Thermal conductivity of inhomogeneous materials

Abstract

The effective thermal conductivity is calculated from the rate of entropy production per unit volume. Thermal conductivity and the temperature field are expressed in terms of Fourier components and these are related. The rate of entropy production is then obtained in terms of the volume-averaged thermal conductivity and the Fourier components of thermal conductivity. A simple expression for the effective thermal conductivity is found. In the case of striations it leads to well-known results. The formalism is applied to solids with inhomogeneously distributed solutes. It is shown that the thermal conductivity is less than the volume-averaged thermal conductivity and that homogenization by diffusion increases the thermal conductivity. Similar results would apply to the electrical conductivity of inhomogeneous alloys.