Porous media diffusion transport mechanisms

In order to simplify the situation, we assume that our porous sample under investigation covers the bottom of an open straight-walled can and fills it to a height d (Figure 1). Such a sample will exhibit the same areal exhalation rate as a free semi-infinite sample of thickness 2d, as long as the walls and the bottom of the can are impermeable and non-absorbant for radon. A one-dimensional analysis of the diffusion of radon from the sample is perfectly adequate under these conditions. To idealize the conditions a bit further we assume that diffusion is the only transport mechanism of radon out from the sample, and that this diffusive transport is governed by Fick s first law. Fick s law applied to a porous medium says that the areal exhalation rate is proportional to the (radon) concentration gradient in the pores at the sample-air interface... 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

The processes of advection, diffusion, and mechanical dispersion transport chemical species in fluids. For a porous medium, the flux, F,-, of species i in the x, y, and z coordinate directions (mol m(rock) s ) can be written as... 

Structural models emerge from the notion of membrane as a heterogenous porous medium characterized by a radius distribution of water-filled pores. This structural concept of a water-filled network embedded in the polymer host has already formed the basis for the discussion of proton conductivity mechanisms in previous sections. Its foundations have been discussed in Sect. 8.2.2.1. Clearly, this concept promotes hydraulic permeation (D Arcy flow [80]) as a vital mechanism of water transport, in addition to diffusion. Since larger water contents result in an increased number of pores used for water transport and in larger mean radii of these pores, corresponding D Arcy coefficients are expected to exhibit strong dependencies on w. 

Diffusion, convection, and dispersion all contribute to the spread of a front. Let us see how much each mechanism contributes to the spread. First, let us see when the diffusion transport is important as compared to the convective transport. We use v2Dot to calculate the spreading distance from a point source 68% of the injected source is within this distance. Table 2.2 shows the results for different time periods compared with the traveled distances during the same time periods by a convective flow of 1 m/day. A typical flow rate in petroleum reservoirs is 1 m/day (interstitial velocity). A typical value of diffusion coefficient of 4 X 10 mVs in a porous medium is used. In the first 5 seconds, the diffusive transport is more important than the convective transport. Soon after, the convective flow becomes the dominant mechanism. 

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. 

We have considered separately the necessary flux equations for the cases of Knudsen diffusion and continuum diffusion. Knudsen diffusion usually dominates when the pore size and the pressure are small, and the continuum diffusion dominates when the pore size and pressure are large. In the intermediate case which is usually the case for most practical systems, we would expect that both mechanisms will control the mass transport in a capillary or a porous medium. In this section, we will consider this intermediate case and present the necessary flux equations. 

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. 

We consider the case of linear isotherm between the gas and solid phases. The mass transport into the particle is assumed to occur by two parallel mechanisms pore and surface diffusions. The mass balance equation describing the concentration distribution in a slab porous medium with these two parallel mechanisms is ... 

If the mechanism of transport into the porous medium is the parallel pore and surface diffusion mechanism, the flux equation can be written in terms of the two individual concentration gradients as follows ... 

Besides electrokinetic transport, chemical reactions also occur at the electrode surfaces (i.e., water electrolysis reactions with production of at the anode and OH at the cathode). Common mass-transport mechanisms like diffusion or convection and physical and chemical interactions of the species with the medium also occur. In a low-permeable porous medium under an electrical field, the major transport mechanism through the soil matrix during treatment for nonionic chemical species consists mainly of electro-osmosis, electrophoresis, molecular diffusion, hydrodynamic dispersion (molecular diffusion and dispersion varying with the heterogeneity of soils and fluid velocity [8]), sorption/ desorption, and chemical or biochemical reactions. Since related experiments are conducted in a relatively short period of time, the chemical and biochemical reactions that occur in the soil water are neglected [9]. 

We examine the problem of diffusion in a porous medium using a homogenization analysis (HA). Diffusion problems have important applications in environmental geosciences. We clarily the mechanism of diffusion, convective transport and adsorption in porous media at both the microscale and macroscale levels. Attention is particularly focused on diffusion processes in bentonite, which is an engineered geological barrier to be used to buffer the transport of radionuclides from deep geologic repositories. 

Type I electrodes, the prevailing type, are three-phase composite media that consist of a solid phase of Pt and electronic support material, an electrolyte phase of ionomer and water, and the gas phase in the porous medium. Gas diffusion is the most effective mechanism of reactant supply and water removal. Yet, CLs with sufficient gas porosity, usually in the range Xp 30-60%, have to be made with thickness of Icl — 10 pm. In this thickness range, proton transport cannot be provided outside of the electrolyte environment. Porous gas diffusion electrodes are, therefore, impregnated with proton-conducting ionomer. The concept of a triple-phase boundary, often invoked for such electrodes, is however inadequate. The amount of the electrochemically active interface is usually controlled by two-phase boundary effects at the interface between Pt and water. 

FIGURE 1 9 Fickian transport by mechanical dispersion as water flows through a porous medium such as a soil. Seemingly random variations in the velocity of different parcels of water are caused by the tortuous and variable routes water must follow. This situation contrasts with that of Fig. 1.8, in which turbulence is responsible for the variability of fluid paths. Nevertheless, as in the case of turbulent diffusion, mass transport by mechanical dispersion is proportional to the concentration gradient and can be described by Fick s first law. 

Modern gas-diffusion medium in low-temperature fuel cells is typically a highly porous carbon paper with porosity in the range of sgdl = 0.6-0.8 and with the mean pore radius in the order of 10 pm (10 cm). By the order of magnitude, the mean free path of molecules in atmospheric pressure air is = l/(A LO-fci ), where Nl = 2.686 10 cm is the Loschmidt number (number of molecules in a cubic centimetre of atmospheric pressure gas at standard temperature) and akin — 10 cm is the molecular cross-section for kinetic collisions. With this data we get 3 10 cm, or 3 10 pm. Obviously, mean pore radius in the GDL is nearly 3 orders of magnitude greater than I f and the physical mechanism of molecule transport is binary molecular diffusion. 

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