Effect of compressibility on gaseous flows in micro-channels Yutaka Asako a, * , Tianqi Pi a , Stephen E. Turner b , Mohammad Faghri b a Department of Mechanical Engineering, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan b Department of Mechanical Engineering and Applied mechanics, University of Rhode Island, Kingston, RI 01882-0805, USA Received 7 August 2002; received in revised form 16 December 2002 Abstract Two-dimensional compressible momentum and energy equations are solved in a parallel plate channel to obtain the effect of compressibility on gaseous flow characteristics in micro-channels. The numerical methodology is based on arbitrary-Lagrangian–Eulerian method. The computations were performed for a wide range of Reynolds number and Mach number and for both Ôno heat conductionÕ and isothermal flow conditions. The results are presented in form of the product of friction factor and Reynolds number (f  Re) expressed as a function of both Reynolds number and Mach number and was compared with available experimental results. It was found that f  Re is mainly a function of Mach number and is different from the incompressible value of 96 for parallel-plate channels. Ó 2003 Elsevier Science Ltd. All rights reserved. 1. Introduction Fabrication of small devices has increased the needs forunderstandingoffluidflowandheattransferinmicro- geometries.SincetheearlyworkofTuckermanandPease [1],manyexperimentalandnumericalinvestigationshave been undertaken. Wu and Little [2] performed experi- mental studies of gaseous flows in micro-channels to obtain friction factor and Nusselt number for nitrogen, argon and helium flows. The channels were etched in silicon and glass ranging in depth from 28 to 65 lm. The relative roughness was estimated to be as high as 0.2–0.3. Measured friction factors were higher than those pre- dicted by the MoodyÕs chart. Pfahler et al. [3] performed severalexperimentalstudiesofliquidandgaseousflowsin shallowmicronandsubmicronchannelstoinvestigatethe flow and heat transfer characteristics of flows in micro- channels. The channels depth was 0.5–38.7 lm and the typicalsurfaceroughnesswasoforderof1%asmeasured by a surface profilometer. Choi et al. [4] performed ex- periments to obtain friction factor and convective heat transfer coefficients for laminar and turbulent gas flow in micro-tubes having the relative roughness in the range of 0.0002–0.0116. The measured friction factors were below those predicted by ModdyÕs chart. As can be seen, there are inconsistencies in the ex- perimental data reported by different researchers. For examples, the product of the friction factor and Rey- nolds number, (f  Re), was higher or lower than the conventional value (f  Re ¼ 96 for a conventional par- allel-plate duct). Measured values of f  Re by Pfahler et al. [3] and Choi et al. [4] are lower than the conven- tional values but the results by Wu and Little [2] are higher. Furthermore, the measured values of f  Re by Pfahler et al. [3] and Choi et al. [4] increased as Re in- creased rather than being a constant. Several factors are attributed to this inconsistency. The rarefaction (the slip on the surface), the surface roughness and the com- pressibility might have significant effect on these results. These effects separately or combined might affect the experimental results. The rarefaction effect can be studied by solving the momentum and energy equations with slip velocity and temperature jump boundary conditions (e.g., [5,6]). This effect is dominant when the characteristic length of the channel is less than about 10 lm. The value of f  Re decreases from the conventional value with decreasing characteristic length. Experimental studies of gaseous flows in micron channels with rough surface [2] and * Corresponding author. Tel.: +81-426-77-2711; fax: +81- 426-77-2701. E-mail address: asako@ecomp.metro-u.ac.jp (Y. Asako). 0017-9310/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0017-9310(03)00074-7 International Journal of Heat and Mass Transfer 46 (2003) 3041–3050 www.elsevier.com/locate/ijhmt