Direct numerical solution of diffusion problems with intrinsic randomness

A hybrid numerical-symbolic manipulation scheme is developed for the analysis of diffusion problems with intrinsic randomness. The scheme is applied in the study of one-dimensional heat conduction in fully-saturated packed beds in order to study the effects of packing disorder on the medium effective conductivity. Conduction is modeled by a parabolic partial differential equation with random local conductivity. Randomness originates from the spatial fluctuation of porosity near solid walls and in the bulk region. Assuming a certain porosity statistical distribution, the steady and unsteady heat conduction problems are solved. The steady-state solution is used to obtain both the mean value and the standard deviation of the effective conductivity for a range of fluid to solid conductivity ratios. The mean and standard deviation are used to interpret the scatter of experimental results found in the literature. The unsteady heat conduction equation is discretized on a finite spatial grid and an explicit integration in time is carried out symbolically for each time step.