Molecule mean free path

In pores that are appreciably smaller than the mean free path, molecules tend to collide with the pore walls rather than with other molecules. Having collided with the wall, the molecules are momentarily retained and then released in a random direction. The coefficient, >, which controls this Knudsen diffusion, considered by Satterheld(31), and in Volume 1, Chapter 3, may be derived from the kinetic theory to give ... 

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

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. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

To be specific let us have in mind a picture of a porous catalyst pellet as an assembly of powder particles compacted into a rigid structure which is seamed by a system of pores, comprising the spaces between adjacent particles. Such a pore network would be expected to be thoroughly cross-linked on the scale of the powder particles. It is useful to have some quantitative idea of the sizes of various features of the catalyst structur< so let us take the powder particles to be of the order of 50p, in diameter. Then it is unlikely that the macropore effective diameters are much less than 10,000 X, while the mean free path at atmospheric pressure and ambient temperature, even for small molecules such as nitrogen, does not exceed... 

In molecular distillation, the permanent gas pressure is so low (less than 0 001 mm. of mercury) that it has very little influence upon the speed of the distillation. The distillation velocity at such low pressures is determined by the speed at which the vapour from the liquid being distilled can flow through the enclosed space connecting the still and condenser under the driving force of its own saturation pressure. If the distance from the surface of the evaporating liquid to the condenser is less than (or of the order of) the mean free path of a molecule of distillate vapour in the residual gas at the same density and pressure, most of the molecules which leave the surface will not return. The mean free path of air at various pressures is as follows —... 

Effusion separator (or effusion enricher). An interface in which carrier gas is preferentially removed from the gas entering the mass spectrometer by effusive flow (e.g., through a porous tube or through a slit). This flow is usually molecular flow, such that the mean free path is much greater than the largest dimension of a traverse section of the channel. The flow characteristics are determined by collisions of the gas molecules with surfaces flow effects from molecular collisions are insignificant. 

Siace the pores ia an aerogel are comparable to, or smaller than, the mean free path of molecules at ambient conditions (about 70 nm), gaseous conduction of heat within them is iaefficient. Coupled with the fact that sohd conduction is suppressed due to the low density, a siUca aerogel has a typical thermal conductivity of 0.015 W/(m-K) without evacuation. This value is at least an order of magnitude lower than that of ordinary glass and considerably lower than that of CFC (chloro uorocarbon)-blown polyurethane foams (54). 

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

Successful operation of the gaseous diffusion process requires a special, fine-pored diffusion barrier, mechanically rehable and chemically resistant to corrosive attack by the process gas. For an effective separating barrier, the diameter of the pores must approach the range of the mean free path of the gas molecules, and in order to keep the total barrier area required as small as possible, the number of pores per unit area must be large. Seals are needed on the compressors to prevent both the escape of process gas and the inflow of harm fill impurities. Some of the problems of cascade operation are discussed in Reference 16. 

Ba.rrier Flow. An ideal separation barrier is one that permits flow only by effusion, as is the case when the diameter of the pores in the barrier is sufficiently small compared to the mean free path of the gas molecules. If the pores in the barrier are treated as a collection of straight circular capillaries, the rate of effusion through the barrier is governed by Knudsen s law (eq. 46) ... 

Molecular distillation occurs where the vapor path is unobstmcted and the condenser is separated from the evaporator by a distance less than the mean-free path of the evaporating molecules (86). This specialized branch of distillation is carried out at extremely low pressures ranging from 13—130 mPa (0.1—1.0 p.m Hg) (see Vacuum technology). Molecular distillation is confined to appHcations where it is necessary to minimize component degradation by distilling at the lowest possible temperatures. Commercial usage includes the distillation of vitamins (qv) and fatty acid dimers (see Dimeracids). 

In small pores and at low pressures, the mean free path of the gas molecule (or atom) is significantly greater than the pore diameter Jpore- Rs magnitude may be estimated from... 

Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by colhsions between molecules. An approximate expression based on Brown, et al. J. Appl. Phys., 17, 802-813 [1946]) for the mean free path is... 

When the size of a particle approaches the same order of magnitude as the mean free path of the gas molecules, the setthng velocity is greater than predicted by Stokes law because of molecular shp. The slip-flow correc tion is appreciable for particles smaller than 1 [Lm and is allowed for by the Cunningham correc tion for Stokes law (Lapple, op. cit. Licht, op. cit.). The Cunningham correction is apphed in calculations of the aerodynamic diameters of particles that are in the appropriate size range. 

FIGt 22-48 Transport mechanisms for separation membranes a) Viscous flow, used in UF and MF. No separation achieved in RO, NF, ED, GAS, or PY (h) Knudsen flow used in some gas membranes. Pore diameter < mean free path, (c) Ultramicroporoiis membrane—precise pore diameter used in gas separation, (d) Solution-diffusion used in gas, RO, PY Molecule dissolves in the membrane and diffuses through. Not shown Electro-dialysis membranes and metallic membranes for hydrogen. 

The mean free path, which is die average distance a molecule navels between collisions, is... 

According to Ktiudsen if a small circular orifice of diameter less than the mean free path of the molecules in a container, is opened in the wall of the container to make a connection to a high vacuum sunounding the container, the mass of gas effusing tlnough the orifice, of area A, is given by an equation derived from the kinetic theoty, where tire pressure is in amiospheres. 

Using cm as unit surface and seconds as unit time, n is the number of molecules falling on 1 cm /sec. The number n thus denotes the number of molecules striking each cm of the surface every second, and this number can be calculated using Maxwell s and the Boyle-Gay Lussac equations. The number n is directly related to the speed of the molecules within the system. It is important to realize that the velocity of the molecules is not dependent on the pressure of the gas, but the mean free path is inversely proportional to the pressure. Thus ... 

Besides these parameters, the properties of a gas ion are sometimes characterized with mean free path A,, which illustrates the mean distance between successive impacts with gas atoms or molecules. The mean free path of ions in air is in the range of 10 to 2 10" m. ... 

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