CHAPTER 1 Numerical modelling of heat transfer in …CHAPTER 1 Numerical modelling of heat transfer...

CHAPTER 1

Numerical modelling of heat transfer in the foodindustry – recent developments and applications

K. Sardi1 & S. Yanniotis21Greek Regulatory Authority for Energy, Athens, Greece.2Department of Food Science and Technology, Agricultural Universityof Athens, Athens, Greece.

Abstract

Numerical modelling in food processing operations is receiving increasing attentionin recent years. Partial differential equations describing momentum, heat and masstransfer coupled with equilibrium and kinetic equations, which usually form amodel for a food processing operation, can be solved easily with today’s computingcapabilities. In this chapter, a brief description of the main numerical techniquesused in heat transfer models in food engineering, i.e. finite differences, finite volumeand finite elements, are briefly presented with advantages and restrictions for eachone, followed by examples of specific applications in the food industry. A briefdescription of the principles of computational fluid dynamics for the solution offluid flows with heat transfer is also presented, together with examples from thefood engineering literature.

1 Introduction

Heat transfer, demonstrated via heating, cooling or freezing is of primary impor-tance in the field of food engineering. Operations involving the heating of foodsare performed for cooking and also with aim to reduce the microbial population,inactivate enzymes, reduce product moisture and modify the functionality of certaincompounds [1]. Operations involving cooling and freezing are performed mainlyto reduce or avoid deteriorative chemical and enzymatic reactions and to inhibitmicrobial growth [2].

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2 Heat Transfer in Food Processing

Understanding and quantifying the driving force of all heat transfer processes, i.e.the temperature gradient, is essential towards attaining better control and avoidingunder- or over-processing which can have a detrimental effect on the food char-acteristics. As heat transfer in the food industry is often a transient process, an apriori determination of heating and cooling times is also of primary importance interms of refrigeration requirements, micro-organism growth inhibition and nutri-tional and sensorial properties preservation [3, 4]. Moreover, heat transfer usuallyoccurs in more than one form either simultaneously or consecutively. For example,in air blast chilling, heat exchange involves forced convective, radiative and evap-orative heat transfer leading to a conductive heat flux within the sample [5, 6]. Onthe other hand, the preparation of sous-vide products involves a time sequence ofprocess steps related to heat transfer such as packaging, fast cooling, cold storageand reheating [4]. To add to the complexity of the problem, foods have neither a reg-ular form nor a homogeneous or isotropic behaviour and a number of their physicalproperties, such as form, size, structure, thermal conductivity, specific heat, den-sity and viscosity, are temperature dependent and of paramount importance on thethermal behaviour of foods [7].

Other transport phenomena such as momentum and mass transfer are also presentin a number of operations concerning food processing including mixing, fluidisa-tion, pneumatic transport, sedimentation, filtration, sterilisation, drying, extrusion,packaging, frying, cooling and freezing and their coupling with heat transfer needsto be understood and accounted for [1]. Some of these processes, such as fluidisa-tion, pneumatic transport and spay drying, also involve two-phase flows [8]. Phasechange, as is the case of ice formation in freezing [9] and crust formation in fry-ing and drying [10, 11], adds to the complexity of the problem. Finally, particularfeatures of food being heated such as non-uniform evaporation of water or openingof pores are of such a complexity that understanding and controlling becomes ahighly demanding and often impossible task [12].

To this end, numerical models and methods offer an efficient and powerful toolfor simulating, comprehending and experimenting with transport processes in thefood industry.

As transport phenomena in food engineering are of increased complexity anddiversity, a number of numerical models accounting for a wide range of processeshave been proposed. No matter the differences between the various models, thestarting point is always a set of differential equations, e.g. the three-dimensionalpartial differential equation of heat conduction for predicting any type of heat treat-ment [3, 13–16], the original and modified Plank equations for predicting freez-ing times [17, and references therein], Fick’s law for predicting mass balance ofwater vapour in a wide range of applications where evaporation occurs [13, 18]and the coupled Navier–Stokes, energy and mass transport equations for predict-ing momentum, heat and mass transfer [19–21]. In addition, kinetic models forpredicting the inactivation of enzymes and the formation of carcinogenic precur-sors [2, 22] and models for two-phase flow formulations are also employed [23].One chooses an appropriate model for the target application and a solution methodis usually designed for a particular set of equations. Trying to produce a general



Numerical Modelling of Heat Transfer 3

purpose solution method, i.e. one that is applicable to all flows, is impractical if notimpossible, and as with most general purpose tools they are unusually not optimumfor any one application [21].

Regardless of the model adopted, a number of common steps are always fol-lowed in the solution of a transport process. In detail, after selecting the mathe-matical model one has to choose a suitable discretisation method, i.e. a methodfor approximating the differential equations by a system of algebraic equations,for the variables in question at some set of discrete locations in space and time.The most common discretisation approaches include the finite difference (FD), thefinite element (FE) and the finite volume (FV) methods [17, 21]. Other approachessuch as boundary elements [24], lattice gas cellular automata and lattice Boltzmanmethods [25] are also gaining increasing attention.

The purpose of this chapter is to review the current state of numerical modellingof heating/cooling processes in the food industry building upon previous work byWelti-Chanes et al. [1, 12], Delgado and Sun [17], Sablani et al. [18], Scott [19], Xiaand Sun [20], Wang and Sun [26], Otero and Sanz [27] and Langrish and Fletcher[28], with emphasis on some of the details of the numerical methods employed ineach application. In this context the chapter is structured according to the discretisa-tion schemes and numerical methods employed, e.g. FDs, FEs and FVs, rather thanthe physical phenomenon modelled, e.g. conduction, convection and evaporation.Each section commences with a brief description of the numerical technique, itsadvantages and restrictions, followed by examples of specific applications in thefood industry.

2 Finite difference (FD) methods

This is the oldest method for the numerical solution of partial differential equations,believed to have been introduced by Euler in the eighteenth century. It is also theeasiest method to use in simple geometries such as spheres, slabs or cylinders. Inthe food industry, FDs are commonly applied in the solution of the heat conductionequation and of Fick’s law for mass transfer:

ρCp∂T

∂t= ∇(k∇T ) + qevap + qi, qevap = ρλ

∂C

∂t(1)

∂Ci

∂t= ∇(Dij∇C) + R (2)

In eqns (1) and (2) T is the temperature of the food sample, C the species concentra-tion (e.g. dry air phase, vapour phase or liquid phase), Dij the diffusion coefficientof phase i to phase j, qevap the heat transport due to vaporisation if present, qi anyother type of heat source (e.g. microwave heating [29]), R the phase change volu-metric rate [30], λ the latent heat of vaporisation, and ρ, Cp and k are the density,heat capacity and thermal conductivity, respectively, of the food sample. Clearlyeqns (1) and (2), written here in their complete three-dimensional time-dependentform, may reduce to simpler formulations (one or two dimensional in space, timedependent or steady state). Note that eqn (2) refers to a group of equations, one per




phase. The actual number of mass transfer equations to be solved depends on thecomplexity of the problem.

Solutions of eqns (1) and (2) are obtained by dividing the computational domain,i.e. the food sample, by a rectangular or cylindrical structured grid – a grid con-sisting of families of grid lines with the property that members of a single familydo not cross each other and cross the members of the other family only once [21].At each grid node the differential equation is approximated by replacing the par-tial derivatives by approximations in terms of the nodal values. The result is onealgebraic equation per grid node in which the variable values at that node, and at acertain neighbour, appear as unknown. Taylor series expansion or polynomial fit-ting is used to obtain approximations to the first and second derivatives [21, 31]. Forexample, the one dimensional in space, time-dependent heat and/or mass transferequation can be easily solved by explicit or implicit one-step FD schemes. Implicitmethods are considered to be more stable than explicit. Thus, unconditionallystable, second-order temporal and spatial accurate solutions can be easily obtainedvia well-established numerical algorithms such as the Crank and Nicolson scheme[31, 32]. Explicit methods can also be employed but care has to be taken in theselection of the appropriate space and time steps. Ansari [33] performed a thor-ough investigation on the numerical issues related to the modelling of heat andmass transfer from the surface of solids exposed to a cooling environment by use ofthe unsteady, normalised, one-dimensional heat transfer equation. The equation wasdiscretised in terms of a Langrangian interpolation scheme with a truncation error ofO(�X)3. A fully explicit and a fully implicit FD scheme and different combinationsof the two schemes with varying weighting factors were studied. Regular-shapedbodies in the form of infinite slab, infinite cylinder and sphere were consideredand the study also included an investigation on the effects of the integration time(in terms of the Fourier number) and the mesh refinement on the solution. It wasconcluded that a simple explicit scheme with a moderately fine mesh and Fouriernumber increments of one-sixth of the square of the space division size can alsoprovide highly reliable and accurate solutions.

Two- and three-dimensional heat and mass transport equations in simple geome-tries are commonly written in terms of an implicit FD formulation and solutionsare obtained by application of the alternating-direction implicit (ADI) method[34, 35] or by the modified Keller box [31] and references therein.

Boundary conditions (BCs) to eqn (1) can be either in the form of a prescribedtemperature value (referred as BC first kind or Dirichlet), a prescribed heat flux (BCsecond kind or Neumann) or most commonly a radiation and/or convection BC (BCthird kind) [36]. The last sets the conduction heat flux from the outer surface of thefood sample equal to the convective heat flux from the surrounding fluid. The heattransfer coefficient, h, required in the evaluation of the convective surface heat fluxis estimated in terms of the process macroscale characteristics in dimensionlessformat, e.g. as a function of the Nusselt and the Reynolds numbers. BCs to eqn (2)usually involve a prescribed concentration value or a zero gradient assumption.

An alternative method to Taylor series expansion or polynomial fitting discretisa-tion for the solution of the heat transport equation involves the application of energy




balances [35]. In the energy balance method, the food sample is also divided by arectangular or cylindrical structured grid. For the one-dimensional approach, devel-opment of the FD equation is accomplished by considering a differential volumeelement �x about each node m and writing the energy balance equation aroundthat node. The method utilises the thermal resistance concept, i.e. the ratio of thedifferential volume element �x to the product of the thermal conductivity and thenode cross-sectional area at distances ±�x/2. An advantage of the method is that itallows for a variation in the thermal conductivity and the cross-sectional area withposition. For two-dimensional heat transfer the approach is similar [37, 38]. Nodesin the energy balance approach can be either capacitance surface nodes (CSNs)or non-capacitance surface nodes (NCSNs). In the former, nodes are separated byequal distances. In the latter, all interior nodes are separated by equal distances, ds,while the distance between the last interior node and the surface node is set to halfthe interior node distance (ds/2). The advantage of the NCSN approach is that itdoes not require knowledge of the surface temperatures to estimate internal temper-atures [37] whereas the time steps for the computation of the unsteady heat transferare larger than in the CSN approach [38]. As in the Taylor series expansion orpolynomial fitting discretisation approach, BCs involve either a prescribed temper-ature value, a prescribed heat flux (BC second kind) or most commonly a radiationand/or convection BC. The use of energy balances allows for an explicit formula-tion of the FD scheme. Solution to the system is obtained by Gauss elimination orGauss–Seidel iteration [36].

The FD method, in terms of Taylor series or energy balance approximations,has been used widely in the modelling of food processes such as heating, frying,drying, sterilisation and cooling. The following paragraphs provide examples ofapplications of the FD method in the modelling of heat transfer processes viaconduction in the food industry. The section concludes with a critical evaluation onthe limitations of the FD schemes.

2.1 FD approaches by Taylor series expansion

The unsteady, zero dimensional in space, heat and mass transfer equations have beenwidely used in simple models of thermal processes under the assumption that thefood sample was homogeneous and isotropic. For example, Delgado and Sun [13]used the one-dimensional heat and mass transfer equations to estimate centre tem-peratures during the thawing of cooked meat modelled as a sphere. In their work, itwas assumed that the food sample was symmetric with a uniform initial temperature.At the surface, third-class BCs were applied [39]. Results revealed a good agree-ment between predicted and measured centre temperatures when Schwartzberg’sequations [40] for the specific heat and thermal conductivity values were used.

The unsteady, zero dimensional in space, heat transfer equation has also been usedin more complex geometries under the assumption that these could be adequatelyapproximated by a family of simple shapes such as cylinders, slabs or bricks. Forexample, Davey and Pham [41] developed a model for predicting the dynamic heatload and weight loss during beef chilling in which the irregular beef geometry was




approximated by a combination of seven cylinders and slabs. The model involvedthe solution of the time-dependent, zero dimensional in space, heat transfer equa-tion in each one of the seven simple shapes, modified by an empirical correctionfactor. Computed values of heat removed and weight loss (calculated from the latentcomponent of the heat load) were shown to overpredict the experimental data byapproximately 12% at early times.

Iguaz et al. [42] also used the unsteady, zero dimensional in space, mass transferequations to predict moisture balance, in the grain and in the air, in a cylindricalrice grain storage bin. Temperature histories along the grain bin were predicted bysolving an enthalpy balance in the grain contained in each control volume (CV).The equation for the enthalpy balance in the air was additionally solved. The modelwas shown to predict the grain temperature with an average error estimate of up to0.38◦C.

The unsteady, one-dimensional heat conduction equation has been widelyemployed to obtain insight on thermal processes in a number of practical prob-lems that could be approximated by assuming that heat transfer is dominant alonga single direction and negligible along all others. In this context, Akterian [43, 44]used the one-dimensional time-dependent heat conduction equation, modified bymeans of an empirical shape factor, to predict the conductive heat transfer in mush-rooms during in-can sterilisation and to assess the thermal sensitivity of sausages.

The coupled one-dimensional time-dependent heat conduction and mass trans-port equations have also been applied in the modelling of a number of thermal pro-cesses such as frying, thawing and freezing. As in the case of the one-dimensionaltime-dependent heat conduction, here a single direction for the temperature gradi-ent is also assumed and special attention is placed in the modelling of weight loss asa result of moisture evaporation and of crust formation via application of a Landau-type transformation or a temperature-enthalpy correction method as suggested byPham [45] and Fikiin [46].

Shilton et al. [47] estimated the temperature and moisture content as a func-tion of time and fat content along a beef patty assumed to be homogeneous andisotropic. The patty was heated by infrared radiation and modelled as an infiniteslab divided into ten nodes. Solution was obtained by a commercial equation solver(DESIRETM). Thermal conductivity and density values were adjusted so as toaccount for both changes in temperature and fat content and an effective convectionterm was added to the thermal conductivity to account for the presence of fat. Com-putations revealed that the addition of the effective convection term enhanced theagreement between experimental data and predicted values of temperature alongthe patty whereas omission of the term resulted in an underestimation of the tem-perature rise in the patty by up to 40%.

Farkas et al. [10] used the time-dependent, one-dimensional heat and mass trans-fer equations to estimate the crust thickness and temperature and moisture distri-butions along an infinite homogeneous half slab with properties corresponding tothose of a rehydrated potato mixture undergoing immersion frying. The movingboundaries between the crust and core regions were handled by a Landau trans-formation and the initial value of the crust thickness was obtained from an order




of magnitude analysis that produced an equation similar to Plank’s equation forfreezing. The physical and thermal properties of the potato mixture were obtainedfrom the literature and the effects of oil flux and accumulation in the sample wereneglected. Excellent agreement with the experimental data was achieved.

The problem of predicting crust formation was also addressed by Hamdami et al.[6], who used a fixed grid FD method to estimate the temperature profile, weightloss rate and moisture content during par-baked bread freezing. The food samplewas modelled as an infinite two-layer cylinder comprising two composites: crumband crust. The sample surface, directly exposed to air was assumed to be cooled byconvection, radiation, evaporation or sublimation. Heat transfer in the material wastreated as heat conduction with phase change. The latter was incorporated in theheat conduction process by means of the specific heat formulation and by applica-tion of the temperature-enthalpy correction method suggested by Pham [45]. Theconduction equation was solved by application of the three level method proposedby Lees [48]. Enthalpy was calculated from an interpolation table using the appar-ent specific heat (literature values) as a function of moisture content. The thermalconductivity was modelled by a Maxwell-type model in three steps consideringa continuous and a dispersed phase in each step and incorporating evaporation–condensation effects. The mass balance was formulated so as to account for thefrozen water, and the mass diffusivity for crumb and crust was modelled by anArrhenius law expression. Computed temperature values at the surface and at thecentre of the food sample were in agreement with the measurements within 0.5◦Cand 1.8◦C respectively. The precision of the computed weight loss was of the orderof 10%.

A similar approach was adopted by Hamdami et al. [9], who used the heat andmass transfer equations to predict the transient thermal and evaporative behaviourof a one-dimensional sponge slab assumed to comprise of dry matter, liquid water,ice and air contained in the pores. The mass balance was formulated so as to accountfor the frozen water and the mass diffusivity for sponge was also modelled by anArrhenius law expression. Results in terms of temperature profiles at the centre andat the edges of the sponge and weight loss estimates as a function of time werein good agreement with measurements obtained from a vegetable sponge of 58%initial moisture content located in an air blast freezer.

The one-dimensional approximation for the solution of the heat and mass transferprocesses inherently involves the assumption of an infinite geometry. Turhan andErdogdu [49] estimated the error due to assuming a finite regular geometry duringtransient heat or mass transfer as a function of the Fourier (Fo) and Biot (Bi)numbers for a number of simple shapes including cylindrical, square and circularrods and square and rectangular slabs. They concluded that the error due to theinfinite geometry assumption increases dramatically with increasing Fo numberand decreases with increasing Bi number. The effect of geometry on the error wasshown to depend on the Bi number for rod-shaped geometries and decreased withdecreasing width over length ratio in slabs.

Two- and three-dimensional solutions of the heat and mass transfer equationsarguably offer increased insight in heat and mass transfer problems. Zorrila and




Singh [50] modelled a hamburger patty as a finite cylinder comprising an innerfrozen region surrounded by an unfrozen region and an outer core in both theaxial and radial directions. The unsteady two-dimensional enthalpy equation andappropriate BCs at the moving interfaces formulated a system of fourteen equa-tions that were solved numerically with an FD scheme. Because the crust thicknesswas small compared to the slab dimensions a linear temperature change in thecrust was assumed. Nodal values of temperatures were related to the calculatedenthalpy values according to the procedure proposed by Mannapperuma and Singh[51]. Following Farkas et al. [10], initial values for the moving boundaries wereset to at least an order of magnitude smaller than the distance travelled by thecrust–core interface in one time step. A combined radiation and convection heattransfer coefficient was assumed at the circumferential surface of the patty. Com-puted centre temperature values were compared to predictions obtained by Zorillaand Singh [50] using a one-dimensional model. It was shown that although noconsiderable differences exist between the two approaches (one and two dimen-sional) with regard to the temperature at the patty centre, the two-dimensionalmodel allows for the estimation of the temperature history at the regions close tothe circumferential edge. The latter was shown to be substantially different thanthat of the centre.

2.2 FD approaches by energy balance

Erdogdu et al. [52] proposed an FD procedure for the estimation of heat conduc-tion in elliptical cylinders using the energy balance approach and the so-calledheat flow lines approach proposed by Eshelman [53] and Pham [54]. The approachreduces an elliptical heat transfer problem from three to two dimensional for a finitelength geometry and from a two to one dimensional for an infinite length geom-etry by assuming that the so-called heat flow lines, which are orthogonal to theelliptic isotherms can be described by simple power curves. The intersection pointsbetween a power curve and the concentric ellipses were determined by using theNewton–Raphson method.Volume elements were defined from the volume betweentwo concentric ellipses and power curves using the NCSN heat transfer analysisdescribed above. The FD difference models were formulated for infinite ellipti-cal and circular cylinders and computational results were compared to analyticalsolutions and experimental data. It was shown that the model could accurately pre-dict centre temperatures in a conduction-heated elliptical cross-sectional cylinderand also in an elliptical infinite or finite cylinder if it was substituted by a circularcylinder of equal surface area to volume ratio.

In a companion article, Erdogdu et al. [55] adapted the previous formulation topredict the temperature distribution during cooking of shrimp. The model used theNCSN approach, the shrimp cross-sectional area was considered circular and vari-able thermal conductivity, specific heat capacity and density were assumed. Mois-ture content reduction and cross-sectional shrinkage were predicted from empiricrelations as a function of temperature. Elements at the same radial distance wereassumed to shrink by the same amount and at each time step the nodal network was




regenerated. Calculated temperature values both at the centreline and close to themodel surface were in good agreement with the experimental data.

Yanniotis and Petraki [56] used a three-dimensional model to calculate the cen-tre temperature in a food sample of cubical shape placed in a freezer, where thetemperature fluctuates around a mean temperature due to the on–off action of thecompressor in the freezer. Schwartzberg’s equations [40] for the specific heat andthermal conductivity were used. The model was solved with an explicit FD schemeand validated experimentally for a cube of ground beef. The effect of heat transfercoefficient, size of the cube, frequency of air temperature fluctuation and amplitudeof air temperature fluctuation on the temperature variation of the cube was studiedwith the model.

2.3 Limitations of the FD approach

The most significant disadvantage of the FD approach is that it is generally limitedto heat conduction problems in simple geometries that can be discretised into simplerectangular or cylindrical structured grids.

Further, in most conduction problems, the surface heat flux is due to convection.The latter requires knowledge of the heat transfer coefficient, h, a parameter that is inturn equal to the ratio of the surface heat flux to the temperature difference betweenthe solid surface and the ambient air at the vicinity of the solid. Thus, the value ofh is related to the surface gradient of the thermal boundary layer in the surroundingfluid. The latter, in the case of a non-stagnant fluid, is in turn largely dependent on thevelocity boundary layer [57, 58] so that h varies along the surface as boundary layersdevelop. In addition, the fluid and thermal boundary layers are influenced by thesolid surface geometry, the flow regime and an assortment of fluid properties suchas density, viscosity and thermal conductivity. Complex aerodynamic phenomenasuch as boundary layer separation and reattachment, vortex shedding and turbulencecan also significantly affect the heat transfer from the fluid to the solid surface.

A number of studies [59–61] have already demonstrated the importance of thelocal distribution of the heat transfer coefficient in a number of food processes.To this end the coupled Navier–Stokes, energy and mass transport equations forpredicting momentum, heat and mass transfer can provide valuable insight andallow for a detailed estimation of the local distribution of h. However, the FDapproach does not enforce conservation unless special care is taken [21] so that itrarely finds application in a complete flow, heat and mass transfer computation.

3 Finite element (FE) methods

The FE method is one of the most general techniques for the numerical solution ofdifferential equations and has been astonishingly successful [62]. It uses the integralform of the conservation equations and the domain is broken in an arbitrary col-lection of sub-domains called FEs. In two-dimensional geometries FEs are usuallytriangles or quadrilaterals whereas in three-dimensional tetrahedra or hexahedraare most frequently used [21]. As long as the FEs are sufficiently small, polynomial




functions can adequately represent the local solution. Jiang [62] based on remarksby Zienkiewicz [63] and Oden [64], offers a comprehensive summary of the mostimportant features of FEs as follows:

• Arbitrary geometries: The FE method is essentially independent of geometryso that it can be applied to domains of complex shape and with arbitrary BCs.

• Unstructured meshes: In FE analyses a global coordinate transformation is notrequired. FEs can be placed anywhere in physical domains and there is norestriction on the number of neighbour elements or nodes. Further, an FE canbe easily added or deleted without the need to modify the global data structure.In contrast to FD approaches, in FEs the element and nodal numbering can bearbitrary without sacrificing efficiency.

• Flexible and general purpose programme formats: The clear structure and ver-satility of the FE method makes it possible to construct general purpose softwarefor a variety of applications.

FEs are also comparatively easy to analyse mathematically and can be shown tohave optimality properties for certain types of equations. Their principle drawbackwhich is also shared by any other method that uses unstructured grids is that theequation matrices are not as well structured as those for regular grids making moredifficult to find efficient solution methods [21].

The FE solution approach can be classified into three major groups: the Rayleigh-Ritz method, the Galerkin method and the least squares method [62]. The Galerkinmethod is the most commonly used in thermal analysis problems and thus in foodengineering so that discussion in the remainder of this section will be restricted toa short presentation of its main features.

In the Galerkin method, the integral of the weighted residue over the elementdomain is set equal to zero by application of appropriate interpolation functions inthe so-called shape functions matrix. Thus, eqn (1) may be written as

∫∫Ve

∫[N]T

[∂

∂x

(kx

∂T

∂x

)+ ∂

∂y

(ky

∂T

∂y

)+ ∂

∂z

(kz

∂T

∂z

)− ρCp

∂T

∂t

]dxdudz = 0

(3)

where [N] = [Ni Nj Nk Nm] is the shape function matrix and Ni, Nj, Nk and Nm arethe interpolation functions for a typical tetrahedron element indicated by four cornernodes i, j, k and m [65–67]. The shape function matrix is evaluated by assuminga pre-defined temperature variation inside each element (e.g. linear [65]). By useof the divergence theorem on the three terms involving a second derivative, afterseveral manipulations and discretisation, eqn (3) reduces to[

[N]

�t+ a[K]

]{T}n+1 =

[[N]

�t− (1 − α)[K]

]{T}n + {P} (4)

where {T} and {P} are the temperature and load vectors and [K] the conductancematrix. A detailed analysis on the derivation of the individual terms of eqn (4) is




offered by Wang and Sun [65]. Also in (4), α is a weighting factor which must bechosen in the interval between 0 and 1. Note that for different values of α differentnumerical schemes are obtained; e.g. for α equal to zero, eqn (4) results to a forwarddifference scheme, for α = 1/2 to a Crank–Nicholson scheme and for α equal tounity to a backward difference scheme.

The FE method is receiving increased attention as a reliable and valuable toolfor predicting temperature distributions in irregularly shaped, heterogeneous andanisotropic samples and also for obtaining insight on the effect of numerous param-eters related to the heating and cooling of foods. However, it is worth pointing outthat the field of FE methods in food engineering is relatively young and modellingis mostly still confined to simple geometries that could have been, in principle,adequately described even by application of FD schemes. It should also be notedthat in contrast to FD schemes, FE methods can be employed for the solution of theNavier–Stokes equations. However, since most current research and commercialcomputational fluid dynamics (CFD) codes utilise the FV approach, FE methodapplications in momentum transfer are scarce. As a consequence, FE methods infood engineering are confined to the solution of heat conduction equation assuminga constant convection heat transfer coefficient as in FD approaches. Some recentexamples of FE applications in the numerical modelling of heating and cooling offoods are summarised in the following paragraphs.

Wang and Sun [68] used in-house software based on a FE method (Galerkinapproximation) to solve the transient, two-dimensional heat conduction equation forthe specific application of cooling a cooked meat sample. The sample was assumedto be infinitely long so that only heat conduction within a cross-sectional area,elliptical in shape, was considered. Due to symmetry, the simulation was carriedout on a fourth of the cross-section. Three heat transfer methods were considered:slow air cooling, air blast and water immersion cooling.Appropriate BCs to accountfor both convection and evaporation were set for each case. It was further assumedthat the meat sample was heterogeneous but isotropic and initial temperature valueswere obtained from measurements. Excellent agreement between calculated andmeasured temperature profiles as a function of temperature at the core, surface andat an arbitrary selected node within the sample was obtained for all cooling methods.Predicted total weight loss for the slow air cooling method was underestimated by2.8% whereas predicted and measured total weight loss values for the case of airblast cooling agreed to within 1%. In a companion article, Wang and Sun [69]also performed a detailed parametric investigation on the potential effect of thesample geometry and the surrounding air and water velocities on the cooling processefficiency. Five two- and three-dimensional simple shapes were considered (infinitecylinder, infinite slab, brick, sphere, cube) and useful conclusions that relate criticaltemperatures to cooling times, sample weight and weight loss were deduced.

Wang and Sun [65] extended the previous approach to model three-dimensionalheat transfer on roasted meat during air blast cooling. The sample was cube shaped,comprising 6000 tetrahedron elements and composed of water, protein, fat, salt andair. The physical and thermal properties were expressed as functions of the con-stituents’ compositions. Temperature and weight loss predictions were generally in




good agreement with experimental data. A maximum deviation (under-prediction)of 2.4◦C between predicted core temperature and the respective measured valuewas identified in the case of a 4.35 kg block-shaped roasted meat having an initialtemperature of 48◦C. The deviation between predicted and measured total weightloss was 2.5%.

Oliveira and Franca [29] also used in-house software based on the Galerkinweighed residuals technique to perform a thorough study on heat transfer by con-duction and microwave heating on cylindrical and irregularly shaped beef, shrimpand pea puree samples irradiated in a frequency range of 900–2800 MHz. In theirapproach, the governing equations describing the electromagnetic field due tomicrowave power absorption (Lambert’s law and/or Maxwell’s equations) weresolved in addition to the two-dimensional heat conduction equation. Note that theuse of Maxwell’s equations depends on the knowledge of the dielectric proper-ties, which are generally available for a wide range of foods. On the other hand,Lambert’s law provides a simpler formulation to the power density term but doesnot offer a complete representation of the electromagnetic field [70]. Computa-tions revealed that microwave heating is significantly dependent on the radiationfrequency, with power absorption and radiation penetration being more effectiveat lower frequencies, on sample size so that in larger samples heating occurredmainly at the outer surface and also on the dielectric properties with the additionof salt leading to an increase in the energy dissipation. The effect of rotation wasalso incorporated in the model through the irradiation BCs and was shown thatrotation led to a decrease in the temperature gradients and thus to a more uniformtemperature distribution on the sample. In the same work, Oliveira and Franca [29]also proposed a correlation to predict the minimum radius of cylindrical samplesabove which Lambert’s law can be employed.

Romano et al. [70] extended the previous work of Oliveira and Franca [29] in thefield of Lambert’s law applicability in cylindrical geometries. In their work, heattransfer and microwave heating of a cylindrical potato sample was investigatedby the application of a commercial FE package (FEMLAB) as a function of thecylinder radius and length. BCs accounted for both convective and evaporative heatloss and results also confirmed that predictions using Lambert’s law improve withincreasing cylinder radius.

An example regarding the potential of FE in phase change modelling can be pro-vided by reference to the work of Sabliov et al. [71]. In their approach, the processesof egg cooling by cryogenic carbon dioxide and the resultant ice formation betweenthe albumen and the shell were considered. The formulation of Voller and Cross[72] was used to reduce the unsteady heat conduction equations for the liquid andsolid phase into a single enthalpy equation with the thermal conductivity and densitybeing a function of temperature. Eggs were assumed to be axisymmetric and ellipti-cal in shape comprising of shell, albumen, air and yolk, each isotropic. The geometrywas divided into a uniform mesh composed of 1247 quadrilateral elements using acommercial mesh generating software (GAMBIT). Due to symmetry, only half ofthe egg was considered. The FE method was based on the Galerkin approximationand solutions to the heat conduction problem were obtained by use of a commercial




FE method software (FIDAP). A convective heat flux on the eggshell surface wasassumed and initial temperatures were considered to be uniform throughout theegg and equal to an experimentally determined value. Simulated centre and shelltemperatures were shown to agree with measurements within 2–3◦C. Differenceswere attributed to both experimental and simulation uncertainties. The latter wererelated to uncertainties regarding the model inputs, i.e. experimentally determinedphysical and thermal properties and environmental conditions.

Commercial FE solvers have been widely employed in the study of heat con-duction problems in food engineering. For example, Zhang et al. [73] used FE toanalyse the pre-cooking and cooling processes of skipjack tuna. In their approach,heat transfer by conduction in the axial direction (head to tail) was assumed to benegligible so that only a cross-section of the fish was modelled. Further, the cross-section was assumed to be elliptical in shape and symmetric in the longitudinaldirection so that only one half was considered in the calculations. The geometrywas divided into a non-uniform mesh comprising 2110 quadrilateral elements. Thetuna body was modelled as non-homogeneous and anisotropic comprising of loin,backbone and viscera. Physical and thermal properties were adjusted accordinglywith values from the literature and by use of empirical models. A convective heatflux was assumed at the sample outer surface with a zero temperature gradientsymmetry BC at the tuna centreline. Weight loss due to evaporation was neglected.Simulation data were in good agreement to measurements from pilot and commer-cial scale experiments for all different regions in the fish (backbone, loin etc.).

In a similar context, Tewkesbury et al. [74] used FE to analyse the coolingof two-dimensional axisymmetric polycarbonate moulds containing chocolate inliquid form. Heat transfer in both the mould and the chocolate was accountedfor. The mesh comprised of rectangular elements and was locally refined at themould–chocolate, mould–air and air–chocolate interfaces. The initial condition wasuniform temperature throughout and BCs were a combination of experimentallymeasured air temperature and two convective heat transfer coefficients for aboveand below the mould. The rheological behaviour of chocolate was modelled asnon-Newtonian and the effect of crystallisation was accounted for by introductionof a variable, effective, specific heat capacity based on the enthalpy method andcomputed as a function of both temperature and cooling rate. Computed temper-atures were in agreement with experimental data within a cooling rate window of0.5–2◦C/min.

Hulbert et al. [75] also used commercial software (ANSYS) to model heat con-duction inside a cross-sectional carrot slice and to estimate the fluid to sampleconvective heat transfer coefficients. Solutions to the two-dimensional heat con-duction equations were obtained by imposing initial and BCs obtained from MRImeasurements.

A more complex approach was adopted by Tattiyakul et al. [76, 77], who studiedheat transfer to a corn starch dispersion contained inside a circularly shaped can. Thecan was assumed to be stationary, continuously rotating or undergoing intermittentagitation. The problem was modelled as two dimensional. The can was meshedby a non-uniform quadrilateral grid comprising 580 elements using the same




commercial software as Tewkesbury et al. [74], Zhang et al. [73] and Sabliov et al.[71]. Various other mesh sizes were tested to ensure a grid-independent solution.The starch was considered to be in liquid form, of constant density and temperature-dependent viscosity. In addition to the heat conduction equation, the continuity andtwo-dimensional laminar momentum (Navier–Stokes) equations were also solvedso as to account for the fluid motion inside the can, again by application of com-mercial software (FIDAP). Computed temperature profiles at the centre of the canwere compared to experimental data for the case of intermittent rotation. It wasshown that simulation results agreed with the measurements within 2% at heatingtimes larger than 500 s but that the model under-predicted the experimental databy as much as 20% at times less than 500 s. Despite this discrepancy, the modelprovided useful insight in the motion of the gelatinised starch layer within the canand it was shown that intermittent rotation can lead to more uniform temperatureand starch gelatinisation distributions.

The FE approach has also been successfully employed in combined simula-tions of heat transfer and enzyme inactivation or carcinogenic precursor formation.Martens et al. [2] used an in-house FE software (CHAMPSPACK) to simultaneouslypredict heat transfer and peroxidase activity on a one-dimensional axisymmetricmodel of a broccoli stem comprising 51 linear elements as a function of tempera-ture. The same model was also applied to predict the inactivation of lipoxygenasein asparagus. The enzyme activity was described by a first-order kinetic model inthe form of an Arrhenius expression:

dA

dt= −krA, kr = kr,ref e−Ea/R(1/T−1/Tref) (5)

In eqn (5), A is the dimensionless enzyme activity at the time t relative to the initialactivity, kr is the rate constant, t the heating time, kr,ref the rate constant at a pre-defined reference temperature Tref, Ea the activation energy and R the universal gasconstant. Both equations (heat conduction and enzyme inactivation) were solvedby use of the Galerkin weighted residual method.

Tran et al. [22] used a similar approach to estimate heat transfer and heterocyclicamine (HA) formation in a pan-fried ground beef patty without flipping. The FEanalysis was carried out by the use of commercial FE software (FlexPDE) and HApredictions were shown to be in good agreement with experimental data.

A more complex approach was adopted by Lian et al. [78], who used commercialsoftware (FIDAP) to solve the continuity equation, the standard Darcy equation [79]for a low Reynolds number fluid flow in a porous media and the volume averagedenergy conservation equation for a packed bed:

∂ρ

∂t+ (ρui)i = 0 (6)

ρ

α

∂ui

∂t+ µ

κui = −pi (7)

(ρc)e∂T

∂t+ ρcuiT, j = (keTj), j + H (8)




and obtained the special distributions of the air flow, temperature and enzymaticoxidation of polyphenols during black tea fragmentation. In eqns (6)–(8) ui is thevolume averaged velocity component,α is the porosity,κ the permeability,µ the vis-cosity, c the air heat capacity, (ρc)e = αρc + (1 − α)ρscs the effective heat capacity,ke = αk + (1 − α)ks the effective conductivity, H the heat generated by the enzy-matic oxidation, p the pressure and i (or j) is the tensor notation. For instance, fora two-dimensional axisymmetric flow ui = (ur , uz) and ui,i = ∂uz/∂r + ∂ur/∂z. Itfollows that T, j = ∂T/∂r + ∂T/∂z and (keTj), j is the second derivative. Twelveenzymatic reactions were used to model the tea fermentation and the rate of deple-tion of catechins was modelled by a Michaeld–Menten equation [80]. Resultsshowed that the air flow through the packed bed exhibits a plug flow type behaviour,the temperature distribution in the radial direction was uniform and that heat trans-fer during fermentation was dominated by the enzymatic reactions and heat removalby the air flow. More importantly, it was also shown the rate of theaflavin accumula-tion was a non-monotonic function of temperature and time. A main conclusion ofthe work was that computational methods can provide not only a means to analysefluid and temperature fields but also a valuable tool in the understanding of complexenzymatic reactions.

FE simulations have also been applied in plate heat exchanges. Fernades et al.[81, 82] used commercial software (POLYFLOW) to model the thermo-rheologicalbehaviour of yoghurt in a three-dimensional corrugated channel. The simulationswere carried out considering stationary, incompressible laminar flow and involvedthe solution of the conservation equations for mass, linear momentum and energyin the form:

div(�u) = 0 (9)

div(‖S‖) + ρb − ρdiv(�u�u) = 0 (10)

‖S‖∇�u + ρh − div(q) = 0 (11)

where �u is the velocity vector, ‖S‖ the total stress tensor and h refers to the heat sup-ply strength of an internal heater.A Herschel–Bulkley model for the yoghurt viscos-ity and an Arrhenius-type term for temperature dependence were used. Numericalresults concerning the difference between the inlet and outlet yoghurt temperaturewere shown to agree with experimental data within 7%. It was further concludedthat the presence of corrugations induced a sinusoidal behaviour in the main flowdirection in the yoghurt velocity, temperature, viscosity and shear rate.

Application of FEs in the food industry for understanding and controlling theheating and cooling of foods remains a field of ongoing research.

4 Finite volume (FV) methods

As in the FE method, the FV method also uses the integral form of the conservationequations and the solution domain is sub-divided into a finite number of contiguouselements, here referred to as CVs. Conservation equations are applied to eachCV [21]. The principal difference between the FE and the FV method is that in




the latter differential equations are integrated directly, i.e. without the applicationof weighting factors. Equation (12) presents the discretised form of the unsteadyconduction equation integrated over a CV of volume �x × 1 × 1 [83].

αpT1P = aE[f T1

E + (1 − f )T0E] + aW [f T1

W + (1 − f )T0W ]

+ [α0P − (1 − f )αE − (1 − f )αW ]T0

P (12)

where

αE = ke

(δx)e(13a)

αw = kw

(δx)w(13b)

α0p = ρcp�x

�t(13c)

αp = f αE + f αW + α0p (13d)

In eqns (12) and (13), superscripts 0 and 1 denote ‘old’ (known) values at timet and ‘new’ (unknown) values at times t + �t, respectively, f is a weighting factorbetween 0 and 1, P is the centroid of the CV where Tp is currently being calculatedand E and W are the centroids of the CVs to the left (east) and right (west) of P.Also in eqns (13a) and (13b), the letters e and w denote CV faces so that (δx)e and(δx)w denote the distance of node P from the respective faces.

From the above it is clear that in the FV method, at the centroid of each CV lies acomputational node at which the variable values are to be calculated. Interpolationis used to express variable values at the CV faces in terms of the nodal (CV-centre) values. Face and volume values are approximated using suitable quadratureformulae. As a result one obtains an algebraic equation for each CV in which anumber of neighbour nodal values appear. The FV method can accommodate anykind of grid so it is suitable for complex geometries [21].

As in FD and FE, the algebraic forms of the conservation equations in FV aresolved with iterative methods such as TDMA or ADI [35, 83].

FV methods are present in most in-house and commercial CFD codes incorporat-ing the two- or three-dimensional transient or steady solution of the Navier–Stokesequations and are increasingly finding applications in the food industry [84–87].

Application of FV methods for the solution of solely heat conduction or speciesconservation equations (eqns (1) and (2)) is generally scarce. One of the rare exam-ples in this direction is provided by the work of Zorilla and Rubiolo [88] whoused the CV approach and a logarithmic mesh for the mathematical modelling ofimmersion chilling of cheese in the shapes of a one-dimensional infinite slab, aninfinite and a finite cylinder, an infinite rectangular, a rectangular parallelepipedand a sphere. The enthalpy method was used to account for the phase change fromliquid to ice and a total of four differential equations (one for enthalpy, one for theice phase and two for the liquid phase) were solved.As expected, multi-dimensional




geometries were shown to improve heat and mass transfer with a penalty regardingCPU time.

Pajonk et al. [89] also used the FV approach to model various Emmental cheesesamples during ripening by application of eqn (1). Overall heat transfer flux at theexternal surface was considered as the sum of convective, evaporative and radiativecontributions. The cheese was cylindrical in shape with thermal properties thatwere experimentally determined as part of the work. Simulated and experimentaltemperature profiles were in excellent agreement when the temperature variationswere small. For temperature changes that exceeded 15◦C discrepancies betweenexperimental and computational results were detected and were attributed to the factthat melting enthalpies of the different fat fractions were neglected in the modellingapproach.

5 Computational fluid dynamics (CFD)

The advantages of CFD as a tool towards obtaining a complete characterisation ofall transport properties during the heating or cooling of foods have been already out-lined. This section aims to provide a brief summary of the principles and challengesof CFD for the solution of laminar and turbulent fluid flows with heat transfer. Aspreviously, the section concludes with appropriate references to specific examplesof the application of CFD in food engineering. Note that the use of CFD by thefood industry is relatively young [19] so that not all of the currently, otherwise,widely available techniques, methods and models have yet been applied. However,for the sake of completeness and also because CFD is rapidly penetrating the field offood engineering (so that techniques, methods and models currently widely avail-able in other scientific and practical areas of engineering and science are indeedexpected to find application in the food industry in the coming years), this sectionincludes a more general approach towards the current, commercial and research,state-of-the-art status of CFD.

CFD by application of FV involves the solution of integrated Navier–Stokesconservation equations in the general form of [83]:

∂

∂t(ρφ) + ∂

∂xj(ρujφ) = ∂

∂xj

(

∂φ

∂xj

)+ Sφ (14)

where ρ is the fluid density, uj the air velocity component, φ any variant (veloc-ity components, enthalpy), a diffusion coefficient and Sφ a source term. For φ

equal to unity, eqn (14) reduces to the continuity equation. On the left-hand sideof the equation, the first term denotes the rate of change and the second the con-vection flux. On the right-hand side, the first term stands for the diffusion flux andthe second for any generation and/or destruction of variable φ. For the momen-tum conservation equations, the term Sφ includes the sum of any generation and/ordestruction term and of the pressure gradient on the particular direction, whereappropriate. Conjugate heat transfer, in the context of coupled fluid-solid tempera-ture and enthalpy calculations, is obtained by simultaneous solution of the FV form




of eqns (1) and (2) together with eqn (14) and application of appropriate BCs at thecommon boundaries.

For laminar flows, integration and discretisation of a group of equations as in(14), i.e. for φ equal to unity, equal to the velocity components and equal to enthalpy,respectively, leads to a system of closed algebraic equations that can be solved bya well-established numerical method such as ADI. Note that for both laminar andturbulent flows that will be subsequently discussed, special attention needs to beplaced on the discretisation scheme employed, in particular for the convection anddiffusion terms, and also on the treatment of the pressure gradient in the momentumequations. A thorough discussion on both issues and descriptions of a number ofwell-established algorithms (e.g. SIMPLE, SIMPLER, SIMPLEC and PISO) areoffered by a number of seminal works, e.g. Ferziger and Peric [21], Patankar [83],Roache [90], Issa [91], Hirsch [92], Vandoormaal and Raithby [93] and Oliveiraand Issa [94].

For turbulent flows, which are indeed the ones most commonly encountered inengineering applications, including food engineering, three levels of CFD compu-tations are identified and may be applied [95, 96].

The first level [Reynolds averaged Navier–Stokes (RANS)] has been historicallythe first possible approach for the solution of turbulent fluid flows. In RANS, theconservation equations for the Reynolds or Favre (mass weighted) time-averagedvalues of variable ϕ are obtained by averaging the respective instantaneous conser-vation equations (i.e. eqn (17)). However, the resulting averaged equations containunclosed terms and call for specific closure rules, namely a turbulence model todeal with fluid dynamics and additional models for energy and chemical speciesconversion. In detail, the RANS approach to turbulence modelling requires that theso-called Reynolds stresses (i.e. the mean of the product of the turbulent veloc-ity fluctuations, u′′

i u′′j ) and the species and enthalpy mean turbulent fluxes (u′′

i Y ′′k

and u′′i h′′

k , respectively) appearing in the RANS averaged momentum, enthalpy andspecies conservation equations, respectively, be appropriately modelled. A com-mon method in this direction employs the Boussinesq hypothesis [97] to relatethe Reynolds stresses and species and enthalpy fluxes to an eddy viscosity andalso to the mean velocity, species and enthalpy gradients. To this end, a numberof turbulence models have been proposed [98–102]. The most popular turbulentmodel in this context remains the so-called standard k−ε model [99, 103] due to itsrobustness, economy and reasonable accuracy for a wide range of turbulent flows.However, the k−ε model is semi-empirical while the derivation of the model equa-tions for the turbulent kinetic energy k and for the dissipation rate ε of the turbulentfluctuations (which are also of the general form of eqn (14)) relies on phenomeno-logical considerations and empiricism [104]. As the strengths and weaknesses ofthe standard model have become known, advancements and modifications havebeen made to the model to improve its performance. Two of these popular variantsinclude the RNG k−ε model and the realisable k−ε model [100, 101].An alternativeapproach to the Boussinesq hypothesis includes the solution of transport equationsfor the Reynolds stresses and the turbulent fluxes [105–107]. An additional scale-determining equation (normally for ε and also for the dissipation χ of the scalar




fluctuations) is also required. This means that at least five additional transport equa-tions are required in two-dimensional flows and at least seven additional transportequations must be solved in three dimensions. Note that in many cases, modelsbased on the Boussinesq hypothesis perform very well, and the additional compu-tational expense of the Reynolds stress model is not justified. However, the RSM isindeed superior for situations in which the anisotropy of turbulence has a dominanteffect on the mean flow. Such cases include highly swirling flows and stress-drivensecondary flow.

Of all models mentioned above only the standard k−ε model has been employedin the field of food engineering. Note that despite the turbulence model employed,the solution of the RANS equations provides averaged quantities corresponding toaverages over time for stationary mean flows, averages over different realisations(or cycles) for periodic flows [95] or averages over a particular time interval fortime evolving mean flows.

The second level of CFD involves the so-called large eddy simulations (LES) andis currently gaining increased attention in a number of turbulent flow calculations[95, 108]. In LES, the larger three-dimensional unsteady turbulent motions aredirectly represented, whereas the effects of the smaller scale motions are modelled.In terms of computational expense, LES lies in-between Reynolds stress models andthe third level of CFD, namely direct numerical simulations (DNS) discussed below[108]. Because large-scale motions are explicitly represented in LES, the approachis expected to be more accurate than RANS (even with a Reynolds stress model)in the case of flows where large-scale unsteadiness is significant, e.g. the flow overbluff bodies. In principle LES can find several applications in food engineering,e.g. during air blast cooling or heating of food samples. However, to the authors’knowledge no publications on LES computations with relation to the food industryare yet available. One reason for the complete absence of LES in the modellingof food products may be attributed to the fact that until quite recently, i.e. the last3–4 years, LES CFD codes were confined to only a few research groups focusingon specialised issues of turbulence, aerodynamics, meteorological applications andcombustion [109–113] and were not available in the form of commercial CFDsoftware. Currently, however, most commercial CFD codes are equipped with theLES approach. For the sake of completeness and also because it is recognised thatit is not unlikely for LES to penetrate the field of food engineering in the comingyears, a short overview on the general concept of the LES approach, divided intofour conceptual steps, is provided [108].

The first step for LES calls for the definition of a filtering operation to decomposethe variable φ of the instantaneous Navier–Stokes conservation equation [eqn (14)]to a sum of a filtered (resolved) component and to a residual (or sub-grid scale,SGS) component. The filtered value which is three dimensional and time dependentrepresents the motion of turbulent eddies.

The second step involves the derivation of the equations for the evolution of thefiltered value of φ from eqn 14. These new equations are of a standard form (i.e.similar to the ones obtained in the RANS approach) and contain the residual stresstensor (or SGS tensor) and the residual species and enthalpy fluxes that arise from




the residual turbulent motions. As in RANS these terms are unclosed and modellingis required.

As in RANS, the third step involves modelling of the unknown terms. Closure isobtained by appropriate approximations, most simply by an eddy viscosity model[114–117] conceptually similar to the one obtained by the Boussinesq hypothesisin RANS.

Finally a numerical solution of the filtered equations that provides a single real-isation of the large-scale turbulent motion is obtained.

Additional details on the application, modelling and numerical issues related tothe LES method are outside the scope of this work. Comprehensive and criticalreviews on the topic may be found in the works of several researchers in the fieldincluding Brandt [118], Pope [108], Poinsot andVeynante [95], Galperin and Orszag[109] and Mason [110]. Research on SGS modelling and other aspects of LES ison-going.

The third and most advanced level of CFD is based on DNS. In DNS, the fullinstantaneous Navier–Stokes equations are solved without any model for turbulentmotions: all turbulence scales are explicitly determined [95]. Thus, DNS is able topredict, for example, all time variations of temperature at a particular location inthe flow exactly like a high resolution sensor would measure them in an experi-ment. In DNS, pseudo-spectral methods [108, 119, 120] are the preferred numericalapproach (instead of the conventional finite differencing schemes employed in allother approaches discussed up to now) and the solution domain is represented byfinite Fourier series. The major drawback of the DNS method is its excessive com-putational cost since the number of grid points required to resolve all scales ofthree-dimensional motion is proportional to the nine-quarter power of the char-acteristic Reynolds number, Re9/4 [96, 121]. Nevertheless DNS studies have beenextremely valuable in supplementing fundamental understanding of turbulent flows[122, 123]. However, with computer times that exceed 200 h on a supercomputerfor flows of low and moderate Reynolds numbers and with 90% of the total effortdevoted to small scales of the dissipation range, rather than on the mean flowand the statistics of the energy-containing motions [108], the wide application ofDNS in practical problems, including those arising in the food industry, remainsquestionable.

Up to now, discussion in this section has been confined to single-phase flows asin the case of modelling of air flow around a particular food sample. Two-phasesituations as in the case of thawing, freezing and frying are treated as conjugateheat transfer problems with, for example, a RANS CFD approach for the fluid flowand by solving eqns (1) and (2) on the food sample.

However, practical problems in food engineering also involve the presence oftwo-phase flows in the sense of a dispersed secondary phase within the primarygaseous phase, e.g. the case of spray drying of milk powder and other instantbeverages [8], the creation of air bubbles inside liquid foods and sedimentation.Computational modelling of two-phase flows is usually defined in either an Eulerianor a Lagrangian frame of reference and has also considerably advanced over thepast 20 years. However, as in other aspects of CFD, applications of two-phase




CFD simulations in food engineering remain limited and mostly confined to theuse of the Lagrangian approach. The following paragraphs, and again for the sakeof completeness, provide a brief overview of two-phase CFD modelling. Specificreferences to applications of CFD in food engineering are provided in the next twosubsections.

In detail, in the Euler–Lagrangian approach, the fluid phase is treated as acontinuum by solving the time-averaged Navier–Stokes equations, while the dis-persed phase is solved by tracking a large number of particles, bubbles or dropletsthrough the calculated flow field. The dispersed phase can exchange momentum,mass and energy with the fluid phase [104] through the addition of extra terms inthe group of eqn (14). A fundamental assumption made in this approach is that thedispersed second phase occupies a low volume fraction, αp, per volume, V , definedas [124]:

αp =∑

i NiVpi

V(15)

where Ni is the number of particles in the size fraction i, having a particle volumeVpi. Since the sum of the volume fraction of the dispersed phase and the continuousphase is unity, the continuous volume fraction is αF = 1 − αP.

The particle or droplet trajectories are computed individually at specified inter-vals during the fluid phase calculation through the solution of ordinary differentialequations that account for the particle instantaneous location, �X and the linear, �upand angular velocities and may be expressed in vector form as [124]:

d�Xdt

= �up (16)

mpd�up

dt=∑ �Fj (17)

Ipd �ωp

dt= �T (18)

where mp is the particle mass, Ip the moment of inertia,∑ �Fj represents the sum

of the different forces acting on the particle (e.g. drag force) and �T is the torque.Equations (16)–(18) are usually solved by FD discretisation and application of thefourth order Runge–Kutta scheme and rotation is commonly neglected.

Additional sub-models accounting for further two-phase flow effects such asevaporation, inter-particle collisions and agglomeration can also be incorporated inthe Lagrangian approach [125–131].

The Lagrangian approach is appropriate for the modelling of spray dryers butquite unsuitable for the modelling of liquid–liquid mixtures, fluidised beds or anyapplication where the volume fraction of the second phase is not negligible. Notethat according to Sommerfeld [128] and Elgobashi [132], two-phase flows withvolume fractions of the order of, or less than, 10−3 are considered as adequatelydilute.




In the Eulerian approach, the different phases are treated mathematically as inter-penetrating continua [104]. Since the volume of a phase cannot be occupied by theother phases, the concept of a phasic volume fraction is introduced. These volumefractions are assumed to be continuous functions of space and time and their sum isequal to unity. Conservation equations for each phase are derived to obtain a set ofequations, which have similar structure for all phases. These equations are closedby providing constitutive relations that are obtained from empirical information,or, in the case of granular flows, by application of kinetic theory. There are threemain categories of Eulerian multi-phase models: (1) the volume of fluid (VOF)model, the mixture model and the Eulerian model; (2) the mixture model and(3) the Eulerian model. The following paragraphs provide an outline of each oneof the three approaches.

The VOF model is a surface-tracking technique applied to a fixed Eulerian mesh[133–135]. It is designed for two or more immiscible fluids where the position ofthe interface between the fluids is of interest. In the VOF model, a single set ofmomentum equations is shared by all fluids, and the volume fraction of each of thefluids in each computational cell is tracked throughout the domain. Applicationsof the VOF model include stratified flows, free-surface flows, filling, sloshing, themotion of large bubbles in a liquid, the motion of liquid after a dam break, theprediction of jet break-up (surface tension) and the steady or transient tracking ofany liquid–gas interface [104].

The mixture model is designed for two or more phases (fluid or particulate). Asin the Eulerian model described below, the phases are treated as inter-penetratingcontinua. The mixture model solves for the mixture momentum equation and pre-scribes relative velocities to describe the dispersed phases. Applications of themixture model include particle-laden flows with low loading, bubbly flows, sedi-mentation and cyclone separators.

Finally, the full Eulerian approach is the most complex of multi-phase models.It solves for each phase a set of conservation equations in the general form of (14).Coupling between phases is achieved through the so-called pressure and inter-phaseexchange coefficients and differs between granular (fluid–solid) and non-granular(fluid–fluid) flows. Applications of the Eulerian approach include bubble columns,risers, particle suspensions and fluidised beds [132, 136–139].

5.1 Single-phase flow problems

This section attempts to provide a review on the application of CFD in food engi-neering. As with FEs, application of FVs and CFD in the food industry remains afield of ongoing research.

Examples of the use of CFD in food engineering for calculating the heat transferrate from the surrounding air flow to specific food samples are provided in the worksof De Baerdemaeker and Nicolai [4], Moureh et al. [140], Olsson et al. [141], Huand Sun [5], Therdthai et al. [142] and Mirade et al. [85].

In detail, De Baerdemaeker and Nicolai [4] used commercial CFD software(CFDS) to calculate the effect of the stacking pattern of one or two packages on the




air flow field inside an oven with application to equipment considerations for sousvide cooking. They concluded that spatial variations in the flow field are indeedpresent and are dependent on the stacking pattern with implications on the heattransfer coefficient.

Moureh et al. [140] also used commercial CFD software (Phoenics) to assessthe effectiveness of insulating devices in reducing the rate of temperature changesduring product transportation on vehicles in the absence of refrigeration. The com-putational domain was three dimensional and comprised of one and a half pelletslocated at the rear end of a vehicle and covered by an insulating canvas. Each pelletwas represented as a succession of air and product layers. Inside the pellet heat wasassumed to be transferred by conduction. Solar flux, external radiative heat fluxand convective heat flux were exchanged between the pellet and the canvas. Themodel was used to estimate the tolerable time for transport or storage of sensitiveproducts and the computations were in excellent agreement with experimental dataalso presented in the article.

Two-dimensional steady-state calculations of slot air jets impinging on cylinder-shaped food products were carried out by Olson et al. [141]. Note that understandingof flow interactions and heat transfer characteristics from multiple slot jets is a topicof great practical implication during the heating, drying, freezing and cooling offoods as the high velocity of the air jets enhances the momentum, heat and masstransport. In the particular work of Olson et al. [141], commercial CFD software(CFX) and the k − ω turbulence model of Menter [102] were used to calculatethe flow field around one and two cylindrical food products. They concluded that theheat transfer to the cylinder surface is uneven, resulting in local dehydration ofthe product and that the use of single, two or three impinging jets and the distancebetween the jets’exit and the food product have a tremendous effect on the localheat transfer.

Emphasis on the local heat transfer rate and coefficients was also placed by Huand Sun [5] who used commercial CFD software (CFX) to predict the turbulentflow field and the heat and moisture transfer in a three-dimensional air blast chillercontaining cooked meats of cylindrical and elliptical shapes. Hu and Sun used fortheir simulations three turbulence models (standard k−ε, low Reynolds numberand RNG k−ε) and assessed the effect of the three models on the accuracy ofthe local heat transfer predictions. Results were compared with experimental datawhich showed that the RNG model provided the best selection due to its easinessin use, comparatively low computation time and good prediction accuracy.

Therdthai et al. [142] used CFD (CFDACE+ code) to simulate a two-dimensionalcross-section of a bread baking oven fitted with a burner and a convection fan. Fewdetails on the CFD simulation and modelling approach are provided in the article sothat it is unclear whether a laminar or turbulent flow was considered. The simulationprovided information on the temperature and air flow pattern in the oven and allowedfor a sensitivity study on the positioning of the controller sensors.

A detailed study on the transient thermal behaviour of large food chillers wascarried out by Mirande et al. [143]. A commercial CFD code (FLUENT) was usedto simulate the flow patterns inside a three-dimensional computational model of a




chiller filled with pork carcasses. Results revealed that the calculated air velocityagreed with measurements and that the use of CFD correctly depicts the effect ofa change in the ventilation level and direction on the air flow pattern. Heat andmass transfer coefficients around the carcass hindquarters with respect to the airvelocity fields were also determined and subsequently used into a chilling modelwhich permitted the calculation of chilling kinetics and carcass weight loss as afunction of the different ventilation conditions.

In addition to the CFD studies that analysed the temperature and flow fields insidefood-related practical applications, a number of CFD simulations were focused onthe particular problem of heat transfer in heat exchangers. For example, Grijspeerdtet al. [144] carried out two- and three-dimensional CFD simulations of the flowof milk between corrugated plates in the FINE-Turbo software environment [145].In all simulations the flow was considered as turbulent of a Reynolds number of4482 (based on a characteristic velocity on 1 m/s) and turbulent shear stresses weremodelled using the Baldwin–Lomax model [146]. They showed that CFD compu-tations can successfully identify regions with the heat exchanger where backflowsand thus higher temperature and fouling may occur and that CFD can provide auseful tool for optimal design of plate heat exchangers.

5.2 Two-phase flow problems

This section summarises previous attempts to predict two-phase flows with par-ticular application to food engineering. Note that here, as discussed in Section 5earlier, the term ‘two phase’ refers to a dispersed secondary phase within the pri-mary gaseous phase rather than the presence of two or more phases on the samefood sample. As already emphasised, most existing work with two-phase flow inthe food industry refers to industrial spray dryers for the drying of dairy and otherfood products. Further, here the focus is mostly on the understanding of the flowpatters inside the dryers rather than on the heat transfer inside the food sample andfor simplicity, most of the existing studies consider either water droplets or solidparticles of arbitrarily selected constant physical properties.

Notably, one of the earliest attempts to predict the flow pattern within a laboratoryspray dryer was reported in the pioneering work of Crowe [147] who used theParticle-Source-in-Cell model to calculate gas flow patterns and particle trajectorieswithin a laboratory spray dryer. Oakley et al. [148] predicted the flow pattern insidea pilot scale co-current spray dryer as a function of the angle of the swirl vanes inthe inlet annulus around the atomiser and reported periodic oscillations in the sizeof the recirculation zones in agreement with experimental data. Similar findingswere reported by Levesley et al. [149] and Langrish et al. [150]. Oakley and Bahu[151] using commercial CFD software (CFDS-FLOW3D) confirmed the existenceof low frequency oscillations for conditions where significant swirl in the inlet airleads to vortex breakdown and the formation of a central recirculation zone witha processing vortex core. They also predicted the trajectories of water dropletsinjected from a hollow-cone nozzle and noted cooling of the central gas jet in theregion below the atomiser and significant evaporation in a recirculation zone located




in the outer regions of the chamber. The existence of a self-sustained quasi-flappingoscillation inside a simple short-form spray dryer configuration was also confirmedby the more recent works of LeBarbier et al. [152] and Guo et al. [153].

Wall deposition in a co-current spray dryer with a rotary atomiser was examinedcomputationally by Goldberg [154] who suggested that powder deposition is greaterin the annular area around the roof and in the region below the height of the atomiserfor medium and larger droplets, respectively. Langrish and Zbicinski [155] alsoexplored the use of CFD towards the reduction of wall deposition by examiningthe effects of the inlet air swirl velocity and the spray cone angle on the particledeposition rate. Kieviet and Kerhof [156] simulated the air flow pattern (no spray)and the temperature and humidity patterns (water spray) in a co-current spray drier.Further, Kieviet [157] and Kieviet et al. [158] correlated wall deposition with theparticle residence time distribution and concluded that the time taken for particlesto slide down the conical wall of a co-current cylinder-on-cone dryer was the mostimportant factor in determining residence times in spray dryers with high walldeposition rates.

In a similar context, Goula andAdamopoulos [159] used a commercial CFD code(FLUENT) to assess the effect of feed concentration oil spray drying of tomato pulpand of pre-conditioned humidity levels. To this end they investigated the trajectoriesof evaporating water droplets inside a pilot scale co-current spray dryer fitted witha two-fluid nozzle atomiser as a function of the drying and atomising air veloci-ties. Their results revealed increased residue accumulation on the dryer walls withdecreasing the velocity of either the drying or the atomising air.

Huang et al. [160] also used commercial CFD software (FLUENT) to investigatethe effects of operating pressure, heat loss and inlet air conditions on the flowpattern, the particle trajectories and the temperature and humidity profiles insidea co-current spray dryer fitted with a pressure atomiser. They concluded that adecrease in the operating pressure or an increase in the ambient air humidity callsfor an increase in the air mass flow rate in order to obtain the same product capacity.In an accompanying article, Huang et al. [161] reported additional results in a co-current spray dryer fitted with a rotary disk atomiser.

The use of CFD to evaluate alternative spray dryer configurations was examinedby Southwell et al. [162] and Huang et al. [163]. In particular, Southwell et al. [162]used commercial CFD software (CFX) to investigate alternative ways of obtaininga uniform flow inside a co-current cylindrical spray dryer with a single off-axis inletpipe. Huang et al. [163] reported gas flow and temperature profiles, droplet trajec-tories, residence times and drying times in four co-current configurations (cylinderon cone, conical, hour glass shaped and lantern shaped) as a function of the particlewall BCs (i.e. trap or reflect). Their results revealed increased and decreased evap-oration intensities for the cylinder-on-cone and conical configuration, respectively,and a strong dependency upon the choice of particle wall BCs.

Closely linked to the effect of wall deposition is that of particle stickiness. Stick-iness is a phenomenon that reflects the tendency of some materials to adhere tocontact surfaces and/or to agglomerate [164]. Although wall particle stickinessis undesirable, the formation of larger particles as a result of droplet/droplet,




droplet/semi-dried particle, droplet/particle or particle/particle collisions is a neces-sity in a number of spray drying applications including milk powder, coffee anddetergents [165]. For these products, a larger and uniform particle size is generallypreferred as it facilitates collection, packaging and post-processing and can enhancesolubility. However, the understanding and successful modelling of the mechanismthat leads to coalescence or agglomeration has proven to be a most challengingtask due to the complex physics of the process and also due to a number of numer-ical difficulties. In particular, coalescence and agglomeration are governed by anumber of major phenomena [166]. These include (1) glass transition temperatureand moisture evaporation effects both of which influence particle stickiness andthus their propensity to agglomerate and (2) turbulent dispersion effects which pro-duce relative velocities between particles so that they approach each other and maypotentially collide.

Collision, coalescence and agglomeration can be modelled either via the Eulerianor the Lagrangian approach described in Section 5. In the former, coalescence ofdroplets is modelled by solving a population balance equation [167, 168]. In thelatter, particles trajectories are calculated by solving the Lagrangian equations ofmass and momentum with collision and coalescence modelled from semi-empiricalrelations stemming from experimental observations [169, 170] that relate the col-lision outcome to the Weber number, the impact angle and the diameter ratio ofthe colliding particles. Two-phase flow modelling and particular issues related tointer-particle interactions are a field of ongoing research.

Yanniotis and Xerodemas [171] used commercial CFD software (FLUENT) tosimulate the operation of a pad humidifier. The flow inside a ‘unit cell’, the repeatinggeometry structure, was solved for a wide range of air velocity magnitudes and forall three directions so that a library of the hydrodynamic losses was established. Thegoverning equations of the momentum, heat and mass exchange between the mainphase (air and water vapour mixture) and secondary phase (water) were solved forthe whole pad humidifier with the dispersed phase model.

6 Conclusions and outlook

The purpose of this work was to provide a general overview of the current state ofnumerical modelling in the food industry with emphasis on some of the details of thenumerical methods. From the selected cases presented herein, which by no meansoffer an extensive listing of the enormous amount of computational work currentlyundertaken by the food engineering R&D sectors, it has become clear that simpleone-dimensional numerical methods for the estimation of heat and mass transfer infood samples have reached an extensive degree of reliability. However these meth-ods are limited to simple geometries and are bound by a number of assumptions andapproximations. On the other hand, the evolution of FEs and CFD coupled with thenow well-established ability of setting up complex and unstructured meshes thatcan accommodate geometries of arbitrary shapes, currently provide a mature, use-ful, reliable and comparatively simple tool to extend knowledge and understandingon most food-related processes including momentum, heat and mass transfer, phasechange and multi-phase flow phenomena. To this end, an effort has been made to




highlight in this work not only methods and models that have already found appli-cation in the food industry (e.g. CFD by application of the k − ε model for thedetermination of the heat transfer coefficients) but also methods and models thathave either reached a certain degree of maturity or are still a subject of ongoingresearch on other disciplines (e.g. LES, DNS, two-phase flow modelling) but yetrelevant in many aspects of food engineering currently and in the near future.

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