Advertisement

Heat and Mass Transfer

, Volume 54, Issue 5, pp 1323–1335 | Cite as

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

  • Chidambara Raja S
  • Karthikeyan P
  • L. A. Kumaraswamidhas
  • Ramu M
Original

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 (keff /kf)

Pr

Prandtl number

ac

Accommodation coefficient

Kn

Knudsen number

X

Parameter for Knudsen number

d

Particle diameter, (m)

Ts

Temperature of the surface (K)

keff

Effective thermal conductivity of two-phase materials, (W/m K)

kf

Fluid or continuous thermal conductivity, (W/m K)

kf

Fluid or continuous thermal conductivity including Knudsen effect, (W/m K)

ks

Solid or dispersed thermal conductivity, (W/m K)

ksf

Equivalent thermal conductivity of a composite layer, (W/m K)

R

Thermal resistance, (m2 K/W)

R2s

Thermal resistance of solid in the square model layer II having cross sectional area (a/2) (l)

R2sf1

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

R2sf2

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

R3s1

Thermal resistance of solid in the cube layer III having length (c/2) and breadth (l/2)

R3s2

Thermal resistance of solid in the cube layer III having length (a/2) and breadth ((a- c)/2)

R3s3

Thermal resistance of solid in the cube layer III having length (c/2) and breadth ((l-a)/2)

R3f1

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

R3f2

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

R3f3

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

R3f4

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

R2s1

Thermal resistance of solid in the cube layer II having length (a/2) and breadth (a/2)

R2f1

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

R2f2

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

R2f3

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

R2f4

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

R1s1

Thermal resistance of solid in the cube layer I having length (c/2) and breadth (c/2)

R1f1

Thermal resistance due to conduction of fluid in the cube layer I having length ((l-c)/2) and breadth (l/2)

R1f2

Thermal resistance due to conduction of fluid in the cube layer I having length (c/2) and breadth ((l-c)/2)

R1f3

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

R1f4

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

hc

Convection heat transfer co-efficient (W/m2 K)

hr

Radiation heat transfer co-efficient (W/m2 K)

l

Length of the unit cell, (m)

h2

Combined Heat transfer co-efficient at layer II,(W/m2 K)

h3

Combined Heat transfer co-efficient at layer III,(W/m2 K)

T

Temperature difference (K)

CON

Convection

CR

Contact Resistance

KN

Knudsen

RAD

Radiation

Greek Symbols

α

Conductivity ratio (ks/kf)`

ε

Length Ratio (a/l)

λ

Contact ratio (c/a)

υ

Concentration

βrad

Non-Dimensional Number (2hrd/kf)

βconv 2

Non-Dimensional Number (h2l/kf)

βconv 3

Non-Dimensional Number (h3l/kf)

β2

Non-Dimensional Number (βconv 2+ βrad)

β3

Non-Dimensional Number (βconv 3+ βrad)

σ

Stephen Boltzman Constant, (W/m2K4)

Ψ

Emissivity of the particle

Subscripts

ana

Analytical

devi

Deviation

cond

Conduction

eff

Effective

exp

Experimental

pre

Predicted

rad

Radiation

References

  1. 1.
    Maxwell JC (1873) A treatise on electricity and magnetism. Clarendon Press, Oxford, p 365MATHGoogle Scholar
  2. 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. 3.
    Wiener O (1904) Lamellare doppelbrechung. Phys Z 5:332–338MATHGoogle Scholar
  4. 4.
    Bruggeman DAG (1935) Dielectric constant and conductivity of mixtures of isotropic materials. Ann Phys (Leipzig) 24:636–679CrossRefGoogle Scholar
  5. 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. 6.
    Raghavan VR, Martin H (1995) Modeling of two-phase thermal conductivity. Chem Eng Process 34:439–446CrossRefGoogle Scholar
  7. 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. 8.
    Deisser RG, Boregli JS (1958) An investigation of effective thermal conductivities of powders in various gases. ASME Trans 80:1417–1425Google Scholar
  9. 9.
    Kunii D, Smith JM (1960) Heat transfer characteristics in porous rocks. Am Inst Chem Eng J 6:71–78CrossRefGoogle Scholar
  10. 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. 11.
    Demirel Y (1996) Heat transfer in rarefied gas at a gas-solid interface. J Energy 21:99–103CrossRefGoogle Scholar
  12. 12.
    Wakao N, Vortmeyer D (1971) Pressure dependency of effective conductivity of packed beds. Chem Eng Sci 26:1753–1765CrossRefGoogle Scholar
  13. 13.
    Kaganer MG (1966) Contact heat transfer in granular material under vacuum. J Eng Phys 11:30–36Google Scholar
  14. 14.
    Masamune S, Smith JM (1963) Thermal conductivity of bed of spherical particles. Ind Eng Chem 2:136–143Google Scholar
  15. 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. 16.
    Calmidi VV, Mahajan RL (1999) ASME J Heat Transf 121:466CrossRefGoogle Scholar
  17. 17.
    Boomsma K, Poulikakos D (2001) Int J Heat Mass Transf 44:827CrossRefGoogle Scholar
  18. 18.
    Wakao N, Kato K (1969) Effective thermal conductivity of packed beds. J. Chem. Eng. Japan 2:24–33CrossRefGoogle Scholar
  19. 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. 20.
    Godbee HW, Ziegler WT (1966) Thermal conductivities of MgO, Al2O3, ZrO2 powders to 850o C-П. Theor J App Phys 37:56–65Google Scholar
  21. 21.
    Bauer R, Schlunder EU (1978) Effective radial thermal conductivity of packings in gas flow. Int Chem Eng 18:181–204Google Scholar
  22. 22.
    Kennard EH (1938) Kinetic theory of gases. McGraw-Hill, New YorkGoogle Scholar
  23. 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. 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. 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

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Chidambara Raja S
    • 1
  • Karthikeyan P
    • 2
  • L. A. Kumaraswamidhas
    • 1
  • Ramu M
    • 3
  1. 1.Department of Mining Machinery EngineeringIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.Department of Automobile EngineeringPSG College of TechnologyCoimbatoreIndia
  3. 3.Department of Mechanical Engineering, Amrita School of EngineeringAmrita UniversityCoimbatoreIndia

Personalised recommendations