Mathematical representation of reaction rate

Factors That Affect Reaction Rate Mathematical Representation of Reaction Rate 8.4 Equilibrium... 

These equations remain valid for bioreactors provided that one employs a suitable mathematical representation of the rate of disappearance of the substrate that is the limiting reagent. In Illustration 13.3 we employ an alternative form of the design equation to determine the holding time necessary to achieve a specified degree of conversion in a strictly batch bioreactor. This illustrative example also indicates how overall yield coefficients are employed as a vehicle for taking the stoichiometry of the reaction into account. Illustration 13.4 describes how one type of semibatch operation (the fed-batch mode) can be exploited to combine the potential advantages of batch and continuous flow operation of a stirred-tank reactor. 

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... 

The interpretation of the elements of the matrix 0 is slightly more subtle, as they represent the derivatives of unknown functions fi(x) with respect to the variables x at the point x° = 1. Nevertheless, an interpretation of these parameters is possible and does not rely on the explicit knowledge of the detailed functional form of the rate equations. Note that the definition corresponds to the scaled elasticity coefficients of Metabolic Control Analysis, and the interpretation is reminiscent to the interpretation of the power-law coefficients of Section VII.C Each element 6% of the matrix measures the normalized degree of saturation, or likewise, the effective kinetic order, of a reaction v, with respect to a substrate Si at the metabolic state S°. Importantly, the interpretation of the elements of does again not hinge upon any specific mathematical representation of specific... 

The development of an adequate mathematical model representing a physical or chemical system is the object of a considerable effort in research and development activities. A technique has been formalized by Box and Hunter (B14) whereby the functional form of reaction-rate models may be exploited to lead the experimenter to an adequate representation of a given set of kinetic data. The procedure utilizes an analysis of the residuals of a diagnostic parameter to lead to an adequate model with a minimum number of parameters. The procedure is used in the building of a model representing the data rather than the postulation of a large number of possible models and the subsequent selection of one of these, as has been considered earlier. That is, the residual analysis of intrinsic parameters, such as Cx and C2, will not only indicate the inadequacy of a proposed model (if it exists) but also will indicate how the model might be modified to yield a more satisfactory theoretical model. 

In this section, methods are described for obtaining a quantitative mathematical representation of the entire reaction-rate surface. In many cases these models will be entirely empirical, bearing no direct relationship to the underlying physical phenomena generating the data. An excellent empirical representation of the data will be obtained, however, since the data are statistically sound. In other cases, these empirical models will describe the characteristic shape of the kinetic surface and thus will provide suggestions about the nature of the reaction mechanism. For example, the empirical model may require a given reaction order or a maximum in the rate surface, each of which can eliminate broad classes of reaction mechanisms. 

Though it is impossible to formulate a complete mathematical representation of the super-rate burning, it is possible to introduce a simplified description based on a dual-pathway representation of the effects of a shift in stoichiometry. Generalized chemical pathways for both non-catalyzed and catalyzed propellants are shown in Fig. 6.26. The shift toward the stoichiometric ratio causes a substantial increase in the reaction rate in the fizz zone and increases the dark zone temperature, a consequence of which is that the heat flux transferred back from the gas phase to the burning surface increases. 

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. 

We now examine the mathematical representation of electrochemical reaction rates in terms of current, rate constant, and electrode potential. The velocity tJ of a chemical reaction may be written 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... 

As outlined in the previous section, there is a hierarchy of possible representations of metabolism and no unique definition what constitutes a true model of metabolism exists. Nonetheless, mathematical modeling of metabolism is usually closely associated with changes in compound concentrations that are described in terms of rates of biochemical reactions. In this section, we outline the nomenclature and the essential steps in constructing explicit kinetic models of metabolic networks. 

We should clearly understand that every conceptual picture or model for the progress of reaction comes with its mathematical representation, its rate equation. Consequently, if we choose a model we must accept its rate equation, and vice versa. If a model corresponds closely to what really takes place, then its rate expression will closely predict and describe the actual kinetics if a model differs widely from reality, then its kinetic expressions will be useless. We must remember that the most elegant and high-powered mathematical analysis based on a model which does not match reality is worthless for the engineer who must make design predictions. What we say here about a model holds not only in deriving kinetic expressions but in all areas of engineering. 

Figure 2 is a good representation of almost all the 112 runs made with 1-octanol. There was curvature on only a few runs, undoubtedly caused by experimental error since they could not be reproduced. This feature was checked carefully after mathematical analysis indicated reasons to expect curvature. The conditions of reactions and values of k0 have been tabulated (Tables I and II). Since k0 depends upon 1-octanol and TMAE, it is called the pseudo-first-order rate constant. 

The Mathematical Model Once a conceptual PBPK model has been created, with some knowledge of the chemical of interest, this representation can be translated into mathematical equations for use in predicting time-course disposition. The fundamental equations utilized arise from chemical species mass balances, which account for the rates at which molecules enter and leave each compartment, as well as other processes (e.g., rates of reactions) that produce or consume the chemical. 

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