Knudsen number, definition

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

The correction of mean free path, hi by the nanoscale effect function results in a smaller mean free path, or a smaller Knudsen number in other word. As a matter of fact, a similar effect is able to be achieved even if we use the conventional definition of mean free path, / = irSn, and the Chapmann-Enskog viscosity equation, /r = (5/16)... 

Taking into account the value of V, (Equation (23)) and the definition of tlie Knudsen number, the Darcy coefficient becomes ... 

The measurements by Harley, Pfahler, and Urbanek not only provide a solid basis for modeling fluid flows in small ducts but also raise a question about the nature of that flow at elevated temperatures. The lower-temperature data justify the use of hydrodynamic theory in simple ducts. Whether this will hold in more complex flow structures needs further study. For gas flow in ducts where the Knudsen number is 0.05 or greater, slip flow is observed. Urbanek s data suggest that there may be increased wall interactions as the temperature approaches the boiling point. A more definitive study is needed to clarify this point. 

Mean Free Path of a Pure Gas Let us start with the simplest case, a particle suspended in a pure gas B. If we are interested in characterizing the nature of the suspending gas relative to the particle, the mean free path that appears in the definition of the Knudsen number is kBB. The subscript denotes that we are interested in collisions of molecules of B with other molecules of B. Ordinarily, air will be the predominant vapor species in such a situation. The mean free path A.br has been defined as the average distance traveled by a B molecule between collisions with other B molecules. The mean speed of gas molecules of B, cB is (Moore 1962, p. 238)... 

Equation (12.34) is based on results for z <directly applicable to light molecules in a heavier background gas. The mean free path included in the definition of the Knudsen number in (12.34) is given by... 

Equations (10) - (12) are by no means definitive. They represent expressions in fairly common use and are supposedly valid for all droplet Knudsen numbers from the continuum to the free molecule regime. The method of analysis is not dependent on the forms of (10) - (12),... 

The central KEMS equation can be derived now that the Knudsen cell vapor source and mass spectrometer have been described. This follows directly from the vapor flux in the molecular beam selected from the distribution of material effusing from the Knudsen cell (molecular beam flux equation) and the definition of the ionization cross section (Equation 48.18). However, in accordance with the aim of identifying factors that affect the measured ion intensity and that are unrelated to sample temperature and composition, it useful to rewrite Equation 48.18 in terms of the number of ions produced per second in the elementary volume dv in the region defined by the intersection of the molecular and electron beams, ni(E) [71,80] (this is prior to the formation of the ion beam) ... 

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