Kinetic simplification principles

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. 

Whilst it is quite straightforward to comprehend the applicability of the previous three basic kinetic simplification principles, the QSSA is not so easy to understand. For example, it may seem strange that the solution of a coupled system of algebraic differential equations can be very close to the system of ODEs. Another surprising feature is that the concentrations of QSS-species can vary substantially over time for example, the QSSA has found application in oscillating systems (Tomlin et al. 1992). The key to the success of the QSSA is the proper selection of the QSS-species based on the error induced by its application. The interpretation of the QSSA and the error induced by the application of this approximation will be discussed fully in Sect. 7.8. 

However, there are other features of the kinetic system of differential equations that may simplify the situation. The application of kinetic simplification principles (see Sect. 2.3) may result in the situation where it is not that the individual parameters have an influence on the solution, but only some combinations of these parameters. A simple example occurs when species B is a QSS-species within the A B C reaction system, and its concentration depends only on ratio kilk2-Also, when the production rate of species C is calculated using the pre-equilibrium approximation (see Sect. 2.3.2) within reaction system A B C, it depends only on equilibrium constant K = kjk2 and does not depend on the individual values of ki and 2-... 

It is in principle possible for a free enzyme to promote reaction in a geochemical system, but enzyme kinetics are invoked in geochemical modeling most commonly to describe the effect of microbial metabolism. Microbes are sometimes described from a geochemical perspective as self-replicating enzymes. This is of course a considerable simplification of reality, as we will discuss in the following chapter (Chapter 18), since even the simplest metabolic pathway involves a series of enzymes. 

There exists a multitude of possibilities to monitor hydrogenations in various pressure ranges. In principle, isochoric and isobaric techniques are feasible. In the latter case, the kinetics allows simplification because the concentration of... 

The analysis of real biological systems may be introduced in idealized simplifications using the principles of physics, chemistry, biology, thermodynamics, and kinetics. The following examples are the simple application of these principles in describing some biological processes. 

Since c% is a thermodynamic quantity, its calculation can be made, in principle, by the methods of statistical thermodynamics. This is an enormous simplification of the kinetic problem. The fundamental assumption of transition-state theory is that if now the products are removed from the system at equilibrium, the rate of the reaction in one direction, A-J-B— C-J-D, is still given by the expression (2.3.1) prevailing at equilibrium ... 

It can be seen that, if there is a rate-determining step, only the rate constants of that step enter into the rate equation. The rate constants of the other steps in quasi-equilibrium appear only as ratios which, as is already known, are equal to the equilibrium constants of these equilibrated steps. This is a considerable simplification of the kinetic problem, since, at least in principle, equilibrium constants are more easily arrived at than rate constants. Even when this is not so, the number of arbitrary constants in the rate equation is reduced considerably. 

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). 

Principle of critical simplification. In accordance with this principle (Yablonsky et al., 2003), the behavior near critical points, for instance ignition or extinction points in catalytic combustion reactions, is governed by the kinetic parameters of only one reaction—adsorption for ignition and desorption for extinction— which is not necessarily the rate-limiting one. 

The principle of critical simplification was first explained by Yablonsky et al. (Yablonskii and Lazman, 1996 Yablonsky et al., 2003) using the catalytic oxidation reaction as an example. The authors presented a dramatic simplification of the kinetic model for this reaction at critical conditions relating to bifurcation points. In this section, results obtained by GoTdshtein et al. (2015) are also used. 

Two basic approaches are often used for fluidized bed reactor modeling. One approach is based on computational fluid dynamics developed on the basis of the mass, momentum, and energy balance or the first principle coupled with reaction kinetics (see Chapter 9). Another approach is based on phenomenological models that capture the main features of the flow with simplifications by assumption. The flow patterns of plug flow, CSTR (continuous-stirred tank reactor). 

We will assume that the water-gas-shift reaction proceeds via the hypothetical sequence of elementary reactions shown below. This sequence is a major simplification of what actually occurs on a comm ial WGS catalyst However, this sequence will illustrate the important principles of catalytic kinetics, without requiring much algebra. 

To gain the analytical solution of the equations of gas kinetics of aerosols rather difficultly in connection with essential complication as differential equations of motion and energy, and equations boundary and entry conditions. It leads to necessity to induct simplification, to replace exact, but difficult communications between magnitudes approached, but more simple. At the heart of such simplifications lie transition from usual variables to generalized and determination of principles or modeling conditions. 

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