Mean transport-pore model

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,... 

Jackson discusses generalizations of the DGM, accounting for the details of the porous structure [19]. Hie Mean Transport Pore Model is also relevant for the present discussion [23]. 

A variety of diffusion cells have been developed for transient measurements. The experimental arrangements and the data analysis methods for the determination of effective diffusivities are described in reviews [1,2] and in the papers cited therein. Interesting applications of a diffusion cell with one compartment closed have been described recently for the investigation of the dynamics of ternary gas mixtures [15,16]. In the last papers the DGM and the mean transport pore model have been used to describe the experiments [17]. 

The obtained transport parameters for the Mean-Transport-Pore-Model (it/, it/) were compared with transport parameters obtained from multicomponent counter-current diffusion in Graham s diffusion cell and from mercury porosimetry pore-size distributions. 

At present two models are available for description of pore-transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[4,5] and the Dusty Gas Model (DGM)[6,7]. Both models permit combination of multicomponent transport steps with other rate processes, which proceed simultaneously (catalytic reaction, gas-solid reaction, adsorption, etc). These models are based on the modified Maxwell-Stefan constitutive equation for multicomponent diffusion in pores. One of the experimentally performed transport processes, which can be used for evaluation of transport parameters, is diffusion of simple gases through porous particles packed in a chromatographic column. 

The Mean Transport Pore Model (MTPM) described diffusion and permeation the model (represented as a boundary value problem for a set of ordinary differential equations) are based on Maxwell-Stefan diffusion equation and Weber permeation law. Parameters of MTPM are material constants of the porous solid and, thus, do not dependent on conditions under which the transport proeesses take place. 

The Dusty Gas Model and the Mean Transport Pore Model described in the following sections are basically extensions of MMS model with pressure gradients. 

Different theoretical models have been proposed to estimate the effective dif-fusivity Dwj- The most commonly used approximations are the mean transport pore model [18] and the random pore model [19]. 

The mean transport pore model uses the expression ... 

It should be mentioned that the mean transport pore model is probably not the best option for SCR catalyst layers, as it assumes a uniform pore size distribution. The actual pore size distribution of the SCR layers is highly bimodal with two distinct maxima micropores and mesopores. The random pore model considers a bi-dispersive washcoat material with two characteristic pore sizes with their respective mean pore sizes and void fractions. The total dififusivity is calculated as a combination of the respective Knudsen dififusivities ... 

Two standard methods (mercury porosimetry and helium pycnometry) together with liquid expulsion permporometry (that takes into account only flow-through pores) were used for determination of textural properties. Pore structure characteristics relevant to transport processes were evaluated fiom multicomponent gas counter-current difhision and gas permeation. For data analysis the Mean Transport-Pore Model (MTPM) based on Maxwell-Stefan diffusion equation and a simplified form of the Weber permeation equation was used. 

Keywords Counter-current gas diffusion. Permeation, Transport parameters. Mean Transport-Pore Model, Maxwell-Stefrn equation, Weber equation... 

As a model we have used the Mean Transport-Pore Model (MTPM) [6] which assumes that the decisive part of the gas transport takes place in transport-pores that are visualized as cylindrical capillaries. The transport-pore radii are distributed around the mean value (first model parameter). The width of this distribution is characterized hy the mean value of the squared transport-pore radii, (second model parameter). The third model parameter is the ratio of porosity, y/i, and tortuosity of transport-pores, qt, q/= Pore diffusion is described by the Maxwell-Stefan diffusion equation extended to account for Knudsen transport [6]. For gas permeation the simplified form of Weber equation [8-10] is used. 

MTPM assumes that the decisive part of the gas transport takes place in transport-pores that are visualised as cylindrical capillaries with radii distributed around the mean value (first model parameter). The second model parameter can be looked upon as ratio of tortuosity, q, and porosity of transport-pores, S, = St t- The third transport parameter,... 

The fine-pore model was developed assuming the presence of open micropores on the active surface layer of the membrane through which the mass transport occurs (10). The existence of these different pore geometries also means that different models have been developed to describe transport adequately. The simplest representation is one in which the membrane is considered as a number of parallel cylindrical pores perpendicular to the membrane surface. The length of each of the cylindrical pores is equal or almost equal to the membrane thickness. The volume flux through these pores can be described by the Hagen-Poiseuille equation. Assuming all the pores have the same radius, then we have... 

Models that are used to predict transport of chemicals in soil can be grouped into two main categories those based on an assumed or empirical distribution of pore water velocities, and those derived from a particular geometric representation of the pore space. Velocity-based models are currently the most widely used predictive tools. However, they are unsatisfactory because their parameters generally cannot be measured independently and often depend upon the scale at which the transport experiment is conducted. The focus of this chapter is on pore geometry models for chemical transport. These models are not widely used today. However, recent advances in the characterization of complex pore structures means that they could provide an alternative to velocity based-models in the future. They are particularly attractive because their input parameters can be estimated from independent measurements of pore characteristics. They may also provide a method of inversely estimating pore characteristics from solute transport experiments. 

The above mentioned parameters influence the degree of external wetting, liquid holdup and transport and reaction mechanisms. The couplings between the extent of pore emptying, partial external wetting, transport and reaction can be assessed by means of the model previously developed by Gabarain et al (1996). 

Let us examine the process of diffusion in clays at the first stage of their consolidation, before the moment when m = 0. We will suppose that the diffusion solution contains cations of the same type, as cations of the exchanging complex of clay meaning that there is no ion exchange reaction. We will also suppose that in every point of environment the equilibrium between the solution in transport pores and the solution between clay particles is established immediately (its parameters we will be marked by the overline). Like this we will successively build the model of diffusion in clays in a local equilibrium approach. The conditions of equilibrium of cations (index 1) and anions (index 2) of two solutions are the equality of the chemical potentials, =. Pi= where... 

UF and RO models may all apply to some extent to NF. Charge, however, appears to play a more important role than for other pressure driven membrane processes. The Extended-Nemst Planck Equation (equation (3.28)) is a means of describing NF behaviour. The extended Nernst Planck equation, proposed by Deen et al. (1980), includes the Donnan expression, which describes the partitioning of solutes between solution and membrane. The model can be used to calculate an effective pore size (which does not necessarily mean that pores exist), and to determine thickness and effective charge of the membrane. This information can then be used to predict the separation of mixtures (Bowen and Mukhtar (1996)). No assumptions regarding membrane morphology ate required (Peeters (1997)). The terms represent transport due to diffusion, electric field gradient and convection respectively. Jsi is the flux of an ion i, Di,i> is the ion diffusivity in the membane, R the gas constant, F the Faraday constant, y the electrical potential and Ki,c the convective hindrance factor in the membrane. 

The simple carrier of Fig. 6 is the simplest model which can account for the range of experimental data commonly found for transport systems. Yet surprisingly, it is not the model that is conventionally used in transport studies. The most commonly used model is some or other form of Fig. 7. In contrast to the simple carrier, the model of Fig. 7, the conventional carrier, assumes that there exist two forms of the carrier-substrate complex, ES, and ES2, and that these can interconvert by the transitions with rate constants g, and g2- Now, our experience with the simple- and complex-pore models should lead to an awareness of the problems in making such an assumption. The transition between ES, and ES2 is precisely such a transition as cannot be identified by steady-state experiments, if the carrier can complex with only one species of transportable substrate. Lieb and Stein [2] have worked out the full kinetic analysis of the conventional carrier model. The derived unidirectional flux equation is exactly equivalent to that derived for the simple carrier Eqn. 30, although the experimentally determinable parameters involving K and R terms have different meanings in terms of the rate constants (the b, /, g and k terms). The appropriate values for the K and R terms in terms of the rate constants are listed in column 3 of Table 3. Thus the simple carrier and the conventional carrier behave identically in... 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

Mass transport inside the catalyst has been usually described by applying the Fick equation, by means of an effective diffusivity Deff a Based on properties of the interface and neglecting the composition effect, composite diffusivity of the multi-component gas mixture is calculated through the simplified Wilke model [13], The effect of pore-radius distribution on Knudsen diffusivity is taken into account. The effective diffusivity DeffA is given by... 

After, the essential features of a mechanical model of adsorption and diffusion to characterize, e.g., the transport of a contaminant with rainwater through the soil will be outlined in particular, the model consists of a fluid carrier of an adsorbate, the adsorbate in the liquid state and an elastic skeleton with ellipsoidal microstructure it means that each pore has different microdeformation along principal axes, namely a pure strain, but rotates locally with the matrix of the material (see [5, 6]). 

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