Abstract: A theoretical model is presented to predict the effective Heat Storage Coefficient (HSC) of fruit system. The system is reduced to a two phase system: Solid phase and continuous fluid phase. The solid is considered as particles having spheroidal shape which are situated at the corners of cubic unit cell. The resistor model is developed to find effective heat storage coefficient from the values of HSC of the constituent phases: protein, fat, carbohydrate, ash and water and their volume fractions. The theoretical calculations of HSC for porous food samples carried out by the proposed model gives an average deviation of 12.8% from experimental values given in literature. A comparison with other models available in the literature has also been made. The theoretical HSC values determined from present model shows least deviation from experimental values.
INTRODUCTION
The thermal characteristics of fruit samples are very important in determining their ability to storage of heat. Theoretical modelling for these substances of agri-food 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 pyro-electric methodology suitable for HSC of high viscosity liquids is proposed by Balderas-Lopez (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 multi-phase 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 x-direction 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 x-axis.
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 z-axis. 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 non-shaded portion represents
the fluid phase. The volume fraction of solid phase is
| Fig. 1: | The resistor model for two-phase system with spheroidal particles |
It is assumed that heat flux is incident normally on the face. Hence, heat storage coefficient of the bar is
(1) |
Where, βs and βf 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
(2) |
(3) |
Hence,
(4) |
Since β″ varies as x changes from 0 to a, therefore, on averaging
(5) |
Combining (Eq. 2, 5) yields the following result
(6) |
Combining (Eq. 1, 6) yields the following result
Therefore
(7) |
For spheroidal particle we have
Thus, from Eq. 7
Therefore,
(8) |
As the quadrants are identical and parallel to heat flow direction, the heat storage coefficient of the complete section is
(9) |
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 βe of the unit cell will be
or
(10) |
The unit cell contains one spheroid that lies inside. Hence fractional volume of the solid phase will be
(11) |
And in the limiting condition c = b, we get
(12) |
Thus, Eq. 10 may also be written as
(13) |
Noting that the expression (13) is based on rigid geometry, which does not represent the true state of affairs of a real two-phase 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
(14) |
Rearranging Eq. 14 we get
(15) |
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
(16) |
Where, constant C1 and C2 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 | = | Semi-minor axis length (m) |
| b | = | Side of the cube (m) |
| c | = | Semi-major axis length (m) |
| C | = | Empirical constants |
| F | = | Formation factor |
| K | = | Thermal conductivity (Wm-1K-1) |
| S | = | Specific heat (kJ kg-1K-1) |
| φ | = | Volume fraction |
| α | = | Thermal diffusivity (m2 sec-1) |
| β | = | Heat storage coefficient (Wm-2C-1sec1/2) |
| 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) |