INTRODUCTION
The thermal characteristics of fruit samples are very important in determining
their ability to storage of heat. Theoretical modelling for these substances
of agrifood bears industrial importance and is a challenging task for
food technologist and physicists. It is required because of the increasing
demand of food substances as processed and preserved and also in drying
of perishable produce.
Thermal conductivity (K), thermal diffusivity (α) and specific heat
(S) are the three parameters cited most often in the literature for describing
the thermal behaviour of the substances. The heat storage coefficient
or effusivity is another important thermophysical parameter for all kinds
of heat transfer processes. Many workers including Babanov (1957), Jacob
(1964), Nerpin and Chudnovskii (1970), Luc and Balageas (1981) have mentioned
the HSC under various names. It is defined as
Lichtenecker (1926) also presented a simple working empirical relation
for porous mixture. In the literature (Ingersoll et al., 1969;
Carslaw and Jaeger, 1959) one finds that the HSC of composites is an additive
property and considering various components as resistors one can take
a combination of these to predict effective HSC. This is a common practice
adopted to predict effective thermal conductivity from the thermal conductivity
of different phases for porous materials. Accepting the similarity, a
geometry dependent resistor model has been proposed for heat storage coefficient
of food materials.
Verma et al. (1990) initiated experimental work and determined
the HSC of metallic powders by using a plane heat source. Thermal heat
storage coefficient or effusivity of drop size insulating liquid has been
measured by pulse transient heat strip technique by Gustavsson et al.
(2003). A new photo pyroelectric methodology suitable for HSC of high
viscosity liquids is proposed by BalderasLopez (2003). This may be used
for characterization for liquids of industrial importance viz vegetables
oil. Measurements of HSC for powdered titania samples by photo acoustic
technique is given by Hernandez Ayala et al. (2005).
The theoretical models for the determination of HSC of porous materials
are also available in literature. Shrotriya et al. (1991) have
proposed a theoretical model for the prediction of HSC of loose granular
substances and compared theoretical values of HSC obtained from the model
with values obtained by experiments performed with plane heat source.
They considered cubic particles in a cubic unit cell. Misra et al.
(1994) proposed a resistor model to determine HSC of two phase systems,
by assuming the grains of the medium as spherical in shape and by replacing
porosity (Φ) by porosity correction factor (Fp). Heat storage characteristic
of soil have also been investigated by Zhang et al. (2007). They
used randomly mixed model to simulate the spatial structure of the multiphase
media and observed, the significant effect of the degree of saturation
on heat storage coefficient.
However, it has been seen that these theoretical models are not suitable
for food substances. Thus in the present study a theoretical model to
predict the effective HSC of fruits is given. Since, the main constituents
of the fruits are protein, fat, carbohydrate, ash and water. The system
may be considered having two phases consisting of water as continuous
phase and other constituents together as discontinuous solid phase. The
arrangement of cubic array has been divided into unit cells. The solid
phase is of spheroidal inclusion in a cubic unit cell and resistor model
is applied to determine effective HSC of unit cell. Since the HSC of two
phase systems also depends upon various factors such as HSC of constituent
phases, porosity, shape factor, size of particles their distribution etc.
and, incorporating all these factors in the prediction of HSC of two phase
system is a complex affair. Therefore a porosity correction term has been
introduced to account for HSC of real two phase systems. The theoretical
values of HSCs obtained from this model are compared with values reported
in literature and these values show a close agreement.
THEORETICAL FORMULATION
In the following analysis we assumed a homogeneous medium with heat flux
in the xdirection and the heat transfer is only by conduction. Let the
solid inclusions be spheroids located at the corners of a cube of side
2b. Their distribution in 2D is shown in Fig. 1(a) and
the 3D geometry of a unit cell is shown in Fig. 1(b).
Let the origin of the coordinate axis be located at the center of the
spheroid having principal axes 2a, 2c and 2a (a < c). The unit cell
can be divided into thin slices by planes perpendicular to the xaxis.
Consider one such slice bounded by two planes at distances x and x + dx.
The section shown in Fig. 1(c) is subdivided into four
quadrants. One such section is shown in Fig. 1(d). Let
us further divide the section by planes perpendicular to the zaxis. It
will divide the section into rectangular bars. One such bar is shown in
Fig. 1(e). Let the length of the bar be b and area of
cross section dxdz. The shaded portion of the element in the Fig.
1(d) represents the solid phase and the nonshaded portion represents
the fluid phase. The volume fraction of solid phase is
and of the fluid phase is

Fig. 1: 
The resistor model for twophase system with spheroidal particles 
It is assumed that heat flux is incident normally on the face. Hence, heat
storage coefficient of the bar is
Where, β_{s} and β_{f} are the heat storage
coefficients of solid and fluid phase, respectively. In reference to the
Fig. 1(d), the heat storage coefficient of the quadrant
will be
Therefore
Hence,
Since β″ varies as x changes from 0 to a, therefore,
on averaging
Combining (Eq. 2, 5) yields the following
result
Combining (Eq. 1, 6) yields the following
result
Therefore
For spheroidal particle we have
Thus, from Eq. 7
Therefore,
As the quadrants are identical and parallel to heat flow direction, the
heat storage coefficient of the complete section is
The sections 1, 2 and 3 in Fig. 1(b) form equivalent
series resistors perpendicular to the direction of heat flow, therefore
the effective heat storage coefficient β_{e} of the unit
cell will be
or
The unit cell contains one spheroid that lies inside. Hence fractional
volume of the solid phase will be
And in the limiting condition c = b, we get
Thus, Eq. 10 may also be written as
Noting that the expression (13) is based on rigid geometry, which does
not represent the true state of affairs of a real twophase system. Thus,
for practical utilization, we have to modify the expression (13) by incorporating
some correction term. Tareev (1975) has shown that, during the flow of
electric flux from one dielectric to another dielectric medium, the deviation
of flux lines in any medium depends upon the ratio of the dielectric constants
of the two media. By the same analogy we can have the concentration of
thermal flux altered from its previous value as it passes through another
medium and that the amount is a function of the heat storage coefficients
of the constituent phases. Considering random packing of phases, non uniform
shape of particles and the flow of heat flux lines not restricted to be
parallel we here replace physical volume fraction of solid phase by porosity
correction term F. F in general should be a function of the physical volume
fraction of the solid phase and the ratio of the heat storage coefficients
of the constituent phases. Therefore, expression (13) may be written as
Rearranging Eq. 14 we get
Where:
RESULTS AND DISCUSSION
We have tested the validity of theoretical model discussed above on two phase
systems for which the characteristics of the constituent phases and the experimental
values are given in literature. Thus the heat storage coefficients of the solid
and fluid phases, porosity and the experimental results for effective heat storage
coefficients have been considered as are given in literature.
Table 1: 
Comparison of effective heat storage coefficient of two phase
systems 

*Composition data from USDA (1996); ASHRAE Refrigeration
Handbook (2002) 

Fig. 2: 
Comparison of experimental and theoretical values of effective
HSC 
The solid phase consists of protein, carbohydrate, fat and ash. The effective
heat storage coefficient of solid phase is calculated from parallel resistor
model because series resistor model results show more deviation from experimentally
measured values (Rahman et al., 1991). The theoretical values of the
heat storage coefficients have been calculated using Eq. 13.These
are compared with experimentally known values which are determined by empirical
relations (Appendix).These are based on extensive experimental data. Since the
deviation between these experimental and theoretical values is appreciable ,
therefore formation factor has been introduced in porosity. The correction factor
introduced for each sample has been computed using Eq. 15
and plotted as a function of .
The curve fitting technique gives the following formation factor for food samples.
Where, constant C_{1} and C_{2} are 0.184, 2.116, respectively.
On applying above equation as the porosity correction in Eq. 14
we have calculated the values of heat storage coefficient for a number of samples
(Table 1). Figure 2 shows a comparison of
the experimental results of heat storage coefficient and calculated values from
Eq. 14. It is seen from this plot that experimental values
and the proposed spheroidal model values show an average deviation of 12.8%.

Fig. 3: 
Comparison of experimental and theoretical values of effective
HSC 

Fig. 4: 
Comparison of experimental and theoretical values of effective
HSC as a function of volume fraction of fluid 
Thus, the spheroidal model with porosity correction can be used successfully
to predict the heat storage coefficients of similar systems when heat storage
coefficients of their constituents phases and the porosity values are known.
Since the samples under study are porous therefore, a comparison with other
models for effective heat storage coefficients for porous materials have also
been made. Thus, HSC using Shrotriya et al. (1991), Misra et al.
(1994) and Lichtenecker model (1926) has been determined. Fig.
3 shows comparison of experimental values of some food samples with these
models. The average deviation in HSC for food materials is 21.57, 43.14, 24.39%,
for Lichtenecker (1926), Misra et al. (1994) and Shrotriya (1991) models,
respectively. However, the proposed model shows only 12.87% deviation. Thus,
the present model gives better results for food samples than the other models.

Fig. 5: 
Comparison of experimental and theoretical values of effective
HSC as a function of volume fraction of solid 
Figure 4 and 5 show a comparative variation
of effective heat storage coefficient as a function of volume fraction of fluid
and solid, respectively and calculated from different models. The results using
present model again show least deviation from the experimental values.
CONCLUSIONS
The effective heat storage coefficient of food systems may be determined
with empirical correction to porosity in the theoretical model. The porosity
correction term in the spheroidal model for prediction of heat storage
coefficient is found to be dependent on the ratio of the HSC of the constituent
phases of the system. And, using the parallel resistor model and the HSC
of constituent phases, the solid phase HSC may be known. The proposed
spheroidal model with porosity correction shows an average deviation of
12.8% from the experimental values. Thus, the values of HSC predicted
by the present model are close to experimental results than obtained from
other models cited in the literature. Thus, using this theoretical model
one can find out the HSC of fruit samples.
ACKNOWLEDGMENT
One of the author DKS is grateful to Council of Scientific and Industrial
Research, New Delhi for providing Senior Research Fellowship.
NOMENCLATURE
a 
= 
Semiminor axis length (m) 
b 
= 
Side of the cube (m) 
c 
= 
Semimajor axis length (m) 
C 
= 
Empirical constants 
F 
= 
Formation factor 
K 
= 
Thermal conductivity (Wm^{1}K^{1}) 
S 
= 
Specific heat (kJ kg^{1}K^{1}) 
φ 
= 
Volume fraction 
α 
= 
Thermal diffusivity (m^{2 }sec^{1}) 
β 
= 
Heat storage coefficient (Wm^{2}C^{1}sec^{1/2}) 
ρ 
= 
Density (kg m^{3}) 
SUBSCRIPTS
av 
= 
average 
e 
= 
effective 
f 
= 
fluid 
s 
= 
solid 
1, 2 respective values 
APPENDIX
Thermal property model for food components 

Source: Choi and Okas (1986) 