Reaction mathematical modeling

Chemistry, Organic—Synthesis. 2. Chemical reactions—Mathematical models. I. Christoph, B. II. Series,... 

Chemical reactions—Mathematical models. 2. Chemical reactions—Computer simulation. 1. Caracotsios, Michael. II. Title. 

I. Multiphase flow - Mathematical models. 2. Chemical reactions - Mathematical models. 3. Transport theory. 4. Dispersion - Mathematical models. 1. Fox, Rodney O., 1959-... 

Chemical reactions — Mathematical models I. Title II. Toth, J. III. Series 541.3 9 0724 QD501... 

Let us consider one more important aspect. Simplification of the reaction mechanism by elimination of unimportant constituents is one of the methods for the reduction of its mathematical dimensionality [35-40]. In other words, it leads to the reduction of the number of differential (kinetic) equations to be integrated, underlying the reaction mathematical model. Thus, it facilitates computational procedures, as well as it analyses the kinetic model, in terms of both its use of predicting the reaction behavior under new conditions and for its control. Not less important is also that at the available accuracy for the rate constants of individual steps, the descriptive capacity of kinetic models may decrease as its complexity increases (this question will be discussed more thoroughly in Section 3.3). 

Obviously, a reaction mathematical model is an equation system... 

The most significant stoichiometric matrix tool, which plays an important role in solving kinetic problems, is its rank. As it is known, a matrix rank defines the number of its linearly independent rows or columns. Using of the notion of a matrix rank allows to reduce the number of differential equations in a reaction mathematical model and, thereby, to make solving the direct and inverse kinetic problems easier. For example, let us consider a reaction scheme ... 

The direct problem of chemical kinetics always has an analytical solution if a reaction mathematical model is a linear system of ordinary first-order differential equations. Sequences of elementary first-order kinetic steps, including ones complicated with reversible and competitive steps, correspond to such mathematical models. Let us mark off the classical matrix method firom analytical methods of solving such ODE systems. 

An R-matrix has a series of interesting matheinatical properties that directly reflect chemical laws. Thus, the sum of all the entries in an R-matrix must be zero, as no electrons can be generated or annihilated in a chemical reaction. Furthermore, the sum of the entries in each row or column of an R-matrix must also he zero as long as there is not a change in formal charges on the corresponding atom. An elaborate mathematical model of the constitutional aspects of organic chemistry has been built on the basis of BE- and R-matriccs [17. ... 

The production of acetic acid from butane is a complex process. Nonetheless, sufficient information on product sequences and rates has been obtained to permit development of a mathematical model of the system. The relationships of the intermediates throw significant light on LPO mechanisms in general (22). Surprisingly, ca 25% of the carbon in the consumed butane is converted to ethanol in the first reaction step. Most of the ethanol is consumed by subsequent reaction. 

Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinefics) are frequently employed in optimization apphcations. These models are conceptually attractive because a gener 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... 

The reacting sohd is in granular form. Decrease in the area of the reaction interface occurs as the reaction proceeds. The mathematical modeling is distinguished from that with flat surfaces, which are most often used in experimentation. 

One mathematical model of the oxidation of nickel spheres was confirmed when it took into account the decrease in the reaction surface as the reaction proceeded. 

In this work the development of mathematical model is done assuming simplifications of physico-chemical model of peroxide oxidation of the model system with the chemiluminesce intensity as the analytical signal. The mathematical model allows to describe basic stages of chemiluminescence process in vitro, namely spontaneous luminescence, slow and fast flashes due to initiating by chemical substances e.g. Fe +ions, chemiluminescent reaction at different stages of chain reactions evolution. 

The mathematical model was based on the scheme utilized in chemiluminescent method that was supplement with the reactions of radicals, formed of inhibitor molecules - AO. 

A numerical value is obtainable by integrating the trend curve for the flow received from the Flow Recorder (FR), from the start of the reaction to a time selected. Doing this from zero to each of 20 equally spaced times gives the conversion of the solid soda. Correlating the rates with the calculated X s, a mathematical model for the dependence of rate on X can be developed. 

To facilitate the use of methanol synthesis in examples, the UCKRON and VEKRON test problems (Berty et al 1989, Arva and Szeifert 1989) will be applied. In the development of the test problem, methanol synthesis served as an example. The physical properties, thermodynamic conditions, technology and average rate of reaction were taken from the literature of methanol synthesis. For the kinetics, however, an artificial mechanism was created that had a known and rigorous mathematical solution. It was fundamentally important to create a fixed basis of comparison with various approximate mathematical models for kinetics. These were derived by simulated experiments from the test problems with added random error. See Appendix A and B, Berty et al, 1989. 

In a continuous reaction process, the true residence time of the reaction partners in the reactor plays a major role. It is governed by the residence time distribution characteristic of the reactor, which gives information on backmixing (macromixing) of the throughput. The principal objectives of studies into the macrokinetics of a process are to estimate the coefficients of a mathematical model of the process and to validate the model for adequacy. For this purpose, a pilot plant should provide the following ... 

In this chapter The background of shock-induced solid-state ehemistry eonceptual models and mathematical models chemical reactions in shock-compressed porous powders sample preservation. 

Horie and his coworkers [90K01] have developed a simplified mathematical model that is useful for study of the heterogeneous nature of powder mixtures. The model considers a heterogeneous mixture of voids, inert species, and reactant species in pressure equilibrium, but not in thermal equilibrium. The concept of the Horie VIR model is shown in Fig. 6.3. As shown in the figure, the temperatures in the inert and reactive species are permitted to be different and heat flow can occur from the reactive (usually hot) species to the inert species. When chemical reaction occurs the inert species acts to ther-... 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

As our first approach to the model, we considered the controlling step to be the mass transfer from gas to liquid, the mass transfer from liquid to catalyst, or the catalytic surface reaction step. The other steps were eliminated since convective transport with small catalyst particles and high local mixing should offer virtually no resistance to the overall reaction scheme. Mathematical models were constructed for each of these three steps. 

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. 

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

C.G. Vayenas, S. Brosda, and C. Pliangos, Rules and Mathematical Modeling of Electrochemical and Chemical Promotion 1. Reaction Classification and Promotional Rules,/. Catal., in press (2001). 

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