Knudsen, diffusion

Knudsen diffusion may take place in a microporous inorganic membrane or through pinholes in dense polymeric membranes. It may also take place in a mixed matrix membrane with insufficient adhesion between the phases. 

Knudsen flow is characterized by the mean free path (A) of the molecules, which is larger than the pore size, and hence collisions between the molecules and the pore walls are more frequent than intermolecular collisions. A lower limit for the significance of the Knudsen mechanism has usually been set at dp 20 A [28]. The classical Knudsen equation for diffusion of gas is 

Here is the probability that a molecule can make a jump in the right direction given the jump length is dp and the velocity is 

Knudsen diffusion [20,30,32,37 ] depending on gas pressure and mean free path in the gas phase applies to pores between 10 A and 500 A in size however, there are examples in the literature where it was observed for much larger pores [41 ]. In this region, the mean free path of molecules in gas phase A is much larger than the pore diameter d. It is common to use the so-called Knudsen number K = X d o characterize the regime of permeation through pores. When 1, viscous (PoiseuiUe) flow is realized. The condition for Knudsen diffusion is 1. An intermediate regime is realized when A), 1. 

The Knudsen diffusion coefficient can be expressed in the following form 

For the flux in the Knudsen regime the following equation holds [42,43]  

Two important conclusions can be made from analysis of Equations (5.16) and (5.17). First, selectivity of separation in Knudsen regime is characterized by the ratio ij = (Mj/Mi) . It means that membranes where Knudsen diffusion predominates are poorly selective. For example, separation factor of separation of O2/N2 pair is 1.07. The highest gas separation selectivity can be observed in separation of lightest and heaviest gases, e.g. hydrogen and butane in this case = 5.4. Another unusual feature of Knudsen diffusion is that increases in temperature result in slight decreases of the flux and permeability coefficient, as J and P depend on temperature as Numerous confirmation of this dependence can be found in the literature. 

For Knudsen diffusion to take place, the lower limit for pore diameter has usually been set to 4 ore 20 A. Gilron and Soffer have, however, discussed thoroughly how Knudsen diffusion may contribute to transport in even smaller pores, and from a model considering pore structure, shown that contributions to transport may both come from activated transport and Knudsen through one specific fiber. It may thus be difficult to know exactly when transport due to Knudsen diffusion is taking place. One way to approach this problem is to calculate the Knudsen number, Knudsen. for the system, which is 2/fi pore where X is the mean free path. If Knudsen 10, then the separation can be assumed to take place according to Knudsen diffusion. Therefore, if the preparation of the carbon membranes has been unsuccessful, one may get Knudsen diffusion. 

The treatment in Section 9.2.4 will first start with some simple limiting cases (Knudsen diffusion and viscous flow in mixtures), followed by a comparison of an extended Pick model with the DGM model derived equations for binary gas mixtures. Subsequently a treatment will be given of a direct application to membrane separation of a set of equations derived from the model of Present and Bethune by Wu et al. [18] and by Eichmann and Werner [19]. 

For Knudsen diffusion collisions between particles are negligible and molecules of different species move entirely independent of each other under the action of their own concentration (or partial pressure) gradient. There is no fundamental difference between flow and diffusion. The resulting expression for the total flux of a mixture with component fluxes and is 

In molecular diffusion the resistance to flow arises from collisions between diffusing molecules. The effect of the pore is merely to reduce the flux as a result of geometric constraints which are accounted for by the tortuosity factor. Molecular diffusion will be the dominant transport mechanism whenever the mean free path of the gas (i.e., the average distance traveled between molecular collisions) is small relative to the pore diameter. However, in small pores and at low pressure the mean free path is greater than the pore diameter and collisions of molecules with the pore walls occur more frequently than collisions between diffusing molecules. Under these conditions the collisions between molecule and pore wall provide the main diffusional resistance and we have what is known as Knudsen diffusion or Knudsen flow. 

When a molecule strikes the pore wall it does not bounce like a tennis ball. Rather the molecule is instantaneously adsorbed and re-emitted in a random direction. The direction in which the molecule is emitted bears no relation to its original direction before the collision and it is this randomness which provides the characteristic feature of a diffusive process. 

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Note the use of a script for the binary pair mutual diffusion coefficient, as distinct from the Roman D already used to represent Knudsen diffusion coefficients. This convention will be adhered to throughout. 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The solute species therefore diffuse independently, rather as in Knudsen diffusion, but with effective diffusion coefficients D, where... 

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. 

The Knudsen diffusion coefficients are given by equations (2.11) in which K. is independent of pressure and proportional to pore diameter a, so that we can write... 

It may seem curious that Knudsen diffusion coefficients still appear in equations (5.18) and (5.19), which supposedly give the flux relations at the limit of bulk diffusion control. However, inspection reveals that only ratios of these coefficients are effectively present, and from equation (2,11) it follows that... 

The first case corresponds to a situation in which all Knudsen diffusion coefficients are equal, and all binary pair bulk diffusion coefficients are equal ... 

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. 

Knudsen diffusion, but the dependence on physical and geometric conditions... 

Che Knudsen diffusion coefficient is independent of composition and pressure,... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

Knudsen diffusion coefficient for the test gas in a micropore. represents the total void fraction and c that part of of the void fraction... 

For an Isothermal pellet with pores sufficiently small that Knudsen diffusion controls, the flux relations are required to take the form (8.1), which can be written... 

It remains to check Chat our supposedly general equations (11.20) -(11.23) reduce to the previously obtained limiting forms at the extremes of bulk diffusion and Knudsen diffusion. The bulk diffusion limit is easily obtained by letting — and assuming that for all r, s,... 

In practice it would not be reasonable to solve the balances at the limit of Knudsen diffusion control by considering the n simultaneous boundary value problems (11.7). All the partial pressures can be expressed in terms of by integrating equations (11,25), with the result... 

In simple cases it is not difficult to estimate the magnitude of the pressure variation within the pellet. Let us restrict attention to a reaction of the form A nB in a pellet of one of the three simple geometries, with uniform external conditions so that the flux relations (11.3) hold. Consider first the case in which all the pores are small and Knudsen diffusion controls, so that the fluxes are given by... 

The above estimates of pressure variations suggest that their magni-tude as a percentage of the absolute pressure may not be very large except near the limit of Knudsen diffusion. But in porous catalysts, as we have seen, the diffusion processes to be modeled often lie in the Intermediate range between Knudsen streaming and bulk diffusion control. It is therefore tempting to try to simplify the flux equations in such a way as to... 

At the limit of Knudsen diffusion control it is not reasonable to expect that any of the proposed approximation methods will perform well since, as we know, percentage variations in pressure are quite large. Nevertheless it is interesting to examine their results, which are shown in Figure 11 4 At this limit it is easy to check algebraically that equations (11.54) and (11.55) become the same, while (11.60) differs from the other two. Correspondingly the values of the effectiveness factor calculated using the approximation of Kehoe and Aris coincide with the results of Apecetche et al., and with the exact solution, ile Hite and Jackson s effectiveness factors differ substantially. 

In chapter 5 we showed that the dusty gas model flux relations could be solved (fjite easily at the limit of Knudsen diffusion control, when they reduce to the form given in equation (5.25), namely... 

In our earlier discussion of the dynamical equations at the opposite limit of Knudsen diffusion control, we obtained a final simplified form, represented by equations (12.15) and (12.16) (or (12.20) and (12.21) for an Irreversible reaction with a single reactant), after introducing certain... 

For the same reaction in a pellet of finely porous structure, where Knudsen diffusion controls, the appropriate dynamical equations sre (12.20) and (12.21) if we once more adopt approximations which are a consequence of Che large size of K. These again have a dimensionless form, which may be written... 

In Chase equaclons Che symbols Z u, A, b and id are noc defined quite as In equations (12.39), since Che bulk diffusion coefficient muse now be replaced by a Knudsen diffusion coefficient. Thus... 

At the limit of Knudsen diffusion control, while the remaining... 

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