Effect of compressibility on velocity profile and friction factor of gaseous laminar flows in a microtube

Laminar flow of nitrogen gas in a microtube was simulated numerically to obtain velocity profile and Fanning friction factor in a quasi-fully developed region. The numerical procedure based on arbitrary-Lagrangian-Eulerian method solved two-dimensional compressible momentum and energy equations. The computations were performed for a wide range of Reynolds number in laminar flow regime with adiabatic wall condition. It was found that the velocity profile deviates from the parabola as Mach number increases, and the product of Fanning friction factor and Reynolds number is not a constant but a function of only Mach number. To explain the compressibility effect, a new theoretical flow model that gives the velocity profile of gaseous laminar flows in a microtube was proposed under the assumption of purely axial flow. The theoretical velocity profile is taking radial-direction density change into account and coincides with the numerically obtained velocity profile. The proposed flow model also shows that the Fanning friction factor of a compressible flow in a microtube is expressed by a quadratic function of Mach number. The coefficient of the Mach squared term is 40% of the numerically obtained correlation. The compressibility effect on friction factor of gaseous laminar flows in a microtube partly results from velocity profile change, which must occur to keep the mass velocity profile when density changes in radial direction. The remainder of the compressibility effect can be considered to result from actual mass transfer in the radial direction whose existence was demonstrated by the numerical results