A New Semiempirical Equation for Gas Flow through Capillaries

Abstract

A new equation for gas flow through capillaries in low and intermediate pressure ranges is derived without attempting to distinguish between Poiseuille flow with slip and Knudsen flow. It is assumed here simply that the total flow has two components, one characterized by an effective mean free path and the other by an effective viscosity, both of which depend on the ratio of the average pressure in the capillary to a characteristic pressure p1 and, in addition, on the mean hydraulic radius of the capillary and properties of the gas. At low pressure, the usual expression for Knudsen flow is obtained, but in a form which separates out a term, due to a viscous effect, that is related to the ``free-molecule viscosity.'' At high pressure, Poiseuille flow with slip is obtained, but the slip term is cast in a new form in terms of the characteristic pressure p1. Also, our equation has a minimum in the relationship between the permeability and the pressure, which is not given by most permeability formulas but which is observed experimentally under some conditions. At this minimum, diffusion flow and viscous flow are equal in magnitude. At pressures below that corresponding to minimum flow, the phenomenon becomes more ``diffusionlike'' and at higher pressures more ``viscouslike.'' Original data of Knudsen and Klose on the flow rates of various gases through capillaries are well fitted by this new equation at low pressure as well as in the ``transition region'' between Knudsen flow and Poiseuille flow. This fit is obtained by setting the adjustable constant α, which defines p1, equal to unity. Elementary theory of the slip-flow correction also indicates that α should be about unity. Results obtained with this equation are compared with those obtained with Knudsen's four-constant formula.