Abstract
Understanding the thermal behavior of a rarefied gas remains a fundamental problem. In the present study, we investigate the predictive capabilities of the regularized 13 and 26 moment equations. In this paper, we consider low-speed problems with small gradients, and to simplify the analysis, a linearized set of moment equations is derived to explore a classic temperature problem. Analytical solutions obtained for the linearized 26 moment equations are compared with available kinetic models and can reliably capture all qualitative trends for the temperature-jump coefficient and the associated temperature defect in the thermal Knudsen layer. In contrast, the linearized 13 moment equations lack the necessary physics to capture these effects and consistently underpredict kinetic theory. The deviation from kinetic theory for the 13 moment equations increases significantly for specular reflection of gas molecules, whereas the 26 moment equations compare well with results from kinetic theory. To improve engineering analyses, expressions for the effective thermal conductivity and Prandtl number in the Knudsen layer are derived with the linearized 26 moment equations.
Highlights
INTRODUCTIONGaseous heat transfer no longer follows Fourier’s law. Gases in microdevices frequently experience rarefaction effects as the molecular mean free path λ is often comparable to the characteristic dimension of the device
At the microscale, gaseous heat transfer no longer follows Fourier’s law
For flows at low speed and not far away from the equilibrium state, as those encountered in microelectromechanical systems (MEMS), both methods are computationally expensive for practical applications
Summary
Gaseous heat transfer no longer follows Fourier’s law. Gases in microdevices frequently experience rarefaction effects as the molecular mean free path λ is often comparable to the characteristic dimension of the device. With a finite number of moments, such as 13 or 26 moments in three-dimensional applications (the number of moments in one- or two-dimensional problems is much reduced), recent studies have shown that both the regularized 13 (R13) and 26 (R26) moment equations are able to capture several well known nonequilibrium phenomena, such as the bimodal temperature profile in force-driven Poiseuille flow, nongradient heat flux in Couette flow, and the Knudsen minimum [5,6,7,8,9]. Additional complications found in more realistic geometries This problem has been extensively studied in kinetic theory by solving the linearized Boltzmann equation [13,14,15,16] and the data provides a useful numerical benchmark for the development of macroscopic models. We show that rarefaction effects can be described by the macroscopic governing equations to good accuracy and the physics of nonequilibrium gas flow is well embedded in the moment equations in the early transition regime
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Physical review. E, Statistical, nonlinear, and soft matter physics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
