalues. The linear multistep method (12.115) is absolutely stable if and only if the roots of the polynomial

l

k

l l i jlt p

=∑ − +

0

[ ( )]α β λ λΔ

are less then one in modulus for all i and j.

Proof Since A has by assumption nnod distinct eigenvalues λi, i = 1,…, nnod, the

eigendecomposition (Equation 12.110) exists. After premultiplication of by H−1 and

postmultiplication by H–T, the following equation is obtained from Equation 12.115

l

k

ll n lt t

=

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟+∑ − Λ ′

02

α βΔ W ( )) ( )+ ′ − Λ+

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟W t tn l

ll

α β2

Δ⎡⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥= 0 (12.128)

′ ′W H� − −1VV H T (12.129)

Equation 12.128 represents an uncoupled set of nnod × nnod equations:

l

kl

l i i j nt W t=

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟ , +∑ − ′0

2

α β λΔ ( lll

l j i j n lt W t) (+ − ′⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟ , +α β λ2

Δ )))⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

55534_C012.indd 33455534_C012.indd 334 10/22/08 12:03:23 PM10/22/08 12:03:23 PM

The following stability definition is now stated.

Definition 2 The linear multistep matrix method (Equation 12.115) is said to

12.108 satisfies

with W′ defined by



or

l

k

l l i j i j n lt W t i j=

+∑ − + ′⎡⎣⎢

⎤⎦⎥ = =

0

0α β λ λΔ ( ) ( ), , , 11, ,… nnod (12.130)

By Equation 12.129, ||V′|| → 0 as n → ∞, if and only if || W′|| → 0 as n → ∞, and

i,j n

12.130 satisfy

|W′i,j (tn)| → 0 if n → ∞, i,j = 1,nnod (12.131)

The solutions of each of the difference equations in Equation 12.130 are given by

i j n

l

k

i j l i j ln

W t c p,

=

, , , ,′ =∑( )

1

(12.132)

where ci,j,l i,j,l, l = 1,…, k are the roots of the

polynomial

l

k

l l i jlt p

=∑ − +

0

[ ( )]α β λ λΔ (12.133)

pi,j,l satisfy

|pi,j,l | < 1

for all i, j and l. This concludes the proof.

Both the implicit and explicit Euler methods are of order k=1, so that the single

root of Equation 12.133 is equal to

pt

ti j

i j

= −− +− +

α β λ λα β λ λ

0 0

1 1

ΔΔ

( )

( )

Substitution of α1 and β1 with the values given in Table 12.3 results in the following

conditions:

Explicit: |1 + Δt(λi + λj)| < 1 (12.134)

Implicit:− +

<1

11

Δt i j( )λ λ (12.135)

for all i and j. Using Lemma 1, the following conclusions regarding the stability of

the implicit and explicit Euler method can be drawn. From Equation 12.135 it follows

55534_C012.indd 33555534_C012.indd 335 10/22/08 12:03:25 PM10/22/08 12:03:25 PM

hence Equation 12.127 is satisfied if and only if all solutions W′ (t ) of Equation

are arbitrary coefficients, and p

Clearly, Equation 12.131 and consequently Equation 12.127 are satisfied if the roots



the eigenvalues are negative and real. On the other hand, the explicit Euler method

will only be stable if and only if the following condition is met:

−2 < Δt(λi + λj) < 0

since the eigenvalues are negative and real. In this case Δt cannot be chosen freely.

12.7 APPLICATION TO THERMAL STERILIZATION PROCESSES

In order to illustrate the above algorithms, we will now analyze a typical thermal

food process with a random variable ambient temperature. The problem consists of a

0

content tomato concentrate with k = 0.542 W/m°C, ρc = 3.89106 J/m3°C. The follow-

ing process conditions were applied: T0 = 65°C, h = 100 W/m2°C. The ambient tem-

perature is now described by means of an AR(1) process with T T∞= =∞

125 1o oC C, σ

and a1 = 0.00277 s–1

region [0,r0] × [0,L/2] is subdivided in 100 axisymmetric linear quadrilateral ele-

ments. The time step is set equal to 36 s.

The Monte Carlo and variance propagation algorithms were programmed on top

In Figure 12.7 the temperature variance at three different positions in the center-

plane of the can are shown as calculated by means of the Monte Carlo method with

100 and 1000 runs, and the variance propagation algorithm. The agreement between

the Monte Carlo method with 1000 runs and the variance propagation algorithm is

good, but the Monte Carlo method with 100 runs is not very accurate. For the mean

shown). The relative CPU time (total CPU time divided by CPU time for determin-

istic simulation) was equal to 74, 242 and 2426, for the variance propagation, Monte

Carlo with 100 runs and Monte Carlo with 1000 runs, respectively.

12.8 CONCLUSIONS

In this chapter some algorithms for stochastic heat transfer analysis are outlined. In

the Monte Carlo method a large number of process samples is obtained by solving

TABLE 12.3

for the Explicit and Implicit Euler Method

Explicit Implicit

α0 −1 −1

α11 1

β01 0

β10 1

55534_C012.indd 33655534_C012.indd 336 10/22/08 12:03:25 PM10/22/08 12:03:25 PM

Coefficients of the Linear Multistep Method

that the stability conditions for the implicit Euler method will always be satisfied as

cylindrical container (radius r =3.41 cm, height L= 10.02 cm) filled with 30% solids

. An implicit Euler finite difference method in the time domain

was used to integrate the differential systems. For the finite element analysis the

of the existing finite element code DOT [44].

value an excellent agreement between the different method was observed (figure not



forward statistical analysis of the simulation results yields the mean values and vari-

ances of the temperature. The variance propagation algorithm is based on stochastic

systems theory and was originally developed for systems of ordinary differential

equations. The formalism is here applied to the spatially discretized heat conduc-

tion equation to yield a system of matrix differential equations which can be solved

numerically.

The Monte Carlo method in general requires a large amount of computer time

to obtain results with an acceptable accuracy. Also, it requires a complete stochastic

only the mean values of the parameters and their covariance matrices must be known.

However, the latter algorithm can provide only limited statistical information such

as the mean value and the variance, whereas the Monte Carlo method can also be

applied to derive other statistical characteristics such as the probability density func-

tion. Also, as the variance propagation algorithm are essentially based on a lineariza-

tion of the governing equations around their mean solution, they are only applicable

ACKNOWLEDGMENTS

The European Communities (FAIR project FAIR–CT96-1192) and the Flemish Min-

NOMENCLATURE

i

c Heat capacity

C Finite element capacity matrix

ε Mean value operator

f Probability density function

f Finite element thermal load vector

g Vector valued function

FIGURE 12.7 Temperature variance as a function of time in a heated A1-can with random

process ambient temperature at three different positions. −: Variance propagation algorithm;

∗: Monte Carlo with 100 runs; +: Monte Carlo with 1000 runs.

0.5

0.4

0.3

0.2

0.1

0.0

0 900 1800 2700 3600

Time (s)

r = 0.0 cm

r = 1.71 cm

r = 3.41 cm

Tem

per

ature

var

iance

(°C

2)

55534_C012.indd 33755534_C012.indd 337 10/22/08 12:03:26 PM10/22/08 12:03:26 PM

the heat transfer model for artificially generated random parameter samples. Straight-

specification of the random parameters, while for the variance propagation algorithm

if the variability is relatively small (coefficient of variation smaller than 0.2).

ister of Science and Technology are gratefully acknowledged for financial support.

a Coefficient of autoregressive process or wave



h 2

h Vector valued function

k Thermal conductivity (W/m°C)

K Finite element stiffness matrix

L Half-height of can

n Outward normal

nnod Number of nodes

nMC Number of Monte Carlo runs

0 Zero matrix

r0 Radius (m)

S Surface

t Time (s)

T Temperature (°C)

T0 Initial temperature (°C)

T∞ Retort temperature (°C)

u Nodal temperature vector

V Covariance function, volume

V Covariance matrix

W White noise process

X Random vector

Y Auxiliary random process/wave

z Position vector

Z Discrete time white noise process

Δt Time step

Γ Convection surface

ρ Density (kg/m3)

σ Standard deviation

τ Separation time

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