Mathematical modelling sensitivity reactors

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. 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

In this paper, we will first illustrate the mathematical models used to describe the coke-conversion selectivity for FFB, MAT and riser reactors. The models also include matrix and zeolite contributions. Intrinsic activity parameters estimated from a small isothermal riser will then be used to predict the FFB and MAT data. The inverse problem of predicting riser performance from FFB and MAT data is straightforward based on the proposed theory. A parametric study is performed to show the sensitivity to changes in coke selectivity and heat of reaction which are affected by catalyst type. We will highlight the quantitative differences in observed conversion and coke-conversion selectivity of various reactors. 

Following the Morbidelli and Varma criterion, several other methods have been proposed in recent years in order to characterize the highly sensitive behavior of a batch reactor when it reaches the runaway boundaries. Among the most successful approaches, the evidence of a volume expansion in the phase space of the system has been widely exploited to characterize runaway conditions. For example, Strozzi and Zaldivar [9] defined the sensitivity as a function of the sum of the time-dependent Lyapunov exponents of the system and the runaway boundaries as the conditions that maximize or minimize this Lyapunov sensitivity. This has put the basis for the development of a new class of runaway criteria referred to as divergence-based approaches [5,10,18]. These methods usually identify runaway with the occurrence of a positive divergence of the vector field associated with the mathematical model of the reactor. 

To minimize these costs it is therefore necessary to maximize the conversion in the reactor and to avoid as far as possible inert substances in the reaction mixture. With irreversible reactions (e.g., partial oxidations) the trend is therefore towards a highly concentrated, approximately stoichiometric feed composition, which may occasionally be in the explosive range. The resulting problems are addressed in Sections 10.1.3.3 and 10.1.4.2) Since fixed-bed reactors constitute one of the most important classes of chemical reactors, much work has been devoted to their proper mathematical modeling as well as to the study of their stability, sensitivity and automatic control. The following standard text books and monographs can be recommended for further reference [1-4]. 

Copolymerization systems offer an even higher degree of d)mamic complexity that involves multiple rates of monomer consumption of the form of eq 2. Pinto and Ray 54) were first to present a comprehensive mathematical model for copolymerization validated with experimental findings. Both experiments and model predictions demonstrated that the stability of such systems is extremely sensitive to small changes in feed composition. For example, when the comonomer feed was switched from pure vinyl acetate to a comonomer mixture containing 0.25 vol% methyl methacrylate, reactor operation changed from stable to oscillatory rapid convergence to stable operation was observed when the feed composition was switched back. 

Experimental data of stearic acid decarboxylation in a laboratory-scale fixed bed reactor for formation of heptadecane were evaluated studied with the aid of mathematical modeling. Reaction kinetics, catalyst deactivation, and axial dispersion were the central elements of the model. The effect of internal mass transfer resistance in catalyst pores was found negligible due to the slow reaction rates. The model was used for an extensive sensitivity study and parameter estimation. With optimized parameters, the model was able to describe the experimentally observed trends adequately. A reactor scale-up study was made by selecting the reactor geometry (diameter and length of the reactor, size and the shape of the catalyst particles) and operating conditions (superficial liquid velocity, temperature, and pressure) in such a way that nonideal flow and mass and heat transfer phenomena in pilot scale were avoided. 

MATHEMATICAL MODELLING AND SENSITIVITY ANALYSIS OF HIGH PRESSURE POLYETHYLENE REACTORS... 

A mathematical model of a wall-catalyzed reactor is used to elucidate the effects of Inlet velocity and concentration, geometry of the duct, and axial conduction on the hysteresis, multiplicity and parametric sensitivity. The monolith reactor for carbon monoxide oxidation is the prototype considered, with wall catalyst being platinum on alumina on a ceramic substrate. 

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