The Reaction Rate Term

Equation (10-37) has an analytical solution only for a first-order reaction, alfliough numerical and/or approximate solutions are available for oflier rate equations. The present discussion will be focused on flie first-order case, wifli careful attention to flie questions of reversibility, and wheflier flie reaction is homogeneous or heterogeneous. 

Homogeneous Reaction Consider a reversible reaction A B, that takes place only in the fluid phase. In fliis discussion of homogeneous reactions, we will assume that any packing in flie reactor is not porous. The reversible reaction obeys the rate equation 

Let a = (Ca/Cao) — P/uCaq)- This transformation makes the above equation completely dimensionless 

Since p/a is the concentration of A at equilibrium, the parameter a can be interpreted as the difference between the dimensionless concentration of A at any point in the reactor (Ca/Cao) and the dimensionless equilibrium concentration of A P/aCao)- The parameter 1 — p/aCAO is the maximum possible change in the dimensionless concentration of A, i.e., the difference between the dimensionless feed and equilibrium concentrations. 

Notethat T (= jV/u) is the octiia/space time for the reactor, i.e., the reactor volume that is not occupied by packing divided by the volumetric flow rate. 

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The above equations are linked by the reaction rate term rA, which depends on concentration and temperature. 

Example 2.13 What is the linearized form of the reaction rate term... 

In this case the reaction rate will depend not only on the system temperature and pressure but also on the properties of the catalyst. It should be noted that the reaction rate term must include the effects of external and intraparticle heat and mass transfer limitations on the rate. Chapter 12 treats these subjects and indicates how equation 8.2.12 can be used in the analysis of packed bed reactors. 

If we assume a series of values of T, the above equation gives the corresponding values of Cxir) At low temperatures will be essentially equal to Cj,o since the reaction rate is very small. As temperature is increased, the reaction rate term becomes larger and larger. This causes to approach zero. Most of the reactant (component A) is consumed in the reaction and there is little of it left in the reactor. 

These solutions to the one-dimensional advection-diffusion model can be used to estimate reaction rate constants Ck) from the pore-water concentrations of S, if and s are known. More sophisticated approaches have been used to define the reaction rate term as the sum of multiple removals and additions whose functionalities are not necessarily first-order. Information on the reaction kinetics is empirically obtained by determining which algorithmic representation of the rate law best fits the vertical depth concentration data. The best-fit rate law can then be used to provide some insight into potential... 

One significant difference between this pair of equations and (6.9) and (6.10) is that the net chemical rate of formation of B is not simply equal to the net chemical rate of removal of A. If (6.9) and (6.10) are added together all the reaction rate terms cancel, but here the term — k2b remains. 

The extension of the two-mode axial dispersion model to the case of fully developed turbulent flow in a pipe could be achieved by starting with the time-smoothed (Reynolds-averaged) CDR equation, given by Eq. (117), where the reaction rate term R(C) in Eq. (117) is replaced by the Reynolds-averaged reaction rate term Rav(C), and the molecular diffusivity Dm / is replaced by the effective diffusivity Dej- in turbulent flows given by... 

The important result is that the two-mode models for a turbulent flow tubular reactor are the same as those for laminar flow tubular reactors. The two-mode axial dispersion model for turbulent flow tubular reactors is again given by Eqs. (130)—(134), while the two-mode convection model for the same is given by Eqs. (137)—(139), where the reaction rate term r((c)) is replaced by the Reynolds-averaged reaction rate term rav((c)). The local mixing time for turbulent flows is given by... 

A somewhat different form of the reaction rate term in Equations (11) and (15) has been widely used for modeling transport of tracers, particularly isotopic tracers (cf. Ogata, 1964 Lassey and Blattner, 1988 Blattner and Lassey, 1989 Bickle, 1992 DePaolo and Getty, 1996 Bickle et ai, 1997) ... 

If the mechanism is not known in detail, the kinetic terms may be replaced by empirically-determined rate laws, i.e., by approximations to the reaction rate term that typically will be some (non-linear) polynomial fit of the observed rate to the concentrations of the major species in the reaction (reactants and products). Such empirical rate laws have limited ranges of validity in terms of the experimental operating conditions over which they are appropriate. Like other polynomial fitting procedures, these representations can rapidly go spectacularly wrong outside their range of validity, so that they must be used with great care. If this care is taken, however, empirical rate equations are of great value. 

This provides a pair of coupled, non-linear (through the Arrhenius temperature dependence) ordinary differential equation for the two variables a and T. If the temperature increases, the reaction rate increases through the increase in k. The consequent increase in T will lead to increases in the heat transfer rates and also to a decrease in the concentration of A, which in turn tends to decrease the reaction rate term ka. To quantify this effect, we can examine the adiabatic case a = 0- In this situation, the temperature rise above the inflow is uniquely linked to the extent of reaction = ao a)/ao through the condition... 

The most difficult term to close in Eq. (5.11) is the reaction rate term. Reaction rates are seldom formulated by considering all the elementary reactions. More often than not, the reactive system is represented by a lumped mechanism, considering only a few species. The case of m components participating in n independent chemical reactions is usually represented by two two-dimensional matrices (m x n) of stoichiometric coefficients and order of reactions and two one-dimensional vectors (n) of frequency factors and activation energy, n chemical reactions are written ... 

The reaction rate term, denoting the net rate of species c mass input per unit volume by chemical reactions is given by ... 

The most difficult term to close in (7.73) is the reaction rate terms denoted by Sc uj). To simulate turbulent reactive flows accurate modeling of this term is very important. For slow reactions (i.e., Dai 1) the turbulent mixing is completed before the reaction can take place thus an adequate closure is available ... 

To + q°Ypo/Cp is the adiabatic flame temperature, R° is the universal gas constant, and E represents the overall activation energy. The limit p -> ao is considered with times a suitable power of p held fixed [see, for example, the expression above equation (5-75)]. As exemplified by equation (5-66), the function co z) depends exponentially on p, and for large P we expect that there will be a narrow reaction zone centered about the position of maximum t. Outside this reaction zone, the reaction-rate term in equation (10) will be exponentially small. Under the assumption that all the fuel is consumed in the reaction zone, we may readily derive the expression rnYpo = w dx from the fuel-conservation equation, where / identifies conditions just downstream from the reaction zone. Use of equations (10) and (11) in this expression gives... 

From the stoichiometry of this reaction, all the reaction rate terms are interrelated. In the case here, the rate at which ethylene chloride is created, that is, the number of moles per second, is equal to the rate at which ethylene is consumed, which is also equal to the rate at yhich chlorine is consumed. That is,... 

For the reactor system without heat eifects, equation (6-53) is just a balance between the reaction-rate term and the convective term for A. Then we let... 

Now, let us put the reaction-rate terms of equation (ix) on the right-hand side such that, after some rearrangement, we obtain... 

Several factors and assumptions are built into equation (7-97). The most noticeable is the time variation of this term will be important if the rate of the deactivation is rapid compared to the rate of the main reaction, that is if k. The most important assumption in equation (7-97), however, is that the deactivation kinetics are separable hence the activity factor s in the reaction rate term. Finally, since A disappears in both of the reactions in the parallel scheme, the overall rate constant is given by the sum shown above. For the series scheme, of course, some changes will have to be made that, by now, one hopes will be obvious. 

So far we have done nothing except manipulate equations to show that the development is another pseudo-homogeneous model. However, where we go from here depends, in large part, on how the reaction rate term, (—r), is handled. So far, we have made no specification for this. To be comfortable, recall an isothermal reaction, first-order with respect to C. We recall that the rate term may be written as... 

Well, yes, Horatio. We finally got you, because you are obviously forgetting the analysis given in Chapter 6, Section 3 For example, the two continuity equations for the catalyst are coupled through the reaction rate term, kC. In nondimensional form this results in the following working relationship. 

The reciprocal of the term kia +kjC is the time constant x for changes in C. For a first-order system, 95% of steady state is reached when t = 3r. In a typical stirred-tank reactor with good gas dispersion, ki d QA sec, and if the reaction rate term kjCg is also about 0.1 sec , the time constant is 5 sec. The solution would reach 95% of the final value of in about 15 seconds. The final value (99.7% of equilibrium), which is really a pseudo-steady-state value is reached after t 5r ... 

When the reaction rate term kiCg is much greater than the mass transfer coefficient k a, the reaction becomes mass transfer controlled, and Eq. (7.20) becomes... 

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