The extended Graetz problem for micro-slit geometries; analytical coupling of rarefaction, axial conduction and viscous dissipation

Abstract

In order to support the recent MEMS and Lab-on-a-chip technologies, we studied heat transport in micro-scale slit channel gas flows. Since the micro convection transport phenomena diverges from conventional macro-scale transport due to rarefaction, axial conduction and viscous heating, an accurate understanding requires a complete coupling of these effects. For such cases, we studied heat transfer in hydrodynamically developed, thermally developing gas flows in micro-slits at various flow conditions. The analytical solution of the energy equation considered both the heat conduction in the axial direction and heat dissipation of viscous forces. Furthermore, updated boundary conditions of velocity slip and temperature jump were applied based on Knudsen number of flow in order to account for the non-equilibrium gas dynamics. Local Nusselt number (Nu) values were calculated as a function of Peclet (Pe), Knudsen (Kn) and Brinkman (Br) numbers which were selected carefully according to possible micro-flow cases. Strong variation of Nu in thermal development length was found to dominate heat transfer behavior of micro-slits with short heating lengths for early slip flow regime. For this instance, influence of axial conduction and viscous dissipation was equally important. On the other hand, high Kn slip flow suppressed the axial conduction while viscous heating in a small surface-gas temperature difference case mostly determined the fully developed Nu and average heat transfer behavior as a function of Kn value.