The effective thermal conductivity of a composite material with spherical inclusions

A new method is presented for calculating the effective thermal conductivity of a composite material containing spherical inclusions. The surface of a large body is assumed kept at a uniform temperature. This body is in contact with a composite material of infinite extent having a lower temperature far from the heated body. Green's theorem is then used to calculate the rate of heat transfer from the heated body to the composite material, yielding $$k_e /k = 1 + \frac{{3(\alpha - 1)}}{{[\alpha + 2 - (\alpha - 1)\phi ]}}\{ \phi + f(\alpha )\phi ^2 + 0(\phi ^3 )\} $$ where k e is the effective thermal conductivity, k is the thermal conductivity of the continuous phase, α is the ratio of the thermal conductivity of the spherical inclusions to k, and φ is the volume fraction occupied by the dispersed phase. The function f(α) is presented in this work. Although a similar result has been found previously by renormalization techniques, the method presented in this paper has merit in that a decaying temperature field is used. As a result, only convergent integrals are encountered, and a renormalization factor is not needed. This method is more straightforward than its predecessors and sheds additional light on the basic properties of two-phase materials.