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

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FOOD

PROCESSING

OPERATIONS

MODELING

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

Design and Analysis

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CRC Press is an imprint of the

Taylor & Francis Group, an informa business Boca Raton London New York

FOOD

PROCESSING

OPERATIONS

MODELING

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

Design and Analysis

E D I T E D B Y

Soojin Jun

Joseph M. Irudayaraj

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v

Preface

The 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

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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 processmodel-ing operations. In providmodel-ing 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, PhD Joseph M. Irudayaraj, PhD

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ix

Editors

Soojin 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 Baerdemaeker

Department of Biosystems Katholieke Universiteit Leuven Leuven, Belgium

Sundar Balsubramanian

Department of Biological and Agricultural Engineering Louisiana State University

Agricultural Center Baton Rouge, Louisiana

Patrick J. Cullen

School of Food Science and Environmental Health Dublin Institute of Technology Dublin, Ireland

Ali Demirci

Department of Agricultural and Biological Engineering

The Pennsylvania State University University Park, Pennsylvania

S. Gunasekaran

Department of Biological Systems Engineering

University of Wisconsin Madison, Wisconsin

Joseph Irudayaraj

Department of Agricultural & Biological Engineering Purdue University West Lafayette, Indiana

Soojin Jun

Department of Human Nutrition, Food and Animal Sciences University of Hawaii Honolulu, Hawaii

Harpreet Kaur Khurana

Department of Human Nutrition Food and Animal Science University of Hawaii Honolulu, Hawaii

Kathiravan Krishnamurthy

Department of Food and Animal Sciences

Alabama A&M University Normal, Alabama

Si-Quan Li

Department 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. Nicolaï

Colm P. O’Donnell

UCD School of Agriculture, Food Science and Veterinary Medicine

University College Dublin Dublin, Ireland

Virendra M. Puri

Department of Agricultural and Biological Engineering

The Pennsylvania State University University Park, Pennsylvania

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K.P. Sandeep

Department of Food Science North Carolina State University Raleigh, North Carolina

Sudhir Sastry

Department of Food, Agricultural and Biological Engineering

The Ohio State University Columbus, Ohio

Nico Scheerlinck

J. Antonio Torres

Department of Food Science & Technology

Oregon State University Corvallis, Oregon

Gonzalo Velazquez

Department of Food Science & Technology, UAM Reynosa-Aztlán Universidad Autónoma de Tamaulipas Tamaulipas, México

Pieter Verboven

Jian Wang

Department of Biological Systems Engineering

Washington State University Pullman, Washington Yifen Wang Department of Biosystems Engineering Auburn University Auburn, Alabama Wade Yang

Department of Food and Animal Sciences

Alabama A&M University Normal, Alabama

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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 55534_C001.indd 1 55534_C001.indd 1 10/22/08 8:19:34 AM10/22/08 8:19:34 AM 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.

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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

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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 E 2 2 2 2 ∂ Φ ∂ 2 ∂ ∂∂ Φ ∂ Φ∂ ∂Φ∂ x + x y+ y + x + ∂∂Φ ∂y+FΦ+ =G 0

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 ∂ u x v y u x v y 0 + + + = The conservative form of the same equation is as follows:

∂

∂x( )+ρu ∂∂y(ρv)=0 or Δ ρ⋅( 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.

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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 formuFormu-lations 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

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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 per-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

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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 difdif-ference between the PDE and the finite difdif-ference

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

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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

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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 extrapolrelax-ation (and is used usually for Laplace’s equrelax-ation 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: ui ju u i 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:

ui j u u k i j k k i j , , _, ++1 − 1 Δx

i + 1,j is the value at the previous location, ui,j. Once ui + 1,j is k

i + 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, , ui ju u u u k i j k k i j k i j i j + ++ − + − ++ k ΔΔx 1.9 INTRODUCTION TO THE FEM 1.9.1 HOWIT 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, u

determined 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.

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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 FORMATIONTO 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].

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1.9.6 SOLUTIONOFTHE SYSTEMOF 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 SUMMARYOFTHE STEPS INVOLVEDINA 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,

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1.11 COMMERCIAL CODES AND RESOURCES AVAILABLE

Commercial codes available these days are often also packaged with pre- and post-processing modules. Pre-post-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.

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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.

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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

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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

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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 bac-death is increasing), and final bac-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

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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 FACTORSAND PROBLEMS ASSOCIATEDWITH 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 conductivviscos-ity 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

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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 founda-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 TYPESOF 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= 0+K γ

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

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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:

μe=μ(1+2.5Φ+14.1Φ2)

Temperature is a major factor that affects the viscosity of Newtonian and non-equation is the most commonly used non-equation to determine the effect of temperature

μ= Be− /EaR Tg

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

N u d n n n n n n GRe 2 3 3 1 = + − − ρ〈 〉 K[( )/ ] 2 NRe=ρ μ ud

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 l ow 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

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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 (N_De

N N

D

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

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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 fl ow 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, flow 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 f low 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

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[8] is: N d D Re 1/2 c=2100 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 N c 2 De 2 2 1 0.0306 288 0.012 i π = ⎛ ⎝ ⎜⎜⎜ ⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ + − NNDe 2 4 288 ⎛ ⎝ ⎜⎜⎜ ⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟

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:

E P fu L d f f 2 =Δ = ρ 2

In the above equation, f is t pipe is given by f_s = 16/N_Re

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 ⎛ ⎝ ⎜⎜⎜ ⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ 4 Re ln . . ε 55534_C002.indd 20 55534_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 tfric-he 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

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

FOOD

PROCESSING

OPERATIONS

MODELING

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

Design and Analysis

FOOD

PROCESSING

OPERATIONS

MODELING

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

Design and Analysis

E D I T E D B Y

Soojin Jun

Joseph M. Irudayaraj

Table of Contents

Preface

Editors

Contributors

1

Introduction to

Modeling and

Numerical Simulation

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

2

Aseptic Processing

of Liquid and

Particulate Foods

K.P. Sandeep and Virendra M. Puri