# Effect of primary and secondary parameters on analytical estimation of effective thermal conductivity of two phase materials using unit cell approach

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## Abstract

Most of the thermal design systems involve two phase materials and analysis of such systems requires detailed understanding of the thermal characteristics of the two phase material. This article aimed to develop geometry dependent unit cell approach model by considering the effects of all primary parameters (conductivity ratio and concentration) and secondary parameters (geometry, contact resistance, natural convection, Knudsen and radiation) for the estimation of effective thermal conductivity of two-phase materials. The analytical equations have been formulated based on isotherm approach for 2-D and 3-D spatially periodic medium. The developed models are validated with standard models and suited for all kind of operating conditions. The results have shown substantial improvement compared to the existing models and are in good agreement with the experimental data.

## List of symbols

- a
Length of the square cylinder and solid cube

- c
Width of the connecting plate in the square cylinder and solid cube.

- K
Non-dimensional thermal conductivity of the two-phase materials (k

_{eff}/k_{f})- Pr
Prandtl number

- a
_{c} Accommodation coefficient

- Kn
Knudsen number

- X
Parameter for Knudsen number

- d
Particle diameter, (m)

- T
_{s} Temperature of the surface (K)

- k
_{eff} Effective thermal conductivity of two-phase materials, (W/m K)

- k
_{f} Fluid or continuous thermal conductivity, (W/m K)

- k
_{f}^{’} Fluid or continuous thermal conductivity including Knudsen effect, (W/m K)

- k
_{s} Solid or dispersed thermal conductivity, (W/m K)

- k
_{sf} Equivalent thermal conductivity of a composite layer, (W/m K)

- R
Thermal resistance, (m

^{2}K/W)- R
_{2s} Thermal resistance of solid in the square model layer II having cross sectional area (a/2) (l)

- R
_{2sf1} Thermal resistance due to conduction of fluid in the square model layer II having cross sectional area ((l-a)/2) (l)

- R
_{2sf2} Thermal resistance due to convection & radiation of fluid in the square model layer II having cross sectional area ((l-a)/2) (l)

- R
_{3s1} Thermal resistance of solid in the cube layer III having length (c/2) and breadth (l/2)

- R
_{3s2} Thermal resistance of solid in the cube layer III having length (a/2) and breadth ((a- c)/2)

- R
_{3s3} Thermal resistance of solid in the cube layer III having length (c/2) and breadth ((l-a)/2)

- R
_{3f1} Thermal resistance due to conduction of fluid in the cube layer III having length ((l-a)/2) and breadth ((l-c)/2)

- R
_{3f2} Thermal resistance due to conduction of fluid in the cube layer III having length ((a-c)/2) and breadth ((l-a)/2)

- R
_{3f3} Thermal resistance due to convection & radiation of fluid in the cube layer III having length ((l-a)/2) and breadth ((l-c)/2)

- R
_{3f4} Thermal resistance due to convection & radiation of fluid in the cube layer III having length ((a-c)/2) and breadth ((l-a)/2)

- R
_{2s1} Thermal resistance of solid in the cube layer II having length (a/2) and breadth (a/2)

- R
_{2f1} Thermal resistance due to conduction of fluid in the cube layer II having length ((l-a)/2) and breadth (l/2)

- R
_{2f2} Thermal resistance due to conduction of fluid in the cube layer II having length (a/2) and breadth ((l-a)/2)

- R
_{2f3} Thermal resistance due to convection & radiation of fluid in the cube layer II having length ((l-a)/2) and breadth (l/2)

- R
_{2f4} Thermal resistance due to convection & radiation of fluid in the cube layer II having length (a/2) and breadth ((l-a)/2)

- R
_{1s1} Thermal resistance of solid in the cube layer I having length (c/2) and breadth (c/2)

- R
_{1f1} Thermal resistance due to conduction of fluid in the cube layer I having length ((l-c)/2) and breadth (l/2)

- R
_{1f2} Thermal resistance due to conduction of fluid in the cube layer I having length (c/2) and breadth ((l-c)/2)

- R
_{1f3} Thermal resistance due to convection & radiation of fluid in the cube layer I having length ((l-c)/2) and breadth (l/2)

- R
_{1f4} Thermal resistance due to convection & radiation of fluid in the cube layer I having length (c/2) and breadth ((l-c)/2)

- h
_{c} Convection heat transfer co-efficient (W/m

^{2}K)- h
_{r} Radiation heat transfer co-efficient (W/m

^{2}K)*l*Length of the unit cell, (m)

- h
_{2} Combined Heat transfer co-efficient at layer II,(W/m

^{2}K)- h
_{3} Combined Heat transfer co-efficient at layer III,(W/m

^{2}K)- ∆
*T* Temperature difference (K)

- CON
Convection

- CR
Contact Resistance

- KN
Knudsen

- RAD
Radiation

## Greek Symbols

- α
Conductivity ratio (k

_{s}/k_{f})_{`}- ε
Length Ratio (a/

*l*)- λ
Contact ratio (c/a)

- υ
Concentration

- β
_{rad} Non-Dimensional Number (2h

_{r}d/k_{f})- β
_{conv 2} Non-Dimensional Number (h

_{2}l/k_{f})- β
_{conv 3} Non-Dimensional Number (h

_{3}l/k_{f})- β
_{2} Non-Dimensional Number (β

_{conv 2}+ β_{rad})- β
_{3} Non-Dimensional Number (β

_{conv 3}+ β_{rad})*σ*Stephen Boltzman Constant, (W/m

^{2}K^{4})- Ψ
Emissivity of the particle

## Subscripts

- ana
Analytical

- devi
Deviation

- cond
Conduction

- eff
Effective

- exp
Experimental

- pre
Predicted

- rad
Radiation

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