Complex reactions mathematical representation

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Section 10.2 describes the MINLP approach of Kokossis and Floudas (1990) for the synthesis of isothermal reactor networks that may exhibit complex reaction mechanisms. Section 10.3 discusses the synthesis of reactor-separator-recycle systems through a mixed-integer nonlinear optimization approach proposed by Kokossis and Floudas (1991). The problem representations are presented and shown to include a very rich set of alternatives, and the mathematical models are presented for two illustrative examples. Further reading material in these topics can be found in the suggested references, while the work of Kokossis and Floudas (1994) presents a mixed-integer optimization approach for nonisothermal reactor networks. 

The kinetics of enzyme-catalyzed reactions can be very complex, and the mathematical representations for the effect of the concentrations of substrate, product, cofactors, and inhibitors are presented in a variety of textbooks in this field [1]. The exact form of this dependence of enzyme activity on these factors might have a profound effect on the behavior of an enzyme biosensor. However, one can delineate general rules of thumb concerning the properties of enzymes for the preliminary design of enzyme-based sensors. 

Polymerization reactors are a specific kind of chemical reactors in which polymerization reactions take place therefore, in principle, they can be analyzed following the same general rules applicable to any other chemical reactor. The basic components of a mathematical model for a chemical reactor are a reactor model and rate expressions for the chemical species that participate in the reactions. If the system is homogeneous (only one phase), these two basic components are pretty much what is needed on the other hand, for heterogeneous systems formed by several phases (emulsion or suspension polymerizations, systems with gaseous monomers, slurry reactors or fluidized bed reactors with solid catalysts, etc.), additional transport and/or thermodynamic models may be necessary to build a realistic mathematical representation of the system. In this section, to illustrate the basic principles and components needed, we restrict ourselves to the simplest case, that of homogeneous reactors in other sections, additional components and more complex cases are discussed. 

Mathematical Representation of Simple and Complex Reactions To the chemist, a simple reaction such as... 

The formation of an enzyme-inhibitor complex reduces the number of enzymes available to bind with the substrate, and as a result, the reaction rate decreases. Equation 4.33 shows a mathematical representation of a competitive inhibition rate ... 

Mathematical representation of simple and complex reactions Independent reactions Rate equations... 

The absence of comprehensive theoretical treatment is likely to be caused by the complexity of these systems. In this section we shall develop a mathematical representation for solid-solid reactions, for systems where the overall rate is controlled by chemical kinetics [25]. We must acknowledge that this treatment will be applicable only to a limited range of practical situations nonetheless it represents a first step, upon which subsequent, more comprehensive studies may be based. 

The formulation of the model as above has the advantage that mathematically it picturizes the bed as an initial value problem in contrast to the more complicated boundary value representation of the Fryer-Potter model. The implications of this reduced complexity become more evident (and considerably more important) when the reactions involved are nonlinear. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

In the mathematical sense, simplification can be defined as model reduction, that is, the rigorous or approximate representation of complex models by simpler ones. For example, in a certain domain of parameters or times, a model of partial differential equations ( diffusion-reaction model) is approximated by a model of differential equations, or a model of differential equations is approximated by a model of algebraic equations, and so on. 

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