Mean-free-path processes

One of the most usefiil applications of the mean free path concept occurs in the theory of transport processes in systems where there exist gradients of average but local density, local temperature, and/or local velocity. The existence of such gradients causes a transfer of particles, energy or momentum, respectively, from one region of the system to another. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

In a cascade process, one incident electron (e ) collides with a neutral atom ((S)) to produce a second electron and an ion ( ). Now there are two electrons and one ion. These two electrons collide with another neutral atom to produce four electrons and three ions. This process continues rapidly and — after about 20 successive sets of collisions — there are millions of electrons and ions. (The mean free path between collisions is very small at atmospheric pressures.) A typical atmospheric-pressure plasma will contain 10 each of electrons and ions per milliliter. Some ions and electrons are lost by recombination to reform neutral atoms, with emission of light. 

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. 

The inelastic collision process is characterized by an inelastic mean free path, which is the distance traveled after which only 1/e of the Auger electrons maintain their initial energy. This is very important because only the electrons that escape the sample with their characteristic Auger energy are usefrd in identifying the atoms in... 

In the discussion so far, the fluid has been considered to be a continuum, and distances on the molecular scale have, in effect, been regarded as small compared with the dimensions of the containing vessel, and thus only a small proportion of the molecules collides directly with the walls. As the pressure of a gas is reduced, however, the mean free path may increase to such an extent that it becomes comparable with the dimensions of the vessel, and a significant proportion of the molecules may then collide direcdy with the walls rather than with other molecules. Similarly, if the linear dimensions of the system are reduced, as for instance when diffusion is occurring in the small pores of a catalyst particle (Section 10.7), the effects of collision with the walls of the pores may be important even at moderate pressures. Where the main resistance to diffusion arises from collisions of molecules with the walls, the process is referred to Knudsen diffusion, with a Knudsen diffusivily which is proportional to the product where I is a linear dimension of the containing vessel. 

The internal structure of the catalyst particle is often of a complex labyrinth-like nature, with interconnected pores of a multiplicity of shapes and sizes, In some cases, the pore size may be less than the mean free path of the molecules, and both molecular and Knudsen diffusion may occur simultaneously. Furthermore, the average length of the diffusion path will be extended as a result of the tortuousity of the channels. In view of the difficulty of precisely defining the pore structure, the particle is assumed to be pseudo-homogeneous in composition, and the diffusion process is characterised by an effective diffusivity D, (equation 10.8). 

In the previous section, the molecular basis for the processes of momentum transfer, heat transfer and mass transfer has been discussed. It has been shown that, in a fluid in which there is a momentum gradient, a temperature gradient or a concentration gradient, the consequential momentum, heat and mass transfer processes arise as a result of the random motion of the molecules. For an ideal gas, the kinetic theory of gases is applicable and the physical properties p,/p, k/Cpp and D, which determine the transfer rates, are all seen to be proportional to the product of a molecular velocity and the mean free path of the molecules. 

The design of the instrument, together with the pumping capacity, ensures a low background pressure (<10 mbar). Under process conditions the pressure directly behind the orifice is about a factor of 10 lower than the process pressure in the mass filter, even a factor of 10. The mean free path of particles that have entered the EQP therefore is several meters. 

Because there are two positive terms in the denominator of equation 4.2.85 (either of which may be associated with the dominant termination process), this equation leads to two explosion limits. At very low pressures the mean free path of the molecules in the reactor is quite long, and the radical termination processes occur primarily on the surfaces of the reaction vessel. Under these conditions gas phase collisions leading to chain breaking are relatively infrequent events, and fst fgt. Steady-state reaction conditions can prevail under these conditions if fst > fb(a — 1). 

As the pressure in the reaction vessel increases, the mean free path of the gaseous molecules will decrease and the ease with which radicals can reach the surfaces of the vessel will diminish. Surface termination processes will thus occur less frequently fst will decline and may do so to the extent that fst + fgt becomes equal to fb oc — 1). At this point an explosion will occur. This point corresponds to the first explosion limit shown in Figure 4.1. If we now jump to some higher pressure at which steady-state reaction conditions can again prevail, similar... 

In recent years, increasing use has been made of in situ methods in EM—as is true of other techniques of catalyst characterization such as IR, Raman, and NMR spectroscopy, or X-ray diffraction. Although the low mean-free path of electrons prevents EM from being used when model catalysts are exposed to pressures comparable to those prevailing in industrial processes, Gai and Boyes (4) reported early investigations of in situ EM with atomic resolution under controlled reaction conditions to probe the dynamics of catalytic reactions. Direct in situ investigation permits extrapolation to conditions under which practical catalysts operate, as described in Section VIII. 

In the relations given earlier, it is assumed that the fluid can be regarded as a continuum and that there is no slip between the wall of the capillary and the fluid layers in contact with it. However, when conditions are such that the mean free path of the molecules of a gas is a significant fraction of the capillary diameter, the flowrate at a given value of the pressure gradient becomes greater than the predicted value. If the mean free path exceeds the capillary diameter, the flowrate becomes independent of the viscosity and the process is one of diffusion. Whereas these considerations apply only at very low pressures in normal tubes, in fine-pored materials the pore diameter and the mean free path may be of the same order of magnitude even at atmospheric pressure. 

The second most apparent limitation on studies of surface reactivity, at least as they relate to catalysis, is the pressure range in which such studies are conducted. The 10 to 10 Torr pressure region commonly used is imposed by the need to prevent the adsorption of undesired molecules onto the surface and by the techniques employed to determine surface structure and composition, which require relatively long mean free paths for electrons in the vacuum. For reasons that are detailed later, however, this so-called pressure gap may not be as severe a problem as it first appears. There are many reaction systems for which the surface concentration of reactants and intermediates found on catalysts can be duplicated in surface reactivity studies by adjusting the reaction temperature. For such reactions the mechanism can be quite pressure insensitive, and surface reactivity studies will prove very useful for greater understanding of the catalytic process. 

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