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

## 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

## References

- 1.Maxwell JC (1873) A treatise on electricity and magnetism. Clarendon Press, Oxford, p 365MATHGoogle Scholar
- 2.Hashin Z, Shtrikman S (1962) A variational approach to the theory of the effective magnetic permeability of multiphase materials. J Appl Phys 33:3125–3131CrossRefMATHGoogle Scholar
- 3.Wiener O (1904) Lamellare doppelbrechung. Phys Z 5:332–338MATHGoogle Scholar
- 4.Bruggeman DAG (1935) Dielectric constant and conductivity of mixtures of isotropic materials. Ann Phys (Leipzig) 24:636–679CrossRefGoogle Scholar
- 5.Zehner P, Schlunder EU (1970) On the effective heat conductivity in packed beds with flowing fluid at medium and high temperatures. Chem Eng Technol 42:933–941Google Scholar
- 6.Raghavan VR, Martin H (1995) Modeling of two-phase thermal conductivity. Chem Eng Process 34:439–446CrossRefGoogle Scholar
- 7.Hsu CT, Cheng P, Wong KW (1995) A lumped parameter model for stagnant thermal conductivity of spatially periodic porous media. ASME J Heat Transf 117:264–269CrossRefGoogle Scholar
- 8.Deisser RG, Boregli JS (1958) An investigation of effective thermal conductivities of powders in various gases. ASME Trans 80:1417–1425Google Scholar
- 9.Kunii D, Smith JM (1960) Heat transfer characteristics in porous rocks. Am Inst Chem Eng J 6:71–78CrossRefGoogle Scholar
- 10.Reddy KS, Karthikeyan P (2009) Estimation of effective thermal conductivity of two-phase materials using collocated parameter model. Heat Transfer Eng 30:998–1011CrossRefGoogle Scholar
- 11.Demirel Y (1996) Heat transfer in rarefied gas at a gas-solid interface. J Energy 21:99–103CrossRefGoogle Scholar
- 12.Wakao N, Vortmeyer D (1971) Pressure dependency of effective conductivity of packed beds. Chem Eng Sci 26:1753–1765CrossRefGoogle Scholar
- 13.Kaganer MG (1966) Contact heat transfer in granular material under vacuum. J Eng Phys 11:30–36Google Scholar
- 14.Masamune S, Smith JM (1963) Thermal conductivity of bed of spherical particles. Ind Eng Chem 2:136–143Google Scholar
- 15.Zhao CY, Lu TJ, Hodson HP, Jackson JD (2004) The temperature dependence of effective thermal conductivity of open- celled steel alloy foams. J Mater Sci Eng A 367:123–331CrossRefGoogle Scholar
- 16.Calmidi VV, Mahajan RL (1999) ASME J Heat Transf 121:466CrossRefGoogle Scholar
- 17.Boomsma K, Poulikakos D (2001) Int J Heat Mass Transf 44:827CrossRefGoogle Scholar
- 18.Wakao N, Kato K (1969) Effective thermal conductivity of packed beds. J. Chem. Eng. Japan 2:24–33CrossRefGoogle Scholar
- 19.Imura S, Takegoshi E (1974) Effect of gas pressure on the effective thermal conductivity of packed beds. Heat Transfer- Jap 3:13–26Google Scholar
- 20.Godbee HW, Ziegler WT (1966) Thermal conductivities of MgO, Al
_{2}O_{3}, ZrO_{2}powders to 850^{o}C-П. Theor J App Phys 37:56–65Google Scholar - 21.Bauer R, Schlunder EU (1978) Effective radial thermal conductivity of packings in gas flow. Int Chem Eng 18:181–204Google Scholar
- 22.Kennard EH (1938) Kinetic theory of gases. McGraw-Hill, New YorkGoogle Scholar
- 23.Song S, Yovanovich MM (1987) In: Imber M, Peterson GP, Yovanovich MM (eds) Correlation of thermal accommodation coefficient for engineering surfaces in fundamentals of conduction and recent developments in contact resistance, vol 69. ASME United Engineering Center, New York, pp 107–116Google Scholar
- 24.Karthikeyan P, Chidambara Raja S, Senthil Kumar AP, Selvakumar B, Prabhu Raja V, Sankaranand R (2012) Influence of contact resistance and natural convection effects on non-dimensional effective thermal conductivity estimation of two-phase materials. Heat Mass Transf 49:451–467CrossRefGoogle Scholar
- 25.Karthikeyan P, Reddy KS (2008) Absolute steady state thermal conductivity measurement of insulation materials using square guarded hot plate apparatus. J Energy Heat Mass Transfer 30:273–286Google Scholar