Effective transport coefficients

Many investigators have studied diffusion in systems composed of a stationary porous solid phase and a continuous fluid phase in which the solute diffuses. The effective transport coefficients in porous media have often been estimated using the following expression ... 

FIG. 27 Effective transport coefficient versus porosity from the model of Trinh et al. (Reproduced with permission from Ref. 399.)... 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

Determination of the effective transport coefficients, i.e., dispersion coefficient and electrophoretic mobility, as functions of the geometry of the unit cell requires an analogous averaging of the species continuity equation. Locke [215] showed that for this case the closure problem is given by the following local problems ... 

Effective transport coefficients for unit cell given in Figure 29. (Reprinted with permis-from Ref. 215, Copyright 1998, American Chemical Society.)... 

Akaimi, KA Evans, JW Abramson, IS, Effective Transport Coefficients in Heterogeneous Media, Chemical Engineering Science 42, 1945, 1987. 

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

The EMA method is similar to the volume-averaging technique in the sense that an effective transport coefficient is determined. However, it is less empirical and more general, an assessment that will become clear in a moment. Taking mass diffusion as an example, the fundamental equation to solve is... 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

A quantity relevant to (diffusional) transport is the effective transport coefficient eE. If the accessible void space would be arranged in parallel layers, then eE equals eA. However, in disordered interdispersions the accessible void space for transport may be tortuous and have inactive dead-ends, so that in general eE eA. 

As an approximation, the nondimensional concentrations 8S and 8P are related to each other by the relation 8P = ax + a2(l-8s), which is derived for stationary states, via the nondimensional affinity A. The accuracy of the solutions of Eqs. (9.150) and (9.151) depends on the availability of reliable data, such as the effective transport coefficients and cross coefficients. Relating the parameter b to degrees of coupling qv and qSt, as shown in Eqs. (9.147) and (9.148), may be helpful in solving these equations, since the degrees of coupling vary between -1 and +1. 

One will often find the flux to or from the surface as written in terms of an effective transport coefficient k s. ... 

The molecular and turbulent transport coefficients may be added as a linear sum of independent contributions determining the effective transport coefficients. 

Having discussed the effective transport coefficients and we now can turn to the main objective of the chapter, an expression of the rate of -reaction for-the-whole eatalyst-pelleV -rp,-in-terms of-the-temperature and -concentrations existing at the outer surface. We start in a formal way by defining an effectiveness factor 17 as follows ... 

Kalnin, J. R., Kotomin, E. Modified Maxwell-Garnett equation for the effective transport coefficients in inhomogeneous media, J. Pbys. A Math. Gen., 31, 7227-7234 (1998). 

Many of the most basic features of time dependence and concentration ranges in the NH4 " pore-water profiles at NWC can be reproduced by the assumptions of this model. Effective transport coefficients of—1-2 x 10" cmVsec result in the best agreement between modeled and observed con-... 

A two-dimensional transport-reaction model incorporating both radial transport into burrows and vertical diffusion is presented. This model is capable of predicting both the form and magnitude of pore-water profiles extraordinarily well at all stations. A one-dimensional model in which an effective transport coefficient is used to account for the influence of reworking and burrow construction on solute movement is far less satisfactory in predicting the observed profiles. 

De Effective transport coefficient E Activation energy F Concentration gradient at depth x = L J, Solute flux at depth x... 

Effective Transport Coefficients. Analytical expressions are available for effective diffusion on a Bethe lattice [113]. For a Bethe lattiee with coordination number the effective diffusion coefficient is found from ... 

Reyes, S. and K. Jensen, Estimation of effective transport coefficient in porous solids based on percolation concepts. Chemical Engineering Science, 1985, 40, 1723-1734. 

Experimental extraction curves can be represented by this type of model, by fitting the kinetic coefficients (mass transfer coefficient to the fluid, effective transport coefficient in the solid, effective axial dispersion coefficient representing deviations from plug flow) to the experimental curves obtained fi om laboratory experiments. With optimized parameters, it is possible to model the whole extraction curve with reasonable accuracy. These parameters can be used to model the extraction curve for extractions in larger vessels, such as from a pilot plant. Therefore, the model can be used to determine the kinetic parameters from a laboratory experiment and they can be used for scaling up the extraction. 

This model represents the catalyst as a pseudo-homogeneous medium in which the effective transport coefficient for the reactant, D ff, is in general a complex function... 

When the reactor contains a solid catalyst the flow pattern is strongly determined by the presence of the solid. It would be impossible to rigorously express the influence of the packing but again the flux of j resulting from the mixing effect caused by its presence is expressed in the form of Pick s law. Consequently, the form of Eq. 7.2.a-6 is not altered, but the effective diffusivity now also contains the effect of the packing. This topic is dealt with extensively in Chapter 11 on fixed bed catalytic reactors. For further explanation of the effective transport coefficients see Himmelblau and Bischoff [3] and Slattery [4]. 

In this case, the approximation would clearly be best for highly turbulent flow, for which the velocity profiles are relatively flat. The discrepancies actually enter into the effective transport coefficients, which have to be empirically measured in any event. Another approximation concerns the reaction rate term ... 

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