New analytical solution for solving steady-state heat conduction problems with singularities

Abstract

A problem of steady-state heat conduction which presents singularities is solved in this paper by using the conformal mapping method. The principle of this method is based on the Schwarz-Christoffel transformation. The considered problem is a semi-infinite medium with two different isothermal surfaces separated by an adiabatic annular disc. We show that the thermal resistance can be determined without solving the governing equations. We determine a simple and exact expression that provides the thermal resistance as a function of the ratio of annular disc radii.

Keywords

Dates

  • Submission Date2012-08-26
  • Revision Date2013-05-20
  • Acceptance Date2013-05-20
  • Online Date2013-06-16

DOI Reference

10.2298/TSCI120826070L

References

  1. Laraqi, N., Thermal impedance and transient temperature due to a spot of heat on a half-space, International Journal of Thermal Sciences, 49 (2010), 3, pp. 529-533
  2. El Ganaoui, M., Hristov, J., Lacroix, M., Analytical and innovative solutions for heat transfer problems involving phase change and interfaces. Comptes Rendus Mécanique, 340 (2012), 7, pp. 463-465
  3. Rashidi, M.M., Laraqi, N., Sadri, S.M., A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-Padé, International Journal of Thermal Sciences, 49 (2010), 12, pp. 2405-2412
  4. Bardon, J.P., Heat transfer at solid-solid interface: basic phenomenon, recent works, Proceedings of Eurotherm, Nancy, France, 4 (1988), pp. 39-63
  5. Baïri, A., Alilat, N., Bauzin, J.G., Laraqi, N., Three-dimensional stationary thermal behavior of a bearing ball, International Journal of Thermal Sciences, 43 (2004), 6, pp. 561-568
  6. Hristov, J., El Ganaoui, M., Thermal impedance estimations by semi-derivatives and semiintegrals: 1-D semi-infinite cases. Thermal Science, 17 (2012), 2, pp. 581-589
  7. Laraqi, N., Baïri, N., Theory of thermal resistance between solids with randomly sized and located contacts, Internatioal Journal of Heat and Mass Transfer, 45 (2002), 20, pp. 4175-4180
  8. Laraqi, N., Alilat, N., Garcia-de-Maria, J.M., Baïri, A., Temperature and division of heat in a pin-on-disc frictional device - Exact analytical solution, Wear, 266 (2009), 7-8, pp. 765-770
  9. Smythe, W.R., The capacitance of a circular annulus, J. Applied Physics, 22 (1951), 12, pp. 1499-1501.
  10. Cooke, J.C., Triple integral equations, Quart. J. Mech. and Appl. Math, 16 (1962), pp. 193-203
  11. Collins, D., On the solution of some axisymmetric boundary value problems by means of integral equations, VIII, Potential problems for a circular annulus, Proc. Edinbourgh Math. Soc., 13 (1963), pp. 235-246
  12. Cooke, J.C., Some further triple integral equation solutions, Proc. Edinbourgh Math. Soc., 13 (1963), pp. 303-316
  13. Fabrikant, V.I., Dirichlet problem for an annular disc, Z. Angew Math. Phys., 44 (1993), 2, pp. 333-347
  14. Laraqi, N., Scale analysis and accurate correlations for some Dirichlet problems involving annular disc, International Journal of Thermal Sciences, 50 (2011), 10, pp. 1832-1837
  15. Carslaw, H.S., Jaeger, J.C., Conduction of Heat in Solid, 2nd edition, Oxford University Press, 1959
  16. Özişik, M.N., Heat Conduction, Second Edition, John Wiley & SONS, INC., 1993
Volume 17, Issue 3, Pages665 -672