Mathematic model of reaction

By assuming a reactor model, it is possible to determine reaction rates from experimental results. Then, various factors affecting yields, selectivities and reaction rates become evident. Experimental rate laws are deduced from results, e.g. in the classical form involving reaction orders and activation energies. At this stage, computers are used for solving numerically the mathematical models of reaction and reactor (Sect. 4) and for making a statistical analysis of experimental results (Sect. 5). 

Our ultimate vision was to automate the reactor scale-up process, from experimental design through the development of mathematical models of reaction kinetics. To achieve this vision, we chose to first demonstrate that a custom, single channel microreactor could be used to generate quantitative scale-up data. 

V.G. Dovi, Use of slack variables in the mathematical modelling of reaction equilibria in complex chemical kinetics, Comput. Chem. Eng., 19 (4), 489-491 (1995)... 

Our suggested method of numerical value analysis for the mechanisms of complex reactions is based on the use of the system of Hamiltonian equations for the mathematical modeling of reactions, with separation of the targeted functional that characterizes the quality of the selected chemical reaction. 

Xie K, Song G, Hou A, Liu Y (2006) Mathematical model of reaction for reactive dyes containing fluorotriazine. Int J Nonlinear Sci Numer Simul 7 117-119... 

V. Van Breusegem and G. Bastin, Reduced Order Mathematical Modeling of Reaction Systems A Singular Perturbation Approach , Lab. d Automatique Dynamique et Analyse des Systemes, Univ. of Louvain, Belgium, 1991. 

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. 

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 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. 

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]. 

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). 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Mathematical models of the reaction system have been developed, enabling prediction of the molecular weight... 

In some cases, the reaction may take place before the substrate is reached while still in the gas phase (gas-phase precipitation) as will be reviewed later. As can be expected, the mathematical modeling of these phenomena can be complicated. 

Operability analysis and control system synthesis for an entire chemical plant Mathematical modeling of transport and chemical reactions of combustion-generated air pollutants... 

C. E. Lapple, Electrostatic Phenomena with Particulates J. R. Kittrell, Mathematical Modeling of Chemical Reactions... 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

Table 20.3 lists the reversible and irreversible processes that may be significant in the deep-well environment.3 The characteristics of the specific wastes and the environmental factors present in a well strongly influence which processes will occur and whether they will be irreversible. Irreversible reactions are particularly important. Waste rendered nontoxic through irreversible reactions may be considered permanently transformed into a nonhazardous state. A systematic discussion of mathematical modeling of groundwater chemical transport by reaction type is provided by Rubin.30... 

This section is concerned with analyses of simultaneous reaction and mass transfer within porous catalysts under isothermal conditions. Several factors that influence the final equation for the catalyst effectiveness factor are discussed in the various subsections. The factors considered include different mathematical models of the catalyst pore structure, the gross catalyst geometry (i.e., its apparent shape), and the rate expression for the surface reaction. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Considering theoretically a copolymerization on the surface of a miniemulsion droplet, one should necessarily be aware of the fact that this process proceeds in the heterophase reaction system characterized by several spatial and time scales. Among the first ones are sizes of an individual block and macromolecules of the multiblock copolymer, the radius of a droplet of the miniemulsion and the reactor size. Taking into account the pronounced distinction in these scales, it is convenient examining the macrokinetics of interphase copolymerization to resort to the system approach, generally employed for the mathematical modeling of chemical reactions in heterophase systems [73]. 

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