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An analytically predictive model for moderately rarefied gas flow

Published online by Cambridge University Press:  05 April 2012

Thomas Veltzke  and
Jorg Thöming
Thomas Veltzke*
Affiliation:
Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, Leobener Strasse, 28359 Bremen, Germany
Jorg Thöming
Affiliation:
Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, Leobener Strasse, 28359 Bremen, Germany
*
†Email address for correspondence: [email protected]
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Abstract

In microducts deviation from continuum flow behaviour of a gas increases with rarefaction. When using Navier–Stokes equations to calculate a flow under slightly and moderately rarefied conditions, slip boundary conditions are used which in turn refer to the tangential momentum accommodation coefficient (TMAC). Here we demonstrate that, in the so-called slip and transition regime, the flow in microducts can be reliably described by a consistently non-empirical model without considering the TMAC. We obtain this equation by superposition of convective transport and Fickian diffusion using two-dimensional solutions of Navier–Stokes equations and a description for the Knudsen diffusion coefficient as derived from kinetic theory respectively. For a wide variety of measurement series found in the literature the calculation predicts the data accurately. Surprisingly only size of the duct, temperature, gas properties and inlet and outlet pressure are necessary to calculate the resulting mass flow by means of a single algebraic equation. From this, and taking the discrepancies of the TMAC concerning surface roughness and nature of the gases into account, we could conclude that neither the diffusive proportions nor the total mass flow rates are influenced by surface topology and chemistry at Knudsen numbers below unity. Compared to the tube geometry, the model slightly underestimates the flow rate in rectangular channels when rarefaction increases. Likewise, the dimensionless mass flow rate and the diffusive proportion of the total flow are distinctly higher in a tube. Thus the cross-sectional geometry has a significant influence on the transport mechanisms under rarefied conditions.

JFM classification

Mathematical Foundations: General fluid mechanics Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 698 , 10 May 2012 , pp. 406 - 422
Copyright
Copyright © Cambridge University Press 2012

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