Chemical kinetics, simplification

Maas, U., Efficient calculation of intrinsic low-dimensional manifolds for the simplification of chemical kinetics, Comput. Visualization Sci. 1 (1998) 69-82. 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

In our approach, we analyze not only the steady-state reaction rates, but also the relaxation dynamics of multiscale systems. We focused mostly on the case when all the elementary processes have significantly different timescales. In this case, we obtain "limit simplification" of the model all stationary states and relaxation processes could be analyzed "to the very end", by straightforward computations, mostly analytically. Chemical kinetics is an inexhaustible source of examples of multiscale systems for analysis. It is not surprising that many ideas and methods for such analysis were first invented for chemical systems. 

In the present chapter we discuss the different ways to represent chemical reactions in modeling reacting flows. The major emphasis is on detailed chemical kinetic models. The chapter deals with issues related to the development and use of reaction mechanisms for analysis of scientific and industrial problems. We attempt to deal with some collective aspects of mechanisms, such as rate-limiting steps, coupled/competitive reactions, and mechanism characteristics and simplifications. Specifically, we are concerned with the following issues ... 

A significant simplification of the algorithm is associated with applying chemical kinetic methods taken from the graphs theory. A graph is a geometrical scheme consisting of a set of points connected by lines. It can be a complex electric scheme, a railway network, a plan of constructional works or finally, a complex chemical reaction. 

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. 

It is generally assumed, for reasons of mathematical simplification, that one of these processes is slower than the rest and that this is the rate determining step. Because the reaction mechanism is rarely known in sufficient detail, steps 2, 3 and 4 are usually considered together. Where the rates of adsorption/reaction/desorp-tion are slower than the rate of diffusion, the true chemical kinetics will be observed. If, however, the reverse is true, diffusion kinetics will be observed and the results will not be characteristic of the chemical reactivity of the system. 

In the strict sense, the real reacting system could be mathematically described by chemical kinetic equations only taking into account the processes of diffusion and convection in self-generated non-uniform temperature fields. However, this problem, as we mentioned above, can be solved only at the expense of significant simplification of the chemical description. It is usually acceptable in studies of combustion and explosion processes where the chemical part may be reduced to a very schematic description, which defines the rate of heat... 

The differential diffusion coefficients are characteristic of any mechanically normal and chemically stable equilibrium mixture they represent properties of state. This reminds us of the necessity in (non-dilute) chemical kinetics of assuming a small change, so that the medium effects will not turn the rate constants into variables of time. This fact regarding the elementary description of a chemical reaction rate is not always explicitly stated in the texts. The reasons may be that a chemical change has often been conveniently measured only in a rather limited concentration range of the reactants and that most experiments have been confined to dilute solutions. If this simplification were not introduced, the kinematics in question would, for instance, contain partial volumes. 

Here we have used the approximation that can be replaced by Dj y and that variations of D y can be ignored within the averaging volume. The fact that only a single tortuosity needs to be determined by equations 1.152 and 1.153 represents the key contribution of this study. It is important to remember that this development is constrained by the linear chemical kinetic constitutive equation given by equation 1.113. The process of diffusion in porous catalysts is normally associated with slow reactions and equation 1.93 is satisfactory however, the first-order, irreversible reaction represented by equation 1.113 is the exception rather than the rule, and this aspect of the analysis requires further investigation. The influence of a non-zero mass average velocity needs to be considered in future studies so that the constraint given by equation 1.97 can be removed. An analysis of that case is reserved for a future study which will also include a careful examination of the simplification indicated by equation 1.117. 

Thermodynamics is important for describing chemical reactions. As seen before, it explains whether a reaction is voluntary and in which direction and in what state equilibrium is. Hence, thermodynamics also describes the mechanisms but without providing information on the exphcit pathway. Many reactions can be parallel and thereby competitive. In quantifying the overall chemical processes (production and/or decay of substances), because of the substantial mathematical simplifications, it is important to delete reactions of minor importance (i. e. those with reaction rates about two orders of magnitude less than the fastest reaction). The task of chemical kinetics is to describe the speed at which chemical reactions occur. The reactions rate is the change in the number of chemical particles per unit of time through a chemical reaction. This is the term ... 

While the issue of chemical kinetics can be avoided in the slow reaction regime (since the rate of reaction is so slow that its actual value needs not be known), and in the instantaneous reaction regime (since the rate of reaction is so fast that, again, its value needs not be known), it cannot be avoided in the case of the fast reaction regime. However, considerable simplifications arise also in this limiting case, and the following simple equation is obtained for the enhancement factor I ... 

Maas, U. (1998). Efficient Calculation of Intrinsic Low Dimensional Manifolds for Simplification of Chemical Kinetics, Comput. Visual Sci. Vol. 1, pp. 69-81. 

Thus, both the small parameters and the multidimensional nature complicate the investigation of mathematical models of chemical kinetics. On the other hand, the small parameters often lead to a simplification of similar problems of classical mechanics by means of asymptotic approximations. From this point of view, the role of the small parameters is positive. 

Among the known asymptotic tools the matching method seems to be the most powerful, because it is applicable to practically any problem where the separation of either time or spatial scales takes place. Used first for hydrodynamic problems, this method was later extended to different fields of mechanics, physics, and mathematics [8]. In particular, the matching method is well suited to the study of problems of chemical kinetics, where separation of the fast and slow processes occurs [9]. The main result of this approach is a simplification of the original problem up to the level where either explicit formulas or standard numerical simulations give valuable results. 

Glassmaker NJ. Intrinsic low-dimensional manifold method for rational simplification of chemical kinetics 1999. p. 1-37. Consulted in July 13,2010, http //www.nd.edu/ powers/nick. glassmaker.pdfindexpdf. 

The vibrational relaxation of molecules is an important process in non-equilibrium chemical kinetics. Out of the many different relaxation phenomena the simplest one involves diatomic molecules for which there is no complicated intramolecular energy transfer. If the degree of vibrational excitation is not too high, the most important processes correspond to one-quantum transitions. This results in considerable simplification of relaxation kinetics which simplifies even further under the condition of a constant translational temperature T. 

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