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MICRO/NANOSCALE HEAT TRANSFER—PART I

# Modeling Micro Mass and Heat Transfer for Gases Using Extended Continuum Equations

[+] Author and Article Information
Manuel Torrilhon

Seminar for Applied Mathematics, ETH Zurich, Zurich 8092, Switzerlandmatorril@math.ethz.ch

Henning Struchtrup

Department of Mechanical Engineering, University of Victoria, PO Box 3055 STN CSC, Victoria BC V8W 3P6, Canadastruchtr@uvic.ca

This definition is most suitable for dimensionless moment equations. Other definitions differ only by factors, e.g., $Kn˜$ in Ref. 3 or (6), which is $Kn˜=(4/5)(8/π)(μθ/p)≈1.277 Kn$, or $k$ used in Ref. 21, which is $k=(4/5)2(μθ/p)≈1.13 Kn$.

J. Heat Transfer 131(3), 033103 (Jan 13, 2009) (8 pages) doi:10.1115/1.3056598 History: Received February 10, 2008; Revised August 16, 2008; Published January 13, 2009

## Abstract

This paper presents recent contributions to the development of macroscopic continuum transport equations for micro gas flows and heat transfers. Within the kinetic theory of gases, a combination of the Chapman–Enskog expansion and the Grad moment method yields the regularized 13-moment equations (R13 equations), which are of high approximation order. In addition, a complete set of boundary conditions can be derived from the boundary conditions of the Boltzmann equation. The R13 equations are linearly stable, and their results for moderate Knudsen numbers stand in excellent agreement with direct simulation Monte Carlo (DSMC) method simulations. We give analytical expressions for heat and mass transfer in microchannels. These expressions help to understand the complex interaction of fluid variables in microscale systems. Additionally, we compare interesting analogies such as a mass flux and energy Knudsen paradox. In particular, the R13 model is capable of predicting and explaining the detailed features of Poiseuille microflows.

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## Figures

Figure 1

General setting for shear flow between two infinite plates. The plates are moving and may be heated.

Figure 2

R13 predicts exponential Knudsen layers for, e.g., parallel heat flux. The upper plot shows a schematic picture for F=0, Kn=0.01, 0.1, and 0.5. These functions lead to typical s-shaped profiles for, e.g., temperature; see the schematic lower plot with Kn=0.01, 0.1, and 0.2.

Figure 3

Averaged mass flow rate in acceleration-driven channel flow. The R13 equations predict the Knudsen paradox.

Figure 4

R13 predicts an energy Knudsen paradox: The normalized total energy content of an externally heated channel shows a nonintuitive minimum when plotted against the Knudsen number. Standard NSF does not show this minimum.

Figure 5

Velocity and temperature profiles in acceleration-driven channel flow for various Knudsen numbers. The symbols in the case Kn=0.068 represent a DSMC result.

Figure 6

Microscale effects, such as parallel heat flux qx and normal stresses σyy, in microchannels as predicted by R13 for various Knudsen numbers. The symbols in the case Kn=0.068 represent a DSMC result.

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