Heat conduction in solids: Temperature-dependent thermal conductivity

Abstract

A variational formulation is derived and applied to problems in heat conduction through solids. This formulation has, as its Euler-Lagrange equation, the general heat equation where the thermal conductivity, density, and specific heat may be temperature-dependent. The variational principle can be applied for steady-state or time-dependent solutions and is valid for boundary conditions where the temperature is specified, being either fixed or time-dependent, or where the heat flux is zero. The macroscopic temperature distribution for two problems is determined using this technique. The first of these is a finite rectangular plate, the initial temperature of the plate being constant and the temperature of the edges being set equal to zero for time greater than zero. In the second example the variational formulation is used to obtain a steady-state temperature distribution for a finite rectangular plate with sides held at fixed but different temperatures. In both of these applications, the thermal conductivity is assumed to vary linearly with temperature and the density and specific heat are assumed to have uniform values. Comparison of the results obtained using the variational formulation with those obtained by finite difference techniques shows an excellent correlation over a range of the temperature-conductivity coefficient.