finite element modeling of heat and mass transfer in food materials during microwave heating

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ELSEVIER

Journal ofFood Engineering 25 (1995) 509-529

Copyright 0 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved

0260~8774/9S/S9.50

0260-8774(94)00032-B

Finite Element Modeling of Heat and Mass Transfer inFood Materials During Microwave Heating - Model

Development and Validation

L. Zhou,” V. M. Puri,” R. C. Anantheswaranb & G. Yehh

“Department of Agricultural and Biological Engineering and “Department of Food

Science, The Pennsylvania State University, University Park, PA 16802, USA

(Received 3 September 1993; revised version received 20 April 1994;

accepted IO June 1994)

ARSTRA CT

A three-dimensional finit e element model (FEM) w as developed to

predict temperature and moist ure distributions in food mat erials during

microwave heating. The FEM w as tested w ith analyt ical solutions andcommercial softw are (TWO DEPEP, AN SYS) calculated values. The

FEM predictions compared favorably w ith analy tical solutions (wit hin

O@i6% of maxim um temperature) and values calculated from commer-

cial softw ares (wit hin 014% of maximum temperature). The three-

dimensional FEM w as also verified using experimental data from

microw ave oven heated cylinder- and slab-shaped potat o specimens. A

,fluoropt ic temperature measurement syst em and the near infrared (NIR )

technique w ere used to measure temperature and moist ure distributions,

respect ively. The FEM predicted temperature in pot ato sam ples agreedw ith measured result s. The absolute m aximum difference for slab

geometry aft er 60 s of heating w as 81°C (or relat ive difference of 155%

from the measured value), w hereas, for the cylindrical geometry, it w as

87°C (or relat iv e dif lerence of 11.4%). The absolut e moist ure di fferences

afrer 60 s of heating betw een FEM predictions and m easured values for

potat o slab and cylinder w ere w ithin 1.97% w et basis (or relativ e differ-

ence o f 24%) and 1.85% w et basis (or relat iv e dif ference of 2*1%), respect -

ively.

NOTATION

A Area (m’)

A, Surface area (m’)

509



5 1 0 L. Zhou et al.

Specific heat (W/m2 “C)Capacitance matrixMoisture diffusivity (m’/s)

Forcing functionEnthalpy &J/kg)Thermal conductivity (W/m “C)Stiffness matrixLatent heat (kJ/kg)Moisture content (kg/kg)Environmental moisture (kg/kg)Initial moisture content (kg/kg)Surface moisture content (kg/kg)Shape function

Power (W )Power at material surface (W/m2)Power in X, y and z directions (W )

Heat generation (W/m3)Radius of cylinder (m)Temperature (“C)Surface temperature of sample (“C)Air temperature in microwave cavity (“C)Initial temperature (“C)Volume (m3)

Distance (m)Convective heat transfer coefficient (W/m2)Surface mass transfer coefficient (m/s)Evaporation rate (kg/s)Unit outside vector of the surfaceVapor pressure (Pa)Heat flux (W/m2)Radial coordinate of cylinder (m)

Time (s)

Weight of sample (kg)

Weight of water (kg)x coordinate value (m)y coordinate value (m)I coordinate value (m)Width of slab (m)Height of slab (m)Height of cylinder or length of slab (m)

Evaporation heat (kJ/kg)

Attenuation factor (dimensionless)IncrementDielectric constant (dimensionless)Dielectric loss factor (dimensionless)Wavelength of microwaveGradient operatorDensity (kg/m3)



Heat and mass transfer during microwave heating 511

INTRODUCTION

Currently, over 80% of American families use microwave ovens. Micro-

waveable foods are becoming increasingly popular in the market place.Nationwide, sales of microwave-packaged foods reached nearly $3000 millionby the end of 1992, up from $900 million in 1987 and $53 million in 1983(Nestleroth & Ciresi, 1992). However, there are problems associated with theuse of microwave energy: (1) unsatisfactory product quality - non-uniformtemperature, rubbery or soggy texture in the end product and unacceptableflavor development; (2) concerns about insufficient microbial destruction due touneven cooking; (3) safety hazards such as over-heating of the center in infantformula bottles. These and other problems stem from the lack of sufficientunderstanding of simultaneous heat transfer, mass transfer, chemical reaction

and biological degradation occurring during microwave heating.Studies related to the modeling of the microwave heating process started with

Ohlsson and Bengtsson (1971). They used the finite difference technique tomodel microwave heating in infinite slabs of salted ham and beef. Temperatureprofiles from numerical model prediction and experimental measurementcompared favorably. Swami (1982) also used the finite difference method todescribe microwave heating of high moisture foods in cylindrical and rectangu-lar shaped food materials. The model predicted temperature distribution was ingood agreement with experimental measurements for gel samples of highmoisture content and different salt concentrations. Heat and mass transfer

analysis during microwave heating was conducted by Tong (1988) and Wei et al.(1985). The test materials used were bread, muffin and biscuit, and sandstone.The model predicted temperatures compared favorably with experimentalvalues, e.g. values were within 5% of each other in Tong’s work. Both Tong( 1988) and Wei et al. (1985) used one-dimensional finite difference formulation.

The finite element method has a number of advantages over the finite differ-ence method (Puri & Anantheswaran, 1993). It has been used to model micro-wave processing of foods. Lin et al . (1995) used the commercial softwareTWODEPEP to predict temperature distribution in agar gels. Center tempera-tures in cylindrical samples were reported to be higher than those at other loca-

tions within the cylinder. Chen et al. ( 1990) derived and incorporated a heatgeneration term into an axisymmetric finite element model to analyse tempera-ture distribution within a cylinder-shaped potato particulate. Pronouncedheating, i.e. temperature rise, was found along the central axis. This agreed withLin’s (1991) finite element results. The finite element method was also used tomodel microwave thawing of pure water and 0.1 M NaCl cylinders (Pangrle et

al., 1991). The electromagnetic field was described by Maxwell’s equations(Ayappa et al., 1991). Non-uniformity of temperature distribution in testsamples was also observed in the work of Pangrle et al. ( 199 1).

The extent of uniform heating in a microwave oven can be provided by either

rotating the food on a carousel or using mode stirrers. However this is con-sidered to be too complex to model mathematically (Datta, 1990). Temperatureprofile in a food sample is determined by three characteristic quantities: thesample size in relation to microwave penetration depth; the boundary and initialconditions; the sample shape.

The literature review revealed the following deficiencies: (1) the numericalmodels were two-dimensional or axisymmetric and commercial software



Heat and mass t ransfer during microw ave heat ing 513

Heat generat ion Q

The power absorbed by food materials during microwave heating is representedby the volumetric heat generation in term Q. Heat generation is a function of

temperature and moisture at a particular location (x, y, z). Power absorbed byfood materials has not been, to date, well understood. Maxwell’s equations(Ayappa et al., 1991) and Lambert’s equation (Von Hippel, 1954; Mudgett,1986; Wei et al., 1985) are generally used to represent the microwave powerintensity. Due to the complexity and difficulty associated with determining theelectromagnetic field within an oven (Datta, 1990), Lambert’s law, as shown ineqn (7), is often used:

P= POexp( - 2ad) (7)

Io=F J{c [(l +tan’ 6)”

7- - lIPI (8)

d=taf’ $0I 1.9)

where a is the attenuation factor, which is a function of dielectric constant E’andloss factor E”. Volumetric heat generation Q can be expressed in terms of powerintensity in three orthogonal directions as shown in eqn ( 10) (Lin et al., 1995):

Equations (l)-( 10) are non-linear and unsteady state equations describingtemperature and moisture change in food materials during microwave heating. Itis not possible to derive generalized analytical solutions for this set of equations.The finite element method, due to its versatility and power to solve non-linearproblems, was used in this study to solve these equations.

Development of the FEM

A three-dimensional finite element program was written in FORTRAN. Thenumerical formulation based on the mathematical model described by eqns( 1)-( 10) is shown in the Appendix. The eight node hexahedral element was usedas the basic element type. The semi-implicit Euler’s method (Dhatt & Touzot,1984) was used to solve non-linear equations.

Experimental methods

Verification of the finite element model includes two steps: (1) testing of thecomputer program; (2) experimental verification. Verification tests wereconducted by running the FEM program, developed herein, with analyticalsolutions and solutions from TWODEPEP (IMSL, 1984) and ANSYS(Swanson Analysis System Inc, 1989). The experimental verification wasconducted using potato as the test material. Potato was used for testing because:




(1) the material’s engineering properties have been published; (2) the materialhas a high moisture content; (3) desired shapes can be easily formed; (4) shapes

can be retained after heating; (5) microwave heating data have been reported

in the literature; (6) the material is inexpensive and readily available.Time-temperature-moisture histories of FEM predicted and experimentalmeasurements were compared to evaluate the finite element model. The micro-wave heating time of 60 s was used for comparisons because this heating timeensured that the product temperature was below 100°C and no phase changeoccurred.

Some of the material properties of the potato sample used in the FEM wereobtained from Chirife ( 1983), Singh and Heldman (1984), Mudgett (1986) andChen ef al. (1990). Surface evaporation &,/A was not readily available in theliterature. A drying experiment was conducted in this study to obtain this value.

Weight losses of the sample at different temperature levels were collected. It wasfound that the surface evaporation rate is a function of temperature. The result-ing regression is given in eqn (11):

n&/A=-9~5946x10-4+5~5062x10-5T-2~5469x10-7T2

- 6.4326 x lo-“’ x T” (11)

where &/A is in g/min cm2 and T is the air temperature in “C.The surface mass transfer coefficient was calculated based on the surface

evaporation rate (McCabe et al., 1985):

h, = &,/A x V, (12)

where V, is humidity volume

v =2*24x(T+273)H

273

H=& (lOl.L5-111

(13)

(14)

p= 0.72658 exp(0.05647 X T )

units of h, and T are in cm/s and “C, respectively.

(15)

The material properties used in the FEM and the initial conditions aresummarized in Table 1. Constant moisture diffusivity was assumed since there isno published data available for addressing its changes with temperature andmoisture content. Additionally, moisture changes for short time microwaveheating are small.

Microwave heating experiments were conducted in a GE 700 W microwave

oven. Test samples were cut into cylindrical and slab shapes with dimensions of50 (diameter) X 40 (height) mm and 64 (length) X 48 (width) X 30 (height) mm,respectively, and heated in the microwave oven. The test sample was placed atthe center of the microwave cavity and on top of a box made from overheadtransparencies. A new box was used for each test. Fluoroptic probes (Model750, Luxtron Corp., Mountain View, CA) were used to measure temperaturesat different locations within test samples. The unit was interfaced to an IBM-PC



Heat and mass t ransfer during m icrow ave heating 515

TABLE 1

Summary of Material Properties and Initial Conditions Used in FEM

Property Value Source

P

CKF

I’0

E’E”h

D

k,lAh,T,,M,

1067 kg/m33.63 kJ/kg “C0.648 W/m “C2.89 W/cm2 for slab3.48 W/cm2 for cylinder581317.85 W/m2 “C1.4 X lOmy m2/s

cqn(l1)eqns(12)-(15)23°Ceqn (16) for slabeqn (17) for cylinder

Chen et al. (1990)Singh et al. (1984)Chen et al. (1990)Present workPresent workMudgett (1986)Mudgett (1986)Chen et al. ( 1990)Chirife (1983)

Present workPresent workPresent workPresent workPresent work

via a RS 232 serial port. Time-temperature data were collected using a BASIC

program on the PC for data analysis. The maximum number of channels fortemperature measurement was limited to four probes. The microwave powerabsorbed by a sample is a function of weight. The experiment for this relation-ship was conducted using distilled water as recommended by Lin (1991). Thepower absorption equation was obtained by regressing the experimental data:

P= 650[1- exp( - 4.6826 W- 0*4572)] ( r2 = 0.991) (16)

where W is the weight of water (kg). The relationship in eqn (16) between powerabsorption and product weight is, clearly, non-linear.

The power absorbed by the potato was calculated using eqn (16) and the

weight of water in the potato sample. The microwave power absorption in foodmaterials is mainly due to the presence of water molecules. Also, water(moisture) content in the potatoes used for these experiments was high (averagemoisture content of potatoes was 85%). Although an approximation, the presentapproach is a reasonable way to obtain power absorption. In addition, no in-expensive instruments were readily available. Experiments were conducted inthe Food Engineering Laboratory for validating this assumption. Validationexperiments were performed by comparing average temperature increments inpotato and water; the error was within 5%.

The surface power POwas approximately calculated by dividing the power

absorbed by the total surface area (Lin, 199 1). The data used for power calcula-tion are listed in Table 2. As can be seen from Table 2, the P,, values for slab andcylinder were 2.89 x lo4 and 3.40 x lo4 W/m’, respectively.

Near infrared reflectance (NIR) spectroscopy (Model 6500, NIR Systems,Silver Spring, MD) was used to measure the moisture content of test samples(Yeh et al., 1994). The NIR is a fast and accurate technique for measuringmoisture content. The reference values for moisture contents were obtained by



516 L. Zhou et al.

TABLE 2

Data Used for Surface Power Calculation

Sam ple W ,(kgX IO-.7 W , (kgxlO_-‘) P(W ) A, (m2X10-4) P,(W /m2x1t i)

Slab 98.33 83.58 371.77 128.64 2.89Cylinder 83,80 71.23 35531 102~10 3.48

using the oven drying method. A comparison showed that the calibrationequation calculated values were within 1% of the reference values. After heatingin the microwave oven, the slab-shaped test specimen was dipped immediatelyinto liquid nitrogen to reduce further evaporation. Following this, the samplewas cut into 36 small elements which were separately wrapped in plastic film.Each element was then brought to room temperature prior to measuring themoisture content using NIR.

In order to measure the moisture content of the test potato cylinder, thespecimens were cut into 15 rings (3 layers of 6 mm thickness X 5 rings of outerdiameters 10,20,30,44 and 50 mm). The oven method was used because of thedifficulty in measuring moisture content for ring-shaped geometry using theNIR.

RESULTS AND DISCUSSION

FEM testing

The FEM program was tested by comparing the predicted values to those calcu-lated using analytical solution and two commercial softwares, TWODEPEP andANSYS. Different specimen dimensions, heat generations and boundary condi-tions were used. Three typical comparisons are discussed below.

( 1) One-dimensional semi-infinit e slabA semi-infinite slab was discretized into 10 equal size cubic elements. The initialtemperature was uniform at 5°C. The semi-infinite slab surface was kept at aconstant temperature of 15°C. Heat generation was assumed to be zero andthermal conductivity was taken to be unity. All nodes at the same distance fromthe surface of the slab were expected to have the same temperature. Tempera-tures at the first and second node layer from the surface, denoted as T 1 and T2,had the most rapid changes. Therefore comparisons between analytical solu-tions and FEM predictions were based on these two nodal temperatures. As canbe seen from Fig. 1, results from the FEM and analytical solution were very

close. After 10 time steps, (i.e. time = 5 s) the difference between the analyticalsolution and the FEM prediction was less than O*Ol”C or 0.066% of themaximum temperature.

(2) Two-dimensional slab

A two-dimensional square slab was assumed to be at a uniform initial tempera-ture of 25°C. Three sides of the slab were insulated and one side was subjected




ANALYTICAL(T1)

cl 2 4 6 6 10 12

TIME (S)

Fig. 1. Comparison of temperature between FEM and analytical solution for a semi-infinite slab (T 1 is the temperature at the first node location: T2 is the temperature at the

second node location).

0 1 2

HEATING TIME (S)

Fig. 2. Comparison of temperature between FEM and TWODEPEP calculated valuesfor a two-dimensional slab (Tl is the temperature at the convective surface; T3 is the

temperature at the third node position from the convective surface).

to the heat convection boundary condition q= 30 - T. Heat generation was

assumed to be constant at 1 (W/cm3). Sixteen equal size elements, i.e. twentynodes, were generated. Because only one side had convection, all nodes which

had the same distance from the convective side were expected to have the sametemperature. This was designed to test whether the program gave the same valueat nodes equidistant from the convective side. Temperatures at the convectionsurface, denoted as T 1, and at the third node layer, denoted as T3, fromthe FEM prediction were compared with the TWODEPEP values. TheTWODEPEP (IMSL, 1984) is a well established commercial two-dimensionalfinite element software. A good agreement was found (Fig. 2) between FEM




predicted and TWODEPEP values. The differences were less than 0.04”C or0~13% of the maximum temperature after 2 s.

(3) Three-dimensional slabA three-dimensional test of the FEM was conducted by comparing the calcu-lated values to those obtained using ANSYS (a commercial finite elementpackage). The software ANSI’S cannot solve field problems involving morethan one dependent variable. A simplified heat transfer case was designed asshown in Fig. 3. It consisted of 96 cubical elements and 175 nodes. Nodenumber 150, as shown in Fig. 3, had the most rapid temperature change. Acomparison of the temperature at node 150 between FEM and ANSYS isshown in Fig. 4. The maximum difference in temperature was 4°C whichoccurred at the first time step. The difference declined quickly with increasing

Fig. 3. Heat

ode 150

Q=O

a=50-T

io=25 c

Cl k=l W/mC

9’ insulated

transfer in a three-dimensional region and the associatedconditions.

35-

35-1 I I I I

0

’ HEAbJG TIhE (S)4 5

boundary

Fig. 4. Comparison of temperature between FEM and ANSYS calculated values for athree-dimensional heat transfer problem (T 150 is the temperature at node 150).




time steps, and after 10 time steps (i.e. time = 2 s) the difference reduced to0.07”C or 0.14% of the maximum temperature.

Additional tests were conducted for different situations, e.g. cylindrical

samples, non-linear material parameters and non-linear boundary conditions. Inall cases, FEM predicted results compared favorably with reference (analyticalor software) values (Zhou, 1993).

Temperature and moisture distributions in potato - slab geometry

Potato was used as the test material for the simultaneous heat and mass transfermeasurement and modeling study. Because of the limitation of potato size, themaximum dimensions of the test sample were 64 (x direction) X 48 (y direc-tion) X 30 (z direction) mm. Temperature and moisture distributions at z = 15

mm after 60 s of heating are shown in Figs 5 and 6, respectively. The trends oftemperature distributions agreed with those reported by Ramaswamy et al .

( 1991) for starch gel, spaghetti and rice, and Lin (1991) for agar gels. Thetemperature decreased away from the corners to the edges, with a furtherdecline to the center. The top surface had a lower temperature than the middlelayer because of a large evaporation. Moisture distribution in the potato slab, ascan be seen from Fig. 6, was not symmetrical because of non-symmetrical initialmoisture distribution. Also, the moisture dropped very rapidly at the cornersand edges, while it was relatively flat in the central region. This moisture distri-bution may be attributed to the low moisture diffusivity and high moisture

evaporation associated with potato.Comparison of temperature histories at different locations within the slab

between the FEM predictions (using values in Table 1) and measurements areshown in Figs 7 and 8. Figure 7 shows the time-temperature history at thegeometric center, whereas Fig. 8 shows the temperature history at the uppercorner. The measured values are the means of 10 replicates. In addition, error

4 8

Fig. 5. Predicted temperature distribution at time = 60 s in a potato slab of dimensions64 (x axis) x 48 (y axis) x 30 (z axis) mm (plot is for x-y plane located at z = 15 mm).



520 L. Zhou et al.

Fig. 6. Predicted moisture distribution at time = 60 s in a potato slab of dimensions 64(x axis) x 48 (y axis) X 30 (z axis) mm (plot is for x-y plane located at z = 15 mm).

I I . I . I I

0 2 0 4 0 2 0

HEATING TIME (S)

Fig. 7. Comparison of measured and FEM predicted temperature at the geometriccenter of a potato slab (64 x 48 X 30 mm).

bars ( + one standard deviation) are also shown. As can be seen, the trends ofFEM predictions and measurements are similar. However some differenceswere also observed. The predicted temperature at the center was lower than

measured values (Fig. 7). The absolute maximum difference of temperature atthe center after 60 s of heating was 8el“C (or relative difference of 15.5% fromthe measured value). The FEM predicted temperatures at the upper corner ofthe potato were higher than measured values (Fig. 8). The reasons for thesedifferences may be explained as follows: ( 1) insufficient accuracy of probe ioca-tion - it was difficult to locate a probe at the exact geometric center or thecomer; (2) insufficient accuracy in the measured surface evaporation rate and




0 20 40 110 00 ’

HEATING TIME (S)

IO

Fig. 8. Comparison of measured and FEM predicted temperature at the upper cornerof a potato slab (64 x 48 x 30 mm).

absorbed power; (3) non-uniform power distribution; (4) insufficient accuracy ofmaterial properties which were obtained from the literature.

A sensitivity analysis was performed to demonstrate the significance of error

in the probe location. After 60 s of heating, the measured temperature, if theprobe position was 4 mm (6.2%) from the center in the x direction, would be2°C higher. The measured temperature would be 3.2”C higher if the probelocation was 4 mm (8%) away from the center in the y direction and 4.8”C

higher if the probe was 4 mm (13%) away from the center in the z direction.These results show that even a small deviation in the probe location from thegeometric center can result in measurements higher than the center value by asmuch as 4.8”C.

Moisture contents measured using the NIR were volume average values.There were 36 elements (i.e. 36 data points) for the entire slab. Each equal size

element had dimensions of 16 x 16 x 10 mm, with 12 elements in each of thethree x-y planes. In order to perform the comparison, initial moisture contentsof each element (as measured) were assigned to its geometrical center. A regres-sion equation was obtained based on the initial moisture contents of 36 smallelements. It is represented as:

MO = 75.85099 + 2.34454 x+ 2.867759 y+ 4.669915 z- 0.39878~’

- 0.621099 y* - 1.61251 z2 - 0.007799 xy+ O-025397 xz+ 0.06062~~

(r2 = 0.973) (1 1)

Equation (17) was used as initial moisture input to the FEM. The FEMpredicted nodal moisture contents at the center of each element were thencompared to measured values. The FEM predicted and the mean measuredmoisture contents values in a potato sample at different locations after 1 min ofheating (number of replicates was six) are compared in Fig. 9. General trends inmoisture distributions between model predictions and measurements were



522 L. Zhou et al.

PREDICTED

Fig. 9. Comparison of measured and predicted moisture content (% w.b.) at differentlocations in a potato slab for a heating time of 60 s.

similar; for example, the moisture content near the center was higher than theexposed edges, and the moisture content near the bottom was higher than thatnear the upper surface. However, differences between predictions andmeasurements were observed. The difference in moisture content ranged from- 0.68% to + 1.97%. In addition, most of the measured values were lower than

P

redicted values. This may possibly be due to the following experimental errors.1) The sample preparation for moisture measurement introduced a certain

amount of moisture loss. The test specimen after microwave heating was cutinto 36 pieces and dipped into liquid nitrogen immediately to reduce furtherlosses due to evaporation. These 36 elements were brought to room tempera-ture before the moisture content was measured by NIR. The whole process tookabout 30 min and each step resulted in some moisture loss. (2) The initialmoisture content distribution was not uniform in potato elemental samples. The

input of initial moisture content values was based on the regression of 36 initialdata points. They were assumed to be the average moisture of each of the 36small elemental volumes. An error was introduced by assuming the moisturecontent at the center of the element to be equal to this average moisture content.(3) The power distribution in the microwave oven was non-uniform, which wasassumed to be uniform in the FEM simulation. (4) The description of materialproperties as a function of temperature and moisture content were, in all like-lihood, not sufficiently accurate.

Temperature and moisture distributions in potato - cylindrical geometry

Temperature and moisture distributions in a potato cylinder 50 mm in diameter(radial direction) and 40 mm in height (z direction) were simulated using theFEM. Surface diagrams of FEM predicted temperature and moisture distribu-tions at z = 20 mm (i.e. half height location) are shown in Figs 10 and 11,respectively. The temperature distribution in a potato cylinder was differentfrom that in the potato slab (Fig. 5). Center heating was observed (Fig. 10) in



Heat and mass t ransfer during m icrow ave heating 523

-25.00 -25.00

Fig. 10. Predicted temperature distribution at time=60 s in a potato cylinder ofdimensions 25 (radius) x 40 (height) mm (plot is for radial plane at z= 20 mm).

‘25.00

-25.00 -25.00

Fig. 11. Predicted moisture distribution at time=60 s in a potato cylinder ofdimensions 25 (radius) X 40 (height) mm (plot is for radial plane at z = 20 mm).



524 L. Zhou et al.

cylindrical samples, i.e. hot spots were located along the central axis of thecylinder. The lowest temperature was located in the region between the centerand the surface. Due to surface evaporation, the temperature at the surface was

lower than its immediate vicinity. This trend agreed well with the results fromLin (1991) and Chen et al. (1990). The temperature distribution pattern canchange with the diameter of cylinder. Moisture distribution in the cylinder wasas expected, i.e. flat in the central region with a rapid drop near the surfacebecause of evaporation (Fig. 11).

Temperature histories were compared between FEM predictions andmeasurements for the center (r= 0) and surface (Y= 25 mm) in z = 20 mm plane.As can be seen from Figs 12 and 13, the differences between measured andpredicted values were within one standard deviation. The maximum difference

0

HEATING TIME (S)

Fig. 12. Comparison of measured and FEM predicted temperature at r= 0 and z= 20mm in a potato cylinder (radius = 25 X height = 40 mm).

-I101

HEATING TIME (S)

Fig. 13. Comparison of measured and FEM predicted temperature at r=25 andz = 20 mm in a potato cylinder (radius = 25 X height = 40 mm).



Heat and mass t ransfer during microw ave heat ing 525

was 8*7”C (or relative difference of 11.4%). The standard deviation of theexperimental data for cylindrical geometry was relatively large compared to theslab geometry. It was believed that the large standard deviation values were

caused by the insufficient accuracy of the location of the probes. In all radialplanes (z = constant), the temperature at the center was the highest. However,along the z direction, the temperature at z = 20 mm was the lowest.

In order to determine the effect of inaccuracy of probe location, a sensitivityanalysis was performed. If the probe position was located at 2.5 mm (5% ofdiameter) away from the center, after 60 s of heating, the measured temperaturevalue would be lower by 139°C. If the probe position was 2.5 mm (6.25% ofheight) above the center in the z direction, the measured temperature valuewould be higher by 2.5”C. Therefore even a small deviation of the probelocation at the center would cause the temperature to vary from - 139“C to

+ 25°C.The FEM predicted temperature at the center was higher than measured

values, as can be seen in Fig. 12. The reason for this is due to the power concen-tration. Lambert’s law assumes that the microwaves are incident perpendicularto the sample surface. This means that every microwave beam passes throughthe central axis. This leads to a high concentration of microwave energy in thecenter. However, the surface of the food sample is not perfectly smooth whichwill cause the microwaves to deviate from the center. This deviation will lead toa reduction in power at the center.

Similar to the slab geometry, a regression equation of initial moisture contents

in a potato cylinder was obtained based on 15 ring elements. It is represented as:

M,, = 84.924607 -0.831661 r+ 1.00 z+ 0.013309 r2 + 2.077561 z*

-0.283148 r3 - 0.625268~~ (r*=0.985) (18)

Equation (18) was used as initial moisture input to the FEM. The FEMpredicted and measured moisture contents at different locations, after 60 s ofheating, were also compared (Fig. 14). The number of replicates for moisture

PREDICTED

Fig. 14. Comparison of measured and predicted moisture content (% w.b.) at differentlocations in a potato cylinder for a heating time of 60 s.



526 L. Zhou et al.

measurements was six. Experimental values were lower than predicted valuesfor the high moisture region located around the center. The maximum differ-ence between FEM predictions and measurements was 1.85% (or a relative

difference of 2.1%). The moisture content was measured using the oven methodin this validation experiment. The specimen after microwave heating was firstcut into three 6 mm thick layers. Each layer was then cut into five rings with fourborers. The outside ring diameters were 10, 20,30,44 and 50 mm. Therefore atotal of 15 small elemental rings were obtained for weighing and drying. Becausethe rings close to the center had high moisture and high temperature, a highmoisture evaporation occurred at the center. This introduced weight loss andled to lower estimates of measured moisture content than FEM predictedvalues, which can be seen in Fig. 14. Some other factors, such as non-uniforminitial moisture distribution and insufficient accuracy of material properties of

test samples, also contributed to the differences between FEM predicted andmeasured values.

SUMMARY AND CONCLUSIONS

A three-dimensional finite element model for microwave heating of food

materials was developed in this study. The FEM predictions comparedfavorably with analytical and commercial software calculated values. The FEMwas also verified using experimental data. In situ temperature measurements

were obtained using the fluoroptic temperature measurement system. Non-destructive moisture measurement was conducted for slab geometry using theNIR technique. Moisture distribution in the potato cylinder was measured usingthe oven method. The following conclusions were drawn from this study.

1. The FEM predicted temperature and moisture distributions duringmicrowave heating compared favorably with the measured values for slaband cylinder geometries. The absolute temperature differences betweenFEM predictions and measured values in potatoes with slab geometry andcylindrical geometry were within 8el”C (or relative difference of 155%

from the measured value) and 8*7”C (or relative difference of 1 l-4%),respectively. The absolute moisture differences between FEM predictionsand measured values were 1.97% w.b. (or relative difference of 2.4%) forslab geometry and 1.85% w.b. (or relative difference of 2.1%) for cylindri-cal geometry.

2. For cylinder-shaped food materials, hot spots occurred along the centralaxis. For slab-shaped materials, a cold spot was located near the geometriccenter.

REFERENCES

ANSYS (1989). Swanson Analysis System Inc. Houston, PA.Ayappa, K. G., Davis, H. T., Crapiste, G., Davis, E. A. & Gordon, J. (1991). Microwave

heating: an evaluation of power formulations. Chem. Engng Sci., 46 (4), 1005- 16.

Chen, D., Haghighi, K., Singh, R. K. & Nelson, P. E. (1990). Finite element analysis oftemperature distribution during microwaved particulate foods. ASAE Paper No.906602. St Joseph, MI.




Chirife, J. (1983). Fundamentals of the drying mechanism during air dehydration of

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Datta, A. K. (1990). Heat and mass transfer in the microwave processing of food. Chem.

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Dhatt, G. & Touzot, G. (1984). The Finit e Element Method Display ed. John Wiley, NY.Lin, Y. E. (1991). Heating characteristics of simulated solid foods in a microwave oven.

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tion during microwave heating. J. Food Engng (in press).McCabe, W. L., Smith, J. C. & Harriott, P. (1985). Unit Operat ions of Chemical Engineer-

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Nestleroth, S. & Ciresi, R. (1992). The heat is on - taking the guesswork out of micro-

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between heating experiments and computer simulation. Microw ave Energy A pplica-

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APPENDIX

Numerical formulation

In order to solve coupled heat and mass transfer equations by the finite element

technique, Galerkin’s method was used to obtain a discretized form of eqns ( 1)

and (2). Temperature and moisture in each element, Tie) and M(‘), were approxi-



528 L. Zhou et al.

mated by algebraic interpolation

P

olynomials relating to element nodaltemperature and moisture ( T} and {M :

Th-9 { ~(4) ‘{ T}(Al)

,J,@)={N@)}T{M) (442)

where {N(‘)} is the shape function matrix. Using an eight node hexahedralelement, eqns (Al ) and (A2) become

Following the usual numerical procedure, the discretized form of heat andmass transfer equations (1) and (2) can be written in matrix notation (Dhatt &Touzot, 1984):

[c,,l{~}+[c,,l{ni}+[K,,l{T}={F,} W)

[c**l~~~+~~**lI~~=I~~2) W)where

K,,l=I,N’PIWN~TdJ’ (A7)

[G,l= I,,dNWIT d r’ w3)

L&,l=K1+K1 (A9)

[K,]=J,K 2 2 ‘dV[ I[ I

(i= l,S;j= 1,3)I I

Wol=W (i= l,S;j= 1,3 ‘) L418)

(AlO)

(All)b-2)

(A13)

(A14)

(AIS)

(A16)

(A17)

(A19)

(A20)



Heat and mass transfer during microw ave heat ing 529

Equations (A5) and (A6) can be further written into a more compact form asfollows:

or

The set of equations given by (A22) has non-linear, unsteady state features.The semi-implicit Euler’s iterative method (Dhatt & Touzot, 1984) was used tosolve the set of eqns (A22). For each iteration, increments A U of nodal value Uwere calculated. Current values of AU were compared to that from the previousiteration, and the difference between these two AU values were used as theconvergence criteria:

W,,1Wl’=VC,~ (~23)

where

K,l= KILL,+ aAWlf;L (A24)

R,l=

{&I= - a[Cli;br{uli;i,-(l - a)[CW},

I&J= -A~a[Klj;~,iU};;~,+(l- a)[W(U},)

{&I=At(o~FIf;k+(l- a){FI,)

At is time step, a is a coefficient between 0 and 1.

We)

(~27)

(A28)

(,A29)

finite element modeling of heat and mass transfer in food materials during microwave heating

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