Effective diffusivity concept

The approach based on the effective diffusivity concept is justified in the region of Knudsen flow, where DeA = DA t + DAsurf. Equation 3.25 with a constant effective diffusivity also follows from the more general DGM equations in the limiting case of dilute mixtures with one species (B) in considerable excess and a negligible pressure gradient. In this case the other species diffuse independently, as in a Knudsen regime, but with the effective diffusion coefficient governed by the equation... 

For liquids, there is no complete theory of multicomponent diffusion yet available. For this reason only rough theoretical approaches, as used for the description of mass transport in the porous particles filled with a liquid are discussed. The effective diffusivity concept just described is the only known approach and... 

A general transient model of diffusion-reaction that uses the effective diffusivity concept described for gas-solid catalytic reactions can be derived here as well, e.g., for a spherical particle ... 

The oldest, simplest, and still widely used methods, pioneered by Hougen and Watson (1947) and by Wilke (1950), employ the concept of an effective diffusion coefficient. The effective diffusivity concept was discussed in detail in Chapter 6 here we show how the effective diffusivity can be used to calculate mass transfer rates. 

In the case of nonequimolal cpunterdiffusion, equation 12.2.6 suffers from the serious disadvantage that the combined diffusivity is a function of the gas composition in the pore. This functional dependence carries over to the effective diffusivity in porous catalysts (see below), and makes it difficult to integrate the combined diffusion and transport equations. As Smith (12) points out, the variation of 2C with composition (YA) is not usually strong, and it has been an almost universal practice to use a composition independent form of Q)c (12.2.8) in assessing the importance of intrapellet diffusion. In fact, the concept of a single effective diffusivity loses its engineering utility if the dependence on composition must be retained. 

The following illustration indicates how the concepts we have developed thus far in this chapter can be used in determining effective diffusivities for use in the analyses we will develop in subsequent sections. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

J.5.2 Implications of the Effectiveness Factor Concept for Kinetic Parameters Measured in the Laboratory. It is useful at this point to discuss the effects of intraparticle diffusion on the kinetic parameters that are observed experimentally. Unless we are aware that intraparticle diffusion may obscure or disguise the... 

Some evidence appeared to support the diffusion concept, since it seemed to best explain the effect of H20 on the experimental flame velocities of CO—02. As described in the previous chapter, it is known that at high temperatures water provides the source of hydroxyl radicals to facilitate rapid reaction of CO and 02. 

Very much more is known about the theory of concentration gradients at electrodes than has been mentioned in this brief account. Experimental methods for observing them have also been devised, based on the dependence of refractive index on concentration (the Schlieren method) by means of interferometry (O Brien, 1986). Nevertheless, the basic concept of an effective diffusion-layer thickness, treated here as varying in thickness with fi until the onset of natural convection and as constant with time after convection sets in (though decreasing in value with the degree of disturbance, Table 7.10), is a useful aid to the simple and approximate analysis of many transport-controlled electrodic situations. A few of the uses of the concept of 8 will now be outlined. 

In the spatially ID model of the monolith channel, no transverse concentration gradients inside the catalytic washcoat layer are considered, i.e. the influence of internal diffusion is neglected or included in the employed reaction-kinetic parameters. It may lead to the over-prediction of the achieved conversions, particularly with the increasing thickness of the washcoat layer (cfi, e.g., Aris, 1975 Kryl et al., 2005 Tronconi and Beretta, 1999 Zygourakis and Aris, 1983). To overcome this limitation, the effectiveness-factor concept can be used in a limited extent (cf. Section III.D). Despite the drawbacks coming from the fact that internal diffusion effects are implicitly included in the reaction kinetics, the ID plug-flow model is extensively used in automotive industry, thanks to the reasonable combination of physical reliability and short computation times. 

This sieve effect cannot be considered statically as a factor that only determines the amount of accessible acid groups in the resin in such a way that the boundary between the accessible and non-accessible groups would be sharp. It must be treated dynamically, i.e. the rates of the diffusion of reactants into the polymer mass must be taken into account. With the use of the Thiele s concept about the diffusion into catalyst pores, the effectiveness factors, Thiele moduli and effective diffusion coefficients can be determined from the effect of the catalyst particle size. The apparent rates of the methyl and ethyl acetate hydrolysis [490] were corrected for the effect of diffusion in the resin by the use of the effectiveness factors, the difference in ester concentration between swollen resin phase and bulk solution being taken into account. The intrinsic rate coefficients, kintly... 

If data are available on the catalyst pore- structure, a geometrical model can be applied to calculate the effective diffusivity and the tortuosity factor. Wakao and Smith [36] applied a successful model to calculate the effective diffusivity using the concept of the random pore model. According to this, they established that ... 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

Experimentally determined effective transport properties of porous bodies, e.g., effective diffusivity and permeability, can be compared with the respective effective transport properties of reconstructed porous media. Such a comparison was found to be satisfactory in the case of sandstones or other materials with relatively narrow pore size distribution (Bekri et al., 1995 Liang et al., 2000b Yeong and Torquato, 1998b). Critical verification studies of effective transport properties estimated by the concept of reconstructed porous media for porous catalysts with a broad pore size distribution and similar materials are scarce (Mourzenko et al., 2001). Let us employ the sample of the porous... 

The extraction of toluene and 1,2 dichlorobenzene from shallow packed beds of porous particles was studied both experimentally and theoretically at various operating conditions. Mathematical extraction models, based on the shrinking core concept, were developed for three different particle geometries. These models contain three adjustable parameters an effective diffusivity, a volumetric fluid-to-particle mass transfer coefficient, and an equilibrium solubility or partition coefficient. K as well as Kq were first determined from initial extraction rates. Then, by fitting experimental extraction data, values of the effective diffusivity were obtained. Model predictions compare well with experimental data and the respective value of the tortuosity factor around 2.5 is in excellent agreement with related literature data. 

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

Here it is assumed that it is possible to use the concept of an effective diffusion coefficient without making too large an error. Hence the effect of micro properties will not be studied here and it is assumed the value of De is known. The discussion is restricted to the impact of the macro properties and reaction properties on the effectiveness factor. Furthermore only simple reactions are discussed. Generalized formulae are provided that enable calculation of effectiveness factor for varying properties of the catalyst or the reacting system. 

Intraparticle diffusion resistance may become important when the particles are larger than the powders used in slurry reactors, such as for catalytic packed beds operating in trickle flow mode (gas and liquid downflow), in upflow gas-liquid mode, or countercurrent gas-liquid mode. For these the effectiveness factor concept for intraparticle diffusion resistance has to be considered in addition to the other resistances present. See more details in Sec. 19. 

Example 11-5 Vycor (porous silica) appears to have a pore system with fewer interconnections than alumina. The pore system is monodisperse, with the somewhat unusual combination of a small mean pore radius (45 A) and a low porosity 0.31. Vycor may be much closer to an assembly of individual voids than to an assembly of particles surrounded by void spaces. Since the random-pore model is based on the assembly-of-particles concept, it is instructive to see how it applies to Vycor. Rao and Smith measured an effective diffusivity for hydrogen of 0.0029 cm /sec in Vycor. The apparatus was similar to that shown in Fig. 11-1, and data were obtained using an H2-N2 system at 25°C and 1 atm. Predict the effective diffusivity by the random-pore model. 

A well-known subacute effect is the growth reduction in algae. Hitherto, only external effect concentrations have been reported for this type of subacute effect, since experimental problems make it difficult to determine those internal effect concentrations, and existing bioaccumulation models for, e. g., fish, do not apply to algae, e.g. [78]. It must be noted that algae and other small organisms are prone to diffusive uptake for contaminants from the ambient environment for which the link between bioconcentration and the internal effect concentration concept would be very promising. 

In the previous sections the concept of the effectiveness factor has been discussed. In this section it is discussed in further detail with the aim of extending the concept to industrially important complex reaction networks. The effectiveness factor is the most widely used man-made factor to account (in a condensed, one number manner) for the effect of different diffusional resistances on the actual (or apparent) rate of reaction for gas-solid catalytic systems. Although the use of the effectiveness factor concept in the simulation of catalytic reactors taxes the solution by extra computations, nevertheless it is a very useful tool to account for the complex interaction between the diffusion and reaction processes taking place within the system. Most of the published work (e.g. Weisz and Hicks, 1962 Aris, 1975a,b) deals with the effectiveness factor for the simple irreversible reaction,... 

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