Kinetic-diffusion approximation description

Several different models were proposed for the slow NOx storage process, while only few details and approximate models are available for the highly transient NOx reduction within the rich phase, lasting only several seconds. The models can be divided into two groups, depending on whether the internal diffusion in the particles of the NOx storage material is considered explicitly, or this effect is included implicitly into the evaluated kinetic parameters. The models can be further differentiated by the level of complexity for the reaction kinetics description, i.e., either (simplified) microkinetic scheme or the global kinetics. 

Although the simple rate expressions, Eqs. (2-6) and (2-9), may serve as first approximations they are inadequate for the complete description of the kinetics of many epoxy resin curing reactions. Complex parallel or sequential reactions requiring more than one rate constant may be involved. For example these reactions are often auto-catalytic in nature and the rate may become diffusion-controlled as the viscosity of the system increases. If processes of differing heat of reaction are involved, then the deconvolution of the DSC data is difficult and may require information from other analytical techniques. Some approaches to the interpretation of data using more complex kinetic models are discussed in Chapter 4. 

A theoretical analysis of the experimental kinetics for Vk centres in KC1-Tl, as well as for self-trapped holes in a-Al203 and Na-salt of DNA, is presented in [55]. The fitting of theory to the experimental curves is shown in Fig. 4.4. Partial agreement of theory and experiment observed in the particular case of Vk centres was attributed to the violation of the continuous approximation in the diffusion description. This point is discussed in detail below in Section 4.3. Note in conclusion that the fact of the observation of prolonged increase in recombination intensity itself demonstrated slow mobility of defects. In the case of pure irradiated crystals, it is a strong... 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

It is instructive to compare the three transport processes (conduction, tracer diffusion and chemical diffusion) by using chemical kinetics and for simplicity concentrating on the electron-rich electron conductor, i.e., referring to the r.h.s. of Fig. 52. The results of applying Eq. (97) are summarized in Table 5 and directly verify the conclusions. Unlike in Section VI.2. ., we now refer more precisely to bimolecular rate equations (according to Eqs. 113-115) nonetheless the pseudo-monomolecular description is still a good approximation, since only one parameter is actually varied. This is also the reason why we can use concentrations for the regular constituents in the case of chemical diffusion. In the case of tracer diffusion this is allowed because of the ideality of distribution. 

We have also cast the DMC model in a set of ordinary differential equations, thus translating it to a mean-field approach with the site-approximation. Only the kinetic oscillations can be modeled in this way. To model the spatio-temporal pattern formations, diffusion terms would have to be added to the mean-field description, in order to account for the spatial dependence of the reactant concentrations. 

It is known that an exact description of transfer processes in the aerosol particles-gas phase system with chemical or phase transformations on the particle surface for arbitrary particle sizes (and correspondingly for arbitrary Knudsen numbers) can be found only by solving the Boltzmann kinetic equation. However, the mathematical difficulties associated with the solution of the given equation lead to the necessity of obtaining rather simple expressions for mass and energy fluxes either on the basis of an approximate solution of the Boltzmann equation or with the use of simpler models. In particular, it is known that the use of the diffusion equation with appropriate boundary conditions on the particle surface leads to the equation that gives correct limiting cases with respect to the Knudsen number [2]. 

Figure 2 shows the comparison of the fractal-layer (solid line a) and two-timescale (solid line b) models with the simulations in terms of effective diffusivity, eq. (13). Both the models furnish a satisfactory level of agreement with simulation data. We may therefore conclude that approximate models based on a Riemann-Liouville constitutive equation are able to furnish an accurate description of adsorption kinetics on fractal interfaces. These models can also be extended to nonlinear problems (e.g. in the presence of nonlinear isotherms, such as Langmuir, Freundlich, etc.). In order to extend the analysis to nonlinear cases, efficient numerical sJgorithms should be developed to solve partied differential schemes in the presence of Riemann-Liouville convolutional terms. 

The first attempt to take into account the two-step kinetic theory of micellisation was made by Fainerman [147]. With that end in view two pairs of diffusion equations (for micelles and monomers) were written down for two situations eorresponding to the fast and slow proeesses. Approximate solutions of the boundary problems for these equations were used subsequently in the course of analysis of experimental data on the adsorption kinetics from micellar solutions [77, 85, 87, 88]. However, as it has been shown by Dushkin et al. [137], this approaeh is equivalent to the PFOR model for the slow proeess and probably eannot be applied to the description of the adsorption kinetics for the fast process. 

Another difficulty arising from this comparison is connected with the mathematical complexity of the corresponding boundary problems even if only linear diffusion equations are used. The mathematical description of the adsorption kinetics from micellar solutions is essentially more complicated in comparison with the case of the adsorption process from sub-micellar solutions. Analytical solutions of the corresponding boundary problems using rather poor approximations have been obtained only for a small number of situations. A sufficiently general solution cannot be obtained analytically and the deficiency of the rather well elaborated numerical methods often compel experimentalists to apply approximate solutions. Therefore, it seems important to consider the main equations proposed for the description of kinetic dependencies of the surface tension and adsorption, and to elucidate the limits of their application before the discussion of experimental results. 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

Solution of this equation requires not only various physicochemical and reaction kinetics data, but also a detailed knowledge of the fluid mechanics near the interface which is however not available in any but artificially simplified agitated system (see, for instance, reference (9) for a detailed review of interphase mass transfer models and description of interfaces). However, the concentration gradient V A and the velocity vector 7 are approximately perpendicular to each other, hence the scalar product of them is always negligible. Furthermore, diffusion is usually unidimensional. Therefore Eqn (1) reduces to ... 

The aim of this lecture is to provide a qualitative description of reversible proton transfer reactions in the excited-state, using the extended theory of diffusion influenced reactions. The complete equations and numerical procedures may be found in the literature [10-14]. Major results include (i) the asymptotic power-law decay and the evidence for diffusive kinetics [10] (ii) The salt effect [11] and the Naive Approximation for the screening function [17, 11] and (iii) an extension [18] of the theory for approximating the effect of competing geminate and homogeneous proton recombination expected atdow pH values. 

The nucleation theory just described is referred to as classical nucleation theory. It relies on the capillarity approximation, in which crystallites of microscopic size are treated as if they are macroscopic, and in which the kinetics is described as the stepwise attachment of single molecules across the crystal-melt interface. In fact, this approximation may not be valid under realistic conditions. A small crystallite may not achieve bulk properties at its center, and its interface may be so strongly curved that the planar value Ysl no longer applies. The interface may be diffuse rather than sharp, so that the description of the kinetics as resulting from addition of solid particles one after another may not be valid instead, a collective fluctuation may result in the simultaneous incorporation of a larger number of molecules in a loosely structured crystallite. 

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