Chemical kinetics methods accuracy

NMR spectroscopy finds a number of applications in chemical kinetics. One of these is its application as an analytical tool for slow reactions. In this method the integrated area of a reactant, intermediate, or product is determined intermittently as the reaction progresses. Such determinations are straightforward and will not concern us further, except to note that the use of an internal standard improves the accuracy. With flow mixing, one may examine even more rapid reactions. This is simply overflow application of the stopped-flow method. 

The task of developing or extending a chemical kinetic model is facilitated since much of the necessary information is readily available. Section 13.3.1 deals with sources of thermodynamic and reaction-specific data. Once an elementary reaction is well characterized (i.e., the rate constant and product channel are known with sufficient accuracy), this information can be used in all reaction mechanisms where the reaction may be important. Large amounts of reaction specific data are now available, and methods for estimating and measuring elementary reaction rates have improved considerably over recent decades. 

Chemical kinetic models require as a minimum thermodynamic and reaction-specific information. If problems involve transport, also proper transport coefficients are necessary. Since the accuracy of a kinetic model is often associated specifically with the chemical reaction mechanism, it is important to note that also the thermodynamic data are essential for the reliability of predictions. Fortunately the quality and quantity of data on thermochemistry of species and on the kinetics and mechanisms of individual elementary reactions have improved significantly over the past two decades, because of advances made in experimental methods. This has facilitated considerably our ability to develop detailed chemical kinetic models [356],... 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Recent developments in chemical kinetics have been in the study of rapid reactions. Chemical events with half-lives as short as 10-7 seconds have been studied with reasonable accuracy. It is obvious that special methods must be used to study such labile reactions. 

There are many ways one can try to reduce the computational burden. Ideally, one would find numerical methods which are guaranteed to retain accuracy while speeding the calculations, and it would be best if the procedure were completely automatic i.e. it did not rely on the user to provide any special information to the numerical routine. Unfortunately, often one is driven to make physical approximations in order to make it feasible to reach a solution. Common approximations of this type are the quasi-steady-state approximation (QSSA), the use of reduced chemical kinetic models, and interpolation between tabulated solutions of the differential equations (Chen, 1988 Peters and Rogg, 1993 Pope, 1997 Tonse et al., 1999). All of these methods were used effectively in the 20th century for particular cases, but all of these approximated-chemistry methods share a serious problem it is hard to know how much error is... 

Simplification of a kinetic mechanism or the kinetic system of ODES is often required in order to facilitate finding solutions to the resulting equations and can sometimes be achieved based on kinetic simplification principles. In most cases, the solutions obtained are not exactly identical to those from the fuU system of equations, but it is usually satisfactory for a chemical modeller if the accuracy of the simulation is better than the accuracy of the measurements. For example, usually better than 1 % simulation error for the concentrations of the species of interest when compared to the original model is appropriate. Historically, simplifications were necessary before the advent of computational methods in order to facilitate the analytical solution of the ODEs resulting from chemical schemes. We begin here by discussing these early simplification principles. In later chapters, we will introduce more complex methods for chemical kinetic model reduction that may perhaps require the application of computational methods. 

Sikalo et al. (2014) compared several options for the application of genetic algorithms to mechanism reduction, exploring the trade-off between the size and accuracy of the resulting mechanisms. Information on the speed of solution was also taken into account, so that, for example, the least stiff system (Sect. 6.7) could be selected. An automatic method for the reduction of chemical kinetic mechanisms was suggested and tested for the performance of reduced mechanisms used within homogeneous constant pressure reactor and burner-stabilised flame simulations. The flexibility of this type of approach has clear utility when restrictions are placed on the number of variables that can be tolerated within a scheme in the computational sense. However, the development of skeletal mechanisms is rarely the end point of any reductiOTi procedure since the application of lumping or timescale-based methods can be applied subsequently. These methods will be discussed in later sections. 

Modem system theory [43] offers mathematical methods, which—on the basis of these variables together with thermodynamics, chemical kinetics and a reactor model—enable the estimation of additional state variables (e.g. Kalman-Bucy filter, Luenberger observer). The more accurate the process model which is assumed as a base, and the more numerous and meaningful the starting measured variables are, the greater is the number and accuracy of the additional, estimated state variables. Consequently, variables which are usually available (e.g. temperature, pressure) should, if possible, be supplemented by further variables which are more meaningful. 

Long before electronic computers were invented, it was realized that mathematical sophistication could be introduced into numerical integration in order to save computational elTort and improve accuracy. Textbooks of numerical analysis are full of ways to do this. The most popular of them, the Runge-Kutta and predictor-corrector algorithms, once were standard methods for numerical solution of the initial value problems of chemical kinetics. They have been replaced, however, by more suitable methods invented for the specific purpose of dealing with chemical kinetics problems. 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

The very basis of the kinetic model is the reaction network, i.e. the stoichiometry of the system. Identification of the reaction network for complex systems may require extensive laboratory investigation. Although complex stoichiometric models, describing elementary steps in detail, are the most appropriate for kinetic modelling, the development of such models is time-consuming and may prove uneconomical. Moreover, in fine chemicals manufacture, very often some components cannot be analysed or not with sufficient accuracy. In most cases, only data for key reactants, major products and some by-products are available. Some components of the reaction mixture must be lumped into pseudocomponents, sometimes with an ill-defined chemical formula. Obviously, methods are needed that allow the development of simple... 

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