(225b) Survival Probability in the Transition and Knudsen Diffusion Regimes in Random Fiber Structures

Random walk simulation results are presented for the molecular survival probability, mean survival distance and time, average step size and collision frequencies in the transition and Knudsen diffusion regime, in random porous media formed by cylindrical fibers randomly distributed in the three-dimensional space (3-d random fiber structures). The numerical data was obtained by modifying earlier random walk simulation algorithms and applying the codes for various values of the porosity (ε) and Knudsen number (Kn). Analytical results are also presented for the Knudsen regime survival probability and mean survival distance and time, derived from a theoretical analysis employing a number of properly defined probability distributions.

The survival probability was found to increase with the porosity and decrease with the Knudsen number for a certain diffusion time, and the same trends were encountered for the dimensionless mean survival distance, as anticipated by intuition. The dimensionless mean survival time was also found to increase with the porosity and depend strongly on the type of diffusion, as expressed by the Knudsen number. It was found to attain constant values at the two extremes of the diffusion spectrum, and increase gradually with the Knudsen number in the transition regime. The values computed for all three parameters in the Knudsen regime are in excellent agreement with the predictions of our probabilistic analysis. Similarly, our simulation results for the bulk diffusion regime practically coincide with those reported by Tomadakis and Robertson (2003) for that type of diffusion.

The dimensionless overall average step size, average insider size, average stopper size and intermolecular collision frequency, were all found to decrease with both the porosity and the Knudsen number, in agreement with an analysis involving the dependence of all these parameters on the mean free path and mean intercept length. In the upper transition and Knudsen number (Kn>3), as well as in the dilute limit (ε→1), the values of all these parameters practically coincide with the theoretical predictions and simulation results of Tomadakis and Sotirchos (1993) for long diffusion times in 3-d random fiber structures. This is found to be the case also in the bulk regime for the average size of insiders and the overall average molecular step size only. These trends are in agreement with an analysis centered on the basic differences of survival trajectories from those involving reflections on the pore surface.

To the best of our knowledge, this is the first investigation of the survival probability distribution and related parameters in the transition regime for any type of porous medium, and only the second in the Knudsen regime, where survival probability results were reported earlier for a dense random packing of hard spheres. Our analytical and numerical results could be used in a variety of applications involving mass transport and heterogeneous reaction in porous media.