Solution of the Steady-State PFR

we begin by solving for the concentrations with linear kinetics, and we do this in complete form. We will do this with DSolve as an exercise, even though the equations are trivial to solve  

Recall that kab is just that is, the reciprocal of the characteristic time for reaction and — is 

When we solved the transient, well-mixed batch reactor with linear kinetics, we obtained the same solution functionally, but instead of kab r, we had kab t as the argument of the differential, that is, in terms of real time instead of holding time. 

We return now to the Langmuir-Hinshelwood kinetics from the CSTR section to see how the PFR will behave and to compare the CSTR and the PFR. As in the case of the steady-state CSTR, we will write a steady-state PFR Module function. Recall that the rate law was  

Therefore, once again invoking the pseudo-homogeneous approximation, the equations we must solve are  

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This feature is illustrated in Fig. 9, which compares the solution of the ideal PFR, ideal CSTR, and the Danckwerts model [Eqs. (146)—(147)] with the twomode convection model [Eqs. (140) (141)] for the case of steady-state and single first-order homogeneous reaction of the form A —> B. The solution of the steady-state Danckwerts model is given by... 

Free Enzymes in Flow Reactors. Substitute t = zju into the DDEs of Example 12.5. They then apply to a steady-state PFR that is fed with freely suspended, pristine enzyme. There is an initial distance down the reactor before the quasisteady equilibrium is achieved between S in solution and S that is adsorbed on the enzyme. Under normal operating conditions, this distance will be short. Except for the loss of catalyst at the end of the reactor, the PFR will behave identically to the confined-enzyme case of Example 12.4. Unusual behavior will occur if kfis small or if the substrate is very dilute so Sj Ej . Then, the full equations in Example 12.5 should be (numerically) integrated. 

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

A PFR can be visualized as a tubular reactor for which three conditions must be satisfied (i) the axial velocity profile is flat (ii) there is complete mixing across the tube, so that all the reaction variables are a function of the axial dimension of the reactor (named z) and (iii) there is no mixing in the axial direction. PFRs have spatial variations in concentration and temperature. Such systems are caUed distributed, and analysis of their steady state performance requires the solution of differential equations. 

Solution With Z>, = 0, a reaction wiU never start in a PFR, but a steady-state reaction is possible in a CSTR if the reactor is initially spiked with component B. An anal5dical solution can be found for this problem and is requested in Problem 4.12, but a numerical solution is easier. The design equations in a form suitable for the method of false transients are... 

For both the CSTR and PFR systems, at DaT = (z0 - z4)/z3 two different manifolds of steady states cross each other, in the combined space of state variables and parameters. According to the bifurcation theory, this is a transcritical bifurcation point Here, an exchange of stability takes place for Da < DaT, the trivial solution... 

While these optimization-based approaches have yielded very useful results for reactor networks, they have a number of limitations. First, proper problem definition for reactor networks is difficult, given the uncertainties in the process and the need to consider the interaction of other process subsystems. Second, all of the above-mentioned studies formulated nonconvex optimization problems for the optimal network structure and relied on local optimization tools to solve them. As a result, only locally optimal solutions could be guaranteed. Given the likelihood of extreme nonlinear behavior, such as bifurcations and multiple steady states, even locally optimal solutions can be quite poor. In addition, superstructure approaches are usually plagued by the question of completeness of the network, as well as the possibility that a better network may have been overlooked by a limited superstructure. This problem is exacerbated by reaction systems with many networks that have identical performance characteristics. (For instance, a single PFR can be approximated by a large train of CSTRs.) In most cases, the simpler network is clearly more desirable. 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

In STR of periodical action (the problem with initial conditions), the solution is always unique and the process is unsteady by definition, while in STR of continuous action (with a constant feed of reactants) there may exist a multiplicity of steady-state solutions and the effects described above for PFR. 

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