Thermal conductivity of polydisperse composites with periodic microstructures

Abstract

A method is developed to treat the effective thermal conductivity of periodic composites whose inclusions are of different sizes or of different thermal conductivities. Based on the Green's function formalism, the effects of induced heat sources on inclusion boundaries are explicitly taken into account, and a set of Rayleigh identities are established. Hence the temperature distribution in a polydisperse composite and its effective thermal conductivity can be evaluated with very high accuracy. The method is applied to two-dimensional polydisperse systems with cylindrical inclusions as well as to three-dimensional ones with spherical inclusions. Illustrative calculations of the effective thermal conductivity are presented for both isotropic and anisotropic systems. The application of the method is further exemplified by an analytical formula for the effective thermal conductivity of a composite which possesses a square symmetry and contains two different kinds of inclusions. The numerical calculations have been carried out for two-dimensional systems.