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Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System



Usha Singh, D.K. Singhal and D.R. Chaudhary
 
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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.

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Usha Singh, D.K. Singhal and D.R. Chaudhary, 2008. Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System. Asian Journal of Agricultural Research, 2: 15-24.

DOI: 10.3923/ajar.2008.15.24

URL: https://scialert.net/abstract/?doi=ajar.2008.15.24

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

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

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 Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System and of the fluid phase is Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System


Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
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

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(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

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

Therefore

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(2)

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(3)

Hence,

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(4)

Since β″ varies as x changes from 0 to a, therefore, on averaging

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(5)

Combining (Eq. 2, 5) yields the following result

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(6)

Combining (Eq. 1, 6) yields the following result

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

Therefore

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(7)

For spheroidal particle we have

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

Thus, from Eq. 7

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

Therefore,

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(8)

As the quadrants are identical and parallel to heat flow direction, the heat storage coefficient of the complete section is

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(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

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

or

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(10)

The unit cell contains one spheroid that lies inside. Hence fractional volume of the solid phase will be

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(11)

And in the limiting condition c = b, we get

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(12)

Thus, Eq. 10 may also be written as

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(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

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(14)

Rearranging Eq. 14 we get

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(15)

Where:

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System

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
Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
*Composition data from USDA (1996); ASHRAE Refrigeration Handbook (2002)

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
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 Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System. The curve fitting technique gives the following formation factor for food samples.


Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
(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%.

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
Fig. 3: Comparison of experimental and theoretical values of effective HSC

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
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.

Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
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
Image for - Theoretical Model for Heat Storage Coefficient of Fruits as a Two Phase System
Source: Choi and Okas (1986)

REFERENCES

1:  ASHRAE, 2002. Ashrae Refrigeration Handbook. Chapter 8.1-8.30.

2:  Babanov, A.A., 1957. Methods for calculation of thermal conduction coefficients of capillary porous materials. Sov. Phys. Tech. Phys., 2: 476-476.

3:  Balderas-Lopez, J.A., 2003. Measurements of the thermal effusivity of transparent liquids by means of a photopyroeleactric technique. Revista Mexicana De Fisica, 49: 353-357.

4:  Carslaw, H.S. and J.C. Jaeger, 1959. Conduction of Heat in Solids. 2nd Edn., Oxford, Clarendon, pp: 75- 88

5:  Choi, Y. and M.R. Okas, 1986. Effect of Temperature and Composition on the Thermal Properties of Foods. In: Food Engineering and Process Applications, LeMaguer, M. and P. Jelen (Eds.). Elsevier Applied Sci. Publishers, London, pp: 93-101

6:  Gustavsson, M., H. Nagai and T. Okutani, 2003. Measurements of thermal effusivity of a drop size liquid using the pulse transient hot strip technique Fifteenth Symposium of Thermophysical Properties , Boulder , Colorado, USA.

7:  Hernandez-Ayala, A., T. Lopez, P. Quintana, J.J. Alvardo-Gil and J. Pacheco, 2005. Time evolution of the thermal properties during dehydration of sol-gel Titania emulsions. Azojomo, 1: 1-11.
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8:  Ingersoll, L.R., O.J. Zobel and A.C. Ingersoll, 1969. Heat Conduction Indian. 1st Edn., Oxford University Press, Calcutta, pp: 91, 110, 118, 156, 244, 246, 254

9:  Jacob, M., 1964. Heat Transfer vol. I. 1st Edn., Wiley, New York, pp: 297-279

10:  Lichtenecker, K., 1926. Physik, 27: 115

11:  Luc, A.M. and D.L. Balageas, 1981. Non stationary thermal behaviour of reinforced composite: A better evaluation of wall energy balance for convective onduction. Proceedings of the Joint Conference on Thermophysical Properties, June 15-18, 1981, Gaithersburg, MD., Overa, Maryland, pp: 1-23

12:  Misra, K., A.K. Shrotriya, N. Singhvi, R. Singh and D.R. Chaudhary, 1994. Prediction of heat storage coefficient of two-phase systems with spherical inclusions. J. Phys. Applied Phys., 27: 1823-1829.
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13:  Nerpin, S.V. and A.F. Chudnovskii, 1970. Physics of soil. 1st Edn., Isrel Program for Scientific Translations Ltd., Jerusalem, pp: 187-194

14:  Rahman, M.S. and R.H. Driscoll, 1991. Thermal conductivity of seafood: Calamari, octopus and prawn. Food, Aust., 43: 356-356.

15:  Shrotriya, A.K., L.S. Verma, R. Singh and D.R. Chaudhary, 1991. Prediction of heat storage coefficient of some loose granular materials on the basis of structure and packing of the grains. Heat Recovery Syst, CHP., 11: 423-430.
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16:  Tareev, B., 1975. Physics of Dielectric Materials. 1st Edn., MIR, Moscow, pp: 128

17:  USDA, 1996. Nutrient database for standard reference. Release 11.US. 1st Edn., Department of Agriculture, Washington, DC.,

18:  Verma, L.S., A.K. Shrotriya, U. Singh and D.R. Chaudhary, 1990. Heat storage coefficient an important thermo-physical parameter and its experimental determination. J. Phys. Applied Phys., 23: 1405-1410.
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19:  Zhang, H.F., X.S. Ge, H. Ye and D.S. Jiao, 2007. Heat conduction and heat storage characteristics of soils. Applied Thermal Eng., 27: 369-373.
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