Multiple reactions independent

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

The reaction of Example 7.4 is not elementary and could involve shortlived intermediates, but it was treated as a single reaction. We turn now to the problem of fitting kinetic data to multiple reactions. The multiple reactions hsted in Section 2.1 are consecutive, competitive, independent, and reversible. Of these, the consecutive and competitive t5T>es, and combinations of them, pose special problems with respect to kinetic studies. These will be discussed in the context of integral reactors, although the concepts are directly applicable to the CSTRs of Section 7.1.2 and to the complex reactors of Section 7.1.4. 

Equilibrium Compositions for Multiple Reactions. When there are two or more independent reactions. Equation (7.29) is written for each reaction ... 

The various energy transfer constraints enter into the analysis primarily as boundary conditions on the difference equations, and we now turn to the generation of the differential equations on which the difference equations are based. Since the equations for the one-dimensional model are readily obtained by omitting or modifying terms in the expressions for the two-dimensional model, we begin by deriving the material balance equations for the latter. For purposes of simplification, it is assumed that only one independent reaction occurs within the system of interest. In cases where multiple reactions are present, one merely adds an appropriate term for each additional independent reaction. 

Transformation replaces a differential equation with independent variable, t, to an algebraic equation in variable, s. The latter relation can be solved algebraically for the transform f s). Then f(t) is found by inversion with Table 1.4. Problems PI.04.01 ff are examples. Cases of multiple reactions are treated this way in problems P2.02.08 and P2.02.10. 

To summarize, we can always find a single concentration variable that describes the change in aU species for a single reaction, and for R simultaneous reactions there must be R independent variables to describe concentration changes. For a single reaction, this problem is simple (use either or X), but for a multiple-reaction system one must set up the notation quite carefully in terms of a suitably chosen set of R concentrations or Xi S-... 

Most textbook discussions of effectiveness factors in porous, heterogeneous catalysts are limited to the reaction A - Products where the effective diffusivity of A is independent of reactant concentration. On the other hand, it is widely recognized by researchers in the field that multicomponent single reaction systems can be handled in a near rigorous fashion with little added complexity, and recently methods have been developed for application to multiple reactions. Accordingly, it is the intent of the present communication to help promote the transfer of these methods from the realm of the chemical engineering scientist to that of the practitioner. This is not, however, intended to be a comprehensive review of the subject. The serious reader will want to consult the works of Jackson, et al. 

Table 12.1 shows that in more than 70% of CYP2C9 reactions, the first option selected by the methodology matches the experimental one. Moreover, in more than 16% of cases, the second atom is that which fits the experimental one. Therefore, in considering the overall ranking list for the single and multiple sites of metabolism, the methodology predicts the site of metabolism for CYP2C9 within the first two atoms selected in approximately 86% of the reactions, independent of the conformer used. 

The concept of extent of reaction can be extended to multiple reactions, only now each independent reaction has its own extent. If a set of reactions takes place in a batch or continuous steady-state reactor and v,/ is the stoichiometric coefficient of substance i in reaction j (negative for reactants, positive for products), we may then write... 

If multiple reactions occur you would calculate the extents of each independent reaction, i, 6, . (Equation 4.6-6 on p. 123). but for such processes you are generally better off using the heat of formation method to be described. 

For multiple reactions, Eq. (2.1-3) is solvable for the full vector T tot if and only if the matrix n has full rank NR, i.e., if and only if the rows of v are linearly independent. If any species production rates Ri,tot are not measured, the corresponding columns of u must be suppressed when testing for solvability of Eq. (2.1-3). Gaussian elimination (see Section A.4) is convenient for doing this test and for finding a full or partial solution for the reaction rates. 

In this chapter we discuss reactor selection and general mole balances for multiple reactions. There are three basic types of multiple reactions series, parallel, and independent. In parallel reactions (also called competing reactions) the reactant is consumed by two different reaction pathways to form different products ... 

Elder [45] has modelled several multiple reaction schemes, including mutually independent concurrent first-order reactions, competitive first-order reactions, mutually independent n-th order reactions, and mutually independent Avrami-Erofeev models with n = 2 or 3. The criteria identified for recognizing the occurrence of multiple reactions were (i) the apparent order of reaction, n, varies with the method of calculation, and (ii) the kinetic parameters, A and vary with the extent of reaction, a. 

The multiple reaction model consists of a large number of independent parallel first-order reactions, all having the same pre-exponential factor, k0, and activation energies in Gaussian distribution with mean E0 and standard deviation o,32 which gives... 

Performing multiple reactions simultaneously in a single step offers possibilities for reduced waste and increased safety as well as manipulation of equilibria. This approach was inspired by the action of enzymes, which constitute interesting examples of multifunctional catalysts as they can promote multi-step reactions. In fact, enzymes immobilize mutually incompatible functional groups in a manner that maintains their independent functionality and, as such, are able to carry out multi-step reaction sequences with functionalities that would not be tolerated together in solution. 

Great caution must be exercised in interpreting multiple-reaction models containing several independent parameters. A four-parameter model will not be unique to a data set (Sposito, 1982). In addition, there is no evidence of... 

As will be discussed later, the rates at which chemical species are being formed (or depleted) depend on all the chemical reactions that actually take place in the reactor (reaction pathways). Hence, to design chemical reactors with multiple reactions, we consider all the chemical reactions that are taking place, including the dependent reactions. Therefore, it is necessary to express the dependent reactions in terms of the independent reactions. Next, we describe how to do so. 

While the use of calculated quantities may seem, at first, cumbersome and even counterproductive, it actually simplifies the analysis of chemical reactors with multiple reactions. In fact, calculated quantities such as enthalpy and free energy are commonly used in thermodynamics resulting in simplified expressions. Here too, by using the extents of independent reactions, we formulate the design of chemical reactors by the smallest number of design equations. 

Equation 4.3.14 is the reaction-based, differential design equation for steady-flow reactors, written for the wth-independent reaction. As will be discussed below, to describe the operation of the reactor with multiple reactions, we have to write Eq. 4.3.14 for each of the independent reactions. 

As discussed in Chapter 4, to describe the operation of a CSTR with multiple reactions, we have to write Eq. 8.1.1 for each independent chemical reaction. The solution of the design equations (the relationships between Z Js and t) provide the reaction operating curves and describe the reactor operation. To solve the design equations, we have to express the rates of the chemical reactions that take place in the reactor in terms of Z s and t. Below, we derive the auxiliary relations used in the design equations. 

The design formulation of nonisothermal CSTRs with multiple reactions follows the same procedure outlined in the previous section—we write the design equation, Eq. 8.1.1, for each independent reaction. However, since the reactor temperature, out> is not known, we should solve the design equations simultaneously with the energy balance equation (Eq. 8.1.14). 

There are four basic types of multiple reactions series, parallel, complex, and independent. These types of multiple reactions can occur by themselves, in pairs, or all together. V en there is a combination of parallel and series reactions. they are often referred to as complex reactions. 

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