Modeling effective diffusion coefficient

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... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

The theory of seaweed formation does not only apply to solidification processes but in fact to the completely different phenomenon of a wettingdewetting transition. To be precise, this applies to the so-called partial wetting scenario, where a thin liquid film may coexist with a dry surface on the same substrate. These equations are equivalent to the one-sided model of diffusional growth with an effective diffusion coefficient which depends on the viscosity and on the thermodynamical properties of the thin film. 

The diffusional transport model for systems in which sorbed molecules can be divided in two populations, one formed by completely immobilized molecules and the other by molecules free to diffuse, has been developed by Vieth and Sladek 33) in a modified form of the Fick s second law. However, if linear isotherms are experimentally found, as in the case of the DGEBA-TETA system in Fig. 4, the diffusion of the penetrant may be described by the classical diffusion law with constant value of the effective diffusion coefficient,... 

In the case that the effective diffusion coefficient approach is used for the molar flux, it is given by N = —Da dci/dr), where Dei = (Sp/Tp)Dmi according to the random pore model. Standard boundary conditions are applied to solve the particle model Eq. (8.1). 

When a two- or higher-phase system is used with two or more phases permeable to the solute of interest and when interactions between the phases is possible, it would be necessary to apply the principle of local mass equilibrium [427] in order to derive a single effective diffusion coefficient that will be used in a one-equation model for the transport. Extensive justification of the principle of local thermdl equilibrium has been presented by Whitaker [425,432]. If the transport is in series rather than in parallel, assuming local equilibrium with equilibrium partition coefficients equal to unity, the effective diffusion coefficient is... 

FIG. 20 Effective diffusion coefficients using Michaels model [241], Eq. (43), versus porosity for various ratios of pore lengths. 

FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

The standard Rodbard-Ogston-Morris-Killander [326,327] model of electrophoresis which assumes that u alua = D nlDa is obtained only for special circumstances. See also Locke and Trinh [219] for further discussion of this relationship. With low electric fields the effective mobility equals the volume fraction. However, the dispersion coefficient reduces to the effective diffusion coefficient, as determined by Ryan et al. [337], which reduces to the volume fraction at low gel concentration but is not, in general, equal to the porosity for high gel concentrations. If no electrophoresis occurs, i.e., and Mp equal zero, the results reduce to the analysis of Nozad [264]. If the electrophoretic mobility is assumed to be much larger than the diffusion coefficients, the results reduce to that given by Locke and Carbonell [218]. 

Gas diffusion in the nano-porous hydrophobic material under partial pressure gradient and at constant total pressure is theoretically and experimentally investigated. The dusty-gas model is used in which the porous media is presented as a system of hard spherical particles, uniformly distributed in the space. These particles are accepted as gas molecules with infinitely big mass. In the case of gas transport of two-component gas mixture (i = 1,2) the effective diffusion coefficient (Dj)eff of each of the... 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

As already said, Taylor s effective model contains a contribution in the effective diffusion coefficient, which is proportional to the square of the transversal Peclet number. Frequently this term is more important than the original molecular diffusion. After his work, it is called Taylor s dispersion coefficient and it is generally accepted and used in chemical engineering numerical simulations. For the practical applications we refer to the classical paper (Rubin, 1983) by Rubin. The mathematical study of the models from Rubin (1983) was undertaken in Friedman and Knabner (1992). 

The linear driving force model has much more physical significance. It has been derived from a two-dimensional model of intra-particle diffusion, solution of which is a series development. The particle size appears explicitly. The effective diffusion coefficient is related to the particle porosity and to the size of the adsorbate molecule. Thus it makes sense to search for correlation of with these properties. However such relations are complex and it is rather difficult to predict for a given carbon and a given molecule. 

The solid lines in Figs. 25 to 28 are the result of model predictions of metal deposition based on porphyrin reaction pathways. Curves are generated using intrinsic kinetic rate parameters and effective diffusion coefficients for the metal species on the order of 10 6 cm2/sec. These values are similar to diffusion coefficients measured in the independent studies referenced. 

The effective diffusion coefficient is calculated according to the model of Ref. 209, which accounts for interfacial instabilities. This model includes a Handlos-Baron-like correlation (210) and one adjustable parameter, C/P ... 

The separation of the caged radical-ion pair can also be rationalized using the model of Konig and Braun, Rajbenbach, and Eirich.103 The probability %(t) that the radical and the ion are at a distance x at a time t (Equation 6.136) depends on the collision diameter a and the effective diffusion coefficient D of the pair, and A is a normalization constant. 

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. 

M. A. Desai and P. Vadgama. Estimation of effective diffusion coefficient of model... 

To estimate the Maxwell-Stefan and effective diffusion coefficients, diffusion data for binary mixtures is necessary. For gas systems under low pressure, the model of Fuller et al. is used most frequently [51]. The method of Wilke and Lee [40] is also valid for low pressures. Both of these methods generally agree with experimental data with an accuracy of up to 10 %, although discrepancies of about 20 % cannot be excluded [40],... 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Hore importantly, the response curves are noticeably affected where one or both of the components is adsorbable, even at low tracer concentrations. The interpretation of data is then much more complex and requires analysis using the non-isobaric model. Figs 7 and 8 show how adsorption of influences the fluxes observed for He (the tracer), despite the fact that it is the non-adsorbable component. The role played by the induced pressure gradient, in association with the concentration profiles, can be clearly seen. It is notable that the greatest sensitivity is exhibited for small values of the adsorption coefficient, which is often the case with many common porous solids used as catalyst supports. This suggests that routine determination of effective diffusion coefficients will require considerable checks for consistency and emphasizes the need for using the Wicke-Kallenbach cell in conjunction with permeability measurements. 

Care is needed in applying the unsteady state pulse technique to a Wicke-Kallenbach cell in order to obtain values for effective diffusion coefficients. For sufficiently small concentrations, where the trace component is of higher diffusivity than the carrier, the commonly used isobaric model is adequate for defining the transport parameters if sufficiently short pulses are used. However, where adsorption of either carrier or trace component occurs or wheipe the trace is of lower diffusivity, then the induced total pressure gradients cause the fluxes to show unusual behaviour and may require analysis by a non-isobaric model. 

The use of the effective diffusion coefficients in situations where a pressure gradient arises from non-equimolal fluxes, such as when chemical reactions occur, should then be based on the non-isobaric equations. Although this means that the models to be used are more complex, the parameters will be consistent. Where the pore size distribution is not monodisperse, the additional structural parameters which influence the effective diffusion coefficient will make the problem even more complex and requires further study. 

Figure 6. Example of two-dimensional 10x10 lattice model used to examine the effects of pore blockage on the effective diffusion coefficient in zeolites. Reprinted with permission from Chem. Eng. Sci., vol. 41, p. 703, W. T. Mo and J. Wei, Effective Diffusivity in Partially Blocked Zeolite Catalyst, copyright 1986 [18], Pergamon Press PLC.

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