Knudsen diffusion free path

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. 

For gas-phase diffusion in small pores at lowpressure, the molecular mean free path may be larger than the pore diameter, giving rise to Knudsen diffusion. Satterfield (Ma.s.s Tran.sfer in Heterogeneous Catalysis, MIT, Cambridge, MA, 1970, p. 43), gives the following expression for the pore dimisivity ... 

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. 

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 bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

By comparing the relative magnitude of the mean free path (z) and the pore diameter (27), it is possible to determine whether bulk diffusion or Knudsen diffusion may be regarded as negligible. Using the principles of the kinetic theory... 

In Figure 2 we presented the permeability coefficient K of oxygen as a function of the mean gas pressure experimentally obtained for a sample of porous material from acetylene black modified with 35% PTFE. The experimental linear dependence is obtained. The intercept with the abscissa corresponds to the Knudsen term DiK. The value obtained is 2,89.1 O 2 cm2/s. The slope of the straight line is small, so that the ratio K,/ Dik at mean gas pressure 1 atm. is small ( 0.1) which means that the gas flow is predominantly achieved by Knudsen diffusion and the viscous flow is quite negligible. At normal conditions (1 atm, 25°C) the mean free path of the air molecules (X a 100 nm) is greater than the mean pore radii in the hydrophobic material (r 20 nm), so that the condition (X r) for the Knudsen-diffusion mechanism of gas transport is fulfilled. 

Collisions of molecules with the walls of the passage provide the resistance to diffusion when the mean free path, A, is appreciably greater than the diameter, d, of the passage. In experimental work the ratio, A/d, is taken as 10 or more to isolate the Knudsen effect. That investigator did experiments with small capillaries and deduced the equation... 

As a rough rule, Knudsen diffusion predominates when the ratio of mean free path to pore radius, A/re 10 or so, and molecular when A/rcs 0.1. For common gases at atmospheric conditions these values correspond to pore radii... 

For S02 in propane at 5 atm and 873 K find (a) the molecular diffusivity (b) the Knudsen diffusivity in a pore whose radius is 2 times the 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 ... 

Of these three mechanisms, i.e. molecular diffusion, laminar flow and Knudsen diffusion, only two are important in pressure-driven separations. These are laminar flow and Knudsen diffusion. These can be qualitatively understood as follows. If the molecules "see each other much more than they see the pore wall (which means the mean free path of the molecules is much smaller than the mean pore radius), laminar ow a molecular diffusion are important. The laminar flow is much larger, howcver, nd thelhdlecular flow can be neglected (Present and de Bethune 1949). If the molecules see the pore wall much more than they see each other, only Knudsen diffusion will occur. Thus, the molecular diffusion can be neglected in all circumstances. From now on it will be assumed, that only laminar flow and Knudsen diffusion occur. 

From an order-of-magnitude analysis, when the mean-free path of a molecule is less than 0.01 times the pore radius, bulk diffusion dominates, and when it is greater than 10 times the pore radius, Knudsen diffusion dominates. This means that Knudsen diffusion is significant when the pore radius is less than about 0.5 fim. For reference, a typical carbon gas-diffusion layer has pores between 0.5 and 20 /rm22 229 in radius, and a microporous layer contains pores between 0.05 and 2 Thus, while Knudsen... 

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

Once particles have been formed, they grow by the combined effect of vapor deposition and continued coagulation. The transport of vapor species to the surfaces of the aerosol particles depends on the size of the aerosol particle relative to the mean free path of the gas molecules. A. that is, it depends on the Knudsen number characteristic of vapor diffusion to the particle, Kn ... 

Knudsen Diffusion Only Is Occurring. For a very fine pore material in which the effective pore diameter is less than the mean free path of the molecules, bulk diffusion and Poiseuille flow do not occur. For this case, the change in volume given when C + CO2 —> 2CO has no influence on the rate of diffusion of carbon dioxide into the rod, and is not dependent on the total pressure in the pores. Considering a wedge of carbon (Fig. Al),... 

Knudsen diffusion This kind of diffusion is observed if the mean free path of the molecules is significantly greater than the pore diameter. In this mode, the intermolecular collisions are much less than those between the molecules and pore walls. 

In liquids, the mean free path is typically of the order of 10 10 m. Hence the Knudsen effect is not important (i.e., diffusing molecules collide with solvent molecules long before they typically arrive at a pore wall). However, diffusion is affected by a different mechanism, the viscous drag caused by the pore walls. This is known as the Renkin effect (Renkin, 1954). In essence, the ratio of pore diffusivity in the liquid-filled pore space and diffusivity in the free liquid, D(pore/D,free, is a function of the nondimensional parameter... 

In industry large pellets of a catalyst were employed (e.g., 6-8 mm in size), and the rate of the process was essentially affected by the slowness of the diffusion of ammonia in the pores of the catalyst the efficiency factor at this size of pellets is about 0.5. The effect of diffusion retardation of the ammonia synthesis was studied both at high pressures (99), when the free path of molecules is much smaller than the radius of catalyst pores so that the bulk diffusion is operative, and at pressures near to 1 atm (116), where there is a transition from the bulk to the Knudsen diffusion. 

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