Heat and Mass Transfer in Food

IMA Journal of Mathematics Applied in Business & Industry (1995) 5, 303-324

Heat and mass transfer in food processing

ROLAND W. LEwis AND K. N. SEETHARAMU

Institute of Numerical Methods in Engineering, University College of Swansea,Swansea SA2 8PP

1. Introduction

Historically, the food industry and its associated technology has been developed ona relatively qualitative and empirical level. It is only in recent years that managementand engineers have come to recognize that an advancement in knowledge wouldrequire a more fundamental and quantitative understanding of the underlyingmechanisms of their process. The food industry is also under continuing pressure toprovide food that is more natural and less processed, and also to provide food witheven higher levels of safety. There is also the issue of reducing energy usage andeffluent production to meet both environmental and economic goals (Johns 1992).Hence, the engineering of food processes is now undergoing a transition towards amore technical approach. The speed of product innovation in the fast movingconsumer-goods business, to which the majority of the food business belongs, isaccelerating. The half-life of product development times has decreased from 10 yearsin 1970 to what will be an estimated 2 to 3 years in the year 2000. The speed of theproduct/process development cycle is therefore of paramount importance (Bruin1992).

Meat and meat products worth in excess of £6000 million per year are sold in theUnited Kingdom, and it is estimated that the use of existing technology in the fieldof refrigeration could reduce the combined drip and evaporative loss by at least onepercent. This would result in a minimum saving to the meat industry of £60,000,000per annum. To achieve further savings, a more fundamental understanding of thefactors influencing heat and mass transfer within meat is required (Malton & James1984). Research has shown that the rates of reducing and subsequently maintainingthe temperature of the meat has important consequences in terms of microbiologicalsafety, eating quality, appearance, weight loss, and overall economies of the processingchain. However, there have been no legislative requirements, for internal trade withinthe United Kingdom, that defined specific meat temperatures. The new Food HygieneRegulations, 1990, were first implemented on 1 April 1991. As part of the process toharmonize legislation throughout the European Community, regulations that currentlyonly cover the export or intervention purchase of foods are being modified, andmostly will apply to internal trade by 1993. Among these regulations are some thatspecify the maximum internal temperature to which carcass meat must be chilledbefore cutting and transportation, maximum internal temperature after the freezingprocess, and maximum surface temperatures during thawing. Others specificallydefine the chilling rates for minced meat and poultry, temperatures during the storage

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304 ROLAND W. LEWIS AND K. N. SEETHARAMU

of offal, and display temperatures for quick-frozen meat. The legal requirements toattain and maintain specific temperature criteria will create an increased demand fordata on the relationship between environmental conditions and the temperaturehistory of meat in carcass, joints, and many other shapes. The penalty for notcomplying with these requirements is £20,000. Problems caused by biologicalvariability make it very time-consuming and expensive to produce all the datarequired by practical experimentation. Increasing predictive modelling techniques,usually in combination with limited experiments, are being used to generate the dataand optimize the process (James 1992).

Many of the problems associated with the manufacture and storage of food aresimilar to those found in a number of other process industries where computationalfluid dynamics (CFD) is already a useful tool. The time is now right to adapt asmuch of the CFD experience as appropriate. This will bring significant benefits fora relatively low investment. CFD can provide benefit to the food-processing industryin the areas of drying, mixing, cooking (baking), refrigeration, and clean-roomconditions. Important examples would be in the drying area, the performance ofspray driers, and also the ability to solve problems of heat and mass transfer in food,along with conjugate problems where it is important to avoid the specification ofunknown surface transfer coefficients, in batch-mixing vessels, in modelling of thephase-change problem of baking, and in the simulation of a cold display cabin forrefrigeration (Quarinip 1992).

Today, consumers are adventurous in their purchases of frozen food, and the trendis for the manufacturer to develop convenient products of high added value. Thistrend has been brought about by significant changes in society such as the increasingproportion of women going out to work and the greater social demand forleisure-time activities. These factors have brought about a general trend towards lessformal meal times, with the replacement of traditionally prepared meals by foodsthat are easily stored and quick and convenient to prepare. These evolving socialpatterns have in some measure been made possible by the corresponding improve-ments in freezing, storage, and distribution technology (Summers 1984). Figure 1shows the worldwide production of processed food (King 1984).

As another example of the food-processing industry, we consider confectionery—inparticular, the manufacture of chocolates. The processes required for the productionof chocolates are very complex. One commonly used technique is moulding, whereconditioned liquid chocolate is cast in moulds carried through a cooler by a chainsystem. The blocks are then demoulded and transferred, at rates up to 2000 perminute, to a battery of wrapping and cartoning machines. Many products consistof centres contained within an envelope of chocolate, these centres being a varietyof confectionery materials such as creams, caramels, or fudges, together with otheringredients such as nuts, raisins, cereals or biscuits. Each type of centre requires itsown special process (Greeves & Knott 1984).

As a further example, beverages, fruits, and vegetables are often stored in cans. Acomputerized mathematical model has been developed to evaluate the thermalprocessing of low-acid foods (e.g. some vegetables) in cylindrical plastic cans. Thesterilizing values are found to be more affected by the wall thickness than the thermaldiffusivity of the wall (Shin Seonggyun & Bownik 1990). Natural convection heating

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HEAT AND MASS TRANSFER IN FOOD PROCESSING 305

- 100

- 10

1Million

01 tonnesper year

0.01

0.001Year 1530 1940 1950 1960 1970 1980 1990 2000

FIG. 1. World-wide production of processed food.

of a canned liquid food during sterilization has also been simulated by thefinite-element method (Kumar & Bhattacharya 1991).

Ohmic heating is the term used to describe the electrical resistance heating of foodby the passage of electric current, usually through a continuous flow of food. Themajor benefit of this process is that heating takes place volumetrically, and theproduct does not experience a large temperature gradient within itself as it heats,even when particulates are present. This obviates the need to process the liquid phaseexcessively to ensure heating of the core of large particulates, as happens whenconventional heat-transfer equipment is used, such as in can heating by pressurecooker. This results in considerably less heat damage to the fluid and the eliminationof overheating and excessive softening of the outside of the particulates. The finishedproduct is therefore of higher quality. Skudder (1991) describes the development andcommercial application of ohmic heater systems, and includes information on theextensive experiments for process validation to ensure that the consumer is providedwith safe and wholesome food.

There has been a growing interest in the use of microwave energy for pasteurizationand sterilization of food. Since its discovery by Fleming during the second world war,distinct advantages have been found for the use of microwave heating as comparedwith conventional heating methods. These have been exploited to a limited extent infood processing. Rapid product heating by immediate internal heat generation anda more uniform heating of homogeneous products may be obtained, in contrast to theoverheating of the surface layers or underheating of the centre region common inconventional heating methods with slow heat penetration. Centre temperaturesnecessary for microbial destruction are more quickly generated. This has led to thedevelopment of microwave food processes with greater retention of heat-labilenutrients (e.g. vitamins) and flavour constituents and with processing times as littleas 5-10% of those obtained in conventional heating. As the differential in capital andenergy cost between microwave and conventional heating equipment continues to

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3 0 6 ROLAND W. LEWIS AND K. N. SEETHARAMU

narrow, the economic incentives for microwave process development will increase(Mudgett & Schwartzberg 1982; Dutta & Liu 1992).

A new concept for the concentration of juice, called diffusion concentration, hasbeen developed in which the juice percolates downwards through a packed columnwith dehumified air flowing counter-currently upwards. This technology avoids thethermal degradation of nutrient compounds and the loss of valuable aromaticvolatiles that occurs at high temperatures. The operating conditions, includingsuitable hydrodynamics modelling the two phases of juice and dehumidified air, havebeen mathematically investigated (Xu et al. 1990).

The cryoflow process is a new process for the production of frozen free-flowingdroplets of liquid food products which are otherwise frozen in plastic pouches, pots,or similar containers having freezing times of the order of 10-20 minutes for productsin quarter-litre plastic pouches. The process involves the injection of food products(fruit juices, yoghurts, dairy cream, etc.) into streams of liquid nitrogen. Thereafter,the food forms pea-sized droplets, which are crust-frozen in less than 10 seconds.Freezing of the product is completed in the gas phase of the nitrogen, resulting in aclosely controlled product output temperature and efficient usage of nitrogen. Thecryoflow process has considerable potential both for producing products of highquality and value and for adding value to some of the byproducts of the food industry.In the baking and confectionery industries, there are many applications for cryoflowproducts. Quality, flexibility, and low capital have always been three of the greatadvantages of liquid-nitrogen freezing systems (Taylor 1984; Rosen 1990).

To achieve considerable savings and to optimize the thermal processing of solidfoods, it is essential to have a more fundamental understanding of the factorsinfluencing heat and mass transfer in porous bodies. Luikov's coupled partialdifferential equations for heat and mass transfer can be used to describe themultiphase distribution in porous media, for both the freezing and drying processes.The formulation of these coupled equations is based on the principles of theconservation of heat and mass transfer and irreversible thermodynamics. A detaileddescription and derivation of the equations involved has been presented by Luikov(1975) and Thomas et al. (1980). Due to the complexity of these equations, anyanalytical solution is mostly restricted to simple one-dimensional forms (Mikhaelov1976; Dural & Hiens 1990; Salvador & Mascharoni 1991). The effective speed ofelectronic computation has approximately doubled every year over the past 30 years,and the trend is expected to continue over the coming decade. As this rapid increasein computational capabilities enables solutions to be obtained for increasinglycomplex problems, numerical methods, especially the finite-element technique, arebecoming evermore widely accepted for solving partial differential equations.

Comini & Lewis (1976) successfully applied the finite-element method to solve thelinear form of the Luikov equations for heat and mass transfer in two dimensions.Some typical examples of the freezing and drying problems are used to demonstratethe feasibility of their solution scheme. Later the fully (and partially) nonlinear formsof these coupled equations were solved by Thomas et al. (1980), with examples ofstress induced in the drying process of timber. Volume changes also occur duringfood processing, and a number of diffusional models take this change of volume intoaccount (Fusco et al. 1991; Hawlder 1991; Rohman & Lai 1990). An implicit

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finite-difference method was used to integrate the partial differential equations(Sereno 1990) in order to obtain both temperature and moisture profiles. Sereno'ssimulation predicted the experimental results for the drying of apple slices and carrotcubes to within 1.1% of moisture content and 12% of the drying rate.

Many solid foods are first chilled and then freeze-dried for storage purposes. Inthe above two processes, convective heat exchange between the surrounding mediumand the food has been the sole method of heat transfer. However, in 'vacuum'thawing, very high heat transfer coefficients are produced by the condensation oflow-pressure steam on the surface of the meat being thawed in an air-free chamber.A mathematical model, including volumetric changes for food freezing or thawing,is obtained by modifying the general heat-conduction equation. The shape of thefood is assumed to be a body of revolution (Sheen & Hayakawa 1991). A numberof procedures are used to predict the freezing time of food, and experimentalmeasurements can be made to test the accuracy, adequacy, and applicability of thevarious mathematical models, e.g. the freezing time of a small cuboidal foodproduct—viz. french fries—see LeBlanc et al. (1990). In part 2 of the same paper,nineteen mathematical models are compared with experimental results. Threeempirical models and one approximate model yielded estimates within 10% of theexperimental freezing time. Other empirical and appoximate models gave answersgreater than 10%. Neither of the two exact models examined were applicable to thefreezing of a small food product when the surface heat-transfer coefficient was finite.

The effective average heat-transfer coefficient is a complex function of manyphysical and experimental variables: flow conditions; velocity profiles; food shapeand orientation; air temperature, velocity, and humidity; radiation; water evaporation;respiration; and phase change—all these have an effect upon the transient heat lossfrom the food product, and hence affect the effective heat-transfer coefficient. Thelocal values vary over a wide range, and thus the average value is only value that ispractical from a design consideration. The value of the heat-transfer coefficient isbased on experimental methods of measuring both the surface and centre temperaturesof the food product, e.g. apples or potatoes (Stewart et al. 1990; Ibrahim 1991).Heat-transfer coefficients are determined in the case of fixed-bed and fluized-bedfreezers with the aid of computer models (Kairullah & Singh 1991). A surface-boilingboundary condition is encountered in the freezing of foods that are immersed inboiling freezants, such as R12. This phenomenon may be incorporated in amathematical model of the freezing process as a surface-temperature-dependentconvective boundary condition (Evans et al. 1991). The determination of accuratethermophysical data for moist food materials under various conditions is animmediate necessity for a better prediction of food processing (Narayana & Murthy1981).

Most foods spent the greatest part of their lives in some form of 'package'—bag,box, sleeve, folded paper/plastic/foil, sealed paper/plastic/foil, can, bottle, and so on,often in multiple layers in single or multiple applications. The interrelationshipbetween the processing and packaging is often ignored, or given little thought,although some major companies are now realizing that the whole system, from theinput of raw material to the point of consumption is, in fact, one long process,requiring a total-integration approach by engineers and scientists of all disciplines

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(Batterly 1984). Emulsive edible film on food products is often used to control masstransfer, moisture movement, oxygen transfer, and preservative migration. Themicrovoid model and micropathway model are used for such mass transfer (Krochta1990). Chemical and colour changes in tomato paste of 26% total solids, as influencedby storage temperature duration and type of packaging material, have also beeninvestigated (Luch & Sharpe 1982). Mathematical models for the prediction of thebehaviour of food products stored in flexible packaging materials have been presentedfor the case of moisture diffusion and adsorption. These models can be used for aprediction of the shelf life of various food products and for the optimization of flexiblepackaging materials (Khanna & Pappas 1982).

In the sections to follow, the Luikov system of partial differential equations, whichcalculates the distribution of temperature, moisture potential, and pressure within acapillary porous body such as food, is given. Followed by this, some examples of thefood processing are presented.

2. Analysis

2.1 Governing system of equations

The governing system of equations which describe the variation of temperature,moisture potential and pressure within a capillary-porous body, as used by Lewis& Ferguson (1990) and Ferguson & Lewis (1991) to solve practical engineeringproblems are

Kl2V2U + Kl3V

2Pl2

(1)

* dt

Cp — = K3lV2T + K32V

2U + K33V2P

whereC, = pocq Cm = p o c m C p = pocp

^ i i = (kq + eXkm6') Kl2 = ekkm K13 = e-i/cp

^23 = kp

(see Table 1 for nomenclature).

2.2 Boundary conditions

The boundary conditions associated with these equations are

T=TW onr\ (2)

dT— ~ -c/ f l) = 0 onf 2 (3)

K21 = M ' K22 = km K23 = kp

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TABLE 1Nomenclature

P total pressure of humid air inside the body, kN/m2

T temperature, ;CU moisture potential, °MV" element deformation velocity (m s " ' )cm moisture capacity, kg/kg °Mcp coefficient of humid air capacity, kg/kg • n • M- 2

cq heat capacity, J /hr • m • kj , density of mass transfer flow

j , dif density of diffusion mass transfer flowj , fil density of filtration mass transfer flow

jq heat fluxkm coefficient of moisture conductivity, m/hr 2

kp moisture filtration coefficient, k g m / s k Nkq coefficient of thermal conductivity, W/m- 'Cau convective mass transfer coefficient, kg/rrr-s-cMat, convective heat transfer coefficient, W/"C-m2

6 thertnogradient coefficient, "M Ke ratio of vapour diffusion to total diffusion

p0 dry density, kg/m3

/. latent heat of vaporisation of water, J/kg

U = UW on T3 (4)

km ^-+jm + kj ^ + au(t/ - 14) = 0 on T4 (5)en en

P = PW. on T5. (6)

Equations (2), (4), and (6) represent the portion of the material boundary where aconstant temperature, moisture content and pressure, respectively are applied.Equations (3) and (5) represent the portion of the boundary to which a specifiedheat flux or moisture flux is applied. This set of boundary conditions can be writtenin the more generalized form

(7)

(8)

(9)

(10)

P = PW on T5 (11)

where

r =

u =

K22

Tw

dT

IT

dU

~dn~ +

,* = o

* = 0

on F,

on F2

on F3

on F4

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3 1 0 ROLAND W. LEWIS AND K. N. SEETHARAMU

A,-——

p~K2I5'*2I5*r

J*. = K? •> I •—

2.3 Sublimation front tracing scheme

Freeze-drying can be closed as a Stefan problem, where instead of melting, sublimationtakes place and the moving boundary is the interface between the frozen materialand the dried material. Initially, when all the material to be freeze-dried is frozen,the moving boundary is situated at the sample surface and as the drying processprogresses with time, the frozen/dried region interface advances through the materialsample. Once the interface has advanced through the sample, when no furthersublimation takes place and only a dried region remains, the sample is further drieduntil the desired residual moisture content is obtained.

Therefore, in order to be able to model the freeze-drying process the exact locationof the sublimation interface must be calculated. This is done by studying theequilibrium of heat energy existing at the advancing sublimation interface. Onassuming that there are no heat sources along the interface, then the heat balanceequation may be expressed as

dT,dried

dx dx~~ / '-PfrozenV int

x = b

(12)

The first term in equation (12) describes the heat flux into the interface from thedried region whilst the second term describes the heat flux from the interface intothe frozen region. The third expression describes the heat given oft" at the interfacedue to the phase change caused by sublimation.

The mesh deformation is accounted for by introducing the finite-element shapefunction as an implicit function of time. Hence, the node movement A^T), is given

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by the following expression:

••N,(X,T). (13)

The approximation function, T(x, x), and its derivative with respect to time are

and

dx [_dr dx

The derivative of the shape function with respect to time, described in equation (15),is used to detect the distortion of the element due to the moving front. Thedeformation of the element in the global coordinates can be mapped into the localcoordinates by using the isoparametric transformation, such as

I i f ) . (16)

The element deformation velocity, V, is given by

The mesh deformation in the global axes, described by equation (17), can be expressedin terms of the element deformation velocity, Vc, and gives:

^ . (18)OX

By substituting equation (18) into equation (15), the time derivative of the approxi-mated function will be

dx ^ dT '

In the absence of mesh deformation, equation (19) will become the conventionalfinite-element equation with the shape function depending on the spatial domain. Theapplication of the finite element discretization will derive a total differential equationof the unknown function, T, with respect to time. When replacing the traditionalshape function defined in equation (13), the resulting approximated function willhave an additional term, equation (19), which can be viewed as the convective effectsdue to mesh deformation.

If the finite-element technique and a suitable weighting procedure are applied toevaluate the set of parabolic equations, the resulting total differential equations willcontain a convective term in the stiffness matrix, which is similar to that derived fromthe convection-diffusion equation.

One of the advantages of using the space-time shape function is that the existingfinite-element program can be easily altered to accommodate the mesh deformation.The moving interface is described by the Stefan boundary condition, equation (12);

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its location can be obtained at each timestep by any finite-difference approximation.Hence

(20)AT

If the front location at the nth iteration is given, the new position can be derivedby iteration according to the following equations:

/•n+l _ fn , A T \U Sfrozen , ^Sried] ,.,.,J —J + ~ "^frozen : K<jdried ' "", \L' >

/.Pol dx ex J

with

'--'dried

dx

?Tv * frozen

= adx

„ frozen__ j f • •

dx

>zen

( 1 -

n

dx

-a ) -

n1

• ^ d r i e d

dx

° 'frozen

dx

1. (22)

This method was used by Lynch & O'Neill (1981) to solve a one-dimensional Stefanproblem where both one-phase and two-phase numerical results were found to bequite accurate in comparison with analytical solutions.

2.4 Finite-element formulation

The variation of the temperature, pressure, and moisture potential throughout thedomain of interest, Q, is approximated in terms of the nodal values, Ts, Us, and Ps.If the approximation given by equations (19) and the approximations for the workingvariables, temperature, moisture potential, and pressure, are substituted into equations(1) a residual is obtained, which is then minimized using the Galerkin method. Thisrequires that the integral of the weighted errors over the domain, £2, must be zero,with the shape functions, Nr, being used as the weighting functions.

AU V-(K,,Vr) + V(Kl2VU) + V(Kl3VP)-Cq?f dfi = 0

Nr\ V-(X21Vr) + V-(K22VU) + V(K13VP)-Cm— dQ = 0Jo L st jC r 3P~\

Nr\V(K3lVT) + V(K32VU) + V(K33VP)-Cp— dft = 0.Jn L ct J ,

The application of Green's theorem (integration by parts) and the introduction ofthe generalized boundary conditions to equations (23) produces a system ofdifferential equations which may be written in matrix form as

(23)

—dt = 0 (24)

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HEAT AND MASS TRANSFER IN FOOD PROCESSING

where K(<t>). C(<D). and C'(<D) are solution-dependent matrices.

K = K. •) [

^ 3 2

C =

C' =

0

0

0

0

T~

V

p

0

cm

0

0

0

0

(

o"0

0.

0

0

J =

Typical matrix elements are

crs= t f c/vr/'•1=1 J n *

r.s=l JoC'/VrK"V/VsdQ

[>-=i J r

4=1 [r=\ J f

313

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2.5 Timesteppiny algorithm

The numerical solution of equation (24) is achieved by using the Lees three-leveltimestepping scheme (Lees 1966), which employs the finite-difference technique intime.

($" +' 4. d>" + fly - ') (fl>"+1 _ d)""1) ((D"+1 -(- cp" + O""1)

3 + 2Ar + C 3 +

(25)

The superscript ;i refers to the time level whilst Ax refers to the timestep. The Leesthree-level timestepping scheme has the advantage of solving for the time level ;i+ 1,by evaluating the coefficient matrices at time level n, which avoids the necessity foran iterative solver. Wood (1978) and Wood & Lewis (1975) showed that this schemewas stable although, when a convective boundary condition was used, oscillationsappeared in the solution. The noise can be dampened to an acceptable level byintroducing a maximum permissible time step.

3. Application

3.1 Freeze-dryiny of coffee solution

The results presented in this section are a comparison of experimental and com-putational results for the freeze-drying of a 3.6% coffee solution (Ferguson et al. toappear). The coffee solution was made up by dissolving coffee granules in water untilthe desired concentration was attained. For a 1 kg coffee solution sample, 36 g(336 cm3) of coffee granules were dissolved into 964 g (964 cm3) of water. The coffeesample to be freeze-dried was placed into the material holding tray within the dryingchamber to a uniform depth of 10 mm. The solution within the tray is idealized as

TABLE 2Material properties for the freeze-drying of a 3.6% coffee solution

Materialproperty

Po

km

ci

CP

6

s

S.I. units

kg/m3

J /hrmKkg/hr-m-N-m"2

m/hr2

J /hrmKkg/kgn-M"2

—°MKW/m2oCkg/m2s-°M

Driedgranular

34.42

39.96

1.7* 10"6

3.72* 10"3

2600.0

8.0* 10"2

0.1

0.01

0.865

8.78* 1 0 ' 3

Frozen

962.0

720.0

2.0* 10 " 8

5.0* 10"6

1900.0

8.0* 10" 2

1.0

0.001

—

—

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B

43 59 75 91 107 123

FIG. 2. Finite-element discretization of the model for the 3.6"; coffee solution.

an infinite sheet, hence, the numerical simulation of the sublimation front becomesa uni-dimensional problem, since edge effects are ignored.

The material properties employed during the numerical simulation of the freeze-drying problem were obtained experimentally and assumed to be constant throughoutthe freeze-drying process. The material properties used are shown in Table 2.

The initial conditions of the frozen coffee sample, before the freeze-dryingoperation commences, are temperatures — 35 3C, moisture content 96.4% and pressure0.5 kN/m2. In order to ensure that sublimation, and not melting, takes place thepressure within the drying chamber must be held below the triple point pressure ofwater, such that the pressure/temperature path crosses the sublimation/ablimation.The heater, in this example, was placed 15 mm away from the upper surface of thefrozen coffee sample and held at a constant temperature of 300cC. A radiative heatingtype boundary condition was applied to temperature and a convective type moistureboundary condition was applied, with a steady-state equilibrium moisture contentof 4%.

The finite-element mesh used in the numerical modelling of this freeze-dryingexample is shown in Fig. 2. The boundary conditions were applied along the faceAB, whilst all other faces, BC, CD, and DA were assumed to be insulatednon-conducting boundaries. Thermocouples were placed at locations within thecoffee sample to measure the temperature, and coincided with nodes 3, 43, 59. 75,91, 107, and 123 of the finite-element mesh.

Figure 5 shows the variation of total moisture content of the coffee sample withtime. Again for the first 25 to 30 hours of the freeze-drying operation the totalmoisture content within the body decreases linearly with time, indicating that thesublimation interface advances with a constant velocity through the coffee sample.It is not until after sublimation has ceased, and drying continues, that the moisturecontent within the sample approaches its residual value. The numerical simulationof the problem compares favourably with the experimental results, although, thenumerical model predicts a slightly faster drying time than was actually achievedexperimentally.

Figures 3 and 4 show the variation of temperature with time at nodes 75 and 123of the finite element mesh. Again, in proceeding through the sample the temperaturetakes longer to reach steady state equilibrium values, thus giving an indication ofthe position of the sublimation front with time. The numerical solutions of temperatureagainst time compare well with the observed experimental values, which on inspection

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60

50

40

O 3 0

« 20

| 10

o ExperimentalComputational

5 10 15 20 25 30 35 40 45 50Time (hours)

FIG. 3. Temperature versus time at node 75.

6050

40

8 30

1 2°| 10

I o-10

-20

-30- 4 0 '

——

—

° Experimental— Computational o o

5 10 15 20 25 30 35 40 45 50Time (hours)

FIG. 4. Temperature versus time at node 123.

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o ExperimentalComputational

0 5 10 15 20 25 30 35 40 45 50Time (hours)

FIG. 5. Total moisture content versus time.

of the figures, tend to be scattered. However, the numerical results compare veryfavourably with the overlying trend of the experimental solution.

3.2 Seafood processing

Seafood items such as fish, shrimps, prawns are shipped from India to most of theEuropean countries. During transit, these items are to be preserved by passing coldair at a temperature of around 5°C and at high relative humidity over the seafood, tomaintain the quality. This involves both the heat and mass transfer between theseafood and surroundings.

Before processing the fish, the head and tail are removed and its trunk is gutted.In order to simulate the actual conditions one such specimen is taken and its

FIG. 6. Discretization of the model.

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longitudinal section investigated. Uniform boundary conditions were assumed alongall the boundary surfaces with constant temperature and moisture potentials. Thethickness of the fish was assumed to be constant. The longitudinal section of thespecimen was then discretized into finite elements and is shown in Fig. 6. The nodesare numbered in such a way that the bandwidth in the global matrix was a minimum.

The values of the thermophysical properties (Khumbar 1984) of fish, which wereassumed to be constant, are given below.

k= 1.00498; A:m = 8.90 x 10 - 3 .

c, = 7.429 x 106; cm = 4.1598 x 106.

The conditions under which the fish was processed are,Initial conditions

Temperature = 35 JC

Moisture potential = 45°C.

Environmental conditions

Case 1 Case 2

Temperature 5°C 0.5°C

Moisture potential 125°M 125.0°M.

The temperature and moisture potentials corresponding to selected nodes on thesurface and interior or the body are plotted as a function of time in Figures 7 and 8.

Figure 7 shows the temperature and moisture potential variation for an environ-ment of 5°C and 120°M. Figure 8 shows the temperature variation for nodes 3, 16

400 10 10000100 1000

Time (seconds)

FIG. 7. Temperature and moisture variations at nodes 3, 16, 10. and 12.

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p

lera

ture

<T

emj

35

30

25

20

15

10

5

0

— V * ^Temperature without ^ / ^

"moisture potential ^ c ^ v

\ \\

3.

Moisture p o t e n t M ^ ^

i i

16 X

10^<

|

t =5"C

\ Temperature withV\ moisture potential

Si -OT -

170

—1150

1 3 0 -

110

90

70

50

30

ial

oten

t

aa>

.2O

10 100 1000Time (seconds)

10000

Flu. 8. Temperature (dashed curves), neglecting moisture migration, compared with temperature curvesaccounting for moisture migration.

(on the surface) and 10, 12 (interior) as a function of time. It is seen from Fig. 7 thatfor a period of 100 sees the temperature drop is small for the surface nodes 3, 16whereas the temperature at the interior nodes 10, 12 remain constant up to 400 sec.After these time limits, the temperature drop is considerable. The temperature valuesat the surface approach the environment temperature after 7000 sec whereas theinterior values remain higher than the surface temperatures by 3CC. Figure 7 showsthe moisture potential variation for the same set of nodes. It can be observed thatthe moisture potential remains almost constant for all nodes up to a period of 100 sec.After this, the moisture potential for the surface nodes increases rapidly with time asexpected and approaches the environmental value in about 4000 sec. The moisturepotential at the interior node remains (node 12) constant for about 4000 sec and thenstarts increasing.

Similar trends are observed in Fig. 8 for the temperature and moisture potentialswhen the environment of the processing is maintained at value of OS'C and 125M,respectively.

When moisture migration is neglected in the analysis the results are shown in Fig.8, indicates that the temperature was lower (after 10 sec). The time required was alsoless compared to the case when moisture migration was accounted for.

3.3 Hamburger processing

As a third example, the processing of a hamburger was considered. Because of thesymmetry only one-fourth section analysed. The initial conditions in the hamburgerwere assumed to be — lS^C and 100cM. The results obtained for the temperatureand moisture potentials in the hamburger are shown in Figs 9 and 10 for two instants

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Temperature at O.Olhrs

x xX X

x x x x x x x x x x x xX X X X X X X

X X X X x x x x x x x x x xx x x x x x x xx x x x x x x x

X X X X X X X X X

X X X X X X Xx x x x x x x x x x x x x

X X X X X X Xx x x x x x x x x x

x x x x x x x x xx x x x x x x x x x x

X X X X XX X X X X X X

X X X X X )x x x x x x x x x

X X X X X X XX X X X X X X

K X X X X X XX X X X X X X

X X X XX X X X X X X X X X X X X XX X X X X X X X X X X X X 3

X X X X X X X X X X X X X X

x x x x x x x x x x x x xX X X X X X X X X X X X

K X X X X X X X X X X X XX X X X X X X X X X X X

X X X XX X X X

x x x x x x x x x xx x x x x x x x x x xX X X X X X X X X X

x x x x x x x x x x x x x x x x x x x x x x xC X X X X

x x x x x x x xx x x x x x x x

X X X X X X X X X X X X X

x x x x x x x x x x x x x x xX X X X X X X X X X X Xx x x x x x x x x x xx x x x x x x x x x

X X X X X X X X X X X X X XX X X X X X X X X X

X X X X X X X X X X XX X X X X

K X X X X X X X X X X X X X X X X X X X X X X XX X X X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X XX X X X X X

x x x x x x x x

X X X >X X X X X

X X X X X >X X X X X X

X X X X XC X X X XX X X X X

Temperature at 0.0341usX X X X

X X X XX X X X 20-C

19'C Initial temperature - - 18'C

FIG. 9.

of time during the processing. Thus, any specified condition, either at the surface orat the centre can be achieved by suitable processing environments.

4. ConclusionsThe importance of the coupled phenomena of heat and mass transfer in foodprocessing is highlighted along with the ensuing commercial aspects. A numericalmodel is presented to calculate the distribution of temperature, moisture, and pressurewithin a capillary-porous body, according to the system of partial differentialequations as defined by Luikov. Examples of the freeze drying of a coffee solution

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Moisture potential at 80.25hrs

Moisture potential at 205.25his

Initial moisture potential - 100'M

FIG. 10.

and the processing of both seafood and reconstituted meat products are given toillustrate the application of the theory.

A versatile three-dimensional heat and mass transfer model of food processing with

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freeze drying, thawing and shrinkage prediction capabilities is an urgent need forachieving a better simulation of such phenomena. The surface heat and mass transfercoefficients can be generated by the application of CFD software for better andeconomic simulation of food processing. There is a need to evaluate the properties offoods under various conditions with a limited number of experiments but gainfullyusing techniques such as inverse methods.

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Heat and Mass Transfer in Food