Porous media diffusion coefficient

In the porous medium, diffusion is affected by the porosity and tortuosity of the medium itself therefore Knudsen diffusion is computed as well as the ordinary diffusion. Eventually, an effective diffusion coefficient is calculated that depends on the ordinary and Knudsen diffusion coefficients and on the ratio between porosity and tortuosity of the medium (Equation (3.58)). 

These must supplement the minimal set of experiments needed to determine the available parameters in the model-It should be emphasized here, and will be re-emphasized later, Chat it is important Co direct experiments of type (i) to determining Che available parameters of some specific model of Che porous medium. Much confusion has arisen in the past frcjci results reported simply as "effective diffusion coefficients", which cannot be extrapolated with any certainty to predict... 

Effective diffusion coefficient, in porous medium at bulk diffusion limit, 14... 

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

Diw is the molecular diffusion coefficient of the chemical in water, x is tortuosity, and aL is the (longitudinal) dispersivity (dimension L). The first term describes molecular diffusion in a porous medium (Eq. 18-57), the second the effect of dispersion (Eq. 22-52). Typical values of the dispersivity aL for field systems with flow distances of up to about 100 m lie between 1 and 100 m. Since aL depends strongly on the scale... 

Here, the length L in (7.38) has been replaced by porous layer thickness d and the surface area Aeff. The effective diffusion coefficient D0,eff characterizes the transport through porous medium and includes both regular diffusion and the Knudsen diffusion coefficient >o,Kn, which has a different temperature dependence from diffusion in bulk. 

Dj is the mass diffusion coefficient, and cgas is the total molar concentration of the gas mixture. Although Equations (3.9a) and (3.9b) can be used for a free-path gas (e.g. gas channel), when a gas is moving within a porous media (i.e. electrode), Equation (3.9) may not be the most appropriate. Different constitutive laws can be employed for describing the diffusive flux within a porous medium. The choice of the most appropriate law depends on the operating conditions and the porous media properties, as further explained in Section 3.3.2. 

Dispersion arises from the fact that, even in a relatively homogenous porous medium, small-scale heterogeneities exist which cause airflow to proceed along various channels at different rates. Barometric pumping causes a significant increase in the coefficient of hydrodynamic dispersion over a pure diffusion-based transport model, thus increasing the overall transport rate. 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

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... 

A good agreement is generally obtained between the models based on transport equations and the SDE for mass and heat molecular transport. However, as explained above, the SDE can only be applied when convective flow does not take place. This restrictive condition limits the application of SDE to the transport in a porous solid medium where there is no convective flow by a concentration gradient. The starting point for the transformation of a molecular transport equation into a SDE system is Eq. (4.108). Indeed, we can consider the absence of convective flow in a non-steady state one-directional transport, together with a diffusion coefficient depending on the concentration of the transported property... 

In the scientific literature, we can find a large quantity of experimental results where the flow characterization inside a porous medium has shown that the value of the dispersion coefficient is not constant. Indeed, for the majority of porous structures the diffusion is frequently a function of the time or of the concentration of the diffusing species. As far as simple stochastic models cannot cover these situations, more complex models have been built to characterize these dependences. One of the first models that gives a response to this problem is recognized as the modd of motion with states having multiple vdodties. 

Here is the molecular diffusion coefficient of the pair C-T and K. 3 (2/3)V(8RgT/7tMT) is the Knudsen constant for the tracer T, Rg is the gas constant, T temperature, and Mt the tracer molecular weight. v t and vi/ are parameters characterizing the porous medium (transport parameters). stands for the integral mean radius of pores through which the... 

To obtain numerically the mass transfer coefficient, a porous medium is stochastically constructed in the form of a sphere pack. Specifically, the representation of the biphasic domains under consideration is achieved by the random deposition of spheres of radius Rina box of length L. The structure is digitized and the phase function (equal to zero for solid and unity for the pore space) is determined in order to obtain the porosity and to solve numerically the convection- diffusion problem. The next for this purpose is to obtain the detailed flow field in the porous domain through the solution of the Stokes equations ... 

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. 

The effective binary pair diffusion coefficient (T>C) in a porous medium is given by... 

In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness 5. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for S, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation, at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to... 

It is emphasized that in this description the gas molecule-dust interactions are accounted for by a set of effective Knudsen diffusion coefficients. Therefore, the dusty gas model contains two types of effective binary diffusion coefficients, the effective binary pair diffusivities in the porous medium and the effective Knudsen diffusivities. The effective diffusivities are related to the corresponding molecular diffusivities by ... 

Diffusion-based methods such as DRYCLEAN are also applicable to cases where the solute of interest, although not having an inherenffy small diffusion coefficient, has an effectively small diffusion coefficient by virtue of its environment (i.e. restricted diffusion such that would occur if the solute were in the intracellular matrix). For example, the intracellular solutes of perfusing cells were able to be selectively observed whilst the extracellular water and solutes in the surrounding medium were suppressed using the stimulated-echo sequence. In a later work Potter et al used the same idea to suppress extracellular water in bacterial suspensions in porous media. 

This section discusses diffusion coefficients in a bulk phase and a porous medium. It also briefly introduces a statistical representation of diffusion. Diffusion is less significant in reservoir flow than dispersion and their mechanisms are different, but the discussion of diffusion provides an analog to the formulation of dispersion. 

Equation 2.1 defines the flux in a bulk liquid or gas phase (with unit porosity or in a straight capillary). The effect of tortuous paths has to be considered in a porous medium. We use the effective diffusion coefficient, D, to replace Do in Eq. 2.1 to consider the effect of tortuosity. The relationship between D, and Do may be defined (Childs, 1969) using Eq. 2.4,... 

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. 

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