Mathematical models chemical Reaction Kinetics

Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications (refer to the examples in Chapters 11 through 16). These models are conceptually attractive because a general model for any system size can be developed even before the system is constructed. A detailed exposition of fundamental mathematical models in chemical engineering is beyond our scope here, although we present numerous examples of physiochemical models throughout the book, especially in Chapters 11 to 16. Empirical models, on the other hand, are attractive when a physical model cannot be developed due to limited time or resources. Input-output data are necessary in order to fit unknown coefficients in either type of the model. 

It is safe to say that most graduate courses in chemical reaction engineering today suffer from an excess of mathematical sophistication and insufficient contact with reality. Because of the complexity of many reaction engineering models, it is essential that students be given a balanced and realistic view of what can and cannot be achieved. For example, they must learn that if the intrinsic kinetics of a reaction are not known accurately, this deficiency cannot be made up by a more detailed understanding of the fluid mechanics. In this connection, it would be useful pedagogically to take a complex model and illustrate its sensitivity to various aspects, such as the assumptions inherent in the model, the reaction kinetics, and the parameter estimates. 

The spatial distribution of composition and temperature within a catalyst particle or in the fluid in contact with a catalyst surface result from the interaction of chemical reaction, mass-transfer and heat-transfer in the system which in this case is the catalyst particle. Only composition and temperature at the boundary of the system are then fixed by experimental conditions. Knowledge of local concentrations within the boundaries of the system is required for the evaluation of activity and of a rate equation. They can be computed on the basis of a suitable mathematical model if the kinetics of heat- and mass-transfer arc known or determined separately. It is preferable that experimental conditions for determination of rate parameters should be chosen so that gradients of composition and temperature in the system can be neglected. 

The premise of molecular biology is that cellular processes are governed by physico-chemical principles, and accordingly, those principles may be used to translate known or hypothetical molecular mechanisms to mathematical equations. In this section, the general principles of chemical kinetics, mass transport, and fluid mechanics used to model chemical reaction systems are reviewed briefly. 

The fundamental concepts used to mathematically represent mechanistic mathematical models are borrowed from chemical reaction kinetics and transport phenomena.1920 However, the application of these concepts in biology requires an understanding of the biological literature. The biological data to build a mechanistic model of mAh action typically include... 

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)... 

Mathematical models of reactor kinetics are used to describe the concentration distributions of various reactants, intermediates and products within the reactor during the course of the reaction. In order to validate these models, concentration distributions must be measured. However, as these chemical reactions, under real process conditions, generally occur within metal reactors enclosed by a heating or pressure mantle, it is not... 

Mathematical modeling of the cure process coupled with the automation of various thermal analytical instruments and Fourier Transform Infrared Spectroscopy (FT-IR) have made possible the determination of quantitative cure and chemical reaction kinetics from a single dynamic scan of the reaction process. This paper describes the application of FT-IR, differential scanning calorimetry (DSC) and dynamic mechanical analysis (DMA) in determining cure and reaction kinetics in some model organic coatings systems. 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

Dente and Ranzi (in Albright et al., eds.. Pyrolysis Theory and Industrial Practice, Academic Press, 1983, pp. 133-175) Mathematical modehng of hydrocarbon pyrolysis reactions Shah and Sharma (in Carberry and Varma, eds.. Chemical Reaction and Reaction Engineering Handbook, Dekker, 1987, pp. 713-721) Hydroxylamine phosphate manufacture in a slurry reactor Some aspects of a kinetic model of methanol synthesis are described in the first example, which is followed by a second example that describes coping with the multiphcity of reactants and reactions of some petroleum conversion processes. Then two somewhat simph-fied industrial examples are worked out in detail mild thermal cracking and production of styrene. Even these calculations are impractical without a computer. The basic data and mathematics and some of the results are presented. 

Ultrasound can thus be used to enhance kinetics, flow, and mass and heat transfer. The overall results are that organic synthetic reactions show increased rate (sometimes even from hours to minutes, up to 25 times faster), and/or increased yield (tens of percentages, sometimes even starting from 0% yield in nonsonicated conditions). In multiphase systems, gas-liquid and solid-liquid mass transfer has been observed to increase by 5- and 20-fold, respectively [35]. Membrane fluxes have been enhanced by up to a factor of 8 [56]. Despite these results, use of acoustics, and ultrasound in particular, in chemical industry is mainly limited to the fields of cleaning and decontamination [55]. One of the main barriers to industrial application of sonochemical processes is control and scale-up of ultrasound concepts into operable processes. Therefore, a better understanding is required of the relation between a cavitation coUapse and chemical reactivity, as weU as a better understanding and reproducibility of the influence of various design and operational parameters on the cavitation process. Also, rehable mathematical models and scale-up procedures need to be developed [35, 54, 55]. 

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. 

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. 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

Dr. Walas has several decades of varied experience in industry and academia and is an active industrial consultant for the process design of chemical reactors and chemical and petroleum plants. He has written four related books on reaction kinetics, phase equilibria, process equipment selection and design, and mathematical modeling of chemical engineering processes, as well as the sections Reaction Kinetics and Chemical Reactors in the seventh edition of Chemical Engineers Handbook. He is a Fellow of the AlChE and a registered professional engineer. 

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