Numerical Evaluation of Spherical Geometry Approximation for Heating and Cooling of Irregular Shaped...

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Numerical Evaluation of Spherical GeometryApproximation for Heating and Cooling ofIrregular Shaped Food ProductsRahmi Uyar and Ferruh Erdogdu

Abstract: Irregular shapes of food products add difficulties in modeling of food processes, and using actual geometriesmight be in expense of computing time without offering any advantages in heating and cooling processes. In thisstudy, a three-dimensional scanner was used to obtain geometrical description of strawberry, pear, and potato, andcoolingheating simulations were carried out in a computational heat transfer program. Then, spherical assumption wasapplied to compare center and volume average temperature changes using volume to surface area ratios of these samplesto define their characteristic length. In addition, spherical assumption for a finite cylinder and a cube was also applied todemonstrate the effect of sphericity. Geometries with sphericity values above 0.9 were determined to hold the sphericalassumption.

Keywords: Food engineering, food processing, heat transfer, mathematical modeling, thermal processing

Practical Applications: Irregular shapes of food products add difficulties in modeling of heating and cooling processes offood products. In addition, using actual geometries are in expense of computational time without offering any advantages.Hence, spherical approximation for irregular geometries was demonstrated under sphericity values of 0.9. This approachmight help in developing better heating and cooling processes.

IntroductionGeometrical modeling can be defined as a process to create

mathematical description of the actual shape of an object. Thisstep is required in mathematical modeling of food processes forfurther design and optimization purposes. Mathematical descrip-tion of a geometrical shape has a greater importance when thefocus is to obtain temperature, concentration, velocity, and pres-sure profiles (Goni and Purlis 2010). Based on this, it is crucialto obtain the actual shape in an accurate way. Various approacheswere applied to deal with this problem to simplify the numer-ical solutions. Smith (1966) was the first to define a geomet-ric factor to approximate ellipsoidal shapes as spheres. Mansonand others (1974) used equivalent cylinders to simulate the ther-mal process in pear-shaped containers where a geometry index,based on Smith (1966), was applied. Cleland and Earle (1982)developed a methodology proposing a concept of equivalent heattransfer dimensions to consider irregular shaped solid geometriesusing equivalent heat transfer dimension concept. Cleland andothers (1987a) assessed the accuracy of numerical methods to pre-dict freezing/thawing times using a comprehensive set of experi-mental data for regular/irregular multidimensional shapes. Clelandand others (1987b) applied geometrical factors to predict freez-ing/thawing times of multidimensional food products. Guemesand others (1988) considered strawberries as spheres for practical

MS 20120268 Submitted 2/22/2012, Accepted 4/11/2012. Authors are withDept. of Food Engineering, Univ. of Mersin, Mersin, Turkey. Direct inquiries toauthor Erdogdu (E-mail: [email protected], [email protected]).

applications since this simplification did not cause any significantdifference in heat transfer analysis. In that study, the proceduredeveloped by Smith and others (1966), where the characteristicdimension calculated with two orthogonal planes passing throughthe thermal center, was used. Hossain and others (1992) defineda geometric factor to predict process time where an analogousellipsoidal model was used to represent an actual object. Noronhaand others (1995) proposed a sphere to represent solid body shapesto simplify numerical solutions to one-dimensional (1-D) analy-sis. Yilmaz (1995) presented equations to predict temperature invarious shapes undergoing heating and cooling. Kim and Teixeira(1997) demonstrated that a numerical heat transfer model for afinite cylinder can be used to predict cold spot temperature incontainers of any shape. The fictitious cylinder model was appliedfor thermal process calculations as long as the process validationwas based upon the temperature change of the coldest point. Theshortest dimension of the given container was considered to bethe height of the phantom cylinder while the longest dimensionrepresented the characteristic length. Lin and others (1996) relatedactual geometric shapes to equivalent ellipsoids using simple geo-metric measurements to develop a simple method for predictionof chilling times. Sahin and others (2002) introduced geometricalshape factors to predict drying times of regular multidimensionalobjects. A procedure to determine the shape factor to use inconduction heat transfer studies was also developed by Bart andHanjelik (2003). As observed in the literature, different method-ologies were applied to make mathematical modeling easier whenthe irregular geometries are involved. However, there does notseem to be an easy procedure to apply to predict the effects ofdeviations that might occur in the process lines. If an easy-to-use

C 2012 Institute of Food Technologists Rdoi: 10.1111/j.1750-3841.2012.02769.x Vol. 0, Nr. 0, 2012 Journal of Food Science E1Further reproduction without permission is prohibited

E:FoodEngineering&PhysicalProperties

Evaluation of spherical approximation . . .

procedure is developed, this might be used as a corrective actionfor heating and cooling processes.Besides these approximations to simplify the irregular geome-

tries, innovative methodologies were also applied to define actual

Table 1 Initial and boundary conditions applied in the three-dimensional simulations and equivalent sphere analyticalsolution.

Initial and boundary conditions

Heat transfer Medium InitialFood coefficienth temperature temperaturematerials (W/m2-K) Tm ( C) Ti ( C)

Pear 21 6.1 24.3Strawberry 29 6.2 20.8Potato 80 104.43 23.0

three-dimensional (3-D) geometry of irregular shaped food prod-ucts. Laser scanning (Crocombe and others 1999), computed to-mography scanning (Borsa and others 2002; Kim and others 2007),computer vision (Scheerlinck and others 2004), reverse engineer-ing method based on surface cross-sectional design (Goni andothers 2007), magnetic resonance imaging (Goni and others 2008),and 3-D scanners (Uyar and Erdogdu 2009) were several of theseinnovative methodologies. Use of 3-D geometries in simulationslead to longer computational times, and applying these modelsin actual processing conditions does not offer additional advan-tages for rapid decision processes to carry out a decision mecha-nism in the process deviations. As Teixeira and others (1999) andSimpson and others (2007) discussed, control of thermal pro-cessing operations require maintaining the specified operatingconditions. Under unexpected variations in the process param-eters, process deviations might occur, and corrective actions

Figure 13-D mesh structures of (A) pear (B) strawberry (C) potato.

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should be taken immediately via the state-of-the-art controlsystems with a heating and cooling simulation model runningbehind.As indicated previously, there have been numerous studies re-

ported in the literature related to approximation of irregular shapesfor simulations and to define their complex geometrical shapes.However, spherical assumptions are known to lead to shorter com-putational times due to the adequacy of a 1-D modeling since itallows the application of an analytical solution in only 1-D in-

stead of the numerical solutions in 3-D required for an irregularshaped solid. With the help of improved digital tools like 3-Dscanners, actual 3-D geometries can be mathematically definedto determine their volume and surface area for further sphericalapproximations. Therefore, the objectives of this study were tovalidate the use of spherical approximation (equivalent sphere) insimulation of heating and cooling processes and to determine therequired conditions for the spherical approximation based on thesphericity value of the irregular shaped food products.

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Figure 2Comparison of three-dimensionalsimulation results with the experimental data (A)for pear cooling (B) strawberry cooling (C) potatoheating.

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Table 2Thermal conductivity and thermal diffusivity values ofthe food materials applied in the three-dimensional simulationsand equivalent sphere analytical solution.

Food Thermal conductivity Thermal diffusivitymaterials (W/m-K) values (m2/s)

Pear 0.52 1.345 107Strawberry 0.57 1.596 107Potato 0.50 1.039 107

Materials and MethodsFor the given objectives, in the 1st stage of the study, strawberry,

pear, and potato samples were selected to demonstrate that theirregular shaped food products can be assumed as a sphere incooling and heating processes. 3-D images of these geometrieswere obtained with a 3-D scanner for further use in mathematicalevaluation of spherical approximation. Surface area and volume ofthe given food samples were determined from the 3-D images.

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Figure 3Comparison of analytical solution fromthe equivalent sphere with the three-dimensionalsimulation (A) for pear (B) for strawberry (C) forpotato.


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The images were then transferred into the computational heattransfer program of Ansys CFX (Ansys Inc., Canonsburg, Pa.,U.S.A.) through Ansys Workbench (Ansys Inc.) to carry out theheating and cooling simulations. Using the analytical solution ofheat transfer, adequacy of spherical approximation was tested forcenter and volume average temperature changes in these products.The required conditions for spherical approximation were alsochecked, and the experiments were carried out to validate thenumerical simulations and analytical solutions. In the 2nd stage,spherical geometry assumption was investigated for regular shapesof a finite cylinder and a cube to relate this approximation with thesphericity value of the regularirregular shaped products. In thispart, the required sphericity value to apply the spherical geometryassumption with its analytical solution was determined.

3-D scanningTo obtain the actual 3-D geometries of the irregular shaped

food samples of pear, strawberry, and potato, a 3-D scanner wasused with its scanning softwareScanCore Studio (NextEngine3D scanner; Next Engine Inc., Santa Monica, Calif., U.S.A.).Scanning procedure was completed in three steps: scanning, align-ing, and fusing. Scanning was to obtain the views of the samples indifferent angles where number of views can be justified dependingupon the complexity of the geometrical shape. Aligning was tobring the images obtained from different angles together, and thelast step, fusing was to combine the aligned surfaces into a singlesurface. After construction of the 3-D surface images, the resultingsurface was converted into a solid volume using SolidWorks 2007(SolidWorks Corp., Concord, Mass., U.S.A.).

Table 3Volume and surface area of the pear, strawberry, andpotato and radius of the equivalent sphere to use in the analyti-cal solution.

Radius ofequivalent

Food Volume Surface area sphere

material V (mm3) A (mm2)V

A(mm) R (mm)

Pear 267930.0 20786.6 12.89 38.67Strawberry 32379.9 5330.5 6.07 18.22Potato 119084.1 12376.5 9.62 28.87

Experimental methodologyTo validate the simulations, experiments were performed with

pear and strawberry for cooling and with potato for heating pro-cess. In cooling and heating experiments, temperature change ofthe pear, strawberry, and potato samples was obtained using aKeithley Integra series 2700 data acquisition system (Keithley In-struments, Inc. Cleveland, Ohio, U.S.A.) and 30 gauge type Tthermocouples. Exact locations of the thermocouples inside thesamples were found by cutting thin slices from the samples afterthe heating and cooling processes were completed. Potato heatingexperiments carried out in steam, and cooling experiments werecarried out in a cold storage room.

Determining the characteristic length for equivalentsphereSurface area and volume of the food samples were obtained from

the Solid Works (SolidWorks Corp.) and used in determining theradius of the sphere to use in the spherical approximation. Forthis purpose, characteristic dimension (Lc), volume (V ) to surfacearea (A) ratio, of the irregular shaped food material was calculated(Eq. 1). As reported by Bart and Hanjalic (2003), volume to sur-face area ratio defines the size of an object universally. Hossain andothers (1992), however, find calculating the surface area computa-tionally complex. Therefore, using a tool like 3-D scanner makesthis calculation rather easy for further approximations.

Lc = Vfood sampleAfood sample (1)

Based on this calculation, characteristic dimension of asphere is:

Lc = VsphereAsphere =43 R33 R2 =

R3

(2)

where R is the radius of the equivalent sphere. To obtain the samevolume to surface area ratio for the equivalent sphere, the radiusof the equivalent sphere (R) was determined as:

R = 3 Vfood sampleAfood sample

(3)

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Figure 4Comparison of timetemperature dataobtained with analytical solutions at the center ofa finite cylinder can and its equivalent sphere.




SimulationsAfter scanning the potato, pear, and strawberry samples with

the NextEngine 3-D scanner (Next Engine Inc.) to obtain theactual 3-D geometrical models, surface images were convertedinto solid volumes with SolidWorks 2007 (SolidWorks Corp.) andtransferred into Ansys Workbench (Ansys Inc.) to carry out thesimulations for heating and cooling processes in Ansys CFX (AnsysInc). Then, the meshing procedure was applied with the result-ing volume elements of 135843, 74986, and 127467 for pear,strawberry, and potato samples. The 3-D images with their sur-face mesh structures are shown in Figure 1. Once meshing stagewas completed, initial and boundary conditions were applied, andsimulations were carried out. The initial and boundary condi-tions were reported in Table 1. In the simulations, thermophysicalproperties were assumed to be isotropic and constant. Thermalconductivity and thermal diffusivity values used in the simulationswere given in Table 2. For pear, the values were used as reportedby Uyar and Erdogdu (2009) where an experimental approachfor density and the empirical equations given by Urbicain andLozano (1997) were applied. Moisture content of the pears, straw-berries and potatoes was experimentally determined to be 6.21,10.5, and 4.90 (kg water/kg dry matter), respectively. Thermo-physical properties of the strawberries and potatoes were obtainedfrom Scheerlinck and others (2004) and Lamber and Hallstrom(1986).Convective heat transfer coefficients were determined experi-

mentally using lumped system methodology with aluminum cast-ings for pear and strawberry where the slope of the experimentaltemperature ratio ( TTmediumTinitialTmedium ) compared to time curve was used:

h = c p VA (4)

where was the slope of the temperature ratio compared totime curve (1/s), h was the convective heat transfer coefficient(W/m2-K)m cp was the heat capacity (950 J/kg-K), was density(2700 kg/m3), and V/A was the characteristic dimension of thealuminum casting. To determine the heat transfer coefficient forpotato heating simulation, the methodology described by Erdogdu(2005) was used.3-D simulations were carried out on an Intel Pentium Quad-

Core, 2.4 GHz with 3 GB RAM PC running on Windows XP 32bit edition. A typical iteration time for a successful convergencein a time step of 1 s in the present hardware configuration was 5to 8 s depending upon the number of volume elements.Five different heating and cooling experiments for each food ir-

regular shape were carried out, but only one of them was reportedsince the objective was just to demonstrate that the 3-D model wasvalidated with experimental results. In different experiments, thethermocouples were placed in different locations. Hence, placinga 2nd thermocouple was not planned. In fact, determining thelocation of thermocouple in a 3-D geometry was not an easytask, and placing the additional thermocouples would bring extradifficulties.

Analytical solutionsAnalytical solutions allow the prediction of transient temper-

ature history for different geometries, and boundary conditionsexist for pure conductive heating foods where the heat transfer isdescribed by Fouriers partial differential equation. In this study,analytical solution of a sphere was used to evaluate the cooling

and heating processes and to compare the results with Ansys CFX(Ansys Inc.) simulations. The sphere geometry is also reported toreflect the higher range of j values (lag factor in heat penetrationcurves) for geometries where the heat transfer is 1-D (Noronhaand others 1995) justifying the spherical approximation.The governing differential equation for 1-D heat conduction in

spherical coordinates is:

1

r 2 r

(r 2 T

r

)= 1

T

t(5)

where = kCp is thermal diffusivity (m

2/s).With the given initialcondition of uniform constant temperature distribution inside thesample (Eq. 6) and the 3rd kind convective boundary condition(Eq. 7) through the surface, the solution to Eq. 6 is given by Eq. 8.

T (r , 0) = Ti (6)

k T (R, t )r

= h [T (R, t ) T] (7)

T(r , t ) TTi T =

n=1

2 (sin n n cos n )n sin n cos n

sin(

n rR

)

n rR

e2n Fo

(8)

where k is thermal conductivity, Ti is the initial temperature,T is the medium temperature, F0 is Fourier number (F0 = tR2 ),( = k

c p ) is thermal diffusivity (m2/s), cp is specific heat (J/kg-K),

is density (kg/m3), and is given by:

Bi = h Rk

= 1 n cot n (9)

where Bi is Biot number.

Results and Discussions

Spherical geometry assumption for irregular shaped foodproductsAverage value from three experiments for the heat transfer co-

efficients were 21 1.0, 29 1.5 and 80 2.5 W/m2-K forpear, strawberry, and potato, respectively. Using the experimen-tally determined heat transfer coefficient values, and given initialconditions and thermophysical properties (Table 1 and 2), the sim-ulations were carried out with Ansys CFX (Ansys Inc.), and thesimulation results were validated. Figure 2 shows the compari-son of the simulation results with the experimental data for pear,strawberry, and potato, respectively. As observed in these figures,the simulation results compared well with the experimental datademonstrating the adequacy of the 3-D simulations. To bettercompare the simulation results with experimental data and analyt-ical solutions, root mean square error (RMSE) values, as suggested


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by Scheerlinnck and others (2004), was used. The RMSE wasgiven by:

RMSE = 1

N

Ni=1

(T Tsimulation)2 (10)

The RMSE values for pear, strawberry, and potato were 0.26,0.38, and 0.49 C, respectively. These values indicated validationof the 3-D simulations results with the experimental data.After the 3-D simulations were validated with the experimental

data, simulation results were compared with the analytical solutionresults of the equivalent sphere. In the analytical solution of thespherical model, the 1st 10000 terms of the Eq. 9 were used to geta perfect solution with reduced errors. Volume and surface area ofthe pear, strawberry and potato samples to determine the radius ofthe equivalent sphere was given in Table 3. Using Eq. 7 with thegiven initial and boundary conditions (Table 1), timetemperaturedata for cooling and heating processes at the center were gener-ated and compared with the results from the 3-D simulations fromthe slowest cooling and heating points. As observed in Figure 3,the 3-D simulation results agreed well with the analytical solutionresults demonstrating the applicability of using equivalent spheri-cal approximation. The RMSE values for 3-D simulation resultswith the sphere analytical solution results for pear, strawberry, andpotato were 0.16, 0.26, and 0.85 C, respectively.Noronha and others (1995) also offered an equivalent sphere to

simplify the heat conduction problem to 1-D heat transfer modelwhere empirical j (lag factor in heat penetration curves) and f (timerequired to reduce the difference between the heating medium temperatureand the product temperature to one-tenth of its value) values were used todefine the apparent position and thermal diffusivity of conductiveheating sphere. Teixeira and others (1999) listed the advantages ofusing a sphere to reduce the required computation time due tothe 1-D solution to further specify the required changes in a givenprocess in the decision mechanisms, for example, intelligent con-trol systems. As Simpson and others (2007) summarized, controlof thermal processing operations require to maintain the specifiedoperating conditions, and under unexpected changes, process de-viations that might occur time to time during the course of theprocess, and correction methodologies should be applied. Thismight be obtained with the state-of-the-art online control systemswhere a simulation model for heating and cooling process is run-ning behind. This simulation model is expected to response to the

process deviations as fast as possible, and hence spherical geometryapproach with its 1-D solution might be presented. Noronha andothers (1995) and Teixeira and others (1999) also discussed the useof analytical spherical models to evaluate the process deviations.

Spherical geometry assumption for finite cylinder and cubeand the effect of sphericityFor a typical #1 can (73 mm in dia and 110 mm in length),

a suggested equivalent spherical model was applied using an in-finite heat transfer coefficient (encountered in canning processes)with constant and uniform initial (20 C) and medium temper-atures (121.1 C). Thermal diffusivity value applied in the ana-lytical solutions (for details see Erdogdu and Turhan 2006) was1.32 107 m2/s. The radius of the sphere (3 VA ) for this casewas 41.11 mm. The objective at this point was to see whetherthe spherical approximation was to hold for regular geometries asdemonstrated for irregular geometries, or there would be anotherfactor required to decide on the suitability of the spherical ap-proximation. Figure 4 shows the comparison of timetemperaturedata obtained at the center of both finite cylinder and its equiv-alent sphere obtained with analytical solution. The RMSE valuefor this comparison was 4.04 C. As observed in this figure, tem-perature increase in the equivalent sphere was higher. The resultswere similar when a lower heat transfer coefficient value was ap-plied. Hence, to better use and apply the spherical model, ther-mal diffusivity value should be adjusted as reported by Noronhaand others (1995). As noted by Simpson and others (2007), themodel developed by Noronha and others (1995) used an apparentthermal diffusivity value with the sphere approximation to pro-duce the same heating rate experienced by the cold spot of theproduct. This demonstrated the limitation of using the sphericalapproach alone. Even though this approach can be easily applied in

Table 4Calculation of the sphericity values for pear, strawberry,potato, finite cylinder can, and cube.

Surface Equivalent EquivalentVolume area radiusa area Sphericity

Material V (mm3) A (mm2) Req (mm) Req (mm) A/Aeq

Pear 267930.0 20786.6 40.0 20106.2 0.97Strawberry 32379.9 5330.5 19.8 4912.8 0.92Potato 119084.1 12376.5 30.5 11705.4 0.95Finite cylinder can 460392.5 33597.8 47.9 28833.9 0.86Cube 48627.1 7993.5 22.6 6442.7 0.81aEquivalent radius was used to obtain the same volume sphere with volume of thematerial.

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Figure 5Comparison of timetemperature dataobtained with analytical solutions at the center ofa cube and its equivalent sphere.




irregular shaped food products, its use for a finite cylinder shapewas limited. Therefore, additional criteria would be required todecide upon the suitability of the spherical approximation.Sphericity is defined to be the ratio of surface area of a sphere

(with the same volume as the given shape) to the surface areaof the given shape. Based on this definition, sphericity values ofpear, strawberry, and potato can easily be determined to be over0.9 while the sphericity value of the given finite cylinder was0.86 (Table 4). In a similar manner, sphericity value of a cube

was determined to be 0.81. Figure 5 shows the comparison oftime temperature data obtained for a cube (36.5 36.5 36.5mm) using the analytical solution (for details see Erdogdu andTurhan 2006) and the same initial and boundary conditions as inthe case of finite cylinder can with thermal diffusivity value of1.32 107 m2/s with respect to its equivalent sphere (obtainedwith analytical solution) with its radius (3 VA ) of 18.25 mm. TheRMSE value, in this case, was 10.61 C. As observed, the differ-ence was worse than the case of finite cylinder shape indicating the

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Figure 6Comparison of volume averagedtimetemperature data obtained with theequivalent analytical solution and thethree-dimensional simulation for the (A) pear (B)strawberry (C) potato.


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significant effect of sphericity. The difference in the temperaturechange at the slowest heating point of the cube and its equivalentsphere was similar when a lower heat transfer coefficient value wasapplied. Based on these, it might be assumed that the sphericityvalue of a given irregular shaped object should be over 0.9 for theequivalent sphere assumption to hold.As demonstrated, for the sphericity values of 0.9 and above, an

irregular shape might be approximated with an ideal sphere as longas their volume to surface area ratios are equal, and temperaturepredictions were required only at the slowest heating or coolingpoint. Teixeira and others (1999) also noted that a product might beassumed in the form of a sphere when the temperature predictionat the cold spot location was required for conduction heating.As explained previously, mathematical models are required for

rapid decision mechanisms and corrective actions when the de-viations in the heating and cooling process lines occur since theaccurate response of the deviation must be known. Application ofthe spherical model with its 1-D analytical solution enables sucha fast corrective action in the heating and cooling process lines.However, in food processing, besides the temperature change atthe slowest heating or cooling point, overall temperature changethrough the volume of the product might also be significant forfurther design and optimization purposes. To test the applicabil-

ity of the equivalent sphere approximation for this purpose, thevolume average temperature changes of the pear, strawberry, andpotato samples were compared with the volume average change ofthe equivalent sphere. The volume average integral of Eq. 8 wasintroduced for this purpose:

T = 1V

V

T (r , t ) d r (11)

This leads to:

T TTi T =

n=1

[63n

(sin n n cos n )2

n sin n cos n e2n Fo

](12)

for sphere where T is the volume average temperature.Figure 6 shows the comparison of volume average temperatures

from 3-D simulations and analytical solutions. As observed, theresults were similar to the slowest cooling point results for pearand strawberry and slowest heating point results for potato. Thesedemonstrated the suitability of the spherical approximation forthe case of volume average temperature change of an irregulargeometry as long as the sphericity value was higher than 0.9. In

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Figure 7Comparison of volume averagedtemperature data of a finite cylinder andequivalent sphere obtained with analyticalsolutions.

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Figure 8Comparison of volume averagedtemperature data of a cube and equivalent sphereobtained with analytical solutions.




the case of finite cylinder can and cube cases, comparison of thevolume average temperature changes were shown in Figure 7 and8. While the comparison for the finite cylinder gave a reasonableresult for the volume average temperature change, due to the effectof lower sphericity value, the comparison results were off for thecase of cube.Biot numbers (Bi = h 3 VAk ) were 4.6, 2.8, and 13.9 for pear,

strawberry, and potato, respectively, while an infinite heat transfercoefficient assumption was applied for testing the finite cylinderand cube cases. In the finite cylinder and cube cases, a finite heattransfer coefficient gave the similar results. Hence, it might beassumed that the given results were independent of Biot numberrange, and irregular or regular geometries with sphericity valuesabove 0.9 might be approximated with spherical approximationfor determining center or volume average temperature changes.Relatively faster analytical solution for sphere in 1-D helps in

saving time and maintaining the specified operating conditions.Via this, corrective methodologies might be applied for processdeviations that might occur time to time during the course of aheating and cooling process.

ConclusionsA 3-D scanner was used to obtain the actual geometrical de-

scription of complex and irregular shapes of strawberry, pear, andpotato samples, and cooling and heating simulations were carriedout. Using the volume to surface area ratios of these samples, cen-ter and volume average time temperature changes were comparedwith the analytical solution results from the spherical geometryassumption, and the geometries with sphericity values above 0.9were determined to hold the spherical assumption.Control of food processing operations requires to maintain the

specified operating conditions under unexpected changes. Whenthe process deviations occur time to time during the course of aheating and cooling process, correction methodologies should beapplied. This might be obtained with the state-of-the-art controlsystems where a simulation model for heating and cooling processis running behind. In the future process lines of food industry, thissimplifying assumption could help developing on-line intelligentcorrection systems that might even include a 3-D scanner.

AcknowledgmentPotato heating experiments were carried out by Rahmi Uyar

and Laura Alessandrini during her visit to the Dept. of FoodEngineering at the Univ. of Mersin in the Spring of 2009. Allhelps and suggestions by Laura are much appreciated.

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