Diffuse layer model distribution coefficient

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

Wellinghoff et al. (1995) derived a correlation directly from a pore diffusion model based on Pick s law and obtained an equation which predicts the effective distribution coefficient of diffusion washing as a function of four dimensionless numbers. The apparent differences between the two approaches vanish upon closer examination. Namely, when neglecting the factors that cannot be arbitrarily influenced due to process inherent restrictions the two approaches are essentially alike. They only differ in that Wellinghoff et al. (1995) additionally consider a dimensionless area which corresponds to the ratio of the pore surface to the surface of the crystal layer. This factor can only be determined experimentally yet which is fairly complicated and thus limits the applicability of the approach. 

Several models have been developed to describe reactions between aqueous ions and solid surfaces. These models tend to fall into two categories (1) empirical partitioning models, such as distribution coefficients and isotherms (e.g., Langmuir and Freundlich isotherms), and (2) surface-complexation models (e.g., constant-capacitance, diffuse-layer, or triple-layer model) that are analogous to solution complexation with corrections for the electrostatic effects at the solid-solution interface (Davis and Kent, 1990). These models have been described in numerous articles (Westall and Hohl, 1980 Morel, Yeasted, and Westall, 1981 James and Parks, 1982 Barrow, 1983 Westall, 1986 Davis and Kent, 1990 Dzombak and Morel, 1990). Travis and Etnier (1981) provided a comprehensive review of the partitioning and kinetic models typically used to define sorption of ions by soils. The reader is referred to the cited articles for details of the models. 

However, this is an important issue in the determination of a diffusion coefficient from experimental data. It seems that the charge transfer resistance, necessary to determine the reaction kinetics, is less affected by a 2D current distribution, and so a simpler model might be used in the kinetic studies. Even simple models based on the Nemst diffusion layer are often used in kinetic studies [180] because of their simplicity. 

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

A better description of the sand distribution can be obtained by allowing the net speed of sand transport to decrease into the Sound. This is consistent with what is known about the current velocities in the region. More detailed models could also be devised to account for variations in the sedimentation rate, the thickness of the layer of mobile sediment, and the effective diffusion coefficient for mobile sand. Another refinement would include the lateral transport of sand that undoubtedly does occur. 

Equations 46 have been directly derived from the full model in [19]. On the other hand, they are almost identical with the relations obtained from the so-called two-compartment model (the only difference is that the numerical coefficient is a little bit lower). The two-compartment model was first developed for sensors with receptors placed on small spheres [23]. In [24-26] it was adapted for the SPR flow cell and in [ 18] it was approved and verified by comparison of munerical results with those obtained from the full model. The two-compartment model approximates the analyte distribution in the vicinity of the receptors by considering two distinct regions. The first is a thin layer around the active receptor zone of effective thickness fiiayer> and the second is the remaining volume with the analyte concentration equal to the injected one, i.e., a. While the analyte concentration in the bulk is constant (within a given compartment), analyte transport to the inner compartment is controlled by diffusion. The actual analyte concentration at the sensor surface is then given by the difference between the diffusion flow and the consump-tion/production of the analyte via interaction with receptors. For the simple pseudo first-order interaction model we obtain ... 

---