Porous media diffusion Knudsen flow

For a given pressure gradient across a porous medium, the mass balance equation can be described as follows, provided that Knudsen diffusion, viscous flow and surface diffusion are additive to the total flux. 

In the DGM model as presented by Mason and Malinauskas [11a] all the different contributions to the transport are taken into accoimt. The wall of the porous medium is considered as a very heavy component and so contributes to the momentum transfer. The model is schematically represented in Fig. 9.12 for a binary mixture (in analogy with an electriccd network). As can be seen from this figure, the flux contributions by Knudsen diffusion /k, and of molecular (continuum) diffusion of the mixture /m,i23re in series and so are coupled. The total flux of component i (i = 1,2) due to these contributions is /j km- Note that /k = /m,i2- The contribution of the viscous flow and of the surface diffusion are parallel with / km J d so are considered independent of each other (no coupling terms, e.g. no transport interaction between gas phase and surface diffusion). 

The same set of transport mechanisms learnt in Chapter 7 is again considered in Chapter 8, but is dealt with in the framework of Maxwell-Stefan. This is the cornerstone in dealing with multicomponent diffusion in homogeneous media as well as heterogeneous media. We first address this framework to a homogeneous medium so that readers can grasp the concept of friction put forwards by Maxwell and Stefan in dealing with multicomponent systems. Next, we deal with diffusion of a multicomponent mixture in a capillary and a porous medium where continuum diffusion, Knudsen diffusion as well as viscous flow can all play an important role in the transport of molecules. 

As we have discussed in the introduction, there are basically four modes of transport of molecules inside a porous medium. They are free molecular diffusion (Knudsen), viscous flow, continuum diffusion and surface diffusion. 

We have discussed so far the Knudsen diffusivity and Knudsen flux for capillary as well as for porous medium. We have said that the flow of one species by the Knudsen mechanism is independent of that of the other species. The question now is in a constant total pressure system, is there a relationship that relates fluxes of all the species when the partial pressures are constrained by the constant total pressure condition. Let us now address this issue. 

In this chapter, we will re-examine these processes, but from the approach developed by Maxwell and Stefan. This approach basically involves the concept of force and friction between molecules of different types. It is from this frictional concept that the diffusion coefficient naturally arises as we shall see. We first present the diffusion of a homogeneous mixture to give the reader a good grasp of the Maxwell-Stefan approach, then later account for diffusion in a porous medium where the Knudsen diffusion as well as the viscous flow play a part in the transport process. Readers should refer to Jackson (1977) and Taylor and Krishna (1994) for more exposure to this Maxwell-Stefan approach. 

In the last sections, you have learnt about the basic analysis of bulk flow, bulk flow and Knudsen flow using the Stefan-Maxwell approach. Very often when we deal with diffusion and adsorption system, the total pressure changes with time as well as with distance within a particle due to either the nonequimolar diffusion or loss of mass from the gas phase as a result of adsorption onto the surface of the particle. When such situations happen, there will be an additional mechanism for mass transfer the viscous flow. This section will deal with the general case where bulk diffusion, Knudsen diffusion and viscous flow occur simultaneously within a porous medium (Jackson, 1977). 

We have shown the essential features of the time lag in Section 12.2 using the simple Knudsen diffusion as an example, and a direct method of obtaining the time lag in Section 12.3. The diffusion coefficient dealt with in the Frisch s method in Section 12.3 is concentration dependent. In this section we will deal with a case where the transport through the porous medium is a combination of the Knudsen diffusion and the viscous flow mechanism. We shall see below that this case will result in an apparent diffusion coefficient which is concentration dependent, and hence it is susceptible to the Frisch s analysis as outlined in the Section 12.3. This means that the results of equations (12.3-21) are directly applicable to this case. 

The steady state Wicke-Kallabach method is usually conducted with a binary system with an aim of determining the binary diffusivity and Knudsen diffusivity in the porous medium. In this binary system, one gas (A) is flowing into and out of one chamber, and the other gas (B) is flowing into and out of the other chamber. 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

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