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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
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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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Evaluation of spherical approximation . . .
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.
0
5
10
15
20
25
30
0 800 1600 2400 3200 4000 4800
Tem
pera
ture
(C)
Time (s)
Pear
Medium
3D simulaon
0
5
10
15
20
25
0 600 1200 1800 2400
Tem
pera
ture
(C)
Time (s)
Strawberry
Medium
3D simulaon
0
20
40
60
80
100
120
0 800 1600 2400 3200 4000 4800
Tem
pera
ture
(C)
Time (s)
Potato
Medium
3D simulaon
A
B
C
Figure 2Comparison of three-dimensionalsimulation results with
the experimental data (A)for pear cooling (B) strawberry cooling
(C) potatoheating.
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Evaluation of spherical approximation . . .
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.
0
5
10
15
20
25
30
0 1200 2400 3600 4800
Tem
pera
ture
(C)
Time (s)
Pear 3-D simulaon
Sphere analycal soluon
0
5
10
15
20
25
0 600 1200 1800 2400
Tem
pera
ture
(C)
Time (s)
Strawberry 3-D simulaon
Sphere analycal soluon
0
20
40
60
80
100
120
0 600 1200 1800 2400 3000 3600
Tem
pera
ture
(C)
Time (s)
Potato 3D simulaon
Sphere analycal soluon
A
B
C
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)
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
Tem
pera
ture
(C)
Time (min)
Finite cylinder
Sphere
Figure 4Comparison of timetemperature dataobtained with
analytical solutions at the center ofa finite cylinder can and its
equivalent sphere.
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Evaluation of spherical approximation . . .
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.
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
Tem
pera
ture
(C)
Time (min)
Cube
Sphere
Figure 5Comparison of timetemperature dataobtained with
analytical solutions at the center ofa cube and its equivalent
sphere.
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Evaluation of spherical approximation . . .
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
0
5
10
15
20
25
30
0 800 1600 2400 3200 4000 4800
Tem
pera
ture
(C)
Time (s)
Pear
Analycal soluon
0
5
10
15
20
25
0 600 1200 1800 2400
Tem
pera
ture
(C)
Time (s)
Strawberry
Analycal soluon
0
20
40
60
80
100
120
0 800 1600 2400 3200 4000 4800
Tem
pera
ture
(C)
Time (s)
Potato
Analycal soluon
A
B
C
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
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
Tem
pera
ture
(C)
Time (min)
Can
Sphere
Figure 7Comparison of volume averagedtemperature data of a
finite cylinder andequivalent sphere obtained with
analyticalsolutions.
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
Tem
pera
ture
(c)
Time (min)
Cube
Sphere
Figure 8Comparison of volume averagedtemperature data of a cube
and equivalent sphereobtained with analytical solutions.
Vol. 0, Nr. 0, 2012 Journal of Food Science E9
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E:FoodEngineering&PhysicalProperties
Evaluation of spherical approximation . . .
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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