Host: The Japan Society of Vacuum and Surface Science
Name : Annual Meeting of the Japan Society of Vacuum and Surface Science 2023
Location : [in Japanese]
Date : October 31, 2023 - November 02, 2023
Calculating the gas flow rate through a cylindrical tube of known geometry is actually complicated because the characteristics of the gas flow depend on pressure, gas species, temperature, tube diameter, and tube length. There are at least six flow regimes to explain the characteristics of the gas flow, such as molecular flow, viscous laminar flow, turbulent flow, critical flow, subcritical flow, and their intermediates including slip flow.
In recent, the author and coworkers have developed the modified Knudsen equation (MK equation) by combining equations of each gas flow, which is applicable to the whole flow regime for arbitrary length of the tubes [1,2]. This equation has two advantages; one is that this equation is used without considering Knudsen number (Kn), Reynolds number (Re), Mach number (Ma) and the length-to-diameter ratio of tube, and the other is that it can be solved in straight forward without an iterative procedure although the other equations sometimes need it. This equation is especially useful when one does not know which flow regime the gas flow is in. The MK equation typically produces results that agree to within 20 % to 30 % of previous results from experiments and computer simulations, but more study is still needed to confirm what range of Kn, Re, and Ma is applicable. Another aim is to improve the MK equation by examining the conditions under which it produces relatively large differences from existing results. As the results of comparison of MK equation with 82 literatures, it is found that the differences can be reduced mostly to below 20 % by introducing the effective length of turbulent flow into the MK equation. On the other hand, it is also found that considerable differences from 30 % to 70 % arise for the gas flow through an orifice when the pressure ratio of downstream to upstream is close to 1 and high-Re flow in short tubes.
Also, as an application of the MK equation, the pump-down time of vacuum chamber is tried to calculate by using the MK equation. Figure 1 shows the comparison of pump-down time calculated by the MK equation and the theory of Senda [3], and the experiments result Yokogawa et. al. [4], where the volume of vacuum chamber is 30 L, the pumping speed of the vacuum pump is 60 L/min and the diameter and the length of the connecting tube are 1.83 mm and 2 m, respectively. The results of MK equation are closer to the experimental results than Senda’s equation until 103 s because the MK equation includes the effect of turbulent flow, but the Senda’s equation does not. The result of MK equation, however, underestimates the pump-down time similar to Senda’s equation at the time longer than 103 s. This is because these calculations do not include the effects of outgassing from the chamber and reduction of pumping speed at lower pressure. Details will be discussed at the presentation.
[1] H. Yoshida, Y. Takei and K. Arai, Vacuum and Surface Science 63 (2020) 373.
[2] H. Yoshida, M. Hirata, T. Hara, Y. Higuchi, Packag Technol Sci. 34 (2021) 557.
[3] Y. Senda, SEI TECHNICAL REVIEW 176, 1 (2010).
[4] K. Yokogawa, K. Seto and T. Fukuda, Vacuum and Surface Science Vol. 61, No. 5 (2018).
