Free-molecule flow

For nonspherical particles, values for the slip correction factor are available in slip flow (MU) and free-molecule flow (Dl). To cover the whole range of Kn and arbitrary body shapes, it is common practice to apply Eq. (10-58) for nonspherical particles. The familiar problem then arises of selecting a dimension to characterize the particle. Some workers [e.g. (H2, P14)] have used the diameter of the volume-equivalent sphere this procedure may give reasonable estimates for particles only slightly removed from spherical, or in near-con-tinuum flow, but gives the wrong limit at high Kn. An alternative approach... 

In the free-molecule flow range, the drag on a sphere is given (Sll), for diffuse molecular reflection (cr = 1), by ... 

Cofn, drag coefficient for free-molecule flow... 

To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures D y eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11], Finally, combining Equations (6.15-6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)) ... 

Fig. 12-12 Three types of flow regimes for a flat plate (a) continuum fiow. ( >) slip flow, (c) free-molecule flow.

When free-molecule flow is encountered, Oppenheim [8] has given convenient charts for calculating recovery factors and heat-transfer coefficients for flow over standard geometric shapes. Figures 12-16 and 12-17 give samples of these charts. The molecular speed ratio S used in these charts is defined by... 

It is well to mention that the equilibrium temperature of a surfac in free-molecule flow is usually influenced strongly by radiation heat transfer because... 

How is a distinction made between continuum, slip, and free-molecule flows in a physical sense ... 

Oppenheim, A. K. Generalized Theory of Convective Heat Transfer in a Free-Molecule Flow, J. Aeron. Sci., vol. 20, p. 49, 1953. 

Here X>iA (r) is a Knudsen diffusivity, which we take as the limiting value for free-molecule flow in a long tube of radius r yfngstroms,... 

In the case of the exhaust plume tests, the exhaust gas must be allowed to fully expand throughout the pertinent region of interest. This test requirement places some unusual demands upon the facility s vacuum pumping system. The requirements for the pumping of a directional free-molecule flow will be discussed in this paper, and a cryopump design technique will be presented which satisfies the exhaust plume test objectives. 

The type of flow encountered when a highly underexpanded nozzle exhausts into a vacuum differs significantly from the random free-molecule flow characterized by the Max-well-Boltzmann distribution function, which is normally encountered in vacuum practice. The difference is illustrated in Fig. 1 and is primarily due to the large directional velocity component given to the gas molecules while still in a dense continuum flow condition in the nozzle. The velocity distribution function for the exhaust gas thus differs from that of the random flow in that it contains both the random thermal velocity components and the large directional velocity component. 

Directional flows are also encountered in random free-molecule flow situations as, for instance, in the case of the flow of a gas through a cylindrical passage. However, in this case... 

Fig. 5. Conductance factors for directional free-molecule flow in cylindrical tubes.

Derjaguin (1946) used a statistical method to follow the motion of a free molecule flowing through a random pore space to obtain the following equation for the Knudsen flux based on total cross-sectional area of the medium ... 

For flow with high Knudsen number, the number of molecules in a significant volume of gas decreases and there could be insufficient number of molecular collisions to establish an equilibrium state. The velocity distribution function will deviate away from the Maxwellian distribution and is non-isotropic. The properties of the individual molecule then become increasingly prominent in the overall behavior of the gas as the Knudsen number increases. The implication of the larger Knudsen number is that the particulate nature of the gases needs to be included in the study. The continuum approximation used in the small Knudsen number flows becomes invalid. At the extreme end of the Knudsen number spectrum is when its value approaches infinity where the mean free path is so large or the dimension of the device is so small that intermolecular collision is not likely to occur in the device. This is called collisionless or free-molecule flows. 

Dahneke, B.E. (1973). Slip correction factors of nonspherical bodies. I. Free molecule flow. J Aerosol Sci 4 147-161. 

---