Kinetic simulations mean-field approach

Real catalytic reactions upon solid surfaces are of great complexity and this is why they are inherently very difficult to deal with. The detailed understanding of such reactions is very important in applied research, but rarely has such a detailed understanding been achieved neither from experiment nor from theory. Theoretically there are three basic approaches kinetic equations of the mean-field type, computer simulations (Monte Carlo, MC) and cellular automata CA, or stochastic models (master equations). 

Despite its severe limitations, the model shows interesting behavior, including kinetic phase transitions of two types continuous (second order) and discontinuous (first order). These phenomena are observed in many catalytic surface reactions. For this reason, the ZGB model has been widely studied and serves as a starting point for many more realistic models. This forms the first reason why we discuss the ZGB model in this section. The second reason is that MC simulations and mean-field (MF) solutions for this model give different results. Cluster approximations to the MF solutions offer a better agreement between the two methods, and then only small discrepancies remain. The ZGB model is therefore a nice example to illustrate the differences between the two approaches. 

The main application that was discussed was a microscopic model for the oxidation of CO, catalyzed by a Pt(lOO) single crystal surface. The simulations show kinetic oscillations as well as spatio-temporal pattern formation in the form of target patterns, rotating spirals and turbulent patterns. Finally, mean-field simulations of the same model were compared with the Monte Carlo simulations. When diffusion is fast and the simulation grids are small, the results of Monte Carlo simulations approach those of the mean-field simulations. 

The balance equations for 0, 9oh, and 6co were formulated and solved with two approaches a mean-field model with nucleation processes on active sites and kinetic Monte Carlo simulations, as illustrated in Figure 3.9. 

These models require information about mean velocity and the turbulence field within the stirred vessels. Computational flow models can be developed to provide such fluid dynamic information required by the reactor models. Although in principle, it is possible to solve the population balance model equations within the CFM framework, a simplified compartment-mixing model may be adequate to simulate an industrial reactor. In this approach, a CFD model is developed to establish the relationship between reactor hardware and the resulting fluid dynamics. This information is used by a relatively simple, compartment-mixing model coupled with a population balance model (Vivaldo-Lima et al., 1998). The approach is shown schematically in Fig. 9.2. Detailed polymerization kinetics can be included. Vivaldo-Lima et a/. (1998) have successfully used such an approach to predict particle size distribution (PSD) of the product polymer. Their two-compartment model was able to capture the bi-modal behavior observed in the experimental PSD data. After adequate validation, such a computational model can be used to optimize reactor configuration and operation to enhance reactor performance. 

In line with the proposed approach, which is outlined in Figure 3.4, a non-reactive CFD simulation is performed first using commercial software (ANSYS Fluent in this case), in which the feed stream consists of an inert gas (e.g., argon and nitrogen) with the same flow rate and temperature of the actual reacting mixture (laminar, stationary, and monocomponent model). In the second phase, the flow field is exported from the commercial code by means of a user-defined function (UDF). In the third and final phase, exported data are introduced into a C-F-F code, which solves the transport equations, taking into account both chemical kinetics and reactor fluid dynamics. 

Gas transport in nano-confinements can significantly deviate from the kinetic theory predictions due to surface force effects. Kinetic theory-based approaches based on the assumption of dynamic similarity between nanoscale confined and rarefied flows in low-pressure environments by simply matching the Knudsen and Mach numbers are incomplete. Molecular dynamics simulations of nanoscale gas flows in the early transition and free-molecular flow regimes reveal that the wall force field penetration depth should be considered as an important length scale in nano-confined gas flows, in addition to the channel dimensions and gas mean free path. 

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