


ARTICLE 

Year : 2012  Volume
: 2
 Issue : 2  Page : 118128 

Estimation of Effective Thermal Conductivity of TwoPhase Materials by Considering the Knudsen Effect: An Analytical Approach
AP Senthil Kumar^{1}, P Karthikeyan^{2}, Prem Paramasivam^{1}, A Ram Kishan^{1}, Azhar A Muhammedh Hasan^{1}
^{1} Department of Mechanical Engineering, PSG College of Technology, Coimbatore, India ^{2} Department of Automobile Engineering, PSG College of Technology, Coimbatore, India
Date of Web Publication  4Aug2012 
Correspondence Address: A P Senthil Kumar Department of Mechanical Engineering, PSG College of Technology, Coimbatore India
Source of Support: None, Conflict of Interest: None  Check 
Abstract   
In this article, the analytic model has been developed to estimate the effective thermal conductivity (ETC) of twophase materials based on the unit cell approach by considering the concentration, conductivity ratio, contact resistance, and Knudsen effect. The derivations of algebraic equations for standard geometry, such as hexagon and octagon cylinder models are developed based on parallel isotherm approach. The developed analytic model has been used to predict the thermal conductivity of various twophase materials (conductivity ratio, α = 3.11310.86 and concentration, ν = 0.05 and 0.74). The present models are validated using the standard models and compared with the experimental data. Further the comparison is made between the present models and existing models. The results are in good agreement. Keywords: Geometrydependent resistance model, effective thermal conductivity, concentration, conductivity ratio, unit cell approach, twophase materials, Knudsen effect
How to cite this article: Senthil Kumar A P, Karthikeyan P, Paramasivam P, Kishan A R, Muhammedh Hasan AA. Estimation of Effective Thermal Conductivity of TwoPhase Materials by Considering the Knudsen Effect: An Analytical Approach. J Eng Technol 2012;2:11828 
How to cite this URL: Senthil Kumar A P, Karthikeyan P, Paramasivam P, Kishan A R, Muhammedh Hasan AA. Estimation of Effective Thermal Conductivity of TwoPhase Materials by Considering the Knudsen Effect: An Analytical Approach. J Eng Technol [serial online] 2012 [cited 2020 Jul 6];2:11828. Available from: http://www.onlinejet.net/text.asp?2012/2/2/118/99300 
1. Introduction   
A twophase material is one in which there are distinct parts of the material that have different chemical or physical structures. The two phases are combined together with the formation of continuous and dispersed phases based on their characteristics. The performance of the twophase material is dependent on the effective thermal conductivity of the materials. There are various twophased systems, such as ceramics, soils, foams, emulsion systems, porous systems, suspension systems, solidsolid mixtures, fiber reinforced materials, and composites. The knowledge of the estimation of effective thermal conductivity (ETC) of twophase materials are becoming increasingly important in many applications such as microelectronic chip cooling, space craft structures, catalytic reactors, heat recovery process, heat exchangers, heat storage systems, petroleum refineries, solar collectors, nuclear reactors, and automotive applications.
The development of effective thermal conductivity of twophase materials has defied an appropriate solution till date. The ETC of twophase materials is influenced by concentration, conductivity of the material, grain size and shape (geometric configuration), contact resistance, radiations, convection, pressure (Knudsen) effect. It is also a matter of concern to engineers and physicists. It is obvious that it is practically impossible to conduct experiments to study the effect of the above parameters on the ETC; hence a theoretic expression is required to predict its value. There are large discrepancies between the thermal conductivities predicted by the different models, and hence poor choice of models could result in highly inaccurate predictions. There are many applicable thermal conductivity models to be found in the literature. However, analytic models are preferred to numerical models in many applications due to its frugal nature, costeffective calculations, and comparatively reasonable accuracy. But a general expression that can predict ETC of most of the twophase systems with the abovementioned parameters, especially like Knudsen effect, convection and radiation is still lacking. The problem is one of the long standing and has been treated in many papers on the basis of unit cell approach by considering primary parameters, such as concentration of the dispersed phase (ν) and conductivity ratio (α) and secondary parameters (contact resistance, heat transfer through radiation and convection, Knudsen effect, and geometric orientations).
Resistance model approach (incorporating Parallel isotherm method) has been applied to determine effective thermal conductivity for various inclusions. Maxwell [1] and Hashin and Shtrikman [2] developed the most restrictive bounds (lower and upper) for estimating the effective thermal conductivity of the twophase materials. The upper and lower limits to the conductivity of twophase materials based on parallel and series resistance were given by Wiener [3]. Bruggeman [4] extended the Maxwell's result for lower concentration of the dispersed phases to the full range of concentration by assuming the mixture to be quasihomogenous. Zehner and Schlunder [5] proposed a model, considering the effect of primary (particle contact) and secondary parameters (thermal radiation, pressure dependence, particle flattening, shape and size distribution) for cylindrical unit cell containing spherical inclusions. An important deficiency in the model is that the deformation of the flux field is taken only as a function of concentration, not as a function of the conductivity ratio.
Thermal conductivity modeling for random distribution of spheres in a continuum of different materials has been carried out by Raghavan and Martin [6]. The model was based on the unit cell approach with constant heat flux conditions. Hsu et al [7] obtained algebraic expressions for effective thermal conductivities of a number of porous media by applying the socalled lumped parameter method, which is based on an electric resistance analogy. The thermal conductivity of a saturated porous medium was calculated for a twolayer model representing as electrical resistance in an electrical circuit [8]. Kunii and Smith [9] proposed a 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. On the basis of the phase averaging of temperature field, Hadley [10] proposed a model for predicting the effective thermal conductivity of the twophase materials. The terms of solid and fluid continuous appeared to have been formulated by Brailsford and Major [11]. The formulation is based on the Maxwell's model by considering solid component as a continuous phase, or the fluid component as a continuous phase.
Numerical study for effective conductivity based on a model made up of spheres in cubic lattice has been carried out by Krupiczka [12]. Samantray et al. [13] proposed a comprehensive conductivity model by considering the primary parameters based on unit cell and field solution approaches. Later, the validity of the model was extended to predict the effective conductivity of various binary metallic mixtures with a high degree of accuracy.
Raed and Gross [14] studied the influence of the poresize distribution on the ETC and modeling of the modification of the ETC of different porous materials (Knudsen and nonKnudsen materials) with various poresize distributions by exchanging the gas atmosphere. The modeling was carried out based on the Kinetic theory and the Knudsen effect of rarefaction of the gas in a wide range of the Knudsen number.
Deqiang Mu et al. [15] used a 3dimensional bond pore network model and simulations to evaluate the effect of pore size and connectivity on the effective diffusion coefficient of random porous media. The simulation results showed that the effective diffusion coefficient is strongly dependent on pore size, when the average pore size, d _{m} < 1 μm. They incurred that for a single pore, the nondimensional effective diffusion coefficient increases with the mean pore diameter when the Knudsen effect is considered.
Demirel [16] has given modified expressions for the heat transfer fluxes at the interface of a solid surface and a gas for different Knudsen number regimes. Based on the experimental data, a general expression had been proposed for the thermal accommodation coefficient (a _{c}).
Wakao and Vortmeyer [17] theoretically computed the effective thermal conductivities of packed beds with stagnant gas and expressed the ETC as an algebraic sum of gassolid conductivity, radiation conductivity, and contact conductivity. They inferred that gassolid and radiation conductivity are pressure dependent. For fine particles, they also showed that the ETC even at normal pressures is still in the pressuredependent region and influenced by the gas accommodation coefficient (a _{c}). They derived an expression for gas thermal conductivity from kinetic theory of gases taking into account the temperature jump effect.
Kaganer et al. [18] measured the thermal conductivity of silica gels at air pressures from one atmosphere to 10^{−2} mmHg. Masaume and Smith [19] measured the thermal conductivity of glass beads at air pressures from one atmosphere to 10^{−2} mmHg.
Reddy and Karthikeyan [20] developed the collocated parameter model based on the unit cell approach for predicting the effective thermal conductivity of the twophase materials. Senthil Kumar et al. [21] estimated the effective thermal conductivity of twophase materials based on the unit cell approach by considering two secondary parameters, namely, contact resistance and geometric configuration in addition to primary parameters.
In this present study, the Knudsen effect has been included, which is largely influenced by the variation of pressure in the ETC of twophase materials, in addition to the primary parameters (concentration and conductivity ratio) and the secondary parameters, namely, geometric configuration and contact resistance. Because pressure is indirectly proportional to the mean free path and hence to the Knudsen number, a very low pressure must exist, which results in either transition regime or the molecular flow regime so that either gaswall collisions or complicated flow can occur. The effective thermal conductivity values of the present models (which includes Knudsen effect) are validated using the standard models and are compared with the experimental data. The comparison with the existing models is also done.
2. Effective Thermal Conductivity Modeling for Various Inclusions Based on Primary and Secondary Parameters   
The development of collocated parameter model for estimating the effective thermal conductivity based on the material's micro and nanostructure is extremely important for thermal design and analysis of twophase systems. The electric resistance analogy leads to algebraic expressions for thermal conductivity of the twophase materials. The resistance method is referred as the collocated parameter model. The main feature of this method is to assume onedimensional 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 twophase system is determined by considering equivalent electrical resistances of parallel and series in the collocated parameter unit cell model. The thermal conductivity of the composite layer is obtained using the series model.
2.1 Hexagon cylinder
The effective thermal conductivity of the twodimensional medium can be estimated by considering a hexagon cylinder with crosssection "a × a0" having a connecting bar width of "c" as shown in [Figure 1]a. The stagnant thermal conductivity of the twodimensional periodic medium is the finite contact between the spheres by connecting plates with "c/a0" denoting the contact parameter. Because of the symmetry of the plates, one fourth of the square crosssection 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 other two rectangular layers consists of solid and fluid phases with a dimension of (l/2) (a√3/2 − c/2) and (l/2) (l/2 − a√3/2), respectively. The model is based on the onedimensional heat conduction in the unit cell. The temperature gradient in the three layers is normal to the direction of heat flow. The effective thermal conductivity of twodimensional hexagon cylinder is calculated for parallel isotherm conditions as follows:
Total resistance offered by the hexagon cylinder in the unit cell is given as follows:
Where,
Resistance offered by the solid layer I,
Total resistance offered by the layer II,
The thermal conductivities of the composite layers can be obtained based on the series model.
The thermal conductivity of the composite layer II is given by
The kinetic theory of gases describes a gas as a large number of atoms or molecules, all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion.
The expression for pressuredependent gas thermal conductivity is given by Kennard (1938):
The molecular mean free path is defined as the average distance a gas molecule travels before it collides with another gas molecule and it is proportional to the gas temperature and inversely proportional to the gas pressure where the velocities of the identical particles have a Maxwell distribution. By kinetic theory of gases, the mean free path can be expressed as [22]
The characteristic length, l_{c} , for the hexagon model is given as,
The term accommodation coefficient, a _{c} used in the expression of gas thermal conductivity represents the heat exchange rate between the gas molecules and the pore walls. It can be obtained by the expression given by Song and Yavanovich [23].
Hence the gas thermal conductivity Eq. (6) can be rewritten as
Using the above relations, Eq. (5) is rewritten as
Similarly, the thermal conductivity of the composite layer III is given by:
The solid phase fraction of the unit cell is represented in terms of concentration (ν), and is given by:
Substituting Eqs (29) in Eq. (1), the total resistance of unit cell is simplified as,
The nondimensional thermal conductivity of twodimensional hexagon cylinder is given as:
2.2 Octagon cylinder
The effective thermal conductivity of the twodimensional medium can be estimated by considering an octagon cylinder with crosssection "a × a" having a connecting bar width of "c" as shown in [Figure 2]a.
The stagnant thermal conductivity of the twodimensional 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 crosssection has been considered as a unit cell and is shown in [Figure 2]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 other two rectangular layers consist of solid and fluid phases with a dimension of {[a/2 + a/√2] − c/2} and (l/2) (l/2 − (a/2 + a/√2)), respectively. The model is based on the onedimensional heat conduction in the unit cell. The temperature gradient in the three layers is normal to the direction of heat flow. The effective thermal conductivity of twodimensional octagon cylinder is calculated for parallel isotherm conditions as follows:
Total resistance offered by the octagon cylinder in the unit cell is given as:
Where,
Resistance offered by the solid layer I,
Total resistance offered by the layer II,
Total resistance offered by the layer III,
The thermal conductivities of the composite layers can be obtained based on the series model. The thermal conductivity of the composite layer II is given by:
Similarly, the thermal conductivity of the composite layer III is given by:
The characteristic length, l_{c} , of octagon is given as:
The solid phase fraction of the unit cell is represented in terms of concentration (ν), and is given by:
Substituting Eqs. (1320) in Eq. (12), the total resistance of unit cell is simplified as,
The nondimensional thermal conductivity of twodimensional octagon cylinder is given as:
3. Results and Discussions   
Nondimensional thermal conductivity of a twophase system mainly depends on concentration, conductivity ratio, geometry, thermal contact between solidsolid, solidfluid interface, and Knudsen effects. The effect of concentration (ν) on the nondimensional thermal conductivity of twodimensional (hexagon and octagon cylinder) shapes have been investigated. The effective thermal conductivity of various models has been developed and it is validated against the standard models (Parallel, Series, Maxwell, and HasinShtrikman).
The influence of concentration on nondimensional thermal conductivity for conductivity ratio (α = 20) and different values of contact ratio (λ) as shown in [Figure 3] and [Figure 4]. The effective thermal conductivity of the present model lies between the standard models (series and parallel models [3]) for α = 20 and λ =00.2. It is noteworthy that at zero concentration, there is no presence of solid phase, so the "Kn" should be taken as zero because the gaswall collision is obsolete. As the concentration increases, the gaswall collision becomes predominant and hence the significance of the Knudsen effect is pronounced.
For hexagon cylinder, the present correlation is valid for concentrations ranging from 0 to 0.8, for further increment in the concentration; the nondimensional thermal conductivity is increasing beyond the upper bound due to the limitations in the shape of the models. Similarly for octagon cylinder, the present correlation is applicable for concentration varying from 0 to 0.8.
The predicted nondimensional thermal conductivity with Knudsen effect increases with the conductivity ratio and contact ratios for various geometries as shown in [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9] and [Figure 10]. For lower concentrations, ν = 0.3, and contact ratio, λ = 0.1 and λ = 0.02, the deviation between all the models is almost same. For higher concentration, ν = 0.8 and contact ratio, λ = 0.1 and λ = 0.02, the effect of contact ratio on nondimensional thermal conductivity is almost negligible.
It can be seen that the contact ratio (λ) is the deterministic parameter when the conductivity ratio (α) is high, whereas concentration is deterministic parameter when α is approaching 1. In most of the cases, for lower conductivity ratios (α < 1), the nondimensional thermal conductivity is insensitive to the contact ratios, but it is sensitive to the higher conductivity ratios (α > 1). From the isoconductance point, a = 1, the nondimensional thermal conductivity approaches to unity for all the models with the same slope. The present model shows a good trend for the concentrations 0.3, 0.5, and 0.8. For low values of conductivity ratio (a), thermal conductivity estimations of all the models are comparable, but they deviate when the conductivity ratio approaches 1000.
Percentage deviation from the experimental value for each model has been determined and an average percentage deviation has been found. A comparison of present models with experimental data for different concentrations has been made for various twophase materials, such as ceramic fiber felt, needle ceramic fiber felt, glass wool, rock wool, rock wool (loose), and mineral wool of different densities. For twophase material, which has been chosen for experimental verification (ν = 0.790.868 and α = 2.02), the hexagon cylinder has good agreement with the experimental data. The range of accuracy appears quite good in consideration with the experimental data and the wide range of shapes included. It is observed that the hexagon cylinder has an average deviation of ±6.35% from an experimental data as against ±22.49% of octagon cylinder, respectively, as shown in [Table 1] and [Table 2].
A comparison of present models with the existing models [21] (geometry, contact resistance) based on average deviation with experimental data has also been made for the same materials of same range of concentration (ν) and conductivity ratio (α). It is observed that the present model with hexagon cylinder has an average deviation of ±6.35% from the experimental data against existing model with hexagon cylinder has an average deviation of ±6.98% as shown in [Table 1]. It is also observed that the present model with octagon cylinder has an average deviation of ±22.49% from an experimental data against existing model with octagon cylinder has an average deviation of ±58.12% as shown in [Table 2].
4. Conclusion   
The collocated parameter models are developed with the effect of various inclusions for estimating the effective thermal conductivity of the twophase materials. The effects of concentration, conductivity, and contact ratios, Knudsen effect on the nondimensional thermal conductivity of various inclusions have been investigated. The thermal conductivity estimations based on the inline touching hexagon cylinder and octagon cylinder models with contact ratio of 0.1 show good results in conformance with the experimental values. The present models are validated against the standard models. Compared with the existing model, the present model shows good agreement with the experimental data. So, the present model can be effectively implemented to estimate the effective thermal conductivity of twophase materials for engineering applications. The predictions of the model could be improved further by including convection and radiation effects.
5. Nomenclature   
a Length of the hexagon and octagon cylinders.
c Width of the connecting plate in the hexagon and octagon cylinders.
K Nondimensional thermal conductivity of the twophase materials (k_{eff} /k_{f} )
k_{eff } Effective thermal conductivity of twophase materials (W/mK)
k_{f } Gas phase thermal conductivity (W/mK)
k_{f} Gas phase thermal conductivity including pressure (W/mK)
k_{s } Solid or dispersed thermal conductivity (W/mK)
k_{g } Gas phase pressure dependent thermal conductivity (W/mK)
Gas phase thermal conductivity (W/mK)
k_{sf2 } Equivalent thermal conductivity of composite layer 2 (W/mK)
k_{sf3 } Equivalent thermal conductivity of composite layer 3 (W/mK)
R Gas Law Constant (8.31451 J/K/mol)
R _{total} Total Thermal resistance (m ^{2} K/W)
R _{1} Thermal resistance for layer 1 (m ^{2} K/W)
R _{eff2} Thermal resistance for layer 2 (m ^{2} K/W)
R _{eff3} Thermal resistance for layer 3 (m ^{2} K/W)
R _{2s} Thermal resistance for layer 2 solid phase (m ^{2} K/W)
R _{3s} Thermal resistance for layer 3 solid phase (m ^{2} K/W)
R _{2sf} Thermal resistance for layer 2 fluid phase (m ^{2} K/W)
R _{3sf} Thermal resistance for layer 3 fluid phase (m ^{2} K/W)
a _{c} Accommodation coefficient
Kn Knudsen number
Pr Prandtl number
L Avagadro's Number (6.0221367 × 10^{−23} /mol)
l _{c}0 Characteristic length (m)
l Length of the unit cell (m)
d _{m} Pore size diameter (μm)
d _{a} Collisional crosssection of gases (m)
P Pressure (Pa)
T Temperature (K)
T _{s} System temperature (K)
T_{0} Reference temperature (K)
M _{g} Molecular mass of gas (g/mol)
M _{s} Molecular mass of solid (g/mol)
M _{g *} Molecular mass for monoatomic gases (1.4 M _{g} for diatomic/polyatomic gases) (g/mol)
x Parameter including Knudsen effect
devi Deviation
knud Knudsen effect
CR Contact resistance
ETC Effective thermal conductivity
Greek Symbols
α Conductivity ratio (k _{s} /k _{f}) _{`}
ε Length ratio (a/l)
ε_{hex} Length ratio of hexagon (a/l)
εoct Length ratio of octagon (a/l)
λ Contact ratio (c/a)
ν Concentration
γ Specific heat ratio
μ Ratio of molecular mass of gas to the molecular mass of solid
Subscripts
Eff Effective
exp Experimental
hex Hexagon
oct Octagon
References   
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12.  R. Krupiczka, 'Analysis of thermal conductivity in granular materials', International Chemical Engineering, 7, 122144, 1967. 
13.  P. K. Samantray, P. Karthikeyan, and K. S. Reddy, 'Estimating Effective Thermal Conductivity of TwoPhase Materials', International Journal of Heat and Mass Transfer, Vol. 49, No. 2122, pp. 42094219, 2006. 
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Authors   
Dr. A. P. Senthil Kumar is working as Faculty in Department of Mechanical Engineering, PSG College of Technology (Govt. Aided Institution), Coimbatore, Tamilnadu, India. He obtained his B.E. degree from MVJ College of Engineering, Bangalore and M.E. degree from Karunya Institute of Technology, Coimbatore. He completed his PhD. degree from Anna University of Technology, Coimbatore. He is having more than a decade of teaching experience. His areas of interest include Heat transfer, and Fluid Mechanics. He has published more than 10 research papers in various national and international journals.
Dr. P. Karthikeyan is working as Faculty in Department of Automobile Engineering, PSG College of Technology (Govt. Aided Institution), Coimbatore, Tamilnadu, India. He completed his PhD. degree from Indian Institute of Technology  Madras, Chennai. He received BOYSCAST fellowship from University of Toledo, USA. His areas of interest include Heat transfer, Fluid Mechanics, and Fuel Cells. He has published more than 20 research papers in various national and international journals.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10]
[Table 1], [Table 2]
