Effective diffusion coefficient volume

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]. 

Cdg Concentration of D in gas phase, moles/volume Cdp Concentration of D in crystallite phase, moles/zeolite pore volume Deff Effective diffusion coefficient, (length) 2/time F Volumetric feed rate to reactor, volume/time H Henry s law-type constant relating gas phase mole fraction to crystallite phase mole fraction... 

Concentration of A Arrhenius constants Arrhenius constant Constant in equation 5.82 Surface area per unit volume Parameter in equation 5.218 Cross-sectional area Concentration of B Stoichiometric constants Parameter in equation 5.218 Concentration of gas in liquid phase Saturation concentration of gas in liquid Concentration of G-mass Concentration of D-mass Dilution rate DamkOhler number Critical dilution rate for wash-out Effective diffusion coefficient Dilution rate for maximum biomass production Dilution rate for CSTF 1 Dilution rate for CSTF 2 Activation energy Enzyme concentration Concentration of active enzyme Active enzyme concentration at time t Initial active enzyme concentration Concentration of inactive enzyme Total enzyme concentration Concentration of enzyme-substrate complex with substance A... 

The above equations for the diffusion coefficients do not take into account the volume fraction of porosity and the tortuous nature of the path through porous bodies. When the transport occurs through a porous body, as in fuel cell electrodes, effective diffusion coefficients accounting for the interaction of gaseous species with the porous matrix must be employed. Different theoretical approaches for the determination of the effective diffusion have been proposed in the literature. The Bruggemann correction allows the evaluation of these coefficients, through the following expression [47] ... 

The characteristics of pore structure in polymers is a key parameter in the study of diffusion in polymers. Pore sizes ranging from 0.1 to 1.0 pm (macroporous) are much larger than the pore sizes of diffusing solute molecules, and thus the diffusant molecules do not face a significant hurdle to diffuse through polymers comprising the solvent-filled pores. Thus, a minor modification of the values determined by the hydrodynamic theory or its empirical equations can be made to take into account the fraction of void volume in polymers (i.e., porosity, e), the crookedness of pores (i.e., tortuosity, x), and the affinity of solutes to polymers (i.e., partition coefficient, K). The effective diffusion coefficient, De, in the solvent-filled polymer pores is expressed by ... 

The parameter / is the characteristic length for a unit cell, E0 is the surface concentration of a carrier protein molecule, and a2, a3, a4, a5 are the reaction rate parameters analogous to that half saturation constants. Table 11.3 displays the experimental effective diffusion coefficients and the volume fraction of intracellular phase A. In the first four sets... 

Because of volume changes due to the reaction, pressure gradients may occur inside the catalyst pellet. This can give rise to two effects. First, it influences the effective diffusion coefficients, since the gas-phase diffusion coefficients depend on pressure. Second, the pressure gradients affect the concentrations (or more accurately, chemical activities), which determine the reaction rate. Hence pressure gradients must directly influence the effectiveness factor. 

Some authors consider diffusion (a), (b) as consecutive processes, and assume the existence of colliding pairs [7-9]. Other models stress the importance of segmental diffusion of the active ends in a common volume of the two colliding macro molecules [10-12]. A common drawback of the mathematical models is the lack of a generally formulated expression for the effective diffusion coefficient of the active end in a coiling chain. Most models try to solve this difficulty by introducing suitable parameters with some physical meaning. 

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