Micro-kinetic Modeling

Micro-kinetic modeling represents the state of the art in describing the kinetics of catalytic reactions. It was pioneered by Stoltze and Norskov in the mid-1980s and was further explored by Dumesic and coworkers in the early 1990s [J.A. Dumesic, 

Apuvicio, J.E. Bekoske and A.A. Trevino, The Microkinetics of Heterogeneous Catalysis (1993), American Chemical Society, Washington DC]. Ideally, as many parameters as can be determined by surface science studies of adsorption and of elementary steps, as well as results from computational studies, are used as the input in a kinetic model, so that fitting of parameters, as employed in Section 7.2, can be avoided. We shall use the synthesis of ammonia as a worked example [P. Stoltze and J.K. Norskov, Phys. Rev. Lett. 55 (1985) 2502 J. Catal. 110 (1988) Ij. 

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As described above, the activity of a catalyst can be measured by mounting it in a plug flow reactor and measuring its intrinsic reactivity outside equilibrium, with well-defined gas mixtures and temperatures. This makes it possible to obtain data that can be compared with micro-kinetic modeling. A common problem with such experiments materializes when the rate becomes high. Operating dose to the limit of zero conversion can be achieved by diluting the catalyst with support material. 

Once the kinetic parameters of elementary steps, as well as thermodynamic quantities such as heats of adsorption (Chapter 6), are available one can construct a micro-kinetic model to describe the overall reaction. Otherwise, one has to rely on fitting a rate expression that is based on an assumed reaction mechanism. Examples of both cases are discussed this chapter. 

The first step in constructing a micro-kinetic model is to identify all the elementary reaction steps that may be involved in the catalytic process we want to describe, in this case the synthesis of ammonia. The overall reaction is... 

Finally, the constructed micro-kinetic model must of course be tested against measurements performed with real catalysts. Figure 7.23 shows a plot of the calculated output from the reactor against experimental values. Apparently, the micro-kinetic model describes the situation very well. This does not prove that the model is correct since models based on another series of elementary steps might also work. 

Overall, catalytic processes in industry are more commonly described by simple power rate law kinetics, as discussed in Chapter 2. However, power rate laws are simply a parameterization of experimental data and provide little insight into the underlying processes. A micro-kinetic model may be less accurate as a description, but it enables the researcher to focus on those steps in the reaction that are critical for process optimization. 

A micro-kinetic model based on 13 elementary steps, of which the first 8 relate to the water-gas shift reaction, describes the process well ... 

A full analysis of the rate expression reveals that all data on the Cu(lOO) single crystal are modeled very well, as shown in Fig. 8.10. Even more important is that the model also describes data obtained on a real catalyst measured under considerably different conditions reasonably well, indicating that the micro-kinetic model captures the most important features of the methanol synthesis (Fig. 8.11). 

Figure S.11. Comparison between the predictions of a micro-kinetic model and measurements on a Cu(lOO) model catalyst with a real methanol synthesis catalyst. The full line represents the ideal match between model and experiment. [Adapted from P.B. Rasmussen, P.M. Holmblad, T. Askgaard,...

The micro-kinetic model also predicts the coverages of the various intermediates on the surface. As shown in Tab. 8.2, the approximation of the surface being dean is quite reasonable. The highest coverages are observed for hydrogen and formate, but the majority of sites are free, even at 50 bar. 

The contribution of different crystal planes to the overall surface area of the particle can thus be calculated and is shown in Fig. 8.12(b). The results have been included in a dynamical micro-kinetic model of the methanol synthesis, yielding a better description of kinetic measurements on working catalysts [C.V. Ovesen, B.S. Clausen, J. Schiotz, P. Stoltze, H. Topsoe and J.K. Norskov, J. Catal. 168 (1997) 133]. 

The low temperature water-gas shift reaction is well described by a micro-kinetic model [C.V. Ovesen, B.S. Clausen, B.S. Hammershoj, G. Sreffensen, T. Askgaard, I. Chorkendorffi J.K. Norskov, P.B. Rasmussen, P. Stoltze and P.J. Taylor,/. Catal. 158 (1996) 170] and follows to a large extent the scheme in Eqs. (23-31). The analysis revealed that formate may actually be present in nonvanishing amounts at high pressure (Fig. 8.18). 

The kinetics of the ammonia synthesis have been discussed as an example of micro-kinetic modeling in Chapter 7. Here we present a brief description of the process, concentrating on how process variables are related to the microscopic details and the optimization of the synthesis. 

Explain the principles of micro-kinetic modelling and its relevance to research in catalysis. 

The link between the microscopic description of the reaction dynamics and the macroscopic kinetics that can be measured in a catalytic reactor is a micro-kinetic model. Such a model will start from binding energies and reaction rate constants deduced from surface science experiments on well defined single crystal surfaces and relate this to the macroscopic kinetics of the reaction. 

If one can understand what the basic parameters of the reactants and the surface are that determine the reaction dynamics (activation barriers etc.) then given a micro-kinetic model one has a knowledge of the factors determining the catalytic activity of the catalyst. 

If the aim is to explore the mechanism of the reaction and understand which are the important parameters of the catalyst determining the activity, then a micro-kinetic model is needed. A micro-kinetic model is based on a detailed mechanism and independent information about the rates of the elementary steps involved and the stability of the intermediates. The micro-kinetic model is the synthesis of all the basic knowledge about a reaction over a given catalyst. 

The kinetic model may be formulated using kinetic equations for all steps or using equilibrium equations for all but the slowest steps. The latter approach reduces the computational effort and leads to a kinetic expression, which is far easier to analyze. However, if a step, which is slow in reality, is modeled by an equilibrium equation, the micro-kinetic model becomes unrealistic and it may in some cases be the simplest to treat a problematic step by a kinetic equation. 

The input into a micro-kinetic model may be measured or calculated. 

In spite of the shortcomings of the modelling, the real strength is that they can be used to understand ( em variations in the catalytic activity from one system to another. The stability of the intermediates and the activation barriers are among the input parameters for the micro-kinetic model, and it is straight forward to calculate the effects of changes in stability for some or all the intermediates. 

The activation enthalpy for the catalytic reaction may be calculated from the micro-kinetic model as... 

Unfortunately, no quantitative data on any of these reaction steps exist, which are obtained under strict control of the experimental variables as in usual surface science experiments with metallic surfaces. Micro-kinetic modeling which could support in a quantitative way the picture derived so far has to await such well-defined experiments. 

Even though the authors could not avoid some adjustment of selected kinetic parameters, what is explicable taking into account the extraordinary complexity of the system. As a result, they succeeded in reproducing in their simulations some important features of the real system and validated their micro-kinetic model against high-pressure spatially resolved experimental data for catalytic partial oxidation of methane. 

A detailed computational model was developed for several different iimovative designs for the preferential carbon monoxide (CO) oxidation reactor using a kinetic mechanism and reaction sequence derived from a micro-kinetic model and literature data for the specific adsorption coefficients and kinetic parameters for a platinum-based catalyst. 

This approach started to be developed in the 70s, when more powerful computers became available and an input in the computer program required all the elementary reactions. Then such programs constructed matrices of mass balance equations, comprising all of the components in the reactions (in the case of heterogeneous catalysis, this included the surface and the adsorbed species) and solved these equations by Newton-Raphson iteration. Later on this approach was refined and coined micro-kinetic modeling. 

Surface segregation of Pd in Pd—Rh catalysts suppresses NOx reduction [61]. De Sarkar and Khanra studied the segregation difference between Pd—Rh and Pt—Rh nanoparticles, and the influence of sulfur in fuel on CO oxidation and NO. They used Monte-Carlo (MC) simulation to predict the surface composition of PtsoRhso and PdsoRhso particles (2406 atoms for 4nm particles). TTiey used a micro-kinetic model to compare the activities of both soHds for reactions of CO -i- O2, CO -I- NO and CO -1- NO -1- O2, and found that Pt and Pd segregate to the particles surface, especially in the Pd catalyst, which is clearly better for CO oxidation, while Pt—Rh is a better catalyst for NO reduction. For both reactions, sulfur poisons the Pd—Rh catalyst more than the Pt—Rh catalyst [62]. 

This paper discusses research efforts towards the prediction of hydrocarbon product distribution for the Fischer-Tropsch synthesis (FTS) on a cobalt-based catalyst using a micro-kinetic model taken fiom the literature. In the first part of the study, a MATLAB code has been developed which uses the Genetic Algorithm Toolbox to estimate parameter values for the kinetic model. The second part of the study describes an ongoing experimental campaign to validate the model predictions of the fixed-bed reactor FTS product distribution in both conventional (gas phase) and non-conventional (near-critical and supercritical phase) reaction media. 

Todic et al. [14] developed a comprehensive micro-kinetic model based on the carbide mechanism that predicts FT product distribution up to carbon number 15. This model explains the non-ASF product distribution using a carbon number dependent olefin formation rate (e term). The rate equations for the olefins and paraffins used in the model are shown in Figure 2. The derivation of the rate equations and physical meaning of the kinetic parameters, as well as their fitted values, can be found in Todic et al. [14]. In the current study, a MATLAB code which uses the Genetic Algorithm Toolbox has been developed, following the method of Todic et al. [14], to estimate the kinetic model parameters. In order to validate our code, model output from Todic et al. [14] was used as the input data to our code, and the kinetic parameter values were back-calculated and compared to the values fi om [14], as shown in Table 1. The model has 19 kinetic parameters that are to be estimated. The objective function to be minimized was defined as... 

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