Heat conduction in a non-uniform composite with spherical inclusions

The process of steady heat conduction in a composite in which equal spheric al inclusions are distributed according to a non-uniform probability distri bution is studied numerically. While the standard results are recovered in the uniform case, the nonuniformity introduces a non-local correction in th e form of a difference between the average temperatures in the two phases. As a consequence, the relation between the mean heat flux and the phase tem peratures must be expressed in terms of two effective conductivities rather than only one as in the uniform case. In other words. contrary to the situ ation encountered with a uniform composite, an effective Fourier law of con duction of the standard form does not hold for a non-uniform composite. In addition to an equation expressing the overall energy balance, closure of t he system of. averaged equations requires a second relation that reveals th e non-local origin of the difference between the phase temperatures. This s econd equation involves two coefficients, dependent on the particle volume fraction. The method used in this paper establishes a relation between the two, but does not determine them individually. The technique developed for the heat-conduction problem studied here can be adapted to the modelling of other disperse physical systems such as non-un iform suspensions.