Mass flow rate measurements in microtubes: From hydrodynamic to near free molecular regime

Abstract

International audienceAn experimental investigation of the reflection/accommodation process at the wall in a single silica microtube and isothermal stationary flow conditions was carried out. Several gases and different diameters were studied through various regimes. Especially for helium, the Knudsen number range was investigated as far as the free molecular regime. This kind of investigation requires a powerful experimental platform to measure mass flow rates, which we have carried out. An analytic expression of the mass flow rate, based on the Navier–Stokes equations with second order boundary condition, was used to yield the tangential momentum accommodation coefficient TMAC in the 0.003–0.3 Knudsen number range. Otherwise, the experimental results of the mass flow rate were compared with theoretical values calculated from kinetic approaches using variable TMAC as parameter over the 0.3–30 Knudsen number range, and an overall agreement appears through the comparison. Finally, whatever the theoretical approach the TMAC obtained from gas nitrogen, xenon, argon, and helium-surface fused silica pairs is similar and lower than unity. A tendency of the TMAC values seems to appear according to the molecular mass of the gases. In addition for each gas, the second order slip coefficient magnitude seems to decrease when the tube diameter increases

Hydrodynamic force measurements under precisely controlled conditions: Correlation of slip parameters with the mean free path

“…The first one focuses on measurements of flow rates in micro-channels (see, for example, Refs. [20][21][22][23][24][25]. Several theoretical approaches are being applied to derive slip and accommodation coefficients from the experimental data.…”

Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation

“…The same arguments as we used in the previous section lead to a system very similar to (11)(12)(13)(14), except that equation (13) now contains a collision term in the right-hand side since gas atom-phonon collisions can occur:…”

“…Then (13)- (14) can be integrated exactly, and the outgoing value F (t, x, y, 0, v x , v z )| vz<0 can be written as a function of the incoming value (14). This outgoing value can be used in the implicit boundary condition (12), that can now be written explicitly as…”

“…Finally, the distribution g of gas atoms in the bulk flow is defined through (11)- (12). Relation (12) means that the distribution of gas atoms that come into the bulk flow from the surface layer is the average on one periodic cell of the distribution F (t, x, 0, v x , v z )| vz<0 of gas atoms that leave the surface layer. In other words, relation (12) can be viewed as an implicit boundary condition for the bulk flow governed by (11), which requires to solve the boundary layer problem (13)- (14).…”

“…In many applications we have to consider rarefied flows on microtubes with a typical length of 1 to 10 cm (see for instance 12 ). In such configurations a reasonable mesh size l * for a simulation based on the Boltzmann equation would be around 10 µm, so that L * l * .…”