Knudsen flow 75 years on: the current state of the art for flow of rarefied gases in tubes and systems

Abstract

Following a brief historical introduction an overview is given relating the most recent studies of rarefied gas flow to the early work of Knudsen. The first paper submitted in October 1908 (published in 1909) initiated a period of intense activity by Knudsen, Smoluchowski and, a little later, by Gaede and Jangmuir. This also covered the transition to the already well established hydrodynamic flow expressed in terms of the ratio of mean free path to critical apparatus dimension-which we now refer to as the Knudsen number. The desorption, evaporation and scattering of molecules from surfaces was described in terms of the Knudsen cosine law of scattering. The Knudsen effusion method for determining vapour pressure, also introduced in 1909, has become the main tool for studies of the reiated problem of the dissociation, chemical bonding and the vaporisation process itself. Clausing developed, as an alternative to conductances, the concept of transmission probability, still referred to as the Clausing factor, and provided a procedure for their more accurate evaluation in long and short tubes. A number of misconceptions of these early efforts have found their way into the literature and current books on vacuum science and technology. However, detailed studies have clarified the problem of gas-surface interactions; the gas flow in tubes has been tackled with Clausing-type integral equations and by statistical computation techniques based on Monte Carlo analysis procedures adaptable to more complex systems. Results have been confirmed experimentally using molecular-impact pressure probe measuring techniques.

An alternative procedure for modeling of Knudsen flow and surface diffusion

“…The direction in which a molecule is leaving the wall is completely independent of the incident one. [19], and [20] give good reviews of the Knudsen flow. It is also noted that the cosine law can be accepted as a correct basis for the evaluation of rarefied gas flow in both simple and complex vacuum systems as well as related fields [19].…”

“…[19], and [20] give good reviews of the Knudsen flow. It is also noted that the cosine law can be accepted as a correct basis for the evaluation of rarefied gas flow in both simple and complex vacuum systems as well as related fields [19]. Nevertheless, it should be noted that under some conditions there are deviations from the expected cosine law [19], e.g., for structured surfaces deviations from Knudsen's law are shown to occur [21].…”

Permeability of Ablative Materials Under Rarefied Gas Conditions

“…The pressure profile was measured at 7 static pressure holes along the length of the cylinder and compared to that from a modified direct simulation Monte Carlo method. The significant result from this work was showing that a Knudsen minimum [14] type phenomenon was found for the permeability of their experimental sample, i.e. there is a distinct minimum in permeability against gas rarefaction, with it being greater both below and above this critical Knudsen number.…”

Benchmark numerical simulations of rarefied non-reacting gas flows using an open-source DSMC code

“…Unfortunately, it is not possible to simulate an aspect ratio of 1000 using DSMC, as the velocities would be too low to obtain a converged solution in a practical time scale. The famous Knudsen minimum [73] can clearly be observed in Fig. 24, where the normalized mass flow rate has a 525 minimum at about Kn = 1.…”