Rogério Martins SALDANHA DA GAMA, Maria Laura MARTİNS-COSTA

The Heat Transfer Problem in a Rigid and Nonconvex Gray Body with Temperature Dependent Thermal Conductivity

This work studies coupled steady-state conduction-radiation heat transfer in a non-convex gray body when the thermal conductivity temperature-dependent. The gray-body assumption is an improvement with respect to the black body model, because in this model a portion of the incident radiant energy can be reflected from the body boundary. The problem is mathematically described by a nonlinear partial differential equation subjected to a nonlinear boundary condition involving a Fredholm operator which arises from the non-convexity of the body. In this problem the absolute temperature distribution is the unknown, as in the case of a black body. The Kirchhoff transformation is employed to linearize the partial differential equation, giving rise to new boundary conditions. The solution of the problem is constructed by a proposed iterative procedure, employing sequences that involve the temperature and the radiosity. The convergence is explicitly demonstrated. Besides, an error estimate, for each element, is presented. It is remarkable that the results obtained for black bodies are a particular case of this work. In other words, the results presented in reference [1] consists of a particular case of this paper, obtained when the emissivity is equal to one.

Keywords:

Fredholm equation, Conduction-radiation, Non-convex gray bodies, Radiosity Solution construction, Temperature-dependent conductivity,

PDF

___

[1] R.M.S. Gama, Existence, uniqueness and construction of the solution of the energy transfer problem in a rigid and non-convex blackbody with temperature- dependent thermal conductivity, Z. Angew. Math. Phys., 66 (2015), 2921-2939. doi:10.1007/s00033-015-0549-3.
[2] R.M.S. Gama, Numerical simulation of the (nonlinear) conduction/radiation heat transfer process in a nonconvex and black cylindrical body, J. Comput. Phys., 128 (1996), 341-350.
[3] R.M.S. Gama, J.A.O. Pessanha, J.A.R. Parise, F.E.M. Saboya, Analysis of a v-groove solar collector with a selective glass cover, Solar Energy, 36 (6) (1986), 509-519.
[4] F. John, Partial Differential Equations, Springer-Verlag, New York,1971.
[5] N. Afrin, Z.C. Feng, Y. Zhang, J. K. Chen, Inverse estimation of front surface temperature of a locally heated plate with temperature-dependent conductivity via Kirchhoff transformation, Int. J. Thermal Sci., 69 (2013), 53-60.
[6] S. Kim, A simple direct estimation of temperature-dependent thermal conductivity with Kirchhoff transformation, Int. Commun. Heat Mass Transfer, 28 (2001), 537-544.
[7] M.L. Martins-Costa, F.B.F. Rachid, R.M.S. Gama, An unconstrained mathematical description for conduction heat transfer problems with linear temperature-dependent thermal conductivity, Int. J. Non-Linear Mechanics, 81 (2016), 310-315. doi:10.1016/j.ijnonlinmec.2016.02.002.
[8] R.J. Moitsheki, T. Hayat, M.Y. Malik, Some exact solutions for a fin problem with a power law temperature-dependent thermal conductivity, Nonlinear Anal. Real World Appl., 11 (2010), 3287-3294.
[9] K.R. Rajagopal, G Saccomandi, Response of an elastic body whose heat conduction is pressure dependent, J. Elast., 105 (2011), 173-185.
[10] S. Chantasiriwan, Steady-state determination of temperature-dependent thermal conductivity, Int. Commun. Heat Mass Transfer, 29 (2002), 811-819.
[11] C.J. Glassbrenner, G.A. Slack, Thermal Conductivity of Silicon and Germanium from 3°K to the Melting Point, Phys. Rev., 134 (1964), 1058-1069.
[12] Yu. Goldberg, M.E. Levinshtein , S.L. Rumyantsev, Properties of Advanced Semiconductor Materials GaN, AlN, SiC, BN, SiC, SiGe, M.E. Levinshtein, S.L. Rumyantsev , M.S. Shur (editors), John Wiley & Sons, Inc., 2001, pp. 93-148.
[13] W. Joyce, Thermal resistance of heat sinks with temperature-dependent conductivity, Solid-State Electronics, 18 (1975), 321–322.
[14] D. Tomatis, Heat conduction in nuclear fuel by the Kirchhoff transformation, Ann. Nuclear Energy, 57 (2013), 100-105.
[15] H.S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, Oxford, Clarendon, 1959.
[16] F. Incropera, P.D.Dewitt, Introduction to Heat Transfer, John Wiley & Sons Inc.,1996.
[17] E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, McGraw-Hill, 1978.
[18] V.S. Arpaci, Conduction Heat Transfer, Addison-Wesley Publishing Company, Inc., Massachusetts,1966.
[19] M.S. Berger, Nonlinearity & Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, London, 1977.
[20] A.E. Taylor, Introduction to Functional Analysis, Wiley Toppan, Tokyo, 1958.