Effective medium conductivity (diffusivity) for a sphere in a cube model

Abstract

A modified analytical solution for the effective medium conductivity (or diffusivity) k(eff) for two-phase mixtures yields estimates similar to 0.2 - 26 % lower than the classic model of a sphere in a cube (Jefferson et al. Ind. Eng. Chem. 1958, 50, 1589) depending on the ratios of phase conductivities and volumes. The difference between the two solutions lies in the assumption of an isothermal plane at the edge of the sphere in the original derivation. The temperature contours obtained with a finite element method simulation of the model confirm the intuitive judgement that this assumption is not valid.