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

S E C O N D

E D I T I O N

FOOD PROCESSING OPERATIONS MODELING Design and Analysis

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S E C O N D

E D I T I O N

FOOD PROCESSING OPERATIONS MODELING Design and Analysis

EDITED BY

Soojin Jun Joseph M. Irudayaraj

Boca Raton London New York

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

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5553-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Table of Contents Preface.....................................................................................................................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

Chapter 10

Table of Contents

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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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 interest in relation to food safety and quality. With rapid adaptation, modification, and infusion of new processes and instrumentation, tomorrow’s consumers will have access to safe, nutritious, high-quality products via novel food processing technologies. High pressure processing (HPP), pulsed electric field (PEF), ohmic heating, ozone 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 computational fluid dynamics (CFD) in which the pressure dependence of latent heat and physical properties of foods can be efficiently interpreted. As described in 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 chemistry and fluid mechanics useful in predicting how real process equipment is likely to respond is discussed in Chapter 9. An introduction to fouling models for heat exchangers accounting for the hydrodynamics and thermodynamics of fluid flow, 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 ranging 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 elaborate on various models available and the influence of different factors on microbial inactivation. In addition, new modeling approaches for infrared heating that include the temperature dependence of spectral distribution and ohmic heating coupled with CFD tools have been addressed. Modeling of multi-phase food products with various electrical conductivities has been introduced in the chapter on ohmic heating. Distortion of electric field due to several factors such as heterogeneous food materials and irregular 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 vii © 2009 by Taylor & Francis Group, LLC

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Preface

of applying the theories to solve practical problems relevant to food process engineering at a higher level. This book is not intended to be a complete book on modeling 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, PhD Joseph M. Irudayaraj, PhD

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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 professor 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, nanotech and applications, biosensors, food packaging, and food safety engineering. Dr. Jun is also a member of the Institute of Food Technologists and American Society of Agricultural and Biological Engineers. Joseph M. Irudayaraj received his PhD from Purdue University in food and bioprocess 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 refereed 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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Contributors Josse De Baerdemaeker Department of Biosystems Katholieke Universiteit Leuven Leuven, Belgium

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

Sundar Balsubramanian Department of Biological and Agricultural Engineering Louisiana State University Agricultural Center Baton Rouge, Louisiana

Kathiravan Krishnamurthy Department of Food and Animal Sciences Alabama A&M University Normal, Alabama

Patrick J. Cullen School of Food Science and Environmental Health Dublin Institute of Technology Dublin, Ireland

Si-Quan Li Department of Research and Development Galloway Company Neenah, Wisconsin

Ali Demirci Department of Agricultural and Biological Engineering The Pennsylvania State University University Park, Pennsylvania

Kasiviswanathan Muthukumarappan Department of Agricultural and Biosystems Engineering South Dakota State University Brookings, South Dakota

S. Gunasekaran Department of Biological Systems Engineering University of Wisconsin Madison, Wisconsin

Bart M. Nicolaï Department of Biosystems Katholieke Universiteit Leuven Leuven, Belgium

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

Colm P. O’Donnell UCD School of Agriculture, Food Science and Veterinary Medicine University College Dublin Dublin, Ireland

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

Virendra M. Puri Department of Agricultural and Biological Engineering The Pennsylvania State University University Park, Pennsylvania xi

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Contributors

K.P. Sandeep Department of Food Science North Carolina State University Raleigh, North Carolina

Pieter Verboven Department of Biosystems Katholieke Universiteit Leuven Leuven, Belgium

Sudhir Sastry Department of Food, Agricultural and Biological Engineering The Ohio State University Columbus, Ohio

Jian Wang Department of Biological Systems Engineering Washington State University Pullman, Washington

Nico Scheerlinck Department of Biosystems Katholieke Universiteit Leuven Leuven, Belgium 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

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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to 1 Introduction Modeling and Numerical Simulation K.P. Sandeep, Joseph Irudayaraj, and Soojin Jun CONTENTS 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Introduction .......................................................................................................1 Classification of Partial Differential Equations ................................................3 Numerical Formulation .....................................................................................3 Classification and Generation of Grids .............................................................4 Boundary and Initial Conditions.......................................................................5 Errors, Consistency, Stability, Compatibility, and Convergence ......................6 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 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 differential 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— differential method, integral method, and stochastic method. The finite difference method falls under the differential method category. Under the integral method we have the variational method, finite volume method (FVM), and method of weighted residuals. The method of weighted residuals can be further divided into four categories—collocation method, subdomain method, Galerkin’s method, and least squares method while the finite volume (or control volume) method can be categorized into two groups—cell-centered schemes and nodal point schemes. The 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 element method in that it uses a similar approach but for the surface or boundary under consideration. It can be used in conjunction with the finite element or FVM. The 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. The FEM and the finite difference methods (FDM) are the most popular techniques used to solve problems associated with food processing. Relatively simple problems can be tackled with ease by commercially available software. Complicated problems require either modification of commercial codes or writing the code from scratch. The FDM has been very popular owing to its simplicity in formulation and ease in modification (especially while introducing different relaxation factors as can be seen 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 computational fluid dynamics (CFD) codes and has been extensively used in many of its applications [1]. This involves the disretization of the equations over the entire solution domain and rigorous conservation of mass and heat flux on each face of the finite volume. For modeling of food processing, it is often difficult to decide which solution 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 rectangular domain then the traditional finite difference methods should be the preferred discretization strategy [2]. When the boundary conditions are irregular then the finite element methods could be adopted. To capture the behavior of the physics and the conservation laws more rigorously, the finite volume techniques are preferred [3].

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1.2 CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS Partial differential equations (PDEs) are classified as linear or non-linear depending 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: 2 2 2 Φ Φ Φ ∂Φ ∂Φ A∂ 2 +B ∂ + FΦ + G = 0 +C ∂ 2 + D +E ∂x ∂ x ∂y ∂y ∂x ∂y

The coefficients 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 the first order PDEs at least for the purpose of classification. This formulation may then be used for solving the system of equations too. A PDE is said to be in conservative form (or conservation form or conservationlaw form or divergence form) if the coefficients of all the derivative terms in the 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

∂u ∂v ∂ρ ∂ρ +ρ +u + v =0 ∂x ∂y ∂x ∂y

The conservative form of the same equation is as follows:  ∂ ∂ (ρu)+ (ρv) = 0 or Δ ⋅ (ρV) = 0 ∂x ∂y 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 Numerical formulations are based on the classification of the governing equation. When dealing with the unsteady state heat equation or the scalar (linear or non-linear)

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Burger’s equation, formulations applicable to parabolic equations are used. When dealing with the wave equation, formulations for hyperbolic equations are used, and when dealing with Laplace’s equation, formulations for elliptic equations are used. Formulations 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 method, Crank–Nicolson method, and the Beta formulation. The five-point and nine-point methods are the commonly used methods to address elliptic problems. Euler’s forward time forward space (FTFS), Euler’s FTCS, first upwind differencing, 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 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 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 methods 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 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 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 can be used to fit irregular, singly-connected and multiply connected domains. For 2-D geometries, the most common method of unstructured grid generation involves discretizing the domain into triangles (the most flexible shape to fit various kinds of boundaries). Advancing front method and the Delaunay method are two of the commonly used techniques for triangulation of the domain.

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The grid system used could be orthogonal—Cartesian, cylindrical, spherical (depending on the boundary configuration of system) or non-orthogonal such as triangular. 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 transformed 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 complex variables. Grid systems are also classified as fixed (independent of solution 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 number of grid points around a specific grid point or around a grid line. It is performed 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 method (where weighting factors are introduced to determine the relative influence of each point near the interface of the coarse and fine grid on the solution variable) and the separate regions method (in which there are two types of grids—interface and non-interface; at the fine grid boundary, interpolation of the values at the coarse grid is performed to obtain values of the variable). One of the easiest ways of obtaining staggered grids is by shifting the grid vertically or horizontally by half a grid space. This technique is used to improve the stability 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 variables being specified on the primary and secondary grids. Consider the example of the incompressible Navier–Stokes equations and consider a grid point in the system 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 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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1.6 ERRORS, CONSISTENCY, STABILITY, COMPATIBILITY, AND CONVERGENCE The errors associated with first-order accurate methods are known as dissipation 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). Truncation error. It is the error introduced by truncating terms in the finite difference formulation. It is the difference between the PDE and the finite difference 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 exact solution of the PDE (without round-off error) and the exact solution of the finite 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 mathematical operations. Consistency. It relates to the extent to which the finite difference formulation 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 applicable to marching problems only. In the solution of finite difference equations two 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 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 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 Once the finite difference equations have been formulated and the stability criteria 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. © 2009 by Taylor & Francis Group, LLC

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Gaussian elimination. It is an efficient means of solving algebraic equations, especially tridiagonal system of equations. Approximately N3 multiplications are required for solving N equations. To improve accuracy, equations are rearranged such that the largest coefficients occupy the diagonal (this process is called pivoting). 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 (e.g. Cartesian); type of domain (e.g. rectangular); size of coefficient matrix, and type of BCs.

1.7.2

ITERATIVE METHODS

Usually an initial solution is guessed, new values computed, and the process continued 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 (usually three unknowns that result in a tridiagonal coefficient matrix), it is called 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 finite difference formulation by tridiagonal formulation). This method applies to two- or three-dimensional cases. Fractional step method for parabolic equations. This technique involves splitting 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 the first iteration step) and the value of the variable at all grid points (at the previous iteration step are used) to solve for the variable at the grid point (i, j) at the new iteration 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 immediately 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 unknowns. The finite difference equation, when processed under the same guidelines as the point Gauss–Siedel method results in a system of linear equations with a tridiagonal coefficient matrix. This method has a faster convergence rate than the point 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 © 2009 by Taylor & Francis Group, LLC

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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 relaxation 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 relaxation 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: Lagging. The coefficient is used at the known value, i. Thus the formulation for u(∂u/∂x) becomes: ui , j

ui+1, j − ui , j Δx

There is only one unknown (ui + 1,j) in this expression and the finite difference formulation is linear. Iterative. This method involves updating the lagged value till convergence is reached. The formulation for this method is: k i, j

u

uik++11, j − u ik, j Δx

For the first iteration, u i + 1,j is the value at the previous location, ui,j. Once ui + 1,j is determined at k + 1, the coefficient u 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: 2uik+1, juik++1,1 j − (u ik+1, j )2 − u ik, j uik++1,1 j Δx

1.9 INTRODUCTION TO THE FEM 1.9.1

HOW IT WORKS

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. 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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boundary nodes, an element may also consist of a few interior nodes. The nodal 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 variable 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. Solution using the finite element technique is obtained predominantly by variational or weighted residual method. The variational approach has its foundations in variational calculus and requires the use of a functional while the weighted residual approach used the governing equations. The finite element procedure consists of the following steps.

1.9.2

DISCRETIZATION

This involves dividing the problem domain into subdomains. Generally for a onedimensional 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. Discretization 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

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 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 connectivity 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 global matrix. The two types of boundary conditions are natural and essential boundary conditions. Natural boundary conditions are convective boundary conditions while essential boundary conditions are constant or specified boundary conditions [7].

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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 values are a function of time and hence an appropriate finite difference time-stepping scheme is generally chosen.

1.9.7

SUMMARY OF THE STEPS INVOLVED IN A TYPICAL FINITE ELEMENT

(a) Discretize the problem domain and construct the finite element mesh. (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 the finite element results by experimental data. Further refinement of the existing finite element procedures will also increase [8]. Appropriate solution procedures 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 areas that will demand attention are adaptive finite elements and the application of parallel processing.

1.10 CFD MODELING The CFD codes provide understanding of the physics of a flow system through nonintrusive flow, thermal, and concentration field predictions. Obtaining accurate CFD 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, either at the user interface or through user-defined codes within the software [9]. The CFD codes required to discretize modeled fluid continuum are numerically obtained 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, and energy because fluid crosses the boundaries of discrete spatial volumes within the 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 are in critical flow regions. The outstanding issues associated with convection scheme, 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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1.11 COMMERCIAL CODES AND RESOURCES AVAILABLE Commercial codes available these days are often also packaged with pre- and postprocessing 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, processing, and post-processing are listed below. Adina R&D (http://www.adina.com) Phoenics (http://www.cham.co.uk) Fluent (http://www.fluent.com), Gambit, FLUENT, FIDAP, POLYFLOW, 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. 2. Turner, I.W., and Perre, P. 1996. A synopsis of the strategies and efficient resolution 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 4. Roache, P.J. 1972. Computational fluid dynamics. NM: Hermosa. 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. 8. Reddy, J.N. 1993. An introduction to the finite element method. NY: McGraw-Hill. 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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Processing 2 Aseptic 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 Convective Heat Transfer Coefficient.................................................. 29 2.4.2 Steam Quality ...................................................................................... 30 2.4.3 Dimensionless Numbers Governing Heat Transfer ............................. 30 2.4.4 Natural (free) and Forced Convection ................................................. 32 2.4.5 Transient Heat Transfer within Particles ............................................. 32 2.4.6 Hydrodynamic and Thermal Entrance Lengths .................................. 33 2.4.7 Heat Transfer Coefficient in Straight Tubes ........................................ 33 2.4.8 Heat Transfer Coefficient in Helical Tubes ......................................... 36 2.4.9 Heating Media and Equipment ............................................................ 37 13 © 2009 by Taylor & Francis Group, LLC

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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 contact heat exchanger), followed by holding it for a specified period of time (in a holding tube), cooling it, and finally packaging it in a sterile container. The use of high temperature for a short period of time (in comparison with conventional canning) in aseptic processing yields a high quality product. The demand for high quality 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 to ensure that all process calculations are performed after the deaeration stage and not based on the initial raw product. Another important part of an aseptic processing system is the back pressure valve which provides sufficient pressure to prevent boiling 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. Sterilization of the processing system, packaging system, and the air flow system prior to processing is of utmost importance. This is what is referred to as presterilization. The recommended heating effect for presterilization (using hot water) of the processing equipment for low-acid foods is the equivalent of 121.1°C for 30 minutes. The 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) used for sterilization. Presterilization of an aseptic surge tank is usually done by saturated steam and not hot water due to the large volume associated with the surge tank. Better product quality (nutrients, flavor, color, texture), less energy consumption, eliminating the need for refrigeration, easy adaptability to automation, use of any

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size package, use of flexible packages, and cheaper packaging costs are some of the advantages of aseptic processing over the conventional canning process. Some of the reasons for the relatively low number of aseptically processed products include slower filler speeds and higher overall cost. Aseptic processing also requires better 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 pasteurization temperatures but not at sterilization temperatures), separation of solids 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. 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 combination does not need to be sterile), and are easier to file with regulatory agencies. 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, puddings, 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 processing 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 commercially sterile product, all pathogenic microorganisms must be destroyed during aseptic processing. Bacteria are the primary organisms of concern in food processing. They multiply by the process called fission wherein one cell splits into two cells. The growth of bacteria 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 death phase (rate of death is increasing), and final death phase (numbers decrease at a constant rate). In order to extend the shelf-life of products, one of the techniques is to prolong the first two phases (lag and accelerated growth phase) of the growth of bacteria. Once bacteria reach the third stage (logarithmic phase), spoilage will occur

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rapidly. Some of the techniques to prolong the first two phases are refrigeration, 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 storage. Some of the other techniques of processing and preservation include hot-fill, 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 determine 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 makes it more energy efficient. In addition, the products are shelf-stable and hence do not require further control like refrigeration of frozen foods. Refrigerated foods (after pasteurization) require careful monitoring of the storage and distribution temperature. 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 viscosity 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 the section that deals with residence time distribution. Heat transfer from the carrier fluid to the particle is a function of the boundary layer surrounding the particle, which in turn is a function of the thermo-physical 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 particle-fluid interface. In addition, the existence of a residence time distribution 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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Knowledge regarding the spread of residence times for the fluid and particles is 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 foundation of aseptic processing in the US as early as 1927 when the HCF (heat, cool, fill) 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 cans and bottles) and the Dole-Martin aseptic process in 1948 (product sterilization 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 wet heat, followed by aseptic filling and sealing of cooled product in a superheated steam environment). The early 1960s was marked with the advent of a form-fill-seal 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 (blank carton) aseptic system. Soon, aseptic filling in drums and bag-in-box fillers 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

The type of fluid, flow characteristics, and fluid properties are some of the fluid mechanics aspects that are important in designing an aseptic process. These parameters and the system configuration, in turn are important factors that determine the choice of the pump to be used. The residence time distribution of the fluid elements, and more importantly that of the particles are the factors that eventually are used in designing holding tubes.

2.3.1

TYPES OF FLUIDS

Time, shear rate, temperature, and particle concentration are some of the factors that affect the viscosity of a fluid or suspension. Fluids are characterized as timedependent or time-independent depending on whether the shear stress experienced by a fluid under a constant shear rate varies as a function of time. If the shear stress 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 two categories, Newtonian and non-Newtonian. Newtonian fluids obey Newton’s . law of viscosity—shear stress (σ) and shear rate (γ) are linearly related, while nonNewtonian fluids do not have a linear relationship of shear stress versus shear rate. The Herschel–Bulkley model, given below, is the most commonly used model to describe the flow behavior of most liquid food products: . σ = σ 0 + K ( γ )n

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In the above equation, σ0 is the yield stress, K is the consistency coefficient, and n is the flow behavior index. For a Newtonian fluid, σ0 = 0, K = μ, n = 1. A pseudoplastic fluid is one for which σ0 = 0, n < 1 while a dilatant fluid is one for which σ0 = 0, n > 1. For a non-Newtonian fluid, the concept of apparent viscosity is introduced since 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 rate since it varies with shear rate. For a pseudoplastic fluid, the apparent viscosity decreases with an increase in shear rate, while for a dilatant fluid, the apparent viscosity increases with an increase in shear rate. When small particles of low concentration (Φ) are suspended in a f luid, the concept of effective viscosity (μe ) comes into picture. One of the equations used to determine the effective viscosity of a suspension is: μ e = μ (1 + 2.5Φ + 14.1Φ 2 ) Temperature is a major factor that affects the viscosity of Newtonian and nonNewtonian fluids. For a Newtonian fluid, the Arrhenius model, given by the following equation is the most commonly used equation to determine the effect of temperature on the viscosity of a fluid: μ = Be−Ea / RgT Thus, to determine the Arrhenius parameters B and Ea, a graph of ln(μ) versus 1/T is made, the slope of which is –Ea /Rg and the intercept is ln(B). Thus, the flow behavior of the fluid as a function of temperature can be modeled.

2.3.2

DIMENSIONLESS NUMBERS GOVERNING FLOW

When a fluid flows through a tube at low velocities, the flow is characterized by 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 generalized Reynolds number (valid for power-law fluids, in addition to Newtonian fluids) is defined as: N GRe =

ρ〈u 〉2−n d n K[(3n + 1)/n]n 2n−3

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

ρud μ

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

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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 coefficient without going into the turbulent regime is by using coiled tubes. Flow in coiled tubes is characterized by flow in the primary (axial) direction and also in the secondary (radial) direction. This is due to the radial pressure gradient that develops due to the centrifugal force. The secondary flow is characterized by two counterrotating vortices in the cross-section of the tube. The strength of these vortices depends on many factors such as the tube to coil diameter ratio, flow rate, viscosity, and pitch of the coil. The non-dimensional number that characterizes flow in a coiled tube is the Dean number (NDe), which is defined as follows: N De = N Re

d 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 attributed to the development of secondary flow. Dean [2] was the first to analyze mathematically the phenomenon of secondary flow in helically-coiled tubes. Dean obtained analytical expressions for the velocity profile valid for large radii of curvature (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 the fluid as a function of position. Truesdell and Adler [4] obtained a numerical solution of the Navier–Stokes equations which are valid over a wide range of curvature and Reynolds numbers. Taylor and Yarrow [5] found that secondary flow could stabilize laminar flow, 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 pressure drop in a tube formed into a helix can be up to four times as great as that in an identical straight tube. They also found that stable laminar flow can be maintained in helices at Reynolds numbers up to 8,000 or more. Both these facts imply the existence of strong secondary flow in helices. Secondary flow is known to increase 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 critical value of Reynolds number that separates laminar and turbulent flow are 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 to flow became 2.9 times that in a straight pipe at NRe = 6,000 (∼ NRe when flow becomes turbulent). When d/D = 1/15, turbulent flow was seen at NRe ∼ 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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laminar flow from turbulent flow. One such equation developed by Srinivasan et al. [8] is: ⎡ ⎛ d ⎞1/2 ⎤ N Rec = 2100 ⎢⎢1 + 12⎜⎜ ⎟⎟⎟ ⎥⎥ ⎝D⎠ ⎣ ⎦ It is important to note that most equations similar to the one developed above have a range of applicability. The limitations may be to the range of Reynolds number, tube to coil diameter ratio, pitch, or other factors. It is known that for a given pressure gradient, the flow rate in a coiled tube is lower than that in a straight tube. Several correlations have been developed to predict the flow rate in a helical tube. One such correlation is presented in a non-dimensional form below [9]: i

⎛ 2 ⎞4 ⎛ N 2 ⎞⎟2 Vc ⎜⎜ De ⎟ + 0.012⎜⎜ N De ⎟⎟ = − 1 0.0306 ⎜⎜ 288 ⎟⎟ ⎜⎜ 288 ⎟⎟ π R 2u ⎠ ⎠ ⎝ ⎝ 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 flow in helical tubes, just like comparisons of Reynolds numbers are made in straight tubes. It can also be seen that decreasing the coils diameter enhances the extent of secondary flow. Optimization is performed to choose the appropriate coil diameter 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

As a fluid flows through a pipe, friction impedes axial flow and creates a pressure drop between the inlet and outlet of the tube. The pressure drop depends on the type 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 (Ef ) and pressure (ΔPf ) due to friction and they are determined as follows: 2

Ef =

ΔPf u L =2f ρ d

In the above equation, f is the friction factor and it varies with the type of pipe, flow conditions and system geometry. The friction factor for laminar flow in a straight pipe is given by fs = 16/NRe and for turbulent flow it is usually determined from the 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 ε 1 1.255 = −4 ln ⎜⎜⎜ + ⎜ f ⎝ 3.7 d N Re f

⎞⎟ ⎟⎟ ⎟⎠

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These are standard methods for determining friction factors in straight pipes. However, 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.

2.3.4

PUMPS AND PUMPING REQUIREMENTS

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 of slippage (if applicable), piping arrangement, fittings present, and the flow behavior of the fluid. The first step in determining the type and rating of a pump required for any food processing operation involves the use of the Bernoulli’s equation: Ep = ΔPE + ΔKE +

ΔP + Ef ρ

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: 2

gZ1 +

2

U1 P U 2 P2 + 1 + Ep = gZ 2 + + + Ef 2Ψ ρ 2Ψ ρ

In the above equation, Ψ 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 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: i

Power = m Ep Once the power of the pump is determined, the next step is to determine the type of pump to be used. Pumps are broadly classified into two categories, centrifugal and 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 experiences maximum pressure and is forced out into the pipeline. For a centrifugal pump, the volumetric flow rate is directly proportional to the pump speed; the total head varies as the square of the pump speed; and the power required varies as the cube of the pump speed. In a positive displacement pump (rotary, reciprocating, axial flow pumps), direct force is applied to a confined liquid to make it move. Some of the factors involved in pump selection are as follows: 1. Flow rate of fluid. 2. Net positive suction head required (NPSHR)—depends on impeller design. It is required to maintain stable operation of pump including avoiding cavitation. © 2009 by Taylor & Francis Group, LLC

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3. Net positive suction head available (NPSHA)—depends on absolute pressure, vapor pressure of liquid, static head of liquid above center line of pump, friction loss in the suction system. 4. Properties of fluid (such as density, viscosity). 5. Characteristic pump curves (graph of head, power consumption, and efficiency versus volumetric flow rate).

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 holding tube is the fact that the residence time in the holding tube should be based on the specific volume of the product at the hold tube temperature and not on the displacement of the pump (since the pump is operating at a different temperature and specific volume varies with temperature). The velocity profile of the fluid in the holding tube 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-deWaale fluids. For a Newtonian fluid flowing under laminar conditions in a straight tube of circular cross-section, the maximum velocity occurs at the center of the holding 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 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 maximum 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 effect) can be calculated based on the knowledge of this minimum residence time, but this will result in an over-processed product. This is where the residence time distribution (RTD) of the particles comes into picture. To understand RTD, we begin with the following equation which describes the velocity profile for flow of a Newtonian fluid under laminar conditions in a pipe of circular cross-section: ⎡ ⎛ r 2 ⎞⎤ u = 2u ⎢⎢1 − ⎜⎜ 2 ⎟⎟⎟⎥⎥ ⎜ ⎟ ⎣ ⎝ R ⎠⎦ Thus, it can be seen that different fluid elements (at different radial locations) spend different amounts of time in the tube. For instance, a fluid element traveling at the center of the tube will travel twice as fast as the average fluid element. The distribution of times spent by various fluid elements within the tube is referred to as the RTD of the fluid elements. Similarly, when different particles are flowing through 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 RTD of the fluid. It also depends on flow rate and viscosity of the carrier medium, 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

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different types of particles (especially particles of different densities) are present in a product, the flow behavior is quite different from the situation when they are each 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 particle is what is of concern and the holding tube length is based on the fastest particle 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 can be used to narrow the RTD of the fluid and particles is the use of helical tubes. When a non-Newtonian (power-law) fluid flows through a straight tube under laminar flow conditions, the velocity profile is given by the following equation: u=

( n+1)/ n ⎤ ⎪ ⎫ 3n + 1 ⎪⎧⎪ ⎡⎢ ⎛⎜ r ⎞⎟ ⎥ ⎪⎬ ⎨u ⎢1 − ⎜ ⎟⎟ ⎥⎪ n + 1 ⎪⎪ ⎣ ⎝ R ⎠ ⎦ ⎪⎭ ⎩

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

3n + 1 u n +1

Hence it can be seen that the maximum velocity in the case of a pseudoplastic fluid is 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 of particles is also narrower when the carrier medium is a pseudoplastic fluid. Studies on RTD of fluid elements and particles in conventional holding tubes 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 tubes is narrower than that in conventional holding tubes. Some of the studies include those of Chen and Jan [18], Tucker and Withers [19], Ahmad et al. [20], and Sandeep et al. [21]. The general trend from these studies is that an increase in flow rate, particle 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 particle 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

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of slightly higher or slightly lower density. It should also be noted that the fastest particle is not always the critical particle (the particle that receives least heat treatment and hence lethality). Slower moving particles of lower thermal diffusivities could very likely receive less heat treatment than faster moving particles of higher thermal conductivities. This factor further complicates determination of process lethality.

2.3.6

FORCES ACTING ON FLUID ELEMENTS AND PARTICLES DURING FLOW

In order to solve for the trajectory and velocity of various particles during flow in a tube, we need to know the fluid flow characteristics. The equations that govern the flow of a fluid are the continuity and momentum equations. As particles flow through 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 system. The density, size, and shape of the particle are important particle characteristics 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. 2.3.6.1

Equations of Motion of the Fluid

The motion of the fluid is described by the continuity equation and three momentum equations. The continuity equation is as follows: ∂ρ f + ∇ ⋅ (ρf u) = 0 ∂t This reduces to: ∇⋅u = 0 for an incompressible fluid. The three momentum equations for the fluid phase are given in vector notation as follows: ρf

Du = ρf g + ∇ : τ Dt

For non-Newtonian fluids, the Ostwald-de Waele model is used [22]: ⎡1 ⎤ ( n−1)/ 2 τ = −m ⎢ ( Δ : Δ)⎥ Δ ⎢⎣ 2 ⎥⎦ where, the second invariant of the strain rate tensor, 1/2(Δ : Δ) is given by: 2 2 2 2⎤ ⎡⎛ 1 ⎢⎜ ∂uf ⎞⎟⎟ ⎛⎜ ∂ vf ⎟⎞⎟ ⎛⎜ ∂ wf ⎞⎟⎟ ⎥ ⎛⎜ ∂ vf ∂uf ⎞⎟⎟ + ( Δ : Δ) = 2 ⎢⎜⎜ ⎟ ⎟ +⎜ ⎟ ⎥ +⎜ ⎟ +⎜ ∂ y ⎟⎠ 2 ⎢⎜⎝ ∂ x ⎟⎠ ⎜⎜⎝ ∂ y ⎟⎠ ⎜⎜⎝ ∂ z ⎟⎠ ⎥ ⎜⎜⎝ ∂ x ⎣ ⎦

⎛ ∂ wf ∂ vf ⎞⎟2 ⎛ ∂ wf ∂uf ⎞⎟2 ⎟⎟ + ⎜⎜ ⎟⎟ + ⎜⎜⎜ + + ⎜⎝ ∂ y ∂ z ⎠⎟ ⎜⎜⎝ ∂ x ∂ z ⎟⎠ © 2009 by Taylor & Francis Group, LLC

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This takes into account the spatial variation of viscosity. Thus, the apparent viscosity of the fluid at any particular shear rate (at any particular location in the tube) can be determined for the corresponding temperature. The effect of temperature and concentration of particles on the effective viscosity of suspensions have been modeled by several researchers [23,24,25]. Thus, the above equations can be used to describe the flow behavior of a power-law fluid containing particles under non-isothermal conditions. 2.3.6.2

Linear Dynamic Equations for Particles

The three linear dynamics equations for the particles (in vector notation) are as follows: ⎛ dV ⎞ mp ⎜⎜⎜ pk ⎟⎟⎟ = ⎜⎝ dt ⎟⎠

∑F

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. Particles suspended in a viscous fluid are subjected to the following forces [26]: 2.3.6.2.1 Magnus Lift Force The 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 = πρ f a 3Ω × (Vp − Vf ) where Ω is the angular velocity of the particle. In the above expression, the difference in velocities Vp − Vf is called the slip velocity or relative velocity (Vr). The vector product Ω × Vr is determined as follows: Ω × Vr = ε ijk ΩjVrk = i(Ω y wr − Ω z wr ) + j(Ω z w r − Ω x wr ) + k(Ω x wr − Ω y wr ) 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: Fx = π a 3ρf [Ωy ( wp − wf ) − Ω( vp − vf )] Fy = πa 3ρf [Ω z (up − uf ) − Ω( wp − wf )] Fz = πa 3ρf [Ωx ( vp − vf ) − Ω(up − u f )] The experimental works of researchers [27,28] indicated that particles migrated radially 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 Force Saffman [29] developed an expression for the lift force acting on a particle during an unbounded shear flow. The shear lift force is independent of the particle rotation © 2009 by Taylor & Francis Group, LLC

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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 magnitude of the vorticity vector. Oliver [27] and Theodore [28] found that if the particle 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 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: ⎛ ⎞⎟1/2 ⎜ v Fs = 6.46ρf a ⎜⎜ ⎟⎟⎟ K × (Vp − Vf ) ⎜⎜ K ⎟⎟ ⎝ ⎠ 2

where K is the curl of the fluid velocity, v is the kinematic viscosity, and a is the radius of the particle. Vp and Vf are the velocities of particle and fluid, respectively. The expression for K in the above equation can be obtained as follows: ⎡ i ⎢ ⎢∂ K = ∇ ×V = ε ijk ⎢⎢ ⎢ ∂x ⎢u ⎢⎣ ⎛ ∂w ∂v ⎞ = i ⎜⎜ − ⎟⎟⎟ + ⎜⎝ ∂y ∂z ⎟⎠

j ∂ ∂y v

k ⎤⎥ ∂ ⎥⎥ ∂z ⎥⎥ w ⎥⎥⎦

⎛ ∂v ∂u ⎞ ⎛ ∂u ∂w ⎞ j ⎜⎜ − ⎟⎟⎟ + k ⎜⎜ − ⎟⎟⎟ ⎜⎝ ∂x ∂y ⎟⎠ ⎜⎝ ∂z ∂x ⎠⎟

We now define the relative or slip velocity, Vr, as follows: Vp − Vf = Vr = iur + jvr + kwr Evaluating the above expressions results in the scalar forms of the Saffman lift force in the x, y, and z direction, respectively as follows: ⎤ ⎛ v ⎞⎟1/2 ⎡⎛ ∂u ∂w ⎞ ⎛ ∂v ∂u ⎞ Fsx = 6.46ρf a ⎜⎜⎜ ⎟⎟ ⎢⎢⎜⎜⎜ − ⎟⎟⎟ ( wp − wf ) − ⎜⎜ − ⎟⎟⎟ ( vp − vf )⎥⎥ ⎜ ⎟ ⎟ ⎜⎝ K ⎠ ⎣⎝ ∂z ∂x ⎠ ⎝ ∂x ∂y ⎠ ⎦ 2

⎤ ⎛ v ⎞⎟1/2 ⎡⎛ ∂v ∂u ⎞ ⎛ ∂w ∂v ⎞ Fsy = 6.46ρf a ⎜⎜⎜ ⎟⎟ ⎢⎢⎜⎜ − ⎟⎟⎟ (up − uf ) − ⎜⎜ − ⎟⎟⎟ ( wp − wf )⎥⎥ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜⎝ K ⎠ ⎣⎝ ∂x ∂y ⎠ ⎝ ∂y ∂z ⎠ ⎦ 2

⎤ ⎛ v ⎞⎟1/2 ⎡⎛ ∂w ∂v ⎞ ⎛ ∂u ∂w ⎞ Fsz = 6.46ρf a ⎜⎜⎜ ⎟⎟ ⎢⎢⎜⎜ − ⎟⎟⎟ ( vp − vf ) − ⎜⎜⎜ − ⎟⎟⎟ (up − uf )⎥⎥ ⎜ ⎟ ⎟ ⎜⎝ K ⎠ ⎣⎝ ∂y ∂z ⎠ ⎝ ∂z ∂x ⎠ ⎦ 2

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

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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 The expression for the drag on a particle in a viscous fluid is given by: 1 Fd = Cdρf π a 2 Vf − Vp (Vf − Vp ) 2 where the drag coefficient, Cd is obtained from the following equation [30]: Cd =

24 (1 + 0.15 Re p0.687 ) for 1 < Rep < 1000 Re p

and the particle Reynolds number is defined by: Re p =

(ρ)(2a) Vf − Vp μ

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: Fb = (4 / 3) π a 3 (ρ f − ρp ) g Substituting the above four equations into the linear dynamic equation for the particle results in: ⎛ ⎞⎟1/ 2 dVp v 2⎜ mp = 6.46ρf a ⎜⎜ ⎟⎟⎟ K × (Vp − Vf ) + π a 3ρf Ω × (Vp − Vf ) ⎜⎜ K ⎟⎟ dt ⎝ ⎠ 1 4 + Cd ρf π a 2 Vf − Vp (Vf − Vp ) + π a 3 (ρf − ρf ) g 2 3 The above equation can be rewritten in the following manner to obtain the expressions 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: ⎤ ⎛ ⎞1/ 2 ⎡⎛ ∂u ∂w ⎞ ⎛ ∂v ∂u ⎞ dVp 2⎜ v ⎟ mp = 6.46ρf a ⎜⎜ ⎟⎟ ⎢⎢⎜⎜⎜ − ⎟⎟⎟ ( wp − wf ) − ⎜⎜ − ⎟⎟⎟ ( vp − vf )⎥⎥ ⎜ ⎟ ⎟ ⎜⎝ K ⎠ ⎣⎝ ∂z ∂x ⎠ dt ⎝ ∂x ∂y ⎠ ⎦ +πρf a 3[Ω y ( wp − wf ) − Ωz ( vp − vf )] 1 + Cdρf π a 2 Vf − Vp (uf − up ) 2 © 2009 by Taylor & Francis Group, LLC

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y-direction: ⎤ ⎛ ⎞1/ 2 ⎡⎛ ∂v ∂u ⎞ ⎛ ∂w ∂u ⎞ dVp 2⎜ v ⎟ mp = 6.46ρf a ⎜⎜ ⎟⎟ ⎢⎢⎜⎜ − ⎟⎟⎟ (up − uf ) − ⎜⎜ − ⎟⎟⎟ (up − uf )⎥⎥ ⎜⎝ ∂y ∂z ⎟⎠ ⎜⎝ K ⎟⎠ ⎣⎜⎝ ∂x ∂y ⎟⎠ dt ⎦ +πρ f a 3[Ω z(up − uf ) − Ωx ( wp − wf )] 1 4 + Cdρf πa 2 Vf − Vp ( vf − vp ) + π a 3 (ρf − ρ f ) g 2 3 z-direction:

mp

⎤ ⎛ v ⎞1/ 2 ⎡⎛ ∂w ∂v ⎞ ⎛ ∂u ∂w ⎞ dVp = 6.46ρf a 2 ⎜⎜⎜ ⎟⎟⎟ ⎢⎢⎜⎜ − ⎟⎟⎟ ( vp − vf ) − ⎜⎜⎜ − ⎟⎟⎟ ( vp − vf )⎥⎥ ⎜⎝ K ⎟⎠ ⎣⎜⎝ ∂y ∂z ⎟⎠ ⎝ ∂z ∂x ⎠ dt ⎦ +πρf a 3[Ω x ( vp − vf ) − Ω y (up − uf )] 1 + Cdρf π a 2 Vf − Vp ( wf − wp ) 2

This accounts for the description of the translation of the spheres. However, the particles 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: ⎛ dΩ ⎞ I ⎜⎜ k ⎟⎟⎟ = ⎜⎝ dt ⎟⎠

∑T

k

where I is the moment of inertia (I = 2/5 mp a2 for sphere) and Tk (k = x, y, z) is the local torque exerted by the viscous fluid on the surface of the particles. 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: dΩx 15μ = dt ρpa 2

⎡ π ⎛ ∂v ∂w ⎞ ⎤ ⎟⎟ − Ω ⎥ ⎢ ⎜⎜ + x ⎟ ⎢ 8 ⎜⎝ ∂z ∂y ⎟⎠ ⎥ ⎣ ⎦

dΩ y 15μ = dt ρpa 2

⎤ ⎡ π ⎛ ∂u ∂w ⎞ ⎟⎟ − Ω y ⎥ ⎢ ⎜⎜ + ⎢⎣ 8 ⎜⎝ ∂z ∂x ⎟⎠ ⎦⎥

dΩ z 15μ = dt ρpa 2

⎡ π ⎛ ∂u ∂v ⎞ ⎤ ⎢ ⎜⎜ + ⎟⎟ − Ω z ⎥ ⎢ 8 ⎜⎝ ∂y ∂x ⎟⎟⎠ ⎥ ⎣ ⎦

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The linear and angular dynamics equations for the particles have to be solved simultaneously along with the four equations of motion for the fluid phase in order to completely describe the flow dynamics of the suspension.

2.3.7

TECHNIQUES TO DETERMINE FLUID AND PARTICLE VELOCITY

The average fluid velocity can be calculated once the volumetric flow rate of the product is known. To determine the distribution of fluid residence times, salt injections, dye tracers, and fine particles are used. Magnetic resonance imaging can also be used under certain circumstances to obtain a fluid flow profile. Fluid flow profiles, though 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 velocimetry, 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 process are the convective heat transfer coefficient, effect of temperature on the physical and thermal properties of the product, mode of heat transfer (conduction or convection), 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 conventional means, the liquid part of the mixture gets heated first and then it transfers heat to the surface of the particles by convection. Further heating of the interior of the particles take place by conduction. The difference between the bulk fluid temperature and the center temperature of particles can be due to the low convective heat transfer coefficient between the fluid and the surface of the particle or due to the low thermal diffusivity of the particles. Since there is not much that can be done to enhance the thermal diffusivity of particles (other than by reformulation), efforts have been geared towards measuring and enhancing the convective heat transfer coefficient between the fluid phase and particles.

2.4.1

CONVECTIVE HEAT TRANSFER COEFFICIENT

Convective heat transfer coefficient has been described [31] as the thermal lag between the particle surface temperature and the fluid temperature for a particle being heated in a fluid. This is mathematically written as follows: Q = hfp Ap (Tps − Tf ) Some of the factors that affect the convective heat transfer coefficient between a fluid and particle (hfp) include the shape of the particle, surface roughness, the position of the particle in the tube, particle concentration, type of flow, and the thermoproperties of the fluid. Heat transfer for laminar flow in a straight tube (heat exchanger or holding tube) is usually low since there is very little mixing in the radial direction. However, flow in a coiled tube is fully three-dimensional and the rate of heat transfer can be much higher than that in a straight tube. This is due to the development of secondary flow © 2009 by Taylor & Francis Group, LLC

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(in the directions normal to the main direction of flow) due to the pressure gradient imposed as a result of the centrifugal forces present in the curved section. The secondary flow serves as a means of redistributing fluid elements in the radial direction, and thereby transferring heat more efficiently between the bulk of the fluid and the fluid elements near the tube wall.

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: i

Q = mst ( Hs − H c ) 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

Just like fluid flow characterization is done by the use of a dimensionless quantity called Reynolds number, convective heat transfer coefficient is represented by a dimensionless quantity called Nusselt number. Nusselt number is a dimensionless temperature gradient at the surface and is given by: N Nu = =

(dT / dy)surface hDc =− (T − Tsurface ) /Dc kf Temperature 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 including Reynolds number under different flow and heat transfer conditions including forced and free convection.

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Grashof number is the ratio of buoyancy to viscous forces and is of importance in free convection only (where buoyancy effects are significant). The expression for the generalized Grashof number (valid for power-law fluids, in addition to Newtonian fluids) is given by: N Gr =

3 n−1 2 gβ f ρ 2f (Tsurface − T∞ ) Dvertical (4 v1−n Dver tical ) n n−1 2 {K [(3n + 1) / n] (2 )}

The quantity β is the coefficient of volumetric thermal expansion and is given by the following expression: β=

1 ⎛⎜ ∂〈V 〉 ⎞⎟ ⎟ ⎜ 〈V 〉 ⎜⎝ ∂T ⎟⎠P

For the case of free convection, Nusselt number is a function of Grashof number. When dealing with Newtonian fluids, the expression for the generalized Grashof number reduces to the following form: N Gr =

3 gβ f ρ 2f (Tsurface − T∞ ) Dvertical μ2

Prandtl number, which is the ratio of momentum diffusivity (μ/ρ) and thermal diffusivity (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 fluids, in addition to Newtonian fluids) is given by: N GPr =

cpf K [(3n + 1) /n]n (2n−1 ) 4 v1−n d n−1kf

For Newtonian fluids, the above expression reduces to the following form: N Pr =

cpf μ f kf

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 involving unsteady state heat transfer and is defined as follows: N Bi =

Internal resistance hDc Dc /ks = = 1/h External resistance ks

Unsteady state heat transfer is characterized by another dimensionless quantity, which is the Fourier number and is defined as follows: N Fo =

αt k (1/Dc ) Dc2 Rate of heat conduction = = 2 ρcp Dc3 /t Rate of heat storage Dc

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Food Processing Operations Modeling: Design and Analysis

NATURAL (FREE) AND FORCED CONVECTION

Convective heat transfer can take place by natural or forced means. In natural convection flow occurs due to the differences in the density of the fluid as it comes into contact with a hot (or cold) surface, thereby resulting in buoyancy forces. The Nusselt number in this case is a function of the Grashof number (NGr) and the Prandtl number and it takes the following form: N Nu =

hLc = c1 ( N Gr N Pr )c2 k

In the above equation, L c is the characteristic length. For internal flows, the characteristic length is given by: ⎛ Cross-sectional area ⎞⎟ ⎟ Lc = 4 ⎜⎜ ⎜⎝ Wetted perimeter ⎟⎟⎠ 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 natural 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 number 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 particles—the first method, called the lumped capacitance method, assumes that the 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 − T∞ hA = exp⎜⎜⎜− ⎜⎝ ρcp V Ti − T∞

⎞ t ⎟⎟⎟ ⎟⎠

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. © 2009 by Taylor & Francis Group, LLC

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33

HYDRODYNAMIC AND THERMAL ENTRANCE LENGTHS

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 the cooling section and hence care has to be taken in designing these components of the aseptic processing system depending on the product. 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 and fully developed region. The entrance region is the region where the velocity and temperature profiles are still developing. The length required for the flow (laminar) to become fully developed (hydraulic length or hydrodynamic entry length, lh) is given by the following equation [32]: lh = 0.05( N Re )(d ) The above equation is also referred to as the Langhaar equation. In the transient 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 (thermal length, lt) is given by the following equation: lt = 0.036( N Pe )(d ) where the Peclet number (NPe) is given by: N Pe =

uf d αf

The Peclet number, given by the above equation (for Newtonian and non-Newtonian fluids) is a product of the Reynolds number and the Prandtl number.

2.4.7

HEAT TRANSFER COEFFICIENT IN STRAIGHT TUBES

While considering transfer of heat between a heating medium (such as water or steam) and a product for flow in a tubular heat exchanger, two convective heat transfer coefficients come into picture—convective heat transfer coefficient between the heating 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 the heat transfer coefficient based on the state of the heating medium (liquid or gas) and whether the flow is in a tube or in an annulus. Some of the correlations used commonly are listed below. The following equation is used to compute the (outside) heat transfer coefficient between steam and the wall of the heat exchanger [33]: ⎞⎟1/ 4 ⎛ kst3.0ρst2.0 gλ ⎜ ⎟ ho( hx ) = 0.725⎜⎜ ⎜⎝ (Tst − Tw( hx ) )do( hx )μ ⎟⎟⎠ © 2009 by Taylor & Francis Group, LLC

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The (inside) heat transfer coefficient between the wall of the heat exchanger and the product is computed using the following equation [33]: 0.14 ⎞1/ 3 ⎛ i ⎛ 3n + 1 ⎞⎟1/ 3 ⎜⎜ m cp( f ) ⎟⎟ ⎛⎜ m ⎞⎟ ⎟⎟ ⎜ ⎟⎟ N Nu = 2.0 ⎜⎜ ⎜ ⎟ ⎝ 4 n ⎟⎠ ⎜⎜ kf Lhx ⎟⎟ ⎜⎝ mw ⎠ ⎠ ⎝

The properties of the fluid are determined at the film temperature (Tfilm), given by the following expression: 1 Tfilm = (Twall + Tfluid ) 2 In the holding tube, the following equation is used to determine the (inside) heat transfer coefficient between the product and the wall of the holding tube [34]: ⎛ 15n 3 + 23n 2 + 9n + 1 ⎟⎞ ⎟ N Nu = 8.0 ⎜⎜ ⎜⎝ 31n 3 + 43n 2 + 13n + 1 ⎟⎟⎠ In the cooling section, the following equations are used to determine the inside and outside heat transfer coefficients [33]: Laminar flow: ⎛ i ⎞1/3 ⎛ 3n + 1 ⎞⎟1/ 3 ⎜⎜ m cp ⎟⎟ ⎟ N Nu = 2.0 ⎜⎜ ⎜ ⎝ 4 ⎟⎟⎠ ⎜⎜ kL ⎟⎟⎟ ⎠ ⎝ Turbulent flow: 8 3 N Nu = 0.023 N G0.Re N G1/Pr

While determining the inside heat transfer coefficient, the inside diameter of the tube and the properties of the fluid undergoing processing have to be used. While determining the outside heat transfer coefficient, the outside diameter of the tube and the properties of the cooling water have to be used. The flow in the cooling section (both internal and external flow) is considered to be laminar if the generalized Reynolds number is less than 2,100). The surface heat transfer coefficient between the fluid and the particle is computed using the following correlation [35]: ⎛ ⎞1.787 233 0.143 ⎜ d p ⎟ N Nu = 2.0 + 28.37 N G0.Re N G Pr ⎜⎜ ⎟⎟ ⎜⎝ d ⎠⎟ 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 presented [36] as a function of Peclet number and the local Graetz number, where the Graetz number (NGz) is given by: © 2009 by Taylor & Francis Group, LLC

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i

m cp N Re N Pr N Gz = = kx x/d The most commonly used equations for determining the Nusselt number for laminar flow in horizontal pipes are given as follows: for NRe NPr (d/L) < 100 N Nu = 3.66 +

⎛ μ b ⎞⎟0.14 0.085[ N Re N Pr (d / L )] ⎜⎜ ⎟ ⎟ 1.0 + 0.045[ N Re N Pr (d / L )]0.66 ⎜⎝ μ w ⎟⎠

for NRe NPr (d/L) > 100 0.14 0.33 ⎛ d⎞ ⎛μ ⎞ N Nu = 1.86 ×⎜⎜ N Re N Pr ⎟⎟⎟ ⎜⎜ b ⎟⎟⎟ ⎝ L ⎠ ⎜⎝ μ w ⎟⎠

Another commonly used correlation for laminar flow heat transfer in pipes is given by the following equation [37]: 0.14 ⎛ 3n + 1 ⎞⎟ 1/3 ⎛⎜ μ b ⎞⎟ ⎜ N Nu = 2.0 ⎜ N ⎜ ⎟ ⎝ 4 n ⎟⎟⎠ Gz ⎜⎝ μ w ⎠⎟⎟

Thus, it can be seen from the above equation that the heat transfer coefficient will be higher for pseudoplastic fluids (n <1) as compared to that for Newtonian fluids. For transition flow, a graph of the Colburn j-factor (jH ) versus the Reynolds number is provided by Perry and Chilton [38] to determine the convective heat transfer coefficient. The statement of the Colburn analogy between heat transfer and fluid friction is given by the following equation: jH =

1 −0.2 f = 0.023 N Re 2

Researchers [39] determined the convective heat transfer coefficients between a fluid and a particle and developed the following correlation to determine the Nusselt number: ⎞0.6272 ⎛ R − r ⎞⎟−0.1142 553 0.2716 ⎛ Lc ⎟ ⎜⎜ N Nu = 2.0 + 8.4703 N G0.Re N G Pr ⎜⎜ ⎟⎟ ⎝d⎠ ⎝ R ⎟⎟⎠ They found that the convective heat transfer coefficient increased with decreasing particle size or viscosity and increasing flow rate. It was also found that convective heat transfer coefficient was higher for a particle near the wall. When dealing with a scraped surface heat exchanger, the convective heat transfer coefficient is determined using an equation such as the one presented below [40]: 0.5 0.33 0.26 N Nu = 1.2 N Re N Pr N b

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A more general equation has been developed [41] which takes into account the speed of rotation and the type of fluid: ⎞0.62 ⎛ D ⎞0.55 ⎛ ρu ( D − D ) ⎞⎟ ⎛ ⎜⎜ s ⎟ C2 ⎜ D ( Ω / 2 π ) ⎟ ⎟⎟ ⎜⎜ s ⎟⎟ ( N )0.53 N Nu = c1 ⎜ ⎟⎟ N Pr ⎜⎜ b ⎟⎟ ⎜⎝ D ⎟⎟⎠ ⎜⎜⎝ ⎜⎜⎝ ⎟⎠ u μ ⎠ 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 the convective heat transfer coefficient should be used. It should also be noted that each of the equations have a range of applicability and also assumptions involved.

2.4.8

HEAT TRANSFER COEFFICIENT IN HELICAL TUBES

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 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. The following correlations have been developed [42] for constant wall heat flux heat transfer in curved tubes: 0.115 0.0108 N Nu = 3.31N De N Pr 0.476 0.2 N Nu = 0.913 N De N Pr

for 20 ≤ N De ≤ 1200 and 0.005 ≤ N Pr ≤ 0.05 for 80 ≤ N De ≤ 1200 and 0.7 ≤ N Pr ≤ 5

For constant wall temperature heat transfer in curved tubes, the following correlation was developed [43]: 0.5 0.1 N Nu = 0.836 N De N Pr for 80 ≤ N De ≤ 1200 and 0.7 ≤ N Pr ≤ 5

Other researchers [44] developed the following Nusselt number correlations for different ranges of Dean numbers: 2 N Nu = 1.7( N De N Pr )1/ 6

for

2 N Nu = 0.9( N Re N Pr )1/ 6

for 20 < N De < 100

0.07 0.43 1/ 6 ⎛ d ⎞ N Nu = 0.7 N Re N pr ⎜⎜ ⎟⎟⎟ ⎝D⎠

2 N De ≤ 20 and (N De N Pr )0.5 > 100

for 100 < N De < 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 2De NPr) > 100, the Nusselt number in the fully developed thermal region was © 2009 by Taylor & Francis Group, LLC

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1/ 6 proportional to N Pr and that for the thermal entry region, the Nusselt number was 1/ 3 proportional to N Pr .

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 injection (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 sterilizing 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 electric field. Plate heat exchangers have a large surface area and hence rapid heat transfer can take place (due to the turbulent flow conditions) to liquid foods and 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 or corrugated tube heat exchangers with flow in the co-current, counter-current or cross-flow mode. With viscous products, the blades of the scraped surface heat 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 particulates. 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 electric fields, irradiation, membrane separation, and high pressure are currently 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 forks), and the fluidized bed system (particles are separately sterilized in a fluidized 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 coefficient in these types of heat exchangers. The total energy transferred to the product can easily be determined by the following equation: •

Q = m cp ( ΔT ) © 2009 by Taylor & Francis Group, LLC

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where ΔT is the rise in temperature of the product. This information is then used to calculate the overall heat transfer coefficient in the heat exchanger making use of the following equation: Q = UA lm ΔTlm where the logarithmic mean temperature difference (ΔTlm) is given by: ΔTlm =

ΔT1 − ΔT2 ln( ΔT1 / ΔT2 )

with ΔT1 and ΔT2 being the difference between the temperatures of the hot and cold fluids at the inlet and exit of the heat exchanger, respectively. The quantity Alm is the logarithmic mean area and is given by: Alm =

Ao − Ai ln( Ao /Ai )

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 on the above outlined procedure to compute the overall heat transfer coefficient. For the same inlet conditions, it can be shown that the amount of energy transferred from the hot to the cold fluid is higher in the case of a counter-current heat exchanger.

2.4.11 GOVERNING HEAT TRANSFER EQUATIONS AND ENERGY BALANCE The energy equation in spherical coordinates is as follows: ρcp

∂T ∂ ⎛⎜ ∂T ⎞ ∂ ⎛⎜ ∂T ⎞⎟ 1 ∂ ⎛⎜ 2 ∂T ⎞⎟ 1 1 ⎟+ S ⎟⎟ + 2 = 2 ⎜k ⎜⎜ kr ⎜⎜ k sin θ ⎟⎟⎟ + 2 2 ∂t r ∂r ⎝ ∂r ⎟⎠ r sin θ ∂θ ⎝ ∂θ ⎟⎠ r sin θ ∂ϕ ⎜⎝ ∂ϕ ⎟⎟⎠

By symmetry, ∂T ∂T = =0 ∂θ ∂ϕ Also, S = 0 for the case where there is no heat source term. Thus, the energy balance equation reduces to: ρcp

∂T 1 ∂ ⎛⎜ 2 ∂T ⎞⎟⎟ = 2 ⎜ kr ⎟ ∂t ∂r ⎟⎠ r ∂r ⎜⎜⎝

For constant properties, the above equation reduces to the following form: α ∂ ⎛⎜ 2 ∂ T ⎞⎟ ∂ T ⎟⎟ − =0 ⎜r r 2 ∂ r ⎜⎝ ∂ r ⎟⎠ ∂ t © 2009 by Taylor & Francis Group, LLC

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This is the equation that has to be then solved to determine the temperature distribution within spherical particles. 2.4.11.1 Energy Balance in the Heat Exchanger 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: i

U hx Ahx (Tst − Tf ) = m f cpf (Tf′ − Tf ) + N phfp AE (Tf − T ps ) In the above equation, T f and T ps are the mean fluid and mean particle-surface temperatures, respectively and are given by: 1 1 Tf = (Tf′ + Tf ) and T ps = (Tps′ + Tps ) 2 2 The terms Tf’ and Tps’ are the fluid and particle-surface temperatures at the new time step (or the new spatial location). Also, the surface area of the heat exchange surface, Ahx, is computed as follows: Ahx = π d hx Δx The following equation is used to determine the overall heat transfer coefficient in the heat exchanger (Uhx): 1 1 1 x hx = + + U hx Alm ( hx ) ho( hx ) Ao( hx ) hi ( hx ) Ai ( hx ) khx Alm(hx) where the logarithmic mean area (Alm) is given by: Alm =

2πL ( Ro − Ri ) ln( Ro / Ri )

The terms 1/ho Ao, Δr/kAlm, 1/hi Ai 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

In the holding tube, the fluid loses heat to the particles and also to the surroundings. The overall energy balance equation in the holding tube is as follows: i

m f cpf (Tf′ − Tf ) = U ht Aht (T f − Tair ) + N phfp Ap (T f − T ps )

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The following equation is used to determine the overall heat transfer coefficient in the holding tube (Uht): x ins 1 1 1 x ht = + + + U ht Alm(ht) ho( hx ) Ao( hx ) hi ( hx ) Ai ( hx ) kht Alm ( ht ) kins Alm ( ht ) A similar approach to that used in the determination of the fluid and particle temperatures in the heat exchanger was then used to determine the temperatures of the fluid and particles in the holding tube. 2.4.11.3 Energy Balance in the Cooling Section In the cooling section, the fluid and particles lose heat to the cooling water. The overall energy balance equation in the cooling section is as follows: i

m f cpf (Tf − Tf′ ) = U cs Acs (T f −Tcw ) + N phfp Ap (T f −T ps ) The overall heat transfer coefficient in the cooling section (Ucs ) is computed using the same equation that was used to determine the overall heat transfer coefficient in the heat exchanger with all parameters for the heat exchanger being replaced by the corresponding parameters for the cooling section.

2.4.12 FOULING AND ENHANCEMENT OF HEAT TRANSFER As a product flows through a system, it tends to stick to the hot surface of the heat 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, the system has to be shut down and cleaned using a CIP solution. This translates to decreased productivity and increased costs. This is more of a problem with viscous, proteinaceous, and starchy foods and is predominant under laminar flow conditions. The use of appropriate flow rate and temperature can minimize fouling, but not eliminate it totally. The use of scraped surface and helical heat exchanger also minimizes fouling. Another detrimental factor associated with fouling is the fact that with increased fouling, the cross-sectional area available for flow decreases, thereby increasing flow velocity and decreasing the residence time (in the heat exchanger 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 The most commonly used method to determine the temperature within a product is the use of thermocouples or RTDs in the flow regime which may be installed in-line with the aid of clamps or compression fittings (such as SwagelokTM). Another technique to determine temperature within a fluid is the introduction of tracer capsules (or data tracers) in the flow, retrieving it at the exit, and downloading the temperature 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

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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 and melting point indicators (that melt at a specific temperature) are some of the other techniques that can be used to determine the temperature within fluids and also within particles. Thermoluminescent markers (that emit a certain wavelength of light depending on the temperature) can be used to determine the temperature within clear fluids on-line.

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 requirements. 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 FDA requires microbiological tests to prove the safety of a process with sufficient 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 acidified and low-acid canned foods.

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 various components in a food. The most direct technique to do this is by measuring the temperature at the desired locations. However, this is not easy in a continuous flow 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 microorganisms 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 withstand 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 © 2009 by Taylor & Francis Group, LLC

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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 nutritious 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 materials 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 processing 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 are first order equations and are given by the following equation: ⎛c⎞ ln ⎜⎜ ⎟⎟⎟ = −kTt ⎜⎝ c0 ⎠ where c0 is the initial concentration of the species under consideration, c is the concentration 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 microorganisms, and the rate of reaction is replaced by the decimal reduction time (DT) to yield the following equation (expressed in base 10): ⎛N⎞ t log⎜⎜ ⎟⎟⎟ = − ⎜⎝ N 0 ⎠ DT with the decimal reduction time (in minutes) being related to the rate of the reaction (in seconds) by the following expression: DT =

2.303 60 kT

When a semi-logarithmic plot is made between the number of viable microorganisms (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, thermal 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: kT = Be _ Ea / RgT 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 © 2009 by Taylor & Francis Group, LLC

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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 reaction on temperature is the quotient indicator method which defines a quotient indicator 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: Q10 =

DT kT +10 = = 1010/z kT D T +10

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: ⎛ D ⎞⎟ T − Tref log⎜⎜ ⎟=− ⎜⎝ Dref ⎟⎠ z 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 reduction 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 microorganisms at any given temperature, is given by: LR = 10 (T −Tref )/ z =

Dref DT

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: t

F=

t

∫ (LR)dt = ∫ 10 0

( T −Tref )/ z

dt

0

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: t

F0 =

∫ 10

( T −250 )/18

dt

0

Another term that is commonly encountered in aseptic processing is lethality. Lethality 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. © 2009 by Taylor & Francis Group, LLC

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An F0 value of 5 minutes indicates that the process is equivalent to a heat treatment 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: t

C=

∫ 10

( T −Tref )/ zc

dt

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 When insufficient data are available regarding cold spots in the product, a very conservative method is generally employed. The conservative approach involves the assumption 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 treatment 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 overcome. 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. The first step is to identify a supplier who can provide high quality 1/2" carrot 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 determine 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

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pump (Model 629A, Marlen Research Corp., KS) will be an appropriate choice. This type of pump will not only result in uniform product flow rate, but also minimal 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 continuous flow radio frequency heater (Model 464, Radio Frequency Co., Millis, MA) as the final heater. The SSHE serves to bring up the temperature of the product to room temperature and a certain elevated temperature and the RF heater is the finisher which has the effect of minimizing the difference in temperature between the fluid and particle since a low frequency (40.68 MHz) translates to a high depth of 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") is used. A helical holding tube results in the development of secondary flow and hence causes mixing of the solid–liquid mixture and hence translates to a relatively 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 cooling medium is chilled water flowing through the system at a high flow rate (20 gpm of chilled water constantly flowing in and out of the system and 200 gpm of water 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 an agitated raque tank (100 gallon PF-2-5-1A agitated filler, Food Systems Inc., 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 Scientific, Logan, UT) and multiplexer capable of handling 32 channels which were run using the software PC208W 3.0. The temperatures of the fluid are monitored 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 with sanitary fittings. 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 control back pressure valve in the vertical section and a wire mesh screen that ensured that only the fluid portion passed through the back pressure valve and the particles 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

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Food Processing Operations Modeling: Design and Analysis

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 change in viscosity of the suspension as a function of time. To aid this, benchtop studies should be conducted to determine the rheological behavior of CMC as a function of shear rate, time, and the high temperatures encountered during processing. This can be done using a controlled stress rheometer (Stresstech, ATS RheoSystems, Bordentown, NJ). During the modeling, conservative estimates of convective heat transfer coefficients and thermal diffusivities are used. Thermal conductivity is measured using the line heat source probe and specific heat is measured using a mixing calorimeter. The first phase in experimental studies is to incorporate tiny magnets (of different magnetic strengths) into several cube-shaped tracer particles in order to determine 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 time. The magnets are of low enough strength to not affect the electromagnetic field created by the RF system. Based on statistics, it has been shown that the residence times of at least 299 particles must be determined in order to have a 95% confidence 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 out into the fluid. If the target for the process was a 5D process, and an initial load of 105 spores per particle is used, a final count of <1 would indicate a safe process. The decimal reduction time of the organisms used is determined by means of thermal 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 the predicted temperatures and lethalities in order to ensure that the model results in a conservative prediction of process lethality. Once verification is successful, all the process and system parameters are noted 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 time, temperatures at various locations, product flow rate, back pressure, and product

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properties. The final step in commercialization of the product involves process filing with the FDA using form 2541C. A comprehensive overview of the procedures and processes involved in process filing for a product such as the one discussed above 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, such as the use of high pressure or pulsed electric field, have to be carefully studied, 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 consumers are becoming more conscious about the nutritive value of foods and leading a healthy lifestyle.

NOMENCLATURE a A B c co cp c1, c2 C Cd d D Dc Ds DT Dvertical Ea Ef Ep f F F0 Fb Fd Frk

Radius of particle, m Surface area, m2 Arrhenius parameter, Pa−s Final concentration of species Initial concentration of species Specific heat, J/kg−K Constants Cook value, min Drag coefficient Diameter of tube, m Diameter of helical coil, m Characteristic dimension, m Diameter of shaft of SSHE, m Decimal reduction time at temperature T, min Vertical dimension, m Activation energy, J/kg−mol Loss in energy due to friction, J/kg Energy to be supplied by pump, J/kg Friction factor Force, N F-value when reference temperature is 250°F and z-value is 18°F, min Buoyancy force, N Drag force, N Magnus lift force, N

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Fs g h Hc Hs Hv I jH k kT K K lh lt L Lc . m mp n N Nb NBi NDe NFo NGGr NGPr NGRe NGr NGz NNu No NP NPe NPr NRe NRec P Q Q10 r R Rep Rg S t T

Food Processing Operations Modeling: Design and Analysis

Saffman lift force, N Acceleration due to gravity, m/s2 Convective heat transfer coefficient, W/m2−K Enthalpy of condensate, J/kg Enthalpy of steam, J/kg Enthalpy of vapor, J/kg Moment of inertia, kg-m2 Colburn j-factor Thermal conductivity, W/m−K Reaction rate constant at temperature T, s−1 Consistency coefficient, Pa-sn Curl of velocity Hydrodynamic entry length, m Thermal entry length, m Length of tube, m Characteristic length, m Mass flow rate, kg/s Mass of particle, kg Flow behavior index Final bacterial count Number of blades Biot number Dean number Fourier number Generalized Grashof number Generalized Prandtl number Generalized Reynolds number Grashof number Graetz number Nusselt number Initial bacterial count Number of particles Peclet number Prandtl number Reynolds number Critical Reynolds number Pressure, Pa Energy transferred, W Quotient indicator Radial location, m Radius of tube, m Particle Reynolds number Universal gas constant, J/kg-mol-K Source term, W/m3 Time, s Temperature, K

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Ti Tk T∞ u –u u, v, w U. V V X z Z

49

Initial temperature, K Local torque exerted by fluid, N-m Free stream temperature, K Velocity, m/s Average velocity, m/s x, y, and z components of velocity respectively, m/s Overall heat transfer coefficient, W/m2 -K Volumetric flow rate, m3 /s Velocity, m/s Volume, m3 Quality of steam Temperature change required for an order of magnitude change in decimal reduction time, °C Height, m

GREEK LETTERS α β ρ ε ϕ, θ σ σ0 . γ λ λc μ μe ΔPf Δr, Δx ΔT ψ Ω Φ

Thermal diffusivity, m2/s2 Coefficient of volumetric thermal expansion, K−1 Density, kg/m3 Roughness of pipe, m Spherical coordinates Shear stress, Pa Yield stress, Pa Shear rate, s−1 Latent heat of vaporization, J/kg Curvature Viscosity, Pa−s Effective viscosity, Pa−s Pressure loss due to friction, Pa Thickness, m Temperature difference, K Constant Angular velocity, rad/s Particle concentration

SUBSCRIPTS b c cs f fp ht hx i ins

Bulk fluid Coiled tube Cooling section Fluid Fluid–particle interface Holding tube Heat exchanger Inside Insulation

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50

lm max o p ps ref s st w

Food Processing Operations Modeling: Design and Analysis

Logarithmic mean Maximum Outside Particle Particle surface Reference temperature Straight tube Steam Wall

REFERENCES 1. David, J.R.D., Graves, R.H., and Carlson, V.R. 1996. Aseptic processing and packaging of food: A food industry perspective. Boca Raton, FL: CRC Press, 21–29. 2. Dean, W.R. 1927. Motion of fluid in a curved pipe. Philosophical Magazine Series 7 4(20): 208–23. 3. Dean, W.R. 1928. The stream-line motion of fluid in a curved pipe. Philosophical Magazine Series 7 5: 673–95. 4. Truesdell, L.C., and Adler, R.J. 1970. Numerical treatment of fully developed laminar flow in helically coiled tubes. AIChE Journal 16: 1010–14. 5. Taylor, G.I., and Yarrow, F.R.S. 1929. The criterion for turbulence in curved pipes. Proceedings of the Royal Society of London A124: 243–49. 6. Koutsky, J.A., and Adler, R.J. 1964. Minimization of axial dispersion by use of secondary flow in helical tubes. The Canadian Journal of Chemical Engineering 42: 239–46. 7. White, C.M. 1929. Streamline flow through curved pipes. Proceedings of the Royal Society A123: 645–63. 8. Srinivasan, P.S., Nandapurkar, S.S., and Holland F.A. 1968. Pressure drop heat transfer in coils. The Chemical Engineer May 46: 113–19. 9. Berger, S.A., Talbot, L., and Yao, L.S. 1983. Flow in curved pipes. Annual Review of Fluid Mechanics 15: 461–512. 10. Moody, L.F. 1944. Friction factors for pipe flow. ASME Transactions 66: 671–84. 11. Colebrook, C.F. 1939. Friction factors for pipe flow. Institute of Civil Engineering 11: 133–156. 12. Manlapaz, R.L., and Churchill, S.W. 1980. Fully developed laminar flow in a helically coiled tube of finite pitch. Chemical Engineering Communications 7: 57–78. 13. Dutta, B., and Sastry, S.K. 1990. Velocity distributions of food particle suspensions in holding tube flow: Experimental and modeling studies on average particle velocities. Journal of Food Science 55(5): 1448–53. 14. Palmieri, L., Cacace, D., Dipollina, G., and Dall’Aglio, G. 1992. Residence time distribution of food suspensions containing large particles when flowing in tubular systems. Journal of Food Engineering 17: 225–39. 15. Sancho, M.F., and Rao, M.A. 1992. Residence time distribution in a holding tube. Journal of Food Engineering 15: 1–19. 16. Sandeep, K.P., and Zuritz, C.A. 1995. Residence times of multiple particles in nonNewtonian holding tube flow: Effect of process parameters and development of dimensionless correlations. Journal of Food Engineering 25: 31–44. 17. Baptista, P.N., Oliveira, F.A.R., Caldas, S.M., and Oliveira, J.C. 1996. Effect of product and process variables in the flow of spherical particles in a carrier fluid through straight tubes. Journal of Food Processing and Preservation 20: 467–86.

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18. Chen, W., and Jan, R. 1993. The torsion effect on fully developed laminar flow in helical square ducts. Journal of Fluids Engineering 115: 292–301. 19. Tucker, G.S., and Withers, P.M. 1994. Determination of residence time distribution of nonsettling food particles in viscous food carrier fluids using hall effect sensors. Journal of Food Process Engineering 17: 401–22. 20. Ahmad, M., Singh, S.N., and Seshadri, V. 1993. Distribution of solid particles in multisized particulate slurry flow through a 90° pipe bend in horizontal plane. Bulk Solids Handling 13(2): 379–85. 21. Sandeep, K.P., Zuritz, C.A., and Puri, V.M. 1997. Residence time distribution of particles during two-phase non-Newtonian flow in conventional as compared with helical holding tubes. Journal of Food Science 62(4): 647–52. 22. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. 1960. Transport phenomena. New York, NY: John Wiley and Sons. 91–103. 23. Fichtali, J., van de Voort, F.R., and Doyon, G.J. 1993. A rheological model for sodium caseinate. Journal of Food Engineering 19(2): 203–11. 24. Ibarz, A., Pagan, J., and Miguelsanz, R. 1992. Rheology of clarified fruit juices. II: Blackcurrant juices. Journal of Food Engineering 15: 63–73. 25. Xuewu, Z., Xin, L., Dexiang, G., Wei, Z., Tong, X., and Yonghong, M. 1996. Rheological models for Xanthum gum. Journal of Food Engineering 27: 203–9. 26. Sastry, S.K., and Zuritz, C.A. 1987. A model for particle suspension flow in a tube. ASAE Paper No. 876537. 27. Oliver, D.R. 1962. Influence of particle rotation on radial migration in the Poiseuille flow of suspension. Nature 194: 1269–1291. 28. Theodore, L. 1964. Sidewise force exerted on a spherical particle in a Poiseuille flow. Engineering Science, Doctoral Thesis. New York University. 29. Saffman, P.G. 1965. The lift on a small sphere in a slow shear flow. Journal of Fluid Mechanics 22(2): 385–400. 30. Clift, R., and Gauvin, W.H. 1971. Motion of entrained particles in gas streams. The Canadian Journal of Chemical Engineering 49: 439–48. 31. Maesmans, G., Hendrickx, N., DeCordt, S., Francis, A., and Tobback, P. 1992. Fluidto-particle heat transfer coefficient determination of heterogeneous foods: A review. Journal of Food Processing and Preservation. 16: 29–69. 32. Christiansen, E.B., and Craig, Jr., S.E. 1962. Heat transfer to pseudoplastic fluids in laminar flow. AIChE Journal 8(2): 154–60. 33. McCabe, W.L., Smith, J.C., and Harriott, P. 1985. Unit operations in chemical engineering. 4th edn. Singapore: McGraw-Hill. 294–354. 34. Beek, N.J., and Eggink, R. 1962. In Developments of Heat Transfer. Cambridge, MA: The M.I.T. Press. Edited by W. M. Rohsenow. p. 334. 35. Zuritz, C.A., McCoy, S.C., and Sastry, S.K. 1990. Convective heat transfer coefficients for irregular particles immersed in non-Newtonian fluid during tube flow. Journal of Food Engineering 11: 159–74. 36. Filkova, I., Koziskova, B., Filka, P. 1986. Heat transfer to a power law fluid in tube flow: An experimental study. In Food Engineering and Process Applications. 1: Transport Phenomena. Elsevier Applied Sciences Publishers. London, England: Edited by Maguer, M.L., Jelen, pp. 259–72. 37. Wilkinson, W.L. 1960. Non-Newtonian fluids. London: Pergamon. p. 104. 38. Perry, R.H., and Chilton, C.H. 1973. Chemical engineer’s handbook. Singapore: McGraw-Hill Book Company. 39. Zitoun, K.B., and Sastry, S.K. 1994. Determination of convective heat transfer coefficient between fluid and cubic particles in continuous tube flow using noninvasive experimental techniques. Journal of Food Process Engineering 17: 209–28. 40. Weisser, H. 1972. Untersuchungen zum Warmeubergang im Kratzkuhler. PhD thesis. Germany: Karlsruhe Universitat. © 2009 by Taylor & Francis Group, LLC

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41. Skelland, A.H.P., Oliver, D.R., and Tooke, S. 1962. Heat transfer in a water-cooled scraped-surface heat exchanger. British Chemical Engineering 7(5): 346–353. 42. Kalb, C.E., and Seader, J.D. 1972. Heat and mass transfer phenomena for viscous flow in curved circular tubes. International Journal of Heat and Mass Transfer 15: 801–17. 43. Kalb, C.E., and Seader, J.D. 1974. Fully developed viscous-flow heat transfer in curved circular tubes with uniform wall temperature. AICHE Journal 20(2): 340–46. 44. Janssen, L.A.M., and Hoogendoorn, C.J. 1978. Laminar convective heat transfer in helical coiled tubes. International Journal of Heat and Mass Transfer 21: 1197–206. 45. Willhoft, E.M.A. 1993. Aseptic processing and packaging of particulate foods. London: Blackie Academic and Professional: 6–7. 46. Simpson, S.G., and Williams, M.C. 1974. An analysis of high temperature short time sterilization during laminar flow. Journal of Food Science 39: 1047–54. 47. CAPPS&NCFST. 1996. Case study for condensed cream of potato soup. Aseptic Processing of Multiphase Foods Workshop. Nov. 14–15, 1995; Mar. 12–13, 1996.

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Moisture 3 Modeling 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

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3.1

Food Processing Operations Modeling: Design and Analysis

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 cause grain kernels to fissure. Fissured or stress-cracked kernels are objectionable 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 can be calculated accurately, better processes can be designed to reduce fissure development. However, such an estimation requires accurate determination of moisture 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 hysteresis 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 moisture 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 primary focus on modeling diffusion of moisture [5,6] and determining moisture diffusivities 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 cumbersome. Mathematical models, based on physical principles, can potentially predict with reasonable accuracy the moisture distribution inside the kernel during adsorption. 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 transport in grain conditioning, storage, deep-bed drying and aeration processes. For efficient processing operations quantitative and predictive models relating to the physical properties of food to transient time-moisture profiles that determine product quality are needed. In this chapter, different mathematical models to determine the moisture diffusivity of individual components of any heterogeneous food grain © 2009 by Taylor & Francis Group, LLC

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are described. As an example, the developed models were validated using the moisture adsorption data in a corn kernel. Moisture diffusivity of individual components of a corn kernel namely pericarp, germ, soft and hard endosperms were determined using the finite difference, analytical, and finite element methods, respectively. The developed finite element model was also used to predict the moisture distribution 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 summarized as water–vapor transport mechanisms and liquid–water transport mechanisms (Figure 3.1). The water–vapor transport mechanism consists of Knudsen diffusion, Stefan diffusion, Mutual diffusion, poiseuille flow, and condensation–evaporation. On the other hand the liquid–water transport mechanisms consists of capillary flow, 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 of diffusion occurs in gas-filled solids with small pores, or under low pressure when the mean free-path of molecules is more than the pore size and the molecules collide 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 density and the Knudsen vapor diffusivity within the product. The size and amount of pores, tortuosity, and the geometry of the solid matrix affect the water vapor flux as represented in the equation column. 3.2.1.2 Stefan Diffusion In the case of Stefan diffusion, the water flux is a function of the vapor pressure, total 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 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 © 2009 by Taylor & Francis Group, LLC

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Type

Knudsen diffusion

Picture

Equations d n w = –ετβDkw ∇ρw1

d

2 Dkw = d 3

( Π2

d n w = –ετDwg

Stefan diffusion

RTk Mw

1/2

)

Mw P ∇Pw P – Pw RTk

Dwg = 2.5 × 10–5 (m 2 /s)

Mutual diffusion

Poiseuille flow

d n w = –ετDwg ∇ρw1

d 2 ρ ∇P d n w = –ετ 32µ

d

Condensation– evaporation

Capillary flow

d n w = nw (∇Tk,·····)

h

n w1 = –ρ χ ∇θ 1

Liquid diffusion

1 n w = –ρ1 Dwg ∇ωw

Surface diffusion

1 n w = –ρ1 Dsu ∇ωw

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.)

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fluid of viscosity μ induced by a constant positive pressure difference or pressure 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 for continuous flow are present. In drying, liquid flow resulting from capillarity applies to liquids not held in solution and to all moisture above the fiber-saturation point, as in textiles, paper, and leather, and to all moisture above the equilibrium moisture content at atmospheric saturation, as in fine powders and granular solids, 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 temperature, this may enhance the total flow of a diffusing component. 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.

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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 diffusivity 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 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 material 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 diffusion 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 the soaking process, causing additional moisture movement due to filtration effect. Coupling the effect of moisture and temperature may be important for accurately modeling 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 process (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) semiempirical 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 diffusion equations for a plane sheet, finite and infinite cylinders, and a sphere and an empirical model during drying of peanuts. They found that the diffusion equation better represented the drying than did the empirical model. Moreover, they concluded that the finite-cylinder model best fit the experimental data. Eckhoff 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 not adequately reflect the true nature of the diffusional processes via the tip cap. 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 the diffusion coefficient that is determined with one geometric shape couldn’t be

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used with another geometric shape. Muthukumarappan and Gunasekaran [22–25] evaluated the effect of different shapes in determining the moisture diffusivity of corn samples and found that the infinite slab model fitted the experimental adsorption data.

3.3 MODELING MOISTURE DIFFUSION IN FOOD GRAINS 3.3.1

THEORETICAL CONSIDERATIONS

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 of Fick’s law of diffusion developed by Crank [26] were used. The differential equation with initial and boundary conditions to describe the system are: ∂M ∂ ⎡⎢ ∂ M ⎤⎥ = Dm m = 1, n ∂t ∂ x ⎢⎣ ∂ x ⎥⎦ ∂M = 0; ∂X

x = 0;

t≥0

(3.1)

(3.2)

M = M e ; x = −l2 , l2 ; t > 0

(3.3)

M = M 0 ; − l2 < x < l2 ; t = 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. Newman [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 with assumption of exponentially varying surface moisture fitted the experimental adsorption curve better than the model with assumption of instantaneous equilibrium surface moisture. Walton et al. [21] assumed a boundary condition, which is a function of the convective mass transfer coefficient in developing a drying model for corn. However, the experimental convective mass transfer coefficient values at

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different temperature and relative humidity (RH) conditions during adsorption are not currently available. This makes it difficult to predict surface moisture content as a function of adsorption time. Following these investigations, the surface moisture content was assumed to vary exponentially with adsorption time. The boundary condition describing the moisture content at the kernel surface (Ms, %) and the surface moisture ratio (MRs) can be written as: M s = [1 − exp(−kt )]×[ M e − M 0 ] + M 0 MR s =

Ms − M0 = [1 − exp(−kt )] Me − M0

(3.5) (3.6)

Moisture ratio (MR) of a multicomponent system can be modeled as: MR i (t ) =

∑X

ij

MR j (t )

(3.7)

i,j

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: ∂M = Δ ( D ΔM ) ∂t

(3.8)

∂M ∂ ⎡⎢ ∂ M ⎤⎥ ∂ ⎡⎢ ∂ M ⎤⎥ D D = + ∂t ∂ x ⎢⎣ ∂ x ⎥⎦ ∂ y ⎢⎣ ∂ y ⎥⎦

(3.9)

In two-dimensions it becomes,

The initial and boundary conditions are: M = M0 , t = 0

(3.10)

M s = [1− exp(−Kt )]×[ M e − M 0 ] + M 0 , t > 0 on Ω

(3.11)

and

where, Ω constitutes the complete boundary surface for the body.

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61

NUMERICAL FORMULATION

The element equations were developed by transforming the governing differential equations using the Galerkin’s weighted residual approach. After the formulation, the element equations can be written in a simplified form as: n

∑

M j [Cij ] +

n

∑ M [K ] = 0 ij

(3.12)

∫ N N dxdy

(3.13)

j=1

j

j=1

where the element moisture capacitance matrix, [Cij ] =

i

j

A

and the element moisture conductance matrix, [ K ij ] =

∫ A

⎡ ∂ N i ∂ Nj ∂ N i ∂ Nj ⎤ ⎢ ⎥ ⎢ ∂ x ∂ x + ∂ y ∂ y ⎥ D dx dy ⎣ ⎦

(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 difference scheme), which is unconditionally stable, was used. The final system of equations incorporating the known boundary conditions, had the following form: ([C ] + Δt [ K ]) M t+Δt = [C ]M t + Δt Ft+Δt

3.4 3.4.1

(3.16)

MOISTURE DIFFUSION IN A CORN KERNEL DURING ADSORPTION CORN STRUCTURE

Corn is a complex cereal grain. The kernel is flattened, wedge-shaped, and relatively broad at the apex of its attachment to the cob. The kernel is composed of germ, soft (floury) and hard (horny) endosperm, and pericarp. The pericarp surrounds the kernel and is strongly adherent to the seed coat. The hard endosperm is found on the

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sides and back of the kernel and bulges in toward the center at the sides. The soft endosperm fills the crown (upper part) of the kernel, extends downward to surround 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 composed 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 [5] verified that liquid diffusivity of rice components did not vary with the initial 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. 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 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 [37] reported the diffusion coefficient as a function of temperature of the fluid 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 that the relative humidity had some influence on moisture diffusivity. Therefore, the diffusivity may depend on moisture content of a material and follow an exponential relationship with fluid temperature. But the dependence of diffusivity of grains on 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 Agricultural 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 © 2009 by Taylor & Francis Group, LLC

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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 University 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 containers 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 computer (Diversified Systems 486) through an Expansion Multiplexer (EXP-16). The 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 voltage 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 sample moisture content reached near-equilibrium with the chamber. Fick’s law of diffusion model considers the geometry of corn germ and corn kernel 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 multivariate secant method [39]. The moisture diffusivity was estimated by minimizing the sum of square deviations (SSD) between the experimental and theoretical corn adsorption data. The characteristic dimensions of the germ and corn kernel were determined 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 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 adsorption 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 infinite slab model. The characteristic dimensions used in the infinite slab model were 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. The surface adsorption coefficient (K) in Equation 3.6 indicates how fast the 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 calculating the time taken to reach MR = 0.5 [26]. Moreover, the effect of varying MR © 2009 by Taylor & Francis Group, LLC

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(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. 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 gave the best fit to the adsorption data. The values of K for all the environmental 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 values were then used to calculate the modified moisture ratio with time. The moisture diffusivity of corn samples with varying surface moisture content was estimated by minimizing the sum of square deviations between the experimental (modified moisture ratio) and the theoretical data using the non-linear, least square multivariate 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 content more rapidly than the corn. This is reflected by the higher moisture ratio for 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 underpredicted 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 frequently. The moisture diffusivity of germ decreased with increasing RH. During the adsorption tests, the concentration gradient is higher at higher RH. This higher concentration 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 moisture diffusivity values of corn germ were about two to five times lower than the effective moisture diffusivity of the composite corn kernels. This could be due to TABLE 3.1 Surface Adsorption Coefficient (K, h −1) Values Used in Equation 3.6 Corn without Pericarp o

Corn with Pericarp

Temperature ( C)

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

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1.0

Modified moisture ratio

0.9 0.8 0.7 0.6 0.5

Corn germ Corn Model predicted

0.4 0.3 0.2 0.1 0.0

0

10

20 Time, h

30

40

FIGURE 3.2 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 moisture content assumption.

TABLE 3.2 Moisture Diffusivities (m2/h) of Pericarp, Germ, Soft and Hard Endosperms of a Corn Adsorption Condition Temperature (oC) 25

30

35

40

Moisture Diffusivity

RH (%)

Pericarp (×10−8)

Germ (×10−7)

Soft Endosperm (×10−7)

Hard Endosperm (×10−7)

Composite (×10−7)

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

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

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

75

0.57

1.18

1.460

0.919

1.40

80

0.53

0.40

1.221

0.687

1.32

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 at different air temperatures, RH, and type of corn were statistically significant at the 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.

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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]: ⎛ B⎞ D = A exp⎜⎜⎜− ⎟⎟⎟ ⎝ Ta ⎟⎠

(3.17)

The model coefficients A and B and the corresponding R 2 (coefficient of determination) 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 densities of 1229 and 1327 kg/m3, respectively. To remove the corn pericarp, preliminary

TABLE 3.3 Coefficients of Arrhenius-type Model (D =A exp(−B/T a)*) for Temperature Dependency of Moisture Diffusivity of Corn Germ, Pericarp, Soft and Hard Endosperms Coefficients Component

Germ

Pericarp

Soft endosperm

Hard endosperm

RH (%)

A

B

R2

75

1.122 × 105

8645.3

0.964

80

3.568

5727.4

0.956

90

0.028

4313.9

0.981

75

4.917 × 10

−7

2116.1

0.991

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

80

2.795 × 10

3127.4

0.911

90

0.418 × 10

6081.5

0.901

75

3.282 × 10−5

1884.8

0.991

80

0.2243

4657.7

0.903

90

1.803×10−2

3862.3

0.978

−3 2

* D = moisture diffusivity (m /h); Ta = absolute temperature (K). 2

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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 predicted 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 2 pericarp

Material 1 Corn without pericarp

D2

D1

j

j–1

Δ l1

j+1

Δ l2

l1 l2

FIGURE 3.3 Schematic of a corn kernel modeled as a slab core (corn without pericarp) and slab shell (pericarp).

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3. Geometrically, the corn kernel is an infinite slab, i.e. the diffusion is onedimensional 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 Equations 3.1 through 3.4 were used with n = 2. The Crank–Nicolson finite difference 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 every point inside the system. Using the finite difference formulations explained in Muthukumarappan [8] the nodal moisture contents were predicted. The average moisture content of the corn can be obtained by volume averaging the nodal moisture content values. Therefore, the moisture ratio of corn during adsorption can be stated as a function of time (t): ∫ v M (x)dv − M 0 = f (t) Me − M0

(3.18)

Two FORTRAN programs were written to solve the finite difference equations using 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 pericarp, 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 pericarp on moisture diffusivity. The surface adsorption coefficient (K) was determined to represent the rate at 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 overpredicted during the first 10 h of adsorption and then underpredicted. But the

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1.0

Moisture ratio

0.9 Without pericarp Experimental 0.8 Predicted 0.7 0.6 0.5

With pericarp Experimental Predicted

0.4 0.3 0.2 0.1 0.0

0

10

20 Time, h

30

40

FIGURE 3.4 Experimental moisture ratio for corn samples (FR27 × MO17) exposed at 35°C and 75% RH and predicted moisture ratio of corn samples using finite difference method.

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 composite 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 pericarp and pericarp were statistically significant at 0.05 level. The mean moisture diffusivity of pericarp varied from 0.30 ×10−9 to 0.57 ×10−9 m 2/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 (desorption). For example, within 10–24% moisture content, reported diffusivity values of pericarp varied from 0.568 ×10−7 to 3.299 ×10−7 m 2/h [6]. The moisture diffusivities 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 m 2/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. 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 corresponding R2 (coefficient of determination) values for corn pericarp 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.

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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 and weighing the component fragments from five kernels for each corn type. Amount 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 endosperm. 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 diffusion 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 component’s total mass as: Mj =

∑ S i X ij ∑ X 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 weighing the component fragments from five kernels from each corn type. Amount 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 =

C1X11 + C2 X 21 X11 + X 21

amount of moisture in soft endosperm =

C1X12 + C2 X 22 X12 + X 22

(3.20)

(3.21)

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amount of moisture in hard endosperm =

C1X13 + C2 X 23 X13 + X 23

71

(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 kernels during soaking. In determining the soft and hard endosperms diffusivity values, 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 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 average value. Assuming constant density, the mass average moisture of a body (M) is defined by Haghighi and Segerlind [14] as M=

∫ V M ( x , y)dm ∫ V dm

for every Δt

(3.23)

A computer program for two-dimensional steady-state field problems written by Segerlind [45] was modified to solve the time-dependent diffusion problem. The modified computer program was written in Fortran77. This program can be used to (1) determine the diffusivity of individual components, and (2) simulate the moisture adsorption of grains. The listing of the program is presented in Muthukumarappan [8]. The diffusivity values were estimated by optimizing the experimental and finiteelement predicted moisture content data of the individual components. A corn kernel without pericarp was considered for the analysis. The cross-section of the corn kernel 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 aluminum 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 corresponding dimensions were determined. A finite-element discretization of the kernel is shown in Figure 3.6. The two-dimensional model in Cartesian co-ordinates consists 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 models were considered simultaneously. For the first model, called ‘hard endosperm 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

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Food Processing Operations Modeling: Design and Analysis 11 10 9 Soft endosperm

8

Hard endosperm

Length, mm

7 6 Hard endosperm

5 4 3

Germ

2 1 0

FIGURE 3.5

0

2 Thickness, mm

4

Cross-section of a corn kernel through its narrowest dimension.

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 endosperms, 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 endosperms 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)

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73

11 10 9 8

Length, mm

7 6 5 4 3 2 1 0

0

2 Thickness, mm

4

FIGURE 3.6 Finite element discretization of the corn kernel.

1.0 0.9 0.8

Moisture ratio

0.7 0.6 0.5 0.4

Soft endosperm

0.3

Hard endosperm Predicted

0.2 0.1 0.0

0

10

20 Time, h

30

40

FIGURE 3.7 Moisture ratio of soft and hard endosperms exposed to air at 35°C and 75% RH using finite element method.

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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 statistically significant at the 0.05 level. 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 followed 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 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 and Gunasekaran [22] compared the diffusivities of corn kernels for three different geometry representations. They found that the diffusivity values of corn kernels 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 allow for a better comparison. Temperature dependency of the diffusivity of corn endosperms were fitted to the Arrhenius-type model shown in Equation 3.17 for corn germ and pericarp. The model coefficients and the corresponding R 2 (coefficient of determination) values for 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 finite-element discretization of the kernel is shown in Figure 3.6. The two dimensional 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 necessary initial and boundary conditions for the finite element simulation. 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 experimental, 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 over-predicted the finite element model in the early stage of adsorption and underpredicted the model in the final stage of adsorption. Mean sum of squares deviation (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

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75

1.0

Moisture ratio

0.8

0.6 Experimental

0.4

Finite element Analytical

0.2

0.0 0

FIGURE 3.8

10

20 Time, h

30

40

Moisture ratio of corn kernels exposed to air at 35°C and 75% RH.

component moisture diffusivities of corn were considered for the FEM while composite moisture diffusivity was considered for the analytical model. Ruan et al. [47] presented 3-D transient moisture profiles of corn kernels during a steeping process using a magnetic resonance imaging technique. From the images they reported that the steepwater moved first into the corn kernel through the space 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 using Surfer software [48]. The moisture profiles for corn samples after 1 h of exposure 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 moisture gradient inside a corn kernel during adsorption at 25°C was lower than at 35°C. The temperature effect on moisture gradient was significant during the early stage 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 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 8 h of adsorption. This shows that the difference in moisture gradient may cause the kernels to fissure. In addition, all the kernels may fissure when the moisture gradient was maximum.

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9.00

11.2

8.00

7.00 9.2

9.2

Kernel length, mm

6.00

5.00

4.00

9.2

3.00

10. 2

2.00

1.00

0.00 0.00

1.00 2.00 3.00 Kernel thickness, mm

4.00

FIGURE 3.9 Moisture profile (% wb) within a corn kernel after 1 h of exposure to air at 35°C and 90% RH during adsorption.

3.5

RECOMMENDATIONS

Future research could be conducted in developing a model to predict possible failures in grains during adsorption and needs to be verified with experimental results. A simultaneous heat and mass transfer model could also be developed to predict

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77

10 25°C 30°C 35°C

Moisture gradient, % wb

8

6

4

2

0

0

10

20

30 40 Time, h

50

60

70

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.

the temperature and moisture profiles inside a grain during adsorption at different temperature and humidity conditions. A storage model may be developed using the thin-layer moisture adsorption models presented in this chapter.

NOMENCLATURE [C] [Cij] Dm D K [K] [Kij] M Mt M0 Me Ms t x,y Δt Ω D1

Global moisture capacitance matrix Element moisture capacitance matrix Mass of an element Diffusivity, m2/h Surface adsorption coefficient, h−1 Global moisture conductance matrix Element moisture conductance matrix Mass average moisture content of a body Moisture content at time t (h), % wb Average initial moisture content of the kernel, % wb Equilibrium moisture content of the kernel, % wb Surface moisture content of the kernel, % wb Adsorption time, h Cartesian co-ordinates Time step, h Boundary surface of the body Diffusivity of corn without pericarp, m2/h

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D2 Dm j−1,j,j + 1 l1 l2 m RH Δl1,Δl2 F {M} {M} Mj MR(t) MRi(t) MRj(t) Si Ta Xij

Food Processing Operations Modeling: Design and Analysis

Diffusivity of pericarp, m2/h Diffusivity of material m, m2/h Spatial nodes defined in Figure 3.1 Half-thickness of corn without pericarp, m Half-thickness of corn, m Number of components included in the adsorption model Relative humidity, % Space interval of corn without pericarp and pericarp, mm Varying boundary condition Nodal moisture values at time t, % Nodal moisture values at time t + Δt, % Amount of moisture in each component j Moisture ratio as a function of time, (M−M0)/(Me−M0) Moisture ratio of corn type i Moisture ratio of the jth component Total amount of moisture in each corn type i Absolute temperature, K 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. 2. GM White, IJ Ross, and CG Poneleit. 1982. Stress crack development in popcorn as influenced 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. 6. MA Syarief, RJ Gustafson, and RV Morey. 1987. Moisture diffusion coefficients for 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 developments. 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 mathematical analysis. Transactions of the ASAE 4(1): 33–36. 11. TB Whitaker, HJ Barre, and MY Hamdy. 1969. Theoretical and experimental studies of diffusion in spherical bodies with a variable diffusion coefficient. Transactions of the ASAE 12(5): 668–72. 12. YI Sharaf-Eldeen, JL Blaisdell, and MY Hamdy. 1978. Factors influencing drying of ear corn-I: Mathematical description of the moisture history of fully-exposed ears of corn. ASAE Paper No. 78–6005. St. Joseph, MI: ASAE.

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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 in an isotropic sphere – A finite element approach. Transactions of the ASAE 31(2): 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 kernels: 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 corn. Transactions of the ASAE 33(3): 855–61. 21. LR Walton, GM White, and IJ Ross. 1988. A cellular diffusion-based drying model for corn. Transactions of the ASAE 31(1): 279–83. 22. K Muthukumarappan, and S Gunasekaran. 1990. Vapor diffusivity and hygroscopic 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 components 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 equations. 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. Transactions of the ASAE 9(1): 52–55. 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. Transactions of the ASAE 35(6): 1955–61. 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. 34. JF Steffe, and RP Singh. 1982. Diffusion coefficients for predicting rice drying behavior. 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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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. 45. LJ Segerlind. 1984. Applied finite element analysis. 2nd ed. New York: John Wiley and Sons, Inc. 46. SLS Jacoby, JS Kowalik, and T. Pizzo. 1972. Iterative methods for nonlinear optimization problems. NJ: Prentice-Hall, Inc. 47. R Ruan, JB Litchfield, and SR. Eckhoff. 1991. Simultaneous and nondestructive measurement of transient moisture profiles and structural changes in corn kernels during 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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Simulation of 4 Computer 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 4.3.1 Definition of 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

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Food Processing Operations Modeling: Design and Analysis

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 ability of food, especially solid food, results in non-uniform and inefficient heating. 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 fields, and foods that are hermetically sealed in microwavable packages [1–5]. Heat is generated within certain materials when the electromagnetic field reverses the polarization 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 between the electromagnetic field and foods [7–10]. 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 heating systems (including generators and applicators) are briefly introduced.

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 voltage rectifier, a tank circuit, and a work circuit, as shown schematically in Figure 4.1. The RF applicator and foods are part of the power generator circuit; a change in the

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+ Load

Variable capacitance applicator

– Main power

High-voltage transformer

Rectifier

Oscillator

Tank circuit

Work circuit

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Variable inductance tuning

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.)

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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 reach high overall efficiency because the load is part of the circuit [13]. The conventional design is also simple to construct and the system is relatively inexpensive. However, during a heating process, variations in applicator separation, food product 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, 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 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 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 output of the power is transferred to a load through the transmission line and impedance matching circuit, which is used to match the impedance of the amplifier and the load to avoid the power reflection [12]. The power amplifier system has the advantage of a stringently controlled operating frequency that meets the requirement of international electromagnetic compatibility. 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 generator. However, the much higher cost and limitation of power output obstructs the wide use of the power amplifier system.

4.2.2

RADIO FREQUENCY APPLICATORS

Whether free running oscillator systems or power amplifier dielectric heating systems are used, the RF applicator should be designed to meet production 50 Line Oscillator

Power amplifier

Impedance matching

Applicator

Load 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 University. Based on Orfeuil, M. 1987. Electric Process Heating. Columbus, OH: Battelle Memorial Institute.)

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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]: 1. Through-field applicator (flat electrodes). A through-field RF applicator 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. 2. Fringe-field applicator (Strayfield electrodes). In this case, several pairs of 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. 3. Staggered through-field (Garland electrodes) applicator. Conceptually, this configuration is a modified through-field applicator with electrodes staggered on either side (Figure 4.3c). It is often used for products of intermediate 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 product. 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. Several frequency bands are reserved for use in industrial, scientific, and medical (ISM) applications according to international agreement [18]. The Federal Communications 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 be RF heaters, and those that operate at 915 and 2450 MHz are usually identified as microwave apparatus.

4.3.1

DEFINITION OF DIELECTRIC PROPERTIES

Most biological materials behave as lossy insulators, tending to both store and dissipate electrical energy in response to an imposed electromagnetic field, in the same fashion as capacitors and resistors [19]. These abilities are defined by dielectric properties, which are normally described in terms of the complex relative permittivity, εr: ε r = ε ’r − jε ’’r

(4.1)

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(a)

Electrodes

Product

VRF

Product (b)

VRF

(c)

VRF

Product

FIGURE 4.3 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. Dielectric drying. Drying Technology 14(5): 1063–98.)

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 in response to an applied electric field. The imaginary part of the relative complex permittivity, ε’’r , known as the relative loss factor, describes the ability of a material to dissipate energy in response to an applied electric field, which typically results in heat generation [16,20–21]. © 2009 by Taylor & Francis Group, LLC

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The corresponding properties of materials that describe their interaction with an applied magnetic field, functioning in the same manner as an inductor, can be expressed in terms of the complex relative permeability μr: μ r = μ ’r − j μ ’’r

(4.2)

Together, εr and μr describe the behavior of a material in an electromagnetic field. They are relative values, since they describe a material’s interaction with electromagnetic field with respect to free space (vacuum), whose permittivity 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 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 [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 in dissipating electromagnetic fields. At microwave frequencies, however, dipolar relaxation often dominates, whereby molecules (typically water molecules) absorb energy due to the repeated reorientation of their polarization in response to the electric field. At even higher frequencies, relaxation of individual atoms can play a role in dielectric heating. The precise contribution of each mechanism by which energy is dissipated is not always easily determined, and is often irrelevant. When relaxation effects are discounted, the loss factor of a material is related to its direct current (DC) conductivity σ by the following Equation: σ = ωε 0 ε ’’r = 2 π f ε 0 ε ’’r

(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 dielectric 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

The transmission properties, which determine energy flow in a material within traveling electromagnetic field, is described in terms of a complex propagation factor (γ) 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 © 2009 by Taylor & Francis Group, LLC

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field as it penetrates a material. The attenuation factor α can be calculated according to: 1

⎡ ⎛ ⎞⎟⎤ 2 ⎛ ε ’’ ⎞⎟2 2 π f ⎢ ⎜⎜ ⎟⎟⎥ ⎜ α= ε ’ 1 + 1 − ⎢ ⎜ ⎟⎟⎥ ⎜⎝ ε ’ ⎟⎟⎠ c ⎢ ⎜⎜⎝ ⎟⎠⎥ ⎣ ⎦

(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. The electric field penetration depth (dp ) of a material, also known as the skin 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 electric field intensity is diminished by a factor of 1/e (e, Naprian base, 2.71828) of its amplitude at the surface [23]. According to Lambert’s law: Ez = E0e−αz

(4.6)

where, E 0 is incident electric field intensity at the surface of a material, Ez is electric field intensity at distance z from the surface in the direction of the electric wave traveling within a dielectric material, and α is the attenuation constant. When E (d p ) = E0 /e , with Equation 4.5 and Equation 4.6, a relation can be derived for dp: dp =

c ⎧⎪ ⎡ ⎛ ε ’’ ⎪ ⎢ 2 πf ⎪⎨ε ’r ⎢ 1 + ⎜⎜ r ⎪⎪ ⎢ ⎝⎜ ε ’r ⎪⎩ ⎣

1

⎞⎟ ⎟⎟ ⎠

2

⎤ ⎫⎪ 2 ⎥⎪ − 1⎥ ⎪⎬ ⎥ ⎪⎪ ⎦ ⎪⎭

(4.7)

This fundamental Equation 4.7, which yields an electric field penetration depth, is normally used in electrical engineering and is not commonly applied in food engineering [24]. The average power dissipation per unit volume in a dielectric subjected to an electromagnetic field can be obtained using the expression: Pav = 2πε 0 f ε’’E 2 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, 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 distance an incident electromagnetic wave can penetrate beneath the surface of a

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material as the power decreases to 1/e of its power at the surface. Since the power is proportional to square of the electric field intensity: Pz = P0e−2α z

(4.10)

where P0 is the incident electric field power density at the surface of a material, Pz is electric field power density at distance z in the direction of electric wave traveling within a dielectric material, and α is the attenuation constant. When P(d p ) = P0 /e , with Equation 4.7 and Equation 4.10, the power penetration depth is given by: dp =

c ⎧⎪ ⎪ 2 2 πf ⎪⎨ε ’r ⎪⎪ ⎪⎩

1

⎡ ⎤ ⎫⎪ 2 ⎛ ε ’’r ⎞⎟2 ⎢ ⎥⎪ ⎜ ⎟⎟ − 1⎥ ⎪⎬ ⎢ 1 + ⎜⎜ ⎝ ε ’r ⎠ ⎥⎪ ⎢ ⎣ ⎦ ⎪⎪⎭

(4.11)

which is precisely half of the amplitude penetration depth. Penetration depth in this 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 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 electric field, which can, in turn, improve heating uniformity. Since the dielectric properties themselves vary with frequency, penetration depth does not vary exactly as 1/f. In general, an electromagnetic field having a short wavelength does not penetrate 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 temperature, 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 contact with the material-under-test. During the measurement, a signal is sent by a vector network analyzer or an impedance analyzer, and reflected back by the material.

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The magnitude and phase change of the reflected wave are used to calculate the 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 accuracy in dielectric constant and low loss factor resolution. Three major error sources for measurement are the cable stability, air gaps, and sample thickness. As the instability and flexing of cable may distort the detective signal sent by network analyzer or impedance analyzer in both amplitude and phase, inaccuracy 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 measurement. The sample surface, which contacts the coaxial probe, should be made as flat and smooth as possible. If the sample is too thin, the detective signal can penetrate the sample and introduce insufficient reflection, thereby influencing the accuracy 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 (mm). εr

Normally, the sample thickness should typically be greater than 1cm. The solid samples must have a flat surface. 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 This method will need the cross-section of a transmission line to be filled by a sample. The transmission line may be either rectangular or coaxial [27]. During the measurement, a vector network analyzer is used to detect the change of the impedance and propagation characteristics due to the fllling of dielectric material. According to 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 fit for the cross-section of the transmission line. The way of measurement makes it difficult for the transmission line method to test liquid and semisolid sample material. 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

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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 dielectric properties of only one frequency for a specific resonance cavity and the cavities are difficult to design and use.

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 phenomena. Those techniques can be classified as experimental, analytical, and numerical. Experimental techniques are expensive, time consuming, and usually do not allow adequate flexibility for parameter variation. Analytical techniques can provide exact electromagnetic field distribution. However, the technique can only solve the problem with very limited and extremely simple configurations. Numerical techniques are used for problems associated with complicated constructions. The finite difference time domain method (FDTD) and finite element method (FEM) are 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 electromagnetic fields are [37]: ∇×H = J +

∇×E = −

∂D ∂t

∂B ∂t

(4.12)

(4.13)

∇ ⋅ D = ρe

(4.14)

∇⋅B = 0

(4.15)

where E (Vm−1) is electric field, H (A m−1) is magnetic field, D (C m−2) is electric flux density, B (Wb m−2) is magnetic field density, J (A m−2) is electric current density, and ρe is electric charge density. E, H, D, B, and J represent the instantaneous field vectors as the function of spatial position and time. However, in many practical systems, the time variations

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are of cosinusoidal form and are referred to as time-harmonic. Under time-harmonic conditions, the Maxwell’s equation can be modified to [38]: ∇ × H = J + jωD = J + jωεE

(4.16)

∇ × E = − jω B = − jωμH

(4.17)

∇ ⋅ D = ∇ ⋅ εE = ρe

(4.18)

∇ ⋅ B = ∇ ⋅ μH = 0

(4.19)

where E, H, D, B, and J represent the corresponding complex spatial forms which are only a function of position. Among a lot of numerical methods the finite element method (FEM) is mainly used to solve the electromagnetic equations [39]. The two most common sub-methods of the finite element method are finite element time domain (FETD) and finite element 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. Recently finite difference time domain (FDTD) method has gained acceptance and is now used in place of FETD to some degree. The FDTD method increases the number of cells, consequently, increases the computational demands at a small matrix. But FEM has superiority over FDTD in handling complicated product configuration.

4.4.2

FINITE-DIFFERENCE TIME DOMAIN METHOD

The finite-difference time-domain (FDTD) method is a convenient, easy-to-use, and efficient method for solving electromagnetic scattering problems. It was first introduced in 1966 by Yee [41] and was developed by Taflove [42–44] to solve Maxwell’s time-dependent curl equations. Several key attributes combine to make the FDTD method a useful and powerful tool. First is the method’s simplicity; Maxwell’s equations in differential form are discretized in space and time in a straightforward manner. Second, since the method 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 geometric flexibility of the method permits the solution of a wide variety of radiation, scattering, and coupling problems [36]. The fundamental scheme for finding the finite difference solution of Maxwell’s equations is the numerical approximation of the derivative of a function f(x). The central-difference formula is [36]: f ′( x 0 ) =

f ( x 0 + Δ x /2) − f ( x 0 − Δ x /2) + O( Δ x 2 ) Δx

(4.20)

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A grid point in a solution region as shown in Figure 4.4 can be defined as [36]: (i, j, k ) ≡ (i Δx , j Δy, k Δz )

(4.21)

and any function of space and time in the solution region can be presented as: F (i, j, k ) = F (iδ, jδ, k δ, nΔt )

(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]:

Z Hz Ey Ex

Ex

Ez

Ey

Ez

Ez

Hy

Hx (i,j,k)

Ex Ey Y

X

FIGURE 4.4 Positions of field components in a unit cell. (From Herve, A.G., Tang, J., Luedecke, L., and Feng, H. 1998. Dielectric properties of cottage cheese and surface treatment using microwaves. Journal of Food Engineering 37(4): 389–410.) © 2009 by Taylor & Francis Group, LLC

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H xn+1/ 2 (i, j + 1/2, k + 1/2) = H xn−1/ 2 (i, j + 1/2, k + 1/22) +

δt μ (i, j + 1/2, k + 1/2)δ

⎡ E yn (i, j + 1/2, k + 1) − E yn (i, j + 1/2, k ) ⎤ ⎢ ⎥ ×⎢ ⎥ n n ⎢+Ez (i, j, k + 1/2) − Ez (i, j + 1, k + 1/2)⎥ ⎣ ⎦ H yn+1/ 2 (i + 1/2, j, k + 1/2) = H yn−1/ 2 (i + 1/2, j, k + 1/22) +

δt μ (i + 1/2, j, k + 1/2)δ

⎡ Ezn (i + 1, j, k + 1/2) − Ezn (i, j, k + 1/2) ⎤ ⎢ ⎥ ×⎢ ⎥ ⎢+E xn (i + 1/2, j, k ) − E xn (i + 1/2, j, k + 1)⎥ ⎣ ⎦ H zn+1/ 2 (i + 1/2, j + 1/2, k ) = H zn−1/ 2 (i + 1/2, j + 1/2, k ) +

(4.23a)

(4.23b)

δt μ (i + 1/2, j + 1/2, k )δ

⎡ E xn (i + 1 / 2, j + 1, k ) − E xn (i + 1/2, j, k ) ⎤ ⎢ ⎥ ×⎢ ⎥ ⎢+E yn (i, j + 1/2, k ) − E yn (i + 1, j + 1/2, k )⎥ ⎣ ⎦

(4.23c)

⎛ σ (i + 1/2, j, k )δt ⎞⎟ n δt ⎟⎟ E x (i + 1/2, j, k ) + E xn+1 (i + 1/2, j, k ) = ⎜⎜⎜1 − ⎟ ⎜⎝ ε (i + 1/2, j, k )δ ε (i + 1/2, j, k ) ⎠ ⎡ H zn+1/ 2 (i + 1/2, j + 1/2, k ) − H zn+1/ 2 (i + 1/2, j − 1/2, k ) ⎤ ⎢ ⎥ (4.23d) ×⎢ ⎥ ⎢+ H yn+1/ 2 (i + 1/2, j, k − 1/2) − H yn+1/ 2 (i + 1/2, j, k + 1/2)⎥ ⎣ ⎦ ⎛ σ (i, j + 1/2, k )δt ⎞⎟ n δt ⎟⎟ E y (i, j + 1/2, k ) + E yn+1 (i, j + 1/2, k ) = ⎜⎜⎜1 − ⎜⎝ ε (i, j + 1/2, k )δ ε (i, j + 1/2, k ) ⎟⎠ ⎡ H xn+1/ 2 (i, j + 1/2, k + 1/2) − H xn+1/ 2 (i, j + 1/2,, k − 1/2) ⎤ ⎢ ⎥ (4.23e) ×⎢ ⎥ n n ⎢+ H z +1/ 2 (i − 1/2, j + 1/2, k ) − H z +1/ 2 (i + 1/2, j + 1/2, k )⎥ ⎣ ⎦ ⎛ σ (i, j, k + 1/2)δt ⎞⎟ n δt ⎟⎟ Ez (i, j, k + 1/2) + Ezn+1 (i, j, k + 1/2) = ⎜⎜⎜1 − ⎜⎝ ε (i, j, k + 1/2)δ ε (i, j, k + 1/2) ⎟⎠ ⎡ H yn+1/ 2 (i + 1 / 2, j, k + 1/2) − H yn+1/ 2 (i − 1/2, j, k + 1/2) ⎤ ⎢ ⎥ ×⎢ ⎥ (4.23f) n n + 1 / 2 + 1 / 2 ⎢+ H x (i, j − 1/2, k + 1/2) − H x (i, j + 1/2, k + 1/2)⎥ ⎣ ⎦ © 2009 by Taylor & Francis Group, LLC

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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. To ensure the stability of the finite difference scheme of equations, the time increment Δt must satisfy the following stability condition [36,45]: 1

− ⎛ 1 1 1 ⎞⎟⎟ 2 ⎜ umax Δ t ≤ ⎜⎜ 2 + 2 + 2 ⎟ ⎜⎝ Δ x Δy Δ z ⎟⎠

(4.24)

where umax is the maximum wave phase velocity within the model. The principle of the FDTD method makes it more effective in finding the 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 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.

4.4.3

FINITE ELEMENT METHOD

Originally finite element method (FEM) was developed and applied in the field of structural analysis [47]. In 1968, the method was applied to electromagnetic problems. Although the concept and programming of FEM is not as simple and easy as finite difference method and method of moment, it is a more powerful and versatile numerical technique for handling problems involving complex geometries and inhomogeneous media [36]. Basically a four-step scheme is applied to solve problems by FEM [48]: 1. Discretizing the solution region into a finite number of sub-regions or 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 effect analysis, besides electromagnetic field investigation, becomes an essential part of the simulation process. During the RF heating two physical factors, temperature and electromagnetic field intensity, will interrelate with each other because the dissipated energy produced by electromagnetic field heats the materials while their dielectric properties change with temperature. The changes in dielectric properties in turn influence the electromagnetic field distribution. The coupling of two physical phenomena is one of the most critical factors for the successful simulation of RF heating processes. Therefore, electromagnetic modeling and thermal modeling must be 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. © 2009 by Taylor & Francis Group, LLC

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Food Processing Operations Modeling: Design and Analysis

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 capability 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 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, Wellesley 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 simulated physical phenomena. The constants and variables were then predefined. © 2009 by Taylor & Francis Group, LLC

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TABLE 4.1 Summary of Some Commercially Available Software in 2004 Company ADINA, Inc. http://www.adina.com

Code Adina-T

Adina-F

Adaptive Research Corp. http://www.adaptive research.com

CFD2000

ALGOR, Inc. http://www.algor.com

Professional Multiphysics Professional CFD

Ansoft, Corp. http://www.ansoft.com ANSYS, Inc. http://www.ansys.com

HFSS CFX

Purpose Heat transfer analysis of solids and field problems Analysis of compressible and incompressible flow Air flow and heat transfer in electronic systems. Radiative and conjugate heat transfer models Multiphsics Heat transfer analysis and fluid flow analysis Electromagnetic field simulation Computational fluid dynamics (CFD) package

Multiphysics

Multiphysics

Design Space

Static structural and thermal, dynamic, weight optimization, vibration mode, and safety factor simulations Structural/thermal analysis

Professional

CD adapco Group http://www.cd-adapco.com

Star-CD

COMSOL http://www.comsol.com

COMSOL Multiphysics

Flomerics Group PLC http://www.flomerics.com

FLO/EMC

Heat transfer, reacting flows, multiphase physics and others Multiphysics

Electromagnetic field analysis

Operating System and requirements Windows, UNIX

Windows

Windows 98/2000/NT/ Me/XP Windows 98/2000/NT/ Me/XP Windows, Linux Solaris UNIX Compaq/HP/ SUN/SGI/IBM Windows NT/2000/XP Linux Windows 2000/XP Linux Compaq/HP/SUN/ SGI/IBM Windows 2000/XP/ NT 4.0 Linux Compaq/ HP/SUN/SGI/IBM

Windows 2000/XP/NT 4.0 Linux Compaq/ HP/SUN/SGI/IBM Windows; UNIX HP/SGI/IBM; Linux Windows 2000 or later, Linux 2.4.x kernel, glibc-2.2.5 or later, Solaris 8, 9, 10 Windows, Sun Solaris (continued)

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Food Processing Operations Modeling: Design and Analysis

(Continued)

Company

Code

Purpose

Flow Science, Inc. http://www.flow3d.com

FLOW-3D

Fluent, Inc. http://www.fluent.com

FLUENT

Fluid modeling, thermal modeling, dielectric phenomena and others Multiphysics

FIDAP

Multiphysics

ThermNet

Standalone or coupled thermal simulation High frequency electromagnetic simulation Coupled thermalstructural interactions and others

Infolytica, Corp. http://www.infolytica.com

FullWave

MSC Software Corp http://www.msc software.com

Marc

MSC Nastran

Heat transfer and others

QWED http://www.qwed.com.pl Remcom, Inc. http://www.remcom.com

QuickWave-3D

Vector Fields Ltd. http://www.vectorfields. com Zeland Software, Inc http://www.zeland.com

Opera 3D

Electromagnetic field analysis Wave electromagnetic solver, temperature rise calculation and others Electromagnetic field and thermal analysis

XFdtd

FIDELITY

Electromagnetic field analysis

Operating System and Requirements Windows NT/XP/ 2000; Linux UNIX DEC/HP/ IBM/Sun/SGI Windows NT/2000/ XP; UNIX SGI/HP/IBM/SUN Windows NT/2000/XP; UNIX SGI/HP/IBM/SUN Windows

Windows NT/2000; Linux UNIX Sun/HP/IBM/ SGI/Compaq Windows NT/2000; Linux; UNIX Sun/HP/IBM/SGI/ Compaq/Cray/ Fujitsu/Nec Windows Windows 2000/XP, Red Hat, Silicon Graphics, Sun Solaris, Mac OSX, HPUX Windows 98/Me/ NT/2000/XP; UNIX SUN/HP/SGI; 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.

After the geometry of the model was built, the sub-domain properties and boundary conditions were assigned. A convergence study was conducted to investigate convergence 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. © 2009 by Taylor & Francis Group, LLC

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Problem analysis Governing equation selection

Constant and variable definition Geometry modeling

Sub-domain and boundary condition setting

Mesh generation

Convergence study Obtain convergent result?

NO

YES Computing solution

Results analysis

FIGURE 4.5 Flow chart of the computer simulation procedure. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

4.4.7.1.1 Assumptions The 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 electromagnetic 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 caused by the electric field. As our research mainly concentrated on the influence of dielectric properties of material on electric field, the propagation of electromagnetic, which is mainly affected by the structure of heating equipment, was overlooked. Without considering the magnetic field, the quasi-static analysis in the current study was acceptable. Secondly, the flow of circulating water inside the pressure-proof vessel 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 layer of the sample trays; the influence from circulating water was small and could be ignored. © 2009 by Taylor & Francis Group, LLC

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4.4.7.1.2 Governing Equations The 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]: ∇ 2V = 0

(4.25)

By solving Equation 4.25, the electric potential (V) and electric field intensity (E) 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 = ωE 2ε 0 ε ’’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 heat transfer phenomena to couple the electromagnetic field with thermal field, and the governing equation for heating transfer by conduction is expressed as: ρCp

∂T − ∇ ⋅ ( k ∇T ) = P ∂t

(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 Model Because 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 differential equations. The software uses the finite element method to model systems of coupled physical phenomena through predefined templates [60]. The geometry 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 specific heat of foods were measured. All of the measured physical properties 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 to perform the simulation. 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 was adequate for obtaining convergent solution, in the model it took about 1 hour for one simulation run. The procedures for computing the solution are shown in the flow chart in Figure 4.8.

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0.7

101

Cavity

0.6 0.4

Upper electrode

0.2

Inductor

0 –0.2

–0.4 –0.6 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.079 0 0.5 0.4 0.2 0

y

z

–0.2

x

–0.4 –0.5

FIGURE 4.6 Model of RF heating system. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

4.4.7.1.4 Simulation Results The hot spot and cold spot were the most critical positions to evaluate the uniformity 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 final heating pattern and temperatures at hot spot and cold spots at the central layer 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 heterogeneous 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.

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0.2

Circulating water 0.1

Aluminum film

0

–0.1 –0.2

0.1 0.079

0.2 0.1 0 –0.1

y

z

x

–0.2

Food Aluminum bottom cover of pressure-proof vessel

FIGURE 4.7 Model of vessel. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

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 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 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 simulate 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 heating heterogeneous food with RF as long as the different components are in close 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 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 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 intensity concentration.

4.5 CONCLUSIONS Assumptions simplified the governing equation and saved computer resources, but affected the simulation accuracy. Therefore, a comprise has to be achieved to

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Assign initial condition

Start time step

FEM EM simulation to obtain electric potential V

Calculate electric field E and time averaged power density P

FEM heat transfer simulation to obtain temperature T

Update dielectric properties based on T

Reach final time step?

NO

YES Solution output FIGURE 4.8 Flow chart of the procedure in solution calculation. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

provide a reasonable indication of electromagnetic field distribution, heating pattern and heating rate with limited computer resources. The results also indicated that computer simulation has the potential to further improve similar simulations and aid the construction and modification of RF heating systems. Computer simulation can also be a useful tool to further improve the RF heating systems.

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Food Processing Operations Modeling: Design and Analysis Position of hot spot 80.0°C 80

70

60

50

40 40.0°C Position of cold spot

FIGURE 4.9 Typical thermal image of the central layer of RF processed food sample. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

80 80

Position of hot spot

75 70 65 60 55 50

Y X

Position of cold spot

45 40 40

FIGURE 4.10 Typical numerical solution of the central layer of processed food sample. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

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105

0.2

Circulating 0.217 0.1 water

Aluminum film 0

Cavity

0.079

0.2

y

z

0.1

Sauce and Cheese x

Meat balls 0

FIGURE 4.11 Model of quarter of packaged heterogeneous food and circulating water. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

0.2

0.217

Plane 1 0.1

0

0.079 Plane 2

x y

z

0

0.1

FIGURE 4.12 Two planes for checking the numerical solutions. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

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(a)

280 (V/m) Noodles

270 260

250 240 230 220 Meat balls y

z

210 x

200 (b)

380 (V/m) 350

300

250

200 Noodles

y

z

150 x

Meat balls

130

FIGURE 4.13 Electric field distribution at (a) plane 1 and (b) plane 2. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

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(a)

123 (°C)

Noodles

122

121

120

119 Meat balls y

z x

118

122.5 (°C)

(b)

122 121.5 121 120.5 120 119.5 119

Noodles y

z

118.5

x

Meat balls

118

FIGURE 4.14 Temperature distribution at (a) plane 1 and (b) plane 2. (From Wang, J. 2007. Study of electromagnetic field uniformity in radio frequency heating applicator. Doctor of Philosophy Dissertation. Department of Biological System Engineering. WA, USA: Washington State University, Pullman.)

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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 applicator 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 dp dp E0 E f k P P0 Pav tmin V A B D E H J Δt α β γ δ, Δx, Δy, Δz εc εr εo ‚ εr ‚‚ εr μr μo ‚ μr ‚‚ μr ρe σ ω

Speed of light in vacuum (m s−1) Electric field penetration depth (m) Power penetration depth (m) Incident electric field intensity (V m−1) Electric field intensity (V m−1) Frequency (Hz) Thermal conductivity (W m−1 K−1) Dissipated power density (W) Incident electric field power density (W) Average power dissipation per unit volume (W m−3) minimum sample thickness (m) Scalar electric potential (V) Magnetic vector potential Magnetic field density (Wb m−2) Electric flux density (C m−2) Electric field intensity (Vm−1) Magnetic field intensity (A m−1) Electric current density (A m−2) Time increment (s) Attenuation factor (Np m−1) Phase constant (Rad m−1) Complex propagation factor Space increment (m) Complex permittivity Relative complex permittivity Permittivity of free space (F m−1) Relative dielectric constant Relative loss factor Relative complex permeability Permeability of free space (H m−1) Real part of relative complex permeability Imaginary part of relative complex permeability Electric charge density (C m−3) Electric conductivity (S m−1) Angular frequency (Rad s−1)

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Radiation for 5 Infrared Food Processing Kathiravan Krishnamurthy, Harpreet Kaur Khurana, Soojin Jun, Joseph Irudayaraj, and Ali Demirci CONTENTS 5.1 5.2 5.3 5.4

Introduction ................................................................................................... 113 Basic Laws of Infrared Radiation ................................................................. 115 Interaction of IR Radiation with Food Components..................................... 116 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 Energy conservation is one of the key factors determining profitability and success 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 heating, 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 material by convection of hot air or by thermal conduction. By exposing an object to 113 © 2009 by Taylor & Francis Group, LLC

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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 electromagnetic (EM) waves and encompasses that portion of the EM spectrum that borders on visible light and microwaves (Figure 5.1). Certain characteristics of IR heating such as efficiency, wavelength, and reflectivity set it apart from and make it more effective for some applications than others. IR heating is also gaining popularity because of its higher thermal efficiency and fast heating rate/response time in comparison to conventional heating. Recently, IR radiation has been widely applied to various thermal processing operations in the food industry such as dehydration, 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]. IR radiation can be classified into three regions, namely, near-infrared (NIR), 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 electronics and allied fields with little practical application in the food processing industry. However, in the last few years significant research efforts have been 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

X rays

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Electromagnetic wave spectrum.

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background on IR heating, the chapter also encompasses application of IR heating in food processing operations such as drying, dehydration, blanching, thawing, 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 summarized in Table 5.1 [1,5]. TABLE 5.1 Basic Laws Pertaining to Infrared Radiation Basic Laws

Aspects Addressed/Explanation Gives spectral blackbody emissive power distribution Ebλ (T , λ).

Planck’s law 2 πhc 2 Ebλ (T , λ) = 2 5 hc / nλ0kT n λ [e 0 −1] Wien’s displacement law 2898 λ max = T Stefan-Boltzmann’s law Eb (T ) = n 2 σT 4 Modified Beer’s law H λ = H λ 0 exp(−σ u) * λ

ρ+α + τ =1

Gives the peak wavelength (λmax), where spectral distribution of radiation emitted by a blackbody reaches maximum emissive power. Gives the total power radiated (Eb(T)) at a specific temperature from an infrared source. Gives the transmitted spectral irradiance (Hλ W/m2 · μm) in nonhomogeneous systems. Reflectivity (ρ): ratio of reflected part of incoming radiation to the 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 medium per unit area (kg/m2); σ*λ spectral extinction coefficient (m2 /kg). 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.

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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 very complex overlapping phenomenon [2]. When radiant electromagnetic energy impinges upon a food surface, it may induce changes in the electronic, vibrational, and rotational states of atoms and molecules. As food is exposed to infrared radiation, it is absorbed, reflected, or scattered (a blackbody does not reflect or scatter), as shown in Figure 5.2. Absorption intensities 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 vibrational state corresponding to wavelength range 2.5–1000 μm (FIR); and (3) changes in the rotational state corresponding to wavelengths above 1000 μm (microwaves) [6]. In general, the food substances absorb FIR energy most efficiently through the 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 absorption 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 absorption 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

Reflected radiation

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L P

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FIGURE 5.3 Principal absorption bands of the main food components compared with water.

TABLE 5.2 The Infrared Absorption Bands for Chemical Groups and Relevant Food Components Chemical Group Hydroxyl group (O-H) Aliphatic carbon-hydrogen bond Carbonyl group (C=O) (ester) Carbonyl group (C=O) (amide) Nitrogen-hydrogen group (-NH-) Carbon-carbon double bond (C=C)

Absorption Wavelength (μm)

Relevant Food Component

2.7–3.3 3.25–3.7 5.71–5.76

Water, sugars Lipids, sugars, proteins Lipids

5.92

Proteins

2.83–3.33 4.44–4.76

Proteins 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.)

infrared absorption bands for chemical groups and relevant food components are summarized in Table 5.2 [4]. Interactions of light with food material and the crucial optical principles such as regular reflection, body reflection, and light scattering were discussed by Birth [7]. Regular reflection takes place at the surface of a material. For body reflection, the light enters the material, becomes diffuse due to light scattering, and undergoes some absorption; and the remaining light leaves the material close to where it enters. Regular reflection produces only the gloss or shine of polished surfaces, whereas body reflection produces the colors and patterns that constitute most of the information obtained visually. For materials with a rough surface, both regular and body 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 is reflected back at the FIR wavelength region [8]. Most organic materials reflect © 2009 by Taylor & Francis Group, LLC

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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 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 measurements [9]. It was experimentally observed that as the thickness of the layer increases, a simultaneous decrease in transmittance and increase in reflection occurs. However, 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 sterilization 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 rehydration rate of the vegetables. Freeze-drying technique is a competitive alternative; however it is comparatively expensive. Application of FIR drying in the food industry is expected to represent a 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 including reduction in drying time, alternate energy source, increased energy efficiency, uniform temperature in the product while drying, better-quality finished products, a reduced necessity for air flow across the product, high degree of process 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, vegetables, fish flakes, and pasta is also done in tunnel infrared dryers. Infrared drying has 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. However, moist porous materials are penetrated by radiation to some depth and their transmissivity depends on the moisture content [22]. Energy and mass balance developed 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 drying in which the heat transfer coefficient would be as high as 200 W/m2 K.

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Factors affecting IR drying kinetics have been studied by several researchers. Masamura et al. [14] confirmed increased drying rates of potatoes with increasing 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 not significant, whereas the drying rate increased monotonically with an increase 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 biological material to IR heat results in swelling and ultimately fracturing of the material [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 in better product quality and energy efficiency [10]. Thus, IR radiation can be considered as surface treatment similar to other radiation technologies. Application of combined electromagnetic radiation and conventional convective heating is considered to be more efficient over radiation or convective heating alone, 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 highvacuum 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 thickness to improve drying efficiency [1]. Hebbar et al. [32] developed a continuous 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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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 freezedrying time of beef was investigated by Burgheimer et al. [37]. The authors concluded 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 liquid and solid foods. Efficacy of microbial inactivation by infrared heating depends 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 stationary phase) and types of food materials. Therefore, several researchers have investigated 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, respectively. Dry heat inactivation of B. subtilis spores by infrared radiation was investigated 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 infrared radiation. Therefore, it is beneficial to investigate the absorption pattern 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 products 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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5.88–6.66 μm using optical bandpass filters for inactivation of Aspergillus niger and Fusarium proliferatum in corn meal. The selected wavelength denatures the protein 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 radiation with short wavelength has relatively higher energy level than FIR radiation with longer wavelength. Hamanaka et al. [42] studied the inactivation efficacy of Bacillus subtilis treated with three infrared heaters (A, B, and C) having different peak wavelengths (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 heaters. 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 22 min, respectively. Therefore, it is obvious that inactivation efficiency is associated 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]. Therefore, 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 different due to their structural and compositional differences. In general, spores are more resistant than vegetative cells. When Bacillus subtilis spores in physiological saline were exposed to infrared heating, a spore population increased up to five times in the first 2 min, followed by subsequent exponential reduction, resulting in shoulder and tailing effects. Upon infrared heat treatment, vegetative cells were inactivated followed by activation of spores. An initial increase in B. subtilis

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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 heating 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 heating) 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 heating, 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 peptidyltransferase 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 © 2009 by Taylor & Francis Group, LLC

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cells clearly verified cell wall damage, cytoplasmic membrane shrinkage, cellular 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 surface 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. With the growing interest in flame-broiling and rapid cooking methods, conveyorized IR broiling is a unique and innovative method. Khan and Vandermey [50] prepared 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 shrinkage, number of servings/h) or sensory quality (appearance, flavor, texture, juiciness and overall acceptability), as compared to conventional gas broiling method. Sakai and Hanzawa [1] reported on the performance of infrared-based systems with conventional ovens for baking rice crackers and for roasting fish pastes. The comparative 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, cooking time of pinto beans was significantly increased. 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 gas-fired heaters. These two types of IR heaters generally fit into three temperature © 2009 by Taylor & Francis Group, LLC

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TABLE 5.3 Inactivation of Pathogenic Microorganisms by Infrared Heating Food/non-food Material

Temperature/energy

Time

Log Reduction*

Reference

Solid Foods 10 s

2.5–5.2 log (estimated)

Wheat or soybean surface

∼50°C++ 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)

1.26 ± 0.14 log10 CFU/10g

Natural bacterial microflora

Wheat surface

2.0 kW

∼24 min 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++

Monilia fructigena

Strawberry

Total bacterial count

92.1 s

4.3 ± 0.4 log10 CFU/cm2

++

103.2 s

+++

6 min

4.5 ± 0.2 log10 CFU/cm2 1.8

80°C Aspergillus niger spores

Corn meal

72°C

+++

[60]

[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

6 min

1.95

Listeria monocytogenes

Oil-browned deli turkey

68°C+++(with an optical filter: 5.45–12.23 μm) 399°C around product surface

75 s

3.7 log10 CFU/mL

[69]

Yeast

Honey

0.2 W/cm2

8 min

∼3.85 log10 CFU/mL+

[48]

(with an optical filter: 5.45–12.23 μm)

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Pathogen

Nonfood Materials 4.2 μW/cm2/nm (peak wavelength: 950 nm) 3.2 μW/cm2/nm (peak wavelength: 1150 nm)

4 min** 22 min** 12 min**

Nutrient agar (depth = 0)

4.36 × 103

6 min

∼2.30–2.48 log10CFU/plate+

(depth = 1 mm from surface)

4.36 × 10

6 min

∼ 0.70 log10 CFU/plate

4.36 × 103 180°C 3.22 kW/m2

6 min

∼ 0.66 log10 CFU/plate

Bacillus subtilis spores E. coli E. coli

(depth = 2 mm from surface) Steel plate Phosphate buffer saline Phosphate buffer

3.2 s** 8 min 2 min

1.8 log10 CFU/mL 0.98 log10 CFU/mL

[39] [43] [40]

Aspergillus niger spores Bacillus subtilis spores

Physiological suspension Physiological suspension

4.0 to 5.0

[47]

E. coli

*

Stainless steel plate at water activity of 0.7

3.7 μW/cm2/nm (peak wavelength: 1100 nm)

61°C 1.0 kW 1.0 kW

3

40 s 10 s

[42]

[87]

∼1.0

In (log10 CFU/mL), unless specified. 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.

Infrared Radiation for Food Processing

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**

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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 charring 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 cleaner form of heat. Electric infrared emitters also provide flexibility in producing the desired wavelength for a particular application. In general, the operating efficiency of an electric IR heater ranges from 40 to 70%, while that of gas-fired IR heaters ranges from 30 to 50% [54]. The spectral region suitable for industrial process 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 seems to be more efficient at the FIR region, while drying of thicker bodies should 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 and color development at the surfaces of foods, such as white bread and wheat flour. 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. Sheridan and Shilton [55] evaluated the efficacy of cooking hamburger patties 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 content, 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. Further, 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 It is crucial and beneficial to investigate the quality and sensory changes occurring 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 © 2009 by Taylor & Francis Group, LLC

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TABLE 5.4 Penetration Depth of NIR (0.75–1.4 μm) into Food Products Product Dough, wheat Bread, wheat Bread, biscuit, dried Grain, wheat Carrots Tomato paste, 70–85% water Raw potatoes Dry potatoes Raw apples

Spectral Peak (μm)

Depth of Penetration (mm)

1.0 1.0 1.0 0.88 1.0 1.0

4–6 11–12 4 12 2 1.5

1.0 1.0 0.88 1.16 1.65 2.36

1 6 15–18 4.1 5.9 7.4

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.

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 indicated 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 deteriorating the food quality [60]. The evaluation of full-fat flour made from IR-heat treated soybeans maintained freshness similar to fresh flour for 1 year. However, untreated samples resulted in rancidity development [61]. Compared to regular freeze-drying, IR-assisted freeze-drying of yam brought about lower color differences as well as faster dehydration. 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 no visible indication [63]. Cell walls of lentils were less susceptible to fracture after 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 broiling in terms of flavor, texture, juiciness, and overall acceptability showed no significant difference between two treatments [50]. However, the appearance of © 2009 by Taylor & Francis Group, LLC

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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 following 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 controls after treatments, refrigerated storage for an hour resulted in no significant 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 deteriorated 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. Table 5.5 briefly summarizes the effect of IR treatment on nutritional quality of various food products. As the literature review substantiates, IR heating does not change the quality attributes of foods significantly, such as vitamins, protein, and 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 explored using the finite element method to explain the phenomenon of heat transfer inside food systems under FIR radiation. The radiation energy driving internal moisture movement during FIR drying of a potato produced the activation energy for diffusion 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 system were solved using Galerkin’s finite element method. The measured temperature distribution in samples was in good agreement with model predictions, permitting control 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,

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TABLE 5.5 Effect of Infrared Treatment on Nutritional Quality of Food Products Food Product Barley Wheat

Lentils

Parameters Effecting Nutritional Quality Germination rate at 55ºC Germination rates (heat treatment for 63 s each) Phytic acid content

Full fat soybeans

Protein solubility

Lentils

Protein solubility

Soymilk

Protein digestibility

Crude canola oil

Phosphorus contents Sulfur contents

Soymilk

Available lysine content

Fried chicken

Thiamine retention

Orange juice

D-values for vitamin C degradation at 75°C

Full fat soybeans

Reduction in urease activity at 140°C and 28% moisture (d.b.) Antioxidant activities (total phenolic compounds in water extract, after 60 min) Radical scavenging activities

Peanut hulls

Effect of Treatment

Reference

25% increase by combination of IR heating and convectional heating Convectional heating: 90–97% Intermittent IR heating: 80–86% Continuous IR heating: 78–85% Untreated: 2.34% High density IR heating (170ºC): 1.06% Infrared heating: 84% Spouted bed drying: 82% Extrusion: 73% Untreated: 74.7% High density IR heating (170ºC): 50.9% Untreated: 83.2% IR heat treated (110–115ºC): 86.5% Untreated canola seeds: 46 ppm IR heat treated (123ºC): 273 ppm Untreated canola seeds: 1.4 ppm IR heat treated (123ºC): 4.4 ppm Untreated: 4.64 g/16g N IR heat treated (110–115ºC): 6.14 g/16g N Reheated by IR heating: 81–84% Convection heating: 86–96% Convectional heating: 27.02 min Ohmic heating: 23.72 min Infrared heating: 23.76 min Infrared heating: 53% Spouted bed drying: 30%

[30]

FIR irradiation: 141.6 μM FIR heating: 90.3 μM

[75]

[70]

[63] [71]

[63] [72] [67] [67] [72] [73] [74]

[71]

FIR irradiation: 48.83% FIR heating: 23.69%

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.

and an approximation of the diffusion model to address the thin-layer infrared drying 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. [77] also reported that Page model adequately fitted the experimental drying data while studying the drying characteristics of high-moisture paddy. © 2009 by Taylor & Francis Group, LLC

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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 which would require the least computing time [78]. However, in a proposal suggested for the solution of heat transfer problems for food materials, it was recommended that if the solution region represents a simple rectangular domain, then the traditional finite difference methods should be the preferred discretization strategy [79]. Tsai and Nixon [80] investigated the transient temperature distribution in a multilayer composite, semi-transparent or transparent, absorbing and emitting medium exposed to a thermal radiative heat flux. The governing conditions with the initial and boundary conditions in consideration of the effects of both thermal radiation and conduction within each layer and convection on both exterior surfaces were solved by a hybrid numerical algorithm, using a fourth-order explicit Runge-Kutta method for the time variable and a finite difference method for the space variable. The experimentally measured temperature distribution of slices of beef during IR frying was successfully predicted by the model developed based on combined infrared radiation and convection heating [81]. Heat conduction equation was solved numerically using the finite difference method. The infinitesimal differentials were replaced by differences of finite size and the degree of accuracy of the representation 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 Cartesian 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 experimental 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 drying behavior of apple, ten different drying models (Newton, Page, modified Page, Wang and Singh, Henderson and Pabis, logarithmic, diffusion approach, simplified Ficks diffusion (SFFD) equation, modified Page Equation-II, and 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]. Sixty-six different model equations relating the temperature and time dependence of infrared drying of apple were derived wherein the model derived from modified Page II had lowest root mean squared error (RMSE), mean bias error (MBE), and chi-square along with highest modeling efficiency and regression coefficient. Moreover, a single equation was derived to predict the moisture ratio change during 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.

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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 of selective heating based on the relation between the optical properties of objects and the spectral distribution of the radiative source. However, the studies did not elaborate on the details or its implementation. 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 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 is a challenge to cut off the entire spectral distribution to obtain a specific bandwidth. 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 importance 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 interior, 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 solids 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 predominantly. 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 amounts to specific food materials. 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 frying process designed consisted of the first stage to aim surface heat transfer by radiant flux with λmax of 3.5–3.8 μm (FIR) and the second stage for greater penetration of heat transfer by radiant flux with a λmax 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, which were quartz tube heaters (Philips 1 kW, type 13195X) whose filament temperature 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 the study that both penetration capacity and reflection increased as the wavelength 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 effects were almost the same due to body reflection. © 2009 by Taylor & Francis Group, LLC

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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 [5] first used a bandpass filter (Optical Coating Laboratory, Inc., type no. L-01436-7) 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 electromagnetic radiation is used for processing. The system had the capability to selectively heat higher absorbing components to a greater extent using optical band pass filters 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: ρCp

∂T ∂2T ∂q =k 2 − r ∂t ∂z ∂z

(5.1)

where, qr is the heat flux (W/m2 ), T is the temperature (°C), ρ is the density (kg/m3 ), Cp is the specific heat of food sample (J/kg⋅°C), k is thermal conductivity (W/m⋅°C), 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 ) = T0

for 0 ≤ z ≤ D; t = 0

(5.2)

where T0 is the initial sample temperature (°C) and D is the sample thickness (m) Infrared radiation

qconv

qrad,out

qabs

Soy protein

Glucose

n = 1 (Boundary 1) n=2 n=3

7 mm

Cylindrical vessel n = N (Boundary 2) 2 mm

Glass plate dq dz

Rotated at 3 rpm by an AC motor (3.8 W)

qcond 0

25.4 mm denotes a thermocouple

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.) © 2009 by Taylor & Francis Group, LLC

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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 k

∂T ∂z

z =0

= hconv (T1 − T∞) − qr (1) + ε (λ) ⎡⎣( Eb (T1 ) − Eb (T∞)⎤⎦ at the top, t > 0

and k

∂T ∂z

= qr ( N ) − z= D

kglass (TN − T∞ ) at the bottom, t > 0 d glass

(5.3)

Here, Eb(T) is the total emissive power at a given source temperature obtained using Planck’s integral, as presented by ∞

∞

Eb (T ) =

∫

Ebλ (T ,λ)d λ =

∫ 0

0

2 πhc02 dλ n λ [e hc0 nλkT –1] 2

5

(5.4)

The IR radiation is not only absorbed by the surface but also penetrates into the food system, causing a local radiative heat flux of qr (n) = qabs exp(–S ⋅ Δz ⋅ (n – 1))

(5.5)

where, qabs is the initial radiant heat flux absorbed by food sample on the surface, S is the extinction coefficient (m−1), and Δz is the grid size (m) [81]. Figure 5.5 shows a simplified schematic of the FIR heating system where boundary 1 denotes an assembly of FIR source lamps, boundary 2 denotes a coneshaped 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)

T1, ε1, q1

T2, ε2, q2 = 0 (insulation)

Waveguide (boundary 2)

Opening (boundary 3)

T4, ε4, q4

T3, ε3, q3 Sample (boundary 4)

FIGURE 5.5 Simplified gray and diffuse enclosure of the heating chamber. (From Jun, S. 2002. Selective far infrared heating of food systems. Ph.D. Dissertation, The Pennsylvenia State University, USA.) © 2009 by Taylor & Francis Group, LLC

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condition that the IR lamps (boundary 1) and the waveguide (boundary 2) are gray and diffuse reflectors, the total heat flux absorbed by food sample can be calculated using the energy balance equation for each surface [91], given as qi − εi

3

∑ j=1

⎞ ⎛1 ⎜⎜ − 1⎟⎟F q = E (T ) − bi i ⎟⎟ i− j j ⎜⎜⎝ ε j ⎠

3

∑F

i− j

Ebj (Ti ), i = 1, 2, 3

j=1

(5.6)

A1q1 + A3q3 = 0 (q2 = 0, for insulation) where, q is the heat flux (W/m2) absorbed or emitted by each boundary (Figure 5.3), F is the view factor, and ε is the emissivity of each boundary. Equation 5.6 for each boundary can be solved simultaneously. The amount of heat flux absorbed by food surface, qabs, is dependent upon the spectral absorptivity (α) of food, the spectral distribution of filtered infrared radiation and the view factor as obtained from the opening (boundary 3) to the food surface (boundary 4). This relationship can be expressed as qabs = F3−4 ⋅ α(λ) ⋅ τ(λ) ⋅ q3

(5.7)

where, τ is the filter transmissivity which is a function of the wavelength (λ). The incoming IR heat fl ux transmitting the boundary 3, q3 is spectral dependent and hence, food absorptivity (α) and filter transmissivity (τ) can be combined into an 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 F3−4 =

{

}

1 a2 + r 2 X − x 2 − 4 , X = 1+ 2 r2

(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. An explicit (forward in time) finite difference method is applied to solve 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 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 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 derivatives in the partial differential equation [80]. The first derivatives of temperature 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, Tnt+1 = Tnt +

Δt k Δt (Tnt+1 − 2Tnt + Tnt−1 ) − (qr (n + 1) − qr (n − 1)) ρCp ( Δz )2 2ρCp Δz

(5.9)

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Hence for n = 1 T1t+1 = T1t + −

⎛ 2 k Δt 2 k Δt 2h Δt ⎞ + conv ⎟⎟⎟ T1t T t − ⎜⎜⎜ 2 2 2 ⎜⎝ ρCp ( Δz ) ρCp Δz ⎟⎠ ρCp ( Δz )

2 Δt Δt (−qr (1) − hT∞ + ε(λ)( Eb (T1 ) − Eb (T∞))) − (qr (2) − qr (1)) (5.10) ρCp Δz ρCp Δz

and n = N

TNt+1 = TNt + +

⎛ 2 k Δt 2kglass Δt ⎞⎟ t 2 k Δt ⎟⎟ TN + T t − ⎜⎜⎜ 2 N −1 2 ⎜⎝ ρCp ( Δz ) d glassρCp Δz ⎟⎠ ρCp ( Δz )

⎞ kglass 2 Δt ⎛⎜ Δt (q ( N ) − qr ( N −1)) T∞ ⎟⎟⎟ − ⎜⎜qr ( N ) + ⎟⎠ ρCp Δz r ρCp Δz ⎜⎝ d glass

(5.11)

The temperature at time, t + 1 is explicitly expressed as a function of neighboring temperatures at an earlier time t. The explicit fi nite difference solution should be restricted by the stability criterion, which otherwise may be diverging and never reach the final solution [20]. The common stability criterion to be satisfied for the explicit solution of Equations 5.9 through Equations 5.11 to be stable is given by Δt ≤

( Δz )2 ρCp 2( k + hΔz )

(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 experimental 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 without the filter. Simulation results from the developed models were consistent 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. Analytical and numerical models of bacterial spores have been developed to predict microbial spore growth during sterilization. Stumbo [94] first validated a model with

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(a) 90 80

Temperature (oC)

70 60 50 Exp. 40

Sim.

Soy protein Glucose

30 20 0

20

40

60 80 Time (sec)

100

120

140

(b) 80

Temperature (oC)

70

60

50

40 Exp.

Sim.

Soy protein

30

Glucose 20 0

50

100

150 Time (sec)

200

250

300

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 heating (a) without filter and (b) with filter. (From Jun S, and Irudayaraj J. 2003, Selective far infrared heating system–design and evaluation (Part I). J Drying Technol 21(1): 51–67.)

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first-order inactivation of uniformly activated spores during a sterilization process. 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 viable spores, and inactivation of activated spores. Jun [90] developed an integrated model that combined the thermal death kinetics 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 with convection-diffusion air flow and heat transfer simulations to investigate the suitability of the method for surface decontamination in strawberries. The model was a powerful tool to evaluate and address complex heating configurations that 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 different 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 including greater energy efficiency, heat transfer rate, and heat flux that results in time-saving as well as increased production line speed. IR heating is attractive primarily for surface heating applications. In order to achieve energy optimum and efficient practical applicability of IR heating in the food processing industry, combination 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 various technological aspects of IR heating for food processing. However, research needs for the upcoming years may include the following:

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1. Selective heating: There is not much literature on selective heating using IR radiation in foods. IR heating can be controlled or filtered to allow radiation within a specific spectral range to pass through using suitable optical band pass filters. Such a controlled radiation can stimulate the 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 flavor compounds and living organisms. 3. Application of catalytic infrared (CIR) heating: CIR heating uses natural gas or propane, which is passed over a mesh catalyst pad to produce thermal radiant energy through a catalytic reaction. This reaction occurs below the ignition temperature of gas so that no flame is produced. The electromagnetic 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 moisture removal. Since CIR directly converts natural gas to radiant energy, it is more energy-efficient than typical infrared emitters using electricity. 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 crucially, 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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54. Hung JY, Wimberger RJ, and Mujumdar AS. 1995. Drying of coated webs. In: Mujumdar AS, editor. Handbook of industrial drying. 2nd ed. New York: Marcel Dekker. 1007–38. 55. Sheridan P, and Shilton N. 1999. Application of far-infrared radiation to cooking of meat products. J Food Eng 41: 203–8. 56. Hashimoto A, Takahashi M, Honda T, Shimizu M, and Watanabe A. 1990. Penetration of infrared radiation energy into sweet potato. Nippon Shokuhin Kogyo Gakkaishi 37(11): 876–93. 57. Hashimoto A, Yamazaki Y, Shimizu M, and Oshita S. 1994. Drying characteristics of gelatinous materials irradiated by infrared radiation. Drying Technol 12: 1029–1052. 58. Ginzburg AS. 1969. Application of infrared radiation in food processing, chemical and process engineering series. London: Leonard Hill. 59. Chua KJ, and Chou SK. 2005. A comparative study between intermittent microwave and infrared drying of bioproducts. Int J Food Sci Technol 40: 23–39. 60. Tanaka F, Verboven P, Scheerlinck N, Morita K, Iwasaki K, and Nicolaı B. 2007. Investigation of far infrared radiation heating as an alternative technique for surface decontamination of strawberry. J Food Eng 79: 445–52. 61. Kouzeh KM, van Zuilichem DJ, Roozen JP, and Pilnik W. 1982. A modified procedure for low-temperature infrared radiation of soybeans.II. Inactivation of lipoxygenase and keeping quality of full fat flour. Lebensm Wiss Technol 15(3): 139–42. 62. Lin YP, Lee TY, Tsen, JH, and King An-Erl V. 2007. Dehydration of yam slices using FIR-assisted freeze-drying. J Food Eng 79: 1295–301. 63. Arntfield SD, Scanlon MG, Malcolmson LJ, Watts BM, Cenkowski S, Ryland D, and Savoie V. 2001. Reduction in lentil cooking time using micronization: Comparison of 2 micronization temperatures. J Food Sci 66(3): 500–5. 64. Galindo FG, Toledo RT, and Sjoholm I. 2005. Tissue damage in heated carrot slices. Comparing mild hot water blanching and infrared heating. J Food Eng 67: 381–85. 65. Huang L. 2004. Infrared surface pasteurization of turkey frankfurters. Innovative Food Sci Emerging Technol 5: 345–51. 66. Maxcy R. 1976. Fate of post-cooking microbial contaminants of some major menu items. J Food Sci 41: 375–78. 67. McCurdy SM. 1992. Infrared processing of dry peas, canola, and canola screenings. J Food Sci 57(4): 941–44. 68. Meeso N, Nathakaranakule A, Madhiyanon T, and Soponronnarit S. 2004. Influence of FIR irradiation on paddy moisture reduction and milling quality after fluidized bed drying. J Food Eng 65(2): 293–301. 69. Muriana P, Gande N, Robertson W, Jordan B, and Mitra, S. 2004. Effect of prepackage and postpackage pasteurization on postprocess elimination of Listeria monocytogenes on deli turkey products. J Food Prot 67(11): 2472–79. 70. Uchino T, Hamanaka D, and Hu W. 2000. Inactivation of microorganisms on wheat grain by using infrared irradiation. Proceedings of International Workshop Agricultural Engineering and Agro-Products Processing toward Mechanization and Modernization in Agriculture and Rural areas. 71. Wiriyaumpaiwong S, Soponronnarit S, and Prachayawarakorn S. 2004. Comparative study of heating processes for full-fat soybeans. J Food Eng 65: 371–82. 72. Metussin R, Alli I, and Kermasha S. 1992. Micronization effects on composition and properties of tofu. J Food Sci 57(2): 418–22. 73. Ang CYW, Basillo LA, Cato BA, and Livingston GE. 1978. Riboflavin and thiamine retention in frozen beef-soy patties and frozen fried chicken heated by methods used in food service operations. J Food Sci 43: 1024–27. 74. Vikram VB, Ramesh MN, and Prapulla SG. 2005. Thermal degradation kinetics of nutrients in orange juice heated by electromagnetic and conventional methods. J Food Eng 69: 31–40. © 2009 by Taylor & Francis Group, LLC

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of Ohmic 6 Modeling 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 6.4.1 Model Verification ............................................................................. 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 generation 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, 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) heating, a high frequency electric field excites the water molecules within the materials while in ohmic heating, ionic motion results in heat generation [2]. Ohmic heating requires electrodes that make good contact with the food; however, microwave heating 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 143 © 2009 by Taylor & Francis Group, LLC

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Electrodes

I

V

R

Electrical analog

FIGURE 6.1

Food

Alternating current power supply

Ohmic heating

The concept of ohmic heating.

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. Modeling ohmic heating is a difficult task due to the unique characteristics and critical factors that distinguish ohmic heating from conventional heating processes. The rate of heat generation in ohmic heating depends strongly on the electrical conductivity of the food. However, heterogeneous foods (e.g. liquid and particulate food materials may have significantly different electrical conductivity) may exhibit different 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. Additional factors contributing to the complexity of the ohmic heating process include possible heat channeling causing hot spots and cold spots, complex coupling between 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 ohmic heating process for sterilizing food products is essential for process validation—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 reliable 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 considered 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].

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6.2 BASIC PRINCIPLES Electric field distribution within an ohmic heater is calculated by solving Laplace’s 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 conduct sufficiently well for ohmic heating to be applied. Non-ionized materials 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: ρCp

∂T = ∇(k ∇T ) + S ∂t

(6.2)

where ρ is the density, Cp is the specific heat, t is the time, and k is the thermal conductivity. The symbol S is the internal energy source term which is generated by electric field. 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 temperature coefficient. Much research has been done on the electrical conductivity 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, but the type of meat cut was not specified. Tulsiyan et al. [22] measured conductivity 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.

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TABLE 6.1 Electrical Conductivity–Temperature Model Parameters Fruits

Chicken

Pork

Beef

σref (S/m)

m (oC)–1

0.089 0.079 0.179 0.124 0.076 0.234 0.663 0.567 0.329 0.428 0.035 0.564 0.527 0.551 0.504 0.456 0.318 0.472

0.049 0.057 0.056 0.041 0.060 0.041 0.020 0.021 0.026 0.024 0.049 0.018 0.020 0.021 0.019 0.023 0.038 0.024

Apple-golden Apple-red Peach Pear Pineapple Strawberry Breast Tender Thigh Drumstick Separable fat Top loin Shoulder Tenderloin Bottom round Chuck shoulder Flank loin Top round

Source: Sarang, S. 2008. Ohmic heating for thermal processing of low-acid foods containing solid particulates. Ph.D. Thesis, The Ohio State University.

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 minimizing 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 manned space flights beyond low earth orbit requires advanced food preservation 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

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problem after use. Solid waste treatment in space for Advanced Life Support (ALS) applications requires that the material be safely processed and stored in a confined 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. None of the past work has dealt with heating of foods in a flexible package using ohmic heating with the package being reusable. The only related work found in the literature involved flexible batteries using food packaging materials, in terms of 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 only minimal charging of electrical double layers were found to significantly inhibit electrochemical reaction [28,29]. Samaranayake [30] observed that pulsed ohmic heating could significantly reduce electrochemical reaction on electrode surfaces, 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. This study was aimed to optimize electrode configurations in a pouch, to yield the most uniform, yet rapid heating thermal profiles in static systems.

6.3.2

PACKAGING

A package developed using flexible pouch materials incorporates a pair of foil electrodes to permit ohmic heating of food materials (Figure 6.2a). Flexible packaging, such as multilayered laminates, provides an alternative to the rigid container. 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 and nylon) on one side for external scratch resistance and a heat sealable polymer (4 mm polyethylene) film on the other side, which becomes the interior after the pouch 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-BipolarTransistor (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 electrochemical 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.

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(a)

Flexible package

Foil electrodes (b) Air assist Cooling water

REG V

Power supply

A REG

Ohmic cell

DAQ DAQ PC REG V A

Data acquisition unit Personal computer Pressure regulator Voltage measurement Current measurement

Water drain PC

FIGURE 6.2 A schematic of flexible package with foil electrodes (a) and pulsed ohmic heating system (b). (From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods inside the flexible package. Journal of Food Process Engineering, 28, 417– 36, 2005. With permission.)

To investigate the uniformity of heating of food materials within the package, three 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 cm in width) on the bottom right of the package, while Pouch B has V-shaped electrodes (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 © 2009 by Taylor & Francis Group, LLC

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(a)

Electrode (b)

(c)

FIGURE 6.3 Different electrode configurations: (a) Pouch A, (b) Pouch B, and (c) Pouch C. (From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods inside the flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With permission.)

and product design, an ohmic heating model was developed to optimize the electrode configuration for temperature uniformity. To develop the 2D transient model for foods under ohmic heating process, the following assumptions are made for simplicity: 1. The temperature profiles are uniform along with the package length, 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. 3. The thermophysical properties of the fluids are considered to be independent 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. © 2009 by Taylor & Francis Group, LLC

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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, ∇V ( x , y, t ) ⋅ → n

=0,

∀t, x and y ∈ [electrodes],

(6.6)

∀t, x and y ∈ [pouch surfaces except electrodes], (6.7)

wall

Thermal boundary condition: k ∇T ( x , y, t ) ⋅ → n

=0,

∀t, x and y ∈ [pouch surfaces],

(6.8)

wall

→

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 defined functions (UDFs) to solve the electric field model, which is not employed in 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 */ long double mag; /* electric field */ 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 this UDS, first increase the number of UDS to 2, compile this group of UDF, then hook up this UDS by choosing v_grad from the list in the UDF function hook up. Define -->User-defined -->Function --> Function hook. At Adjust function, choose

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v_grad. Since UDS-1 is used only to store the values of the voltage gradient, don't solve this UDS. Solve only the flow, energy, and uds-0 Equations. */ 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 electrode configurations were constructed using the GAMBIT 2.0 pre-processor.

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Through mesh refinement study, the optimum numbers of mesh elements for Pouches A, B, and C were 468, 1048, and 924, respectively (Figure 6.3).

6.3.4

MODEL VALIDATION

For model verification, temperature values were measured at different locations inside the package with one unsealed end. Temperature values of thermocouple probes (Ttype, Omega, TMQSS-062U-6) installed at seven different locations (Figure 6.4) were isolated to eliminate the signal interference with the electric field before being transmitted to the data acquisition unit (Campbell Scientific, 21X Micrologger). 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 allocated to determine the likely cold and hot zones for different configurations of electrodes. A critical issue was to install and maintain the thermocouple probes at fixed locations during heating since the packaging material is flexible. The 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 L

Y 3 2 1

X 1

2

3

4

5

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 flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With permission.) © 2009 by Taylor & Francis Group, LLC

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TABLE 6.2 Formulations and Thermal Properties for Chicken Noodle Soup and Black Beans Chicken Noodle Soup Ingredient

Black Beans

Percentage (%)

Fettuccine Filling aid starch Medium toasted flour Modified food starch Salt Black pepper Parsley Poultry seasoning Water (slurry) Water Chicken broth Half and half Chicken Carrots Celery Onions Total

7.68 1.77 0.58 0.35 0.25 0.06 0.03 0.03 10.25 13.62 22.13 3.41 22.13 8.85 4.43 4.43 100

Density (kg/m3)

1033 3977

Ingredient

Percentage (%)

Black beans Crushed tomatoes Cumin Oregano Black pepper Salt Green chillies Onions Garlic

74.66 11.35 0.25 0.15 0.1 0.09 3.55 9 0.85

100 Thermal Properties

Specific heat (J/kg°C) Thermal conductivity (W/m°C) Net weight (g)

0.553

1157 3388 0.486 170

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]. Changes in electrical conductivity values with field strength and temperature were 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 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 V/m, Pouch B between 1 and 2457 V/m, and Pouch C between 24 and 4985 V/m. © 2009 by Taylor & Francis Group, LLC

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Electrical conductivity (S/cm)

0.018 9.18 V/cm 15.8 V/cm 28.4 V/cm

0.015 0.012 0.009 (b) Black beans 0.030

0.025

0.020 8.4 V/cm 14.2 V/cm

0.015

0.010 10

20

30

40

50 60 Temperature (oC)

70

80

90

FIGURE 6.5 Changes of electrical conductivity values with field strength and temperature for (a) chicken noodle soup and (b) black beans. (From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods inside the flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With permission.)

TABLE 6.3 Coefficients of Electrical Conductivities for Chicken Noodle Soup and Black Beans

Chicken noodle soup

Black beans

Field Strength (V/cm)

σ0

m

9.18 15.8 28.4 8.4 14.2

0.0054 0.0067 0.0092 0.0068 0.0058

0.044 0.030 0.015 0.041 0.049

Source: From Jun, S. and Sastry, S. Modeling and optimizing of pulsed ohmic heating of foods inside the flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With permission.

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(a)

Electrode V/m (b)

(c)

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

FIGURE 6.6 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 is more desirable than the others. However, it should be noted that the electric 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. Understanding the pouch geometry, the area adjacent to the electrode edge, which is 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 key factor dominating the thermal performance inside the package. As verification of the thermal behavior of chicken noodle soup in a 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 the existence of the overshoot of electric field strength near the electrode edges,

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Temperature (oC)

80 60 40 20 0 X

3

1

2

Y

2

3

4

5

1

(b)

Temperature (oC)

80 60 40 20 0

3

X 1

2

Y

2

3

4

5

1 : Experimental data

(c)

: Simulation data

Temperature (oC)

80 60 40 20 0 X

1

Y 3 2

3

2 4

5

1

FIGURE 6.7 Comparison of temperature profiles of chicken noodle soup between simulation 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 inside the flexible package. Journal of Food Process Engineering, 28, 417–36, 2005. With permission.)

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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 (X5, Y2), increased significantly with the maximum prediction error of 14°C. The observed under-prediction error may be attributed to local non-homogeneities of chicken noodle soup, filled with solid chunky particulates. 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). There was little discrepancy of simulated electric field distributions between 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%. Defining the cold zone as the area covering the temperature distribution between 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) oC

80

40

12

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.

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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. 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 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 of internal heat contents with respect to the area, being determined by mass, specific 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 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 food heating in the package.

6.3.5

DELIVERABLES

0.8 60 55 50 45

Cold zone area ratio

Temperature variation (oC)

65

55

300

50

250

45 0.6 40 0.4 35 0.2

30

40

Average temperature (oC)

1.0

70

200 150

Power (W)

The electrode configuration could be optimized to ensure uniformity in food temperature during reheating process. Unlike electrodes in parallel, the pouch

100 50

35

0.0 0.05

0.10 0.15 0.20 0.25 Dimensionless electrode width

25 0.30 Cold zone area ratio Average temperature (oC) Power (W) Temperature variation (oC)

FIGURE 6.9 Optimization of electrode width with respect to cold zone ratio, power consumption, average temperature, and temperature variation.

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electrodes were found to induce the overshoot of voltage gradient on the edges. The field overshoots and non-homogeneities of food characterized by various electrical 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, permitting optimization of the electrode configuration to minimize cold zones. 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 sealended corner of the pouch may be lower than expected, resulting in localized undertreatment of food materials. We have noted the presence of a so-called ‘shadow area’ of electric field in the cross-sectional domain of 2D model [36]. 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 potential gradient were coded and coupled to solve the electric field model. The 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. By using solution-adaptive refinement of meshes in Fluent, grid points could be 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 Ttype 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 Technologies, Inc., Palo Alto, CA) via a signal conditioning module, to eliminate signal interference with the electric field. Due to geometric symmetry of the pouch, sensor 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 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 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).

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(a)

Symmetry

M

B

L

C

R

E

z

x (b) L

C

y=0

R

y

Sensor tip x Thermocouple wires

Taping

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, 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.)

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 composition data [33,34], and were: density of 1020 kg/m3, viscosity of 3000 × 10 –6 Pa·s, specific heat of 4020 J/kg°C, and thermal conductivity of 0.6 W/m°C. Sample electrical conductivity was characterized as a function of temperature, increasing with a linear correlation, σ (S/m) = 0.032*T (°C) + 0.98. © 2009 by Taylor & Francis Group, LLC

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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 modeling 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 elements. 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. 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 0.0002 and 2050 V/m. The potential patterns appeared to be perpendicular to the x direction; however, the electric field strength near the edges of electrodes overshoot getting close to the maximum value, as observed in our previous work [25]. On the 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 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

E

A R B C Y X

Z

L M

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. 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.)

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FIGURE 6.12 Simulated 3D (a) electric potential and (b) field distributions inside the pouch. (From Jun, S. and Sastry, 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.)

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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 long term space mission: 3D ohmic model for verification of sterility efficacy. Journal of Food Engineering, 80(4), 1199–1205, 2007. With permission.)

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 thermal distribution in the middle of the pouch appeared to be a valley with elongated 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 noted previously. Secondly, even colder zones were noted at the corners of the pouch © 2009 by Taylor & Francis Group, LLC

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(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, the field overshoot is unlikely to occur in a rectangular shaped domain in which two electrodes are arranged parallel to one another, and electrodes and insulating layers cross at right angles. However, redesigning also involves numerous other considerations including practicality, mechanical strength and economics. Figure 6.14 compares model-predicted temperature values with experimental 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 temperature 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 periods of the simulation. The greatest error (around 6°C) was associated with this location, and represented an underprediction. This error implies that the expected value of heat transfer coefficient was not valid at the edge locations; indeed, the V-shape 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 regarding 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 temperatures 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 underprediction was observed. Simulations suggest the presence of significant hot and 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 the edge of the pouch, where current densities are lowered, are identified as points of 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 conductivities has been reported. Distortion of electric field due to several factors such © 2009 by Taylor & Francis Group, LLC

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FIGURE 6.14 Comparison of predicted and measured transient temperature profiles of tomato soup in the pouch. (From Jun, S. and Sastry, 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.)

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 performance of ohmic heaters. Sterilization of solid–liquid mixtures by ohmic heating requires the assurance that all parts of the food or biomaterial in question are treated adequately to ensure inactivation of pathogenic spore formers. The © 2009 by Taylor & Francis Group, LLC

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fundamental problem in continuous flow sterilization of solid–liquid mixtures is our lack of knowledge of (and inability to measure) temperature at the slowest-heating location within the entire system. Ohmic heating poses even greater challenges in measurement than conventional heat exchange processes due to the presence of an electric field. This necessitates the use of mathematical modeling to predict cold-spot temperatures; indeed modeling is a prerequisite to success in this process. The work done by Sastry and Salengke [37] has identified worstcase scenarios as being associated with a single solid piece (inclusion particle) of substantially different electrical conductivity than its surroundings. Two potentially hazardous situations were modeled, both involving an inclusion particle, but one involving a static medium surrounding the solid; and the other involving a mixed fluid, with a circuit theory analysis for the electric field [38]. The governing equations for the mixed fluid and static models were solved iteratively using the Galerkin–Crank–Nicholson algorithm (Galerkin three-dimensional finite element method in space; Crank–Nicholson finite difference scheme in time) by 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 experimental part of this study. Results indicate that the mixed fluid model provides a more conservative (preferable slightly underpredicting temperature) prediction of mixture cold-spot temperatures 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). However, 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 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, except when the solid size becomes sufficiently large to intercept a large fraction 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 conductive, 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 fluid model, eventually becomes lower than) the medium cold spot. Under such condition 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 potentially 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 transfer coefficient may not necessarily relate to the worst case, since fluid motion tends to moderate heating in such situations. If a fat-rich (low conductivity) phase is aligned to significantly intercept the current, it is possible for such a zone to heat faster than the surrounding fluid. In any case, care must be taken in establishing the process. Due to the complexity and great number of possible situations in food

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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 conductive as the fluid; (a) well-mixed fluid (b) static fluid.

processing, it may be prudent to investigate all potentially likely scenarios in process evaluation. The models compared in this work each have their respective merits, enabling large-system, high solids content simulation and low computation cost in the case of the circuit model, and rigorous, fine-detail predictions in the case of the Laplace model.

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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 conductive as the fluid; (a) well-mixed fluid (b) static fluid.

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CONCLUSION

Ohmic heating is one of the successful alternative food processing methods because it provides fast, energy efficient, and volumetric heating of food materials. Heating 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 predominant and it seems that its adoption is inevitable and progressive and aimed for comprehensive numerical solutions. Not necessary for fluid motion, the CFD is maturing into a powerful and pervasive tool to efficiently quantify the dynamic processes, heat 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 heating 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 particles with different electrical and thermal properties in fluid to the electric field needs to be incorporated.

NOMENCLATURE Cp k k0 m T t S V σ σ0 ρ

Specific heat (J/kg°C) Thermal conductivity (W/m°C) Pre-exponential factor Temperature coefficient Temperature (°C) Time (s) Internal energy source (W/m2) Voltage (V) Electrical conductivity (S/cm) Referencing electrical conductivity (S/cm) Density (kg/m3)

REFERENCES 1. Salengke, S. 2000. Electrothermal effects of ohmic heating on biomaterials: Temperature monitoring, heating of solid-liquid mixtures, and pretreatment effects on drying rate and oil uptake. Ph.D. Dissertation, The Ohio State University. 2. Sarang, S., and Sastry, S. K. 2007. Diffusion and equilibrium distribution coefficients of salt within vegetable tissue: Effects of salt concentration and temperature. Journal of Food Engineering, 82(3), 377–82. 3. Fryer, P. J., and Davies, L. J. 2001. Modeling electrical resistance (“ohmic”) heating of foods. In Irudayaraj, J. (Ed.) Food Processing Operations Modeling. 1st ed. New York: Marcel Dekker, 225–30.

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4. Rahman, M. S. 1999. In Rahman, M. S. (Ed.) Handbook of Food Preservation. New York: Marcel Dekker, 521–32. 5. Lima, M., and Sastry, S. K. 1999. The effect of ohmic heating frequency on hot-air drying rate and juice yield. Journal of Food Engineering, 41, 115–19. 6. Wang, W. C., and Sastry, S. K. 2000. Effects of thermal and electrothermal pretreatments on hot air drying rate of vegetable tissue. Journal of Food Process Engineering, 23, 299–319. 7. Zhong, T., and Lima, M. 2003. The effect of ohmic heating on vacuum drying rate of sweet potato tissue. Bioresource Technology, 87, 215–20. 8. de Alwis, A. A. P., and Fryer, P. J. 1990. A finite element analysis of heat generation and transfer during ohmic heating of foods. Chemical Engineering Science, 45, 1547–59. 9. Sastry, S. K., and Palaniappan, S. 1992a. Influence of particle orientation on the effective electrical resistance and ohmic heating rate of a liquid-particle mixture. Journal of Food Process Engineering, 5(3), 213–27. 10. Sastry, S. K., and Palaniappan, S. 1992b. Mathematical modeling and experimental studies on ohmic heating of liquid-particle mixtures in a static heater. Journal of Food Process Engineering, 5(4), 241–61. 11. Kim, H. J., Choi, Y. M., Yang, T. C. S., Taub, I. A., Tempest, P., Skudder, P., Tucker, G., and Parrott, D. L. 1996. Validation of ohmic heating for quality enhancement of food products. Food Technology, 50(5), 253–55. 12. Ye, X., Ruan, R., Chen, P., and Doona, C. 2004. Simulation and verification of ohmic heating in static heater using MRI temperature mapping. Lebensmittel-Wissenschaft und-Technologie, 37, 49–58. 13. Samprovalaki, K., Bakalis, S., and Fryer, P. J. 2007. Ohmic heating: Models and measurements. In Yanniotis, S., and Sunden, B. (Eds) Heat Transfer in Food Processing. UK: WIT Press, 159–64. 14. Palaniappan, S., and Sastry, S. K. 1991. Electrical conductivity of selected solid foods during ohmic heating. Journal of Food Process Engineering, 14, 221–36. 15. Icer, F., and Ilicali, C. 2005. Temperature dependent electrical conductivities of fruit purees during ohmic heating. Food Research International, 38, 1135–42. 16. Castro, I., Teixeira, J. A., Salengke, S., Sastry, S. K., and Vicente, A. A. 2004. Ohmic heating of strawberry products: Electrical conductivity measurements and ascorbic acid degradation kinetics. Innovative Food Science and Engineering Technologies, 5, 27–36. 17. Mitchell, F. R. G., and de Alwis, A. A. P. 1989. Electrical conductivity meter for food samples. Journal of Physics, E., 22, 554–56. 18. Castro, I., Teixeira, J. A., Salengke, S., Sastry, S. K., and Vicente, A. A. 2003. The influence of field strength, sugar and solid content on electrical conductivity of strawberry products. Journal of Food Process Engineering, 26, 17–29. 19. Saif, S. M. H., Lan, Y., Wang, S., and Garcia, S. 2004. Electrical resistivity of goat meat. International Journal of Food Properties, 7(3), 463–71. 20. Kim, S. H., Kim, G. T., Park, J. Y., Cho, M. G., and Han, B. H. 1996. A study on the ohmic heating of viscous food. Foods and Biotechnology, 5(4), 274–79. 21. Halden, K., de Alwis, A. A. P., and Fryer, P. J. 1990. Changes in the electrical conductivity of foods during ohmic heating. International Journal of Food Science and Technology, 25(1), 9–25. 22. Tulsiyan, P., Sarang, S., and Sastry, S. K. 2007. Electrical conductivity of multicomponent systems during ohmic heating. International Journal of Food Properties, 11(1), 233–41. 23. Shirsat, N., Lyng, J. G., Brunton, N. P., and McKenna, B. 2004. Ohmic processing: Electrical conductivities of pork cuts. Meat Science, 67, 507–14.

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24. Sarang, S. 2008. Ohmic heating for thermal processing of low-acid foods containing solid particulates. Ph.D. Thesis, The Ohio State University. 25. Perchonok, M., and Bourland, C. 2002. NASA food systems: Past, present, and future. Nutrition, 18, 913–20. 26. Peterson, B. V., Hummerick, M., Roberts, M. S., Krumins, V., Kish, A. L., Garland, J. L., Maxwell, S., and Mills, A. 2004. Characterization of microbial and chemical composition of shuttle wet waste with permanent gas and volatile organic compound analyses. Advances in Space Research, 34, 1470–76. 27. Jansen, A. N., Amine, K., Newman, A. E., Vissers, D. R., and Henriksen, G. L. 2002. Low cost, flexible batter packaging materials. JOM, 54(3), 29–32. 28. Amatore, C., Berthou, M., and Hebert, S. 1998. Fundamental principles of electrochemical ohmic heating of solutions. Journal of Electoanalytical Chemistry, 457, 191–203. 29. Wu, H., Kolbe, E., Flugstad, B., Park, J. W., and Yongsawatdigul, J. 1998. Electrical properties of fish mince during multi-frequency ohmic heating. Journal of Food Science, 63(6), 1028–32. 30. Samaranayake, C. P. 2003. Electrochemical reactions during ohmic heating. Ph.D. Dissertation, The Ohio State University. 31. Jun, S., and Sastry, S. 2005. Modeling and optimizing of pulsed ohmic heating of foods inside the flexible package. Journal of Food Process Engineering, 28, 417–36. 32. Samaranayake, C. P., Sastry, S. K., and Zhang, Q. H. 2005. Pulsed ohmic heating — a novel technique for minimization of electrochemical reactions during processing. Journal of Food Science, 70(8), E460–65. 33. Rahman, R. 1995. Food properties handbook. Boca Raton, FL: CRC Press. 34. Singh, R. P., and Heldman, D. R. 2001. Introduction to food engineering. 3rd ed. San Diego, CA: Academic Press. 35. Shames, P., Sun, P. C., and Fainman, Y. 1996. Modeling and optimization of electrooptic phase modulator. SPIE Proceedings: Physics and Simulation of Optoelectronic Devices IV, 2693, 787–96. 36. Jun, S., and Sastry, S. 2007. Reusable pouch development for long term space mission: 3D ohmic model for verification of sterility efficacy. Journal of Food Engineering, 80(4), 1199–205. 37. Sastry, S. K., and Salengke, S. 1998. Ohmic heating of solid–liquid mixtures: A comparison of mathematical models under worst-case heating conditions. Journal of Food Process Engineering, 21, 441–58. 38. Salengke, S., and Sastry, S. K. 2007. Models for ohmic heating of solid–liquid mixtures under worst-case heating scenarios. Journal of Food Engineering, 83, 337–55. 39. Center for Food Safety and Applied Nutrition (CFSAN). 2000. Kinetics of Microbial Inactivation for Alternative Food Processing Technologies: Ohmic and Inductive Heating, http://www.cfsan.fda.gov/~comm/ift-ohm.html. Accessed December, 2007.

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Pressure 7 Hydrostatic Processing of Foods J. Antonio Torres and Gonzalo Velazquez CONTENTS 7.1 7.2

Introduction ................................................................................................... 173 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. Consumers 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 content 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]. 173 © 2009 by Taylor & Francis Group, LLC

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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 food industry [4] inactivates microorganisms without causing significant flavour and 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 significant chemical changes in foods. Particularly important are thermal degradation reactions leading to off-flavours, destruction of nutrients and other product quality 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 treatments (e.g. 1 s at 89ºC, 0.1 s at 96ºC or 0.01 s at 100ºC) lowers flavour quality 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 times are not financially viable. HPP treatments (586 MPa for 3 and 5 min) at moderate temperature (55°C) extend the refrigerated shelf life of milk to over 45 days [10] while retaining milk volatile profiles similar to those observed after conventional 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 process induces strong ‘cooked’ off-flavour notes [12,13] thus limiting its consumer 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 of low-acid foods (pH under 4.5). PATP conditions are sufficiently severe to achieve 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 quality factors and constituents with important health benefits to consumers. It may also 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 development 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

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changes in food quality as compared to conventional technologies [6,7,16]. Pressure acts by disrupting mainly hydrogen bonds without affecting covalent bonds. Therefore, 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 flavours [4,17,18]. Research has confirmed that the sensory characteristics of HPP 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 uniform and nearly instantaneous throughout the food and thus independent of food geometry and equipment size. This has facilitated the scale-up of laboratory findings to full-scale production. The key HPP equipment technologies are the pressure vessels and the high hydrostatic pressure generating pumps or pressure intensifiers. Oil at ∼20 MPa is fed on the high-pressure oil side of the main pump piston which 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 ∼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 pump piston and the high-pressure fluid exits on the other pump side. The casting 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 current definition of low cost operations such as oyster shucking requiring 200–400 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 technology 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

Main piston

High pressure fluid piston

Outlet

Inlet

High pressure oil

Low pressure oil

High pressure fluid

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.)

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Yoke

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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.)

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 disinfestation (Figure 7.3). 7.2.1.1

Niche Opportunities for HPP Foods

HPP is an alternative processing technology that has reached consumers with a variety of new products. The successful introduction of a new technology demands the identification of specific competitive advantages over existing ones. In the case of HPP, an additional constraint is the large capital investment which is overcome by operating HPP plants at full capacity. Therefore, the processing of seasonal commodities requires identifying a product mix achieving maximum utilization of the equipment investment. The following sections provide examples of the many opportunities in which HPP has a clear competitive advantage. 7.2.1.1.1 Consumer Demand for Fresh Foods The 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 min. HPP avocado is now in national distribution because there was an unsatisfied consumer demand for products with acceptable shelf-life, convenient to use and free from chemical additives. Refrigerated fresh-cut fruit salads that consumers demand

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Atm 0

1000 2000 Potential: Disinfestation

0

100

3000 Low cost e.g., oysters

4000

5000 6000 High cost e.g., avocado

400

500

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Technology step

MPa 200

300

600

700

Pasteurization

800

Sterilization

kPsi 0

10

20

Low pressure

30

40

50

Moderate pressure

60

70

80

High pressure

90

100

110

120

Research

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.)

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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 on long moderate-pressure processes, typically 5–15 min at ∼400 MPa, reflected the 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 16month-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 regulations 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 including 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 investigated [36]. Non-carbonated and carbonated juices subjected to conditions ranging from 200 to 600 MPa pressure, 30–300 s dwell time and 15–50°C final processing 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 antioxidant levels of pressure-pasteurized apple and pulp-free orange juice show no significant differences between pressure-treated and control samples (Figure 7.4) [19]. Triangular test sensory evaluations using 101 apple and 221 orange juice consumers show that fresh-squeezed and HPP-treated juice are indistinguishable. These studies show that pressure treated juices are safe and similar to fresh-squeezed samples [19,37–44]. 7.2.1.1.2 Pressure Processing Effect is Unique The 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 hand-shucking is the most significant development in the oyster industry in the past 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

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Control

Apple juice

179

HPP

Orange juice

30

80 60

20 40 10 20

L

a

b

L

a

b

0

6

6

4

4

2

2

0

ORAC

FRAP

Ascorbic acid

ORAC

FRAP

mM Trolox equiv./L

mM Trolox equiv./L

0

0

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 opportunities and research challenges in the high pressure processing of foods. Journal of Food Engineering 67, 95–112, 2005.)

microbial risk to consumers by inactivating Vibrio parahemolyticus, V. vulnificus, 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. Scanning electron microscopy (SEM) shows major changes in the microstructure of Cheddar cheese immediately after HPP treatment (Figure 7.5). Transmission electron microscopy (TEM) shows similar microstructure changes (Figure 7.6) in Cheddar 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 presence 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 refrigerated storage savings of over $30/1000 kg cheese and a simplified handling of

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(a) Milled Control

10 µm

345 MPa/3 min

10 µm

Control

10 µm

345 MPa/3 min

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.)

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 observation 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 Producer This 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 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% incidence [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 (Table 7.1). These recalls are extremely costly in terms of financial and reputation losses. HPP could reduce this risk with a process and product formulation developed to minimize protein damage and thus changes in product texture.

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181

(a)

10 µm (b)

1 µm (c)

1 µm

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.)

A related example is the production of restructured fish products from underutilized species such as arrowtooth flounder (Atheresthes stomias). Using minced raw fish has the risk of food borne diseases associated with microbial contamination during post-capture manipulation and processing, particularly during deboning (or

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345Mpa-3min

Control

8

Milled

345Mpa-7min

VO

Day 1

6

A

483Mpa-3min 8

Stirred

SS

483Mpa-7min VO

Day 1

6

A

4

4

2

2

0

0

SS

ML

ML TO

TO

UL

PC VO 8 6

A

UL

PC VO 8

Day 27 SS

Day 27

6

A

4

4

2

2

0

0

SS

ML TO

TO

PC VO = Visual oiliness SS = Smooth surface ML = Mean length

UL UL = Uniformity of length PC = Presence of crumbles

ML

PC

UL TO = Tactile oiliness A = Adhesiveness

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 comercial. 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.)

filleting and mincing), protein solubilizing with salt, product forming and packaging. HPP can reduce this microbial load and could be an alternative to induce gel formation without heat to obtain products closer to raw fish. Arrowtooth flounder paste treated at 400–600 MPa for 5 min yields products with appropriate mechanical and functional properties [51]. These high pressure conditions are appropriate to inactivate parasites and most spoilage microorganisms. © 2009 by Taylor & Francis Group, LLC

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TABLE 7.1 Examples 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

Highland Crest finest smoked Scottish salmon

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]

Denaturation/aggregation of myofibrillar proteins induced by high pressure depends on pressure level, holding time [51] and temperature [52]. Gels obtained from myofibrillar proteins previously denatured/aggregated by freezing, dehydration, freeze-drying or thermal abuse show poor mechanical properties [53]. However, pressure treating fish paste at 300 MPa for 30 min at 5°C did not deteriorate the 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 gels when compared with gels obtained by heating untreated fish paste [51]. Pressure induces a protein denaturation/aggregation different from that resulting from freezing, drying or heating where denaturation is immediately followed by irreversible aggregation affecting negatively the mechanical properties of gels. Differential 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 denaturation 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 inducing changes in protein conformation favoring a protein aggregation reaction with © 2009 by Taylor & Francis Group, LLC

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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 characterized by k3 >>(k1 + k2). Evidence of this type of aggregation has been reported in pressure-induced fish proteins gels [54]. Because of this, pressure-induced aggregation improves the mechanical properties of heat-induced gels, a situation similar to Preferred agregation: side-to-side interaction of near-native proteins

Native protein HH2O

H2O

HPP k3

HPP

k1

Denaturated protein

Crosslinking of denatured proteins H2O

H2O

HPP k2

Protein structure showing intramolecular interactions Denaturated protein structure with broken intramolecular interactions Pressure induced crosslinking protein interactions Pressure effect on the protein system

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 on mechanical and functional properties of restructured products from arrowtooth flounder (Atheresthes stomias). Journal of the Science of Food and Agriculture 84(13), 1741–49, 2004.)

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the favorable protein aggregation induced at low temperature by transglutaminase in the so-called setting or suwari phenomenon. Setting induces protein aggregation only for partially denatured myofibrillar proteins (mostly myosin). Fully denatured myofibrillar proteins aggregate immediately inhibiting the transglutaminase effect [56]. Sugars and polyols at concentrations below 8% are used to inhibit protein interactions. Sorbitol inhibits the aggregation of myofibrillar proteins of raw pressuretreated samples while sucrose and trehalose are unable to inhibit this aggregation in low-temperature, pressure-treated arrowtooth flounder mince [51]. Sorbitol shows a higher stabilizing effect on the ATPase activity of fish myosin than lactitol, sucrose 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 possibility 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 Lability Biologically 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 worldwide opportunity. A major trend that ensures continuing growth in the demand for 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 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 Bacteria Mechanistic studies now emerging in the literature show that HPP inactivates microorganisms by interrupting cellular functions responsible for reproduction and survival (Figure 7.10). HPP can damage microbial membranes and thus affect transport phenomena involved in nutrient uptake and disposal of cell waste. Intracellular fluid compounds have been found in the cell suspending fluid after pressure treatment demonstrating that leaks occur while cells are held under pressure [62]. Membrane damage occurs later than cell death and this suggests that dye exclusion measurements assessing this pressure effect can be used to characterize microbial pressure inactivation [63]. Knowledge of cell damage and repair mechanisms could lead to new HPP applications [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].

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Native proteins H2O

H2O

HPP

Stabilizing agent Stabilized proteins against side-to-side aggregation

Native proteins H2OO

H2O

HPP

Protein structure showing intramolecular interactions Stabilizing agent molecule Pressure induced crosslinking protein interactions Pressure effect on the protein system

FIGURE 7.9 Mechanism of stabilization of sugars and polyols on myofibrillar proteins 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 functional properties of restructured products from arrowtooth flounder (Atheresthes stomias). Journal of the Science of Food and Agriculture 84(13), 1741–49, 2004.)

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 suspensions treated at 200 or 300 MPa did not differ significantly from the control, whereas cells treated at >400 MPa had decreased cell wall hydrolase activity. However, cells treated at 100 MPa released significantly more reducing sugar than all other samples indicating that this pressure activates cell wall hydrolase activity or increases cell wall accessibility to the enzyme. © 2009 by Taylor & Francis Group, LLC

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187

(a) Nutrients Membranes

Waste Leakage

(b)

Denaturation

Active enzyme

Enzymes

Inactive enzyme

Renaturation

FIGURE 7.10 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.)

Increasing our knowledge of the behaviour of bacterial membrane proteins subjected to pressure under different conditions (e.g. pH or aw) will lead to effective hurdle preservation technologies. For example, electrophoretic profiles of the outer 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 iodide (PI) staining and changes in membrane potential by flow cytometry. SEM studies reveal no significant changes in cellular morphology while PI staining followed by flow cytometry shows a small population proportion with membrane integrity loss 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). Morphological changes increasing with pressure correlate with a progressive decrease of the pH differential, intracellular potassium, and ATP concentration [71,72]. © 2009 by Taylor & Francis Group, LLC

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The outer membrane (OM) providing a protective barrier to Gram-negative bacteria is susceptible to pressure-mediated permeabilization. The kinetics of OM and cytoplasmic membrane permeabilization induced by pressure treatments can be determined by staining pressure-treated cells with the fluorescent dyes propidium iodide (PI) and 1-N-phenylnaphtylamine (NPN), respectively [73]. PI fluorescence increases only slightly even after pressure treatments resulting in a >6 log decrease in viable cell counts while increased NPN fluorescence, indicating OM permeabilization, is observed prior to cell death. Reversible OM damage occurs rapidly and is in thermodynamic 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 exponentialphase cells, pressure resistance increased with greater membrane fluidity, whereas in stationary-phase cells, no simple relationship between membrane fluidity and pressure resistance has been observed. 7.2.1.2.2 Bacterial Spores Although the application of 400–800 MPa inactivates pathogenic and spoilage bacteria [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. Spores of six Bacillus species showed no significant inactivation when pressurized at 980 MPa for 40 min at room temperature [78]; however, combining temperatures 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 achieved five decimal reductions (DRs) at neutral pH [79]. However, subjecting spores of Clostridium sporogenes, considered a non-toxigenic equivalent to proteolytic 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 and 6.5 pH) on spores of five isolates of Clostridium perfringens type A carrying the gene that encodes the C. perfringens enterotoxin (cpe) on the chromosome (Ccpe), 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

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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 bacterial spores more efficiently. A focus on a mechanistic understanding and process 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 germination 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 identified for C-cpe spores. At 30–50°C, spores of C-cpe isolates germinate slowly 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 germination of any C. perfringens type A food poisoning isolate as a function of spore germination 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) conditions 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 acid model solution pressure influences the thermally induced formation of Maillardderived compounds [86]. Another study reported that reported that although HPP does not lower the consumer acceptability of orange juice it could change its volatile 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 milk [11]. Moreover, at the same temperature, a principal component analysis of the volatile profile in milk under higher hydrostatic pressure was displaced in a direction 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]: dc = kc n dt

(7.1)

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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 − = kt c c0

(7.4)

The regression curve obtained by simple linear regression of the concentration compared with time at constant pressure and temperature with the best correlation coefficient (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 ‘pseudoinitial concentration’. Multiple linear regression analysis is used to test the difference between the linearized intercepts at the same temperature for each pressure level tested to confirm that all regression lines start at the same pseudo-initial concentration 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 represent 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 vessel (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 microextraction and gas chromatography with pulsed-flame photometric detection (HSSPME/GC-PFPD) for eight volatile sulphur compounds [89]. Dimethyl sulfide and 19 other volatiles were analyzed using a headspace solid-phase microextraction and gas chromatography with flame ionization detection (HS-SPME-GC/FID) technique [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 wellbuffered food system, reaction rates may be affected by the temporary pH shift induced by pressure [93]. As pressure increases, the columbic field of ions produces an alignment of water molecules resulting in a more compact arrangement around charged species [94–98]. This temporary pH shift cannot be measured experimentally as pH probes are currently unavailable for measurements at high pressure levels. The strategy of a pressure-independent buffer system used in studies of reaction

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TABLE 7.2 Effect of Hydrostatic Pressure and Temperature on the First-Order Reaction Rate Constant for the Formation of Various Straight-Chain Off-Flavour Aldehydes in Milk* Hexanal †

Heptanal

T (°C)

‡

P (MPa)

k

45

482

0.010a

0.87

0.006a

586

0.046

b

0.99

b

620

0.084c

0.99

655

d

0.99

d

0.105

§

1.58

c0 55

a

482

0.012 0.049

b

620

0.091

c

655

0.106d

586

0.97

Octanal 2

k

0.81

0.001a

0.92

0.003a

0.049

0.94

0.038

b

0.92

0.023

b

0.92

0.004

0.085c

0.99

0.035b

0.91

0.034c

0.93

0.008b

0.91

0.99

c

0.86

d

0.98

c

0.91

0.34 a

0.011

b

0.054

0.92 0.96

0.25

0.045

0.005 0.043

b c

0.98

0.048

0.90

0.065d

0.97

0.102

0.98

0.062

0.97

0.130d

0.99

0.079d

0.80 0.98

0.44

0.82

2.69

a

c

0.37

R

R2

K

0.049

k

Decanal 2

R

0.127

R

Nonanal 2

k 0.002a

0.017

ab

0.85 0.80

3.69

0.007

a

0.84

0.006a

0.84

0.037

b

0.93

0.009ab

0.95

c

0.98

0.013

b

0.94

0.98

0.019c

0.97

2.91

5.77

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

d

0.97

d

0.99

d

0.97

d

0.97

c

0.90

0.108

4.63

c0 75

0.89

2.88

c0 60

R

2

a

482

0.029 0.084

b

620

0.099

c

655

0.109d

586

c0 craw|| #

c75°C/655 MPa/ 10 min

0.96 0.99

0.134

0.56 a

0.033

b

0.088

0.93 0.96

0.098

0.77

0.067

4.22

0.019

a

0.119

c b

0.93

0.071

0.95

0.082d

0.98

c

0.125

0.98

0.107

0.97

0.138d

0.98

0.128d

0.92 0.96

0.023

5.88

0.014

a

0.96

0.016a

0.91

0.057

b

0.98

0.021b

0.92

c

0.99

0.025

c

0.93

0.98

0.030d

0.90

12.41

2.10

1.15

8.03

7.46

1.51

0.26

0.24

2.54

3.42

144.5

36.9

18.3

49.2

16.1

a, b, c, d Different letters mean significant difference (α = 0.05) of the rate constant for different hydrostatic pressure values at constant temperature. * k = Rate constant (min−1) with R2 = correlation coefficient. † T = Temperature. ‡ P = Pressure. § c = Pseudo-initial concentration (μg/kg), i.e. concentration in raw milk plus changes due to sample 0 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 volatile formation in milk subjected to pressure-assisted thermal treatments. Journal of Food Science 72(7), E389–98, 2007.

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TABLE 7.3 Effect of Hydrostatic Pressure and Temperature on the Zero-Order Rate Constants for the Formation of Various Off-Flavour Compounds in Milk* 2-Methylpropanal T† (°C)

P‡ (MPa)

45

R2

k

482

0.016

a

586

0.018a

620

0.017

a

0.016

a

655 c0§

2,3-Butanedione R2

k

0.88

0.009

a

0.86

0.010a

0.81

0.010

a

0.013

b

0.91 0.73

55

482

0.018a

586

0.023

b

620

0.021

b

655

0.022b

0.83

0.112

a

0.90

0.84

0.165b

0.94

0.85

0.205

c

0.96

0.208

c

0.93

0.97

0.44

1.62

0.009a

0.90

0.119a

0.90

0.83

0.014

a

0.94

0.227b

0.89

0.91

0.022

b

0.94

0.221

b

0.85

0.87

0.024b

0.86

0.242b

0.88

482

0.019

a

0.83

0.014

a

0.94

0.140

a

0.87

586

0.024b

0.83

0.020b

0.89

0.236b

0.94

620

0.023

b

0.934

0.040

c

0.92

0.275

c

0.91

655

0.021a

0.93

0.053d

0.98

0.280c

0.94

482

0.023

a

0.82

0.029

a

0.88

0.217

a

0.84

586

0.030a

0.90

0.045b

0.97

0.314b

0.82

620

0.027

a

0.92

0.055

c

0.96

0.371

c

0.80

655

0.030a

0.92

0.082d

0.99

0.394c

0.86

0.54

0.37

0.48

c0 75

R2

k

0.85

c0 60

Hydrogen Sulfide

8.57

0.41

14.56

c0

0.38

0.55

16.65

craw||

0.75

0.36

1.03

0.64

1.24

23.2

#

c75°C/655 MPa/ 10 min

a, b, c, d Different letters mean significant difference (α = 0.05) of the rate constant for different hydrostatic pressure values at constant temperature. * k = Rate constant (min−1) with R2 = correlation coefficient. † T = Temperature. ‡ P = Pressure. § c = Pseudo-initial concentration (μg/kg), i.e. concentration in raw milk plus changes due to sample han0 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.

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Hydrostatic Pressure Processing of Foods

193

kinetics in model systems or the inactivation of bacterial spores [81] cannot be used in studies with milk. All pseudo-initial concentration c 0 values shown in Table 7.2 and Table 7.3 for compounds following 1st and zero order kinetics, respectively, except those for 2methylpropanal and 2,3-butanedione, increased significantly with test temperature 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 linear regression with 95% confidence) confirming that regardless of the pressure 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 hydrostatic 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 constant) are calculated as follows. − Ea

k = k0e RT

ln( k ) = ln( k0 ) −

(7.5) Ea RT

(7.6)

A quantity derived from the pressure dependence of the rate constant k (Equation 7.7) is the partial activation volume (ΔV*), is defined as the difference between the 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 integration constant. Values for ΔV* as a function of temperature are then calculated by linear regression of lnk versus pressure p. ⎛ ∂ ln k ⎞⎟ ⎟ ΔV * = −RT ⎜⎜ ⎜⎝ ∂p ⎟⎟⎠ T

(7.7)

( ΔV * ) p RT

(7.8)

ln k = ln A −

The 1st-order kinetic constants observed for straight chain aldehydes in milk processed under PATP fitted well (R2 > 0.9) the Arrhenius model with activation energy (Ea) values decreasing significantly with pressure (Table 7.4 and Table 7.5) [15]. 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 zero order kinetic models (Table 7.3) showed also a good fit to the Arrhenius model

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194

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Other Compounds (Zero Order Reaction Kinetics)

Straight Chain Aldehydes (First Order Reaction Kinetics) Hexanal P† (MPa)

Ea

R2

Heptanal

Octanal

Nonanal

Decanal

2-Methyl-propanal

2,3-Butanedione

Hydrogen Sulfide

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) with R2 = correlation coefficient. P = Pressure. 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 of Agricultural and Food Chemistry 54(24), 9184–92, 2006a. *

†

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Food Processing Operations Modeling: Design and Analysis

TABLE 7.4 Effect of Hydrostatic Pressure on the Activation Energy for the Formation in Milk of Various Off-Flavour Compounds*

Hydrostatic Pressure Processing of Foods

195

TABLE 7.5 Relationship 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

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 = −0.001x +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

2

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. * †

(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-methylpropanal 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 analyzed 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 under pressure was a most interesting finding. High hydrostatic pressure inhibited the 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 temperature 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 experiments 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,

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TABLE 7.6 Effect of Temperature During Sample Loading (3.5 min) into the Pressure Vessel on the Average Pseudo-Initial Concentration* (c0) of Various OffFlavour Compounds in Milk Temperature (°C) Compound

45

Ethyl acetate (μg/kg)

55

60

0.19a

0.21a

a

b

Relationship Intercept (I) vs. Temperature (T)

75

0.27b

R2

0.29b

I = 0.003T − 0.03

0.854

c

2-Methylbutanal (μg/kg)

0.14

0.24

0.24

0.28

I = 0.004T − 0.02

0.817

2-Pentanone (μg/kg)

0.14a

0.15a

0.15a

0.18b

b

a

a

0.09

3-Methyl-1-butanol (μg/kg)

0.07

bc

0.07

I = 0.001T − 0.07

0.925

ab

NR

NR

c

0.08

2-Hexanone (μg/kg)

0.07

0.13

0.15

0.28

I = 0.007T–0.25

0.982

2-Furaldehyde (μg/kg)

0.29a

0.43b

0.44b

0.47b

NR

NR

a

b

c

d

a

b

b

2-Heptanone (μg/kg)

1.14

1.44

2.05

4.01

I = 0.002T − 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

a

a

2

2-Nonanone (μg/kg)

0.47

0.48

0.57

0.64

I = 0.006T–0.18

0.893

2-Decanone (μg/kg)

0.62ab

0.69b

0.58ab

0.55a

NR

NR

ab

c

b

NR

b

I = 0.004T − 0.40T +15.04 0.999

0.30

2-Undecanone (μg/kg)

a

0.52

a

ab

a

0.22

a

b

0.38

NR

Methanethiol (μg/kg)

5.20

5.21

5.59

7.76

Carbon disulfide (ng/kg)

21.87c

19.16b

20.01bc

14.42a

NR

NR

a

NR

a

a

a

2

Dimethyl sulfide (μg/kg)

3.97

3.95

3.91

3.89

NR

Dimethyl disulfide (ng/kg)

24.84a

33.39b

22.08a

34.72b

NR

Dimethyl trisulfide (ng/kg)

30.83a

43.96b

36.82ab 122.83c

ab

b

a

c

NR

I = 0.141T − 14.05T +378.32 0.968 2

Dimethyl sulfoxide (mg/kg)

0.66

0.71

0.57

1.29

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 concentration (Tukey HSD 95%). * c = Pseudo-initial concentration (μg/kg), i.e. concentration in raw milk plus changes due to sample 0 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 volatile formation in milk subjected to pressure-assisted thermal treatments. Journal of Food Science 72(7), E389–98, 2007.

was chosen to approach conditions achieving microbial inactivation including bacterial spores [15]. Further confirmation of the inhibitory effects of pressure on several chemical reactions in PATP milk can be obtained by comparing the effect of thermal treatments 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 © 2009 by Taylor & Francis Group, LLC

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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

Hydrostatic Pressure Processing of Foods

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TABLE 7.7 Comparison between the Effect of HPP and Heat Treatment at Atmospheric Pressure on the Formation of Off-Flavour Compounds in Milk*

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 treatments. Journal of Food Science 72(7), E389–98, 2007. * †

197

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198

Food Processing Operations Modeling: Design and Analysis

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 heating during pressurization of milk initially at 75°C. Straight-chain aldehydes were present 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. These findings support previous conclusions that the effect of pressure-assisted thermal treatments on the volatile profile of milk is different to that of an equivalent heat 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 concentration of saturated aldehydes is thought to cause the stale off-flavour in milk [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 are needed because the aroma profiles obtained are very different from those produced during conventional thermal treatments [11]. It has been reported that volatile sulphur compounds are mainly responsible for the development of the cooked flavour defect in heated milk [14,102,108,109]. Methanethiol is probably the most powerful sulphur-containing aroma compound in heated milk [11] with a low sensory 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 an unpleasant eggy, sulphury aroma, but a recent analysis using improved quantification 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 development of stale-heated flavour in UHT milk [111]. Although their high sensory thresholds 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 impart a perceptible flavour [112]. 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), whereas ΔV* values defined as the difference between the partial molar volumes 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 negative 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 © 2009 by Taylor & Francis Group, LLC

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(c) [Activated state]* ln k

E

Ea decreases If ΔV*< 0 0.1 and 650 MPa No change in Ea If ΔV*= 0 650 MPa 650 MPa

Ea>0

Reagents

Ea increases If ΔV*> 0

[Products]

Absolute temperature–1

Reaction pathway (b)

[Activated state]* w/ΔV* > 0

(d) 482–655 MPa

ln k

V* ΔV*

ΔV*< 0

[Products] ΔV = 0

Reagents ΔV*

Reaction w/ΔV*= 0 ΔV*> 0 [Activated state]* w/ΔV*< 0 Reaction pathway

ΔV*> 0 Pressure

199

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FIGURE 7.11 Analysis of pressure and temperature effects on reaction kinetics. (a) Definition of energy of activation Ea. (b) Definition of activation 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 pressure-assisted thermal treatments. Journal of Food Science 72(7), E389–98, 2007.)

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(a)

200

55534_C007.indd 200

Hexanal T (°C)

ΔV*

Heptanal 2

R

ΔV*

Octanal 2

R

ΔV*

Nonanal 2

R

ΔV*

R

Decanal 2

ΔV*

2,3-Butanedione 2

R

ΔV*

2

R

Hydrogen Sulfide 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) with R2 = correlation coefficient. 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.

∗

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Food Processing Operations Modeling: Design and Analysis

TABLE 7.8 Effect of Temperature on the Activation Volumes for the Formation of Various Off-Flavor Compounds in Milk*

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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 consistent 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 they did not fit the Arrhenius model (R 2 < 0.648). No changes in the concentration 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

ln k 482–655 MPa ΔV*< 0

Pressure

Minimum rate for detectable formation within 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 VazquezLandaverde, 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.)

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meeting current consumer demand for foods with minimal effects of processing. Most significantly, alternative reaction formation mechanisms are not likely involved 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 technology 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 constituents with important health benefits to consumers, but may inhibit also the 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 addition, 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. Reports of larvae survival to these hydrothermal treatments [120] appear to reflect heat-conducting differences among fruits of different maturity, fruits containing larvae 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 hydrothermal 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 the Mediterranean fruit fly (Ceratitis capitata), one of the most destructive fruit pests [121]. When detected in Florida and California, each infestation has required intensive and massive eradication and detection procedures. Its larvae feed and develop on many deciduous, subtropical, and tropical fruits and some vegetables. The Caribbean 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 eggs under the skin of fruit just as they begin to ripen, often where some break in

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TABLE 7.9 Inactivation of Mediterranean Fruit Fly (Ceratitis capitata) Eggs by Relative Low Hydrostatic Pressure (LHP) Treatments Temperature (oC) 0

12.5

25

32.5

40

Time (mn) Pressure (Pa)

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. Source: Adapted from Butz, P, and Tauscher, B. Inactivation of fruit fly eggs by high pressure treatment. Journal of Food Processing and Preservation 19, 161–64, 1995.

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, and at least one peer-reviewed published report on the Mediterranean fruit fly [115], 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 fly. As shown in Table 7.9, inactivation was almost independent of pressure holding time and treatment temperature. Unfortunately, this is the only inactivation data reported to assist in the design of LHP treatments. At present, the pressure inactivation conditions for the eggs of insect of commercial importance in California and the Pacific Northwest (Table 7.10) are being determined at Oregon State University in cooperation with the California Kearney Agricultural Center. Preliminary observations of the quality of LHP-treated dry fruits, including raisins, figs and apricots are encouraging.

7.5 CONCLUSIONS The application of high hydrostatic pressure for the processing of foods was first reported over a century ago when a 5–6 log-cycle total count reduction was achieved © 2009 by Taylor & Francis Group, LLC

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TABLE 7.10 Infestation of Commercial Concern in California and the Pacific Northwest Common Name

Scientific 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

Dried fruit beetle

Carpophilus hemipterus

Figs, raisins

Infests stored products at least on NW Oregon

Ten-lined June beetle

Polyphylla decemlineata

Almonds, walnuts

Statewide

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

Olive fruit fly

Bactorcerus oleae

Olives

Not known to occur

Walnut husk fly

Rhagoletis complete

Walnuts

Statewide

European red mite

Panonychus ulmi

Almonds, stone & pome fruit

Statewide

Pacific spider mite

Tetranychus pacificus

Almonds, stone & pome fruit

Statewide

Two spotted spider mite

Tetranychus urticae

Almonds, stone & pome fruit

Statewide

Coleopterans

Hemipterans

Diptera

Acari

*

Provided by Rick Westcott, Rich Worth, and Kathleen Johnson at the Oregon Dept. of Agriculture, July 10, 2007.

without using heat by treating milk at 670 MPa for 10 min. HPP technology is nowadays a well-established food technology with commercial success stories all over the world [124, 125]. Unlike thermal processing and most other preservation technologies, HPP effects are uniform and nearly instantaneous throughout the food and thus

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independent of food and equipment geometry and size. This has facilitated the scaleup of laboratory findings to full-scale production and is a key factor explaining the rapid commercialization of HPP processing technology. The next generation of high hydrostatic pressure units will allow the combination of pressure and thermal treatments. It is interesting to note that it is again experimental research on milk that has demonstrated the promising advantages of PATP technology. Only thermal degradation reactions leading to the formation of aldehydes were accelerated by high pressure. The formation of most volatiles reported to be factors of the consumer rejection of ‘cooked’ milk flavour were actually fully 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 t k n c0 Ea R k0 ΔV* A p R2

Concentration, g L−1 Time, min Reaction rate constant, units vary with reaction order Reaction order Pseudo-initial concentration Activation energy, kJ mol−1 Universal gas constant, 8.314×10-3 kJ mol−1 K−1 Pre-exponential rate constant Partial activation volume Integration constant Pressure, MPa Regression coefficient

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81. DG Paredes-Sabja, M Gonzalez, MR Sarker, and JA Torres. 2007. Combined effects of hydrostatic pressure, temperature and pH on the inactivation of spores of Clostridium perfringens Type A and Clostridium sporogenes in buffer solutions. Journal of Food Science 72: M202–M206. 82. VK Juneja, and HM Marks. 1999. Proteolytic Clostridium botulinum growth at 12–48 °C simulating the cooling of cooked meat: Development of a predictive model. Food Microbiology 16: 583–92. 83. S Smith, and DW Schaffner. 2004. Evaluation of a Clostridium perfringens predictive model, developed under isothermal conditions in broth, to predict growth in ground beef during cooling. Applied Environmental Microbiology 70(5): 2728–33. 84. VK Juneja, L Huang, and HH Thippareddi. 2006. Predictive model for growth of Clostridium perfringens in cooked cured pork. International Journal of Food Microbiology 110(1): 85–92. 85. J Collado, A Fernandez, M Rodrigo, and A Martinez. 2006. Modelling the effect of a heat shock and germinant concentration on spore germination of a wild strain of Bacillus cereus. International Journal of Food Microbiology 106(1): 85–89. 86. T Hofmann, F Deters, I Heberle, and P Schieberle. 2005. Influence of high hydrostatic pressure on the formation of Maillard-derived key odorants and chromophores. Annals of the New York Academy of Sciences 1043: 893. 87. IA Baxter, K Easton, K Schneebeli, and FB Whitfield. 2005. High pressure processing of Australian navel orange juices: Sensory analysis and volatile flavor profiling. Innovative Food Science and Emerging Technologies 6(4): 372–88. 88. P Taoukis, and T Labuza. 1996. Summary: Integrative concepts. In: OR Fennema, editor. Food Chemistry. 3rd ed. New York: Marcel Dekker Inc., 1018–23. 89. PA Vazquez-Landaverde, JA Torres, and MC Qian. 2006b. Quantification of tracevolatile sulfur compounds in milk by solid-phase microextraction and gas chromatography-pulsed flame photometric detection. Journal of Dairy Science 89: 2919–27. 90. PA Vazquez-Landaverde, G Velazquez, JA Torres, and MC Qian. 2005. Quantitative determination of thermally derived volatile compounds in milk using solid-phase microextraction and gas chromatography. Journal of Dairy Science 88: 3764–72. 91. AE Harvey, AP Peskin, and SA Klein.1996. NIST/ASME steam program. Physical and Chemical Properties Div., Natl. Inst. of Standards and Technology, US. Dept. of Commerce, Boulder, CO. 92. E Ting, VM Balasubramaniam, and E Raghubeer. 2002. Determining thermal effects in high-pressure processing. Food Technology 56(2): 31–56. 93. N Datta, and HC Deeth. 1999. High pressure processing of milk and dairy products. Australian Journal of Dairy Technology 54: 41–48. 94. A Distche. 1972. Effects of pressure on the dissociation of weak acids. Symposia of the Society for Experimental Biology 26: 27–60. 95. BS El’yanov, and SD Hamann. 1975. Some quantitative relationships for ionization reactions at high pressure. Australian Journal of Chemistry 28: 945–54. 96. DE Johnston, BA Austin, and RJ Murphy. 1992. Effects of high hydrostatic pressure on milk. Milchwissenschaft 47: 760–63. 97. E Morild. 1981. The theory of pressure effect on enzymes. Advances in Protein Chemistry 34: 93–165. 98. RC Neuman, W Kauzmann, and A Zipp. 1973. Pressure dependence of weak acid ionization in aqueous buffers. Journal of Physical Chemistry 77(22): 2687–91. 99. AD McNaught, and A Wilkinson. 1997. Compendium of chemical terminology: IUPAC recommendations. 2nd ed. Malden, MA: Blackwell Science. 100. D Mussa, H Ramaswamy. 1997. Ultra high pressure pasteurization of milk: Kinetics of microbial destruction and changes in physico-chemical characteristics. LebensmittelWissenschaft und -Technologie 30: 551–57.

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101. MM Calvo, and L de la Hoz. 1992. Flavour of heated milks. A review. International Dairy Journal 2: 69–81. 102. KR Christensen, and GA Reineccius. 1992. Gas chromatographic analysis of volatile sulfur compounds from heated milk using static headspace sampling. Journal of Dairy Science 75: 2098–104. 103. G Contarini, M Povolo, R Leardi, and PM Toppino. 1997. Influence of heat treatment on the volatile compounds of milk. Journal of Agricultural and Food Chemistry 45: 3171–77. 104. RA Scanlan, R Lindsay, LM Libbey, and EA Day. 1968. Heat-induced volatile compounds in milk. Journal of Dairy Science 51: 1001–7. 105. T Shibamoto, S Mihara, O Nishimura, Y Kamiya, A Aitoku, and J Hayashi. 1980. Flavor volatiles formed by heated milk. In: G Charalambous, editor. The analysis and control of less desirable flavors in foods and beverages. New York: Academic Press Inc., 260–63. 106. S Rerkrai, IJ Jeon, and R Bassette. 1987. Effect of various direct ultra-high temperature heat treatments on flavor of commercially prepared milks. Journal of Dairy Science 70: 2046–54. 107. M Rychlik, P Schieberle, and W Grosch. 1998. Compilation of odor thresholds, odor qualities and retention indices of key food odorants. Garching, Germany: Deutsche Forschungsanstalt fur Lebensmittelchemie and Institut fur Lebensmittelchemie der Technischen Universitat Munchen. 108. N Datta, AJ Elliot, ML Perkins, and HC Deeth. 2002. Ultra-high-temperature (UHT) treatment of milk: Comparison of direct and indirect modes of heating. Australian Journal of Dairy Technology 57(3): 211–27. 109. M Simon, and AP Hansen. 2001. Effect of various dairy packaging materials on the shelf life and flavor of ultrapasteurized milk. Journal of Dairy Science 84: 784–91. 110. G Fenaroli. 1995. Fenaroli’s handbook of flavor ingredients. 3rd ed. New York: CRC Press. 111. G Contarini, and M Povolo. 2002. Volatile fraction of milk: Comparison between purge and trap and solid phase microextraction techniques. Journal of Agricultural and Food Chemistry 50: 7350–55. 112. JE Langler, and EA Day. 1964. Development and flavor properties of methyl ketones in milk fat. Journal of Dairy Science 47: 1291–96. 113. RE Paull, and JW Armstrong. 1994. Insect pests and fresh horticultural products: Treatments and responses. Wallingfroth, UK: CAB International, United Press. 114. JL Sharp, and GI Hallman. 1994. Quarantine treatments for pests of food plants. Boulder, CO: Westview Press, Inc. 115. P Butz, and B Tauscher. 1995. Inactivation of fruit fly eggs by high pressure treatment. Journal of Food Processing and Preservation 19: 161–64 116. KC Shellie, MJ Firko, and RL Mangan. 1993. Phytotoxic response of ‘Dancy’ tangerine to high-temperature, moist, forced-air treatment for fruit fly disinfestation. Journal of the American Society of Horticultural Science 118(4): 481–85. 117. J Jaczynski, Y-C Chen, G Velazquez, and JA Torres. 2008. Procesamiento de productos pesqueros con haz de electrones. In: I Guerrero-Legarreta, MR Rosmini, RE Armeneta-López, editors. Tecnología de productos de origen acuático, volumen. Tecnología de pescado y mariscos. LIMUSA, S.A. de C.V., México D.F., Mexico. 118. KK Jacobi, EA MacRae, and SE Hetherington. 2001. Postharvest heat disinfestation treatments of mango fruit. Scienta Horticulturae 89(2001): 171–93. 119. KC Shellie, and RL Mangan.2002. Hot water immersion as a quarantine treatment for large mangoes: Artificial versus cage infestation. HortScience 37(3): 430–34. 120. DB Thomas, and KC Shellie. 2000. Heating rate and induced thermotolerance in Mexican fruit fly (Diptera: Tephritidae) larvae, a quarantine pest of citrus and mangoes. Journal of Economic Entomology 93(4): 1373–79.

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121. WC Mitchell, and SH Saul. 1990. Current control methods for the Mediterranean fruit fly, Ceratitis capitata, and their application in the USA. Review of Agricultural Entomology 78: 923–30. 122. RV Dowell, and CJ Krass. 1992. Exotic pests pose growing problem for California. California Agriculture 46(1): 6–8, 10–12. 123. AE Sloan. 2008. Mega markets and nasty niches. The next generation of nutraceutical markets. Nutraceuticals World Consulted July 21, 2008 (http://www nutraceuticalsworld. com/articles/2001/11/mega-markets-and-nasty-oniches-the-next-generations). 124. C Pérez Lamela, and JA Torres. 2008. Pressure-assisted thermal processing: 1. A promising future for high flavour quality and health-enhancing foods. AgroFOOD Industry Hi-Tech 19(3): 60–62. 125. C Pérez Lamela, and JA Torres. 2008. Pressure-assisted thermal processing: 2. Microbial inactivation Kinetics and pressure and temperature effects on chemical changes. AgroFOOD Industry Hi-Tech 19 (In Press).

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Electric Field 8 Pulsed (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 Efficacy of PEF Processing ............................... 221 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 Acidified Food ................................................... 229 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

Pulsed electric field (PEF) processing involves subjecting food products to certain controllable pulsed electric fields to inactivate microorganisms and enzymes and to 213 © 2009 by Taylor & Francis Group, LLC

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modify the properties of the food components. The number one benefit of applying 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 technology are fresher in flavor, more nutritious and richer in heat-labile bioactive 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 heatinduced toxic compounds in food products. Consumers showed significantly higher 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]. Successful 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]. Genesis became the first company to introduce a PEF product in an emotional debut 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 see that myself”. Diversified Technologies, Inc. (DTI, the builder of the high voltage pulser) and Genesis were nominated for 2007 IFT Industrial Achievement Award to recognize their contribution in introducing PEF technology into food industry practices. The PEF processed juices helped the company win their customers back and attracted many new customers due to the significantly fresher flavor of the products. 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 dramatically 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 researchers since the 1960s. In 1960, a patent was issued to Doevenspeck [3] who creatively applied uniform electric fields for inactivating microorganisms. Doevenspeck found that the intensity of the electric fields applied onto the microbial cells have different inactivation 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 pasteurization. The authors reported that electric pulses of high voltage electric fields up to 25 kV inactivate Escherichia coli, Staphylococcus aureus, Micrococcus lysodeikticus, Sarcina lutea, Bacillus subtillus, B. cereus, B. megatherium, Clostridium welchii, Saccharomyces cerevisiae, and Candida utilis. Electrical fields up to 25 kV × cm−1 were applied as a series of direct current pulses from 2 to 20 μs to the suspensions of microorganisms. Temperature increase during pulsed electric field treatment for ten pulses of 20 μs at 19.5 kV was 10°C. The contribution of thermal inactivation to the total inactivation observed was literally negligible. Electric field strength and the total treatment time are the two most important parameters responsible for the inactivation of microorganisms.

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However, different cell structural properties, such as size of the cell, properties of the 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 inactivation effects by PEF. Generally cells with larger size, such as yeast and molds, are more sensitive to the electric fields than their smaller peers like bacterial cells and virus. Gram negative microorganisms are more sensitive to PEF treatment than Gram positive microorganisms. 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 established 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 introduction of this new technology has the potential to provide consumers with microbiologically 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 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 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 in model foods such as simulated milk ultrafiltrate [9]. The cell suspension temperature was maintained below the lethal temperature. The results suggest that the microbial inactivation effect of PEF treatment is not due to thermal effect. Stepwise PEF treatment on E. coli at electric field strength ranged from 35 to 75 kV × cm−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 each pulse, measured with a fiberoptic temperature probe, was only 0.3°C. Since then many other microbial inactivation experimental results have been reported [12–20]. Pothakamury and co-workers [10] reported their studies about influences of microorganism growth stage and processing temperature on inactivation effects of E. coli by PEF. The authors reported that at 3–40°C processing temperature, squarewave showed higher inactivation effects on E. coli suspended in simulated milk ultrafiltrate (SMUF) than exponential decaying wave pulses at same electric field strength of 36 kV × cm−1. Logarithmic-phase cells were more sensitive than stationary and lag-phase cells. This result implies that when we treat food samples not only the PEF parameter should be counted as the influential factor for the inactivation effects but also the microorganisms themselves. Significant breakthrough in PEF hardware development was reported [21] with the introduction of the new generation of continuous, automatically controlled PEF processing systems, OSU-4, using co-field flow chamber systems. The innovative

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PEF system using H-bridge electronic pulse generating circuit and solid state high speed switch devices allows researchers to treat samples continuously with accurate temperature control. Electric field strength can be up to 50 kV × cm−1 with a treatment time up to 400 μs. System adjustability in pulse duration time and delay time as well as options in pulse polarity provides researchers with tremendous flexibility to 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 commercial scale PEF processing system were assembled at The Ohio State University, Ohio, USA in 2001 and 2003, respectively. The up-scaled PEF processing facilities at The Ohio State University provided the necessary tools for evaluation of PEF efficacy 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 conventional thermal pasteurization on single strength orange juice, PEF processing showed 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 soymilk product [25], while thermal process at the same pasteurization power caused over 80% immunoglobulin G activity loss. Later studies confirmed that loss of immunoglobulin 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 co-workers proposed the shape factor concept to define the changes in IgG activity. 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 understanding of this non-thermal technology. Potential applications in food industry and possible trends of this technology will be briefly discussed at the later part of this chapter.

8.2

PULSED ELECTRIC FIELD (PEF) PROCESSING TECHNOLOGY AND MECHANISMS FOR MICROBIAL INACTIVATION

8.2.1 8.2.1.1

PEF TECHNOLOGY AND HARDWARE DEVELOPMENT Basics of PEF Technology

PEF processing involves the application of pulses of high voltage (typically 20–80 kV × cm−1) to foods between two electrodes, in a finely controlled manner to inactivate © 2009 by Taylor & Francis Group, LLC

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microorganisms and enzymes existing in food products, and/or to modify certain properties 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 temperature 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 minimizing the heat impairment on food quality [28]. Food products processed using PEF technology may be fresher, more nutritious, with more heat sensitive physiologically bioactive components and better flavor compared to those processed by conventional approaches [24,25,28]. (a)

(b) PLC A

B

C

D

A

B

C

D

Step motors

Syringes

Thermocouple readers

Oscilloscope

Pulse generator Water bath

Marked area A

FIGURE 8.1 OSU-4J PEF system (a) and schematic diagram of a typical lab scale OSU-4 PEF system (b).

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Chamber 4

Chamber 2 Chamber 3

HV Gap distance GRD PEF zone Sample In

T1

T2

T4

T3

Sample Out

Cooling coils

Thermocouple readers

FIGURE 8.2 Schematic diagram of a typical co-field flow treatment chamber system coupled with OSU-4 PEF system. GND

HV

GND

PEF Zones

FIGURE 8.3 ment zones.

Modified co-field flow PEF chamber system with two compactly paired treat-

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 developments in electronic technology and innovation in treatment chambers, such as the compactly paired co-field flow chamber system (Figure 8.3) ensuring operational safety and improving PEF efficacy. PEF technology is also seeking more applications in medicine, breeding and weed control industries. The world’s very first commercial 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 vegetable 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 System In a typical PEF processing system, the following subunits are critical to the overall systematic functionality: a. Power supplier PEF processing requires stable power supplies, particularly stable voltage supply. An effective power supplier is always needed to provide primary power needs. However, © 2009 by Taylor & Francis Group, LLC

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due to the nature of high intensity of the electric fields needed for the expected effects 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 electric field strength. b. Pulse generator The pulse generator includes a signal generator, pulse generating electric circuits and solid phase high speed switching devices. A signal generator is the commanding 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 trigger 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 processing is in continuous flow mode. 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 Semicron. IGCT devices have been successfully used in many OSU-4 PEF systems. Nevertheless, 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 over flow of system memory. Cooling of switching devices and the circuits connected is critical to ensure high efficiency and stable performance of pulse generating systems. Maintaining system temperature stable and close to room temperature is as Rc

U3

U1 + –

V

C1 Transformer U4

U2 PEF chambers

FIGURE 8.4 H-bridge electric circuits for generating square waveform pulses (Circuit diagram 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). © 2009 by Taylor & Francis Group, LLC

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critical as the fundamental design of the system, because the switching devices will drop their efficiency and accuracy under an elevated temperature. It is particularly important when the system is operated under higher electric field strength for pasteurization purposes. Currently both water cooling and forced air cooling are in use in the PEF processing systems with built-in pulse generators. Interruption of communication between subunits and temperature control issues are the most commonly technical challenges encountered by users in practice. c. Control and monitoring system Control 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 the observed electric field strength. Different designers may choose to measure the 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 diffusion and voltage drop from transformer to treatment chamber is avoided. Attention is also needed to closely watch the temperature fluctuation which will impact the normal function of the controlling and monitoring system. An effective cooling system is needed to provide optimal temperature condition for the controlling and monitoring system to sustain stable performance. Forced air cooling and fluid cooling are commonly used in many different versions of PEF equipment, depending on the heat load generated during operation. d. Fluid handling system A PEF fluid handling system includes PEF chamber system, pump(s), pressure control units, delivering units, receiving units and monitoring systems such as temperature monitors and flow meter (Figure 8.1b). PEF chamber system (Figure 8.2) is where the delicately generated electric pulses meet with food products, and is the center of a fluid handling system and the heart of the whole PEF processing system. All other subsystems serve the purpose of either generating delicately controlled electric pulses at designed field strength, accurate pulse width and stable pulse replication rate or making sure the food product passes through the treatment chamber at a uniform 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 PEF treatment zone, ensuring the uniformity through every single portion of the chamber volume. PEF treatment chambers were first designed in static mode used for laboratory application, and now are still in wide use for kinetic studies. The advantages of using static chambers include the uniform distribution of electric fields in most cases, and

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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 replication rate is too fast, a significant amount of pulses were applied during experiments. Static chambers have also found themselves of difficult use in industrial practices due to their limited production capacity. However, on the other hand, continuous flow PEF chambers have found lots of potential industrial applications, such as liquid food pasteurization. There have been many different designs in pursuit of a successful application in the food industry, 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 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 packaging Aseptic 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 recontamination. Thanks to its sanitary continuous flow characteristics, many different models of aseptic filling/packaging systems can be easily hooked up with a PEF processing system. A Banco® aseptic packaging system, with in-line thermoforming of multilayer cups, adjustable in cup size, was hooked together with the world first commercial scale PEF processing system at The Ohio State University. The assembly had been in successful operation for research purposes and for industry plant tests. 8.2.1.1.2 Critical Parameters Determining the Efficacy of PEF Processing Table 8.1 illustrates the primary and secondary factors that have significant impacts on PEF inactivation effects against microorganisms. Primary parameters that determine 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 electrodes divided by their gap distance, L. E = V × L–1

(8.1)

The nominal average electric field strength is generally simplified as ‘electric field strength’ and widely accepted by different groups of researchers. However, readers should notice that, in many cases, electric field strength is not uniform throughout the whole treatment zone due to the nature of many different chamber designs. For instance, in the case of disc type static chamber design electric field strength could 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

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TABLE 8.1 Some Important Factors Determining the Efficacy of a PEF Processing Primary Parameters

Secondary Factors

Electric field strength, E

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.

different phases within sample matrix. On the other hand, in the case of co-field flow 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 happens at the central axis. PEF treatment time is defined as the time when food is actually exposed to the 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 used (n), pulse width (τ), pulse replication rate (prr) and the volumetric flow rate (fr) of food matrix. The total PEF treatment time can be calculated using Equation 8.2: t = 0.25π × d 2 × L × fr –1 × prr × n × τ

(8.2)

When monopolar pulses are used, pulse replication rate is the same as pulse frequency, 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 prr = 2 × fp

(8.3)

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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 pulse duration time are important parameters that may have significant influences 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. Other than primary parameters, there are also a group of factors contributing significantly 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 compared with the primary factors, E and t, but may also show significant impact on 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, electrode erosion and geometry characteristics are commonly in the list of important secondary factors and significantly affect the overall effect of PEF. Failure to closely 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 investigated by many researchers since PEF was first suggested for cell destruction. 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 breakdown of cell membrane. The membrane itself is a dielectric, flexible and 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 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. 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 charges are induced on membrane surfaces along the direction of the 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 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

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Food Processing Operations Modeling: Design and Analysis (a) 10000 8000 6000 4000 2000 0 Voltage (V) –2000 –4000 –6000 –8000 –10000 –12000 0

60 40 20 0 Current (A) –20 Voltage

–40 5

10 Time (µs)

15

20

Current

–60

(b) Tek Run: 50MS/s Sample 3.54 v –5.44 v Ch2 period 6.714 µs Low signal amplitude

2*

1

Ch1

1v

Ch2

1v

M

1 µs Ext

–440 mV 18 Jun 1999 09:48:59

FIGURE 8.5 Two types of typical electric pulses: (a) square and (b) exponential decaying.

and negative charges. Since the membrane is flexible and deformable to external 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 and the cell survives the electric stress well. Further increasing electric field strength to such a point that the transmembrane potential reaches 1 V, at which cellular membrane 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 external electric field. At this point, membrane conductivity and permeability can be dramatically increased due to reduction in membrane thickness and deformation in membrane structure. Along with further increasing external electric field strength 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

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Stages:

a

b

+

–

+

–

Cytoplasm +

– Medium

+

–

+

–

+ + + + + + + + +

225

c – – – – – – – –

+ + + + + + + + +

d – – – – – – – – –

+ + + + + + + + +

– – – – – – – – –

Big pores allowing cytoplasm escape Cytoplasm fluxes into medium

FIGURE 8.6 Illustration of cell membrane reversible and irreversible dielectric breakdown. (a) Cell membrane with a potential Vm when there is no external electric field; (b) membrane compression when cell is exposed to an external electric field; (c) small pore formation at critical transmembrane potential, reversible breakdown; (d) large pores are formed, irreversible dielectric breakdown. (Redrawn from U Zimmerman. 1986. Electric breakdown, electropermeabilization, and electrofusion. Review of Physiology, Biochemistry, and Pharmacology 105: 175–256.)

and bigger. As a result, cytoplasma of the cell leak out via the induced pores, and thus resulting in death of the cells. Sale and Hamilton [5] first reported this phenomenon with cell suspension at 10 kV × cm−1 electric field strength. The authors reported that 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 correlation 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), suggested that cell inactivation and membrane damage have a very high positive association. 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 electric field strength, will be induced. The critical electric field strength for cell inactivation is approximately 1 V or higher, depending heavily on the different species of microorganisms and their growth conditions. This is because different microorganisms have different membrane composition and structure. Some cell membranes need higher transmembrane voltage to form pores since they may be less flexible in structure and not sensitive to electric field treatment. But for most cells, 1 V of transmembrane 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 induced under certain electric field. For spherical cells surrounded by non-conducting membrane, the induced potential is given by the equation: Vm = f × r × Ec

(8.4)

where Vm refers to the transmembrane potential induced by the external electric field 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

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TABLE 8.2 Staphylococcus 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

Source: AJH Sale, and WA Hamilton. 1967. Effect of high electric fields on microorganisms, I. Killing of bacteria and yeast. Biochemistry and Biophysics Acta 148: 781–88.

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 − 0.33d )

(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: Vm = [L × (1 − 0.33d )−1 ]× r × Ec

(8.6)

where Ec is external electric field strength applied, and r refers to the diameter of the cells. So we know the critical voltage for cell inactivation is the induced transmembrane 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 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 destruction 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 concentration 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.

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Pore initiation

227

Membrane rupture

Electric field

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 fields hurdles. Food Research International 29(2), 117–21, 1996. With permission.)

Another well-known theory is the electroporation mechanism which was proposed 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 electric field became permeable to small molecules such as water. The elevated permissibility of cellular membranes to small molecules caused the influx of water 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 breakdown 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 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 formation of the pores is most likely to happen at the two polar areas of the cell along the electric field direction—if the pore formation process indeed happens.

8.2.3

MODELING OF PEF INACTIVATION OF MICROORGANISMS

Sensoy and co-workers [36] reported their kinetic studies about the inactivation of Salmonella dublin by PEF. The authors used an electric field strength of 15–40 kV × cm−1 and treatment time of 12–127 microseconds in a co-field flow high intensity pulsed electric field treatment system. The medium temperature of 10–50°C were 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

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the treated cells based on the electric field strength (model 1 and 2) and total treatment time (model 3 and 4). Model 1. s = e −(E−Ec )×K c −1

(8.7)

where s is survival fraction, E is applied electric field strength (kV × cm–1), Ec is critical electric field strength (kV × cm–1), kc is constant factor (kV × cm–1). Model 2 is Peleg’s [37] model in the form of: Model 2. s = 1 / {1 + e [(E−Ec )/ ke ]}

(8.8)

where t

ke (t ) = ke0 e k1

(8.9) t

Ec (t ) = Ec0 e −k2

(8.10)

where s is the survival fraction, E is the applied electric field strength (kV × cm–1), Ec is critical electric field strength (kV × cm–1), kc, ke0, Ec0, k1(μs–1), k2 (μs–1) are constant factors. Model 3. s = e(tc −t )/ kt

(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 (Ec −E )/ k ′

Model 4. s = {t × tc−1}

(8.12)

where s is the survival fraction, t is the total treatment time, tc is critical treatment time, E is the applied electric field strength (kV × cm–1), Ec is the critical electric field strength (kV × cm–1) for the targeted microorganisms, k′ is a constant factor (kV × cm–1). All these models were developed to explain the relationship between cell survival fraction or ratio and electric field strength or the treatment time, which are the 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 using a lab scale continuous system. Although significant differences exist between the mechanisms for microbial inactivation and those for enzyme inactivation, the

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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 8.2.4.1

TRENDS IN PEF RESEARCH Hardware Development has a Long Journey Ahead

It is always challenging, exciting and a must-do to continue further hardware development to improve the production capacity and system reliability for PEF facilities. 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 and new development in electronic and semiconductor technology, more efficient capacitors and more capable high speed switching devices will be available to allow engineers to design PEF systems for higher electric field strength with high tolerance in electric current flowing through the treatment chambers. High tolerance of 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 current 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 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, more uniform treatment and more reliable system behavior. One of the major limitations preventing PEF technology from industrial practice is the current small inner diameter in treatment chambers, limited by the overall system electric current tolerance, particularly with products with large particles or high viscosity. Efforts will also be invested to redesign current chamber geometry, although it will be extremely difficult, for uniform electric field distribution. Electrode erosion will be investigated 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 challenging due to the high intensity of applied electric fields. The current method measures sample temperature before and after PEF treatment zone with a distance for safety purpose and to avoid possible interruption from operating PEF. Significant improvement in temperature monitoring is also needed for better understanding of PEF inactivation 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 Pasteurization of High Acid or Acidified Food PEF first found its commercial application in juice processing. In the near future, PEF will continue to focus on applications in high acid food processing. Fruit juices,

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vegetable juices with acidification and high acid beverages including low alcoholic 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 improve food safety with enhanced quality attributes. PEF may also find successful applications in medical and other biological industries, such as enzyme manufacturing and purification. 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 process for shelf stable food products, even for high acidic foods. However, when combined 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 microorganisms and no significant quality changes [23]. Using heat treatment alone 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 advantages of significantly improved flavors. 8.2.4.2.3 Research and Applications in PEF Assisted Food Processing Another big area in the near future would be PEF assisted food processing. Preliminary studies show that low intensity PEF treatment can significantly increase the production 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 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 shortening processing time, and to increase yield rate. The selective inactivation of PEF processing 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 pasteurization to improve the quality attributes, including fresher flavor profile and more heat-labile functional compounds, of processed food products. The fresher flavor of the processed food products attracts more customers with significant numbers 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 IgG, the more health benefits consumers can get from consumption of such products. More importantly, PEF is an effective and efficient technology for food pasteurization, particularly pumpable high acid foods, via a mechanism named cell membrane dielectric breakdown. The proposed mathematic models, Hulsheger, Fermi and other

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modified ones, can fit well with the experimental results and can be used for prediction of microbial pasteurization, even enzyme inactivation by PEF. However, PEF is a complicated system consisting of many subsystems including power supplier, pulse generator, chamber systems, fluid handling system, controlling and monitoring system, and aseptic packaging system. To make things more complicated, while electric field strength and total PEF treatment time are the two critical parameters, there are lots of secondary parameters, such as pH, ion strength, processing temperature, growth phase and nutrient conditions of microbes, etc. contributing significantly 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 t fr prr f Vm Ec kc ke0, Ec0 k1 k2 tc kt k′

Nominal or average electric field strength, kV.cm−1 Total PEF treatment time, μs Volume flow rate, cm3.s−1 Pulse replication rate, s−1 Form factor, m2 Transmembrane potential, volts critical electric field strength, kV.cm−1 Constant factor, kV.cm−1 Constant factors, kV.cm−1 Constant factor, μs−1 Constant factor, μs−1 Critical treatment time, μs Constant factor, μs Constant factor, kV × cm−1

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6. JE Dunne, and JS Pearlman. 1987. Methods and apparatus of extending the shelf life of fluid food products. U.S. Pat. 4,695,472. 7. AJ Castro, GV Barbosa-Canovas, and BG Swanson. 1993. Microbial inactivation of foods by pulsed electric fields. Journal of Food Processing and Preservation 17: 47–73. 8. S Jayaram, GSP Castle, and A Pargaritis. 1992. Kinetics of sterilization of Lactobacillus brevis cells by the application of high voltage pulses. Biotechnology and Bioengineering 40: 1412–20. 9. UR Pothakamury, A Monsalve-Gonzalez, GV Barbosa-Canovas, and BG Swanson. 1995. Inactivation of Escherichia coli and Staphylococcus aureus by pulsed electric field technology. Food Research International 28(2): 167–71. 10. UR Pothakamury, H Vega, QH Zhang, GV Barbosa-Canovas, and BG Swanson. 1996. Effect of growth stages and processing temperature on the inactivation of E. coli by pulsed electric field. Journal of Food Protection 59(11): 1167–71. 11. Q Zhang, BL Qin, GV Barbosa-Canovas, and BG Swanson. 1994. Inactivation of E. coli for food pasteurization by high-intensity short-duration pulsed electric fields. Journal of Food Processing and Preservation 19: 103–18. 12. Q Zhang, GV Barbosa-Canovas, and BG Swanson. 1994. Engineering aspects of pulsed electric field pasteurization. Journal of Food Engineering 25: 268–81. 13. Q Zhang, A Monsalve-Gonzalez, B Qin, GV Barbosa-Canovas, and BG Swanson. Inactivation of Sacchromyces cerevisiae by square wave and exponential-decay pulsed electric field. Journal of Food Processing Engineering 17: 469–78. 14. Q Zhang, FJ Chang, GV Barbosa-Canovas, and BG Swanson. 1994. Inactivation of microorganisms in semisolid foods using high voltage pulsed electric fields. Food Science and Technology (LWT) 27: 538–43. 15. Q Zhang, A Monsalve-Gonzalez, GB Barbosa-Canovas, and BG Swanson. 1994. Inactivation of E. coli and S. cerevisiae by pulsed electric fields under controlled temperature conditions. Transaction ASAE 37: 581–87. 16. GA Evrendilek, QH Zhang, and ER Richter. 1999. Inactivation of Escherichia coli O157: H7 and Escherichia coli 8739 in apple juice by pulsed electric fields. Journal of Food Protection 62(7): 793–96. 17. B Qin, UR Pothakamury, H Vega, O Martin, GV Barbosa-Cannovas, and BG Swanson. 1995. Food pasteurization using high-intensity pulsed electric fields. Food Technology 49(12): 55–60. 18. LD Reina, ZT Jin, QH Zhang, and AE Yousef. 1998. Inactivation of Listeria monocytogenes in milk by pulsed electric fields. Journal of Food Protection 61(9): 1203–6. 19. SQ Li, and QH Zhang. 2004. Inactivation of E. coli 8739 by pulsed electric fields and extension of microbial shelf life stability of enriched soymilk. Journal of Food Science 69(7): M169–74. 20. GA Evrendilek, S Li, WR Dantzer, and QH Zhang. Pulsed electric field processing of a highly carbonated beverage: Microbial, sensory and quality analyses. Journal of Food Science 69(8): M228–232. 21. X Qiu, S Li, and QH Zhang. 1999. Design and construction of a bench scale automatic PEF units of OSU-4 series. IFT Annual Meeting. Chicago, IL. July. 22. S Min, ZT Jin, and QH Zhang. 2003. Commercial scale pulsed electric field processing of tomato juice. Journal of Agricultural and Food Chemistry 51(11): 3338–44. 23. SQ Li, QH Zhang, TZ Jin, EJ Turek, and MH Lau. 2005. Elimination of Lactobacillus plantarum in model ranch salad dressing by pilot scale pulsed electric fields combined with mild heat and achievement of shelf stability at room temperature. Innovative Food Science and Emerging Technologies 6(2): 125–33. 24. HW Yeom, CB Streaker, QH Zhang, and DB Min. 2000. Effects of pulsed electric fields on the quality of orange juice and comparison with heat pasteurization. Journal of Agricultural and Food Chemistry 48(10): 4597–605.

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Models for 9 Fouling 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 industry is thermal treatment. Heat exchangers offer an effective and efficient means of heat utilization through heat recycling and better heat transfer. During heat treatment the food products undergo structural and chemical changes. Owing to changes occurring in the food product some of the constituents like proteins and minerals precipitate resulting in film-like deposits which adhere to the food processing equipment surface. These deposits are generally referred to as foulants and this phenomenon 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 transfer coefficient. 2. Increase in pressure drop across the heat exchanger unit thereby lowering the overall system performance. 235 © 2009 by Taylor & Francis Group, LLC

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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 operations 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 consumption, 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]. Specifically, the costs incurred due to fouling includes increased cost due to oversized 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 fouling 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 process is completed, disposal of the effluents in an environmentally friendly manner further adds to the cost of production. Hence, prolonging the operation of the heat exchanger unit by reducing the rate of fouling could be beneficial in reducing 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 denaturation 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 equipment surface that have an influence on the fouling process. They observed the 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. 2. Surface charge; the cleaning process and the industrial finishing conditions during manufacture dictate the nature of the surface charge.

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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. Other factors that have an influence on the fouling process are the product composition, 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 seasonal variations are some of the factors that have an influence on the extent of 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-lactoglobulin 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 9.1 Characteristics of Type A and Type B Fouling Deposit Formed During Heating of Milk Deposit 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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owing to the importance of thermal processing in the dairy industry. There are various factors that contribute to the fouling process during processing; similar to these mentioned previously for milk. These include, the product composition, temperature, pH, surface geometry of the processing equipment, presence of air bubbles, and the mixing intensity which is dependent on both the fluid flow rate and the 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 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 useful in building reliable models that can shed light on the fouling occurring in different food equipment surfaces. With the advent of improvements in the field of computational fluid dynamics (CFD) the detailed geometry of the heat exchanger surface can be taken into account thus developing models that could closely analyze the interactions between processing surface geometry and fluid flow. An optimum model that can closely predict the fouling behaviour under various operating conditions like temperature, residence time, and flow rate will help in optimizing the 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 In order to model the fouling process it is imperative to first understand the underlying 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 mass transfer takes place between the bulk fluid containing the proteins and the thermal boundary layer. The aggregated proteins formed then adhere to the equipment 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 is subjected to hydrodynamic forces from the moving fluids resulting in possible removal of the deposits [3]. There has been extensive research conducted on the fouling in milk processing 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-Lactoglobulin) 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-lactalbumin) to form aggregates. It is here that the fouling mechanism becomes debatable by researchers. Some believe that the denatured proteins form the first fouling layer and others believe that it is the aggregated proteins that are involved in forming the fouling layer. Hence, researchers modeling the fouling mechanism follow

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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 ambiguity 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 adsorption isotherms will not be an ideal fit for biopolymers [20].

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 thermodynamic flow patterns occurring within the heat exchanger. Food engineers 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 chemical 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

The thermal processing of non-Newtonian food fluids in continuous plate heat exchangers was experimentally and theoretically discussed by Rene et al. [21]. An experimental model was developed by them that could predict the temperature profiles inside each channel of the PHE. The developed model defined the calorific factor which was used to estimate the calorific temperatures of the cold and hot fluids. 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 proposed models is expected to lead to significant enhancements in the food industry. The transient response of a multi-pass PHE was studied and a model based on the axial heat dispersion in the fluid was developed [22]. This model took into consideration the deviation from ideal plug flow. The fluid flows in PHEs are different from shell-and-tube heat exchangers because of the phase lag at the entry and the successive channels. The phase lag increases with increase in number of flow channels 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 studies have been extended to include the phase lag effect in multi pass PHE units.

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The developed algorithms also took into consideration the mixing of fluids between 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]. i. The flow rate and temperature profiles are uniform across the channel and plate width. ii. Heat transfer only takes place between channels and not between channels and ports or through the seals and gaskets. iii. The thermal and physical properties of the fluids are not dependent on pressure and temperature. 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. v. The loss of heat to the environment is negligible. Negligible radiation heat losses are encountered. 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 of solid plate Figure 9.1, the energy balance over these control volumes taking into account the above mentioned assumptions gives rise to the following fluid flow equations related to the channel and plate [26]: ⎛ ∂T ∂T ⎞ giρciCpci ⎜⎜⎜ ci + vi ci ⎟⎟⎟ = Ui (Tp(i−1) + Tpi − 2Tci ) ⎝ ∂t ∂x ⎠ Channel i–1

Channel i

(9.1)

Channel i+1

vi

x

Δx Tci

Plate i–1

FIGURE 9.1

Tpi

Plate i

Fluid control volume

One dimensional view of control volume of fluid inside the PHE channel.

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⎡ ∂T ⎤ δ piρpiCppi ⎢ pi ⎥ = Ui (Tci + Tc (i+1) − 2Tpi ) ⎢⎣ ∂t ⎥⎦

(9.2)

where t = time; x = axial position; Tci = temperature of fluid in ith channel formed between plate i and i + 1; Tpi = temperature of ith plate; Cpci = specific heat at constant pressure of fluid in ith channel; Cppi = specific heat at constant pressure of fluid in ith plate; ρci = fluid density in ith channel; ρpi = fluid density in ith plate; gi = gap between plates i and i + 1; δpi = thickness of ith plate; Ui = overall heat transfer coefficient in the ith channel; v = average fluid velocity which can be positive or negative depending upon the direction of flow. 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. The overall heat transfer coefficient (U) can be calculated using the dimensionless 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 0.662 − 3.2)Pr 0.4

(9.3)

where the Re and Pr numbers can be derived from the following relationships; Re =

ρvDe Cpμ , Pr = , De = 2 gi μ k

(9.4)

where k = thermal conductivity of the fluid; h = convective heat transfer coefficient; μ = viscosity of the fluid; and De = equivalent diameter. From the Nu number, the convective heat transfer coefficient (h) can be calculated using the relation, Nu =

hDe k

(9.5)

The overall heat transfer coefficient (U) can now be calculated by; δ 1 1 1 = + + p U hhot hcold kp

(9.6)

where hhot = convective heat transfer coefficient of the hot fluid stream; hcold = convective heat transfer coefficient of the cold fluid stream; and kp = thermal conductivity 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

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and is related to the rate of deposition of the aggregated proteins by the following expression [26]: ∂Bi = βkmCA∗ ∂t

(9.7)

where Bi = Biot number; β = constant; km = mass transfer coefficient; and CA∗ = concentration of aggregated protein in the thermal boundary layer. It can be shown that the Biot number and the overall heat transfer coefficient under fouling condition, Uf, are related by the Equation Uf =

U0 1 + Bi

(9.8)

where, U0 is the heat transfer coefficient under no fouling conditions. Biot numbers are considered to be important in the design of heat exchangers by engineers. A fouling 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. The first and last channel in a PHE transfers heat to one adjacent fluid channel and to the terminal block. Thus, the equations for these two channels for a PHE having N plates can be written as follows [30]: ⎛ ∂T ∂T ⎞ g1ρ1Cpc1 ⎜⎜⎜ c1 + v c1 ⎟⎟⎟ = U1 (Tp1 − Tc1 ) + U∞ (T∞− Tc1 ) ⎝ ∂t ∂x ⎠

(9.9)

⎛ ∂T ∂T ⎞ gN ρN CpcN ⎜⎜⎜ cN + v cN ⎟⎟⎟ = U N (Tp( N −1) − TcN ) + U∞ (T∞− TcN ) ⎝ ∂t ∂x ⎠

(9.10)

1 1 1 δ = + + b U∞ hhot hcold k b

(9.11)

where k b = 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 characterize the geometrical changes in the corrugated plate profiles is the pressure 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 the pressure drop varying across the PHE because of over simplified flow streams. A quadratic temperature profile model has been developed [31] that could provide an estimate of the milk temperature along the plate height. The estimated temperature 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

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studying the hydrodynamic performance of heat exchangers. Models have also been developed to estimate the temperature in each channel in the PHE [28]. These empirical 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 experimental 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 two-dimensional models by considering the velocity vectors of the flow in x and y directions. In order to compute the two-dimensional models incorporating the velocity and pressure distribution of the flow, the Navier–Stokes (N–S) equations have to be solved. The assumptions made in the case of two-dimensional modeling are that the plate surface is flat and smooth. The two-dimensional N–S flow equation given by Kays [32] include the continuity and momentum equations as given below: Continuity equation:

∂u ∂v + =0 ∂x ∂y

(9.12)

x-momentum:

⎡ ∂ 2u ∂ 2u ⎤ ∂u ∂u ∂u 1 ∂P +u + v =− +ϑ⎢ 2 + 2 ⎥ ⎢⎣ ∂x ∂t ∂x ∂y ρ ∂x ∂y ⎥⎦

(9.13)

y-momentum:

⎡ ∂2 v ∂2 v ⎤ ∂v ∂v ∂v 1 ∂P +u + v =− +ϑ⎢ 2 + 2 ⎥ ⎢⎣ ∂x ∂t ∂x ∂y ρ ∂y ∂y ⎥⎦

(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. The transient energy equation for a two dimensional incompressible flow as noted by Ozisik [33] which is essentially an extension of Equation 9.1 and Equation 9.2 are given as follows: ⎛ ∂T ∂T ∂T ⎞ giρciCpci ⎜⎜ ci + ui ci + vi ci ⎟⎟⎟ = Ui (Tp(i−1) + Tpi − 2Tci ) ⎜⎝ ∂t ∂x ∂y ⎟⎠

(9.15)

⎡ ∂T ⎤ δ piρpiCppi ⎢ pi ⎥ = Ui (Tci + Tc (i+1) − 2Tpi ) ⎢⎣ ∂t ⎥⎦

(9.16)

where subscripts i, i + 1, i − 1, c and p refer to the plate i, plate i + 1, plate i − 1, fluid and plate, respectively, and have been defined previously. The solving of the flow equations for the two-dimensional model is much more involved than solving one-dimensional model equations. To simplify the process 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 the momentum equations. A vorticity-stream function approach applicable to

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two-dimensional modeling is usually used [33]. This approach defines the vorticity vector and streamline functions for a two-dimensional Cartesian coordinate system and use these terms to transform the two-dimensional flow equations to a simpler form. The vorticity vector is given by: ω=

∂v ∂u − ∂x ∂y

(9.17)

and the streamline function (ψ) for the velocity vectors, u and v is given by: ∂ψ = u, ∂y

∂ψ = −v ∂x

(9.18)

From Equation 9.17 and Equation 9.18, the vorticity vector can be transformed into the following relationship ∂2 ψ ∂2 ψ + 2 = −ω ∂x 2 ∂y

(9.19)

From the streamline function definition it is obvious that the relationship is identical to the continuity equation (Equation 9.12). Transformation of the dependent variables 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 transformation leads to the following relationship: ⎡ ∂ 2ω ∂ 2ω ⎤ ∂ω ∂ω ∂ω +u +v =ϑ⎢ 2 + 2 ⎥ ⎢⎣ ∂x ∂t ∂x ∂y ∂y ⎥⎦

(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: ⎡ ∂u ∂v ∂u ∂v ⎤ ∂2 P ∂2 P ⎥ + 2 = 2ρ ⎢ − 2 ⎢⎣ ∂x ∂y ∂y ∂x ⎥⎦ ∂x ∂y

(9.21)

Reducing this equation by including the streamline function (ψ) the differential equation for the pressure term can be given by: ⎡⎛ ∂2 ψ ⎞⎛ ∂2 ψ ⎞ ⎛ ∂2 ψ ⎞2 ⎤ ∂2 P ∂2 P ⎟⎟ ⎥ + = 2ρ ⎢⎢⎜⎜ 2 ⎟⎟⎟⎜⎜ 2 ⎟⎟⎟ − ⎜⎜ ⎟⎥ ∂x 2 ∂y 2 ⎢⎣⎜⎝ ∂x ⎟⎠⎝⎜ ∂y ⎟⎠ ⎜⎝ ∂x ∂y ⎟⎠ ⎥⎦

(9.22)

Using finite-difference approximation and Gauss-Seidel iterative solver, the pressure distribution at various locations on a grid for the entire flow field can be determined 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

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plate large prediction errors (up to 43.2%) were observed. With the use of the twodimensional 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 the two-dimensional model was based on the assumption of a flat plate surface geometry the error observed during prediction could be due to the fact that the actual plate 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 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 estimating the average energy balance of mass flow. By being able to predict the temperatures 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 twodimensional 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 to friction, changes in velocity, changes in direction of flow and changes in height 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: ΔP =

4 fm 2 L 2ρDh

(9.23)

where ΔP = pressure drop; f = friction factor; m = mass flow rate per cross sectional 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 Re− y

(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 the turbulent, three-dimensional fluid and heat transfer flow between two parallel Chevron plates was first reported in the late 1990s [36]. An algorithm proposed by 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 © 2009 by Taylor & Francis Group, LLC

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potential of using finite element analysis to have a better understanding of the fouling phenomenon occurring between a flow channel. However, the results from using 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. A detailed two-dimensional and three-dimensional study on the flow pattern of milk between two corrugated plates was studied by Grijspeerdt et al. [38] using CFD (Computational Fluid Dynamics). The three-dimensional calculations showed the virtual flow velocity fields which were not possible using two-dimensional calculations. This clearly showed the limitations of two-dimensional calculations for designing new plate configurations that could reduce fouling. The three-dimensional calculations help to identify regions of turbulent backflows that could cause elevated temperature regions. These regions of elevated temperatures are potential fouling locations on the plate surface. Eliminating such occurrence is essential to minimizing fouling and that could be done by better plate configuration design. Thus CFD models have immense potential in optimizing the design of plates for heat exchangers. So far the CFD based studies on fouling behavior have been concentrated towards the thermodynamic and hydrodynamic aspects of fluid flow. Incorporating the physical 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 flow equations describing the three-dimensional model can be derived from the continuity and momentum equations by extending the two-dimensional model which was described in detail earlier. The continuity equation in this case is defined by ∂u ∂v ∂w + + =0 ∂x ∂y ∂z

(9.25)

The equations for the momentum in the x, y and z directions are now described by the following Equations: x-momentum:

⎡ ∂ 2u ∂ 2u ∂ 2u ⎤ ∂u ∂u ∂u ∂u 1 ∂P +u + v + w =− + ϑ ⎢ 2 + 2 + 2 ⎥ (9.26) ⎢⎣ ∂x ρ ∂x ∂t ∂x ∂y ∂z ∂y ∂z ⎥⎦

y-momentum:

⎡ ∂2 v ∂2 v ∂2 v ⎤ ∂v ∂v ∂v ∂v 1 ∂P +u + v + w =− +ϑ⎢ 2 + 2 + 2 ⎥ ⎢⎣ ∂x ρ ∂y ∂t ∂x ∂y ∂z ∂y ∂z ⎥⎦

z-momentum:

(9.27)

⎡ ∂2 w ∂2 w ∂2 w ⎤ ∂w ∂w ∂w ∂w 1 ∂P +u +v +w =− + ϑ ⎢ 2 + 2 + 2 ⎥ (9.28) ⎢⎣ ∂x ρ ∂z ∂t ∂x ∂y ∂z ∂y ∂z ⎥⎦

where w is the velocity component of the flow in the z direction.

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For a three-dimensional incompressible flow the transient energy equation can be written as [39]: ∂T ∂T ∂T ∂T k +u +v +w = ∂t ∂x ∂y ∂z ρCp

⎡ ∂ 2T ∂ 2T ∂ 2 T ⎤ ⎢ 2+ 2 + 2⎥ ⎢⎣ ∂x ∂y ∂z ⎥⎦

(9.29)

where T is the temperature of the fluid. The above Equation can be simplified by substituting 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 ⎡ ∂ 2T ∂ 2T ∂ 2T ⎤ ∂T ∂T ∂T ∂T +u +v +w =α⎢ 2 + 2 + 2 ⎥ ⎢⎣ ∂x ∂t ∂x ∂y ∂z ∂y ∂z ⎥⎦

(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 configuration of the PHE which was not possible with the two-dimensional model. 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 new surface configurations that can help in minimizing fouling. This is not possible 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 essentially 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 hydrodynamic and thermodynamic factors of fouling with the physical and chemical contributing factors of fouling it is imperative to understand the protein denaturation process. The dynamic fouling models study the denaturation of β-LG and its relationship to the fouling observed. Delplace et al. [40] studied the complex flow arrangements in a PHE consisting 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 numerical determination of temperature profile for each channel. The simulated quantity of denatured β-LG was then compared with the actual amount obtained from experiments through measurements using immunodiffusion methods. From the developed © 2009 by Taylor & Francis Group, LLC

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model they could predict the native β-LG at the outlet of the PHE with an experimental error of less than 10%. The model for predicting the β-LG was given by C (t ) =

C0 1 + kC0 t

(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 second 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 state conditions, predicted temperature profiles and the amount of heat denaturation 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 of pressure and temperature do not change significantly. These later change when the fouling period begins resulting in increased pressure drop and decreased heat transfer coefficients. Most models developed deal with the fouling period. The fouling 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. dm = θD − θR dt

(9.32)

where m is the mass deposited, θD and θR are the mass flow rates per unit surface 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 equation for the fouling deposition process based on the above Equation 9.32: ⎡ E 1⎤

⎢− ⎥ dBi ⎢ ⎥ = kd e ⎣ R Tfi ⎦ − kr Bi dt

(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 the process fluid. The general equation described above (Equation 9.32) can be modified appropriately to include various factors relevant to fouling such as chemical reaction, mass transfer, fluid shear, and bond resistance. The models developed using this relation result in linear (constant rate), falling rate or asymptotic fouling growth © 2009 by Taylor & Francis Group, LLC

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Linear fouling Falling rate fouling Fouling resistance

Induction period

Fouling period

Asymptotic fouling

Time

FIGURE 9.2

Fouling curves.

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 decrease in the heat transfer coefficient. This leads to rapid increase in pressure drop and blockage of the passage of fluid flow due to the foulants [43]. In the study of calcium sulfate scale deposition on heat transfer surfaces a falling rate of fouling growth model was observed [44,45]. Owing to the non-uniform heat flux at various 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 and the removal term was estimated based upon the hydrodynamics of flow and 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 foreign body, just as in the case of heat exchanger surface. The heterogeneous nucleation can be calculated using the term [44]: ⎛ 16 πNV 2σ 3 ⎞ H N = B’exp⎜⎜⎜− 3 3 m 2 ⎟⎟⎟ ⎜⎝ 3 R Ti (ln S ) ⎟⎠

(9.34)

where B′ is the pre-exponential factor, N is the Avagadro’s number, Vm is the molar volume, σ is the specific interfacial energy, R is the universal gas constant, Ti is the solid–liquid interface temperature, and S is the supersaturation. A low energy surface (having high contact angle) will require higher supersaturation for nucleation to occur than for a surface with high energy (having low contact angle). Surface modification techniques like those that use coatings to alter the surface roughness can help © 2009 by Taylor & Francis Group, LLC

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in delaying the nucleation occurrence by altering the surface energy [47]. The supersaturation 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 salt followed by curve-fitting techniques to obtain a standard equation in relation to 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]: ⎡ ⎢1⎛ β ⎞ dm = β ⎢ ⎜⎜ ⎟⎟⎟ + (cb − cs ) − ⎢ 2 ⎜⎝ kr ⎠ dt ⎢⎣

⎪⎫⎪ ⎤⎥ ⎪⎧⎪ 1 ⎛ β ⎞2 ⎛ β ⎞ ⎨ ⎜⎜⎜ ⎟⎟⎟ + ⎜⎜⎜ ⎟⎟⎟ (cb − cs )⎬ ⎥ ⎪⎪ ⎥ ⎪⎪ 4 ⎝ kr ⎠ ⎝ kr ⎠ ⎭ ⎥⎦ ⎩

(9.35)

The term kr is the rate constant, and the mass transfer coefficient β can be obtained from the Sherwood number, ⎛ 6d ⎞ Sh = 0.023 Re 0.8 Sc 0.33 ⎜⎜1 + h ⎟⎟⎟ ⎝ x ⎠

(9.36)

where Sc is the Schmidt number and dh is the hydraulic diameter at position x. The Sherwood number is given by: Sh =

βd h α

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 processes. In contrast, the three and four phase model deals with surface reaction process. 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.

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The rate of disappearance and formation of these protein phases are given by the relation [18,19]: −

dCN = kUCNa − k NCUb dt

(9.38)

dCU = kUCNa − k NCUb − k ACUc dt

(9.39)

dCA = k ACUc dt

(9.40)

where C is the protein concentration. The suffix a, b, c, pertain to the orders of the 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 to be a = 1, b = 1, and c = 2. Jun and Puri [39] have described a simplified model with 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 process 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: −

dCN = kUCNa dt

(9.42)

dCU = kUCNa − k ACUc dt

(9.43)

dCA = k ACUc dt

(9.44)

The main mechanism in the fouling process of skim milk is a reaction-controlled adsorption of the unfolded β-LG protein [39]: FR = kDCU1.2

(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: E − n RT

kn = An e

(9.46)

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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 denaturation 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 β-LG played a less significant role in the fouling process, while the unfolded β-LG 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)

+M U ⎯→ ⎯ D → D∗

(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 voluminous in occurrence than protein fouling could have a lesser impact on the pressure drop and thermal resistance encountered during the fouling process. Also, the first 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 (temperatures 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 surface, the solvent and any other solute present in the fluid system [54,55]. Though the actual mechanism of calcium phosphate fouling is complex to understand and has not been fully understood, Visser and Jeurnink [9] have postulated a possible pathway for the fouling mechanism. According to them, as a pre-cursor to the fouling phenomenon, the calcium and phosphate ions first form a colloidal calcium phosphate 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 final product formed, namely HAP is the thermodynamically stable form of calcium 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.

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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]. According to their model the pathway for β-LG denaturation is given by: −

dCN = k UC N dt

(9.50)

dCU = kUCN − k ACU2 − kDCU dt

(9.51)

dCA = k ACU2 dt

(9.52)

dCD = kDCU dt

(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 (Equation 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 concentration of aggregated protein in the thermal boundary layer. Figure 9.3 and Figure 9.4 schematically represent the flow and reaction model dynamics proposed. Basically 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: i. Proteins react in both the bulk and thermal boundary layer in fluid milk. Unfolded proteins are formed by a first order reaction from native proteins and 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 resistance 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. It would be beneficial to have a comprehensive model that can encompass the reactions 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

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Channel i

y x y

Height Δx

Plate i–1

Width Plate i

FIGURE 9.3

Two-dimensional view of control volume of fluid inside the PHE channel.

Reaction N

U

A Bulk fluid

Mass transfer

N*

U*

A* 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.)

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 temperature 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 it dictates the amount of resources and time spent, not to mention the influence

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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 product processed and the processing conditions. Altering the composition of the liquid food material being processed has a significant influence in the fouling profiles observed. The aim of an appropriate cleaning model developed is to optimize the time involved with reference to the flow velocity, chemical concentration of 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 information 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 significant after 3 h. The same study also focussed on the adhesive strength versus the 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 stages of cleaning. This could be the time to decrease the flow rates and cleaning chemical concentration. Similar studies on the removal of milk proteins revealed that the deposits could not be removed completely with water alone. Chemical 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 proteins decreased with time. Also, the order of use of the acid and base chemicals during cleaning had an influence on the milk foulants removal [66]. These studies 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 appropriate 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].

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The minimum adhesive energy between the surface and the deposits can be expressed in terms of the surface energy by the relationship [69]: ⎡1⎤ LW LW ⎤ γ surface = ⎢ ⎥ ⎡⎢ γ LW foulant + γ fluid ⎥ ⎦ ⎢⎣ 2 ⎥⎦ ⎣

(9.54)

LW LW where γ surface , γ LW foulant , γ fluid are the Lifshitz van der Waals (LW) surface free energies of the surface, foulants, and the fluid, respectively. This relationship indicates that if 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:

Wremoval ∝

∇x deposit Sd

(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 adhesive = ω s + ψ v x deposit

(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 surface energies a different method of breakdown of the deposits could be possible. A model incorporating the concentration of the chemicals for cleaning was proposed 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 © 2009 by Taylor & Francis Group, LLC

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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 directional change (CDC) values of more than 92% has been developed using neural networks 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 cleaning 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 it measured the heat flux directly from the plate heat exchanger unit and the neural 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: Cf =

1 (Cp + Cc + Cu ) Tf + Tc

(9.57)

where Cf = total costs due to fouling; Cp = costs due to changes in engine performance; 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 relationship after taking into consideration the power requirements, and the engine performance: Tf = Tc2 +

2(Cu + Cc ) − Tc a.efu + b.ee

(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 increase in energy consumption due to fouling is not significant. With high emphasis 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. © 2009 by Taylor & Francis Group, LLC

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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 fouling will be beneficial to food processors, which is likely to result in substantial savings. There has been a dearth in comprehensive models that can explain the fouling 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. Understanding the fouling phenomena will help in optimizing the process conditions, and timely scheduling of cleaning operations that will cut down costs, and increase performance efficiency. There is a long way to go to address this issue, and with the 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 in its infancy, it would be prudent to borrow the findings from sectors such as automotive, aerospace, chemical, petroleum, and marine industries, where the issue of fouling has been addressed for a long time.

REFERENCES 1. HM Muller-Steinhagen. 2000. Handbook of Heat Exchanger Fouling: Mitigation and Cleaning Technologies. Essen, Rugby: Publico Publications, Institution of Chemical Engineers. 2. HM Muller-Steinhagen, MR Malayeri, and AP Watkinson. 2005. Fouling of heat exchangers—new approaches to solve an old problem. Heat Transfer Engineering 26: 1–4. 3. B Bansal, and XD Chen. 2006. A critical review of milk fouling in heat exchangers. Comprehensive Reviews in Food Science and Food Safety 5: 27–33. 4. BA Garett, P Ridges, and NJ Noyes. 1985. Fouling of Heat Exchangers: Characteristics, Cost, Prevention, Control, and Removal. 1st ed. NJ: Prentice Hall. 5. PJ Williams, and PA Anderson. 2006. Operational cost savings in dairy plant water usage. International Journal of Dairy Technology 59: 147–54. 6. PJ Fryer, PT Robbins, C Green, PJR Schreier, AM Pritchard, APM Hasting, DG Royston, and JF Richardson. 1996. A statistical model for fouling of a plate heat exchanger by whey protein solution at UHT conditions. Transactions of the Institute of Chemical Engineers 74: 189–99. 7. KD Rausch, and GM Powell. 1997. Dairy processing methods to reduce water use and liquid waste load. Department of Agricultural and Biological Engineering Report # MF2071, Cooperative Extension Service, Kansas State University, Manhattan, Kansas. 8. WJ Rebello, SL Richlen, and F Childs. 1988. The cost of heat exchanger fouling in the US industries. Report no. EGG-M-39187. Department of Energy, Washington DC. 9. J Visser, and ThJM Jeurnink. 1997. Fouling of heat exchangers in the dairy industry. Experimental Thermal and Fluid Science 14: 407–24. 10. P de Jong. 1997. Impact and control of fouling in milk processing. Trends in Food Science and Technology 8: 401–5. 11. S Prakash, N Datta, and HC Deeth. 2005. Methods of detecting fouling caused by heating of milk. Food Reviews International 21: 267–93. 12. PK Nema, and AK Datta. 2006. Comparative study of heat induced fouling of various types of milk flowing over a heated metal surface. International Journal of Food Engineering 2: 1–11.

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Treatment 10 Ozone of Food Materials Kasiviswanathan Muthukumarappan, Colm P. O’Donnell, and Patrick J. Cullen CONTENTS 10.1 10.2

Introduction ................................................................................................. 263 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 microbiological strains including Listeria monocytogenes and virulent strains of Escherchia coli. Consumer preference for minimally processed foods free of chemical preservatives, recent outbreaks of foodborne pathogens, identification of new food pathogens, 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 significant economic losses [1]. The Center for Disease Control and Prevention estimates the yearly cost of foodborne diseases in the US is $7–8 billion [2]. 263 © 2009 by Taylor & Francis Group, LLC

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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 paramount in recent years. It has become obvious that current systems of food production contain inadequate bacterial interventions (kill or reduction steps). Today, the only bacterial interventions for meat and poultry are antibacterial rinses during the slaughter process, and the final cooking stage. However, with rapidly changing lifestyles 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 constitute a major share of this category of food products. With this change in cooking habits, the RTE foods may not be attaining sufficiently high temperatures and hence, 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 emerging 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 pathogens in food would reduce the likelihood of future foodborne disease outbreaks. There are several processing methods available for inactivation of microorganisms in foods namely thermal, high pressure, pulsed electric field, oscillating magnetic field, irradiation, and ozonation. Ozonation treatment of food materials for microbial safety in solid and liquid food and mathematical modeling in liquid food are emphasized 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 recognized 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, © 2009 by Taylor & Francis Group, LLC

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TABLE 10.1 Estimated Annual Food Borne Illnesses, Hospitalization, and Deaths Due to Selected Pathogens, US, 2005 Disease/Agent Bacterial Campylobacter spp. Clostridium perfringens Escherichia coli O157:H7 Listeria monocytogenes Salmonella, nontyphoidal Staphylococcus

Illness

Hospitalization

Deaths

1,963,141

10,539

99

248,520

41

7

62,458

1,843

52

2,493

2,298

499

1,341,873

15,608

553

185,060

1,753

2

Vibrio cholerae, toxigenic

49

17

0

Vibrio vulnificus

47

43

18

112,500

2,500

375

Parasitic Toxoplasma gondii

Source: PS Mead, L Slutsker, V Dietz, LF McCaig, JS Bresee, C Shapiro, PM Griffin, and RV Tauxe. 2005. Food-related illness and death in the United States. Emerging Infectious Diseases 11 (5): 607–25.

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 agricultural 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 following 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]. © 2009 by Taylor & Francis Group, LLC

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It is a relatively unstable gas at normal temperatures and pressures, is partially soluble 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 (E 0, 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 O3 (g) + 2H+ + 2e− ⇔ O 2 (g) + H 2O{E 0 = 2.075V}

10.2.1

(10.1)

PRODUCTION OF OZONE

Ozone is generated by the exposure of air or another gas containing normal oxygen to a high-energy source. High-energy sources such as a high voltage electrical 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 most widely used commercial methods, have relatively low efficiencies (2–10%) and 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 electrodes 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 10.2 Physical Properties of Ozone Physical Properties Boiling point, °C

Value −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

Specific gravity

1.658

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The electrodes are typically either concentric metallic tubes or flat, plate-like 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 electrical discharge field to be formed. In excess of 80% of the applied energy is converted 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 decomposition, 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 decomposition 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

Electrode (high tension) Dielectric O2

Discharge gap

O3 Electrode (low tension)

He

FIGURE 10.1 Ozone generation by corona discharge method.

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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 10.3.1

MODELING OZONE IN FOOD MATERIALS MODELING OZONE DIFFUSION IN LIQUID FOOD

Predicting the ozone profile in a bubble column and contact chambers is important for determination of the log reduction and bromate formation in any ozone application in liquid such as water treatment, microbial inactivation in fruit juices, etc. For improvement of operational management of ozonation by model control, the model must be able to predict the ozone profile for changes in different control parameters 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 of quality change such as UV254 nm absorbance in the influent and the effluent of 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: ∂c ∂c u 6 = −u + kL ⋅ RQ ⋅ ⋅ ⋅ (α ⋅ kD ⋅ cg − c) − kO3 c − kr c u g db ∂t ∂x ∂cg ∂c k 6 = −ug g + L ⋅ ⋅ (α ⋅ kD ⋅ cg − c) ∂t ∂x α db 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 coefficient (m/s); RQ = gas to liquid flow ratio (Qg /Qw) (−); ug = velocity of gas phase through reactor (m/s); db = bubble diameter (m); α = pressure and temperature correction factor α = (Pg/P0)·(T0 /Tg); P0, Pg = standard and actual pressure (Pa); T0 , Tg = standard and actual temperature (K); kD = distribution coefficient (−), depending on nature of gas and temperature; cg = concentration of ozone in gas under standard pressure and temperature (g/Nm3); kO3 = first order constant for slow decay (s−1); and kr = first order constant for rapid decay (s −1). In the equation for the ozone concentration in liquid, the first term on the right 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

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ug

db h

FIGURE 10.2

Schematic of ozone analysis in a bubble column.

occurs [15]. Therefore one can describe the rapid decay as function of the degradation of UV254, resulting in the following Equation: kr = c

kUV ⋅ (UV − UVs ) Y

∂(UV) ∂(UV) = −u − kr ⋅ Y ⋅ c ∂t ∂x where kUV = decay coefficient of UV254 (l/(mg·s)); Y = yield factor, gives the relation 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 coefficient, k L-value. This expression gives a range for the k L-value depending on the density of the bubbles in the column varying from a kL-value for a single bubble to © 2009 by Taylor & Francis Group, LLC

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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 followed. 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 ozone behaves when introduced to fluids. Experimentally determined ozone bubble size distributions for the ozone processing of liquid foods would be of significant 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 capture 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 diameter 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; diffuser characteristics, ozone flowrates, static mixing, etc.

10.4 MICROBIAL INACTIVATION OF FOOD MATERIALS When a cell becomes stressed by viral, bacterial or fungal attack, its energy level is reduced by the outflow of electrons, and becomes electropositive. Ozone possesses 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, proteins, 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) resulting 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 application, and the species. Susceptibility varies with the physiological state of the culture, pH of the medium, temperature, humidity and presence of additives, such

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as, acids, surfactants, and sugars [19]. The antimicrobial spectrum and sanitary applications of ozone in food industry are summarized in Table 10.3. Sheldon and Brown [20] investigated the efficacy of ozone as a disinfectant for poultry carcasses. The microbial counts of ozone treated carcasses stored at 4°C were significantly lower than carcasses chilled under non-ozonated conditions. 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 different 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 of gaseous ozone on five species of fish bacteria and reported that ozone in relatively

TABLE 10.3 Antimicrobial Spectrum and Sanitary Applications of Ozone in Food Industry Sanitation

Dosage

Animal

>100 ppm

Black berries

0.3 ppm

Susceptible Microorganisms HVJ/TME/Reo type 3/murine hepatitis virus Botrytis cinerea

Cabbage

3

7–13 mg/m

Shelf life extension

Carrot

5–15 mg/m3

Shelf life extension

60 μl/L

Botrytis cinerea/Scerotinia sclerotiorum

Dairy

5 ppm

Alcaligens faecalus/P. fluorescens

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.

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low concentration (< 0.27 × 10 −3 g/l) was an effective bactericide of vegetative cells. Kaothien, Jhala, Henning, Julson and Muthukumarappan [24] evaluated the effectiveness of ozone in controlling Listeria monocytogenes in cured ham. There was a significant (p > 0.05) reduction (about 90%) in bacterial population, with ozone concentration 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 vegetables after harvest using water held at approximately 1°C can ozonate the water to prevent product contamination. Cooling fruits and vegetables helps slow product respiration, 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 washing process. Ozone can also be injected or dissolved in process waters of all kinds to provide chilling, fluming, rinsing or washing of meat, poultry, seafood, and eggs 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 Cranston [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 content 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 populations 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 blackberries 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]. 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 age wine and spirits [40], to reduce aflatoxin in peanut and cottonseed meals [41], to sterilize bacon, beef, bananas, eggs, mushrooms, cheese and fruit [42,43], to preserve 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 2 log reduction of total counts and significant reduction of spoilage and potentially 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

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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, additives (surfactants, sugars, etc.), and the amount of organic matter surrounding the 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 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 availability during treatment [53]. Efficacy of ozone is demonstrated more readily when targeted microorganisms are suspended and treated in pure water or simple buffers (with low ozone demand) than in complex food systems where it is difficult to predict 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 combination with antimicrobials such as dimethyl dicarbonate (DMDC; 250 or 500 ppm) or hydrogen peroxide (300 or 600 ppm) was evaluated by Williams, Summer and Golden [57]. In their first study they found no combination of treatments resulted 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 concluded 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 treatment on quality and physiology of various kinds of food products have been evaluated by various researchers. Ozone is not universally beneficial and in some cases

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may promote oxidative spoilage in foods [59]. Surface oxidation, discoloration or development of undesirable odors may occur in substrates such as meat, from excessive use of ozone [12,60]. Richardson [61] reported that ozone helps to control odor, flavor and color while 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 and Forney [47]. Ozonated water treatment resulted in no significant difference in 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 activity. However, ozone washing treatment was reported to have no effect on the final phenolic content of fresh-cut iceberg lettuce [31]. Contradictory reports are found in the literature regarding ascorbic acid, with decomposition of ascorbic acid in broccoli florets reported after ozone treatment by Lewis, Zhuang, Payne and Barth [63]. Conversely Zhang, Lu, Yu and Gao [62] reported no significant difference between 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 treatments 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 the volatile compounds. In spite of its efficacy against microorganisms both in the 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 qualities 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 sedimentation, lower sucrose content and a decrease in soluble solids by day 21 [69]. Recently researchers in Spain evaluated the effects of continuous and intermittent applications of ozone gas treatments, applied during cold storage to maintain postharvest quality during subsequent shelf life, on the bioactive phenolic composition 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 development, its application increased the total flavan-3-ol content at any sampling time. Continuous 0.1 μL L−1 O3 application also preserved the total amount of hydroxycinnamates, while both treatments assayed maintained the flavonol content 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.

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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 exposure 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 recommended 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 environments. It has been reported that ozone decreased surface flora by 3 log10 units when tested in wineries for barrel cleaning, tank sanitation, and clean-inplace processes [72]. Water containing low concentrations of ozone can be sprayed onto processing equipment, walls or floors to both remove and kill bacteria or other organic matter 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 10.4 Approved 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.

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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 ability for in-line testing of efficacy. 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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48. M Achen, and AE Yousef. 2001. Efficacy of ozone against Escherichia coli O157:H7 on apples. Journal of Food Science 66: 1380–84. 49. JG Kim, and AE Yousef. 2000. Inactivation kinetics of foodborne spoilage and pathogenic bacteria by ozone. Journal of Food Science 65 (3): 521–28. 50. EA Bryant, GP Fulton, and GL Budd. 1992. Disinfection alternatives for safe drinking water. NY: Van Nostrand Reinhold. 51. EL Stover, and RW Jarnis. 1981. Obtaining high level wastewater disinfection with ozone. Journal of Water Pollution Control Federation 53: 1637–47. 52. L Restaino, E Frampton, J Hemphill, and P Palnikar. 1995. Efficacy of ozonated water against various food related micro-organisms. Applied and Environmental Microbiology 61: 3471–75. 53. A Pascual, I Llorca, and A Canut. 2007. Use of ozone in food industries for reducing the environmental impact of cleaning and disinfection activities. Trends in Food Science and Technology 18: S29–35. 54. M Cho, H Chung, and J Yoon. 2003. Disinfection of water containing natural organic matter by using ozone-initiated radical reactions. Applied and Environmental Microbiology 69: 2284–91. 55. ZB Guzel-Seydim, AK Grene, and AC Seydim. 2004. Use of ozone in food industry. Lebensmittel Wissenschaft und Technologie 37: 453–60. 56. LL Dock. 1995. Development of thermal and non-thermal preservation methods for producing microbially safe apple cider. Thesis, Purdue University, West Lafayette, IN. 57. RC Williams, SS Sumner, and DA Golden. 2005. Inactivation of Escherichia coli O157:H7 and Salmonella in apple cider and orange juice treated with combinations of ozone, dimethyl dicarbonate, and hydrogen peroxide. Journal of Food Science 70 (4): M197–201. 58. FDA. 2004. Guidance for Industry: Recommendations to Processors of Apple Juice or Cider on the Use of Ozone for Pathogen Reduction Purposes. http://www.cfsan.fda. gov/~dms/juicgu13.html. Accessed July 22, 2008. 59. RG Rice, JW Farguhar, and LJ Bollyky. 1982. Review of the applications of ozone for increasing storage times of perishable foods. Ozone Science and Engineering 4: 147–63. 60. J Fournaud, and R Lauret. 1972. Influence of ozone on the surface microbial flora of frozen boot and during thawing. Indian aliment. Agriculture 89: 585–89. 61. SD Richardson. 1994. Drinking water disinfection by-products. In: RA Meyers, ed. The encyclopedia of environmental analysis and remediation. NY: John Wiley & Sons. Vol. 3: 1398–1421. 62. L Zhang, Z Lu, Z Yu, and X Gao. 2005. Preservation fresh-cut celery by treatment of ozonated water. Food Control 16: 279–83. 63. L Lewis, H Zhuang, FA Payne, and MM Barth. 1996. Beta-carotene content and color assessment in ozone-treated broccoli florets during modified atmosphere packaging. In 1996 IFT Annual Meeting Book of Abstracts. Chicago: Institute of Food Technologists. 64. MWF Luwe, U Takahama, and U Heber. 1993. Role of ascorbate in detoxifying ozone in the apoplast of spinach (Spinacia oleracea L.) leaves. Plant Physiology 101: 969–76. 65. A Ranieri, G D’Urso, C Nali, G Lorenzini, and GF Soldatini. 1996. Ozone stimulates apoplastic antioxidant systems in pumpkin leaves. Physiologia Plantarum 97: 381–87. 66. AG Perez, C Sanz, JJ Rios, R Olias, JM Olias. 1999. Effects of ozone treatment on postharvest strawberry quality. Journal of Agricultural and Food Chemistry 47: 1652–56. 67. A Nadas, M Olmo, and JM Garcia. 2003. Growth of Botrytis cinerea and strawberry quality in ozone-enriched atmospheres. Journal of Food Science 68 (5): 1798–802.

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68. JG Kim, AE Yousef, and MA Khadre. 2001. Microbiological aspects of ozone applications in food: A review. Journal of Food Science 66 (9): 2035–52. 69. LH Choi, and SS Nielsen. 2005. The effects of thermal and nonthermal processing methods on apple cider quality and consumer acceptability. Journal of Food Quality 28: 13–29. 70. F Artes-Hernandez, E Aguayo, F Artes, and FA Tomas-Barberan. 2007. Enriched ozone atmosphere enhances bioactive phenolics in seedless table grapes after prolonged shelf life. Journal of the Science of Food and Agriculture 87: 824–31. 71. H Urano, and S Fukuzaki. 2001. Facilitation of alumina surfaces fouled with heattreated bovine serum albumin by ozone treatment. Journal of Food Protection 64 (1): 108–12. 72. BC Hampson. 2000. Use of ozone for winery and environmental sanitation. Practical Winery and Vineyard (Jan/Feb): 27–30.

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Pasteurization 11 UV of Food Materials Kathiravan Krishnamurthy, Joseph Irudayaraj, Ali Demirci, and Wade Yang CONTENTS 11.1 11.2

Introduction ................................................................................................. 281 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 governmental regulations and potential risk of costly recalls, the food industry has been forced to ensure that their food products are free from pathogenic microorganisms. Furthermore, increased consumer awareness about minimally processed foods and industries’ thirst to reduce the total cost of food processing, propels researchers to investigate the efficacy of alternative food processing technologies to effectively pasteurize the food material while preserving the quality. Owing to the increased consumer demand for wholesome and fresh-like products, application of novel food processing technologies such as pulsed electric field, high pressure processing, ultraviolet (UV) light, pulsed 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 281 © 2009 by Taylor & Francis Group, LLC

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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 intermittent pulses in a lamp filled with inert gases such as xenon. This leads to the 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 11.2.1

UV AND PULSED UV LIGHT PROCESSING 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 = hυ =

hc λ

(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 contribution 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 actually absorbed by the food material, whereas the rest of the energy is reflected back,

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TABLE 11.1 Characteristics of UV, Visible, and Infrared Regions of Electromagnetic Spectrum Wavelength (nm)

Frequency (Hz)

Photon Energy (eV)

Molar Photon Energy (kJ/mol)

Vacuum UV

100–200

3.00 × 1016–3.00 × 1015

UV-C

124–12.4

11975–1197

200–280

3.00 × 1015–1.07 × 1015

12.40–4.43

1197–427

UV-B

280–315

1.07 × 10 –9.52 × 10

14

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 × 10 –4.28 × 10

14

3.10–1.77

299–171

Near infrared

700–1400

4.28 × 10 –2.14 × 10

14

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 × 10 –3.00 × 10

0.41–0.12

39.9–12.0

Region

15

14 14

13

13

Source: K Krishnamurthy. 2006. Decontamination of milk and water by pulsed UV light and infrared heating. PhD dissertation, Pennsylvania State University, University Park, PA.

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 0e− x

(11.2)

where T is the transparency coefficient of the food material, I is the intensity of the 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 dissipated 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 penetration limitation. Futhermore, good mixing can aid in uniform exposure. UV light is absorbed and penetrates into the microorganism depending upon the chemical composition, size of the microorganism, wavelength of interest, and medium of introduction 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 photons in the UV range. Therefore, it is clear that UV light has sufficient energy to break most of these chemical bonds. Hence, UV light can cause cleavage in organic

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TABLE 11.2 Percent Transmission to the Center of Selected Cells and Viruses Percent Transmission at Selected Wavelengths (%) Biological Sample

Diameter (μm)

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

69

97

100

1.6

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.

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 ionization 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 temperature 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, broadspectrum 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 nanoseconds to microseconds) and transferred through a lamp filled with inert gas (xenon or 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 approximately 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 © 2009 by Taylor & Francis Group, LLC

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TABLE 11.3 Strength of Common Bonds in Biomolecules Chemical Bond Type

Wavelength

Bond 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.

light can ionize the molecules, whereas visible light results in vibration of molecules 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 years because it can provide sufficient antimicrobial inactivation and commercial 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 efficiency of UV light follows a bell shaped curve with the maximum inactivation occurring between 254 and 264 nm (Figure 11.1). As a typical conventional mercury UV lamp produces UV light at 254 nm, this wavelength is often used for comparison of the disinfection efficiencies of conventional UV lamps. © 2009 by Taylor & Francis Group, LLC

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80

60

40

254

Relative bactericidal effect

100

20

220

260

300

nm

FIGURE 11.1 Germicidal efficiency of UV light. (From WJ Masschelein. 2002. Ultraviolet Light in Water and Wastewater Sanitation. Boca Raton: W.H. Freeman Publishers.)

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, thymine 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-linking 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.

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TABLE 11.4 Photoproducts Produced in DNA because of Absorption of UV Light Percentage of Total Photoproducts Photoproduct

UV-C

UV-B

Cyclobutyl pyrimidine dimer

77

78

Pyrimidine pyrimidinone[6–4]-photoproduct

20

10

Dewar pyrimidinone

0.8

10

Adenine–thymine heterodimer Cytosine photohydrate Thymine photohydrates

0.2 −2.0

– −2.0

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.

Though UV light can damage the microorganisms, some of them can repair themselves by photo-reactivation or dark repair. As the name indicates, photoreactivation 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 thymine 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 H 2O2 [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 dihydrothymine 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, inactivation is expected to be predominately caused by the UV light portion of the broad

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TABLE 11.5 UV Light Exposure (at 254 nm) Required for 4-log10 Reduction of Pathogens in Drinking Water Exposure 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

50

210

100

320

Microorganism

Vibrio cholerae wild isolate Yersinia enterocolitica

Source: Hoyer. 1998. Testing performance and monitoring of UV systems for drinking water disinfection. Water Supply. 16: 424–29.

spectrum. Krishnamurthy examined the pulsed UV light treated cells with transmission 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/cytoplasmic 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 higher flux densities (0.5 J/cm2 ), leading to cell rupture. Localized heating of bacteria 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 steam flow causing membrane destruction [29]. Takeshita, Shibato, Sameshima, 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 © 2009 by Taylor & Francis Group, LLC

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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 categorized into: a. Photo-chemical effect: Thymine dimer formation and other photochemical 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 decades, especially for water disinfection. Chang, Ossoff, Lobe, Dorfman, Dumais, Qualls, and Johnson investigated the efficacy of UV light on inactivation of Escherichia coli, 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 using a 1.0-μm nucleopore polycarbonate membrane and the filtrate was treated with 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 occurring microfloras of lamb (mostly Pseudomonas, Micrococcus, and Staphylococcus 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 CFU/lettuce reductions, respectively [33]. A thin film 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 population. 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]. © 2009 by Taylor & Francis Group, LLC

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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 decontamination 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 continuous 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 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 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]. Milk, artificially inoculated with S. aureus cells were pulsed UV light treated 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 [4]. A continuous flow-through system was developed for inactivation of S. aureus in milk. Milk was treated at 5, 8, or 11 cm distance from UV light strobe at 20, 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 efficiency. Log10 reductions varied from 0.55 to 7.26 log10 CFU/ml. Complete inactivation 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 efficacy of pulsed UV light for inactivation of E. coli O157:H7 and Listeria monocytogenes Scott A on salmon fillets was also investigated [41]. The authors demonstrated 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.

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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 reduction 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. sporogenes. 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

The dose–response curves for dispersed or free-floating microorganisms by UV light can be represented by first-order kinetics [44–46], resulting in a sigmoidal curve with 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, resistance of bacteria are some of the factors influencing the shoulder effect. Tailing effect can be attributed to shielding of external particles, clumping of bacteria, and resistant microorganisms. The first order inactivation equation [46] can be represented as follows: N = N 0 exp−( k *I *t )

(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); k = first order inactivation coefficient (cm2 /J); I = intensity of UV light energy applied (J/cm2); and t = treatment time (sec). Equation 11.3 can also be represented as [44]: N = N 0 exp(−kD ) = 10

⎛ D ⎟⎞ ⎟⎟ −⎜⎜⎜ ⎜⎝ D10 ⎟⎠

(11.4)

where D = I*t = UV dose delivered or fluence rate (J/cm2); and D10 = UV dose required to achieve 90% reduction in microbial population. Therefore, ⎛N ⎞ D log⎜⎜ 0 ⎟⎟⎟ = (11.5) ⎜⎝ N ⎟⎠ D10 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, N = exp(−kD ) N0

(11.6)

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Therefore, ⎛N ⎞ ⎛ k ⎞⎟ 1 log⎜⎜ 0 ⎟⎟⎟ = kD * =⎜ ⎟* D ⎝N⎠ ln(10) ⎜⎝ 2.303 ⎟⎠

(11.7)

where log(N0 / N) is the log10 reduction of microbial population and (K / 2.303) is the slope of the fitted straight line for the plot of log10 reduction versus available UV dosage. When the microorganism exhibits a shoulder effect, Equation 11.6 can be modified as N = N 0 (1 − (1 − exp(−kD ) )d

(11.8)

where d is the intercept of the exponential phase of the dose–response curve with the y-axis [44]. Similarly, Equation 11.6 can be modified to take into account the tailing effect as follows: N = N 0 e(−kD ) + N pe(−kp D )

(11.9)

where N0 is the concentration of dispersed microorganisms present; Np is the concentration 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% of the total energy). Therefore, the first order kinetics (Equation 11.4) can be used for pulsed UV light inactivation modeling. However, due to the contribution of visible light and infrared heating, first order kinetics may not be able to explain all the variation 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 estimate 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. β ⎛ N ⎞⎟ 1 ⎛⎜ t ⎞⎟ ⎜ log10 ⎜⎜ ⎟⎟ = − ⎜ ⎟ 2.303 ⎜⎝ α ⎟⎟⎠ ⎝ N 0 ⎟⎠

(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). © 2009 by Taylor & Francis Group, LLC

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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 products can also be achieved by UV light exposure. Jasinghe and Perera successfully fortified edible mushrooms with vitamin D2 [49]. The Shitake, oyster, button and 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 various 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 inactivate 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 beverages 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, particularly 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 discoloration. Super oxide radials can further induce carbohydrate cross-linking, protein cross-linking, protein fragmentation, peroxidation of unsaturated fatty acids, and loss of membrane fluidity function. Water molecules absorb UV photons and produces OH- and H+ radicals, which in turn interacts with other food components. Furthermore, UV radiation may also denature proteins, enzymes, and amino acids (especially

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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. UV light may also cause flavor and color changes in food [54]. During UV light treatment, oxygen radicals are formed. These oxygen radicals lead to the formation of ozone, especially between 185 and 195 nm, causing off-flavors in the food 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 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 of methionine from methanol which leads to a burnt protein-like, burnt feathers-like, or medicinal-like flavor. Prolonged treatment with UV light can also result in discoloration 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 and thus induce temperature related quality changes such as cooked flavor, change 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 researchers. In general, moderate treatments did not induce significant changes in the sensory attributes, while severe treatments lead to objectionable changes. Choi and Nielsen investigated the efficacy of UV light for pasteurization of apple cider [56]. A consumer panel sensory evaluation with 40 panelists suggested that there was no significant difference (p < 0.05) in the sensory scores for the odor, color, cloudiness, sweetness, acidity, overall flavor, and overall product ratings for untreated, pasteurized, and UV light treated apple ciders stored for 4 days. Furthermore, the pasteurized apple cider received less acceptable rating for color, cloudiness, 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 panelists significantly preferred UV light treated apple cider over thermally pasteurized apple cider for color, cloudiness, sweetness, acidity, overall flavor, and overall likeness 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 cabbage 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 consumption 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-package 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

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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 quality. Further modification and optimization of the UV light and pulsed UV 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 competitive 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 processing 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 inactivation of pathogenic microorganisms [4]. Dunn, Bushnell, Ott, and Clark estimated that a 4-J/cm2 pulsed UV light treatment 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 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, 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 selective food products.

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TABLE 11.6 Coefficient of Absorption of Various Liquid Food Products at 254 nm Liquid Food Product

Coefficient of Absorption (cm −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 11.7 Wavelength 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.

UV light can easily penetrate through transparent liquids such as water. However, 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 © 2009 by Taylor & Francis Group, LLC

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light lamp in a photo-reactor, which can be utilized to enhance the effectiveness of absorption by target material (Figure 11.2). A falling film photo-reactor design might be essential for effective penetration in milk and other opaque food products, as a thin film of food material can be maintained. Other designs which may reduce the 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 will be essential to treat the food product as a thin film in order to avoid temperature 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

Q

Q

RV

R

RV

R

Q L

L

L

L

L

RV Flow of medium A

Flow of medium B

Tubular flow-through

Flow of medium C

Flow of medium D

Falling film flat bed CFBR

FIGURE 11.2 Arrangements of lamps for flow-through systems. Cross sectional and topview 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 tube. (From T Oppenlander. 2003. Photochemical Purification of Water and Air. Weinheim, Germany: Wiley-VCH.)

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UV light must also be clearly identified in order to shed more light on the efficacy of pulsed UV light. One of the drawbacks of pulsed UV light technology is the temperature buildup during prolonged treatments, as it can lead to quality deterioration in temperature sensitive products. Therefore, the infrared region of the spectrum needs to be filtered out for treatment of temperature sensitive food products. This will also minimize the quality deterioration due to temperature increase. Provision of a heat sink may be beneficial in avoiding excessive temperature build-up. Pulsed UV light also emits radiation from 330 to 480 nm, which is responsible for photoreactivation (a mechanism to repair the DNA damage caused by UV light). In order to avoid, photo-reactivation, it may be beneficial to filter out this wavelength spectrum. A report submitted by Institute of Food Technologists (IFT) to the FDA identified the following research needs for UV light [47]: 1. Effects of individual parameters, such as suspended and dissolved solids concentration. 2. Identification of the pathogens most resistant to UV light. 3. Identification of surrogate microorganisms for pathogens. 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]: 1. Data on pulsed light effectiveness for specific commodities. 2. Comparison of resistance of specific pathogens, including bacteria, viruses, and parasites exposed to pulsed light. 3. Identification of critical process factors and their effect on microbial activation. 4. Optimization critical process factors and development of protocols to monitor 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 treatment, 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 © 2009 by Taylor & Francis Group, LLC

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is the limiting factor for both pulsed UV light and UV light, food products have to be treated as a thin film, especially for opaque foods. Further investigation on the inactivation mechanisms of pulsed UV light can help us understand the process better and may lead to a better and effective equipment design.

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40. RR Sharma, and A Demirci. 2003. Inactivation of Escherichia coli O157:H7 on inoculated alfalfa seeds with pulsed ultraviolet light and response surface modeling. Journal of Food Science 68: 1448–53. 41. NP Ozer, and A Demirci. 2005. Inactivation of Escherichia coli O157:H7 and Listeria monocytogenes inoculated on raw salmon fillets by pulsed UV light treatment. International Journal Food Science and Technology 40: 1–7. 42. SL Hillegas, and A Demirci. 2003. Inactivation of Clostridium sporogenes in clover honey by pulsed UV light treatment. Agricultural Engineering International: the CIGR Journal of Scientific Research and Development. Manuscript FP 03-009. Vol. V. 7. 43. S Jun, J Irudayaraj, A Demirci, and D Geiser. 2003. Pulsed UV light treatment of corn meal for inactivation of Aspergillus Niger spores. International Journal Food Science and Technology 38: 883–88. 44. EPA. 2003. UV disinfection guidance manual. EPA document no. 815-D-03-007. Washington, DC: Environmental Protection Agency. 45. H Liu, Y Du, X Wang, and L Sun. 2004. Chitosan kills bacteria through cell membrane damage. International Journal of Food Microbiology 95: 147–55. 46. BF Severin, MT Suidan, BE Rittmann, and RS Engelbrecht. 1984. Inactivation kinetics in a flow-through UV reactor. Journal of Water Pollution Control Federation 56: 164–69. 47. CFSAN-FDA. 2000. Ultraviolet light. In: Kinetics of Microbial Inactivation for Alternative Food Processing Technologies. Atlanta, GA: Center for Food Safety and Applied Nutrition – Food and Drug Administration. Available at: http://www.cfsan.fda. gov/~comm/ift-uv.html. Accessed July 28, 2008. 48. KL Bialka, A Demirci, and VM Puri. 2008. Modeling the inactivation of Escherichia coli O157:H7 and Salmonella enterica on raspberries and strawberries resulting from exposure to ozone or pulsed UV light. Journal of Food Engineering 85: 444–49. 49. VJ Jasinghe, and CO Perera. 2006. Ultraviolet irradiation: The generator of Vitamin D in edible mushrooms. Food Chemistry 95: 638–43. 50. AE Yousef, and EH Marth. 1985. Degradation of aflatoxin M1 in milk by ultraviolet energy. Journal of Food Protection 48: 697–98. 51. S Chung, W Yang, and K Krishnamurthy. 2008. Effects of pulsed UV-light on peanut allergeus in extracts and liquid peanut butter. Journal of Food Science. 73(5): C400–C404. 52. P Tomasik. 2004. Chemical modifications of polysaccharides. In: P Tomasik, ed. Chemical and Functional Properties of Food Saccharides. New York: CRC Press, 123–130. 53. A Kolakowska. 2003. Lipid oxidation in food systems. In: ZE Sikorski, A Kolakowska, eds. Chemical and Functional Properties of Food Lipids. New York: CRC press, 133–68. 54. T Ohlsson, and N Bengtsson. 2002. Minimal processing of foods with non-thermal methods. In: Ohlsson, T. and N. Bengtsson, eds. Minimal Processing Technologies in the Food Industry. New York: CRC Press, 34–57. 55. M Cuvelier, and C Berset. 2005. Phenolic compounds and plant extracts protect paprika against UV-induced discoloration. International Journal Food Science and Technology 40: 67–73. 56. LH Choi, and SS Nielsen. 2005. The effects of thermal and non-thermal processing methods on apple cider quality and consumer acceptability. Journal of Food Quality 28: 13–29. 57. VM Gomez-Lopez, F Devileghere, V Bonduelle, and J Debevere. 2005. Intense light pulses decontamination of minimally processed vegetables and their shelf-life. International Journal of Food Microbiology 103(1): 79–89.

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58. J Rice. 1994. Sterilizing with light and electrical impulses: Technological alternative to hydrogen peroxide, heat, and irradiation. Food Processing 7: 66. 59. KT Higgins. 2001. Fresh today, safe next week. Food Engineering 73(44–46): 48–49. 60. F Taghipour. 2004. Ultraviolet and ionizing radiation for microorganism inactivation. Water Research 38: 3940–48. 61. J Dunn, A Bushnell, T Ott, and W Clark. 1997. Pulsed white light food processing. Cereal Food World 42: 510–515. 62. D Lander. 1996. Microbial kill with pulsed light and electricity – fruitful possibilities. Fruit Processing 6(2): 50–51. 63. G Shama. 1999. Ultraviolet light. In: RK Robinson, C Batt and P Patel, eds. Encyclopedia of Food Microbiology. pp. 2212, San Diego: Academic Press. 64. H Burton. 1951. Ultraviolet irradiation of milk. Dairy Science Abstracts 13(3): 229–44. 65. T Oppenlander. 2003. Photochemical Purification of Water and Air. Weinheim, Germany: Wiley-VCH. 66. NACMCF. 2006. (National Advisory Committee on Microbiological Criteria for Foods). Requisite scientific parameters fro establishing the equivalence of alternative methods of pasteurization. Journal of Food Protection Supplement 69: 1190–216.

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Finite Element 12 Stochastic 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 temperature 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 303 © 2009 by Taylor & Francis Group, LLC

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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, including finite differences [1,3,4], and finite elements [2] have been applied successfully by numerous investigators to the numerical solution of the Fourier equation. While commercial codes (mostly based on the finite element or finite volume method) are now widely available, they require that the product and process parameters are accurately 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 quantities, which must be characterized by statistical means. Nicolaï et al. [9] reviewed the sources of uncertainty in thermal sterilization processes. The coefficient of variation (CV) of the f-value, and, hence, the thermal 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% confidence level. Sheard and Rodger [12] compared the time required for vacuum-packed 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 variations 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 maximum 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 heat transfer coefficient in thermal food processes. Martens [13] used a CV of 10 and 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 © 2009 by Taylor & Francis Group, LLC

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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 parameter fluctuations in space and/or time [22–25]. In this chapter the use of stochastic finite element methods to calculate statistical characteristics of the temperature field inside conduction heated foods will be described. The main features of the Monte Carlo and variance propagation algorithms 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 conditions is governed by the Fourier equation k ∇ 2T + Q = ρc k

∂T ∂t

∂ T = h (T∞ − T ) on Γ ∂n T = T0

at t = t0

(12.1) (12.2) (12.3)

where T is the temperature (ºC), k the thermal conductivity (W/mºC), ρc the volumetric heat capacity (J/m3 ºC), T∞ the (known) process temperature (ºC), n the outward normal to the surface, h the convection coefficient (W/m2 ºC), Γ the boundary 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 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 a very flexible and accurate method for solving partial differential equations such as the Fourier equation. In the framework of the finite element method the continuum is subdivided in elements of variable size and shape which are interconnected in a finite number nnod of nodal points. In every element j the unknown temperature is approximated by a low order interpolating polynomial T

u j (t ) = φ j u j (t )

(12.4)

where uj(t) is the approximate temperature in element j, u j(t) is the vector containing the nodal temperatures in element j, and φ j is the vector of shape functions corresponding to element j. The application of a suitable spatial discretization technique 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

d u + Ku = f dt

(12.5)

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u(t = 0) = u0

(12.6)

with u = [u1 u2 unnod ]I the overall nodal temperature vector, C the capacitance matrix and K the conductance matrix, both nnod × nnod matrices, and f a nnod × 1 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 contribuj j j tions of each element (the ‘element matrices’) C , K and f Cj =

∫ ρcφ φ j

jT

(12.7)

dV

Vj

Kj =

∫ kB B j

V

fj =

jT

dV +

j

∫ hφ φ j

S

jT

∫ hT φ dS +∫ Qφ dV ∞

S

j

j

j

V

dS

(12.8)

j

(12.9)

j

with Bj =

∂φ j , ∂z

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 advantageously for reducing the CPU time required for the solution of Equation 12.5. The finite element method has been successfully used in a number of thermal food processing applications such as sterilization of baby food jars [2], cooling of broccoli stalks [27] and tomatoes [28].

12.3

DESCRIPTION OF UNCERTAINTY

In the finite element method it is assumed that all parameters are deterministic and known. However, in reality this is certainly not always the case. In the stochastic finite element method, knowledge about the uncertainty of the material and process 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 extensions such as random process, field and wave, will be introduced. For a more precise 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.

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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 probability density function f(x) and statistical moments such as the mean value X and the variance σ2, if existent and known. X  ε (X )

(12.10)

∞



∫ xf (x)dx

(12.11)

−∞

σ 2  ε ( X − X )2

(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 random vector X. Similar to the univariate case we can then define the mean value X and covariance matrix V of the random vector as X  ε( X)

(12.14)

V  ε ⎡⎢( X − X)( X − X)T ⎤⎥ ⎣ ⎦

(12.15)

The i-th diagonal entry of V is the variance σ 2Xi of random variable Xi; the (i, j)-th entry of V is the covariance σ Xi , X 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 often used to describe bivariate random variables. It is defined as 2 ⎧⎪ ⎡⎛ ⎪⎪ (1 − RX21 ,X2 )−1/2 ⎢⎜ X1 − X1 ⎞⎟⎟ −1 f ( X1 , X 2 ) = exp ⎨ ⎢⎜⎜ ⎟⎟ 2 2 πσ X1 σ X2 ⎪⎪⎪ 2(1 − RX1 ,X2 ) ⎢⎢⎜⎝ σ X1 ⎟⎠ ⎣ ⎩

⎛ X − X ⎞⎟⎛ X − X ⎞⎟ ⎛ X − X ⎞⎟2 ⎤⎥ ⎪⎪⎫ 1 ⎟⎜ 2 2⎟ ⎜ 2 2⎟ ⎪ − 2 RX1 ,X2 ⎜⎜⎜ 1 ⎟⎜ ⎟+⎜ ⎟ ⎥⎬ ⎜⎝ σ X1 ⎟⎟⎠⎜⎜⎝ σ X2 ⎟⎟⎠ ⎜⎜⎝ σ X2 ⎟⎟⎠ ⎥ ⎪⎪ ⎥⎦ ⎪⎭

(12.16)

R is called the correlation coefficient, and −1 ≤ R ≤ 1.

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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 defined by X = ε( X )

(12.17)

∞



∫ x f (x, t)dx

(12.18)

−∞

V (τ) = ε{[ X (t ) − X ][ X (t + τ) − X )]}

(12.19)

The covariance function describes how much the current value of the random function will affect its future values. By definition of a stationary random process, the 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 normalization of the covariance function: R (τ) = V (τ)/σ2

(12.20)

A Gaussian stationary white noise process W with covariance VW ,W (τ) = σ W2 ,W δ(τ),

(12.21)

where δ is the Dirac delta, can be used to describe very rapid unpredictable fluctuations. Sample values of W are uncorrelated no matter how close together in time they are. However, white noise does not exist in reality as it has an infinite energy content and variance. Autoregressive processes provide a tool to incorporate fluctuations which change more smoothly as a function of time. An autoregressive random process of order m is defined by the following stochastic differential equation dm d m−1 X (t ) + a1 m−1 X (t ) +  + am X (t ) = W (t ) m dt dt

(12.22)

where a1, a2,…, am are constants, m ≥ 1, and W(t) is a stationary Gaussian white noise process with W = am X . The time scale of the fluctuations is dependent on the coefficients a1,…,a m, and their high frequency content decreases with increasing order m. The (Gaussian) random variable initial condition corresponding to the stochastic differential Equation 12.22 is defined as ε[ X (t0 )] = X

(12.23)

ε[ X (t0 ) − X ]2 = σ 2

(12.24)

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Note that a random variable parameter X can be modeled as a trivial case of an AR(1) process: d X=0 dt

(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 single measure, Vanmarcke [29] introduced the concept scale of fluctuation, which is defined as +∞

θ=

∫ R(τ)d τ −∞

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 autocovariance function and the scale of fluctuation are given for AR(1) and AR(2) processes. For the latter, the characteristic polynomial ξ m + a1ξ m−1 +  + am = 0

(12.26)

has two real or two complex conjugate roots, resulting in non-oscillating or oscillating correlation functions, respectively. In Figure 12.1 some correlation functions and corresponding realizations of an AR(2)-process with different scales of fluctuation 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 12.1 Autocovariance Function and Scale of Fluctuation of AR(1) and AR(2) Processes AR(1)

Vx , x (τ) = σ 2x e−a1 τ θ = 2 / a1

AR(2) real roots

(

)

Vx , x (τ) = σ 2x ξ 2 e ξ1 τ − ξ1e ξ2 τ [(ξ 2 − ξ1 )]−1 with ξ1 and ξ 2 the roots of ξ 2 + a1ξ + a2 = 0 θ = 2a1 / a2

⎛ − 1 a1 τ AR(2) complex Vx , x (τ) = σ 2x ⎜⎜⎜e 2 ⎜⎝ roots

⎤⎞ ⎡ ⎢ cos( p τ ) + a1 sin( p τ )⎥ ⎟⎟⎟ ⎢ ⎥⎟ 2p ⎦ ⎟⎠ ⎣

with p = (a1 − a12 / 4)1/ 2 θ = 2a1 / a2

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Rx,x( Δ t)

(a)

–1000

–500

0 Δ t (s)

500

1000

(b) θ = 100 s

x (t)

θ = 10 s

θ= 1 s

0

200

400 600 Time (s)

800

1000

FIGURE 12.1 Correlation functions (a) and corresponding realizations (b) of a AR(2)process with different scales of fluctuation.

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are completely correlated. In this case the random process concept is far too sophisticated to describe the physical quantity since all the meaningful probabilistic features of the quantity can be captured by a simple random variable model. The correlation functions and corresponding realizations of different types of random processes are shown in Figure 12.2. Clearly the scale of fluctuation is a 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 following state space form [31]: d X(t ) = AX(t ) + BW (t ) dt

(12.27)

where ⎤ ⎡0⎤ ⎥ X ⎥ ⎢ ⎥ ⎥ ⎢0⎥ dX / dt ⎥⎥⎥ ⎢ ⎥ ⎥ ⎢⎥ ⎥ B = X=  ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ d m−2 X / dt m−2 ⎥⎥⎥ ⎢0⎥ ⎥ ⎢ ⎥ ⎥ m − 1 m − 1 d X /dt ⎢⎣ 1 ⎥⎦ ⎥⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

A=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢

0

 0 −am

1 0 −am−1

   

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1 ⎥⎥⎦

 1 −a

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 properties of heterogeneous materials such as foods, hydraulic properties of soils, elastic properties of construction materials, etc. The random field concept provides a convenient mathematical framework to describe such phenomena [29]. A parameter which fluctuates both in space and time can be described by means of random waves. The random wave model is a straightforward extension of the 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 is referred to the literature [29].

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Rx,x( Δ t)

(a)

–1000

–500

0 Δ t (s)

500

1000

(b) AR(2) complex roots

x (t)

AR(2) real roots

AR(1)

0

200

400 600 Time (s)

800

1000

FIGURE 12.2 Correlation functions (a) and corresponding realizations (b) of several types of autoregressive processes with the same scale of fluctuation.

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12.4 THE MONTE CARLO METHOD 12.4.1 DESCRIPTION The Monte Carlo method was introduced by John von Neumann and Ulam during World War II for studying random neutron diffusion problems in fissionable material 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. In the Monte Carlo finite element method, samples of the random parameters are generated by means of a random generator. For every parameter set the heat conduction 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 1 T = n 2 σˆ =

n

∑T

j

(12.28)

j=1

1 n −1

n

∑ (T

j

− T )2

(12.29)

j=1

where Tj is the solution in the j-th Monte Carlo run and the symbol “ˆ” means “estimate 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 2 the confidence intervals for T and σˆ are given by [33] σˆ σˆ T − t0.975 ≤ Ti ≤ T + t0.975 n n

(12.30)

χ 20.025 σˆ 2 χ 20.975 ≤ ≤ n −1 σ 2 n −1

(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 2χ 2 − 2n − 1 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 temperature T is linearly dependent on the random parameter(s). Even if this is not so, 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 parameters is not too large.

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For n = 100 and 1000, Equation 12.31 becomes ˆ2 0.74 < σ2 < 1.28 for n    = 100 σ

(12.32)

ˆ2 0.91 < σ2 < 1.09 for n    = 1000 σ

(12.33)

This means that, even for n = 1000, the relative confidence band is almost 20% 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 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 generate the parameter samples can outweigh by far the actual CPU time required to solve the finite element problem. A careful choice of the algorithms to generate 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 parameter set must be completely specified in the probabilistic sense, including (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. The modulo (‘mod’) is defined as the remainder of the integer division, e.g., 5 mod 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 not cause integer overflow on most systems. It is emphasized here that the implementation of a good random number generator 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

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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 recursively by time discretization of the corresponding differential equation. For example, for an AR(1) process we have that d X + a1X = W dt

(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)

X(t + Δt) = (1 + a1Δt)−1 (X(t) + ΔtZ (t + Δt))

(12.39)

or

0.175 0.150

f(u)

0.125 0.100 0.075 0.050 0.025 0.000 –3.6 –2.8 –2.0 –1.2 –0.4 U

0.4

1.2

2.0

2.8

FIGURE 12.3 Histogram of a measurement experiment of a random variable U. The full line is the limiting probability density function.

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with Z discrete time white noise (a sequence of Gaussian random numbers). It can be shown [36] that the variance σ 2Z of Z is equal σ 2Z /Δt . The algorithm is bootstrapped 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 variance of the generated sample will be smaller than the target variance. Methods to generate 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 temperature field. The variance propagation algorithm is an alternative to the Monte 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 following 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 the surface heat transfer coefficient h at the solid–liquid interface. In the lumped 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, satisfies the following constraint Bi =

hL < 0.1 k

(12.40)

where L is the characteristic length of the solid which in the case of a sphere is usually defined as L = 3r0 [38]. It is easy to show that applying an overall energy balance leads to the following differential equation: ρc

d 3h T = (T∞− T ) dt r0

(12.41)

After integration the following formula for the temperature course is found ⎛ 3h ⎞⎟ T = T∞+ (T0 − T∞ ) exp⎜⎜− t⎟ ⎜⎝ ρcr0 ⎟⎟⎠

(12.42)

For simplicity we will assume that T∞ is an AR(1) random process described by means of the following differential equation: d T∞ + a1T∞ = W dt

(12.43)

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with W a white noise process with mean W = a1 T ∞ . We can combine Equation 12.43 and Equation 12.41 into the following global system: d x = g( x) + h(W − W )(t ) dt

(12.44)

with ⎡T ⎤ x = ⎢⎢ ⎥⎥ ⎣T∞⎦

(12.45)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

3h 3h ⎤⎥⎥ T∞ − T ρcr0 ⎥⎥⎥ g = ρcr0 ⎥ W − a1T∞ ⎥⎥⎦

(12.46)

⎡0⎤ h = ⎢⎢ ⎥⎥ ⎣1⎦

(12.47)

It can be shown [36] that first order approximate expressions for the mean vector and the covariance matrix of the solution of Equation 12.44 are given by d x = g[ x ] dt

(12.48)

⎛ ∂g ⎞T d ∂g Vx ,x = Vx ,x + Vx ,x ⎜⎜⎜ ⎟⎟⎟ + hσ W2 hT ⎝ ∂x ⎠ dt ∂x

(12.49)

with ∂g ∂g[ x(t ), t ]  ∂x ∂x(t ) x (t ) 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:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

3h − d Vx ,x = ρcr0 dt 0

d 3h (T∞ − T ) T= ρcr0 dt

(12.50)

d T∞ = 0 dt

(12.51) ⎡

⎢ 3h ⎤ ⎢− 3h ⎥⎥ ⎢ ⎢ ρcr0 ⎥ ρcr0 ⎥⎥ Vx ,x + Vx ,x ⎢⎢⎢ ⎥ ⎢ 3 h ⎢ −a1 ⎦⎥⎥⎥ ⎢ ⎢ ρcr 0 ⎣⎢

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1⎥ ⎥ ⎥⎦

⎡0 + ⎢⎢ ⎣0 −a

0 ⎤⎥ σ W2 ⎥⎦

(12.52)

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with ⎡ ⎢

σ T ,T∞ ⎤⎥⎥ ⎥ σT2∞ ⎥⎥⎥⎦

σ T2 ⎢ σ T ,T∞ ⎢⎣

Vx ,x = ⎢⎢⎢

(12.53)

and σ T ,T∞ the covariance of T and T∞. The initial conditions are given by T (t = 0) = T0

(12.54)

T∞ (t = 0) = T∞

(12.55)

⎡ σ T20 Vx ,x (t = 0) = ⎢⎢ ⎢⎣ 0

0 ⎤⎥ σ T2 ∞ ⎥⎥⎦

(12.56)

Equation 12.50 expresses that the mean solution can be found by solving the original differential equation for the mean value of the random parameter. Equation 12.51 confirms that the mean value of the random parameter is constant (which we expected since an autoregressive process is stationary). Equation 12.52 can be elaborated further to yield d 2 6h 2 6h σT + σT = − σ T ,T∞ ρcr0 ρcr0 dt ⎛

(12.57)

⎞

⎟⎟ d 3h 2 ⎜⎜⎜⎜ 3h ⎟ σ T ,T∞ = σ T∞− ⎜⎜ + a1 ⎟⎟⎟⎟⎟ σ T ,T∞ ⎜⎜ ρcr ρcr0 dt ⎝ ⎠⎟⎟ 0

(12.58)

d 2 σ T = −2a1σ T2∞+ σ W2 dt ∞

(12.59)

As T∞ is stationary, σ T2∞ not a function of time so that Equation 12.59 reduces to σ T2∞ = σ W2 / 2a1

(12.60)

The solution of Equation 12.58 can readily be found through direct integration

σ T2 =

⎞⎟ ⎤ ⎪ ⎫ ⎡ ⎛⎜⎜ 3h ⎪⎧ ⎟⎟ 3h / ρcr0 σ T2∞⎪⎨1 − exp ⎢⎢−⎜⎜⎜⎜ + a1 ⎟⎟⎟⎟⎟ t ⎥⎥ ⎪⎬ ⎜ ⎪ ⎟⎟⎠ ⎪ 3h / ρcr0 + a1 ⎢⎣ ⎜⎝ ρcr0 ⎥⎦ ⎪⎭ ⎪⎩

(12.61)

After substitution of Equation 12.61 in Equation 12.57 and subsequent integration we can derive the following expression for σ T2

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σ T2 =

319

⎛ 6h ⎞⎟ 3h / ρcr0 3h / ρcr0 σ T2∞exp⎜⎜⎜− t ⎟⎟ σ T2∞+ 3h / ρcr0 + a1 3h / ρcr0 − a1 ⎝ ρcr0 ⎟⎠ −

⎞⎟ ⎤ ⎡ ⎛⎜⎜ 3h ⎟⎟ 18h 2 / ρ2c 2r02 σ T2∞exp ⎢⎢−⎜⎜⎜⎜ + a1 ⎟⎟⎟⎟⎟ t ⎥⎥ ⎜ ⎟⎟⎠ (3h / ρcr0 + a1 )(3h / ρcr0 − a1 ) ⎢⎣ ⎜⎝ ρcr0 ⎥⎦

(12.62)

In the special case of a random variable, we can simplify the above expression by putting a1 = 0, so that we have ⎡ ⎛ 3h ⎞ ⎤ ⎛ 6h ⎞⎟ ⎟⎟ t ⎥ σ T2 = σ T2∞+ σ T2∞exp⎜⎜⎜− t ⎟⎟ − 2σ T2∞exxp ⎢⎢−⎜⎜⎜ ⎟⎟ ⎥ ρ cr ⎠ ⎥⎦ ⎝ ρcr0 ⎟⎠ ⎝ 0 ⎢⎣

(12.63)

A sample of the random process ambient temperature and the corresponding temperature 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. A Crank–Nicolson finite difference scheme in the time domain was used to solve Equation 12.41. The high frequency fluctuations are smoothed because of the 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 variance 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 mean value and 95% confidence interval for the temperature prediction in 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, as defined by the following state space equations d x T∞ (t ) = A T∞x T∞ (t ) + BT∞WT∞ (t ) dt

(12.64)

d x h (t ) = A h x h (t ) + BhWh (t ) dt

(12.65)

d xQ (t ) = AQ xQ (t ) + BQWQ (t ) dt

(12.66)

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Temperature (°C)

70 60 50 40 30 20 0

FIGURE 12.4 in sphere.

20

40

60 Time (min)

80

100

120

Realization of AR(1) ambient temperature and corresponding temperature

6

Variance of temperature (°C2)

5 4 3 2 1 0 –1 0

20

40

60 Time (min)

80

100

120

FIGURE 12.5 Temperature variance in sphere subjected to random process ambient temperature. —: variance propagation; ∗: Monte Carlo (nMC=1000); +: Monte Carlo (nMC=100); O: Monte Carlo (nMC=5000).

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 d d k = ρc = 0 dt dt

(12.67)

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90 80

Temperature (°C)

70 60 50 40 30 20

0

20

40

60 Time (min)

80

100

120

FIGURE 12.6 Mean value (−) and 95% confidence interval (∗) for temperature prediction in sphere subjected to random process ambient temperature.

with appropriate initial conditions. Obviously, T0 is modeled as a random variable as well. As with the lumped capacitance problem, a first step is to write the stochastic 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 ⎡u ⎤ ⎢ ⎥ ⎢ x T∞ ⎥ ⎢ ⎥ ⎢ xh ⎥ ⎥ x = ⎢⎢ ⎥ ⎢ xq ⎥ ⎢ k ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ρc ⎥⎦

(12.68)

⎡ C−1 (−Ku + f ) ⎤ ⎢ ⎥ ⎢A x + B W ⎥ T T∞ ∞⎥ ⎢ T∞ T∞ ⎢ ⎥ ⎢ A h xh + Bh W h ⎥ ⎥ g=⎢ ⎢ AQ xQ + BQ W Q ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎢⎣ ⎥⎦

(12.69)

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⎡ 0 ⎢ ⎢B ⎢ T∞ ⎢ 0 h = ⎢⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

⎤ ⎥ ⎥ ⎥ ⎥ 0⎥ ⎥ BQ ⎥ ⎥ 0⎥ ⎥ 0 ⎦⎥

0 0

0 0

Bh 0 0 0

(12.70)

⎡WT∞ − W T∞ ⎤ ⎢ ⎥ w = ⎢⎢ Wh − W h ⎥⎥ ⎢W − ⎥ ⎢⎣ Q W Q ⎥⎦

(12.71)

with 0 null vectors of appropriate dimension, and

Vw,w (τ)  ε[ W(t ) W T (t + τ)] = δ(τ)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

σ W2 ,T∞

0

0

σ W2 ,h

0

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 ⎥⎥ W ,Q ⎥ ⎦

0

0

σ

(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 d u = C−1 (−Ku + f ) dt

(12.73)

⎡ d ∂K ∂C d u T Vu,u = C−1 ⎢⎢−KVu,u − uVu,T k − Vu,ρc dt ∂k ∂ρc dt ⎢⎣ +

⎛ ∂f ∂K ⎞⎟ T ∂f T ∂f T Vu, T∞ + ⎜⎜ − u⎟ V + V ⎜⎝ ∂h ∂h ⎟⎟⎠ u, h ∂Q u, Q ∂T∞

⎡ ∂K ∂C d u T Vu,ρc + ⎢⎢−KVu,u − uVu,T k − c dt ∂ k ∂ρ ⎢⎣ T ⎛ ∂f ∂K ⎞⎟ T ∂f T ⎤⎥ −T ∂f T ⎜ ⎟ V u + V V C + − u, T∞ + ⎜ ⎜⎝ ∂h ∂h ⎟⎟⎠ u, h ∂Q u, Q ⎥ ∂T∞ ⎦

⎞ ⎛ d ∂f Vu,xT = C−1 ⎜⎜−KVu,xT + VT∞,xT ⎟⎟⎟ + Vu ,xT ATT∞ ⎜⎝ ∞ ∞ ∞⎟ ∞ dt ∂T∞ ⎠ ⎤ ⎡⎛ d ∂f ∂K ⎞⎟ Vu,xh = C−1 ⎢⎢⎜⎜−KVu,xh + − u⎟⎟ Vh ,xh ⎥⎥ + Vu ,xh ATh dt ∂h ∂h ⎟⎠ ⎥⎦ ⎢⎣⎜⎝

(12.74)

(12.75)

(12.76)

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⎛ ⎞ d ∂f Vu,xQ = C−1 ⎜⎜−KVu ,xQ + VQ ,xQ ⎟⎟⎟ + Vu ,xQ AQT ⎜ ⎟⎠ dt ∂Q ⎝

(12.77)

⎛ d ∂K ⎞⎟ Vu,k = C−1 ⎜⎜−KVu ,k − σ 2k u⎟ ⎜⎝ dt ∂k ⎟⎟⎠

(12.78)

⎞ ⎛ d 2 ∂C d Vu,ρc = C−1 ⎜⎜−KVu,ρc − σ ρc u⎟⎟⎟ ⎜⎝ dt ∂ρc dt ⎟⎠

(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(t = 0) = u 0

(12.80)

Vu ,u = σ T20 I

(12.81)

where I is an nnod × nnod unity matrix. Further, since the initial temperature is uncorrelated with k, ρc, h, T∞, and Q, the other initial conditions are equal to null matrices 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 nonlinear heat conduction with temperature dependent thermal properties since Equation 12.48 and Equation 12.49 are applicable to general nonlinear systems. The corresponding 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 of the finite element formulation of the (nonlinear) heat conduction equation, it can however be expected to be sufficiently accurate for smooth nonlinear heat conduction 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 d V(t ) = AV(t ) + V(t ) AT + B(t ) dt 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 d V(t ) = AV(t ) + V(t )B + C(t ) dt

(12.82)

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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

d V(t ) + K V(t ) = h(t ) dt

− − with C and Κ the finite element matrices, and V and h vectors of dimension nnod. 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 VxT∞,xT∞, Vxh,xh, and VxQ ,xQ in Equation 12.75 through Equation 12.77 can be computed by straightforward application of the variance propagation algorithm to the Equation 12.64 through Equation 12.66, respectively, which yields for example for T∞ d Vx ,x = A T VxT ,xT + VxT ,xT ATT∞+ BT∞σ T2∞BTT∞ ∞ ∞ ∞ ∞ ∞ dt T∞ 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 T VxT ∞

,x ∞ T∞

+ VxT

,x ∞ T∞

ATT∞+ BT∞σ T2∞ BTT∞ = 0

(12.84)

Similarly, A h Vx h ,xh + Vx h ,xT ATh + Bh σ T2∞ BTh = 0

(12.85)

AQ Vx Q ,xQ + Vx Q ,xT AQT + BQ σ T2∞ BQT = 0

(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 algorithm as in the case of a deterministic problem. For an implicit Euler finite difference algorithm the following recursive relationship can be used ⎛C ⎞ ⎜⎜ + K ⎟⎟ u + Δt − C u = f t t+Δ t ⎟⎟ t ⎜⎝ Δ t Δt ⎠

(12.87)

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TABLE 12.2 Variance Propagation Algorithm u

Step 1

Compute

Step 2

Solve the Lyapunov matrix Equation 12.84 through Equation 12.86

Step 3

Compute

Step 4

Compute Vu,u (t) by solving the Lyapunov matrix differential Equation 12.74

from Equation 12.73 with initial condition (Equation 12.80)

Vu ,xT∞, Vu ,xh , Vu ,xQ , Vu ,k and Vu ,ρc

from Equation 12.75 through Equation 12.79

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 can be calculated by means of the first order perturbation algorithm which was 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 initial condition and thermal conductivity. This fact can be exploited advantageously, as the perturbation algorithm involves only the solution of vector differential equations. Proof The first order perturbation algorithm for heat conduction problems with random 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] d u + Ku = f dt

(12.88)

d ⎛⎜ ∂u ⎞⎟ ∂u ∂K =− ⎜ ⎟+ K dt ⎜⎝ ∂k ⎟⎟⎠ ∂k ∂k

(12.89)

d ⎛⎜ ∂u ⎞⎟ ∂u ⎟+ K =0 ⎜ dt ⎜⎝ ∂T0 ⎟⎟⎠ ∂T0

(12.90)

C

C

C

subject to the initial conditions ∂u =0 ∂k ∂u = [111]T ∂T0 © 2009 by Taylor & Francis Group, LLC

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The covariance matrix is then calculated from

Vu, u =

T T ∂u ⎛⎜ ∂u ⎞⎟ 2 ∂u ⎛⎜ ∂u ⎞⎟ 2 σ T0 σ + ⎟ ⎟ k ⎜ ⎜ ∂k ⎜⎝ ∂k ⎟⎠ ∂T0 ⎜⎝ ∂T0 ⎟⎠

(12.91)

Differentiation of Equation 12.91 with respect to time yields T T ⎛ ∂u ⎞ d ⎛ ∂u ⎞T d d ⎛ ∂u ⎞⎛ ∂u ⎞ d ⎛ ∂u ⎞⎟⎛⎜ ∂u ⎞⎟ 2 Vu , u = ⎜⎜⎜ ⎟⎟⎟⎜⎜⎜ ⎟⎟⎟ σ 2k + ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ σ 2k + ⎜⎜ ⎟⎟⎜ ⎟⎟ σ T0 ⎝ ∂k ⎠ dt ⎝ ∂k ⎠ dt dt ⎝ ∂k ⎠⎝ ∂k ⎠ dt ⎜⎝ ∂T0 ⎠⎜⎝ ∂T0 ⎠

⎛ ∂u ⎞⎟ d ⎛ ∂u ⎞⎟T 2 + ⎜⎜ ⎟ ⎜ ⎟ σ ⎜⎝ ∂T0 ⎟⎠ dt ⎜⎜⎝ ∂T0 ⎟⎠ T0

(12.92)

Further, Equation 12.89 and Equation 12.90 can be rearranged as

d ⎛⎜ ∂u ⎞⎟ ∂u ∂u ∂K −1 u − C−1 K − C−1 ⎜⎜ ⎟⎟⎟ = −C K ∂k dt ⎝ ∂k ⎠ ∂k ∂k

(12.93)

d ⎛⎜ ∂u ⎞⎟ ∂u ⎟ = −C−1 K ⎜ dt ⎜⎝ ∂T0 ⎟⎟⎠ ∂T0

(12.94)

Substitution of Equation 12.93 and Equation 12.94 in Equation 12.92 results in

⎛ ∂u ∂K ⎞⎟⎛ ∂u ⎞⎟T 2 ∂u ⎛ ∂u ∂K ⎞⎟ −T 2 d u⎟ C σ k Vu , u = −C−1 ⎜⎜⎜K + u⎟⎜⎜ ⎟ σ k − ⎜⎜⎜K + ⎝ ∂k dt ∂k ⎟⎠⎜⎝ ∂k ⎟⎠ ∂k ⎝ ∂k ∂k ⎟⎠ − C−1 K

T T ∂u ⎛⎜ ∂u ⎞⎟ 2 ∂u ⎛⎜ ∂u ⎞⎟ T −T 2 K C σ T0 − σ ⎟ ⎟ T0 ⎜ ⎜ ∂T0 ⎜⎝ ∂T0 ⎟⎠ ∂T0 ⎜⎝ ∂T0 ⎟⎠

(12.95)

The perturbation algorithm is based on the following first order Taylor expansion of u( k,T0 ) around u u ≅ u+

∂u ∂u Δk + ΔT0 ∂k ∂T0

(12.96)

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where Δk = k − k ΔT0 = T0 − T0

From Equation 12.96 it follows that

Vu ,k = ε[(u − u)( k − k )] =

∂u 2 σk ∂k

Vu,T0 = ε[( u − u )(T0 − T0 )] =

∂u 2 σ T0 ∂T0

(12.97)

(12.98)

After substitution of Equation 12.91, Equation 12.97 and Equation 12.98 in Equation 12.95 the following Equation is obtained

T ⎛ ⎞ ⎛ d ∂K ∂K T ⎞ ⎟⎟ C−T Vu,u = −C−1 ⎜⎜⎜KVu,u + uVu,T k ⎟⎟⎟ − ⎜⎜⎜KVu,u + uVu,k ⎟⎠ ⎝ ⎠ ⎝ dt ∂k ∂k

(12.99)

Further, right multiplication of Equation 12.93 by σ 2k and using Equation 12.97 gives d ∂K 2 Vu,k = −C−1 KVu ,k − C−1 uσ k dt ∂k

(12.100)

Equation 12.99 and Equation 12.100 are equivalent to Equation 12.74 and Equation 12.78 for the given stochastic specifications. This concludes the proof.

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 computed by differentiation of the element matrices and subsequent incorporation in the global derivative matrices. The following expressions are easily derived from Equation 12.7 through Equation 12.9.

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∂C j = ∂ρC

∫φφ j

V

12.6

jT

dV

j

∂K j = ∂k

∫

∂K j = ∂h

∫φφ

T

B j B j dV

Vj j

jT

dS

Sj

∂f j = ∂h

∫T

∂f j = ∂T∞

∫ hφ dS

∂f j = ∂Q

∫ φ dV

∞

φ j dS

Sj

j

Sj

j

Vj

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 V(t ) = A(t )V(t ) + V(t )B(t ) + C(t ) dt

(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 be applied for the numerical solution of matrix differential equations. A first order explicit algorithm is obtained by substitution of the differential operator in Equation 12.101 by a first order forward difference operator: 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.

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An unconditionally stable implicit algorithm is obtained by substitution of the differential operator in Equation 12.101 by a first order backward difference operator: ⎡1 ⎤ ⎡1 ⎤ B (t + Δt )⎥ ⎢ I − Δt At (t + Δt )⎥ V(t + Δt ) + V (t + Δt ) ⎢ I − ΔtB ⎥⎦ ⎢⎣ 2 ⎥⎦ ⎣⎢ 2 = V(t ) + ΔtC (t + Δt )

(12.103)

where the equality V = 12 V + 12 V is used. Equation 12.103 can be written as DX + XE = F

(12.104)

with ⎡1 ⎤ D = ⎢ I − Δt A(t + Δt )⎥ ⎢2 ⎥ ⎣ ⎦ ⎡1 ⎤ E = ⎢ I − Δt B (t + Δt )⎥ ⎢2 ⎥ ⎣ ⎦ F = V(t ) + ΔtC (t + Δt ) X = V(t + Δt ) 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 transformation U: ⎡ D1′,1 ⎢ ⎢ D′ D′ = U T DU = ⎢⎢ 2,1 ⎢  ⎢ ′ ⎢⎣ Dr ,1

0 D′2,2 Dr′ ,2

   

0 ⎤⎥ 0 ⎥⎥  ⎥⎥ ⎥ Dr′ ,r ⎦⎥

where the diagonal submatrices D′i ,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′: ⎡ E1,1 ′ ⎢ ⎢ 0 E ′ = U ′T EU ′ = ⎢⎢ ⎢  ⎢ ⎢⎣ 0

′ E1,1 E ′2,2 0

   

E1,′ s ⎤⎥ E ′2,s ⎥⎥  ⎥⎥ ⎥ E ′s ,s ⎥⎦

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where again E′i ,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)

F′ = UT FU′

(12.106)

X′ = UT XU′

(12.107)

with

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 backtransformed 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 stochastic heat conduction problem with random variable initial temperature and random 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 random process ambient and initial temperature is described elsewhere [41]. From Equation 12.74 it follows that the variance propagation algorithm for linear heat conduction problems with random field initial temperature is given by the following Lyapunov equation: d V(t ) = AV(t ) + V(t ) AT dt V(t) = V0

at

t=0

(12.108) (12.109)

with A = −C−1 K A and V are square matrices of dimension nnod × nnod. Note that A is constant if the surface heat transfer coefficient does not change over time. Equation 12.108, 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

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popular methods for the solution of differential equations. The convergence and stability theory of linear multistep methods is well established for scalar and vector 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 W = ΛW + WA dt

(12.111)

W  H−1 VH−T

(12.112)

where

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 V(t ) = F(t , V) dt with an appropriate initial value. A general linear multistep matrix method of order k is defined as k

∑ l =0

k

α l V ′(tn+l ) = Δt

∑ β F′(t l

n+l

)

(12.114)

l =0

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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 ); and αl and βl are the coefficients of the method. For both the explicit and the implicit Euler method k = 1. The values of the coefficients α1 and β1 are given in Table 12.3. An 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 κ

k

∑

α l V ′(tn+l ) = Δt

l =0

∑ β ⎡⎣⎢ AV′(t l

n+l

l =0

) + V ′(tn+l ) AT ⎤⎥ ⎦

(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 exact solution V(t), t ∈[0,tf]. The time final t f is hereby kept constant, so that at the same time n → ∞. This can be stated more precisely as follows: Definition 1 The linear multistep method defined by Equation 12.115 is said to be convergent if lim V′(tn ) = V(tn ) Δt → 0

(12.116)

tn fixed

holds for all n and for all starting values V′(tl) for which lim V′(tl ) = V(t l ), l = 0, k –1 Δt→ 0

(12.117)

The conditions for convergence of the linear matrix multistep method are summarized in the following theorem. Theorem 1 The method defined by Equation 12.115 is convergent if and only if it is consistent k

∑

k

α l = 0;

l =0

∑

k

lα l =

l =0

∑β

(12.118)

l

l =0

and zero-stable, which means that no root ξi of the characteristic polynomial k

∑α ξ = 0 l l i

(12.119)

l =0

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 k

∑ l =0

k

α l V′* (tn+l ) = Δt

∑ β A′ (t *

l

n+l

)

(12.120)

l =0

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where ⎡ V1′ ⎤ ⎢ ⎥ V′ = ⎢⎢  ⎥⎥ ⎢ V′ ⎥ ⎢⎣ nnod ⎥⎦ *

⎡ ⎢ AV1′ + ⎢ A′* = ⎢⎢ ⎢ ⎢ AV′ + ⎢⎣ nnod

⎤ Vi′A1,i ⎥ ⎥ ⎥  ⎥ ⎥ nnod Vi′A nnod,i ⎥⎥ i=1 ⎦

∑

(12.121)

nnod

i=1

(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 lim V′* (tn ) = V* (tn ) Δt → 0

(12.123)

tn fixed

for all tn, n = 1,2,…, provided that the initial sequence is chosen such that lim V∗ (tl ) = V′∗ (tl ), l = 0, …, k − 1

Δt →0

(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 Δt tends to zero, it is also interesting to investigate whether for a fixed time step the 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−1 K are real and negative Proof Let λ and x be an eigenvalue–eigenvector pair of A. Then, by definition, Ax = −C−1 Kx = λx Left multiplication of both sides by x T C yields −x T Kx = x T Cxλ Since K and C are both positive definite, the following relations hold for any vector x (real or imaginary) and thus also if x is an eigenvector x T Kx > 0

(12.125)

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x T Cx > 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 The following stability definition is now stated. Definition 2 The linear multistep matrix method (Equation 12.115) is said to be absolutely stable if, for a given Δt, the approximate solution V′(tn) of Equation 12.108 satisfies ||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 k

∑ [α − Δtβ (λ + λ )] p l

l

i

l

j

l =0

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 k

∑ l =0

⎡⎛ ⎢⎜⎜ ⎢⎜⎜ ⎢⎜⎜ ⎢⎜ ⎢⎜⎝ ⎣

⎛

⎞

⎞⎤

⎟⎟ ⎟⎟⎥ ⎜⎜ α αl − Δtβl Λ ⎟⎟⎟⎟⎟⎟ W′(tn+l ) + W′(tn+l ) ⎜⎜⎜⎜ l − Δtβl Λ ⎟⎟⎟⎟⎟⎟⎥⎥⎥ = 0 ⎜⎜ 2 ⎟⎠ ⎟⎠⎥ 2 ⎝ ⎦

(12.128)

with W′ defined by W′  H−1V ′H−T

(12.129)

Equation 12.128 represents an uncoupled set of nnod × nnod equations: k

∑ l =0

⎛⎜ ⎞⎟ ⎞⎟ ⎤ ⎡⎛⎜⎜ α l α ⎢⎜⎜⎜ − Δtβl λ i ⎟⎟⎟⎟⎟ W ′i, j(tn+l ) + ⎜⎜⎜⎜ l − Δtβl λ j ⎟⎟⎟⎟⎟ W ′i, j(tn+l ))⎥ = 0 ⎟⎠ ⎟⎠ ⎜⎝ 2 ⎥⎦ ⎢⎣⎜⎝ 2

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or k

∑ ⎡⎣⎢α − Δtβ (λ + λ )W ′ l

l

i

j

i, j

l =0

(tn+l )⎤⎦⎥ = 0,

i, j = 1, …, nnod

(12.130)

By Equation 12.129, ||V′|| → 0 as n → ∞, if and only if || W′|| → 0 as n → ∞, and hence Equation 12.127 is satisfied if and only if all solutions W′i,j (tn) of Equation 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 k

W ′i, j(tn ) =

∑c

i, j,l

pin, j,l

(12.132)

l=1

where ci,j,l are arbitrary coefficients, and pi,j,l, l = 1,…, k are the roots of the polynomial k

∑ [α − Δtβ (λ + λ )] p l

l

i

l

j

(12.133)

l =0

Clearly, Equation 12.131 and consequently Equation 12.127 are satisfied if the roots 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 p=−

α 0 − Δtβ 0 (λ i + λ j ) α1 − Δtβ1 (λ i + λ j )

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)

1 <1 1 − Δt (λ i + λ j )

(12.135)

Implicit:

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

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TABLE 12.3 Coefficients of the Linear Multistep Method for the Explicit and Implicit Euler Method Explicit

Implicit

α1

−1 1

−1 1

β0

1

0

β1

0

1

α0

that the stability conditions for the implicit Euler method will always be satisfied as 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 cylindrical container (radius r0 =3.41 cm, height L= 10.02 cm) filled with 30% solids content tomato concentrate with k = 0.542 W/m°C, ρc = 3.89106 J/m3°C. The following process conditions were applied: T0 = 65°C, h = 100 W/m2°C. The ambient temperature is now described by means of an AR(1) process with T∞= 125o C, σ T∞ = 1o C and a1 = 0.00277 s–1. An implicit Euler finite difference method in the time domain was used to integrate the differential systems. For the finite element analysis the region [0,r0] × [0,L/2] is subdivided in 100 axisymmetric linear quadrilateral elements. The time step is set equal to 36 s. The Monte Carlo and variance propagation algorithms were programmed on top of the existing finite element code DOT [44]. In Figure 12.7 the temperature variance at three different positions in the centerplane 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 value an excellent agreement between the different method was observed (figure not shown). The relative CPU time (total CPU time divided by CPU time for deterministic 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 © 2009 by Taylor & Francis Group, LLC

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Temperature variance (°C2)

0.5

337

r = 3.41 cm

0.4 0.3 0.2

r = 1.71 cm

0.1

r = 0.0 cm

0.0 0

900

1800 Time (s)

2700

3600

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.

the heat transfer model for artificially generated random parameter samples. Straightforward statistical analysis of the simulation results yields the mean values and variances 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 conduction 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 specification of the random parameters, while for the variance propagation algorithm 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 function. Also, as the variance propagation algorithm are essentially based on a linearization of the governing equations around their mean solution, they are only applicable if the variability is relatively small (coefficient of variation smaller than 0.2).

ACKNOWLEDGMENTS The European Communities (FAIR project FAIR–CT96-1192) and the Flemish Minister of Science and Technology are gratefully acknowledged for financial support.

NOMENCLATURE ai c C ε f f g

Coefficient of autoregressive process or wave Heat capacity Finite element capacity matrix Mean value operator Probability density function Finite element thermal load vector Vector valued function

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h h k K L n nnod nMC 0 r0 S t T T0 T∞ u V V W X Y z Z Δt Γ ρ σ τ

Food Processing Operations Modeling: Design and Analysis

Surface heat transfer coefficient (W/m2 °C) Vector valued function Thermal conductivity (W/m°C) Finite element stiffness matrix Half-height of can Outward normal Number of nodes Number of Monte Carlo runs Zero matrix Radius (m) Surface Time (s) Temperature (°C) Initial temperature (°C) Retort temperature (°C) Nodal temperature vector Covariance function, volume Covariance matrix White noise process Random vector Auxiliary random process/wave Position vector Discrete time white noise process Time step Convection surface Density (kg/m3) Standard deviation Separation time

REFERENCES 1. A.A. Teixeira, J.R. Dixon, J.W. Zahradnik, and G.E. Zinsmeister. 1969. Computer optimization of nutrient retention in thermal processing of conduction-heated foods. Food Technology, 23(6): 137–40. Cited by Teixeira et al. (1989). 2. D. Naveh, I.J. Kopelman, and I.J. Pflug. 1983. The finite element method in thermal processing of foods. Journal of Food Science, 48: 1086–93. 3. A.K. Datta and A.A. Teixeira. 1987. Numerical modeling of natural convection heating in canned liquid foods. Transactions of the ASAE, 30(5): 1542–51. 4. A.K. Datta and A.A. Teixeira. 1988. Numerically predicted transient temperature and velocity profiles during natural convection heating of canned liquid foods. Journal of Food Science, 53(1): 191–95. 5. D. Naveh. 1982. Analysis of transient conduction heat transfer in the thermal processing of foods using the finite element method. PhD thesis, University of Minnesota, USA, September. 6. A.K. Datta, A.A. Teixeira, and J.E. Manson. 1986. Computer-based control logic of on-line correction of process deviations. Journal of Food Science, 51: 480–83. 7. J.R. Banga, J.M. Perez-Martin, J.M. Gallardo, and J.J. Casares. 1991. Optimization of the thermal processing of conduction-heated canned foods: Study of several objective functions. Journal of Food Engineering, 14: 25–51. © 2009 by Taylor & Francis Group, LLC

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