Modeling Heterogeneous Processes

In this section the modeling procedures that can be used for a true heterogeneous system are briefly outlined. 

It is possible to apply the concept of pseudohomogeneity in cases where the diameter of small particles is slightly less than the film thickness d, which, in the two-film theory, corresponds to the transport resistance. This situation is illustrated in Fig. 4.29. In this example of a three-phase process, the S phase particles are suspended in the L phase. Five different concentration profiles are shown at the G [ L interface (cf. curves a-e) for various ratios of the reaction rate coefficient to the transport coefficient /cjr. In the case of a L S reaction with a large dp, only profiles a-c apply (cf. Fig. 4.28). 

All of these cases of coupling between kinetics (kj and transport (kj ) phenomena can be handled and solved with a uniform scheme, which is shown in Fig. 4.30 with Equs. 4.48 through 4.59. One uses the concept of an effectiveness factor, rj, which is regarded as a function of the ratio In individual 

For model building when the problem is one of mass transport with simultaneous reaction, the conservation of mass equation (Equ. 2.3a-c) is again utilized. In the one-dimensional stationary case it has the form 

The boundary conditions for fluid (gas or liquid)-solid processes are  

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Analysis of such a correlation may reveal the significant variables and interactions, and may suggest some model, say of the L-H type, that could be analyzed in more detail by a regression process. The variables Xi could be various parameters of heterogeneous processes as well as concentrations. An application of this method to isomerization of /i-pentane is given by Kittrel and Erjavec (Ind. Eng. Chem. Proc. Des. Dev., 7,321 [1968]). 

Shindell, D. T., and R. L. de Zafra, Limits on Heterogeneous Processing in the Antarctic Spring Vortex from a Comparison of Measured and Modeled Chlorine, J. Geophys. Res., 102, 1441-1449(1997). 

In all these models, knowledge of parameters such as q0 (LSPP model), E0 (PSSE model), or I0 and yL (LL model) are necessary to determine the photolysis rate of M. These parameters are determined experimentally by actinometry experiments [86]. It is noteworthy to mention that the use of these theoretical models (LSPP or PSSE models) implies that all radiation incident into the solution is absorbed without end effects, reflection, or refraction. In experimental photoreactors, it is not usual to fulfill all these assumptions because of the short wall distance of the photoreactor. For instance, to account for such deviations, Jacob and Dranoff [114] introduced a correcting equation, as a function of position. Another important disadvantage is the presence of bubbles that leads to a heterogeneous process as, for example, in the case of 03/UV oxidation. In this case, photoreactor models should be used [109]. This is the main reason for which the LL model is usually applied in the laboratory for the kinetic treatment of photochemical reactions. In the LLM,... 

And finally, the coke burning is a heterogeneous process. Its modeling includes description of the processes of molecule adsorption on the surface, surface reactions, desorption of reaction products, diffusion through the pores and diffusion to the particle surface. At present the majority of these processes for coke are relatively poorly known. The key distinction of surface reactions from reactions at the gas phase consists in the necessity to attract for description of their rates such notions as surface active centers and adsorbed particles. And in the kinetic models a different nature of active centers (different energy of dislocations) necessitates consideration of the same particles adsorbed on them as different compounds because of... 

Abstract In this chapter, photochemical processes in natural aquatic systems involving iron in one way or another are reviewed. Homogeneous and heterogeneous processes are examined with attention given to both simple model systems in which the species distribution is generally well understood and to complex systems more typical of natural or treatment environments. Insights into mechanistic aspects are provided where possible and implications to natural and treatment systems assessed. 

Recently, kinetic models have been combined with the equilibrium data of the interfacial processes, taking into account that soils and rocks are heterogeneous and consequently have different sites. These models are called nonequilibrium models (Wu and Gschwend 1986 Miller and Pedit 1992 Pedit and Miller 1993 Fuller et al. 1993 Sparks 2003 Table 7.2). These models describe processes when a fast reaction (physical or chemical) is followed by one or more slower reactions. In these cases, Fick s second law is expressed—that the diffusion coefficient is corrected by an equilibrium thermodynamic parameter of the fast reaction (e.g., by a distribution coefficient), that is, the fast reaction is always assumed to be in equilibrium. In this way, the net processes are characterized by apparent diffusion coefficients. However, such reactions can be equally well described using Equation 1.126. 

It should be noted that Equations 1, 2, 3, 4, and 5 imply a homogeneous kinetic system. Coking in tubular reactors results from a combination of homogeneous and heterogeneous processes. As the kinetics of these processes are not well understood and as the quantitative yield of coke is several orders of magnitude smaller than other pyrolysis products, it is more convenient to model coke formation separately based on commercial operating data. 

Although DMS is only moderately soluble in rainwater, there has been some interest in the oxidation in the liquid phase, where modeling suggests that the multiphase reactions can be important. Ozone seems to be the most important oxidant, where there is a predicted lifetime for DMS of a few days in clouds. The oxidation reactions offer the potential for these heterogeneous processes to yield more soluble oxidized sulfur compounds such as DMSO and DMSO2 (Betterton, 1992 Campolongo e/u/., 1999). 

For some of the less complex ec processes, cyclic voltammetry (i-E) waves have been used to show, qualitatively, the effects of the follow-up reaction (reaction 2). The model first developed for this scheme consider only pseudo first-order conditions where the heterogeneous process was either reversible (6,7) or irreversible (7,8). Accordingly, when the homogeneous rate constant was very large, the i-E scan appeared similar to a conventional polargraphic wave without any peaks. The complications of Nicholson and Shain (7) provide cyclic i-E data that can be compared directly to experimental results. While these data can be used to fit the case represented by equations 1 and 2 (n - 1), more complex schemes... 

Reaction rates may be determined by the ease of intracrystalline transport to the surface, or by the chemical change on the surface. These surface reactions often resemble behaviour described in modelling heterogeneous catalytic processes and are usually reversible so that decomposition rates are sensitive to any gases present. Behaviour of reactants of the present group is similar to that of the oxides (Chapter 9), which are in the same class. Little information is available for other binary compounds, fluorides, chlorides, etc., which usually melt rather than decompose. 

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