Diffusion, bulk

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

Chapcer 4. GAS MOTION IN A LONG TlfBE AT THE LIMIT OF BULK DIFFUSION AND VISCOUS FLOW... 

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. 

This determines the diffusion fluxes at the limit of bulk, diffusion control. 

This determines the total flux at the li/nic of viscous flow. Equations (5.18 and (5.19) therefore describe the limiting form of the dusty gas model for high pressure or large pore diameters -- the limit of bulk diffusion control and viscous flow,... 

Ac this point It is important to emphasize that, by changing a and p, it is not possible to pass to the limit of viscous flow without simultaneously passing to the limit of bulk diffusion control, and vice versa, since physical estimates of the relative magnitudes of the factors and B... 

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... 

At the limit of bulk diffusion control this reduces to... 

There is no doubt that the coefficient of the third term on the right hand side of this equation is much larger than the coefficient of the second term at the bulk diffusion limit, and this justifies our original form (5,29). However, on constructing from and using equation (5,32) and its... 

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

The second case follows from a suggestion of Bird [27] to Che effect that binary pair bulk diffusion coefficients might be approximated by expressions of the form... 

Thus his experiments were the first to indicate the surprising result that relation (6,1) remains valid even in conditions where bulk diffusion resistance is completely dominant. Accordingly (6.1), perhaps the most important single experimental result on diffusion in porous media, will be referred to as Graham s relation. 

Now when the pressure is high enough that bulk diffusion controls... 

Remick and Geankoplis made flux measurements for both species in the isobaric diffusion of nitrogen and helium through their tube bundle. Pressures spanned the interval from 0.444 nim, Hg to 300,2 ram Hg, which should cover the whole range between the limits of Knudsen streaming and bulk diffusion control. Then, since K and K, are known in this case, the form of the proposed flux relations could be tested immediately by plotting the left hand side of equation (10.15) against... 

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. 

Here L is the thickness of the porous septum and jS the length of each dead-end micropore, the effective binary bulk diffusion coefficient... 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. 

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,... 

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

As a consequence of this, i enever bulk dlffusional resistance domin ates Knudsen diffusional resistance, so that 1, it follows that fi 1 also, and hence viscous flow dominates Knudsen streaming. Thus when we physically approach the limit of bulk diffusion control, by increasing the pore sizes or the pressure, we must simultaneously approach the limit of viscous flow. This justifies a statement made in Chapter 5. 

At the opposite limit, where all the pores are sufficiently large that bulk diffusion controls, a similar calculation can be performed. In this case the appropriate flux relations are equation (5.29) and its companion obtained by interchanging the suffixes. For the symmetric systems considered here these may be written in scalar form ... 

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... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

---