Transient heat conduction in a porous medium

This paper is concerned with modeling time-dependent effects due to the diffusion processes in a medium containing multiple circular (in two dimensions) or spherical (in three dimensions) cavities (pores). The cavities may have different sizes provided that they do not overlap. The application of interest is for transient heat conduction in a porous material, and the aim is to devise a method that is capable of accurately computing the temperature and heat flux at any point and any time, without the need to consider a series of discrete time steps, as in conventional numerical solution procedures involving finite elements and finite differences. The approach is based on the use of the analytical solution to a corresponding problem of a single cavity in an infinite domain and superposition. Application of the analytical Laplace transform and its inversion results in a semi-analytical solution for the case of multiple cavities in the form of a truncated Fourier series (in two dimensions) or a series of surface spherical harmonics (in three dimensions). The limiting case of large time is investigated and the asymptotic formula that describes the behavior of the solution for this case is obtained by using the analytical solution in the Laplace transform domain. The use of the asymptotic formula increases the effectiveness of the method and further reduces the cost of the computations.