Methods for Solving Kinetic Equations

When equilibrium is reached, the forward and reverse reactions have the same rates. This corresponds to a formulation of the principle of microscopic reversibility. Thus, at equilibrium, the concentrations of reactant and product do not change with time 

The same ideas can be applied to reversible reactions that occur in various steps. The principle of microscopic reversibility can help resolve more complicated systems of kinetic equations when the system is in equilibrium. 

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METHODS FOR SOLVING KINETIC EQUATIONS 4.3.1 Laplace transforms... 

Methods for Solving Kinetic Equations by the product, as described by the scheme... 

E. A. Novikov, Numerical methods for solving differential equations of chemical kinetics. Mathematical methods for chemical kinetics, Nauka, Novosibirsk, Russia, 1990, pp. 53-68 (Russian). 

An analogous method for solving the inverse problem allows us to determine the constants in a kinetic equation using other experimental data for example, the results of measurements of the T(t) dependence at different points on an article (or model sample) can be successfully used for this purpose. 

For electron transfer processes with finite kinetics, the time dependence of the surface concentrations does not allow the application of the superposition principle, so it has not been possible to deduce explicit analytical solutions for multipulse techniques. In this case, numerical methods for the simulation of the response need to be used. In the case of SWV, a semi-analytical method based on the use of recursive formulae derived with the aid of the step-function method [26] for solving integral equations has been extensively used [6, 17, 27]. 

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

The initial concentrations of each of the three species then can be calculated by measuring C, at three reaction times and solving the resulting three simultaneous equations. A method for the kinetic determination of five peroxides by reaction with selected sulfides has been devised that involves five equations and five unknowns. 

Mathematical solution of kinetic equations often requires numerical methods for solving simultaneous differential equations. In a few cases, exact analytical solutions exist. 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

Application of numerical methods have been rather seldom in studies of adsorption kinetics from micellar solutions. The main difficulties are probably connected with the large number of independent parameters. The first work belongs to Miller [146]. Fainerman and Rakita also published numerical results of the solution of the boundary value problem (5.236), (5.237), (5.245) [85]. Recently Danov et al. proposed an original method for solving the boundary value problem for the diffusion of micelles and monomers [92]. The system of equations was reduced to a system of ordinary differential equations by using a model concentration profile in the bulk phase. The obtained results agree better with dynamic surface tensions of micellar solutions than equation (5.248). 

A.17 One method for solving Newton s equations of motions for a mechanical system employs a function called the Lagrangian L, commonly defined as the kinetic energy K minus the potential energy U ... 

In reactive distillation, chemical reactions are assumed to occur mainly in the liquid phase. Hence the liquid holdup on the trays, or the residence time, is an important design factor for these processes. Other column design considerations, such as number of trays or feed and product tray locations, can be of particular importance in reactive distillation columns. Moreover, because chemical reactions can be exothermic or endothermic, intercoolers or heaters may be required to maintain optimum stage temperatures. Column models of reactive distillation must include chemical reaction equilibrium or kinetic equations along with the material and energy balance equations and the phase equilibrium relations. These models and methods for solving them are discussed in Chapter 13. 

Numerical Solution of the Direct Problem in Chemical Kinetics Table 3.1 Numerical methods for solving ordinary differential equations in Maple suite ... 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

The kinetics of hydrogenation of phenol has already been studied in the liquid phase on Raney nickel (18). Cyclohexanone was proved to be the reaction intermediate, and the kinetics of single reactions were determined, however, by a somewhat simplified method. The description of the kinetics of the hydrogenation of phenol in gaseous phase on a supported palladium catalyst (62) was obtained by simultaneously solving a set of rate equations for the complicated reaction schemes containing six to seven constants. The same catalyst was used for a kinetic study also in the liquid phase (62a). 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

In this chapter we consider how to construct reaction models that are somewhat more sophisticated than those discussed in the previous chapter, including reaction paths over which temperature varies and those in which species activities and gas fugacities are buffered. The latter cases involve the transfer of mass between the equilibrium system and an external buffer. Mass transfer in these cases occurs at rates implicit in solving the governing equations, rather than at rates set explicitly by the modeler. In Chapter 16 we consider the use of kinetic rate laws, a final method for defining mass transfer in reaction models. 

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