Kinetic modeling mathematical principles

In this section we will see how all these rules can be described mathematically by a single and simple kinetic model based on fundamental thermodynamic and catalytic principles."... 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

The replenishment of the vacancy can be directly from the gas phase or indirectly from the catalyst. In the latter case, the oxygen mobility within the catalyst is so large that bulk oxygen can diffuse to the vacancy. Then oxygen from the gas phase reoxidizes the lattice on sites which differ from hydrocarbon reaction sites. In a steady state, the rate of catalyst oxidation will be equal to the rate of reduction by the substrate. The steady state degree of reduction, equivalent to the surface coverage with oxygen, is determined by the ratio of these two rates. Kinetic models based on these principles are called redox models, for which the simplest mathematical expression is... 

Mathematical Models Diffusion principles have been traditionally recognized as the most important determinants in skin absorption. Thus, Pick s law of diffusion provided the mathematical basis for early kinetic descriptions of percutaneous absorption. Pick s law simply states that... 

The science of mechanics constitutes a vast number of sub-disciplines commonly considered beyond the scope of the standard chemical engineering education. However, when dealing with kinetic theory-, granular flow- and population balance modeling in chemical reactor engineering, basic knowledge of the principles of mechanics is required. Hence, a very brief but essential overview of the disciplines of mechanics and the necessary prescience on the historical development of kinetic theory are given before the more detailed and mathematical principles of kinetic theory are presented. 

Several methods and mathematical tools have been suggested and elaborated to reduce complex kinetic models to be more concise. We believe that it would be important to mention briefly the main guiding principles and criteria for such reduction. First of all, again we must stress that the tools and ways are caused by the goal—in this case the goal of reduction. 

The va/Mg-based approach significantly improves the effectiveness of procedures of controlling chemical reactions. Optimal control on the basis of the value method is widely used with Pontryagin s Maximum Principle, while simultaneously calculating the dynamics of the value contributions of individual steps and species in a reaction kinetic model. At the same time, other methods of optimal control are briefly summarized for a) calculus of variation, b) dynamic programming, and c) nonlinear mathematical programming. 

The methodology to answering these parameter estimation and set-based questions relies on different mathematical approaches. In principle, the parameter identification of chemical kinetic models can be posed as classical statistical inference [17,19-21] given a mathematical model and a set of experimental observations for the model responses, determine the best-fit parameter values, usually those that produce the smallest deviations of the model predictions from the measurements. The validity of the model and the identification of outliers are then determined using analysis of variance. The general optimizations are computationally intensive even for well-behaved, well-parameterized algebraic functions. Further complications arise from the highly ill-structured character... 

To start, we consider one-dimensional diffusion model. This is not only because water diffusion into polymer matrix is the first step but the mathematics of two- and three-dimensional diffusion is more complicated. In fact, the results obtained from one-dimensional diffusion model has been practically used whenever diffusion kinetics under investigation. The kinetic models obtained from one-dimensional theory are enough to cover the kinetic models often used by most researchers in the world. Two- or three-dimensional diffusion follows the same principle. 

Simplification not only is a means for the easy and efficient analysis of complex chemical reactions and processes, but also is a necessary step in understanding their behavior. In many cases, to understand means to simplify. Now the main question is Which reaction or set of reactions is responsible for the observed kinetic characteristics The answer to this question very much depends on the details of the reaction mechanism and on the temporal domain that we are interested in. Frequently, simplification is defined as a reduction of the original set of system factors (processes, variables, and parameters) to the essential set for revealing the behavior of the system, observed through real or virtual (computer) experiments. Every simplification has to be correct. As a basis of simplification, many physicochemical and mathematical principles/methods/approaches, or their efficient combination, are used, such as fundamental laws of mass conservation and energy conservation, the dissipation principle, and the principle of detailed equilibrium. Based on these concepts, many advanced methods of simplification of complex chemical models have been developed (Marin and Yablonsky, 2011 Yablonskii et al., 1991). 

Is it possible to mathematically derive the kinetic outcome (i.e., the TOF) of a gigantic and anarchic mechanistic network In principle it is, as long as we have all the kinetic rate constants or, equivalently, the Gibbs energies of each transition state and intermediate. This may not be really insightful, but can serve as a stepping stone to understand simpler kinetic models. 

The experimentalist often formulates a mathematical model in order to describe the observed behavior. In general, the model consists of a set of equations based on the principles of chemistry, physics, thermodynamics, kinetics and transport phenomena and attempts to predict the variables, y, that are being measured. In general, the measured variables y are a function of x. Thus, the model has the following form... 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

This intermediate scale affords a preliminary validation of the intrinsic kinetics determined on the basis of microreactor runs. For this purpose, the rate expressions must be incorporated into a transient two-phase mathematical model of monolith reactors, such as those described in Section III. In case a 2D (1D+ ID) model is adopted, predictive account is possible in principle also for internal diffusion of the reacting species within the porous washcoat or the catalytic walls of the honeycomb matrix. 

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