PFRs and CSTRs in Series

When a brand-new reactor system is being designed, it is tmcommon (but not impossible) to encounter a situation that calls for using CSTRs and PFRs in series. However, if existing equipment is being used to satisfy an interim need, to establish production quickly, or to expand an existing plant, there may be good reason to consider such combinations. 

If CSTRs and PFRs are to be used in series, one obvious question is which type of reactor should come first For a given feed rate, inlet concentration, and inlet temperature, and for fixed reactor volumes, will the final conversion depend on how the reactors are ordered  

Since CSTRs and PFRs represent the extremes of mixing, the best order wUl depend on whether it is better to mix when the reactant concentrations are high or when they are low. Stated differently, is it better to mix early in the reaction (when the conversion is low) or to mix late in the reaction (when the conversion is high) The answer depends on the rate equation. 

If the effective order of the reaction is less than 1 (n 1), mix as soon as possible. Drop the reactant concentration as low as possible as soon as possible. In the above example, for n 1, the large CSTR should be first, followed by the small CSTR, with the PFR last. 

Iftheeffectiveorderisexactlyl (n= 1), then the earliness or lateness ofmixing does not matter. The order of the reactors will not affect the final conversion. 

---

In addition to the one-parameter models of tanks-in-series and dispersion, many other one-parameter models exist when a combination of ideal reactors is to model the real reactor. For example, if the real reactor were modeled as a PFR and CSTR in series, the parameter would be the fi action,/, of the total reactor volume that behaves as a CSTR Another one-parameter model would be the fi action of fluid that bypasses the ideal reactor. We can dream up many other situations which would alter the behavior of ideal reactors in a way that adequately describes a real reactor. However, it m be that one parameter is not sufficient to yield an adequate comparison between theoiy... 

To demonstrate these ideas, let us consider three different schemes of reactors in series two CSTRs, two PFRs, and then a combination of PFRs and CSTRs in series. To size these reactors, we shall use laboratory data that gives the reaction rate at different conversions. 

Figure 10-12 The exit-age distribution for various combinations of PFRs and CSTRs in series.

Simple combinations of reactor elements can be solved direc tly. Figure 23-8, for instance, shows two CSTRs in series and with recycle through a PFR. The material balances with an /i-order reaction / = /cC are... 

This is a recursion formula for the exact case. We would like to be able to apply this to any number n of CSTRs in series and find an analytical and then quantitative result for comparison to the exact PFR result. To do this weneedrecursive programming. There are threeprogrammingstylesin Mathematica Rule-Based,Functional,and Procedural.Wewill attackthisprobleminrecursionwith Rule-Based,Functional,and Procedural programming. WecanbeginbylookingattherM/e-tosed recursioncodesforCaandCbinanynCSTRs. 

Consider the series combination of PFR and CSTR s shown in Figure 8.19. In terms of the fundamental design equations for these idealized... 

Some multiple-vessel configurations and consequences for design and performance are discussed previously in Section 14.4 (CSTRs in series) and in Section 15.4 (PFRs in series and in parallel). Here, we consider some additional configurations, and the residence-time distribution (RTD) for multiple-vessel configurations. 

Packed beds usually deviate substantially from plug flow behavior. The dispersion model and some combinations of PFRs and CSTRs or of multiple CSTRs in series may approximate their behavior. 

Erlang with time delay expC-t Vd + t2s/n)n The last item is of a PFR and an n-stage CSTR in series. More complex combinations are the subject of problems P5.01.33, P5.03.10, P5.03.02 and... 

CSTRs in series. The latter is often normalised by dividing by the volume of an ideal PFR required to perform the same duty. Different charts are required for each reaction rate expression. Figure 12 refers specifically to first-order kinetics, but other charts are available in, for instance refs. 17, 18 and 26. Figure 12 re-emphasises many of the points we have made already. In particular, the performance of the N CSTRs in series tends to that of a PFR of the same total volume as N becomes large and the PFR volume required to achieve a certain conversion for a first-order reaction is always smaller than the total volume of any array of CSTRs which perform the same duty. Charts in the form of Fig. 12 are particularly useful when performing approximate design calculations. 

One might intuitively expect that infinite recycle rates associated with a system as described by eqn. (61) would produce a completely well-mixed volume with concentration independent of location. This is indeed so and under these conditions, the performance tends to that of an equal sized CSTR. At the other extreme, when R is zero, PFR performance pertains. Fractional conversions at intermediate values of R may be determined from Fig. 14. The specific form of recycle model considered is thus seen to be continuously flexible in describing flow mixing between the PFR and CSTR extremes just as was the tanks-in-series model. The mean and variance of this model are given by eqns. (62) and (63) and these may be used for moments matching purposes of the type illustrated in Example 6. 

A deep stirred vessel is provided with two impellers on a single shaft. Feed is between the impellers, A plausible model for such a vessel is two CSTR s partly in parallel and partly in series, followed by a PFR, as indicated on the sketch. Let a be the fraction of the total feed that goes to the first CSTR, and let p and y be the fractions of the volume occupied by each of the CSTRs. Find the transfer function of the whole vessel and the response to impulse input. 

Remark 2 Note that the approximation of PFRs with a cascade of equal volume CSTRs is improved as we increase the number of CSTRs. By doing so however we increase the number of binary variables that denote their existence, and hence the combinatorial problem becomes more difficult to handle. Usually approximations of a PFR with 5-10 CSTRs are adequate. Kokossis and Floudas (1990) studied the effect of this approximation and their proposed approach could deal with approximations of PFRs with 200 equal volume CSTRs in series. 

The data are from Ref. 31. The objective for optimization is the maximization of the effluent concentration of component B. The performance limit of the system is identihed with each stochastic run requiring an average of only 120 CPU sec on an HP 9000-C100 workstation. Numerous designs are obtained from the stochastic search that perform close to the performance target, mostly variations of series arrangements of PFRs and CSTRs. A detailed discussion of this and other studies is given in Ref. 31. 

Equation (19-22) indicates that, for a nominal 90 percent conversion, an ideal CSTR will need nearly 4 times the residence time (or volume) of a PFR. This result is also worth bearing in mind when batch reactor experiments are converted to a battery of ideal CSTRs in series in the field. The performance of a completely mixed batch reactor and a steady-state PFR having the same residence time is the same [Eqs. (19-5) and (19-19)]. At a given residence time, if a batch reactor provides a nominal 90 percent conversion for a first-order reaction, a single ideal CSTR will only provide a conversion of 70 percent. The above discussion addresses conversion. Product selectivity in complex reaction networks may be profoundly affected by dispersion. This aspect has been addressed from the standpoint of parallel and consecutive reaction networks in Sec. 7. 

Aris (1991a) has analyzed the case of M CSTRs in series, each one endowed with the same residence time TIM. This is one of the homotopies spanning the range between a PFR and a CSTR When M = 1, one has a CSTR, and when M approaches oo one has a PFR. The result is... 

Aris (1991a), in addition to the case of M CSTRs in series, has also analyzed two other homotopies the plug flow reactor with recycle ratio R, and a PFR with axial diffusivity and Peclet number P, but only for first-order intrinsic kinetics. The values M = 1(< ), R = >(0), and P = 0( o) yield the CSTR (PFR). The M CSTRs in series were discussed earlier in Section IV,C,1. The solutions are expressed in terms of the Lerch function for the PFR with recycle, and in terms of the Niemand function for the PFR with dispersion. The latter case is the only one that has been attacked for the case of nonlinear intrinsic kinetics, as discussed below in Section IV,C,7,b. Guida et al. (1994a) have recently discussed a different homotopy, which is in some sense a basically different one no work has been done on multicomponent mixture systems in such a homotopy. 

P14-1b Make up and solve an original problem. The guidelines are given in Problem P4-1. However, make up a problem in reverse by first choosing a model system such as a CSTR in parallel with a CSTR and PFR [with the PFR modeled as four small CSTRs in series Figure P14-l(a)] or a CSTR with recycle and bypass [Figure P14-l(b)]. Write tracer mass balances and use an ODE solver to predict the effluent concentrations. In fact, you could build up an arsenal cf tracer curves for different model systems to compare against real reactor RTD data. In this way you could deduce which model best describes the real reactor. 

From Figure 2-6. wc note a very important observation The total volume to achieve 80% conversion for five CSTRs of equal volume in series is roughly the same as the volume of a PFR, As wc make the volume of each CSTR smaller and increase the number of CSTRs, the total volume of the CSTRs in series and the volume of the PFR will become identical. That is, we can model a PFR with a large number of CSTRs in series. This concept of using many CSTRs in series to model a PFR will be used later in a number of situations, such as modeling catalyst decay in packed-bed reactors or transient heat effects in PFRs. 

---