Heat Transport Between Parallel Plates in a Rarefied Gas of Rigid Sphere Molecules

Abstract

The linearized Boltzmann integral equation is solved for the problem of conduction of heat between parallel plates in a rarefied gas of rigid sphere molecules. The solution is obtained as an expansion in terms of the eigenfunctions of the Boltzmann collision operator for a Maxwell gas. The temperature distribution between the plates has been determined for Knudsen numbers X0 = d/2l ranging from 0.01 to 500, d denoting the distance between the plates and l the mean free path. The temperature varies linearly through most of the region, except for a ``Knudsen layer,'' of a thickness of a few mean free paths, near the wall, where the temperature rises above the straight line. Near the wall there is also a temperature jump ΔTw whose magnitude becomes negligibly small as the distance between the plates exceeds a few mean free paths. In the linear region, the gradient of the temperature can be determined by assuming a virtual extension of the spacing between the plates by a distance Δ which is equal to 2l in the Knudsen limit, and reaches an asymptotic value of around 7l in the Clausius limit of a dense gas. In the Knudsen limit of extreme rarefaction, when the spacing between the plates is only a small fraction of l, the temperature in the gas is nearly constant, and the deviation from the average temperature of the plates takes place entirely as a temperature jump ΔTw at each wall.