Study on Knudsen flow through membrane pores using analytical approximation and Monte Carlo method

Abstract

In this study, Knudsen diffusion of low-pressure gases of infinite mean free path through various tubes is studied using the integral equation theory, standard diffusion theory, and Monte Carlo simulations. We investigated the transmission probabilities (TPs) of linearly diverging-converging, sinusoidally bulging, and periodic tubes as compared with TPs of conventional straight cylinders. An exact analytic solution for the TP through the straight cylindrical tube was developed using the standard diffusion theory with a linear concentration approximation. Integral equation theories for the TPs through the divergingconverging and bulging tubes were developed. Monte Carlo simulation techniques were applied to calculate TPs through all the tube types whose azimuthal symmetry was held with tube radius changing only along the axial coordinate (z). The equivalent radii of these tubes are calculated as root-mean-squares of the z-dependent radius to have the same internal void volume. The linearly diverging-converging and sinusoidally bulging tubes provide noticeably higher TPs than those of the equivalent straight tubes. Periodic tubes show that if the tube length scaled by the equivalent diameter is on the order of or greater than the periodicity coefficient (equal to the number of peaks on the tube wall), the TP of the periodic tube is larger than that of the equivalent straight tube. In general, an opening at the tube inlet significantly enhances the Knudsen diffusion.