Conjugate Heat Transfer in a Microchannel Simultaneously Developing Gas Flow: A Vorticity Stream Function-Based Numerical Analysis

A simultaneously developing microchannel gas flow is analyzed numerically, using the vorticity-stream function form of the Navier-Stokes equation, together with the fluid energy equation and the solid wall heat conduction equation. Rarefaction, shear work, viscous dissipation, pressure work, axial conduction, and conjugate effects on heat transfer characteristics are investigated. The shear work contribution to the wall heat flux is evaluated in both the developing and the fully developed flow regions and compared with the conductive wall heat flux. The assumption of hydrodynamically fully developed, thermally developing flow-normally used in the analysis of channel heat transfer-is assessed and compared with the simultaneously developing flow case. Analytical expressions for the fluid flow and heat transfer parameters under fully developed conditions are also derived and compared with the numerical results for verification. The analysis presented shows that the shear work and the combined viscous dissipation and pressure work result in extending the thermal entrance length by far. Heat conduction in the wall also contributes to increase the thermal entry length. The results presented also demonstrate the shear work contribution to heat transfer in the slip flow regime, although minor in the very first portion of the thermal entrance length, and it becomes progressively more significant as the flow thermal development conditions are approached and turns out to be exactly equal in magnitude to the conductive wall heat flux in the thermally fully developed region, resulting in a zero Nusselt number, as verified by both the exact and numerical solutions.