The dusty gas model

The most fundamental model of a porous structure is provided by the dusty gas model as critically reviewed and described by Jackson [19]. The basis of the theory, as suggested by Maxwell as early as 1860, is to suppose that the action of the porous material is similar to that of a number of particles fixed in space and obstructing the motion of molecules. Thus, if a number of gaseous species are diffusing through three-dimensional space, the imposition of one further species of much 

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To obtain equations describing the dusty gas model, equations (3.1) must be applied to a pseudo mixture of (n+1) species, in which the extra species, numbered n+1, represents the dust. We must also require... 

These are the flux relations associated with the dusty gas model. As explained above, they would be expected to predict only the diffusive contributions to the flux vectors, so they should be compared with equations (2.25) obtained from simple momentum transfer arguments. Equations (3,16) are then seen to be just the obvious vector generalization of the scalar equations (2.25), so the dusty gas model provides justification for the simple procedure of adding momentum transfer rates. 

The relation between the dusty gas model and the physical structure of a real porous medium is rather obscure. Since the dusty gas model does not even contain any explicit representation of the void fraction, it certainly cannot be adjusted to reflect features of the pore size distributions of different porous media. For example, porous catalysts often show a strongly bimodal pore size distribution, and their flux relations might be expected to reflect this, but the dusty gas model can respond only to changes in the... 

Thermal transpiration and thermal diffusion effects have been neglected in developing the dusty gas model, and will be neglected throughout the rest of the text. The physics of these phenomena and the justification for neglecting them are discussed in some detail in Appendix I. 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

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,... 

Finally, before leaving our exploration of the dusty gas model, we must compare the large pore (or high pressure) limiting form of its flux relations with the corresponding results derived in Chapter 4 by detailed solution of the continuum equations in a long capillary. The relevant equations are (4,23) and (4,25), to be compared with the corresponding scalar forms of equations (5.23) and (5.24). Equations (4.25) and (5.24).are seen to be identical, while (4,23) and (5.23) differ only in the pressure diffusion term, which takes the form... 

It is interesting to note that the dusty gas model equations also... 

To appreciate the questions raised by Knudsen s results, consider first the relation between molar flow and pressure gradient for a pure gas flowing through a porous plug, rather than a capillary. The form predicted by the dusty gas model can be obtained by setting = 1, grad = 0 in equation... 

One flux model for a porous medium—the dusty gas model- has already been described in Chapter 3. Although it is perhaps the most important and generally useful model currently available, it has certain shortcomings, and other models have been devised in attempts to rectify these. However, before describing these, we will review certain general principles to which all reasonable flux models must conform. 

When a model is based on a picture of an interconnected network of pores of finite size, the question arises whether it may be assumed that the composition of the gas in the pores can be represented adequately by a smooth function of position in the medium. This is always true in the dusty gas model, where the solid material is regarded as dispersed on a molecular scale in the gas, but Is by no means necessarily so when the pores are pictured more realistically, and may be long compared with gaseous mean free paths. To see this, consider a reactive catalyst pellet with Long non-branching pores. The composition at a point within a given pore is... 

These are equivalent to the dusty gas model equations, but are valid only for isobaric conditions, and this fact severely limits the capability of the model to represent Che behavior of systems with chemical reaction. To see this we need only remark that (8,7) and (3.8) together imply that ... 

One of Che earliest examples of a properly conceived experimental investigation of the flux relations for a porous medium is provided by the work of Gunn and King [53] on the dusty gas model equations, and the following discussion is based largely on their work. Since all their experiments were performed on binary mixtures, the appropriate flux relations are (5.26) and (5,27). Writing... 

In summary, a combination of the plot based on equation (10.6), using any single substance, and determination of the asymptote (10.14), using any pair of substances, provides a sound means of evaluating the parameters K, tC and. Having found these, further experimental points on (10.6) and (10.15), and possibly also (10.7), provide a check on the adequacy of the dusty gas model. Provided attention is limited to binary mixtures, this check can be quite comprehensive. In their published paper Gunn and King... 

Though the solution procedure sounds straightforward, if tedious, practice difficulty is encountered immediately because of the implicit nature of the available flux models. As we saw in Chapter 5 even the si lest of these, the dusty gas model, has solutions which are too cumbersc to be written down for more than three components, while the ternary sol tion itself is already very complicated. It is only for binary mixtures therefore, that the explicit formulation and solution of equations (11. Is practicable. In systems with more than two components, we rely on... 

Hugo s approach can be extended without difficulty to apply throughout the whole range of pore sizes, but to accomplish this a specific and complete flux model must be used. To be definite we will assume that the dusty gas model is adequate, but the same reasoning could be applied to certain other models if necessary. The relevant flux relations are now equations (5.4). Applied to the radial flux components In one of our three simple geometries they take the form... 

T-Jhile the stoichiometric relations have rendered the above problem tractable by permitting an explicit solution of the dusty gas model flux relations, it should be pointed out that they do not lead to equally radical simplifications with all flux models. In the case of the Feng and Stewart models [49- for example, Che total flux of species r is formed by in-... 

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 ... 

Hite s treatment is based on equations (5.18) and (5.19) which describe the dusty gas model at the limit of bulk diffusion control and high permeability. Since temperature Is assumed constant, partial pressures are proportional to concentrations, and it is convenient to replace p by cRT, when the flux equations become... 

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... 

Apart from the Knudsen limit equations (12.12), the only other reasonably compact solution of the dusty gas model equations is that given by equations (5.26) and (5.27), corresponding to binary mixtures. Consequently, if we are to study anything other than the Knudsen limit, attention will... 

Equations (12.13) and (12.14) are themselves approximations, since in Chapter 3 we eliminated thermal transpiration and diffusion from consideration before constructing the dusty gas model flux relations. See Appendix 1. 

When developing the dusty gas model flux relations in Chapter 3, the thermal diffusion contributions to the flux vectors, defined by equations (3.2), were omitted. The effect of retaining these terms is to augment the final flux relations (5.4) by terms proportional to the temperature gradient. Specifically, equations (5.4) are replaced by the following generalization... 

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