PFR as a Series of CSTRs

The steady-state solutions for one CSTR are shown here  

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 we need recursive programming. There are three programming styles in Mathematica Rule-Based, Functional, and Procedural. We will attack this problem in recursion with Rule-Based, Functional, and Procedural programming. We can begin by looking at the rule-based recursion codes for Ca and Cb in any n CSTRs. 

The seeds for these rules are the solutions for the first CSTR, and then these exit concentrations become the inlet concentrations to the second CSTR, whose exit concentrations become the inlet concentrations for the third CSTR, and so it goes on through to n CSTRs. We can have Mathematica assemble the equations and the variables that we will need for the Solve routine. This is illustrated with n = 3  

Chapter 9 Continuous Stirred Tank and the Plug Flow Reactors 

General spelll Possible spelling error new symbol name Cdf is similar to existing symbol CDF . 

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To improve the overall selectivity, reduce the [B] in the feed and/or increase flie [A] in the feed, if possible. Operate in a PFR or a series of CSTRs since the conversion of B is so low that the [B] actually is higher in the effluent from the single CSTRofthis example than it is in the feed. Since Ais hydrogenated faster than B, the ratio [B]/[A] increases as the reaction proceeds. Examine the feasibility of operating with a series of PFRs or CSTRs with fresh feed added between reactors. 

The liver will be modeled as a number of CSTRs in series to approximate a with a volume of l.l dm-. Approximating a PFR with a number of CSTRs in t was discussed in Chapter 2. The total volume of the liver Ls divided into CSTRs. 

A question arises how to reduce this difference The solution consists of using a series of CSTRs instead a single PFR. Since the reaction rate is higher in each intermediate volume, finally a much higher productivity than in a single equivalent reactor is obtained (Fig. 8.6). At limit, an infinite series of CSTRs behaves as a single PFR of the same volume. 

To exempUly the modeling of visbreaker reactors, in this section it is assumed that the coil behaves as a PFR and can be modeled either with the design equation of PFR or as a series of equal volume CSTR. The soaker behaves as an adiabatic CSTR. A schematic representation of these two reactors is presented in Figure 3.7. 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

The representation of different types of reactor units in the approach proposed by Kokossis and Floudas (1990) is based on the ideal CSTR model, which is an algebraic model, and on the approximation of plug flow reactor, PFR units by a series of equal volume CSTRs. The main advantage of such a representation is that the resulting mathematical model consists of only algebraic constraints. At the same time, however, we need to introduce binary variables to denote the existence or not of the CSTR units either as single units or as a cascade approximating PFR units. As a result, the mathematical model will consist of both continuous and binary variables. 

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. 

Using the approximation of a PFR with a cascade of equal volume CSTRs in series, we allow for all feed streams (i.e., fresh feeds, streams from the outlets of other reactors) to be distributed to any of the CSTRs that approximate the PFR unit. In addition, potential by-passes around each CSTR that approximates the PFR unit are introduced so as to include cases in which full utilization of these CSTRs is not desirable. 

It is obvious from the table that the conversion of the series of CSTRs increases with the number of CSTRs (total volume is constant), approaching an upper limit that is the value of the conversion of a PFR with the same volume as the total volume of the CSTR system. The PFR itself can be considered as an infinite number of differential homogeneous slices, each one of which behaves like a CSTR. The increase in conversion with the number of CSTRs is at first very rapid, levelling off later and hence, in practice, a number of CSTRs between 10 and 20 has an overall conversion similar to the equivalent total volume PFR. How... 

After completing this chapter you will be able to size CSTRs and PFRs given the rate of reaction as a function of conversion and to calculate the Overall conversion and reactor volumes for reactors arranged in series. 

Unlike the TS-PFR and the TS-CSTR, the TS-SSR normally requires that a broad range of temperature ramping rates be possible. In this the two TS-SSRs are similar to the TS-RR and all the issues regarding the speed and frequency of analysis described in the section on the TS-BR apply here too. The option of varying feed flow rate in a series of runs at constant ramping rate, as is done in a standard TSR experiment, is also available if ramping rates can not be varied adequately. 

A tubular flow reactor exhibits a residence-time distribution which can be modeled by a sequence of CSTRs in series, all of the same volume. The nominal residence time in the tubular reactor is 20 s. Compare, for a Type III reaction, the conversion, selectivity, and yield obtained in the reactor (as modeled by the CSTR sequence) with that which would be obtained in a true PFR of the same residence time. The rate constants are =0.1 s and ki = 0.05 s. ... 

The tanks-in-series model, in which a sequence of CSTRs is used to simulate various degrees of partial mixing, was also anticipated and briefly considered in Chapter 10. Equation 10.18 for a first-order reaction shows the approach to PFR of a CSTR as the number of tanks N is increased from one to infinity. This is illustrated in Figure 13.5 (Levenspiel, 1993) as plots of (Fn/ pfr) versus [y4f,j]/[y4]Q for different values of N. Notice that increases by almost a factor... 

Structures (c) and (e) are both similar, as both structures involve combinations of a CSTR and PFR in series. It is known that the final approach to the extreme points of the AR take place as a result of the union of PFR trajectories, and thus we should expect that final fundamental reactor type of any optimal reactor structure on the AR boundary is a PFR. We may conclude that structure (e) does not produce an effluent concentration that is an exposed point on the AR boundary (although the effluent concentration may still lie on the AR boundary, the point will not be exposed). The CSTR feeding the PFR in (c) must therefore produce a concentration that is a point on the AR boundary. 

When a steady stream of fluid flows through a vessel, different elements of the fluid spend different times within it. The time spent by each fluid element can be identified by an inert tracer experiment, where a pulse or a step input of a tracer is injected into the flow stream, and the concentration of the pulse in the effluent is detected. As the reader may quickly infer, the tracer must leave the PFR undisturbed. On the other hand, a step pulse may give rise to an exponential distribution in a CSTR. In the beginning of this chapter, we already demonstrated that PFR behavior approaches that of a CSTR under infinite recycle. It follows that infinite CSTRs in series behave like a PFR. Thus, we conclude that any nonideal reactor can be represented as a combination of the PFR and MFR to a certain degree. First, let us show a representative pulse response curve for each of the ideal reactors in Figure 3.5. As seen in the figure, the response to a step input of tracer in a PFR is identical to the input function, whereas the response in a CSTR exhibits an exponential decay. The response curves as shown in Figure 3.5 are called washout functions. The input function of the inert tracer concentration [/] can be mathematically expressed as... 

Figure 7-6 The maximum value of the concentration of the intermediate product, R, in the series reactions A R S, as a function of the ratio of the rate constant for the second reaction kz) to the rate constant for the first reaction ( i). Both reactions are first order and ineveisible. There is no R in the feed. The reactors are isothermal. Solid line— PFR dashed line—CSTR.

Thus, the limit gives the same result as a piston flow reactor with mean residence time t. Putting tanks in series is one way to combine the advantages of CSTRs with the better yield of a PFR. In practice, good improvements in yield are possible for fairly small N. 

Develop the E(t) profile for a 10-m laminar-flow reactor which has a maximum flow velocity of 0.40 m min-1. Consider t = 0.5 to 80 min. Compare the resulting profile with that for a reactor system consisting of a CSTR followed by a PFR in series, where the CSTR has the same mean residence time as the LFR and the PFR has a residence time of 25 min. Include in the comparison a plot of the two profiles on the same graph. 

Consider a reactor system made up to two vessels in series a PFR of volume Vpp and a CSTR of volume EST, as shown in Figure 17.5. In Figure 17.5(a), the PFR is followed by the CSTR, and in Figure 17.5(b), the sequence is reversed. Derive E(d) for case (a) and for case (b). Assume constant-density isothermal behavior. 

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