CSTR sequences

A summary of results for several simple-order rate equations is given in Table 4.2. The effect of volume changes on reaction is presumed negligible in these results. Note that in the general th-order case and the second-order reaction between different reactants, it is not possible to obtain an explicit solution for the exit concentration. This leads to some difficulty in the analysis of CSTR sequences, as seen below. 

There are some other points, however, to be considered in determining the length of a CSTR sequence. Obviously, any given conversion can be obtained in a single large reactor as well as in a series of smaller ones (think of irreversible... 

Figure 4.21 Comparison of a PFR and a CSTR sequence conversion for a first-order...

A comparison in terms of CSTR sequences rather than individual reactors is given in Figure 4.21, where we show the conversion obtained in a single PFR of set residence time with various CSTR sequences, differing in the number of reactors but with the same total residence time as the PFR, that is, tcsTR = hFRl > with n the number of CSTR. For the first-order example illustrated, the conversion steadily increases as n increases in the CSTR sequence, and indeed will approach the PFR value in the limit of very large n. This can be shown quantitatively by comparison of the series expansions of the conversion expressions for the two types of... 

For an -unit equally sized CSTR sequence we may obtain Ca by modification of equation (4-89)... 

For the -unit, equally sized CSTR sequence, we may derive the following results. 

Since the reaction is essentially irreversible at this temperature, we have for the CSTR sequence... 

Now remember that the holding time per vessel in the CSTR sequence, 0.5 h, must be multiplied by the number of vessels to give the PFR residence time. Thus,... 

The PFR does give a small increase in the conversion compared to the CSTR sequence. Keep in mind that according to Figure 4.21, however, the six-tank sequence is already well along the path to where CSTR series PFR. 

The transcendental nature of equation (4-135) for T, precludes the development of any generalized treatment of nonisothermal or adiabatic CSTR sequences as... 

A second approach is to use various combinations of the ideal reactor models in simulation of nonideal behavior. This may seem a bit contradictory at first, but hopefully our later discussion will be sufficient to illuminate the reasoning behind this method. A hint of this approach is given by the discussion in Chapter 4 on the comparisons between the conversions in PFR and CSTR sequences of the same total... 

One can see immediately that there is trouble The generation of analytical F t) or E t) responses is not going to be a fruitful exercise, since the concentration of tracer downstream, C +i, appears in the equation for the determination of C However, there is an interesting relationship between these CSTR sequence mixing models, both with and without backmixing, and the dispersion models to be discussed in the following, so we shall keep both in mind for the present. 

In Chapter 4 it was pointed out that the performance of a CSTR sequence approached that of a single PFR of equivalent total residence time as the number of units in a sequence approached infinity. This result is also obeyed by the F 6) and E 6) curves computed from the mixing-cell model reported in Figure 5.3. Since the plug-flow model represents one limit of the dispersion model (that when D 0), it is reasonable to assume that there is an interrelationship between mixing-cell and dispersion models that can be set forth for the more general case of finite values... 

Figure 5.12 Variation of conversion with recycle ratio in a three CSTR sequence.

Mixing-cell models were discussed extensively in Chapter 4 under the guise of the analysis of CSTR sequences. It is a good time to revisit some of this analysis from the specific point of view of modeling nonideal reactors. 

Figure 5.14 (a) Effect of nonideal exit-age distribution on conversion as modeled by a CSTR sequence. 

Use of the CSTR sequence as a model for nonideal reactors has been criticized on the basis that it lacks certain aspects of physical reality, such as the absence of backward communication between the individual mixing cell units. Such may be the case nonetheless the mathematical simplicity of the approach makes it very attractive, particularly for systems with complex kinetics, nonisothermal effects, or other complicating factors. 

It is apparent from the shapes of the curves given in Figure 5.21 that interpolation or extrapolation for various orders and values of Np, is difficult. Indeed, if any degree of accuracy is required, individual numerical solutions, will probably be needed. Again, the numerical convenience of the CSTR sequence beckons. 

For the CSTR sequence, the general mass balance with chemical reaction in the unsteady-state becomes... 

In many cases this can prove to be a very restrictive assumption, so that in general the evaluation of transients for CSTR sequences with non-first-order kinetics is a numerical problem. Various other cases of unsteady-state operation are summarized in the papers of Piret and coworkers cited in Chapter 4, and in the text by Cooper and Jeffries [A.R. Cooper and G.V. Jeffries, Chemical Kinetics and Reactor Design, Prentice-Hall, Inc., Englewood Cliffs, NJ, (1973)]. 

A comparison of these transient forms for first-order kinetics, equation (5-115) for the CSTR sequence and (5-116) for the axial dispersion model, on the basis of their steady-state equivalence in terms of n and Npe, is suggested in the exercises. The basic equation to be answered is whether the equivalence of the two models in the... 

Derive an expression for the response of an -unit CSTR sequence to a step-function decrease in inlet concentration. 

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. ... 

Make a comparison of the selectivity behavior of a CSTR sequence and approach to PFR behavior, similar to that of Figure 5.14b, for a Type III system in which... 

Similar procedures yield differential equations giving higher-order responses for each stage i. The model here is formally analogous to the CSTR-sequence mixing model. 

The difficulty in this is the awkward form of equation (6-140) with respect to 7). One likes to compute in sequence through the series of cells, but here we face the implicit form of Ti as a function of r, i. This is the same basic difficulty that limits the utility of the CSTR sequence as an analytical model for nonisothermal reactors. For the case here though, where we employ a relatively large value of the index n in approximation of a plug-flow reactor, and where the solution will be via numerical methods anyway, we will strong-arm the problem with the approximation T,- r,- ] in the exponentials, so that... 

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