# Numerical evaluation of spherical geometry approximation for heating and cooling of irregular shaped food products.

**ABSTRACT** Irregular shapes of food products add difficulties in modeling of food processes, and using actual geometries might be in expense of computing time without offering any advantages in heating and cooling processes. In this study, a three-dimensional scanner was used to obtain geometrical description of strawberry, pear, and potato, and cooling-heating simulations were carried out in a computational heat transfer program. Then, spherical assumption was applied to compare center and volume average temperature changes using volume to surface area ratios of these samples to define their characteristic length. In addition, spherical assumption for a finite cylinder and a cube was also applied to demonstrate the effect of sphericity. Geometries with sphericity values above 0.9 were determined to hold the spherical assumption. PRACTICAL APPLICATIONS: Irregular shapes of food products add difficulties in modeling of heating and cooling processes of food 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 approach might help in developing better heating and cooling processes.

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E: Food Engineering &

Physical Properties

Numerical Evaluation of Spherical Geometry

Approximation for Heating and Cooling of

Irregular Shaped Food Products

Rahmi Uyar and Ferruh Erdogdu

Abstract:

might be in expense of computing time without offering any advantages in heating and cooling processes. In this

study, a three-dimensional scanner was used to obtain geometrical description of strawberry, pear, and potato, and

cooling–heating simulations were carried out in a computational heat transfer program. Then, spherical assumption was

applied to compare center and volume average temperature changes using volume to surface area ratios of these samples

to define their characteristic length. In addition, spherical assumption for a finite cylinder and a cube was also applied to

demonstrate the effect of sphericity. Geometries with sphericity values above 0.9 were determined to hold the spherical

assumption.

Irregular shapes of food products add difficulties in modeling of food processes, and using actual geometries

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

Practical Applications:

food 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 approach

might help in developing better heating and cooling processes.

Irregular shapes of food products add difficulties in modeling of heating and cooling processes of

Introduction

Geometrical modeling can be defined as a process to create

mathematical description of the actual shape of an object. This

step is required in mathematical modeling of food processes for

further design and optimization purposes. Mathematical descrip-

tion of a geometrical shape has a greater importance when the

focus is to obtain temperature, concentration, velocity, and pres-

sure profiles (Goni and Purlis 2010). Based on this, it is crucial

to obtain the actual shape in an accurate way. Various approaches

were 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. Manson

and 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 heat

transfer dimensions to consider irregular shaped solid geometries

using equivalent heat transfer dimension concept. Cleland and

others (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. Cleland

and others (1987b) applied geometrical factors to predict freez-

ing/thawing times of multidimensional food products. Guemes

and others (1988) considered strawberries as spheres for practical

MS 20120268 Submitted 2/22/2012, Accepted 4/11/2012. Authors are with

Dept. of Food Engineering, Univ. of Mersin, Mersin, Turkey. Direct inquiries to

author Erdogdu (E-mail: ferruherdogdu@yahoo.com, ferruherdogdu@mersin.edu.tr).

applications since this simplification did not cause any significant

difference in heat transfer analysis. In that study, the procedure

developed by Smith and others (1966), where the characteristic

dimension calculated with two orthogonal planes passing through

the thermal center, was used. Hossain and others (1992) defined

a geometric factor to predict process time where an analogous

ellipsoidal model was used to represent an actual object. Noronha

and others (1995) proposed a sphere to represent solid body shapes

to simplify numerical solutions to one-dimensional (1-D) analy-

sis. Yilmaz (1995) presented equations to predict temperature in

various shapes undergoing heating and cooling. Kim and Teixeira

(1997) demonstrated that a numerical heat transfer model for a

finite cylinder can be used to predict cold spot temperature in

containers of any shape. The fictitious cylinder model was applied

for thermal process calculations as long as the process validation

was based upon the temperature change of the coldest point. The

shortest dimension of the given container was considered to be

the height of the phantom cylinder while the longest dimension

represented the characteristic length. Lin and others (1996) related

actual geometric shapes to equivalent ellipsoids using simple geo-

metric measurements to develop a simple method for prediction

of chilling times. Sahin and others (2002) introduced geometrical

shape factors to predict drying times of regular multidimensional

objects. A procedure to determine the shape factor to use in

conduction heat transfer studies was also developed by Bart and

Hanjelik (2003). As observed in the literature, different method-

ologies were applied to make mathematical modeling easier when

the irregular geometries are involved. However, there does not

seem to be an easy procedure to apply to predict the effects of

deviations that might occur in the process lines. If an easy-to-use

C ?2012 Institute of Food TechnologistsR ?

doi: 10.1111/j.1750-3841.2012.02769.x

Further reproduction without permission is prohibited

E166Journal of Food SciencerVol. 77, Nr. 7, 2012

Page 2

E: Food Engineering &

Physical Properties

Evaluation of spherical approximation...

procedure is developed, this might be used as a corrective action

for 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-

dimensionalsimulations and

solution.

equivalentsphere analytical

Initial and boundary conditions

Heat transfer

coefficient—h

(W/m2-K)

Medium

temperature—

Tm(◦C)

Initial

Food

materials

temperature—

Ti(◦C)

Pear

Strawberry

Potato

21

29

80

6.1

6.2

24.3

20.8

23.0104.43

three-dimensional (3-D) geometry of irregular shaped food prod-

ucts. Laser scanning (Crocombe and others 1999), computed to-

mographyscanning(Borsaandothers2002;Kimandothers2007),

computer vision (Scheerlinck and others 2004), reverse engineer-

ing method based on surface cross-sectional design (Goni and

others2007),magneticresonanceimaging(Goniandothers2008),

and 3-D scanners (Uyar and Erdogdu 2009) were several of these

innovative methodologies. Use of 3-D geometries in simulations

lead to longer computational times, and applying these models

in 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) and

Simpson and others (2007) discussed, control of thermal pro-

cessing operations require maintaining the specified operating

conditions. Under unexpected variations in the process param-

eters, process deviations might occur, and corrective actions

Figure 1–3-D mesh structures of (A) pear (B) strawberry (C) potato.

Vol. 77, Nr. 7, 2012rJournal of Food Science E167

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E: Food Engineering &

Physical Properties

Evaluation of spherical approximation...

should be taken immediately via the state-of-the-art control

systems with a heating and cooling simulation model running

behind.

As indicated previously, there have been numerous studies re-

ported in the literature related to approximation of irregular shapes

for simulations and to define their complex geometrical shapes.

However,sphericalassumptionsareknowntoleadtoshortercom-

putational times due to the adequacy of a 1-D modeling since it

allows the application of an analytical solution in only 1-D in-

stead of the numerical solutions in 3-D required for an irregular

shaped solid. With the help of improved digital tools like 3-D

scanners, actual 3-D geometries can be mathematically defined

to determine their volume and surface area for further spherical

approximations. Therefore, the objectives of this study were to

validate the use of spherical approximation (equivalent sphere) in

simulation of heating and cooling processes and to determine the

required conditions for the spherical approximation based on the

sphericity value of the irregular shaped food products.

0

5

10

15

20

25

30

08001600 2400320040004800

Temperature (°C)

Time (s)

Pear

Medium

3D simula?on

0

5

10

15

20

25

0600120018002400

Temperature (°C)

Time (s)

Strawberry

Medium

3D simula?on

0

20

40

60

80

100

120

080016002400320040004800

Temperature (°C)

Time (s)

Potato

Medium

3D simula?on

A

B

C

Figure 2–Comparison of three-dimensional

simulation results with the experimental data (A)

for pear cooling (B) strawberry cooling (C) potato

heating.

E168 Journal of Food SciencerVol. 77, Nr. 7, 2012

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E: Food Engineering &

Physical Properties

Evaluation of spherical approximation...

Table 2–Thermal conductivity and thermal diffusivity values of

the food materials applied in the three-dimensional simulations

and equivalent sphere analytical solution.

Food

materials

Thermal conductivity

(W/m-K)

Thermal diffusivity

values (m2/s)

Pear

Strawberry

Potato

0.52

0.57

0.50

1.345 × 10−7

1.596 × 10−7

1.039 × 10−7

Materials and Methods

For the given objectives, in the 1st stage of the study, strawberry,

pear, and potato samples were selected to demonstrate that the

irregular shaped food products can be assumed as a sphere in

cooling and heating processes. 3-D images of these geometries

were obtained with a 3-D scanner for further use in mathematical

evaluation of spherical approximation. Surface area and volume of

the given food samples were determined from the 3-D images.

0

5

10

15

20

25

30

0120024003600 4800

Temperature (°C)

Time (s)

Pear 3-D simula?on

Sphere analy?cal

solu?on

0

5

10

15

20

25

0600 12001800 2400

Temperature (°C)

Time (s)

Strawberry 3-D

simula?on

Sphere analy?cal

solu?on

0

20

40

60

80

100

120

06001200 1800 2400 3000 3600

Temperature (°C)

Time (s)

Potato 3D simula?on

Sphere analy?cal

solu?on

A

B

C

Figure 3–Comparison of analytical solution from

the equivalent sphere with the three-dimensional

simulation (A) for pear (B) for strawberry (C) for

potato.

Vol. 77, Nr. 7, 2012rJournal of Food Science E169

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E: Food Engineering &

Physical Properties

Evaluation of spherical approximation...

The images were then transferred into the computational heat

transfer program of Ansys CFX (Ansys Inc., Canonsburg, Pa.,

U.S.A.) through Ansys Workbench (Ansys Inc.) to carry out the

heating and cooling simulations. Using the analytical solution of

heat transfer, adequacy of spherical approximation was tested for

center and volume average temperature changes in these products.

The required conditions for spherical approximation were also

checked, and the experiments were carried out to validate the

numerical simulations and analytical solutions. In the 2nd stage,

spherical geometry assumption was investigated for regular shapes

of a finite cylinder and a cube to relate this approximation with the

sphericity value of the regular–irregular shaped products. In this

part, the required sphericity value to apply the spherical geometry

assumption with its analytical solution was determined.

3-D scanning

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

food samples of pear, strawberry, and potato, a 3-D scanner was

used with its scanning software—ScanCore Studio (NextEngine

3D 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 in

different angles where number of views can be justified depending

upon the complexity of the geometrical shape. Aligning was to

bring the images obtained from different angles together, and the

last step, fusing was to combine the aligned surfaces into a single

surface. After construction of the 3-D surface images, the resulting

surface was converted into a solid volume using SolidWorks 2007

(SolidWorks Corp., Concord, Mass., U.S.A.).

Table 3–Volume and surface area of the pear, strawberry, and

potato and radius of the equivalent sphere to use in the analyti-

cal solution.

Radius of

equivalent

sphereFood Volume—Surface area—

material

V (mm3)

A (mm2)

V

A(mm)

12.89

6.07

9.62

R (mm)

Pear

Strawberry

Potato

267930.0

32379.9

119084.1

20786.6

5330.5

12376.5

38.67

18.22

28.87

Experimental methodology

To 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 of

the pear, strawberry, and potato samples was obtained using a

Keithley Integra series 2700 data acquisition system (Keithley In-

struments, Inc. Cleveland, Ohio, U.S.A.) and 30 gauge type T

thermocouples. Exact locations of the thermocouples inside the

samples were found by cutting thin slices from the samples after

the heating and cooling processes were completed. Potato heating

experiments carried out in steam, and cooling experiments were

carried out in a cold storage room.

Determining the characteristic length for equivalent

sphere

Surface area andvolume ofthe foodsamples wereobtained from

the Solid Works (SolidWorks Corp.) and used in determining the

radius of the sphere to use in the spherical approximation. For

this purpose, characteristic dimension (Lc), volume (V) to surface

area (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 and

others (1992), however, find calculating the surface area computa-

tionally complex. Therefore, using a tool like 3-D scanner makes

this calculation rather easy for further approximations.

Lc =Vfoodsample

Afoodsample

(1)

Based on this calculation, characteristic dimension of a

sphere is:

Lc =Vsphere

Asphere

=

4

3· π · R3

3 · π · R2=R

3

(2)

where R is the radius of the equivalent sphere. To obtain the same

volume to surface area ratio for the equivalent sphere, the radius

of the equivalent sphere (R) was determined as:

R = 3 ·Vfoodsample

Afoodsample

(3)

0

20

40

60

80

100

120

140

02040 6080 100120

Temperature (°C)

Time (min)

Finite cylinder

Sphere

Figure 4–Comparison of time–temperature data

obtained with analytical solutions at the center of

a finite cylinder can and its equivalent sphere.

E170 Journal of Food SciencerVol. 77, Nr. 7, 2012

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Physical Properties

Evaluation of spherical approximation...

Simulations

After scanning the potato, pear, and strawberry samples with

the NextEngine 3-D scanner (Next Engine Inc.) to obtain the

actual 3-D geometrical models, surface images were converted

into solid volumes with SolidWorks 2007 (SolidWorks Corp.) and

transferred into Ansys Workbench (Ansys Inc.) to carry out the

simulations for heating and cooling processes in Ansys CFX (Ansys

Inc). 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 stage

was completed, initial and boundary conditions were applied, and

simulations were carried out. The initial and boundary condi-

tions were reported in Table 1. In the simulations, thermophysical

properties were assumed to be isotropic and constant. Thermal

conductivity and thermal diffusivity values used in the simulations

were given in Table 2. For pear, the values were used as reported

by Uyar and Erdogdu (2009) where an experimental approach

for density and the empirical equations given by Urbicain and

Lozano (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 obtained

from 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 experimental

temperature ratio (

T−Tmedium

Tinitial−Tmedium) compared to time curve was used:

h = −β · ρ · cp·V

A

(4)

where β was the slope of the temperature ratio compared to

time curve (1/s), h was the convective heat transfer coefficient

(W/m2-K)m cpwas the heat capacity (950 J/kg-K), ρ was density

(2700 kg/m3), and V/A was the characteristic dimension of the

aluminum casting. To determine the heat transfer coefficient for

potato 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 32

bit edition. A typical iteration time for a successful convergence

in a time step of 1 s in the present hardware configuration was 5

to 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 reported

since the objective was just to demonstrate that the 3-D model was

validated with experimental results. In different experiments, the

thermocouples were placed in different locations. Hence, placing

a 2nd thermocouple was not planned. In fact, determining the

location of thermocouple in a 3-D geometry was not an easy

task, and placing the additional thermocouples would bring extra

difficulties.

Analytical solutions

Analytical solutions allow the prediction of transient temper-

ature history for different geometries, and boundary conditions

exist for pure conductive heating foods where the heat transfer is

described by Fourier’s 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 to

reflect the higher range of j values (lag factor in heat penetration

curves) for geometries where the heat transfer is 1-D (Noronha

and others 1995) justifying the spherical approximation.

The governing differential equation for 1-D heat conduction in

spherical coordinates is:

1

r2·

∂

∂r

?

r2·∂T

∂r

?

=1

α·∂T

∂t

(5)

whereα =

condition of uniform constant temperature distribution inside the

sample (Eq. 6) and the 3rd kind convective boundary condition

(Eq. 7) through the surface, the solution to Eq. 6 is given by Eq. 8.

k

ρ·Cpisthermaldiffusivity(m2/s).Withthegiveninitial

T (r,0) = Ti

(6)

−k ·∂T (R,t)

∂r

= h · [T (R,t) − T∞]

(7)

T(r,t) − T∞

Ti− T∞

=

∞

?

n=1

⎡

⎢

?λn· r

λn· r

R

⎢

⎣

2 · (sinλn− λn· cosλn)

λn− sinλn· cosλn

?

· e−λ2

·

sin

R

n·Fo

⎤

⎥

⎥

⎦

(8)

where k is thermal conductivity, Ti is the initial temperature,

T∞is the medium temperature, F0is Fourier number (F0=α·t

(α =

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

R2),

k

ρ·cp) is thermal diffusivity (m2/s), cpis specific heat (J/kg-K),

Bi =h · R

k

= 1 − λn· cotλn

(9)

where Bi is Biot number.

Results and Discussions

Spherical geometry assumption for irregular shaped food

products

Average value from three experiments for the heat transfer co-

efficients were 21 ± 1.0, 29 ± 1.5 and 80 ± 2.5 W/m2-K for

pear, strawberry, and potato, respectively. Using the experimen-

tally determined heat transfer coefficient values, and given initial

conditions and thermophysical properties (Table 1 and 2), the sim-

ulations were carried out with Ansys CFX (Ansys Inc.), and the

simulation 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 data

demonstrating the adequacy of the 3-D simulations. To better

compare the simulation results with experimental data and analyt-

ical solutions, “root mean square error (RMSE)” values, as suggested

Vol. 77, Nr. 7, 2012rJournal of Food Science E171

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E: Food Engineering &

Physical Properties

Evaluation of spherical approximation...

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

given by:

?

i=1

The RMSE values for pear, strawberry, and potato were 0.26,

0.38, and 0.49◦C, respectively. These values indicated validation

of 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 solution

results of the equivalent sphere. In the analytical solution of the

spherical model, the 1st 10000 terms of the Eq. 9 were used to get

a perfect solution with reduced errors. Volume and surface area of

the pear, strawberry and potato samples to determine the radius of

the equivalent sphere was given in Table 3. Using Eq. 7 with the

given initial and boundary conditions (Table 1), time–temperature

data for cooling and heating processes at the center were gener-

ated and compared with the results from the 3-D simulations from

the slowest cooling and heating points. As observed in Figure 3,

the 3-D simulation results agreed well with the analytical solution

results demonstrating the applicability of using equivalent spheri-

cal approximation. The RMSE values for 3-D simulation results

with the sphere analytical solution results for pear, strawberry, and

potato 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 model

where empirical j (lag factor in heat penetration curves) and f (time

required to reduce the difference between the heating medium temperature

and the product temperature to one-tenth of its value) values were used to

define the apparent position and thermal diffusivity of conductive

heating sphere. Teixeira and others (1999) listed the advantages of

using a sphere to reduce the required computation time due to

the 1-D solution to further specify the required changes in a given

process in the decision mechanisms, for example, intelligent con-

trol systems. As Simpson and others (2007) summarized, control

of thermal processing operations require to maintain the specified

operating conditions, and under unexpected changes, process de-

viations that might occur time to time during the course of the

process, and correction methodologies should be applied. This

might be obtained with the state-of-the-art online control systems

where a simulation model for heating and cooling process is run-

ning behind. This simulation model is expected to response to the

RMSE =

?

?

?1

N·

N

?

(T − Tsimulation)2

(10)

process deviations as fast as possible, and hence spherical geometry

approach with its 1-D solution might be presented. Noronha and

others (1995) and Teixeira and others (1999) also discussed the use

of analytical spherical models to evaluate the process deviations.

Spherical geometry assumption for finite cylinder and cube

and the effect of sphericity

For 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) was

1.32 ×10−7m2/s. The radius of the sphere (3 ·V

was 41.11 mm. The objective at this point was to see whether

the spherical approximation was to hold for regular geometries as

demonstrated for irregular geometries, or there would be another

factor required to decide on the suitability of the spherical ap-

proximation. Figure 4 shows the comparison of time–temperature

data obtained at the center of both finite cylinder and its equiv-

alent sphere obtained with analytical solution. The RMSE value

for this comparison was 4.04◦C. As observed in this figure, tem-

perature increase in the equivalent sphere was higher. The results

were 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 Noronha

and others (1995). As noted by Simpson and others (2007), the

model developed by Noronha and others (1995) used an apparent

thermal diffusivity value with the sphere approximation to pro-

duce the same heating rate experienced by the cold spot of the

product. This demonstrated the limitation of using the spherical

approach alone. Even though this approach can be easily applied in

A) for this case

Table 4–Calculation of the sphericity values for pear, strawberry,

potato, finite cylinder can, and cube.

Surface

area—

A (mm2)

Equivalent Equivalent

radiusa—

Req(mm)

40.0

19.8

30.5

47.9

22.6

Volume—

V (mm3)

area—

Req(mm)

20106.2

4912.8

11705.4

28833.9

6442.7

Sphericity—

A/Aeq

0.97

0.92

0.95

0.86

0.81

Material

Pear

Strawberry

Potato

Finite cylinder can

Cube

aEquivalent radius was used to obtain the same volume sphere with volume of the

material.

267930.0

32379.9

119084.1

460392.5

48627.1

20786.6

5330.5

12376.5

33597.8

7993.5

0

20

40

60

80

100

120

140

0510 1520 2530

Temperature (°C)

Time (min)

Cube

Sphere

Figure 5–Comparison of time–temperature data

obtained with analytical solutions at the center of

a cube and its equivalent sphere.

E172 Journal of Food SciencerVol. 77, Nr. 7, 2012

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Physical Properties

Evaluation of spherical approximation...

irregular shaped food products, its use for a finite cylinder shape

was limited. Therefore, additional criteria would be required to

decide 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 area

of the given shape. Based on this definition, sphericity values of

pear, strawberry, and potato can easily be determined to be over

0.9 while the sphericity value of the given finite cylinder was

0.86 (Table 4). In a similar manner, sphericity value of a cube

was determined to be 0.81. Figure 5 shows the comparison of

time temperature data obtained for a cube (36.5 × 36.5 × 36.5

mm) using the analytical solution (for details see Erdogdu and

Turhan 2006) and the same initial and boundary conditions as in

the case of finite cylinder can with thermal diffusivity value of

1.32 × 10−7m2/s with respect to its equivalent sphere (obtained

with analytical solution) with its radius (3 ·V

RMSE value, in this case, was 10.61◦C. As observed, the differ-

ence was worse than the case of finite cylinder shape indicating the

A) of 18.25 mm. The

0

5

10

15

20

25

30

08001600 2400 3200 4000 4800

Temperature (°C)

Time (s)

Pear

Analy?cal solu?on

0

5

10

15

20

25

06001200 18002400

Temperature (°C)

Time (s)

Strawberry

Analy?cal solu?on

0

20

40

60

80

100

120

08001600 2400 3200 4000 4800

Temperature (°C)

Time (s)

Potato

Analy?cal solu?on

A

B

C

Figure 6–Comparison of volume averaged

time–temperature data obtained with the

equivalent analytical solution and the

three-dimensional simulation for the (A) pear (B)

strawberry (C) potato.

Vol. 77, Nr. 7, 2012rJournal of Food Science E173

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Physical Properties

Evaluation of spherical approximation...

significant effect of sphericity. The difference in the temperature

change at the slowest heating point of the cube and its equivalent

sphere was similar when a lower heat transfer coefficient value was

applied. Based on these, it might be assumed that the sphericity

value of a given irregular shaped object should be over 0.9 for the

equivalent 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 long

as their volume to surface area ratios are equal, and temperature

predictions were required only at the slowest heating or cooling

point.Teixeiraandothers(1999)alsonotedthataproductmightbe

assumed in the form of a sphere when the temperature prediction

at 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 the

accurate response of the deviation must be known. Application of

the spherical model with its 1-D analytical solution enables such

a fast corrective action in the heating and cooling process lines.

However, in food processing, besides the temperature change at

the slowest heating or cooling point, overall temperature change

through the volume of the product might also be significant for

further design and optimization purposes. To test the applicabil-

ity of the equivalent sphere approximation for this purpose, the

volume average temperature changes of the pear, strawberry, and

potato samples were compared with the volume average change of

the equivalent sphere. The volume average integral of Eq. 8 was

introduced for this purpose:

?

V

T =1

V·

T (r,t) · dr

(11)

This leads to:

T − T∞

Ti− T∞

=

∞

?

n=1

?6

λ3

n

·(sinλn− λn· cosλn)2

λn− sinλn· cosλn

· e−λ2

n·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, the

results were similar to the slowest cooling point results for pear

and strawberry and slowest heating point results for potato. These

demonstrated the suitability of the spherical approximation for

the case of volume average temperature change of an irregular

geometry as long as the sphericity value was higher than 0.9. In

0

20

40

60

80

100

120

140

0 20 4060 80100 120

Temperature (°C)

Time (min)

Can

Sphere

Figure 7–Comparison of volume averaged

temperature data of a finite cylinder and

equivalent sphere obtained with analytical

solutions.

0

20

40

60

80

100

120

140

05 1015 20 2530

Temperature (°c)

Time (min)

Cube

Sphere

Figure 8–Comparison of volume averaged

temperature data of a cube and equivalent sphere

obtained with analytical solutions.

E174 Journal of Food SciencerVol. 77, Nr. 7, 2012

Page 10

E: Food Engineering &

Physical Properties

Evaluation of spherical approximation...

the case of finite cylinder can and cube cases, comparison of the

volume average temperature changes were shown in Figure 7 and

8. While the comparison for the finite cylinder gave a reasonable

result for the volume average temperature change, due to the effect

of lower sphericity value, the comparison results were off for the

case of cube.

Biot numbers (Bi =

strawberry, and potato, respectively, while an infinite heat transfer

coefficient assumption was applied for testing the finite cylinder

and cube cases. In the finite cylinder and cube cases, a finite heat

transfer coefficient gave the similar results. Hence, it might be

assumed that the given results were independent of Biot number

range, and irregular or regular geometries with sphericity values

above 0.9 might be approximated with spherical approximation

for 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 process

deviations that might occur time to time during the course of a

heating and cooling process.

h·3·V

k

A

) were 4.6, 2.8, and 13.9 for pear,

Conclusions

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

scription of complex and irregular shapes of strawberry, pear, and

potato samples, and cooling and heating simulations were carried

out. Using the volume to surface area ratios of these samples, cen-

ter and volume average time temperature changes were compared

with the analytical solution results from the spherical geometry

assumption, and the geometries with sphericity values above 0.9

were determined to hold the spherical assumption.

Control of food processing operations requires to maintain the

specified operating conditions under unexpected changes. When

the process deviations occur time to time during the course of a

heating and cooling process, correction methodologies should be

applied. This might be obtained with the state-of-the-art control

systems where a simulation model for heating and cooling process

is running behind. In the future process lines of food industry, this

simplifying assumption could help developing on-line intelligent

correction systems that might even include a 3-D scanner.

Acknowledgment

Potato heating experiments were carried out by Rahmi Uyar

and Laura Alessandrini during her visit to the Dept. of Food

Engineering at the Univ. of Mersin in the Spring of 2009. All

helps and suggestions by Laura are much appreciated.

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