COMSOL 6.4 - Molecular Flow Through a Microcapillary

Computing molecular flows in arbitrary geometries produces complex integral equations which are very difficult to compute analytically. Analytic solutions are therefore only available for simple geometries. One of the earliest problems solved was that of gas flow through tubes of arbitrary length, which was first treated correctly by Clausing (Ref. 1). Later the integral expressions he derived were computed more accurately by Cole (Ref. 2). These authors derived values for the transmission probability of molecules incident on a tube of arbitrary length, which is independent of the pressure, provided that the Knudsen number is much greater than one (that is in the molecular flow regime).

This model uses COMSOL Multiphysics to compute the transmission probability for molecular flow of molecules through a microcapillary with different ratios of the length to the radius. The results are compared with the accurate solution due to Cole (Ref. 2).

Model Definition

Nitrogen gas flows through a microcapillary tube of diameter 200 μm, from a reservoir at a fixed pressure of 10−3 mBar to a reservoir at high vacuum. The flux of molecules entering the tube from the high vacuum end is assumed to be negligible. At this pressure the mean free path of the nitrogen is approximately 100 mm, so the Knudsen number of the flow is much greater than 1. The ratio of the flux of molecules entering the tube to the flux of molecules leaving the tube is computed for tube lengths of 0.4 mm, 0.8 mm, 1.2 mm, 1.6 mm, and 2.0 mm. The model is axially symmetric and is built in the 2D axially symmetric interface. The geometry is shown in Figure 1 for the case of the 2.0 mm long tube. The lower boundary is connected to the reservoir at low vacuum, the upper boundary is connected to the reservoir at high vacuum and the nonaxial vertical boundary is the capillary wall.

Figure 1: Model geometry.

The probability, χ, of molecules that strike the tube passing through the tube can be computed as the following integral:

where J is the outgoing flux, G is the incident flux, r is the radial distance, RS is the boundary adjacent to the reservoir, and HV is the boundary adjacent to high vacuum.

Results and Discussion

The incident flux on the surfaces of the tube is shown in Figure 2 for the case of the 2 mm long tube. A line plot of the pressure along the wall is shown in Figure 3. As expected, the incident flux decreases steadily from the reservoir end to the end at high vacuum, as does the pressure. Note that the incident flux (in contrast with the emitted flux) is not constant along the two reservoir boundaries, as it results from reflections within the geometry. The transmission probability for the tube is plotted against length to radius ratio in Figure 4. Good agreement is obtained between the calculated results and the probabilities calculated by Cole (Ref. 2).

Figure 2: The incident flux inside the tube.

Figure 3: The pressure along the tube wall.

Figure 4: The transmission probability for the pipe as a function of the length to radius ratio. The results due to Cole (Ref. 2) are shown for comparison.

Notes About the COMSOL Implementation

The model is straightforward to set up using the Molecular Flow interface. The capillary walls are assigned the wall boundary condition, the reservoir boundary condition is used for the bottom opening, and the top opening is assigned the total vacuum boundary condition. The geometry is parameterized, and a parametric solver is used to vary the capillary dimensions.

References

1. P. Clausing, “Über die Strömung sehr verdünnter Gase durch Röhren von beliebiger Länge,” Ann. Physik, vol. 404, no. 8, pp. 961–989, 1932. English translation available as: “The Flow of Highly Rarefied Gases Through Tubes of Arbitrary Length,” J. Vacuum Science and Technology, vol. 8. no. 5, pp. 636–646, 2009.

2. R.J. Cole, “Complementary Variational Principles for Knudsen Flow Rates,” IMA J. Appl. Math., vol. 20, pp. 107–115, 1977.

Molecular_Flow_Module/Benchmarks/vacuum_capillary

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