Analysis of Kinetic Models for Chemical Reaction Systems

Analysis of Kinetic Models for Chemical Reaction Systems 

The basis of the value approach is a Hamiltonian systematization of the mathematical model of a conq)lex reaction with the extraction of the characteristics (functional) of a target reaction and with the kinetic comprehension of conjugate variables. Regardless of a reaction s conplexity, pivotal in this approach is a universal numerical determination of kinetic trajectories of value units (the kinetic significance) of individual steps and chemical species of a complex reaction, according to the selected target reaction characteristic. 

In this and next Chapters fundamentals on the value analysis of chemieal reaction systems are stated. Identifieation of kinetie significance of the reaction steps and chemical species is of great importance in such an approach. In its turn this task is related with solving the following key issues cited in Chapter 2 and presented below schematically 

Within the fi-amework of the value approach, the kinetic significance of individual steps is determined by the value magnitudes. Specific feature of these values is that they are aimed at identifying an influence of the rate variation of steps or the the rate-of-production of reaction species on the magnitude of the output reaction parameter. 

Quantitative changes in the concentration of chemical species with time in a spatially homogeneous reaction system under isothermal conditions may be described by a system of ordinary differential equations 

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Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

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. 

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component... 

This chapter is aimed to show the importance of determining the kinetic significance of individual chemical steps to illuminate the essence of the reaction mechanism. A short overview is presented and a critical analysis is given for the existing methods that assess the significance of chemical species and individual steps in the kinetic models of chemical reaction systems. 

In many respects the diagram in Figure 3.1 determines the subsequent structure of contents. Before moving to interprete the main issues, let us mention briefly the principal statements serving as the basis for the development of a new analysis method for the kinetic models of chemical reaction systems. 

In a continuous-flow chemical reactor, the concern is not only with probabilistic transitions among chemical species but also with probabilistic liansitions of each chemical species between the interior and exterior of the reactor. Pippel and Philipp [8] used Markov chains for simulating the dynamics of a chemical system. In their approach, the kinetics of a chemical reaction are treated deterministically and the flow through the system are treated stochastically by means of a Markov chain. Shinnar et al. [9] superimposed the kinetics of the first order chemical reactions on a stochastically modeled mixing process to characterize the performance of a continuous-flow reactor and compared it with that of the corresponding batch reactor. Most stochastic approaches to analysis and modeling of chemical reactions in a flow system have combined deterministic chemical kinetics and stochastic flows. 

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