Arbitrary reaction kinetics

Example 15.13 Solve Zwietering s differential equation for arbitrary reaction kinetics and an exponential residence time distribution. 

This equation is the basic relation for the mean residence time in a plug flow reactor with arbitrary reaction kinetics. Note that this expression differs from that for the space time (equation 8.2.9) by the inclusion of the term (1 + SAfA) and that this term appears inside the integral sign. The two quantities become identical only when 5a is zero (i.e., the fluid density is constant). The differences between the two characteristic times may be quite substantial, as we will see in Illustration 8.5. Of the two quantities, the reactor... 

Arbitrary Reaction Kinetics. If the Thiele modulus is generalized as follows [see Froment and Bischoff (1962)]... 

This holds for arbitrary reaction kinetics and arbitrary catalyst geometries. For example, for n-th order kinetics this yields... 

These formulae give the Aris numbers for arbitrary reaction kinetics and intraparticle temperature gradients. Thus the dependency of the conversion rate on the temperature can also be of any arbitrary form. 

The estimate A.17 holds for arbitrary reaction kinetics, and only for the... 

A comparison shown in Figure 4.38 reveals that the laminar flow predicts a value for the reactant concentration that resides between those predicted for a PFR and a completely backmixed reactor. Corresponding comparisons can be performed for reactions with arbitrary reaction kinetics Equation 4.140 needs to be solved, either analytically or numerically, as a function of t and, consequently, the average value is calculated using the numerical integration of Equation 4.138. 

For arbitrary reaction kinetics, the balance equation for a catalyst particle. Equation 5.29, must be solved numerically with the boundary conditions. Equations 5.31 and 5.32. The effectiveness factors can subsequently be obtained from Equation 5.50 or 5.55. For some limited cases of chemical kinetics, it is, however, possible to solve the balance Equation 5.29 analytically and thus obtain an explicit expression for the effectiveness factor. We shall take a look at a few of these special cases. 

S.2.3.2 Arbitrary Reaction Kinetics Diffusion Resistance in the Gas Film as the Rate-Determining Step... 

In theory, an arbitrary number of scalars could be used in transported PDF calculations. In practice, applications are limited by computer memory. In most applications, a reaction lookup table is used to store pre-computed changes due to chemical reactions, and models are limited to five to six chemical species with arbitrary chemical kinetics. Current research efforts are focused on smart tabulation schemes capable of handling larger numbers of chemical species. 

We have used the first-order irreversible reaction as an example, but this is easy to generalize for any reaction, irreversible or reversible, with any kinetics. In a PFTR the mass-balance equation for an arbitrary reaction becomes... 

The quantity njv, often called the electron number, on the other hand, is not arbitrary and is characteristic of the electrode reaction kinetics. It indicates the number of electrons gained or lost by the electrode in the completion of the rate-determining step. 

To assess the generality of the decomposition solution method we have applied it to several arbitrary reaction orders, for example, n = 1.73, 0.67, -0.5, -1.0 etc.10,11 These reaction orders have been chosen to represent typical cases of kinetics in heterogeneous catalysis and electrocatalysis where adsorption phenomena play a major role. Values of effectiveness of a plane catalyst pellet for the different reaction orders are shown in Figure 5. Clearly all of the data for positive reaction orders show the expected trend of a decrease in effectiveness with increase in Thiele modulus. Effectiveness values determined for reaction... 

The desire to formulate reaction schemes in terms of molecular processes taking place on a catalyst surface must be balanced with the need to express the reaction scheme in terms of kinetic parameters that are accessible to experimental measurement or theoretical prediction. This compromise between mechanistic detail and kinetic parameter estimation plays an important role in the use of reaction kinetics analysis to describe the reaction chemistry for a catalytic process. Here, we discuss four case studies in which different compromises are made to develop an adequate kinetic model that describes the available observations determined experimentally and/or theoretically. For convenience, we selected these examples from our work in this field however, this selection is arbitrary, and many other examples could have been chosen from the literature. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

Hie Aris numbers An0 and An, are much alike. This is illustrated in Table 6.4 where the formulae for An0 and An, are given for arbitrary kinetics, for n-th order kinetics and for first-order kinetics. In practice, reaction kinetics do not differ too much from first-order kinetics, and hence the values of An0 and An, will remain very close to each other (as also the geometry factor T is close to one). In that case both Aris numbers will be roughly equal to the square power of the shape-generalized Thiele modulus of Aris [6]. 

The above analytical relations have been derived for irreversible first-order kinetics. Arbitrary kinetics do not generally ciUow an analytical solution. However, a generalization can be made for arbitrary kinetics. Without going into details (see for instance Refs. [9,10]), the following result for a generalized Thiele modulus, which is independent of the reaction kinetics and defined such that the limits expressed in Eqns. (8.71) and (8.72) hold, is obtained ... 

Consider an arbitrary network of chemical reactors called the grand network. Let us denote the set of achievable concentrations associated with this network by X, where X must contain positive values. The convex hull of the points in X is given by conv(X). The complement principle describes a relationship between sections of the grand network to the reaction kinetics of the system. 

In the application of chemical kinetics, a formal kinetic evaluation method has been proposed (Schmid and Sapunov, 1982). An operation scheme is illustrated in Fig. 5.16 it uses two properties of c/t curves as decision criteria, called invariance I and invariance II. These properties concern the linear transformation capability of first- and second-order reactions. Kinetic curves with various initial concentrations Cj o can be superimposed over arbitrary standard curves (cj o)s by multiplying ordinates by ratios (cj o)s/Ci,o tbe case... 

In the literature of stochastic reaction kinetics it was often assumed that the stationary distributions of chemical reactions were generally Poissonian (Prigogine, 1978). The statement is really true for systems containing only first-order elementary reactions, even when inflow and outflow are taken into account (i.e. for open compartmental systems see Cans, 1960, p. 692). If the model of open compartmental systems is considered as an approximation of an arbitrary chemical reaction near equilibrium, then in this approximation the statement is true. 

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