Predicting enhanced mass flow rates in gas microchannels using nonkinetic models

Different nonkinetic approaches are adopted in this paper towards theoretically predicting the experimentally observed phenomenon of enhanced mass flow rates accompanying pressure-driven rarefied gas flows through microchannels. Our analysis utilizes a full set of mechanically consistent volume-diffusion hydrodynamic equations, allowing complete, closed-form, analytical solutions to this class of problems. As an integral part of the analysis, existing experimental data pertaining to the subatmospheric pressure dependence of viscosity were analyzed. The several nonkinetic approaches investigated were (1) pressure-dependent viscosity exponent model, (2) slip-velocity models, and (3) volume diffusion model. We explored the ability to predict the gas's mass flow rate over the full range of Knudsen numbers, including furnishing a physically sound interpretation of the well-known Knudsen minimum observed in the mass flow rate. Matching of a pressure-dependent viscosity model, one that follows the standard temperature-viscosity power law and its supporting single momentum diffusion mechanism, did not allow an accurate interpretation of the data. Rather, matching of this model with the flow rate was found to mismatch the experimental pressure dependence of the viscosity. An additional transport mechanism model, one based on volume diffusion, offered a comprehensive understanding of the Knudsen minimum, while also resulting in excellent agreement with experimental data well into the transition regime (up to a Knudsen number of 5).

Figure 1

Comparison of theoretical mass flow rate predictions using the variable viscosity model with experiments. Note that “$\text{second}$-order slip varying viscosity” as given by Eq. (28) gives the best fit in this case.

Figure 2

Comparison between viscosity values fitting Ewart et al.'s [8] experimental data and the true experimental viscosity data: “Fitting viscosity values” are those viscosities obtained using Eq. (32) with Eq. (33), whereas “experimental viscosity” data are those given by Ref. [44].

Figure 3

Volume diffusion hydrodynamic predictions of mass flow rate compared with experiments.

Figure 4

Comparison of slip and volume diffusion contribution to the mass flow rate.