Effective thermal conductivity of composite spheres in a continuous medium with contact resistance

The self-consistent field concept is extended to the determination of the effective thermal conductivity of mixtures containing composite spheres randomly distributed in a continuous medium. Contact resistance is allowed for at the interface between the spheres and the continuous phase. A simple, yet exact, analytical solution is derived by first determining the temperature distribution for particle/continuous phase spheres as they are surrounded by an extensive effective medium. Special cases of the general solution are developed which correspond to various types of ideal particle/continuum behaviors. One category-perfect contact with negligible core gas heat conduction-is a useful model for many syntactic foams. Under these conditions, it is shown that the ratio of the effective conductivity of the medium to the conductivity of the continuous phase depends on a single parameter. The 'critical value' of this parameter is determined for which this ratio is unity. The result is consistent with prior studies of critical conditions in composite media.