Continuous Porous Model

In the preceding sections the impedance of a single pore or the distribution of simple pores of known geometry was treated. However, in real cases of porous electrodes used in, for example, lithium or hydrogen storage batteries or fuel cells, one does not 

At the interface solution-electrode at x = 0 there is only an ionic current, and the total current I = i2, the electronic current = 0, and the potential in solution 1)2 = 0 (by definition). On the other hand, at the electrode-solid contact interface at X = / there is only an electTMiic current, I = i,ionic current, j 2 = 0, and the potential in the electrode is As x increases from zero to /, the ionic ciurent decreases and the electronic current increases, and inside the porous electrode ij + 12 = /. In this model, a continuous variation of the potential I i in the solid and 2 in the solution is assumed. 

The impedances may be evaluated by applying a small ac perturbation that generates a small perturbations of the parameters around their dc values, adding the charging current [456, 457], 

This method was also applied by other authors to hydrogen absorption in a porous electrode consisting of spherical particles of ABs-type material [458] to describe a whole polymer electrolyte fuel cell with gas diffusion in pores [459], alkaline fuel cells [460], and intercalation electrodes [461]. Several authors attempted to fit the experimental impedances to their models [456-461]. 

It is evident that a single-pore theory cannot describe more complex cases of real porous electrodes containing particles of different sizes that are randomly distributed. Nevertheless, numerical simulations demand some additional information about the electrode, particle size distribution, possible heterogeneity of the material, conductivity of different phases, etc., and the electrode studied should be composed of a uniform packing of the particles with their size being much smaller than the electrode thickness. 

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In the industrial applications of electrochemistiy, the use of smooth surfaces is impractical and the electrodes must possess a large real surface area in order to increase the total current per unit of geometric surface area. For that reason porous electrodes are usually used, for example, in industrial electrolysis, fuel cells, batteries, and supercapacitors [400]. Porous siufaces are different from rough surfaces in the depth, /, and diameter, r, of pores for porous electrodes the ratio Hr is very important. Characterization of porous electrodes can supply information about their real surface area and electrochemical utilization. These factors are important in their design, and it makes no sense to design pores that are too long and that are impenetrable by a current. Impedance studies provide simple tools to characterize such materials. Initially, an electrode model was developed by several authors for dc response of porous electrodes [401-406]. Such solutions must be known first to be able to develop the ac response. In what follows, porous electrode response for ideally polarizable electrodes will be presented, followed by a response in the presence of redox processes. Finally, more elaborate models involving pore size distribution and continuous porous models will be presented. 

There are, however, two broad classes of exceptions to this conclusion. The first comes with the slow reaction of a gas with a very porous solid. Here reaction can occur throughout the solid, in which situation the continuous reaction model may be expected to better fit reality. An example of this is the slow poisoning of a catalyst pellet, a situation treated in Chapter 21. 

Two types of model have been used to describe the charge transport through electroactive films continuous models, considering ionic transport in a compact film based on the Nernst-Planck equations, and porous models, whose transport is described by transmission line equivalent circuits. 

Any interpretation of the Type I isotherm must account for the fact that the uptake does not increase continuously as in the Type II isotherm, but comes to a limiting value manifested in the plateau BC (Fig. 4.1). According to the earlier, classical view, this limit exists because the pores are so narrow that they cannot accommodate more than a single molecular layer on their walls the plateau thus corresponds to the completion of the monolayer. The shape of the isotherm was explained in terms of the Langmuir model, even though this had initially been set up for an open surface, i.e. a non-porous solid. The Type I isotherm was therefore assumed to conform to the Langmuir equation already referred to, viz. 

Recently, many experiments have been performed on the structure and dynamics of liquids in porous glasses [175-190]. These studies are difficult to interpret because of the inhomogeneity of the sample. Simulations of water in a cylindrical cavity inside a block of hydrophilic Vycor glass have recently been performed [24,191,192] to facilitate the analysis of experimental results. Water molecules interact with Vycor atoms, using an empirical potential model which consists of (12-6) Lennard-Jones and Coulomb interactions. All atoms in the Vycor block are immobile. For details see Ref. 191. We have simulated samples at room temperature, which are filled with water to between 19 and 96 percent of the maximum possible amount. Because of the hydrophilicity of the glass, water molecules cover the surface already in nearly empty pores no molecules are found in the pore center in this case, although the density distribution is rather wide. When the amount of water increases, the center of the pore fills. Only in the case of 96 percent filling, a continuous aqueous phase without a cavity in the center of the pore is observed. 

The models derived for continuous oxide layers remain valuable when porous oxides are formed they provide a frame of reference against which deviations may be examined and give a basis for understanding the factors governing the location of new oxide. In many cases, however, the experimentally derived rate laws no longer have a unique interpretation. For example, the linear rate law relating the thickness of oxide, x, to the time, t... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The search for models of biological membranes among porous membranes continued in the twenties and thirties. Here, Michaelis [67] and Sollner (for a summary of his work, see [90] for development in the field, [89]) should be mentioned. The existence and characteristics of Donnan membrane equilibria could be confirmed using this type of membrane [20]. The theory of porous membranes with fixed charges of a certain sign was developed by Teorell [93], and Meyer and Sievers [65]. 

Berkowitz B, Emmanuel S, Scher H (2008) Non-Fickian transport and multiple rate mass transfer in porous media Water Resour Res 44, D01 10.1029/2007WR005906 Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42, W01202, D01 10.1029/2005WR004578 Blunt MJ (2000) An empirical model for three-phase relative permeability. SPE Journal 5 435-445... 

As summarized above, there are many transport models and flow mechanisms describing reverse osmosis. Each requires some specific assumptions regarding membrane structure. In general, membranes could be continuous or discontinuous and porous or non-porous and homogeneous or non-homogeneous. One must be reasonably sure about the membrane structure before he analyzes a particular set of experimental data based on one of the above theories. Since this is difficult, in many cases, it would be desirable to develop a model-independent phenomenological theory which can interpret the experimental data. 

The model proposed above is analogous to a continuous, unsteady state filtration process, and therefore may be called "Filtration Model". In this model, the concentration of the filtrate, viz. the concentration of the solute remained in the treated solid is one s major concern. This is given by the rate of Step 3, which may be expressed by an equation similar to Pick s Law including a transmission coefficient D for the porous medium, viz. the P.S.Z. and the concentration difference Aw across the P.S.Z. as the driving force, and the thickness of the P.S.Z. as the distance Ax. 

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