Heat conduction from a heated particle placed in a two-phase medium with a dispersed phase size comparable to the particle size

The steady rate of heat conduction Q from a heated sphere maintained at a constant temperature different from that of a surrounding two-phase medium, which consists of an array of spherical particles of thermal conductivity αk and a matrix of conductivity k, is determined by a method of multipole expansion to O(φ⁴), where φ is the volume fraction of the particles in the array, and by a lubrication analysis for highly conducting and closely packed arrays. The results of the two analyses are combined to obtain estimates of Q that are expected to be accurate over the complete range of α and φ. These results are compared with the predictions of an effective-medium theory, and it is shown that the predictions of this theory are quite accurate over a rather wide range of α, 0<α<10 ³, for the case of a simple cubic array, even at its maximum allowable volume fraction, but differ by as much as 60% for the case of a face-centered cubic array.