Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for laminar flows

Micro-scale fluid systems are becoming common in many applications ranging from electronic cooling to refrigeration systems and more. One-dimensional numerical models represent a simple and fast tool for the design of such devices, yet they struggle to accurately predict the flow characteristics in compressible micro-flows. Under the adiabatic assumption, the elegant theory developed by Fanno allows models for the viscous compressible flow in constant cross-section channels to be easily built. Although reasonably accurate, these models suffer from drawbacks inherent to their being one-dimensional, as such they cannot take into account the local profiles of quantities like the velocity and the temperature. In cascade, this results into incorrect evaluations of other dependent quantities, such as the dynamic pressure and the fluid thermophysical properties. The mismatch turns large when the fluid compressibility becomes important. As the Mach number grows, the velocity profile changes, and so the friction factor, even though a reliable model for predicting this change is still missing. In fact, a constant friction factor throughout the channel is generally assumed, following the incompressible flow theory. Here, a set of correlations is proposed improving the 1D theory accuracy by taking into account the effects of the non-uniform velocity and temperature profiles in a quasi-2D fashion. A detailed analysis of the velocity profiles at different Mach numbers coming from a large set of CFD simulations results in a model for assessing the impact of compressibility on friction and other quantities. The numerical model proposed, being able to properly account for the compressibility effects, offers an improved tool for the design of micro-scale fluid systems. Extending the analysis to include heat transfer is not difficult as the effect of heat flux will be analogous to the effect of pressure drop due to friction.