The effect of cross-sectional shape on the molecular flow in long tubes

Abstract

The flow of gases in uniform long tubes with cross-sections other than circular is of considerable practical interest—for example, in particle accelerating systems, where chambers with approximately elliptical cross-sections are used to fit into the space of the focusing magnet gaps. A theoretical analysis of conductances and transmission probabilities in the molecular flow regime has revealed large discrepancies with published formulae (increasing with ellipticity). The widely used Knudsen-Dushman relation, involving the ratio of cross-sectional area to length of perimeter, was demonstrated to have no physical significance. In contrast to this, the conductance of rectangular and elliptical tubes with equal cross-sectional areas are shown to agree to within 1.2% for side or axes ratios a/b up to 1000:1. The cross-sectional area is found to be the most important parameter (as for the thin aperture) while the distortion of the cross-section from circular (or square) to ellipical (rectangular) has a small (<10%) effect, for a/b up to 4:1, but becomes significant for larger a/b ratios.