Mean free length

Mean free length The mean free length of a particle (pm) used in particle scrubbing equations. 

In principle, the Boltzmann transport equation (BTE) can cover the regime where die lengdi and time scales are larger than carrier mean free time rand mean free length A. However, tremendous computational efforts are required in practice when the system length scale L and the process time scale t are getting larger. The BTE is, thus, usually... 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

It ls not surprising chat such a relation should hold at the Limit of Knudsen diffusion, since Che Knudsen diffusion coefficients are themselves inversely proportional to the square roots of molecular weights, but the pore diameters in Graham s stucco plugs were certainly many times larger chan the gaseous mean free path lengths at the experimental conditions. 

Knudseci s very careful experiments on a long uniform capillary show that N L/ Pj -p ) passes through a marked minimum when plotted as a function of (P +P2)/2, at a value of the mean pressure such that the capillary diameter and the mean free path length are comparable. At higher values of the mean pressure, N L/(pj " 2 rises linearly, as in the case of a porous medium. 

Example 4. For a given lattice, a relationship is to be found between the lattice resistivity and temperature usiag the foUowiag variables mean free path F, the mass of electron Af, particle density A/, charge Planck s constant Boltzmann constant temperature 9, velocity and resistivity p. Suppose that length /, mass m time /, charge and temperature T are chosen as the reference dimensions. The dimensional matrix D of the variables is given by (eq. 55) ... 

Figure 6 Mean free path lengths as a function of KE, determined for (a) metals and (b) inorganic compounds. ...

M. P. Seah and W. A Dench. Surf. Interface Anal. 1,1, 1979. Of the many compilations of measured mean free path length versus KE, this is the most thorough, readable, and useful. 

This discussion of geometric effects ignored the attenuation of radiation by material through which the radiation must travel to reach the receptor. The number of particles, dN, penetrating material, equals the number of particles incident N times a small penetration distance, dx, divided by the mean free path length of the type of particle in the type of material (equation 8.3-8). Integrating gives the transmission coefficient for the radiation (equation 8.3-9). 

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