Combinatory Models for Predicting the Effective Thermal Conductivity of Frozen and Unfrozen Food Materials

Abstract

A model to predict the effective thermal conductivity of heterogeneous materials is proposed based on unit cell approach. The model is combined with four fundamental effective thermal conductivity models (Parallel, Series, Maxwell-Eucken-I, and Maxwell-Eucken-II) to evolve a unifying equation for the estimation of effective thermal conductivity of porous and nonporous food materials. The effect of volume fraction (ν) on the structure composition factor (ψ) of the food materials is studied. The models are compared with the experimental data of various foods at the initial freezing temperature. The effective thermal conductivity estimated by the Maxwell-Eucken-I + Present model shows good agreement with the experimental data with a minimum average deviation of ±8.66% and maximum deviation of ±42.76% of Series + Present Model. The combined models have advantages over other empirical and semiempirical models.

1. Introduction

The model for estimation of effective thermal conductivity of heterogeneous mixtures is of interest in several engineering applications particularly food storage and food processing. The heterogeneous materials are bivariant compositions considered in the continuous and dispersed phases. The two-phase porous systems and food materials are heterogeneous media that share a few features as well as a several differences. The two-phase materials are generally made of solid-fluid composition with verity of shapes and sizes. The foods are heterogeneous capillary-porous colloidal materials consist of carbohydrates, protein, fat, minerals, vitamins, and ash along with water and air voids.

Numerous models were developed to find out the effective conductivity of the mixtures, but one of the major limitations of the models is its suitability for specific applications. The Eucken [1] and Maxwell [2] models were the most restrictive bounds for lower and higher volume fractions of the mixtures. But the models are only valid for vanishingly small and higher volume fractions of the dispersed phase of the mixtures (spherical shape). Meredith and Tobias [3] presented a review on conduction in heterogeneous systems and modified the earlier models of Rayleigh [4] for interactions of higher order between particles. A review of various methods for predicting the effective thermal conductivity of composite systems was proposed by Progelhof et al. [5]. They suggested that the semi-empirical approach may be more appropriate to characterize the effective thermal conductivity of structural materials such as low-density foams. Wang et al. [6] proposed a unifying equation for heterogeneous mixtures for different combinations of fundamental models, which form the basis of many of the more complex models available in the literature. The modeling of complex materials as composition of the five basic structural models using combinatory rules (equal structure volumes and equal structure effective thermal conductivities) based on the structure volume factor was proposed. Carson et al. [7] developed a model for predicting the thermal conductivity of frozen and unfrozen food materials. The effective thermal conductivity models that are functions only of the components thermal conductivities and volume fractions could not be accurate for both granular type (external porosity) and foam type porous foods (internal porosity). The structure composition factor is an extra parameter, required for predicting the thermal conductivity more accurately than the earlier proposed models. Levy [8] developed a model (modified Maxwell-Eucken) for predicting the thermal conductivity of two-component mixtures. It is one of the accurate models for predicting the thermal conductivity of heterogeneous materials, but the limitation of this model is that it was derived solely by algebraic manipulation. It was based on mathematical rather than physical arguments.

Series (De Vries [9]) and asymptotic (Fernandes et al. [10], Batchelor and O'Brien [11]) approaches have been introduced to evaluate thermal conductivity of regular arrays of spheres. The assumptions made for parallel and series structures of the composite mixtures are parallel and perpendicular to the heat flow. The stagnant thermal conductivity of a porous medium can be estimated based on a two-layer model analogous to resistances in an electrical circuit (Deissler and Boegli [12]). The maximum and minimum values of the thermal conductivities are obtained when the thermal resistances are in parallel and series, respectively, to the direction of the temperature gradient (normal to the heat flux). Kunii and Smith [13] proposed the unit cell model consisting of spherical particles contacting each other with point to point. They assume that the temperature gradient is applied along the direction of point contact of the spheres. A lumped parameter model for predicting the thermal conductivity of the porous medium was described by Hsu et al. [14]. Numerical simulations of finite difference or finite element methods were used to predict the effective conductivity of the unidirectional fibrous composite materials (Rocha and Cruz [15], Buonanno and Carotenuto [16]) and packed beds (Christon et al. [17]). Samantray et al. [18] proposed a comprehensive model to estimate the effective thermal conductivity of two-phase materials. The model was adapted for predicting the effective conductivity of various binary metallic mixtures with a high degree of confidence (Karthikeyan and Reddy [19]). Reddy and Karthikeyan [20] developed the collocated parameter model based on the unit cell approach for predicting the effective thermal conductivity of the two-phase materials.

Thermal conductivity measurement of food materials using guarded hot plate apparatus was carried out by Willix et al. [21]. The thermal conductivity of various food materials was measured for the temperature range of −40 to ${40}^{°}\text{C}$ . The thermal conductivity of 16 food products (eleven kinds of cheese, four kinds of yogurt and butter) at various temperatures ( ${15}^{°}\text{C}$ and ${30}^{°}\text{C}$ ) was carried out by Tavman and Tavman [22]. The effect of parameters such as water, fat, and protein content on the thermal conductivity was investigated by the hot wire technique. It was observed that the thermal conductivity increases linearly with increase in water content whereas it decreases linearly with increase in fat and protein contents. Several two-phase porous geometries used in civil engineering are heterogeneous with different particle contact density. The model developed by Fricke [23], as extended by De Vries [9], has been relatively successful in the estimation of the effective thermal conductivity of geometries such as soils. The soil thermal conductivity models were extended to frozen foods for predicting the thermal conductivity by Tarnawski et al. [24]. They considered eight thermal conductivity models, tested against experimental data of 13 meat products in the temperature range from 0 to $-{40}^{°}\text{C}$ .

The primary effects influencing the effective conductivity of the heterogeneous mixtures include conductivity ratio (α), structure volume fraction (ν), and structure composition factor (ψ). At present, there is no satisfactory solution for all ranges of α and ν, because the neighbour interactions on the field produced higher order effects, which is difficult to model. In addition to the primary effects, the secondary parameters such as contact resistance, radiation, convection, Knudsen effect, and particle geometrical characteristics influence the effective conductivity. In this paper, an extended approach to predict the effective thermal conductivity of two-dimensional spatially periodic two-phase systems based on the unit cell approach with isotherm has been attempted and unifying equations are solved for frozen and unfrozen foods (multicomponent material). The unifying equations from four fundamental structure models combined with present model have been validated with experimental data for various food materials to evolve a better multistructure model. The reliability of the combinatory models has been checked by comparing the present model predictions with the experimental data and Levy's model.

2. Thermal Conductivity Estimation of Two-Phase System

2.1. Isotherm-Based Unit Cell Model

The earliest model of the unit cube was prescribed by Krischer [25]. The upper and lower limits to the conductivity of two-phase mixtures were described by Wiener [26]. The electric resistance analogy leads to algebraic expressions for stagnant thermal conductivity of the two-phase materials. The resistance method is referred to as the collocated parameter model. The main feature of the method is to assume one-dimensional heat conduction in a unit cell. The unit cell is divided into three parallel layers, namely, solids, fluid, or composite layers normal to the temperature gradient. The effective thermal conductivity of two-phase system is determined by considering equivalent electrical resistances of parallel and series in the unit cell model. The thermal conductivity of the composite layer is obtained using the series model.

The effective thermal conductivity of the two-dimensional medium can be estimated by considering a square cylinder with cross-section “ $a\times a$ ” having a connecting bar width of “c” as shown in Figure 1(a). The stagnant thermal conductivity of the two-dimensional periodic medium is the finite contact between the spheres by connecting plates with “c/a” denoting the contact parameter. Because of the symmetry of the plates, one fourth of the square cross-section has been considered as a unit cell and is shown in Figure 1(b). The unit cell consists of three rectangular layers normal to the direction of heat flow. The thermal conductivity of the solid and fluid layer is obtained based on a series model. The first rectangular layer is fully occupied by the solid with a dimension of ( $l/2$ ) ( $c/2$ ), and the other two rectangular layers consist of solid and fluid phases with a dimension of $(l/2)((a-c)/2)$ and $(l/2)((l-a)/2)$ , respectively. The model is based on the one-dimensional heat conduction in the unit cell. The temperature gradient in the three layers is normal to the direction of heat flow. Based on the resistance approach, for two-phase systems of $0<\alpha <1000$ and $0<\nu <1$ , the effective conductivity is calculated under the isotherm condition as follows.

Two-Dimensional spatial periodic two-phase system. (a) Touching square cylinder. (b) Unit cell of square cylinder.

The total resistance offered by the unit cell is given as

$R_{\text{total}}=R_1+R_{\text{eff}2}+R_{\text{eff}3},$

(1)

where, resistance offered by layer 1 is, $R_1=c/k_2{l}^2$ , resistance offered by layer 2 is, $R_{\text{eff}2}=(a-c)/(l[k_{2}a+k_1(l-a)]),$ resistance offered by layer 3 is, $R_{\text{eff}3}=(l-a)/(l[k_{2}c+k_1(l-c)]).$

Also,

$R_{\text{total}}=\frac{\epsilon }{l}\left[\frac{\zeta }{k_2}+\frac{(1-\zeta )}{k_2\epsilon +k_1(1-\epsilon )}+\frac{(1-\epsilon )}{{\epsilon }^2{k}_2+k_1(1-\epsilon )}\right].$

(2)

The concentration is defined as the solid-phase fraction of the unit cell and is given by

$\begin{array}{cc}\hfill {\nu }_2& =\frac{\text{Volume of the solid}}{\text{Total volume of the unit cell}}\hfill \\ \hfill & =\frac{[cl+a(a-c)+c(l-a)]}{l^2}.\hfill \end{array}$

(3)

Equation (3) can be rewritten as

${\nu }_2=\left[{\epsilon }^2\left(1-2\zeta \right)+2\epsilon \zeta \right],$

(4)

where $\alpha =k_2/k_1,\epsilon =a/l,\zeta =c/a,\epsilon \zeta =c/l$

$k_{\text{eff}}=\frac{1}{R_{\text{total}}l}.$

(5)

The effective thermal conductivity of two-phase mixtures is given as:

$k_{\text{eff}}=\epsilon {\left[\frac{\zeta }{k_2}+\frac{(1-\zeta )}{k_2\epsilon +k_1(1-\epsilon )}+\frac{(1-\epsilon )}{{\epsilon }^2{k}_2+k_1(1-\epsilon )}\right]}^{-1}.$

(6)

2.2. Other Basic Structure Models

The popularly reported basic structure models for estimation of effective thermal conductivity of two-phase materials include Parallel, Series, ME-I (Rocha and Cruz [15]; Buonanno and Carotenuto [16]), and ME-II (Rocha and Cruz [15]; Buonanno and Carotenuto [16]). In Series and Parallel models, it is assumed that the layer of the components of physical structures is oriented either perpendicular or parallel to the heat flow.

In the parallel model, the effective thermal conductivity is expressed as

$k_{\text{eff}}=k_1{\nu }_1+k_2{\nu }_2.$

(7)

In the series model, the effective thermal conductivity is given by

$k_{\text{eff}}=\frac{1}{\left({\nu }_1/k_1\right)+\left({\nu }_2/k_2\right)}.$

(8)

The effective thermal conductivity of low concentration two-phase systems is given by Maxwell-Eucken-I (Rocha and Cruz [15]; Buonanno and Carotenuto [16])

$k_{\text{eff}}=\frac{k_1{\nu }_1+k_2{\nu }_2\left[3k_1/\left(2k_1+k_2\right)\right]}{{\nu }_1+{\nu }_2\left[3k_1/\left(2k_1+k_2\right)\right]}.$

(9)

For high concentration two-phase system, the effective thermal conductivity is expressed by Maxwell-Eucken-II (Rocha and Cruz [15]; Buonanno and Carotenuto [16])

$k_{\text{eff}}=\frac{k_2{\nu }_2+k_1{\nu }_1\left[3k_2/\left(2k_2+k_1\right)\right]}{{\nu }_2+{\nu }_1\left[3k_2/\left(2k_2+k_1\right)\right]}.$

(10)

2.3. Combinatory Models for Heterogeneous Materials

Krischer [25] developed a model based on the complex structure, which could be approximated by the mixture of simple structures. The effective conductivity of the structure is the combined effect of parallel and series models and is given as.

$k_{\text{eff}}=\frac{1}{\left(f/k_{\text{series}}\right)+\left(1-f\right)/\left(k_{\text{parallel}}\right)}.$

(11)

The complex physical structures of the heterogeneous materials can be modeled by using simple combinatory rules. Wang et al. [6] developed a new approach of unifying equation for fundamental models by selecting suitable multicomponent parameters. The method proposed by them is adapted to combine the present model with four fundamental structure models to evolve unifying equations for heterogeneous two-phase materials. The thermal conductivity of two-component materials is expressed by Wang et al. [6]:

$k_j=\frac{{\sum }_{i=1}^m{k}_i{\nu }_i{\psi }_{ij}\left(d_{i}k/(d_i-1)k+k_i\right)}{{\sum }_{i=1}^m{\nu }_i{\psi }_{ij}\left(d_{i}k/(d_i-1)k+k_i\right)}.$

(12)

Equation (12) is used in the series and parallel models when d _i = 1 and $d_i=\infty ,$ respectively. For the Maxwell-Eucken–I and Maxwell-Eucken-II (Rocha and Cruz [15]; Buonanno and Carotenuto [16]) the above expression is applicable when d _i = 3 (assuming spherical particle in the dispersed phase).

The schematic of two-component materials as uniform mixtures of two-structural models is shown in Figures 2(a)–2(d). The heterogeneous model assumed that the half of the mixture is represented by one model and the remaining half by another model. Figure 2(a) shows that half of the volume has parallel structure with component 1 as continuous phase, whereas the other half of the volume has the two-dimensional spatially periodic square cylinder structure (present model) with component 2 as continuous phase. Similarly other structural models can be combined with present model for effective thermal conductivity prediction of food materials (Figures 2(b)–2(d)). The selected binary structure models for food materials are discussed in the following sections.

Two-component materials as uniform mixtures of two-structural models (a) Parallel + Present Model. (b) ME-I + Present Model. (c) ME-II + Present Model. (d) Series + Present Model.

For modeling of thermal conductivity, the food materials may be classified as four types:

1. unfrozen nonporous foods ( $k_{\text{water}}/k_{\text{solid}}\le 3$ ),

2. unfrozen porous foods $(k_{\text{water}}/k_{\text{gas}}\le 20)$ ,

3. frozen nonporous foods $(k_{\text{ice}}/k_{\text{solid}}\approx 12),$

4. frozen porous foods $(k_{\text{ice}}/k_{\text{gas}}\approx 100)$ .

2.3.1. Parallel + Present Model

Substituting (6) and (7) in (12) and equating the Parallel and Present models yield effective thermal conductivity as

$\begin{array}{cc}\hfill k_{\text{eff}}& =\frac{k_1{\nu }_1{\psi }_{11}+k_2{\nu }_2{\psi }_{21}}{{\nu }_1{\psi }_{11}+{\nu }_2{\psi }_{21}}\hfill \\ \hfill & =k_2\left[\left(1-\epsilon \right)k_1{\psi }_{12}+{\epsilon }^2{k}_2{\psi }_{22}\right]\left[k_2\epsilon +k_1\left(1-\epsilon \right)\right]/\hfill \\ \hfill & \hspace{1em}[\left\{\epsilon \zeta \left[\left(1-\epsilon \right)k_1{\psi }_{12}+\epsilon k_2{\psi }_{22}\right]\left[{\epsilon }^2{k}_2+k_1\left(1-\epsilon \right)\right]\right\}\hfill \\ \hfill & \hspace{1em}\hspace{1em}+\epsilon k_2\left(1-\zeta \right)\left[\left(1-\epsilon \right)k_1{\psi }_{12}+{\epsilon }^2{k}_2{\psi }_{22}\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+\epsilon k_2\left(1-\epsilon \right)\left[\left(1-\epsilon \right)k_1{\psi }_{12}+\epsilon k_2{\psi }_{22}\right]].\hfill \end{array}$

(13)

The solution of (13) is obtained as

${\psi }_{11}{\psi }_{12}A+{\psi }_{12}{\psi }_{21}B+{\psi }_{11}{\psi }_{22}C+{\psi }_{21}{\psi }_{22}D=0,$

(14)

where

$\begin{array}{cc}\hfill A& ={k_1}^2{k}_2{\nu }_1\left[{\epsilon }^3\left(\epsilon \zeta -\zeta -1\right)+\epsilon \zeta \left(1-\epsilon \right)+4{\epsilon }^2-4\epsilon +1\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^2{\nu }_1\epsilon \left(1-\epsilon \right)+{k_1}^3{\nu }_1\left\{\epsilon \zeta \left(2\epsilon -{\epsilon }^2-1\right)\right\},\hfill \\ \hfill B& ={k_1}^2{k}_2{\nu }_2\left[\left\{\epsilon \zeta \left(2\epsilon -{\epsilon }^2-1\right)\right\}+{\epsilon }^2-2\epsilon +1\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^2{\nu }_2\left[\left\{\epsilon \zeta \left({\epsilon }^3-{\epsilon }^2-\epsilon +1\right)\right\}-\epsilon \left({\epsilon }^2-2\epsilon +1\right)\right],\hfill \\ \hfill C& =k_1{k_2}^2{\nu }_1\left[{\epsilon }^3\zeta \left(1-\epsilon \right)-{\epsilon }^3\right]+{k_1}^2{k}_2{\nu }_1\left[{\epsilon }^2\zeta \left(\epsilon -1\right)\right]\hfill \\ \hfill & \hspace{1em}+{k_2}^3{\nu }_1{\epsilon }^3,\hfill \\ \hfill D& =k_1{k_2}^2{\nu }_2\left[{\epsilon }^2\zeta \left(\epsilon -1\right)+{\epsilon }^2\left(1-\epsilon \right)\right]\hfill \\ \hfill & \hspace{1em}+{k_2}^3{\nu }_2\left[{\epsilon }^3\zeta \left(1-\epsilon \right)+{\epsilon }^2\left(\epsilon -1\right)\right].\hfill \end{array}$

(15)

The volume composition factor for multicomponent material may be written as

$\begin{array}{c}{\psi }_{11}{\nu }_1+{\psi }_{21}{\nu }_2=0.5,\end{array}$

(16)

$\begin{array}{c}{\psi }_{12}{\nu }_1+{\psi }_{22}{\nu }_2=0.5.\end{array}$

(17)

Based on the structure composition factor for multicomponent material can be expressed as

$\begin{array}{c}{\psi }_{11}+{\psi }_{12}=1,\end{array}$

(18)

$\begin{array}{c}{\psi }_{21}+{\psi }_{22}=1.\end{array}$

(19)

From (16)–(19), substituting the values ${\psi }_{12},{\psi }_{21}$ and ψ₂₂ in terms of ψ₁₁ in (14) and after simplification, we have

$\begin{array}{cc}\hfill & {{\psi }_{11}}^2\left[-4{v_2}^{2}A+4{\nu }_1{\nu }_{2}B+4{\nu }_1{\nu }_{2}C-4{\nu }_1^{2}D\right]\hfill \\ \hfill & \hspace{1em}+{\psi }_{11}(4{\nu }_2^{2}A-4{\nu }_1{\nu }_{2}B-2{\nu }_{2}B+4{\nu }_2^{2}C-2{\nu }_{2}C+2{\nu }_{1}D\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-4{\nu }_1{\nu }_{2}D+2{\nu }_{1}D)+2{\nu }_2(B+D)-D=0.\hfill \end{array}$

(20)

The solution of (20) is obtained as

$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{11}& =\frac{{\nu }_{2}B\left[1+\left({\nu }_1/2\right)\right]-{\nu }_{2}C\left[{\nu }_2-1\right]-2{{\nu }_2}^{2}A-{\nu }_{1}D\left[2-\left({\nu }_2/2\right)\right]}{\left(4{\nu }_2\left[{\nu }_{1}B+{\nu }_{1}C-{\nu }_{2}A\right]-4{{\nu }_1}^{2}D\right)}\hfill \\ \hfill & \hspace{1em}\pm \frac{\sqrt{{\left({\nu }_{2}C\left[{\nu }_2-1\right]+4{{\nu }_2}^{2}A-{\nu }_{2}B\left[1+\left({\nu }_1/2\right)\right]+{\nu }_{1}D\left[2-\left({\nu }_2/2\right)\right]\right)}^2-\left(\left[4{\nu }_2\left[{\nu }_{1}B+{\nu }_{1}C-{\nu }_{2}A\right]-4{{\nu }_1}^{2}D\right]\left\{2{\nu }_2\left[B+D\right]-D\right\}\right)}}{\left(4{\nu }_2\left[{\nu }_{1}B+{\nu }_{1}C-{\nu }_{2}A\right]-4{{\nu }_1}^{2}D\right)}.\hfill \end{array}$

(21)

2.3.2. Series + Present Model

Combining the Series model and Present model yields an effective thermal conductivity; from (12) we have

$\begin{array}{cc}\hfill k_{\text{eff}}& =\frac{{\nu }_1{\psi }_{11}+{\nu }_2{\psi }_{21}}{\left({\nu }_1{\psi }_{11}/k_1\right)+\left({\nu }_2{\psi }_{21}/k_2\right)}\hfill \\ \hfill & =k_2\left[\left(1-\epsilon \right)k_1{\psi }_{12}+{\epsilon }^2{k}_2{\psi }_{22}\right]\left[k_2\epsilon +k_1\left(1-\epsilon \right)\right]/\hfill \\ \hfill & \hspace{1em}[\left\{\epsilon \zeta \left[\left(1-\epsilon \right)k_1{\psi }_{12}+\epsilon k_2{\psi }_{22}\right]\left[{\epsilon }^2{k}_2+k_1\left(1-\epsilon \right)\right]\right\}\hfill \\ \hfill & \hspace{1em}\hspace{1em}+\epsilon k_2\left(1-\zeta \right)\left[\left(1-\epsilon \right)k_1{\psi }_{12}+{\epsilon }^2{k}_2{\psi }_{22}\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+\epsilon k_2\left(1-\epsilon \right)\left[\left(1-\epsilon \right)k_1{\psi }_{12}+\epsilon k_2{\psi }_{22}\right]].\hfill \end{array}$

(22)

Equation (22) is simplified as

${\psi }_{11}{\psi }_{12}A+{\psi }_{11}{\psi }_{22}B+{\psi }_{12}{\psi }_{21}C+{\psi }_{21}{\psi }_{22}D=0,$

(23)

where

$\begin{array}{cc}\hfill A& ={k_1}^2{k_2}^2{\nu }_1\left[\epsilon \zeta \left(1-\epsilon -{\epsilon }^2+{\epsilon }^3\right)+1-4\epsilon +4{\epsilon }^2-{\epsilon }^3\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^3{\nu }_1\epsilon \left(1-\epsilon \right)+{k_1}^3{k}_2{\nu }_1\left(2{\epsilon }^2\zeta -{\epsilon }^3\zeta -\epsilon \zeta \right),\hfill \\ \hfill B& ={k_1}^2{k_2}^2{\nu }_1\left[{\epsilon }^2\zeta \left(\epsilon -1\right)\right]+k_1{k_2}^3{\nu }_1\left[{\epsilon }^3\left(\zeta -\epsilon \zeta -1\right)\right]\hfill \\ \hfill & \hspace{1em}+{\epsilon }^3{\nu }_1{k_2}^4,\hfill \\ \hfill C& ={k_1}^2{k_2}^2{\nu }_2\left[\epsilon \zeta \left({\epsilon }^3-{\epsilon }^2-\epsilon +1\right)+\epsilon \left(2\epsilon -{\epsilon }^2-1\right)\right]\hfill \\ \hfill & \hspace{1em}+{k_1}^3{k}_2{\nu }_2\left[{\epsilon }^2-2\epsilon +1+\epsilon \zeta \left(2\epsilon -{\epsilon }^2-1\right)\right],\hfill \\ \hfill D& ={k_1}^2{k_2}^2{\nu }_2\left[{\epsilon }^3\left(\zeta -1\right)+{\epsilon }^2\left(1-\zeta \right)\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^3{\nu }_2\left[{\epsilon }^3\left(1+\epsilon \right)-{\epsilon }^2-{\epsilon }^4\zeta \right].\hfill \end{array}$

(24)

2.3.3. Maxwell-Eucken-I + Present Model

Combining the Maxwell-Eucken-I and present model yields an effective thermal conductivity:

$\begin{array}{cc}\hfill k_{\text{eff}}& =\frac{k_1{\nu }_1{\psi }_{11}+k_2{\nu }_2{\psi }_{21}\left[3k_1/\left(2k_1+k_2\right)\right]}{{\nu }_1{\psi }_{11}+{\nu }_2{\psi }_{21}\left[3k_1/\left(2k_1+k_2\right)\right]}\hfill \\ \hfill & =k_2\left[\left(1-\epsilon \right)k_1{\psi }_{12}+{\epsilon }^2{k}_2{\psi }_{22}\right]\left[k_2\epsilon +k_1\left(1-\epsilon \right)\right]/\hfill \\ \hfill & \hspace{1em}[\left\{\epsilon \zeta \left[\left(1-\epsilon \right)k_1{\psi }_{12}+\epsilon k_2{\psi }_{22}\right]\left[{\epsilon }^2{k}_2+k_1\left(1-\epsilon \right)\right]\right\}\hfill \\ \hfill & \hspace{1em}\hspace{1em}+\epsilon k_2\left(1-\zeta \right)\left[\left(1-\epsilon \right)k_1{\psi }_{12}+{\epsilon }^2{k}_2{\psi }_{22}\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+\epsilon k_2\left(1-\epsilon \right)\left[\left(1-\epsilon \right)k_1{\psi }_{12}+\epsilon k_2{\psi }_{22}\right]].\hfill \end{array}$

(25)

Equation (25) is rearranged as

${\psi }_{11}{\psi }_{12}A+{\psi }_{12}{\psi }_{21}B+{\psi }_{11}{\psi }_{22}C+{\psi }_{21}{\psi }_{22}D=0,$

(26)

where

$\begin{array}{cc}\hfill A& =[{k_1}^2{k_2}^2{\nu }_1\left[\epsilon \zeta \left({\epsilon }^3-{\epsilon }^2-\epsilon +1\right)-\epsilon \left({\epsilon }^2-2\epsilon +2\right)+1\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+{k_1}^3{k}_2{\nu }_1\left[2{\epsilon }^3\zeta \left(\epsilon -1\right)+2\left(2{\epsilon }^2-4\epsilon +1\right)\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+k_1{k_2}^3{\nu }_1\left[\epsilon \left(1-\epsilon -{\epsilon }^3\zeta \right)\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+{k_1}^4{\nu }_1\left[2\epsilon \zeta \left(2{\epsilon }^2-\epsilon -1\right)\right]],\hfill \\ \hfill B& ={k_1}^2{k_2}^2{\nu }_2\left[3\epsilon \zeta \left({\epsilon }^3-{\epsilon }^2-\epsilon +1\right)-3\epsilon \left({\epsilon }^2-2\epsilon +1\right)\right]\hfill \\ \hfill & \hspace{1em}+{k_1}^3{k}_2{\nu }_2\left[3\left({\epsilon }^2-2\epsilon +1\right)\left(1-\epsilon \zeta \right)\right],\hfill \\ \hfill C& ={k_1}^2{k_2}^2{\nu }_1\left[{\epsilon }^2\zeta \left(3\epsilon -2{\epsilon }^2-1\right)-2{\epsilon }^3\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^3{\nu }_1\left[{\epsilon }^2\left(2\epsilon \zeta +\epsilon -1\right)\right]\hfill \\ \hfill & \hspace{1em}+{k_1}^3{k}_2{\nu }_1\left[2{\epsilon }^2\zeta \left(\epsilon -1\right)\right]+{k_2}^4{\epsilon }^3,\hfill \\ \hfill D& ={k_1}^2{k_2}^2{\nu }_2\left[3{\epsilon }^2\zeta \left(\epsilon -1\right)-3{\epsilon }^2\left(\epsilon -1\right)\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^3{\nu }_2\left[3{\epsilon }^2\left(2\epsilon -{\epsilon }^2\zeta -1\right)\right].\hfill \end{array}$

(27)

2.3.4. Maxwell-Eucken-II + Present Model

The Maxwell-Eucken-II and the present model are combined which yields an effective thermal conductivity:

$\begin{array}{cc}\hfill k_{\text{eff}}& =k_2\left[\left(1-\epsilon \right)k_1{\psi }_{11}+{\epsilon }^2{k}_2{\psi }_{21}\right]\left[k_2\epsilon +k_1\left(1-\epsilon \right)\right]/\hfill \\ \hfill & \hspace{1em}[\left\{\epsilon \zeta \left[\left(1-\epsilon \right)k_1{\psi }_{11}+\epsilon k_2{\psi }_{21}\right]\left[{\epsilon }^2{k}_2+k_1\left(1-\epsilon \right)\right]\right\}\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}+\epsilon k_2(1-\zeta )[(1-\epsilon )k_1{\psi }_{11}+{\epsilon }^2{k}_2{\psi }_{21}]\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}+\epsilon k_2(1-\epsilon )[(1-\epsilon )k_1{\psi }_{11}+\epsilon k_2{\psi }_{21}]]\hfill \\ \hfill & =\frac{k_2{\nu }_2{\psi }_{22}+k_1{\nu }_1{\psi }_{12}\left[3k_2/\left(2k_2+k_1\right)\right]}{{\nu }_2{\psi }_{22}+{\nu }_1{\psi }_{12}\left[3k_2/\left(2k_2+k_1\right)\right]}.\hfill \end{array}$

(28)

Also

${\psi }_{11}{\psi }_{12}A+{\psi }_{11}{\psi }_{22}B+{\psi }_{12}{\psi }_{21}C+{\psi }_{21}{\psi }_{22}D=0,$

(29)

where

$\begin{array}{cc}\hfill A& =3{k_1}^2{k_2}^2{\nu }_1\left[\epsilon \zeta \left({\epsilon }^3-{\epsilon }^2-\epsilon +1\right)-\left({\epsilon }^3-4{\epsilon }^2+4\epsilon -1\right)\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^3{\nu }_1\left[3\epsilon \left(1-\epsilon \right)\right]+3{k_1}^3{k}_2{\nu }_1\left[2{\epsilon }^2\zeta -{\epsilon }^3\zeta -\epsilon \zeta \right],\hfill \\ \hfill B& =[{k_1}^2{k_2}^2{\nu }_2\left[\epsilon \zeta \left({\epsilon }^3-3{\epsilon }^2+3\epsilon -1\right)+\left(4{\epsilon }^2-{\epsilon }^3-5\epsilon +2\right)\right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+k_1{k_2}^3{\nu }_2\left[2\epsilon \zeta \left({\epsilon }^3-{\epsilon }^2+\epsilon -1\right)+4{\epsilon }^2-2{\epsilon }^3-2\epsilon \right]\hfill \\ \hfill & \hspace{1em}\hspace{1em}+{k_1}^3{k}_2{\nu }_2\left[\epsilon \zeta \left(2\epsilon -{\epsilon }^2-1\right)+\left({\epsilon }^2-2\epsilon +1\right)\right]],\hfill \\ \hfill C& =3{k_1}^2{k_2}^2{\nu }_1\left[{\epsilon }^2\zeta \left(\epsilon -1\right)\right]+3k_1{k_2}^3{\nu }_1\left[{\epsilon }^3\left(\zeta -{\zeta }^2-1\right)\right]\hfill \\ \hfill & \hspace{1em}+3{\epsilon }^3{\nu }_1{k_2}^4,\hfill \\ \hfill D& ={k_1}^2{k_2}^2{\nu }_2\left[{\epsilon }^2\zeta \left(\epsilon -1\right)-{\epsilon }^2\left(\epsilon -1\right)\right]\hfill \\ \hfill & \hspace{1em}+k_1{k_2}^3{\nu }_2\left[{\epsilon }^2\left(3\epsilon \zeta -{\epsilon }^2\zeta -2\zeta +1\right)\right]\hfill \\ \hfill & \hspace{1em}+{k_2}^4{\nu }_2\left[2{\epsilon }^2\left(\epsilon \zeta -{\epsilon }^2\zeta +\epsilon -1\right)\right].\hfill \end{array}$

(30)

From (16)–(19), substituting the value of ${\psi }_{12},{\psi }_{21},$ and ψ₂₂ in terms of ψ₁₁ in (23), (26), and (29) we obtained the expressions for ψ₁₁.

3. Parametric Analysis of Combined Models

The nondimensional conductivity of a heterogeneous system mainly depends on volume fraction (ν), conductivity ratio (α), thermal contact between solid-solid interface and structure composition (ψ). The variation of nondimensional conductivity with volume fraction (ν₂) for unfrozen nonporous food $(k_1/k_2=3)$ is shown in Figure 3. The two limits of Maxwell-Eucken-I and Maxwell-Eucken-II lines lie between the extreme limits of parallel and series models. All the models predicted closely for low conductivity ratio $(k_1/k_2=3)$ . The combinatory model, Maxwell-Eucken- I + Present predicted well as compared to other models. The two limits of Maxwell-Eucken-I and Maxwell-Eucken-II lines lie between the extreme limits of parallel and series models. All the models predicted closely for low conductivity ratio $(k_1/k_2=3)$ . However, ME-I + Present model predicted well as compared to other models. The variation of nondimensional conductivity with volume fraction (ν₂) for $k_1/k_2=12$ , 20, and 100 is shown respectively in Figures 4, 5 and 6. For frozen nonporous foods ( $k_{\text{ice}}/k_{\text{solid}}=12$ ), the Parallel + Present model and Series + Present model lie in between ME-I and ME-II. The ME-I + Present model and ME-II + Present model are lie in between Parallel + Present and Series + Present models. The predictions of combinatory models for unfrozen porous and nonporous foods are comparable. Therefore, all the models predict well within the most restrictive bounds proposed by Eucken [1] and Maxwell [2] for lower and higher volume fraction of the mixtures. In case of frozen and unfrozen porous foods, the deviations in the bounds are significant due to the presence of gaseous dispersion phase in the food materials.

Figure 3

The variation of nondimensional conductivity with volume fraction of the disperse phase for unfrozen nonporous foods ( $k_{\text{water}}/k_{\text{solid}}-3$ ).

Figure 4

The variation of nondimensional conductivity with volume fraction of the disperse phase for frozen nonporous foods ( $k_{\text{ice}}/k_{\text{solid}}-12$ ).

Figure 5

The variation of nondimensional conductivity with volume fraction of the disperse phase for unfrozen porous foods ( $k_{\text{water}}/k_{\text{gas}}-20$ ).

Figure 6

The variation of nondimensional conductivity with volume fraction of the disperse phase for frozen porous foods ( $k_{\text{ice}}/k_{\text{gas}}-100$ ).

The effect of volume structure factor on the volume composition factor for different heterogeneous models has been investigated for lower ( $k_1/k_2=3$ ) and higher conductivity ratio ( $k_1/k_2=100$ ). The effects of volume fraction on structure composition factor of lower conductivity ratio ( $k_{\text{water}}/k_{\text{solid}}-3$ ) for various combinatory models are shown in Figures 7, 8, 9, and 10. It is observed that the structure composition factors ψ₁₁ and ψ₁₂ are mirror images to each other for Parallel + Present model and Series + Present model. This is because the conductivity tends to upper and lower limits for Parallel and Series models, respectively. Similarly for ME-I and ME-II + Present models, the structure composition factor ψ₁₁ and ψ₁₂ are mirror images to each other, as the dispersed phase is used as a continuous phase for ME-II Model. These combined models may not be applicable when material structure is defined mathematically. In such cases, the present models are not in closer agreement with the material conductivity. Nevertheless, for the natural occurring materials, which are the combination of soil characteristics and biological materials, the models can be predicted with higher accuracy.

Figure 7

The variation of structure composition factor with volume fraction for unfrozen nonporous foods ( $k_{\text{water}}/k_{\text{solid}}-3$ ) for Parallel + Present Model.

Figure 8

The variation of structure composition factor with volume fraction for unfrozen nonporous foods ( $k_{\text{water}}/k_{\text{solid}}-3$ ) for Series + Present Model.

Figure 9

The variation of structure composition factor with volume fraction for unfrozen nonporous foods ( $k_{\text{water}}/k_{\text{solid}}-3$ ) for Maxwell Eucken-I + Present Model

Figure 10

The variation of structure composition factor with volume fraction for unfrozen and nonporous foods ( $k_{\text{water}}/k_{\text{solid}}-3$ ) for Maxwell Eucken-II + Present Model.

4. Effective Thermal Conductivity of Foods

The effective thermal conductivity of various foods has been estimated using combinatory models. The food materials are essentially consisting of water, protein, fat, carbohydrate, salt, and ash. A lumped conductivity has been considered using parallel approach for a dispersed phase consisting of more than one component. The thermal conductivity and density of food components are given in Table 1. In the frozen foods, ice, water plus ice, water, and fat contents are considered as a continuous phase whereas food solid components, food solid components plus water, and food solid components plus ice, food solid components plus ice plus water are considered as the dispersed phase. The major constituents of the various food materials are illustrated in Table 2. The experimental conductivity data were considered at initial freezing temperature of the food materials for the comparison of four unifying equations. It is evident that the unifying equations are not a function of temperature.

Table 1:

Properties of food components [27].

Component	Water	Ice	Protein	Fat	Ash	Carbohydrate
k(W/mK)	0.56	2.21	0.2	0.18	0.33	0.2
ρfood (kg/m3)	1000	917	1380	930	2424	1600

Table 2:

Composition of food materials.

Food Materials	Compositions
	Water	Protein	Fat	Ash	Salt	Carbohydrate
Lean beef [21]	0.75	0.23	0.009	0.011	0	0
Beef Mince [21]	0.64	0.193	0.17	0.009	0	0
Boneless Chicken [21]	0.751	0.236	0.008	0.012	0	0
Pork Sausage Meat [21]	0.637	0.117	0.217	0.026	0.017	0
Trim Pork Mince [21]	0.716	0.189	0.095	0.01	0	0
Veal Mince [21]	0.685	0.19	0.095	0.01	0	0
Venison [21]	0.754	0.225	0.0075	0.015	0	0
Vension (Thawed) [21]	0.748	0.234	0.0055	0.012	0	0
Beef dripping [21]	0.003	0.002	0.995	0	0	0
Pork dripping [21]	0.002	0.001	0.997	0	0	0
Pork fat [21]	0.132	0.04	0.817	0.002	0	0
Gurnard Fillets [21]	0.797	0.197	0.0055	0.0135	0	0
Lemon fish fillets [21]	0.77	0.261	0.001	0.013	0	0
Snapper fillets [21]	0.782	0.204	0.0035	0.0135	0	0
Tarakihi fish fillets [21]	0.779	0.198	0.015	0.0115	0	0
Trevally fillets [21]	0.777	0.216	0.009	0.012	0	0
Coxs orange appeals [21]	0.873	0.0035	0.003	0.003	0	0
Fiesta apples [21]	0.871	0.003	0.0035	0.002	0	0
Royal Gala Apples [21]	0.876	0.002	0.0004	0.0017	0	0
Cheddar cheese [21]	0.363	0.235	0.341	0.036	0.0145	0
Edam Cheese [21]	0.401	0.248	0.314	0.031	0.0099	0
Mozzarella Cheese [21]	0.437	0.257	0.251	0.0304	0.0107	0
Salted butter [21]	0.146	0	0.824	0.0145	0.0132	0
Unsalted butter [21]	0.152	0	0.827	0.001	0	0
Tylose [21]	0.763	0	0.0004	0.001	0	0
Butter [22]	0.151	0.0118	0.836	0.0012	0	0
Hamburger Cheese [22]	0.41	0.2058	0.2478	0.0584	0	0.078
Old Kashkaval Cheese [22]	0.41	0.2656	0.2655	0.0388	0	0.0201
Tulum Cheese [22]	0.41	0.248	0.289	0.0376	0	0.0154
Fresh Kashkaval Cheese [22]	0.4379	0.2616	0.2275	0.0299	0	0.0431
Buffet Kashkaval Cheese [22]	0.4984	0.3197	0.1427	0.036	0	0.0056
Fresh Cream Cheese [22]	0.5632	0.0749	0.2354	0.0258	0	0.1007
Labne [22]	0.6913	0.0562	0.2094	0.0099	0	0.0332
Low fat Labne [22]	0.7465	0.0899	0.1025	0.0131	0	0.048
Spreadable Cheese [22]	0.606	0.1538	0.1625	0.0432	0	0.0345
Strained Yogurt [22]	0.7423	0.0959	0.0742	0.0131	0	0.0745
Light Yogurt [22]	0.8195	0.069	0.0155	0.0129	0	0.0831
Pasteurized Yogurt [22]	0.8248	0.0566	0.0412	0.0214	0	0.056
Extra light Yogurt [22]	0.8681	0.0598	0.0019	0.003	0	0.0672
Leg Muscle II Parallel [24]	0.779	0.158	0.054	0.005	0	0.005
Leg Muscle Minced [24]	0.783	0.148	0.051	0.004	0	0.013
Hearts [24]	0.728	0.117	0.138	0.005	0	0.011
Hearts Minced [24]	0.715	0.116	0.156	0.004	0	0.009
Livers [24]	0.739	0.156	0.072	0.007	0	0.027
Livers Minced [24]	0.738	0.17	0.043	0.009	0	0.039
Brains [24]	0.818	0.08	0.085	0.006	0	0.011
Kidneys [24]	0.838	0.114	0.035	0.006	0	0.008
Thymus [24]	0.818	0.111	0.071	0	0	0
Thymus Minced [24]	0.786	0.105	0.103	0.006	0	0
Leg Muscle Per [24]	0.762	0.152	0.082	0.004	0	0.001
Fat [24]	0.126	0.023	0.851	0	0	0
Fat Minced [24]	0.105	0.021	0.874	0.001	0	0

Levy [8] proposed a model based on the Maxwell-Eucken Model for multicomponent materials. The Levy's model predictions are better for fat and other types of food materials, when the entire range of compositions and temperatures is considered. The Maxwell-Eucken-I + Present model estimations are comparable to the Levy's model for frozen, nonfrozen porous, and nonporous food materials. Therefore, the physical model of Levy must be the same as the ME-I + Present model for a heterogeneous mixture (Macroscopic scale) for conductivities of both equal component structures. The comparison of Levy's model with Maxwell-Eucken-I + Preset model for different types of lower ( $k_1/k_2=3$ ) and higher ( $k_1/k_2=100$ ) conducting porous materials is shown in Figure 11. For lower conductivity ranges ( $k_1/k_2=3$ ), the Maxwell-Eucken-I + Present work showed better results as compared to Levy's model, as the higher order and secondary effects are negligible. For high conductivity ratio ( $k_1/k_2=100$ ), the higher order and secondary effects play an important role and Levy's results deviate from the present model. Levy's model was derived solely on algebraic manipulation, with lack of physical justification (Wang et al. [6]). Subsequently, the researchers (Willix et al. [21]; Pham and Willix [28]) have been reluctant to recommend its use for estimating thermal conductivity of food materials. The present model predictions have been compared with the experiment value of various heterogeneous materials available in the literature (Willix et al. [21]; Tavman and Tavman [22]; Tarnawski et al. [24]; Pham and Willix [28]; Choi and Okos [27]). The Maxwell-Eucken-I + Present model show better results as compared with other models. The average deviation from the experimental data to the ME-I + Present model and Levy's model at the initial freezing temperature of the food materials is varying from ±8.66% to ±9.16% as shown in Figures 12 and 13. The comparison of four combinatory models with Present model against the experimental data is given in Table 3. For other models like Parallel, Series, and Maxwell-Eucken-II + Present models, the average deviations from the experimental data are ±11.59%, ±42.76% and ±35.57%, respectively and these are shown in Figures 14, 15, and 16. In Figures 15 and 16, the deviation of the predicted thermal conductivity from the experimental data is high (42% and 35%). This is due to the fact that the present model is combined with Series and Maxwell-Eucken II Models. The series model always acts as a lower bound for thermal conductivity of homogeneous two-phase materials. Similarly, the dispersed phase is used as a continuous phase for Maxwell-Eucken II model which acts as an upper bound for thermal conductivity of homogeneous two-phase materials. This is the reason for higher deviation from the experimental data for frozen and unfrozen foods. But essentially, this study focuses on how basic structure models are combined with present isotherm unit cell model to evolve in the model of unifying equations for the prediction of the effective thermal conductivity of various food materials.

Table 3:

Comparison of present and Levy's models with experimental data for various food materials.

Food mater	$({\nu }_2)$	$k_1/k_2=\alpha$	k exp	$k(\text{Pa}+\text{Pre})$	$k(\text{Se}+\text{Pre})$	$k(\text{ME-I}+\text{Pre})$	% Devi $k_{\exp }-k(\text{ME-I}+\text{Pre})/k_{\exp }$	k(ME-II+Pre)	k Levy
Lean beef [21]	0.23	2.8	0.467	0.535	0.316	0.412	11.78	0.365	0.453
Beef Mince [21]	0.193	8.092	0.395	0.461	0.156	0.346	12.41	0.199	0.398
Boneless Chicken [21]	0.236	2.8	0.477	0.529	0.315	0.404	15.3	0.368	0.45
Pork Sausage Meat [21]	0.22	9.032	0.377	0.404	0.132	0.345	8.49	0.19	0.372
Trim Pork Mince [21]	0.189	10.2	0.425	0.432	0.133	0.407	4.24	0.209	0.388
Veal Mince [21]	0.19	10.182	0.423	0.431	0.132	0.397	6.15	0.197	0.387
Venison [21]	0.225	2.8	0.458	0.541	0.316	0.419	8.52	0.364	0.455
Vension (Thawed) [21]	0.234	2.8	0.468	0.531	0.315	0.439	6.2	0.366	0.451
Beef dripping [21]	0.997	3.111	0.18	0.181	0.18	0.181	0.56	0.18	0.181
Pork dripping [21]	0.998	3.111	0.169	0.18	0.18	0.181	7.1	0.18	0.181
Pork fat [21]	0.817	3.613	0.209	0.194	0.158	0.196	6.22	0.162	0.205
Gurnard Fillets [21]	0.197	2.8	0.499	0.559	0.333	0.462	7.41	0.376	0.467
Lemon fish fillets [21]	0.261	2.8	0.507	0.504	0.314	0.488	3.75	0.381	0.44
Snapper fillets [21]	0.204	2.8	0.46	0.543	0.324	0.451	1.96	0.383	0.464
Tarakihi fish fillets [21]	0.198	2.8	0.461	0.549	0.33	0.46	0.22	0.376	0.466
Trevally fillets [21]	0.216	2.8	0.455	0.552	0.318	0.432	5.05	0.366	0.459
Coxs orange appeals [21]	0.127	2.8	0.448	0.552	0.385	0.483	7.81	0.441	0.498
Fiesta apples [21]	0.129	2.8	0.427	0.557	0.395	0.452	5.85	0.434	0.497
Royal Gala Apples [21]	0.124	2.8	0.457	0.542	0.404	0.46	0.66	0.442	0.5
Cheddar cheese [21]	0.341	4.198	0.35	0.367	0.189	0.338	3.43	0.176	0.363
Edam Cheese [21]	0.314	5.278	0.35	0.366	0.163	0.331	5.43	0.155	0.355
Mozzarella Cheese [21]	0.257	5.913	0.354	0.407	0.166	0.352	0.56	0.142	0.378
Salted butter [21]	0.824	3.111	0.195	0.216	0.189	0.2	2.56	0.187	0.228
Unsalted butter [21]	0.827	3.111	0.178	0.215	0.19	0.199	11.8	0.186	0.227
Tylose [21]	0.237	3.111	0.459	0.518	0.296	0.388	15.47	0.36	0.441
Butter [22]	0.836	3.111	0.233	0.213	0.191	0.199	14.59	0.186	0.224
Hamburger Cheese [22]	0.247	6.557	0.398	0.403	0.154	0.339	14.82	0.139	0.376
Old Kashkaval Cheese [22]	0.265	5.176	0.384	0.409	0.181	0.341	11.2	0.268	0.384
Tulum Cheese [22]	0.289	5.283	0.377	0.387	0.17	0.354	6.1	0.236	0.369
Fresh Kashkaval Cheese [22]	0.262	5.996	0.403	0.401	0.166	0.353	12.41	0.22	0.374
Buffet Kashkaval Cheese [22]	0.32	6.253	0.409	0.345	0.186	0.339	17.11	0.163	0.337
Fresh Cream Cheese [22]	0.235	9.777	0.434	0.373	0.158	0.393	9.45	0.166	0.357
Labne [22]	0.21	11.429	0.463	0.381	0.218	0.519	12.1	0.11	0.366
Low fat Labne [22]	0.102	15.41	0.542	0.56	0.339	0.489	9.78	0.215	0.447
Spreadable Cheese [22]	0.163	9.343	0.494	0.491	0.231	0.409	17.21	0.359	0.413
Strained Yogurt [22]	0.1	16.807	0.539	0.546	0.359	0.479	11.13	0.279	0.447
Light Yogurt [22]	0.069	33.755	0.583	0.48	0.358	0.499	14.41	0.178	0.468
Pasteurized Yogurt [22]	0.0566	29.899	0.593	0.601	0.316	0.536	9.61	0.248	0.485
Extra light Yogurt [22]	0.0598	45.528	0.596	0.492	0.127	0.474	20.47	0.201	0.476
Leg Muscle II Parallel [24]	0.158	13.659	0.45	0.435	0.119	0.392	12.89	0.207	0.399
Leg Muscle Minced [24]	0.148	14.737	0.466	0.441	0.116	0.405	13.09	0.203	0.405
Hearts [24]	0.117	11.2	0.39	0.51	0.178	0.408	4.62	0.312	0.443
Hearts Minced [24]	0.116	10.769	0.407	0.531	0.181	0.409	0.49	0.292	0.446
Livers [24]	0.156	12.727	0.417	0.453	0.126	0.4	4.08	0.198	0.404
Livers Minced [24]	0.17	2.8	0.425	0.541	0.356	0.462	8.71	0.391	0.479
Brains [24]	0.182	18.667	0.494	0.394	0.084	0.498	0.81	0.185	0.364
Kidneys [24]	0.114	2.8	0.507	0.554	0.407	0.541	6.71	0.438	0.504
Thymus [24]	0.111	16.471	0.497	0.514	0.129	0.442	11.07	0.213	0.436
Thymus Minced [24]	0.105	14	0.487	0.529	0.168	0.497	2.05	0.248	0.447
Leg Muscle Per [24]	0.152	12.444	0.421	0.467	0.13	0.374	11.16	0.209	0.408
Fat [24]	0.851	3.544	0.219	0.189	0.162	0.175	20.09	0.164	0.198
Fat Minced [24]	0.874	3.457	0.212	0.188	0.165	0.18	15.09	0.168	0.196
Average Deviation (%)							8.66

Figure 11

Comparison of Levy's model with Maxwell Eucken-I + Present model for frozen and unfrozen porous and nonporous materials.

Figure 12

Comparison of experimental results with Maxwell Eucken-I + Present model for food materials.

Figure 13

Comparison of Levy's model with experimental results.

Figure 14

Comparison of Parallel + Present model with experimental results.

Figure 15

Comparison of Series + Present model with experimental results.

Figure 16

Comparison of Maxwell-Eucken-II + Present model with experimental results.

5. Conclusion

The various combinatory models are developed to evolve unifying equation to predict the effective thermal conductivity of the food materials. The effects of volume fraction on nondimensional conductivity are responsible for the physical corrections of the model. The structure composition factor and structure volume fraction are important tools to predict the thermal conductivity of the multicomponent materials. The combinatory models are compared with the experimental data for various food materials. The results show that the ME-I + Present model is superior to the other models for wide variety of food materials. The minimum deviation is found to be ±8.66% for ME-I + Present model. This is because measurement errors in the composition of foods, temperature dependence of thermal conductivity of ice, and nature were the limitations of the predictive models. The unifying equations can effectively be used to predict the thermal conductivity of various multicomponent materials.