Rate of heat conduction from a heated sphere to a matrix containing passive spheres of a different conductivity

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Abstract
Using a multipole expansion scheme, we calculate the rate of heat conduction Q from a heated sphere with constant surface temperature to a matrix of conductivity k, containing a dilute dispersion of equal-sized passive spheres of conductivity αk, and whose temperature far away from the heated sphere vanishes. Values are presented for Q for various choices of α and λ, the ratio of the radius of a passive sphere to that of the heated sphere, and asymptotic expressions are derived for the cases λ≪1 and λ≫1, respectively. When the passive spheres are very small compared to the heated sphere, the composite can be viewed as an effective continuum on the length scale of the latter with an effective thermal conductivity relative to the matrix given by Maxwell’s expression, k*=1+3βc, where β=(α−1)/(α+2) and c is the inclusion volume fraction assumed to be small. We find, however, that this effective continuum approach can still be applied with a surprising degree of accuracy over the whole range of α even when λ is as large as O(10), provided that the effective thermal conductivity is allowed to vary with r, the distance from the center of the heated sphere, according to the expression k*≡1+3βρ(r), where ρ(r) denotes the probability that a point in the composite lies within a passive sphere. It is shown, furthermore, that this effective continuum approach becomes exact to O(α−1) for all λ.