Hyperbolic Heat Conduction in Composite Materials

A hyperbolic heat conduction (HHC) equation has been proposed to replace Fourier heat conduction equation in cases heat transfer takes place in a very short period of time or at extremely low temperature. There is a growing interest in the investigation of HHC problem in recent years, but to the author’s knowledge, HHC in composite media in multidimension has not been studied up to date. This paper presents a numerical method to solve two-dimensional HHC in composite materials. The numerical method used in the paper is free of oscillation in predicting the solution of hyperbolic systems encountering discontinuities. The key to solve HHC in composite media is to keep both temperature and heat flux continuous at the interfaces. The HHC equation is non-dimensionalized in a way, so that the HHC problem in composite media can be solved in nondimensional form. Nomenclature ] [A , ] [B Jacobian matrix c thermal wave speed p C specific Heat E , E flux vector F , F flux vector g heat source k thermal conductivity q heat flux ] [R , 1 ] [ − R matrix S , S source vector t time T temperature Tr reference temperature U , U Vector of unknown varibles W characteristic variable W vector of characteristic variables x , y coordinates variables * Graduate Student + Professor, Member AIAA Copyright  2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Greek Symbols α thermal diffusivity ( ) /( p C k ρ ) ∆ variable difference λ eigenvalue ξ , η coordinates direction ρ density τ relaxation time ( 2 / c α ) ψ limiter function Superscripts i i th characteristic n time level n u upwind scheme ∗ non-dimensionlized variable Subscripts A , B matrixes ] [A and ] [B x , y coordinates direction ξ , η coordinates direction Introduction In situations dealing with transient heat transfer in an extremely short period of time and at a temperature near absolute zero, the wave nature of heat propagation can be observed. Peshkov 1 performed one of the earliest experiments to detect thermal waves in liquid helium in 1944. In order to account for the wave phenomenon of heat transportation, a modified heat flux equation was proposed by Vernotte 2 and Cattaneo, 3 instead of Fourier’s law of heat conduction. In 1984, thermal shocks were observed reflecting from the liquid-vapor interface in superfluid helium by Torczybski, et al. using a conventional schlieren system. Frankel, et al. 5 gave a theoretical investigation of the observed reflections of thermal waves. They presented a general one-dimensional temperature and heat flux formulation for hyperbolic heat conduction in composite materials, as well as a general solution by using a generalized finite integral transform technique. Present work studies two-dimensional HHC in composite media using characteristics. The method of characteristics has been used to solve one-dimensional and two-dimensional HHC in homogeneous materials, but not