Significance of Upstream Wall Conditions in Characterizing the Heat Transfer Phenomena of Rarefied Flows

Abstract

Slip flows in small-scale flow networks involve simultaneous presence of multiple factors governing the flow field. In addition, conditions of upstream wall need to be clearly defined for quantifying the total heat that fluid receives from the wall. The present work addresses these aspects by analyzing the heat transfer aspects of slip flow of gaseous nitrogen through a circular pipe, undergoing either heating or cooling. The complete form of the governing equations is solved numerically while retaining property variation. The thermal field is found to exhibit two distinct asymptotic regions, with the first one representing fully developed heat transfer and the second one representing isothermal states. The fully developed Nusselt number (Nufd) is found to rise first, before dropping continuously with rise in Knudsen number (Kn). The pair of Kn and maximum Nufd is found to be dependent on Peclet number (Pe) of the system. Local Nu is found to drop to a minimum, lower than Nufd for Kn∼O(10−3) due to a significant radial advection. The presence of an adiabatic upstream wall reveals that heat may propagate up to the inlet for Kn≳0.015. An analytical solution is developed to approximate this limiting value of Kn, and it agrees well with the numerical results. The observed flow behavior leads to the categorization of flow regime into three types: (i) Kn<0.001, possessing dependence on change in Pe only, (ii) 0.001≤Kn<0.01, possessing concurrence of effects due to change in Pe and Kn, and (iii) 0.01≤Kn<0.1, possessing dependence on change in Kn only. Further, Pe is shown to represent Nubulk for the flow, where in the range 0.01≤Kn<0.1, Nutot≈Nubulk as Kn approaches 0.01 and Nutot≈Nuin as Kn approaches 0.1. A convenient approach is proposed to evaluate Nutot for any condition of the upstream wall. These outcomes indicate the necessity to clearly define the condition of the upstream wall and to evaluate the total heat transfer in small-scale heat exchangers, which may be much larger than what fluid carries downstream.