Porous matrix model

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

Migration of fluids in a porous matrix with solid-liquid fractionation results in a process much similar to the chromatographic separation of elements (DeVault, 1943 Korzhinskii, 1970, Hofmann, 1972). This mechanism has recently been revived in the context of mantle metasomatism by Navon and Stolper (1987), Bodinier et al. (1990), Vasseur et al. (1991), in the context of hydrothermal systems by Lichtner (1985) and, for stable isotopes, by Baumgartner and Rumble (1988). Only a simplified account of this model will be given here. Let

solid matrix and melt, respectively, and vHq the fluid velocity relative... 

On the other hand, model calculations varying parameters, show clearly that the smaller diameter capillaries, representative of the porous matrix, do play a controlling role In the separation factor behavior. 

Rate equation analyses for classical size exclusion chromatography have been based on treating the porous matrix as a homogeneous, spherical medium within which radial diffusion of the macromolecular solute takes place (e.g. (28,30,31)) or If mobile phase lateral dispersion Is considered Important, a two dimensional channel has been used as a model for the bed (32). In either case, however, no treatment of the effects to be expected with charged Brownian solute particles has been presented. As a... 

The regions of the polymer domain that are not occupied by either the framework or bound molecules will contain water and counterions to the charged residues. It is evident from the model that the polymer forms a very open, porous matrix into which water molecules have easy access. 

Abstract A simplified quintuple model for the description of freezing and thawing processes in gas and liquid saturated porous materials is investigated by using a continuum mechanical approach based on the Theory of Porous Media (TPM). The porous solid consists of two phases, namely a granular or structured porous matrix and an ice phase. The liquid phase is divided in bulk water in the macro pores and gel water in the micro pores. In contrast to the bulk water the gel water is substantially affected by the surface of the solid. This phenomenon is already apparent by the fact that this water is frozen by homogeneous nucleation. 

Taking into account the aforementioned effects of ice formation in porous materials, a macroscopic quintuple model within the framework of the Theory of Porous Media (TPM) for the numerical simulation of initial and boundary value problems of freezing and thawing processes in saturated porous materials will be investigated. The porous solid is made up of a granular or structured porous matrix (a = S) and ice (a = I), where it will be assumed that both phases have the same motion. Due to the different freezing points of water in the macro and micro pores, the liquid will be distinguished into bulk water ( a = L) in the macro pores and gel water (a = P, pore solution) in the micro pores. With exception of the gas phase (a = G), all constituents will be considered as incompressible. 

In this paper we consider a pore scale model for crystal dissolution and precipitation processes. We follow the ideas in [3], where the corresponding macroscopic model was introduced. Let Q C (d > 1) denote the void region. This region is occupied by a fluid in which cations (Mi) and anions (M2) are dissolved. The boundary of Q has an internal part (IV ), which is the surface between the fluid and the porous matrix (grains), and an external part, which is the outer boundary of the domain. In a precipitation reaction,... 

In the grain model, it is assumed that the CaO consists of spherical grains of uniform size distributed in a porous matrix. The rate of reaction is controlled by the diffusion of SO2 through the porous matrix and the product CaSO layer formed on each grain (11). Allowance can be made for a finite rate of the CaO/SC reaction (12). The models have been found to describe experimental data for many limestones (13) by adjusting the constants in the model, most notably the diffusivity through the product layer. 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

The essence of the above model is the assumption that pore segments of radius r in which the liquid is above the hysteresis temperature rH(/ i) cannot cause delayed desorption. This assumption is immediately plausible if 7h were to coincide with the pore critical temperature, as in this case the pore fluid is in a supercritical state above Tn, and thus the mass transport is not retarded by a gas/liquid meniscus. Some aspects of our model are remeniscent of the tensile strength hypothesis, although the concept of a pore critical temperature was not discussed at the time when that hypothesis was proposed. On the other hand, the present picture does not imply that the locus of lower closure points (p// o)L should be independent of the nature of the porous matrix. We conjecture that (/>/ o)l is given by the locus of pore hysteresis points of the fluid in open-pore systems of uniform pore size. A more comprehensive discussion of this model will be presented elsewhere."... 

Membranes have also been used in reactors where their permselective properties are not important. Instead their well-engineered porous matrix provides a well-controlled catalytic zone for those reactions requiring strict stoichiomeuic feed rates of reactants or a clear interface for multiphase reactions (e.g., a gas and a liquid reactant fed from opposing sides of the membrane). Functional models for these types of membrane reactors have also been developed. The conditions under which these reactors provide performance advantages have been identified. 

To illustrate the principles of the shrinking core model, we shall consider the removal of carbon from the catalyst particle just discussed. In Figure 11-15 a core of unreacted carbon is contained between r = 0 and r = R. Carbon has been removed from the porous matrix between r = R arid r = R. Oxygen diffuses from the outer radius Ro to the radius R, where it reacts with carbon to form carbon dioxide, which then diffuses out of the porous matrix. The reaction... 

Model templated structures can be assembled from Monte Carlo simulations of binary mixtures of matrix and template particles [55-57]. Upon removal of the template from the quenched equihbrated structure, a porous matrix is recovered with an enhanced accessible void volume for adsorption. GCMC simulation studies have established that the largest enhancement of adsorption uptake occurs when the template particles used to fashion the porous matrix are the same size as the adsorbate molecules for which the adsorbent is intended [55]. The enhanced adsorption capacity of the templated material relative to a nontemplated matrix is noticeable even for modest template particle densities [55]. 

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. 

If one then fixes the thermodynamic state such that the bulk mixture is a gas [represented by in the inset in Fig, 4.15.(a)], confinement to a relatively wide pore (i.e., z — 12) may first cause capillary condensation to a mixed liquid mixture analogous to ordinary capillary condensation in pure fluids. If the fluid is confined to a narrower pore (z = 6), however, decomposition into A-rich and B-rieh liquid phases is triggered by confinement upon condensar tioii. Thus, by choosing an appropriate pore width, one can either promote condensation of a gas to a mixed liquid phase or, alternatively, initiate liquid liquid phase separation in the porous matrix where both processes are solely confinement-driven because the pore walls are nonselective for molecules of either species in our present model. 

Bobba, A.G. 1989. Numerical model of contaminant transport through conduit-porous matrix system. Math. Geol. 21 861-890. 

In this model the porous media is assumed to be made of a porous matrix of complex interconnected capillary network structure. Flux relations are obtained using momentum transfer laws and kinetic theory of gases in a single pore (single capillary) for different types of transport modes. The problem of the geometrical structure is overcome in two ways ... 

As in the case of hydrophilic (swelling) matrix systems and reservoir (membrane) systems, drug release profiles from insoluble (porous or non porous) systems are most of the time described on a basis of the diffusion theory. This is not true for every situation and we have for example shown that the release from porous matrix systems is dissolution-controlled above the solubility limit of the drug [150]. A simplified equation for the model proposed and for values of kd t > 4, is ... 

A different type of a multiple membrane reactor system was proposed and modeled by Kim and Datta [5.68]. Their membrane reactor consists of liquid-phase catalytic layer supported on a porous matrix, which is sandwiched in between two different membranes. They considered a simple irreversible A- B reaction. The membrane, which is in contact... 

Our model about the porous matrix includes, as a particular case, previous void theories which does not anticipate any size effects in torsion on the other hand the virtual inertia considered in our immiscible mixture allows the presence of an inviscid drag term depending on changes of the radius of the material elements, other than on the relative velocities, a term which is usually absent in previous theories (see Giovine (1990)). 

In certain papers (Aroutiounian and Ghulinyan, 2000 Aroutiounian et al., 2000), a fractal model of a porous layer formation was proposed.The consideration of the time-dependent pore growth process has allowed us to calculate important parameters of the porous matrix, such as the formed surface area, and the surface and volume porosity values. We have theoretically shown that the formed surface area is strongly dependent on the difference between the pore size growth velocities parallel and perpendicular to the surface (i.e. the crystallographic orientation of the silicon surface). The volume and surface porosity values and the formed porous surface area are linear functions of the density of the anodization current. These results are in agreement with other theoretical and experimental data. 

At D k, the reaction zone extends across the whole transverse section of the polymer film. Therefore, regulation of the ratio between various parameters (viscosity of solution, temperature, concentration of reagents, etc.), makes it possible to obtain materials with different model schemes. It is possible to obtain particles of different chemical content by the reduction of metal ions depending on the nature of the polymer matrix. For example, in a swelling matrix (PVS, cellulose, etc), copper oxide is formed by reduction of Cu, while copper(O) forms mainly in a porous matrix (PE, PTFE). 

The mechanical behavior of porous-matrix composites has been modeled, but the compromises between matrix properties and toughness are not yet fully quantified. Initial estimates indicated that a matrix porosity of 30% is needed for crack deflection within a porous matrix [4, 43]. Subsequent efforts have been aimed at better quantifying the transition between porous and dense CMC failure behavior [44] however, further investigation is essential, especially considering the temperature dependent nature of the porous microstructure. 

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