Kinetic models integral equation

The values of the rate constants are estimated by fitting equations 1.4a and 1.4b to the concentration versus time data. It should be noted that there are kinetic models that are more complex and integration of the rate equations can only be done numerically. We shall see such models in Chapter 6. An example is given next. Consider the gas phase reaction of NO with 02 (Bellman et al. 1967) ... 

Initially, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A and B. However, given that all the reaction stoichiometric coefficients are unity, and the initial reaction mixture has equimolar amounts of A and B, it seems sensible to first try to model the kinetics in terms of the concentration of A. This is because, in this case, the reaction proceeds with the same rate of change of moles for the two reactants. Thus, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A. In principle, there are many other possibilities. Substituting the appropriate kinetic expression into Equation 5.47 and integrating gives the expressions in Table 5.5 ... 

Watanabe and Ohnishi [39] have proposed another model for the polymer consumption rate (in place of Eq. 2) and have also integrated their model to obtain the time dependence of the oxide thickness. Time dependent oxide thickness measurement in the transient regime is the clearest way to test the kinetic assumptions in these models however, neither model has been subjected to experimental verification in the transient regime. Equation 9 may be used to obtain time dependent oxide thickness estimates from the time dependence of the total thickness loss, but such results have not been published. Hartney et al. [42] have recently used variable angle XPS spectroscopy to determine the time dependence of the oxide thickness for two organosilicon polymers and several etching conditions. They did not present kinetic model fits to their results, nor did they compare their results to time dependent thickness estimates from the material balance (Eq. 9). More research on the transient regime is needed to determine the validity of Eq. 10 or the comparable result for the kinetic model presented by Watanabe and Ohnishi [39]. 

Several rate equations, some of them in the integrated form, have been used for a kinetic description of cracking. A critical comparison of them yielded the consistent kinetic model [206] discussed below. 

Due to a similarity of reaction stages in the Lotka and Lotka-Volterra models the equations for the pop and pop remain the same as in Section 8.2. Other kinetic equations are slightly simplified, a number and multiplicity of integrals are reduced. 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

Sufficient DO data were not obtained from basalt-synthetic Grande Ronde groundwater experiments to allow determination of a definitive rate law. A first order kinetic model with respect to DO concentration was assumed. Rate control by diffusion kinetics and by surface-reaction mechanisms result in solution composition cnanges with different surface area and time dependencies (32,39). Therefore, by varying reactant surface area, determination of the proper functional form of the integrated rate equation for basalt-water redox reactions is possible. 

Finally, the enzyme deactivation in the EMR could be modeled by a first-order linear dependence with respect to H2O2 addition rate [19]. The integration of this equation in the kinetic model allows simulation of the process efficiency as a function of H2O2 addition rate, HRT, and Orange II concentration in the influent, and helps determine the best operational conditions. 

Initial reaction rates obtained with a pure feed in which only reactants are present can be used for the discrimination between rival kinetic models, i.e. to identify whether adsorption, desorption, or surface reactions are the rate-determining steps. When pure A is fed to an integral reactor, for example, initial rates are observed at the inlet, where the product concentration is still zero. Comparing possible rate equations, which are often simpler in case of absence of products, with experimental data obtained at different concentrations of A, helps to reveal the appropriate [33,35]. 

There are a number of possible approaches to the calculation of influences of finite-rate chemistry on diffusion flames. Known rates of elementary reaction steps may be employed in the full set of conservation equations, with solutions sought by numerical integration (for example, [171]). Complexities of diffusion-flame problems cause this approach to be difficult to pursue and motivate searches for simplifications of the chemical kinetics [172]. Numerical integrations that have been performed mainly employ one-step (first in [107]) or two-step [173] approximations to the kinetics. Appropriate one-step approximations are realistic for limited purposes over restricted ranges of conditions. However, there are important aspects of flame structure (for example, soot-concentration profiles) that cannot be described by one-step, overall, kinetic schemes, and one of the major currently outstanding diffusion-flame problems is to develop better simplified kinetic models for hydrocarbon diffusion flames that are capable of predicting results such as observed correlations [172] for concentration profiles of nonequilibrium species. 

The establishment of a detailed kinetic model provides an opportunity for the numerical prediction of the behaviour of a chemical system under conditions that may not be accessible by experimental means. However, large-scale models with many variables may require considerable computer resource for their implementation, especially under non-isothermal conditions, for which stiffness of the system of differential equations for mass and energy to be integrated is a problem. Computation in a spatial domain, for which partial differential expressions are appropriate, becomes considerably more demanding. There are also many important fluid mechanical problems in reactive systems, the detailed kinetic representation of the chemistry for which would be highly desirable, but cannot yet be computed economically. In such circumstances there is a place for the use of reduced or simplified kinetic models, as discussed in Chapter 7. Thus,... 

To reversely check the kinetic model, the integral rate equation (for non-isothermal conditions) describing the As release during pyrolysis of CCA treated wood is used in combination with the measured temperature profiles T(t) in order to calculate the corresponding As content of the pyrolysis residues. The calculated arsenic content of the pyrolysis residues is compared with the experimental values labscale and TGA experiments) in the parity plot, presented in Figure 5. 

Using equations (3) and (4), equations (2a-2c) can be written as depending on only one variable. This variable should be easily monitored experimentally. Such a variable is the mass fraction of the gas G or the mass fraction of the solid residue (1 - G). The three equations (2a-2c) can be initially used to calculate the values for k-, k2 and a-). This can be done using a best-fit technique for experimental data that are assumed to be described by equations (2) in isothermal conditions. Once the values for k-, k2 and a- are known, the kinetics equations can be integrated and solved for any time t. This model has been successfully applied, for example, to describe the pyrolysis of cellulose and of pine needles [8]. In anal ical pyrolysis this model can be used to determine the amount of gas generated during pyrolysis. Also, analytical pyrolysis data can be used to fit the kinetics model for use in other practical applications. 

The most realistic description of the kinetics of a unimolecular reaction is given by the RRKM method,3,16 which has been successfully used in the investigations of a wide variety of reaction systems. However, the general equation of the RRKM method is quite complex, and values of the rate constant at a given temperature and pressure are obtained by numerical evaluation of a complicated integral expression. This is an important limitation of the RRKM method because kinetic modeling studies require a simple expression, best in an analytical form, convenient for estimating the rate constant under any experimental conditions of pressure and temperature. 

EROS handles concurrent reactions with a kinetic modeling approach, where the fastest reaction has the highest probability to occur in a mixture. The data for the kinetic model are derived from relative or sometimes absolute reaction rate constants. Rates of different reaction paths are obtained by evaluation mechanisms included in the rule base that lead to partial differential equations for the reaction rate. Three methods are available that cover the integration of the differential equations the GEAR algorithm, the Runge-Kutta method, and the Runge-Kutta-Merson method [120,121], The estimation of a reaction rate is not always possible. In this case, probabilities for the different reaction pathways are calculated based on probabilities for individual reaction steps. 

Note The integral forms of these kinetic models are shown differential forms are given by Galwey and Brown (29). These are the most frequently tested rate equations a limited number of alternatives infrequently appear in the literature. ID, 2D, and 3D refer to the number of dimensions in... 

A proper fit of the time-courses of some batch reactor experiments at different starting concentrations represents an appropriate test of the rate equation. This implies that numerical integration of the rate equation (e. g. by the Runge Kutta method11121), yielding a simulated time-course, has to fit the data of the measured time-course over the whole range of conversion (compare to Fig. 7-17 B). Examples of these methods will be given after the presentation of the basic kinetic models. 

The kinetic modelling of complex multiphase catalytic reactions needs a careful consideration of various complexities of adsorption and desorption of reactants and products. In such cases the kinetic model developed based on the initial rate data may not be adequate to explain the integral batch reactor performance. Hence it was thought appropriate to use mainly the integral rate data for developing a suitable kinetic model. Different rate equations were derived based on various Langumir-Hinshelwood mechanisms and a few of them are given below. 

Here D is the diffusion coefficient, t is the time, t is a dummy integration variable. Using Equation (8), respective T(t) dependencies can be obtained, while the Equations (l)-(7) serve as boundary condition for the diffusion model. This set of equations yield a quasi-equilibrium diffusion model which means that at a given surface pressure the composition of the surface layer under dynamic conditions is equal to that in the equilibrium. Another regime of adsorption kinetics, called kinetic model, can also be described by assuming compositions of the adsorption layer that can differ from the equilibrium state. The deviation of the adsorption layer from the equilibrium composition is the result of the finite rate of the transition process between the adsorption states. In case of two adsorption states we have6... 

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