6 Modelling and simulation of thermal...

6.1 Introduction

Thermal food processes are one of the few production processes in industrywhich rely on a mathematical model to ensure the safety of the process.1 For aproper process design, the food centre temperature must be known during thesubsequent stages of the preparation process, so that the effect of the thermaltreatment on the microbiological and sensory quality can be evaluated using well-established methods. Mathematical process models and simulation softwarerepresent a powerful alternative to the traditional, time-consuming temperaturemeasurements and quantitative microbiological and food quality analyses.

The objective of this chapter is to give an overview of the existing models forconduction and convection heat transfer during food manufacture. As thesemodels can only be solved for very simple problems, numerical solution isusually mandatory. A wide variety of numerical techniques and correspondingsoftware is now available to solve these models.

The outline of the chapter is as follows. In Section 6.2 the Fourier equationfor conduction heat transfer will be introduced, along with its correspondingboundary and initial conditions. Some analytical solutions will be given. TheNavier-Stokes equations which describe convective transport phenomena will bedescribed in Section 6.3. It will be shown how the transport equations can bemodified to take into account turbulence effects. Several types of boundaryconditions which are relevant to food processes will be described as well. InSection 6.4 several numerical methods to solve heat and mass transfer problemswill be introduced, including the finite difference, finite element and finitevolume methods. An overview will be given of commercially available softwarepackages. In section 6.5 some illustrative examples will be given.

6

Modelling and simulation of thermalprocessesB.M. Nicolaı, P. Verboven and N. Scheerlinck, KatholiekeUniversiteit, Leuven

6.2 Modelling of conduction heat transfer: the Fourier equation

6.2.1 DerivationHeat transport in solids is governed by lattice waves induced by atomic motionand is called conduction. Transient heat conduction in an isotropic object � withboundary � is governed by the Fourier equation2

�c�T�t

� �k�T � Q on � �6�1�where � is density (kg m�3), c is heat capacity (J kg�1 ºC�1), k is thermalconductivity (W m�1 ºC�1), Q is volumetric heat generation (W m�3), T istemperature (ºC), t is time (s), and xi is coordinate (m).

The thermophysical parameters k, �, and c may be temperature dependent sothat the problem becomes nonlinear. Thermophysical properties of variousagricultural and food products are compiled in various reference books (e.g., thecompilation made by the ASHRAE).3 Further, equations have been publishedwhich relate the thermophysical properties of agricultural products and foodmaterials to their chemical composition. In general, both the heat capacity andthe density can be calculated with sufficient accuracy, but the models for thethermal conductivity require some assumptions about the orientation of thedifferent main chemical constituents with respect to the direction of heat flowwhich is not always obvious.

In conventional thermal food processes the heat generation Q is zero.However, in the case of volumetric heating techniques such as microwave andohmic heating, Q is the driving force of the heat transfer. The modelling of thesetechniques is a very active research area.4–7

The initial condition for the Fourier equation can be described as a spatialdependent function at time t � 0:

T�x� y� z� t� � T0�x� y� z� at t � 0 �6�2�At the boundary � of the heated or cooled object, fixed temperature (Dirichlet),convection or radiation conditions may apply:

T�x� y� z� t� � f �x� y� z� t�k

�

�nT � h�T � T� � ��T4

� T4� on � �6�3�with f (x, y, z, t) a known function (e.g., it was measured, or it is known fromcontrol procedures), n the outward normal to the surface, h the convectioncoefficient (W/m2 ºC), T the (known) ambient temperature, � the emissioncoefficient, and � the Stefan-Boltzmann constant. The surface heat transfercoefficient h must be considered as an empirical parameter.

6.2.2 Analytical solutionsEquation [6.1] can be solved analytically under a limited set of initial andboundary conditions for simple geometries only. Several solution techniques

92 Thermal technologies in food processing

such as separation of variables, Green functions and variational methods arediscussed in the many books on partial differential equations.10, 11 A largenumber of analytical solutions of the Fourier equation were compiled byCarslaw and Jaeger.12

Usually the Fourier equation is rewritten in dimensionless coordinates byintroducing a dimensionless temperature � and a dimensionless time Fo which iscalled the Fourier number

� � T � TT0 � T

�6�4�

Fo � kt��cL2 �6�5�with L a characteristic length, e.g., the half-thickness of a slab. For differentgeometries such as slab, cylinder and sphere, it can be shown that there exists alinear relationship between the logarithm of � and Fo. For example, for a slab ofhalf-thickness L subjected to convection boundary conditions, � is given by

��n�1

4 sin�n�2�n � sin�2�n� exp��2

n Fo�cos��x� �6�6�

and the discrete values of n are positive roots of the transcedental equation

n tan �n� � Bi �6�7�where the Biot number Bi is defined as

Bi � hLk

�6�8�For Fo � 0�2, it can be shown that the infinite series in equation [6.6] can beapproximated by the first term of the series. The graphical representation of theresulting relationship is commonly known as a Heissler chart and can be foundin any standard textbook on heat transfer.13

6.3 The Navier–Stokes equations

6.3.1 Conservation equationsIn fluids, transport of heat and mass is more complicated than in solid foods, asbesides diffusion also convective transport of liquid particles may take place.The driving force behind convective transport is a pressure gradient in the caseof forced convection, e.g. due to a fan in an oven, or density differences becauseof, e.g. temperature gradients. Navier and Stokes independently derived theequations for convective transport which now bear their names. For simplicitywe will restrict the discussion to a single Newtonian fluid system. This meansthat we will only consider fluids for which there is a linear relationship betweenshear stress and velocity gradient, such as water or air. More complicated fluidssuch as ketchup, starch solutions, etc., are so-called non-Newtonian fluids, andthe reader is referred to standard books on rheology for more details.14

Modelling and simulation of thermal processes 93

When we apply the conservation principle to a fixed infinitesimal controlvolume dx1dx2dx3 we obtain the continuity, momentum and energy equations,written in index notation for Cartesian coordinates xi (i� 1, 2, 3 for the x-, y- andz-direction, respectively), and whenever an index appears twice in any term,summation over the range of that index is implied (for example, ��uj��xj

becomes ��u1��x1� � ��u2��x2� � ��u3��x3�):��

�t� ��uj

�xj� 0 �6�9�

��ui

�t� ��ujui

�xj� �

�xj�

��ui

�xj� �uj

�xi

��

�xi

�p � 2

3��uj

�xj

�� fi �6�10�

��H�t

� ��ujH�xj

� �

�xj

�k�T�xj

�� p

�t� Q �6�11�

where ui (i� 1, 2, 3) Cartesian components of the velocity vector U (m s�1), T istemperature (ºC), H is static enthalpy (J kg�1), p is pressure (Pa), � is density(kg m3), k is thermal conductivity (W m�1 ºC�1), � is dynamic viscosity(kg m�1 s�1), fi is external body forces (N m�3), and Q is heat source or sink(W m�3). For a full derivation of these equations we refer to any textbook onfluid mechanics.15, 16

The system of five equations (Eq. [6.9]–[6.11], three equations for thevelocity components plus the continuity and the energy equation) contains sevenvariables (u1, u2, u3, p, h, T, �). We therefore need additional equations to closethe system. The thermodynamic equation of state gives the relation between thedensity � and the pressure p and temperature T. The constitutive equation relatesthe enthalpy h to the pressure and the temperature. For an ideal gas we can usethe following equations:

� � pMRT

�6�12�

c ��H�T

�p

�6�13�

with M the molecular weight of the fluid (kg mol�1) and R the universal gasconstant (J mol�1 K�1). When the heat capacity is assumed constant, theconstitutive equation reduces to a linear relation between H and the differencebetween the actual temperature T and a reference temperature. Since onlyrelatively low velocities are encountered in the food processes underconsideration, the flow is often assumed incompressible and these equationscan be applied.

For isothermal fluids we can assume that the density � is constant so that thecontinuity equation vanishes. In the case of non-isothermal flows the Boussinesqapproximation is often applied, in which it is assumed that density is the onlyparameter which depends on the temperature.16


6.3.2 TurbulenceMany heat transfer processes in food operations often involve turbulent flow ofair or water. Turbulence can be induced by the presence of flow obstructionssuch as baffles, shelves and the foods themselves. Turbulence is a state of theflow which is characterised by fluctuations of the flow variables (eddies) over alarge range of scales, both in time and space. This complex pattern of motionenhances heat transfer rates considerably but also causes additional pressuredrops which must be taken into account in the design of the equipment.Turbulence must therefore be incorporated in the governing models unless alaminar flow regime can be guaranteed.

Although the Navier–Stokes equations are general conservation equationswhich are equally well applicable to turbulent flow, the large variation of spatialscales introduces severe numerical problems, and only for simplified cases andlow Reynolds numbers is it currently possible to perform such direct numericalsimulations on supercomputers.17 Simulation shortcuts are possible at differentlevels of complexity and approximations. The least approximations are neededin large eddy simulations, in which case the largest eddies are resolved but theeffects of smaller eddies are estimated by additional models.16 This approach isnow being used more widely, since it is almost within reach of current computerpower.

The most popular approach is based on the Reynolds Averaged Navier–Stokes (RANS) equations, which are obtained from averaging out the governingequations (Eq. 6.10) and including the effect of the turbulent fluctuations byadditional models for the new terms appearing in the RANS equations. In theBoussinesq approach, the turbulence is accounted for by a turbulent ‘viscosity’which is incorporated in the viscous and thermal diffusion transport terms. In K–� models, originally proposed by Jones and Launder, the turbulent viscosity isobtained as a function of the turbulent variables K, which represents theturbulent kinetic energy associated with the fluctuating components of the flowvelocities, and �, the turbulent energy dissipation rate:18

�t � �C�K2

��6�14�

The constant C� may be assumed constant for equilibrium conditions, where theturbulence production nearly equals the turbulence dissipation.

Additional transport equations have been derived for these turbulent flowvariables. Several undefined constants appear in the model equations, whichtogether with several assumptions and the specific near-wall treatment renderthis model empirical. There are three popular K–� models, namely the standardk–� model, a RNG (Renormalisation Group) k–� model and a LRN (LowReynolds Number) k–� model.18–20 Verboven et al. compared these threeturbulence models for a typical forced convection heating process of complexlyshaped foods, and concluded that the boundary layers are badly represented bythe wall function approach and the departure from local equilibrium is notaccounted for.21 A correction function can be added to correct for the latter


behaviour in conjunction with the Low Reynolds Number model (see the workof Yap).22 Nevertheless, it was found that experimental input for thesecorrections is needed in order to determine important constants.

More complex closures for the RANS models are based on dynamicequations for the Reynolds stresses and fluxes themselves in the RANSequations. In addition to the equations for the mean flow, this approach results inseven more partial differential equations. These models are believed to be moreaccurate but require a better insight into the process of turbulence and care mustbe taken with their numerical solution. Finally, it must be noted that newturbulence models are constantly proposed and tested.

6.3.3 Initial and boundary conditionsUnlike the diffusion equations, there are no conclusive general rules for theimplementation of boundary conditions for the Navier–Stokes equations in orderto have a well-posed problem because of their complex mathematical nature. Fora full account, we refer to Hirsch.23 For incompressible and weaklycompressible flows, it is possible to define Dirichlet boundary conditions (fixedvalues of the variables, mostly upstream), Neumann boundary conditions (fixedgradients, mostly downstream) and wall boundary conditions (a wall functionreflecting the behaviour of the flow near the wall). Initial values must beprovided for all variables.

Difficulties arise when the exact conditions are unknown. This is especiallytrue in turbulent flows, where the exact values of the turbulence energy andenergy dissipation rate are often unknown at the inlet, and need to be guessedusing information about the velocity and the flow geometry. The direction of theflow at boundaries may be difficult to specify, but may have considerableinfluence when the flow contains swirls. The effect of the pressure resistance(e.g. in a cool room) on the fan flow rate may be considerable and cannot alwaysbe taken into account appropriately. In any case a sensitivity analysis can beuseful to obtain an error estimate associated with approximate or guessedboundary conditions.

6.3.4 Additional equationsIn the case of an air flow through a bulk of products (e.g. cooling of horticulturalproducts), or a flow of multiple fluids (e.g. the dispersion of disinfectants in acool room or the injection of water for air humidification), the separate phasesneed to be considered. Depending on the flow conditions, a multi-phasemodelling or a mixed-fluid modelling approach can be applied. The reader isreferred to the literature for more details.24

When the problem contains chemical kinetics (e.g. microbial inactivation), anadditional transport equation must be introduced. Further, the chemical reactionmust be solved. Therefore the reactions rates, property changes and heat releasesmust be calculated as part of the solution. The heat of reaction can be calculated


from the heats of formation of the species and depends on temperature. Thereaction leads to sources/sinks in the species conservation and energy equations.For more details, see reference 25.

6.4 Numerical methods

6.4.1 Numerical discretisationFor realistic – and thus more complicated – heat and mass transfer problemsusually no analytic solution is available, and a numerical solution becomesmandatory. For this purpose the problem is reduced significantly by requiring asolution for a discrete number of points (the so-called grid) rather than for eachpoint of the space-time continuum in which the heat and mass transfer proceed.The original governing partial differential equations are accordingly transformedinto a system of difference equations and solved by simple mathematicalmanipulations such as addition, subtraction, multiplication and division, whichcan easily be automated using a computer. However, as a consequence of thediscretisation the obtained solution is no longer exact, but only an approxima-tion of the exact solution. Fortunately, the approximation error can be decreasedsubstantially by increasing the number of discretisation points at the expense ofadditional computer time.

Various discretisation methods have been used in the past for the numericalsolution of heat conduction problems arising in food technology. Among themost commonly used are the finite difference method, the finite elementmethod, and the finite volume method. It must be emphasised that – particularlyin the case of nonlinear heat transfer problems – the numerical solution mustalways be validated. It is very well possible that a plausible, convergent butincorrect solution is obtained. At least a grid dependency study must be carriedout to verify whether the solution basically remains the same when thecomputational grid is refined.

6.4.2 The finite difference methodPrincipleThe finite difference method is the oldest discretisation method for thenumerical solution of differential equations and had been described as long agoas 1768 by Euler. The method is based on the approximation of the derivativesin the governing equations by the ratio of two differences. For example, the firsttime derivative of some function T(t) at time ti can be approximated by

dTdt

��ti

� T�ti�1� � T�ti��t

�6�15�

with �t � ti�1 � ti. This expression converges to the exact value of the deriva-tive when �t decreases. The power of �t with which the so-called truncation


error decreases is called the order of the finite difference approximation, and canbe obtained from a Taylor series approximation of T at time ti.

Equation [6.15] is called a forward difference as it uses the future value of thefunction and it is of order 1. A backward difference of order 1 is given by

dTdt

��ti

� T�ti� � T�ti�1��t

�6�16�

Equations [6.15] and [6.16] are also called forward and backward Eulerschemes. Likewise, finite difference formulas can be established for secondorder derivatives. The so-called central difference formula is of order 2 and isdefined by

d2Tdt2

��ti

� T�ti�1� � 2T�ti� � T�ti�1��t2

�6�17�

The finite difference method will be illustrated for a two-dimensional heatconduction problem. For this purpose the computational domain is subdivided ina regularly spaced grid of lines which intersect at common nodal points (Fig.6.1). Subsequently, the space and time derivatives are replaced by finitedifferences. For example, if central differences are used it is easy to see that thefollowing expression is obtained for the Fourier equation:

�Ti� j

�t� k�t

�c

�Ti�1� j � 2Ti� j � Ti�1� j

��x�2 � Ti� j�1 � 2Ti� j � Ti� j�1

��y�2

��6�18�

Similar equations can be established for all interior nodes of the grid, andspecial procedures are available to discretise the boundary conditions in thenodes which are on the boundary of the grid. The large number of equations

Fig. 6.1 Finite difference grid of a two-dimensional rectangular region. The nodeswhich are involved in the computation of the temperature at position (i,j) are indicated by

dots.


(equal to the number of nodal points) can conveniently be ordered into adifferential system of the general form

Cddt

u � Ku � f �6�19�with u� [u1 u2 uN]T the nodal temperature vector. This vector differentialequation can be discretised in time, and typically leads to a system of algebraicequations which must be solved by appropriate means. The system matricescontain many zeros, and this feature can be exploited advantageously to reducethe required number of computations.

6.4.3 The finite element methodPrincipleIn the finite element method,26 a given computational domain is subdivided as acollection of a number of finite elements, subdomains of variable size and shape,which are interconnected in a discrete number of nodes. The solution of thepartial differential equation is approximated in each element by a low-orderpolynomial in such a way that it is defined uniquely in terms of the(approximate) solution at the nodes. The global approximate solution can thenbe written as a series of low-order piecewise polynomials with the coefficientsof the series equal to the approximate solution at the nodes. Substitution of theapproximate solution in the differential equation produces in general a non-zeroresidual. In the Galerkin method, the unknown coefficients of the low-orderpiecewise polynomials are then found by orthogonalisation of this residual withrespect to these polynomials. This results in a system of algebraic or ordinarydifferential equations which can be solved using the well-known techniques.

The Galerkin finite element methodA first step in the construction of a finite element solution of a partial differentialequation is the subdivision of the computational domain in a grid of finiteelements, which are interconnected at a discrete number of common nodalpoints. The elements may be of arbitrary size and shape. A large number ofelement shapes have been suggested in the literature and are provided in mostcommercial finite element codes. Typical 2D and 3D element shapes are shownin Figs 6.2 and 6.3.

The unknown solution is expressed in each element as a piecewise continuouspolynomial in the space coordinates with the restrictions that (i) continuitybetween elements must be preserved and (ii) any arbitrary linear function couldbe represented.26 In general, the unknown temperature field T (x, y, z, t) can thenbe approximated by

T�x� y� z� t� � NT�x� y� z�u�t� �6�20�with N a vector of so-called shape functions and u a vector containing thetemperatures at the nodes of the finite element grid. In general the approximate


temperature field T is not identical to T, and when T is substituted in the heatconduction equation, a non-zero residual r is obtained:

r � �c�T�t

��k�T � Q �6�21�

This residual is subsequently orthogonalised with respect to the shape functionsN: �

�

N��c

�T�t

��k�T � Q

�d� � 0 �6�22�

It can be shown that after the application of Green’s theorem and some matrixalgebra a system of the form [6.19] is obtained.26–28 C and K are now called thecapacitance matrix and the stiffness matrix, respectively; f is the thermal loadvector. The matrices C, K and f are constructed element-wise. As in the case ofthe finite difference method, the system [6.19] is solved using traditional finite

Fig. 6.2 Typical 2D and 3D finite element shapes.


difference methods. Note that K and C are positive definite, symmetric andbanded. These very important features can be exploited advantageously tosignificantly reduce the computational effort and memory requirements.

Special attention has been paid recently to stochastic finite element methodswhich were developed to take into account random variability of product andprocess parameters.29, 30, 31

6.4.4 The finite volume methodPrincipleThe finite volume method of discretisation is most widely used in commercialCFD (computational fluid dynamics) codes at the moment. It owes its popularityto the fact that it obeys the clear physical principle of conservation on thediscrete scale. The concepts of the method are easy to understand and havephysical meaning.

The system of general conservation equations can be written in coordinate-free notation and integrated over a finite control volume V with surface A.Applying Gauss’s theorem to obtain the surface integral terms, the equationshave the following form, with the transported quantity:�

V

��

�tdV �

�A��U � ndA �

�A�� ndA �

�V

S dV �6�23�

This equation states the conservation principle on a finite scale for all relevantquantities in the system when the surface integrals are the same for volumes

Fig. 6.3 3D finite element grid for a food container. Because of symmetry reasons onlya quarter of the food container needs to be modelled.


sharing a boundary. Moreover, the finite volume form of the model becomesindependent on the coordinate system. When the physical domain is subdividedinto control volumes, a grid only defines the boundaries of the volumes. This isadvantageous for modelling complex geometries.

The volume integrals are approximated in terms of the volume-centered valueof . The values at the volume faces are required for solving the surfaceintegrals in equation [6.23]. This requires interpolation in terms of volume-centered values. Some interpolation schemes may be highly accurate, butproduce unbounded solutions when grids are too coarse. Others areunconditionally stable, but have a low accuracy and produce erroneous resultscalled numerical or false diffusion. The reader is referred to the literature for amore elaborate discussion about the limits and benefits of differentapproximating formulas.15, 16 The time discretisation in the control volumemethod is carried out using finite differences in the time domain, explainedabove.

Solution of the discretised equationsDiscretisation results in the following set of equations, in matrix-vector notation:

A� � Q �6�24�where A is a square sparse matrix containing the coefficients resulting from thediscretisation, � is a vector containing the unknowns at the control volumecentres and Q is a vector containing the variable-independent source terms.Equation [6.24] is still non-linear: the flow variables appear in the coefficients.An iterative method is therefore required in which the non-linear terms have tobe linearised. The least expensive and most common approach is the Picarditeration. In this method coefficients are updated using the most recent solutionof the system. This approach requires more iterations than Newton-like methods,which use a Taylor series expansion, but do not involve the computation ofcomplex matrices and are found to be much more stable.

The solution of the linearised equations can be performed by direct methods,which are computationally very costly and generally do not benefit from themathematical properties of the linear system. It is therefore advantageous to usean iterative method.

The iterative method should have certain properties in order to guarantee avalid solution. The main requirement for convergence of the solution is that thematrix A be diagonally dominant, which has been shown by Scarborough:32

� �Anp��AP�

� 1 at all P� 1 at all P at least

�6�25�

where np are the neighbouring nodes of the node P. Several iterative solvers areavailable. A detailed discussion is given by Ferziger and Peric.16

To verify the validity of the mathematical solution, the solution changeduring the iterative procedure should be monitored. One can then stop the


iteration process, based on a predefined convergence criterion and be assured ofa convergent solution of the discretised equations. The convergence error � n

c canbe defined as:16

� nc � � � �n �6�26�

where � is the converged solution of equation [6.24] and �n is the approximatesolution after n iterations. It is not possible to obtain � n

c directly and it is evenhard to calculate a suitable estimation of the value. In practice, the residual rn

can be used to test for convergence:

A�n � Q � rn �6�27�When the residual goes to zero, the convergence error will be forced to decreaseas well, because:

A� nc � rn �6�28�

The reduction of the norm of the residual is a convergence criterion to stop theiterations. The residual should be reduced by three to five orders of magnitude.It may happen that the residual decreases much faster than the actualconvergence error, in which case care should be taken and the iterationprocedure continued.

6.4.5 Commercial softwareSome general characteristics of finite difference, finite element and finitevolume methods are compared in Table 6.1. Because of their generality, mostCFD codes are currently based on finite element and finite volume methods.

Most commercial CFD codes for fluid flow analysis are available on UNIX aswell as NT platforms. Parallel versions are often available as well. Some of the

Table 6.1 Characteristics of the finite difference, finite element and finite volumemethod

Finite Finite element Finite volumedifference

Geometry of problem simple complex moderately complexBoundary conditions difficult easy easyNonlinearities difficult easy easyComplexity low high highMain application area general mechanical computational fluid dynamics

engineeringPhysical background none depends on physical balances

discretisation satisfiedAvailability of commercial few many manycodesPrice low high very high


commercial codes dedicated to CFD analysis are described below. Somegeneral-purpose numerical codes, like ANSYS (Ansys Inc, Swansee, USA) alsoinclude CFD features, but are mainly intended for structural and conduction heattransfer analysis. The main packages are CFX/TASCflow (AEA, Harwell,U.K.), Fluent/FIDAP (Lebanon, NH, USA), PHOENICS (Cham Ltd., London,U.K.), STAR-CD (Computational Dynamics Ltd., London, U.K.). The basicfunctionality of these packages is very similar.

6.5 Applications

6.5.1 Design of thermal processesThe design of the thermal sterilisation process is well-established and is basedon the analysis of the heat penetration in the sterilised food and the kinetics ofthermal inactivation of microorganisms. Teixeira et al.33 suggested solvingnumerically the Fourier equation by means of the finite difference method, andto use the computed centre temperature as an input for the calculation of theprocess lethality by numerical integration. As a further improvement, the use oftime-varying retort temperature profiles was considered34 in order to maximisethe retention of thiamin while safeguarding the required process value. Thiseventually led to the STERILMATE software package for computer-aideddesign of sterilisation processes.35 More elaborate computer-aided optimisationprocedures have been described in the literature.36, 37

Applications of the finite element method include the simulation ofconduction heat transfer in foods with complicated geometrical shapes such aschicken legs,38 a baby food jar,39 broccoli stalks,40 tomatoes,41 and lasagna.42

Integrated software packages have appeared which incorporate heat transfersimulation models with microbial kinetics and food process engineeringknowledge. The CookSim package43 was essentially a knowledge-based systemto guide the user towards a safe thermal process design by automatically solvingthe mathematical models underlying the heat transfer process and the associatedmicrobial kinetics. A related approach was followed in the development of theChefCad package for computer-aided design of complicated recipes consistingof consecutive heating/cooling steps.44–46 The data and knowledge basecontains the declarative and procedural knowledge of the system. Thedeclarative knowledge encompasses all the data in the system, including thecurrent recipe, a list of food ingredients (the complete food table is in thesystem), species of microorganisms and the parameters of their growth/inactivation models, equipment types such as ovens and refrigerators, etc. Theprocedural knowledge base contains finite element routines for the numericalsolution of 2D heat conduction problems, an automatic finite element gridgenerator, routines to calculate the thermophysical properties from the chemicalcomposition of the food, routines to calculate the surface heat transfercoefficient of the heating/cooling fluid, differential equation solvers for themicrobial growth/inactivation and texture changes. The inference engine is the


core of the system. It is a part of the programming environment and containsprocedural knowledge for making logical inferences. It is not immediatelyaccessible to the programmer. The inference engine processes the user requestswhich arrive through the user interface. The necessary declarative data arefetched from the data and knowledge base, and passed to the calculation routineswhich are then fired. The calculation results are then transferred back to the userinterface for visualisation. Also, a microbial safety diagnosis of the recipe ismade by inferencing appropriate rules. The main window of the package isshown in Fig. 6.4.

6.5.2 Design of forced convection ovensNon-homogeneous heating of foods may cause considerable microbial risks anda large non-uniformity of the food quality. Industrial appliances for theconvection heating of foods have been shown to produce a stronglyinhomogeneous distribution of the main processing parameters, such as theprocessing temperature and the surface heat transfer coefficient (see, forexample, the work of Sheard and Rodger47). The heat transfer implications of anairflow that varies in direction as well as in magnitude may be large, especiallyat low velocities: it has been shown that when the surface heat transfercoefficient is small, small deviations in its value may result in large deviations inthe food product temperature.48 It is expected that this non-uniformity can beattributed to a bad distribution of the heating medium, which results from an

Fig. 6.4 Main window of the ChefCad package.


improper oven design. For example, even at a mild value of the air velocity of1.5 m/s and a turbulence intensity of 38%, a velocity deviation of 5% will lead toa typical product centre temperature deviation of 1ºC, if a food is heated from5ºC to 90ºC, using a first order sensitivity analysis of the Fourier equation forheat conduction in foods with fluctuating boundary conditions.49

Even more pronounced is the effect of an improper distribution of moisture inthe air. When, due to an inhomogeneous flow distribution, local humidity levelsdiffer, very large temperature deviations can be expected, as given in Fig. 6.5.This effect is due to different rates of moisture condensation onto the productsurface at different humidity levels. This was a study performed by means of amodel for combined airflow, heat and mass transfer during heating of a flatsurface of a food (thickness 5 cm, length 30 cm, heating from 5ºC to 60ºC, airvelocity 4 m/s). The model was implemented and solved using the commercialCFD code CFX (AEAT, Harwell, UK).50

As a consequence of these findings, it is required to have a comprehensivetool to optimise the airflows and consequently the design of ovens. A CFDmodel of an oven was developed and solved using the commercial code CFX(AEAT, Harwell, UK).51, 52 The model consists of the Reynolds-averagedconservation equations of mass, momentum and energy, a k–� turbulence modeland additional equations to describe the fan operation, including its rotation, andheat transfer from the heating coils. Model calculations were validated againsthot film velocimetry (TSI, St Paul, MN) and temperature measurements bymeans of thermocouples. The model was solved by means of the commercialCFD code CFX (AEAT, Harwell, UK). The code employs a finite volumeformulation for the numerical solution of the conservation equations. Hybrid

Fig. 6.5 Calculated time-temperature profiles at the food centre, during condensationheating with different values for the air humidity and an ambient temperature of 60ºC.50


differencing is used for the convection terms, central differencing for thediffusion terms. Time stepping is performed using the implicit backwardscheme. A fine body-fitted structured grid with 55,944 rectangular volumes(15� 20� 70) was used. The mesh has control volumes of dimensionsapproximately equal to 0.025 m by 0.025 m by 0.01 m in each direction.

The flow pattern and the temperature distribution are shown in Fig. 6.6.Recognise the hot spots (light grey contours) and the regions of lower velocity(small arrows). Figure 6.7 shows the temperature response of polymer brickssubject to forced convection heating in the oven, compared to experiments.Obviously, in quantitative terms, the CFD model lacks some accuracy, due tocomplex features such as turbulence and fan rotation. However, the cold and hotspots are correctly predicted by means of this CFD model.

Fig. 6.6 Calculated airflow and temperature distribution in an oven cavity, arrows: airvelocity vectors (white: � 6 m/s, black: 0 m/s), contours: temperature distribution (grey:

� 100ºC, white: � 105ºC).


6.6 Conclusions

Because of the increasing power of computers, food engineers now have tools tosolve complicated heat transfer problems involving a variety of heat transfermechanisms. The reliability of the numerical solution largely depends on theavailability of suitable thermophysical properties and the complexity of thegoverning models. The numerical solution of convective transport problemsdescribed by the Navier–Stokes equations remains a difficult task, particularlywhen turbulence is involved. The empirical constants involved in the mostpopular turbulence models necessitate a careful validation of the resultsobtained.

The currently available finite difference, finite element or finite volumecodes can only be used appropriately by highly trained engineers who have aformal knowledge of both heat transfer theory and numerical discretisationmethods. There is, hence, a need for more user friendly software packages whichcan also be used by operators who have less knowledge about the physicalmathematical details so that they can concentrate on the actual application.These software packages should combine heat transfer simulations withprediction of microbial growth/inactivation and quality changes, and HACCP.That this is at least possible for relatively simple heat conduction problems isshown by the ChefCAD package.

Fig. 6.7 Measured (black symbols) and calculated (clear symbols) centre temperatureresponses of polymer bricks at different positions in the oven: top (❏ ), bottom (❍ ).


6.7 Acknowledgements

The authors wish to thank the European Union (projects FAIR-CT96-1192 andINCO IC15 CT98 0912) and the Flemish Government (COF 99-003) forfinancial support.

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