On a tracer flow through irregular packed bed

The concentration of a tracer injected into a packed bed is studied as a function of the time and of the axial position in the frame-work of a phenomenological model of irregular packed bed. For that we consider a one-dimensional Fick diffusion equation with a stochastic porosity. To perform an explicit analysis we model the porosity by a telegraph process natural for packing by a dust-type particles. We show that, under reasonable physical conditions, a self-averaging occurs, namely that far away from the inlet of the column the measured concentration will approach the average, with respect to the stochastic process, of the solutions. We than deduce closed equations for this averaged quantity and show that they differ from the original diffusion equation by an additional time-diffusive term (with time-delay in the full general situation).