Effective diffusion coefficient definition

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

It should be mentioned here that a different definition of the diffusion coefficient is often used in chemical engineering problems, which is more appropriate for the description of reactant or tracer transport. It takes into account the fact that the total fluid contained in a porous substance of porosity e is reduced by this factor relative to the bulk, so that an effective diffusion coefficient D of the reactants is defined such that... 

Treatment of class (c) membranes, on the other hand, presents a considerably more complicated problem. Here, S and DT in Eqs. (1) and (2) are functions of the spatial coordinates. The problem becomes much more acute if S and DT are also dependent on C 4,5). Under these conditions, transformation of Eqs. (2) into (3) is not generally possible and there are no standard methods, as in the previous cases, of fully characterizing the membrane-penetrant system 3 "5). There is usually no difficulty in determining an overall or effective solubility coefficient but the definition of useful effective diffusion coefficients is a more difficult matter, which, not surprisingly, is a major concern of current research in the field. 

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. 

The pores in the pellet are not straight and cylindrical rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas. It would not be fimitfiil to describe diffusion within each and every one of the tortuous pathways individually consequently, we shall define an effective diffusion coefficient so as to describe the average diffusion taking place at any position r in the pellet. We shall consider only radial variations in the concentration the radial flux will be based on the total area (voids and solid) normal to diffusion transport (i.e., 4TTr ) rather than void area alone. This basis for is made possible by proper definition of the effective diffusivity D. ... 

The effect of the solid obstructing the gas transport is determined by the porosity s and tortuosity t of the GDL. Tortuosity describes the elongation of the direct transport distance x by the solid structure of the GDL, and definition of the porosity is obvious. Liquid water may occupy part of the pore space, decreasing porosity and increasing tortuosity. This effect depends on the amount of liquid water present in the GDL, which is given as the saturation s, the fraction of the pore space occupied. The effective diffusion coefficient is therefore defined as ... 

As shown in Fig. 9, the effective diffusion coefficient increases linearly with the original HCI content. This phenomenon is quite obvious when taking the interpretation of Her [11] and Falcone [12] into account they reported a definite effect of the pH environment on the structure of silica gels. 

This dimensionless number indicates the relative magnitude of the nutrient uptake rate over the nutrient diffusion rate. For this definition, L is the characteristic length of the scaffold [35,131], D is the effective diffusion coefficient of the molecule in the tissue-filled scaffold, and Cj is a reference concentration of the molecule (depends on problem formulation). 

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. 

In this project, the definition of effective diffusion coefficient, from the Wagner solution is adapted and bulk oxygen diffusivity data are taken from the open literature and experimental test results available. The grain boundary diffusivity values are approximated to yield 100 times the bulk values. 

The effects of longer pores and smaller areas are often lumped together in the definition of a new, effective diffusion coefficient >efr... 

The limitation of using such a model is the assumption that the diffusional boundary layer, as defined by the effective diffusivity, is the same for both the solute and the micelle [45], This is a good approximation when the diffusivities of all species are similar. However, if the micelle is much larger than the free solute, then the difference between the diffusional boundary layer of the two species, as defined by Eq. (24), is significant since 8 is directly proportional to the diffusion coefficient. If known, the thickness of the diffusional boundary layer for each species can be included directly in the definition of the effective diffusivity. This approach is similar to the reaction plane model which has been used to describe acid-base reactions. 

Internal resistance relates to the diffusion of the molecules from the external surface of the catalyst into the pore volume where the major part of the catalyst s surface is found. To determine the diffusion coefficients inside a porous space is not an easy task since they depend not only on the molecules diffusivity but also on the pore shape. In addition, surface diffusion should be taken into account. Data on protein migration obtained by confocal microscopy [8] definitely demonstrate that surface migration of the molecules is possible, even though the mechanism is not yet well understood. All the above-mentioned effects are combined in a definition of the so-called effective diffusivity [7]. 

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

Another explanation of the lithium gap in the Hyades could be found in terms of turbulent diffusion and nuclear destruction. Turbulence is definitely needed to explain the lithium abundance decrease in G stars. If this turbulence is due to the shear flow instability induced by meridional circulation (Baglin, Morel, Schatzman 1985, Zahn 1983), turbulence should also occur in F stars, which rotate more rapidly than G stars. Fig. 2 shows a comparison between the turbulent diffusion coefficient needed for lithium nuclear destruction and the one induced by turbulence. Li should indeed be destroyed in F stars This effect gives an alternative scenario to account for the Li gap in the Hyades. The fact that Li is normal in the hottest observed F stars could be due to their slow rotation. 

It is not unreasonable to use the left-hand side of this equation as the definition of the effective diffusion constant K, the more so as it will be shown that any distribution tends to normality. With this definition K is the sum of the molecular diffusion coefficient, D, and the apparent diffusion coefficient k = oP-U2I 48D, which was discovered by Taylor in his first paper (Taylor 1953, equation (25)). Equation (26), however, is true without any restriction on the value of p, or on the distribution of solute. The constant 1/48 is a function of the profile of flow, and for so-called piston flow with x — 0 this constant is zero and K = D as it should. 

It has long been a mystery why diffusion coefficients of polymer-diluent systems, especially when the diluent is a good solvent for a given polymer, exhibit so pronounced a concentration dependence that it looks extraordinary. Several proposals have been made for the interpretation of this dependence. Thus Park (1950) attempted to explain it in terms of the thermodynamic non-ideality of polymer-diluent mixtures, but it was found that such an effect was too small to account for the actual data. Fujita (1953) suggested immobilization of penetrant molecules in the polymer network, which, however, was not accepted by subsequent workers. Recently, Barrer and Fergusson (1958) reported that their diffusion coefficient data for benzene in rubber could be analyzed in terms of the zone theory of diffusion due to Barrer (1957). Examination shows, however, that their conclusion is never definitive, since it resorted to a less plausible choice of the value for a certain basic parameter. 

Undeniably, the speed vector, by its size and directional character, masks the effect of small displacements of the particle. Another difference comes from the different definition of the diffusion coefficient, which, in the case of the property transport, is attached to a concentration gradient of the property it means that there is a difference in speed between the mobile species of the medium. A second difference comes from the dimensional point of view because the property concentration is dimensional. When both equations are used in the investigation of a process, it is absolutely necessary to transform them into dimensionless forms [4.6, 4.7, 4.37, 4.44]. 

We also feel that portions of the material in this book ought to be taught at the undergraduate level. We are thinking, in particular, of the materials in Section 2.1 (the Maxwell-Stefan relations for ideal gases). Section 2.2 (the Maxwell-Stefan equations for nonideal systems). Section 3.2 (the generalized Fick s law). Section 4.2 (estimation of multicomponent diffusion coefficients). Section 5.2 (multicomponent interaction effects), and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of mass transfer in binary mixtures that is normally included in undergraduate courses. 

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