Types of kinetic models

Recently, Praharso et al also developed a Langmuir-Hinshelwood type of kinetic model for the SR kinetics of i-Cg over a Ni-based catalyst. In their model, it was assumed that both the hydrocarbon and steam dissociatively chemisorb on two different dual sites on the catalyst surface. The bimolecular surface reaction between dissociated adsorbed species was proposed as the ratedetermining step. The following generalized rate expression was proposed ... 

A reverse kinetic problem consists in identifying the type of kinetic models and their parameters according to experimental (steady-state and unsteady-state) data. So far no universal method to solve reverse problems has been suggested. The solutions are most often obtained by selecting a series of direct problems. Mathematical treatment is preceded by a qualitative analysis of experimental data whose purpose is to reduce drastically the number of hypotheses under consideration [31]. 

A redox type of kinetic model was developed in which the rate of reaction for any of the N species, Rjj is expressed in the form... 

The allowance for the short-range effects has been carried out in two types of kinetic models (Kuchanov, 1992). In the first of them, the reactivity of a macroradical is presumed to be dependent on the types of n monomeric units preceding the terminal one. Here the mathematical formalism differs from that used in the case of the ideal model only in one point. The states of the Markov chain are associated in the framework of these models with monomeric units, each supplied by the label specifying the type of sequence U, of units acting upon the reactivity of the active center. 

The shape of the mass loss and mass loss rate curves determine the types of kinetic models that can be applied to fit the data. Simple kinetic models, which only include one reaction, may be applied to describe the pyrolysis of wheat straw, but cannot describe the two peaks observed in the pyrolysis of washed wheat straw. 

For any industrial reacting system, the relevant parameters appearing in the rate expression (Eq. (5.14)) need to be obtained by carrying out experiments under controlled conditions. It is necessary to ensure that physical processes do not influence the observed rates of chemical reactions. This is especially difficult when chemical reactions are fast. It may sometimes be necessary to employ sophisticated mathematical models to extract the relevant kinetic information from the experimental data. Some references covering the aspects of experimental determination of chemical kinetics are cited in Chapter 1. It must be noted here that in the above development, the intrinsic rate of all chemical reactions is assumed to follow a power law type model. However, in many cases, different types of kinetic model need to be used (for examples of different types of kinetic model, see Levenspiel, 1972 Froment and Bischoff, 1984). It is not possible to represent all the known kinetic forms in a single format. The methods discussed here can be extended to any type of kinetie model. 

Let us consider now the other type of kinetic model, nan iy the system of differential equations, and the ways of utilizing this model for evaluating the complexity of linear and non linear mechanisms. The system of kinetic differential equations is given in matrix fclosed system at constant volume ... 

The approach adopted in NANCY will be discussed in the following, this possesses both an algorithmic aspect, using an iterative method based on a systematic activation of generic reactions, and an expert system aspect, as the user can define the type of kinetic model which he or she wishes to generate. 

The type of kinetic model to be used depends on the type of reaction considered. For a homogeneous reaction occurring in the bulk of the fluid, a power-law kinetic model is often appropriate (see, e.g., [79]). In such models the rate of a certain reaction depends on a product of powers of the species concentration. On the other hand, heterogeneously catalyzed reactions are often conducted in microreactors. In a strict sense, power-law kinetics does not capture the dynamics of such processes over the full range of pressure, temperature and concentrations. Rather, a more complicated kinetic model of, e.g., Langmuir-Hinshelwood type [80] would have to be used. Nevertheless, power-law kinetics is frequently applied to heterogeneously catalyzed processes in a limited parameter range to simplify the description. 

The simulator is able to treat different heterogeneous reaction kinetics, depending on the reaction rate and its character. For example, a detailed model for the heterogeneous catalyst mass-transfer efficiency can be used, which is based on the approach of [99]. When applying this type of kinetic model, the intrinsic kinetics data are needed (see Section 10.3.1). Another way is the pseudohomoge-neous approach with effective kinetics expressions, by which the kinetics description is introduced as source terms into the balance equations (see Eqs. (10.3) and (10.4)). 

This method is known as the Walker plot method (Walker and Schmidt, 1944), and it is often used in the area of biological waste water treatment (Wilderer, 1976). The Walker plot has the advantage that, for various types of kinetic models (power law equations with different reaction orders as well... 

There are several alternatives to a graphical solution for the type 2 kinetic equations (Fig. 4.12) that form the basic type of kinetic model. In Fig. 4.24c, the commonly used Lineweaver-Burk (1934) linearization is presented by Equ. 4.42 ... 

This assumption is implicitly present not only in the traditional theory of the free-radical copolymerization [41,43,44], but in its subsequent extensions based on more complicated models than the ideal one. The best known are two types of such models. To the first of them the models belong wherein the reactivity of the active center of a macroradical is controlled not only by the type of its ultimate unit but also by the types of penultimate [45] and even penpenultimate [46] monomeric units. The kinetic models of the second type describe systems in which the formation of complexes occurs between the components of a reaction system that results in the alteration of their reactivity [47-50]. Essentially, all the refinements of the theory of radical copolymerization connected with the models mentioned above are used to reduce exclusively to a more sophisticated account of the kinetics and mechanism of a macroradical propagation, leaving out of consideration accompanying physical factors. The most important among them is the phenomenon of preferential sorption of monomers to the active center of a growing polymer chain. A quantitative theory taking into consideration this physical factor was advanced in paper [51]. 

Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications (refer to the examples in Chapters 11 through 16). These models are conceptually attractive because a general model for any system size can be developed even before the system is constructed. A detailed exposition of fundamental mathematical models in chemical engineering is beyond our scope here, although we present numerous examples of physiochemical models throughout the book, especially in Chapters 11 to 16. Empirical models, on the other hand, are attractive when a physical model cannot be developed due to limited time or resources. Input-output data are necessary in order to fit unknown coefficients in either type of the model. 

When a simple, fast and robust model with global kinetics is the aim, the reaction kinetics able to predict correctly the rate of CO, H2 and hydrocarbons oxidation under most conditions met in the DOC consist of semi-empirical, pseudo-steady state kinetic expressions based on Langmuir-Hinshelwood surface reaction mechanism (cf., e.g., Froment and Bischoff, 1990). Such rate laws were proposed for CO and C3H6 oxidation in Pt/y-Al203 catalytic mufflers in the presence of NO already by Voltz et al. (1973) and since then this type of kinetics has been successfully employed in many models of oxidation and three-way catalytic monolith converters... 

Solutions for this type of kinetics can only be achieved numerically. Model calculations with constant kinetic parameters have been made [H. Wiedersich, et al. (1979)], however, the modeling of realistic transport (diffusion) coefficients which enter into the flux equations is most difficult since the jump rate vA vB. Also, the individual point defects have limited lifetimes which determine the magnitude of correlation factors (see Section 5.2.2). Explicit modeling for dilute or non-dilute alloys can be found in [A.R. Allnatt, A.B. Lidiard (1993)]. 

The kinetic model proposed by Ng and Manas-Zloczower (1989) for the copolymerization of unsaturated polyester (UP) with styrene (S) is one example of this type of hybrid model. Relevant species considered in the analysis are C=C double bonds of the UP (E) styrene (S) initiators (In 1,2,..., N) and an inhibitor (Z). The model is based on the following hypotheses ... 

Kinetic models referred to as adsorption models have been proposed, especially for olefin polymerisation with highly active supported Ziegler-Natta catalysts, e.g. MgCl2/ethyl benzoate/TiCU AIR3. These models include reversible processes of adsorption of the monomer (olefin coordination at the transition metal) and adsorption of the activator (complexation via briding bonds formation). There are a variety of kinetic models of this type, most of them considering the actual monomer and activator concentrations at the catalyst surface, m and a respectively, described by Langmuir-Hinshelwood isotherms. It is to be emphasised that M and a must not be the same as the respective bulk concentrations [M] and [A] in solution. Therefore, fractions of surface centres complexed by the monomer and the activator, but not bulk concentrations in solution, are assumed to represent the actual monomer and activator concentrations respectively. This means that the polymerisation rate equation based on the simple polymerisation model should take into account the... 

Bonn6 [1] studied reaction kinetics of tetraphenylporphyrin model compounds over several catalysts and concluded that hydrogenation and hydrogenolysis occur on different sites. Therefore the reaction mechanism of VO-TPP hydrodemetallisation is described by a two-site model. It was also found that a Langmuir-Hinshelwood type of kinetics applies with small inhibition by hydrogen. 

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