Heat diffusion in inhomogeneous media Steady state conduction

The system under examination consists of a mix of N discrete particles embedded in an otherwise continuous matrix. We will assume that the particles are spheres. The spheres each have a radius Ri, thermal conductivity κi, and position ri = (xi, yi, zi) relative to a common origin, for i = 1, 2, . . . , N . Assume that the radii are dimensionless (scaled with an appropriate length, e.g., the average sphere radius) and that the conductivity of the matrix is unity. In this respect κ is the ratio of the particle/matrix conductivities.