In porous media diffusion

As for Illustrative Example 18.2a (diffusivity of CFC-12 in air), these values agree fairly well with each other, except for the Stokes-Einstein relation, which was not meant to be a quantitative approximation but an expression to show qualitatively the relationship between diffusivity and other properties of both molecule and fluid. 

In this section, we consider a solute or vapor diffusing through fluid-filled pores of a porous medium (note that both liquids and gases are called fluids). There are several reasons why in this case the flux per unit bulk area (that is, per total area occupied by the medium) is different from the flux in a homogeneous fluid or gas system. 

Due to the small dimensions of the channels in porous media, viscous forces usually suppress turbulence. Hence, diffusion through the pore space occurs by molecular motions. If the size of the pores is small, molecular motions are reduced. In gas-filled pores, this is the case if the pore size is similar to or smaller than the 

The following is a short overview of how these effects are quantitatively described. Then we assess their consequences for the two Fickian laws. 

The diffusive flux equation per unit bulk area for chemical i can be written as  

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Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Proposed flux models for porous media invariably contain adjustable parameters whose values must be determined from suitably designed flow or diffusion measurements, and further measurements may be made to test the relative success of different models. This may involve extensive programs of experimentation, and the planning and interpretation of such work forms the topic of Chapter 10, However, there is in addition a relatively small number of experiments of historic importance which establish certain general features of flow and diffusion in porous media. These provide criteria which must be satisfied by any proposed flux model and are therefore of central importance in Che subject. They may be grouped into three classes. 

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. 

Chapter 8. MODELS OF FLOW AND DIFFUSION IN POROUS MEDIA... 

A number of different approaches have been taken to describing transport in porous media. The objective here is not to review all approaches, but to present a framework for comparison of various approaches in order to highlight those of particular interest for analysis of diffusion and electrophoresis in gels and other nanoporous materials. General reviews on the fundamental aspects of experiments and theory of diffusion in porous media are given... 

Saez, AE Perfetti, JC Rusinek, I, Prediction of Effective Diffusivities in Porous Media Using Spatially Periodic Models, Transport in Porous Media 6, 143, 1991. 

Evans, R. B., G. M. Watson and E. A. Mason. 1961. Gaseous diffusion in porous media at uniform pressure. J. Chem. Phys. 35 2076-83. 

Systematic experiments on diffusion in porous media with aim of isolating various modes of transport. 

The thermal movement of molecules often serves as a prototype of random motion. In fact, molecular diffusion is the result of the random walk of atoms and molecules through gaseous, liquid, solid, or mixed media. This section deals with molecular diffusion of organic substances in gases (particularly air) and in aqueous solutions. Diffusion in porous media (i.e., mixes of gases or liquids with solids) and in other media will be discussed in the following section. 

Studies of metal compound diffusion in porous media have consistently demonstrated that the rate of diffusion within the microporous material is less than would be observed in an unrestricted medium. This discrepancy, observed for all liquid diffusion processes in pores of small diameter is related to hydrodynamic phenomena. The proximity of the molecule to the pore wall increases the frictional drag on the diffusing species when the... 

S. Nomura, Y. Yang, C. Inoue and T. Chida, Observation and evaluation of proton diffusion in porous media by the pH-imaging microscopy using a flat semiconductor pH sensor, Anal. Sci., 18 (2002) 1081-1084. 

DIFFUSION IN POROUS MEDIA 5.8.1 Transport Mechanisms in Porous Media... 

For diffusion between soil and sediment particles, n represents the interstitial porosity. For diffusion within soil and sediment particles, n represents the in-traparticle porosity. Values of m close to 1 are common for diffusion in porous media [29-31]. However, in low porosity materials this value can increase [32]. A more thorough treatment of this topic can be found in Grathwohl [33]. 

The ordinary diffusion equations have been presented for the case of a gas in absence of porous medium. However, in a porous medium, whose pores are all wide compared to the mean free path and provided the total pressure gradient is negligible, it is assumed that the fluxes will still satisfy the relationships of Stefan-Maxwell, since intermolecular collisions still dominate over molecule-wall collisions [19]. In the case of diffusion in porous media, the binary diffusivities are usually replaced by effective diffusion coefficients, to yield... 

Feng, Kostrov and Stewart (1974) reported multicomponent diffusion data for gaseous mixtures of helium (He), nitrogen (N2) and methane (CH4) through an extruded platinum-alumina catalyst as functions of pressure (1 to 70 atm), temperature (300 to 390 K), and terminal compositions. The experiments were designed to test several models of diffusion in porous media over the range between Knudsen and continuum diffusion in a commercial catalyst (Sinclair-Engelhard RD-150) with a wide pore-size distribution. 

Many numerical models make additional assumptions, valid if only some specific questions are being asked. For example, if one is not interested in the start-up phase or in changing the operation of a fuel cell, one may apply the steady state condition that time-independent solutions are requested. In certain problems, one may disregard temperature variations, and in the free gas ducts, laminar flow may be imposed. The diffusion in porous media is often approximated by an assumption of isotropy for the gas diffusion or membrane layer, and the coupling to chemical reactions is often simplified or omitted. Water evaporation and condensation, on the other hand, are often a key determinant for the behaviour of a fuel cell and thus have to be modelled at some level. 

Currie, J.A., 1960. Gaseous diffusion in porous media. Brit. J. Appl. Physics, 11 318. 

KeU et al. [46,47] discussed the modeling of diffusion and reaction in porous media. For a successful modeling of these phenomena, a variety of difficult problems must be solved, such as multicomponent diffusion in porous media, surface... 

An important application of multicomponent mass transfer theory that we have not considered in any detail in this text is diffusion in porous media with or without heterogeneous reaction. Such applications can be handled with the dusty gas (Maxwell-Stefan) model in which the porous matrix is taken to be the n + 1th component in the mixture. Readers are referred to monographs by Jackson (1977), Cunningham and Williams (1980), and Mason and Malinauskas (1983) and a review by Burghardt (1986) for further study. Krishna (1993a) has shown the considerable gains that accrue from the use of the Maxwell-Stefan formulation for the description of surface diffusion within porous media. 

M.2 The Dusty Gas Model (DGM) (Mason and Malinauskas, 1983) is commonly used for describing diffusion in porous media. In this model the medium is modeled as giant molecules (dust) held motionless in space (u ugt = 0). Derive the DGM equations using the treatment given in Chapter 2 by taking the medium as the ( + l)th species in the mixture. Compare the results of your derivations with that in Mason and Malinauskas (1983). You may also refer to Wesselingh and Krishna (1990) for more information. 

The bulk diffusion processes within the pores of catalyst particles are usually described by the Wilke model formulation. The extended Wilke equation for diffusion in porous media reads ... 

This expression clearly is not symmetric in the two components. References [17,19-21] contain representative examples of applications of the MG model to diffusion in porous media. 

Geometrical models of flow and diffusion in porous media... 

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