Effective diffusion coefficient continuum

The physicochemical reason why two regimes occur is as follows. In the free molecule region, the effective diffusion coefficient is determined by collisions of the molecules with the catalyst pore walls. Hence, the gas composition will not influence the effective diffusion coefficient. However, in the continuum region mutual collisions between the molecules are determining and the dependence of the effective diffusion coefficient on the concentration becomes most pronounced. 

The study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

Diffusion in a continuum Effective diffusion coefficient Geometric descriptions of porous materials Percolation descriptions of porous materials... 

Descriptions of the absorption of solutes into porous materials can be based on continuum formulations by defining an effective diffusion coefficient, Deff. With an effective diffusivity, Fick s second law ... 

In porous materials, diffusion of a solute is complicated by the geometric constraints of the pore structure. Since they are easily solved, continuum expressions have been used as the basis for many studies of diffusion in porous structures. In most cases, the continuum approach is parametric a numerical value for Deff is selected so that the solution of equation 16 fits a particular set of experimental transport measurements. By this method, correlation between effective diffusion coefficients obtained for different solutes or different porous materials is difficult. In this section, descriptions of porous geometries are used to examine the influence of pore microstructure on effective diffusion coefficients. These descriptions will have value only under certain conditions for example, if the size of a characteristic pore is much less than the thickness of the slab and the pore structure is well connected. 

With this definition of M, equation 13 can be empirically fit to experimental sorption data an effective diffusion coefficient is obtained from this parametric fit. The fraction of accessible pores has often been neglected in applying modifications of continuum formulations to real porous media. In cases where is less than unity, this leads to erroneously high predictions for the effective diffusion coefficient. In searching for the physical determinants of effective diffusivity in porous materials, it is important to separate the confounding effect of inaccessible porosity from real reductions in the diffusivity of solute. 

MD and BD are widely nsed numerical tools to calculate diffusion coefficients in bulk materials. Given the fact that the former is computationally expensive, MD cannot consider dispersed phase in a continuum. On the condary, BD shows its advantages in calculating the effective diffusion coefficient of heterogeneous material. In this section, both methods are introduced in sequence in each subsection. 

Diffusion in heterogeneous media with dispersed impermeable domains had been described in several publications. MaxwelP solved the problem of a suspension of spheres in a continuum and obtained an expression for the effective diffusion coefficient of the composite medium. Cussler et al. solved the problem of a suspension of impermeable flakes oriented perpendicular to the diffusion and obtained the following relation (11.7) for the effective diffusion coefficient ... 

A simple estimate of the diffusion coefficients can be approximated from examining the effects of molecular size on transport through a continuum for which there is an energy cost of displacing solvent. Since the molecular weight dependence of the diffusion coefficients for polymers obeys a power law equation [206], a similar form was chosen for the corneal barriers. That is, the molecular weight (M) dependence of the diffusion coefficients was written as ... 

The derivation of (7.16) is based on the assumption (hat the diffusion coefficients of the colliding particles do not change as the particles approach each other. That this is not correct can be seen qualitatively from the discussion in Chapter4 of the increased resistance experienced by a particle as it approaches a surface. The result is that the term (AH- A ) to decrease as the particles approach each other. This effect is countered in the neighborhood of the surface because the continuum theory on which it is based breaks down about one mean free path ( 0.1 /i m at NTP) from the particle surface in addition, van der Waals forces tend to enhance the collision rate as discussed later in this chapter. For further discussion of the theory, the reader is referred to Batchelor (1976) and Alam (1987). Experimental support for (7.16) is discussed in the next section. 

From the measurements of the effective uptake coefficient, the aqueous phase reaction rate constant can be calculated, as long as the gas phase and liquid phase diffusion coefficients and the Henry constant are known. As mentioned, a gas transfer into droplets is characterized by the continuum regime. The unsteady-state diffusion flux (it means that depends on t as well as on x) of species A along the x-axis to the stationary droplet (Fig. 4.20) was described by Seinfeld and Pandis (1998), where c x, t) is the concentration, depending on time and location ... 

Diffusion in the continuum regime (A/rp)< 1 can be described by the usual molecular diffusion coefficient. However, the treatment of multi-component mixtures is very tedious except when they can be approximately considered as binary. Diffusion in the transition regime (A/r 1) has been described by effective diffusivities which are functions of the molecular and the Knudsen diffusivity [34]. 

In random walk MC simulations, trajectories of an ensemble of particles are obtained by a sequence of random numbers and used to calculate quantities such as the photocurrent and charge recombination transients. For electron transport in DSSC, MC is more realistic than continuum transport models because the continuity equations assume aU electrons have the same diffusion coefficient, whereas MC can address transport in disperse systems where the diffusion coefficient is a poorly defined quantity as shown in more detail in Sect. 3.1. MC can also be implemented for complex geometries and so can separate out the effect of morphology on electron transport noted in Sect. 1.1. Section 3.2 describes simulations that explicitly consider the morphology of the oxide. 

The porous structure filled with electrolyte is considered as the superposition of two continua, the electrode matrix and the solution in the unoccupied spaces within the matrix. The two phases, which complement one another, are supposed to be homogeneous and isotropic. Effective parameters rather than actual parameters are used for the description of the properties like pore size, conductivity, diffusion coefficient etc. The problem is treated as a one-dimensional one. This is equivalent to the assumption that the penetration depth of the current is larger than the size of the structural units (grains, holes) of the porous electrode. Continuum models were analyzed under different assumptions in references 1,4,9,12 to 14,16 to 20. Comprehensive treatments with strict derivations were published by Tobias and coworkers [21, 22] and by Micka [23]. 

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