# Interdisciplinary Applied Mathematics

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From a system designer’s point of view, the flowrate obtained from a filter is one of its most important characteristics. It is necessary to know when and to what extent simple analytical formulas are adequate, when the Navier-Stokes solvers are valid, or when the DSMC approach is required. Here, DSMC results are compared with the analytical results for rarefied long-channel flows in order to gain insight into validity of simple analytical formulas. Volumetric flowrate at the inlet of the channels is found using the mass flowrate equation (4.6) by considering only the first-order slip effects (see Section 4.1.1). Predictions from this analytical relation as well as the volumetric flowrates obtained from the DSMC simulations are presented in Table 6.2.

For filters 1, 2, and 4, the simulated flowrates are smaller than the analytical estimates by 50.8%, 7.3%, and 0.9%, respectively. The analytical formula will tend to overestimate the flowrate because it ignores the entrance effects, and to underestimate the flowrate because it assumes a fully developed flow. For filters 1, 2, and 4, the analytical estimates of flowrate are larger than that of the DSMC predictions. This indicates that the entrance effects are dominant, and they result in a decreased flowrate. Inspection of the Knudsen number at their outlets show that filters 3 and 5 are in the transition and free-molecular flow regimes, respectively. For filters 3 and 5,    the    flowrates obtained    by    the    DSMC    are    larger    than    those

calculated using the analytical formula by 4.8% and 57.9%, respectively. This difference can be attributed to the rarefaction effects. The slip velocity and the rarefaction effects are underestimated by the two-dimensional analytical model. For Kn > 0.10, disagreements between the continuum and DSMC predictions are expected. These simulations support the result that for Kn < 0.10, the analytical approach is adequate for membranes with long channels, and that in other cases, Navier-Stokes solvers with slip boundary conditions will be adequate. For Kn > 0.10, DSMC simulations are necessary to capture the rarefaction effects in short channels. In the case of long channels, the unified model outlined in Section 4.2 can be used.

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