Porous media diffusion models

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

Today two models are available for description of combined (diffusion and permeation) transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[21,22] and the Dusty Gas Model (DGM)[23,24]. Both models enable in future to connect multicomponent process simultaneously with process as catalytic reaction, gas-solid reaction or adsorption to porous medium. These models are based on the modified Stefan-Maxwell description of multicomponent diffusion in pores and on Darcy (DGM) or Weber (MTPM) equation for permeation. For mass transport due to composition differences (i.e. pure diffusion) both models are represented by an identical set of differential equation with two parameters (transport parameters) which characterise the pore structure. Because both models drastically simplify the real pore structure the transport parameters have to be determined experimentally. 

The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes simply combine additively with the diffusion fluxes in the gaseous phase. 

Though a porous medium may be described adequately under non-reactive conditions by a smooth field type of diffusion model, such as one of the Feng and Stewart models, it does not necessarily follow that this will still be the case when a chemical reaction is catalysed at the solid surface. In these circumstances the smooth field assumption may not lead to appropriate expressions for concentration gradients, particularly in the smaller pores. Though the reason for this is quite simple, it appears to have been largely overlooked,... 

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... 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

The simple pore structure shown in Figure 2.69 allows the use of some simplified models for mass transfer in the porous medium coupled with chemical reaction kinetics. An overview of corresponding modeling approaches is given in [194]. The reaction-diffusion dynamics inside a pore can be approximated by a one-dimensional equation... 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

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 constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

Model a process of diffusion and chemical reaction A -> B inside a porous catalyst. Assume that the reaction takes place on the catalytic surface within the porous medium. 

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

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. 

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. 

For the tortuous and irregular capillaries of porous media, it has been reported theoretically and experimentally that a minimum in the permeability of adsorbates at low pressures is not expected to appear. In our study of n-hexane in activated carbon, however, a minimum was consistently observed for n-hexane at a relative pressure of about 0.03, while benzene and CCI4 show a monotonically increasing behavior of the permeability versus pressure. Such an observation suggests that the existence of the minimum depends on the properties of permeating vapors as well as the porous medium. In this paper a permeation model is presented to describe the minimum with an introduction of a collision-reflection factor. Surface diffusion permeability is found to increase sharply at very low pressure, then decrease modestly with an increase in pressure. As a result, the appearance of a minimum in permeability was found to be controlled by the interplay between Knudsen diffusion and surface diffusion for each adsorbate at low pressures. 

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 DGM has two independent parts, a diffusion part, consisting of a set of Maxwell-Stefan diffusion equations and a viscous-flow part consisting of an equation of motion. The structure of the porous medium is treated as a completely independent problem. Although the DGM model is not an exact approach and it... 

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... 

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