Effective diffusivity definition

The only advantage of the effective diffusivity definitions is their simplicity in computation no matrix functions need be evaluated. The primary disadvantage of the use of D- is that these parameters are not, in general, system properties except for the limiting cases noted in Chapter 6. 

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. 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

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... 

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]. 

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. 

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. 

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 definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds e = Dg eI may be obtained from Maxwell s equation... 

The available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

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. ... 

Simplify the definition of the effective diffusivity given by Eq. 6.1.7 for the special case when two molar fluxes are zero, = N2 0. What is the relationship between the effective diffusivity method and the method of Section 8.5.2 for this special case. 

The t values (without subscripts) in Eq. (E) refer to those calculated from Eq. (14-21) and correspond to the dashed lines in Fig. 14-4. Y- was evaluated at 477°C by finding what value would give the best agreement of with the experimental data, making the calculations with Eq. (F). The solid line for 477°C (Fig.i4-4) shows the curve for Y- = 0.66. There appears to be some deviation at the highest conversion, but the agreement for all other x values is good. The effective diffusivity is then readily obtainable from the definition of Y and Eq. (C),... 

Bridging effects. A definite interaction is involved in this process that may include mutual diffusion or "alloying" between the substance of the particle and the surface. Liquid/solid bridging may be involved at the interface that invokes capillary forces. 

Consider a problem on definition of collision frequency of small spherical particles executing Brownian motion in a quiescent liquid. In Section 8.2, Brownian motion was considered as diffusion with a effective diffusion factor. It was supposed that suspension is sufficiently diluted, so it is possible to consider only the pair interactions of particles. To simplify the problem, consider a bi-disperse system of particles, that is, a suspension consisting of particles of two types particles of radius ai and particles of radius a2. In this formulation, the problem was first considered by Smolukhowski [59]. 

Small particles and drop transfer in turbulent flow can be considered as diffusion with an effective diffusion factor [19]. Before defining it, let us consider basic laws of motion of liquid in turbulent flow. In case of developed turbulence, these laws are well studied and described in works [19, 33-35], therefore we will limit here to the aspects concerning definition of particle and drop collisions. 

One last useful definition is the tortuosity t. The tortuosity relates the effective diffusivity in the pores - effective to 6 molecular diffusivity in free solution, Dnjoiecuiar... 

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