A Modified Temperature-Jump Method for the Transition and Low-Pressure Regime

Abstract

For modeling the gas heat conduction at arbitrary Knudsen numbers and for a broad range of geometries, we propose a modified temperature-jump method. Within the modified approach, we make a distinction between an inner convex surface and an outer concave surface enclosing the inner surface. For problems, where only a single geometric length is involved, i.e., for large parallel plates, long concentric cylinders and concentric spheres, the new method coincides at any Knudsen number with the interpolation formula according to Sherman, and therefore also with the known solutions of the Boltzmann equation obtained by the four momenta method. For the general case, where more than one geometric length is involved, the modified temperature method is trivially correct in the limit of high pressure and identical with Knudsen’s formula in the limit of low pressure. For intermediate pressure, where there is a lack of known solutions of the Boltzmann equation for general geometries, we present experimental data for the special two-dimensional plate-in-tube configuration and compare it with results of the modified temperature-jump method stating good agreement. The results match slightly better compared to the standard temperature method and significantly better compared to the interpolation formula according to Sherman. For arbitrary geometries and Knudsen numbers, the modified temperature method shows no principal restrictions and may be a simple approximative alternative to the solution of the Boltzmann equation, which is rather cumbersome.