Gaseous Diffusion in Porous Media. II. Effect of Pressure Gradients

Abstract

A previously proposed model for the diffusion of gases in porous media at uniform pressure is extended to allow for pressure gradients. The porous medium is visualized as a collection of ``dust'' particles constrained to remain stationary in space. As before, the entire range of intermediate mechanisms from the Knudsen to the normal diffusion region can be covered by varying the mole fraction of real gases. The effect of pressure gradients is to introduce into the fundamental kinetic theory equations both a pressure diffusion term and an external force term, which is needed to keep the porous medium from being pushed along by the pressure gradient. Somewhat surprisingly, there is a considerable cancellation of terms, and the final diffusion equation has the same form as in the uniform pressure case. No additional parameters beyond those necessary to define a diffusing system at uniform pressure are thus required to compute the diffusion rates when pressure gradients are present. The coefficients of the diffusion equation, however, now depend on position through their dependence on pressure, and the net flux J of all molecules is an undetermined quantity. A complete solution requires also a forced flow equation giving J as a function of the pressure gradient. A forced flow equation is derived on the basis of the dusty gas model, but one parameter must be made disposable in order to compensate for the fact that the model permits only a diffusive mechanism for flow, never a viscous mechanism. Another forced flow equation is derived by analogy with Poiseuille flow, in which the porous medium is visualized as a group of capillaries. The two models are shown to be not inconsistent in viscous flow regimes, and complement rather than contradict each other. The Poiseuille model is mathematically very cumbersome, however, and has been used only at one especially simple point (at which J=0) to give an independent evaluation of the disposable parameter occurring in the dusty gas model. Agreement of the calculations with available experimental data is good. As in the uniform pressure case, the model does not give an a priori characterization of any porous medium, but rather permits prediction of the behavior of different gas mixtures under a variety of conditions on the basis of three parameters which can be obtained from only a few experimental measurements.