Tanks-in series model

Consider a non-ideal reaction vessel of volume V through which inert fluid flows at a volumetric flow rate q. The mean residence time 0 is 

-- Cj are the tracer concentrations in N ideal CSTRs. The outlet tracer concentration of the Nth ideal CSTR is the effluent tracer concentration of the non-ideal reactor. 

Taking the unsteady-state tracer balance in the /th CSTR, we get 

Writing the equation in the form of standard first-order differential equation [ dy/dx) + P(x)y - Q(x)], we have 

It may be verified that Equation 3.298 for N = 1 reduces to E(d) = e V0/ which is the exit age distribution equation for an ideal CSTR. The E(0) versus 0 plots for different 

The relationship between N and the variance according to the cell model is given by 

The mean residence time obtainable from the first moment of the distribution corresponds to the space time, if the density of the fluid is constant. 

Several useful ideas and conclusions can be obtained by studying the flow of an unreactive tracer, such as a coloured dye, through a series of tanks each of which is assumed to be ideally mixed. So that we may readily compare the effects of increasing the number of tanks, let us consider a volume V which can be made up of one tank only or two tanks each of volume V/2 or three tanks each of volume K/3 or, in general, i equally sized tanks each of volume V/i (Fig. 2.7). Through this series of tanks, a pure liquid flows at a volumetric flowrate v. Suppose that at time t - 0 when all the tanks are full and the whole system is in a state of steady flow, a shot of 0 moles of tracer is injected into the feed. The tracer is assumed to be 

At any time t, let tank 1 contain moles of tracer, whence the concentration in the tank C, = nJ V/i). 

Applying a material balance to the tracer over tank 1, and noting that as far as tracer concentration is concerned, the system is not in a steady state  

The progress of a tracer through a series of tanks can be followed using these equations. Taking the case of three tanks as an example, Fig. 2.7c, the curves shown 

As the basis for further comparison, let us return to the case of a fixed volume V which may be a single tank (Fig. 2.7a) or two tanks (Fig. 2.7b) or three tanks and so on. In the general case there will be i tanks and the concentration of the tracer leaving the last tank will be C,. If we now plot C,/C0, where C0 = n0/V, against the reduced time (vt/V), the family of curves shown in Fig. 2.9a is obtained. The curves are reduced C (i.e. outlet concentration) curves, as already indicated in Section 2.1.2. 

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Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Equations 8-109, 8-110, 8-111, and 8-112 are redueed to an ordinary tanks-in-series model when N = i and h = 0. For the equivalent number of ideal CSTRs, N is obtained by minimizing the residual sum of squares of the deviation between the experimental F-eurve and that predieted by Equation 8-109. The objeetive funetion is minimized from the expression... 

The Tanks-in-Series Model. A simple model having fuzzy first appearance times is the tanks-in-series model illustrated in Figure 15.2. The washout function is... 

FIGURE 15.2 The tanks-in-series model (a) physical representation (b) washout function. 

Solution Equations (15.27) and (15.28) give the residence time functions for the tanks-in-series model. For A =2,... 

The limits for part (b) are at the endpoints of a vertical line in Figure 15.14 that corresponds to the residence time distribution for two tanks in series. The maximum mixedness point on this line is 0.287 as calculated in Example 15.14. The complete segregation limit is 0.233 as calculated from Equation (15.48) using/(/) for the tanks-in-series model with N=2 ... 

The Stirred Tanks in Series Model Another model that is frequently used to simulate the behavior of actual reactor networks is a cascade of ideal stirred tank reactors operating in series. The actual reactor is replaced by n identical stirred tank reactors whose total volume is the same as that of the actual reactor. 

The performance of a CSTR can be brought closer to that of a PFR, if the CSTR is staged. This is considered in Chapter 20 in connection with the tanks-in-series model. 

Apply the tanks-in-series model to the following kinetics scheme involving reactions in parallel ... 

I 478 I ex19-3.msp I paramc iter estimation for tanks-in-series model... 

As explained in Section 4.4.4, there exists an equivalency between tubular dispersion models and stagewise or tank in series models. The stagewise model, used in CHROMPLATE considers the chromatographic column to consist of a large number of well-mixed stirred tanks, arranged in series and thus represents an alternative modelling approach to that of the tubular dispersion model CHROMDIFF. The same two-component separation process is modelled and simulated in both cases. 

CSD Functions. From the RTD function based on the tank-in-series model, both the number basis and weight basis probability density functions of final product... 

The tanks-in-series model is a flexible one-parameter model which amounts to characterising a system in terms of the general transfer function of the equation... 

Expressions for the mean, Mi, and the variance, T2, of the tanks in-series model in both unnormalised and normalised forms [eqns. (43) and (45)]... 

Fig. 11. Normalised residence time distributions E(0), eqn. (46), for the tanks-in-series model plotted for various specified numbers of tanks, N.

The conversion achieved in most reactors lies between that which would be expected from a PFR or CSTR of the same size. The tanks-in-series model can be used to predict this level of conversion once tracer test data have been recorded and processed. The following extimple illustrates typical calculations. 

A first-order liquid-phase reaction takes place in a baffled stirred vessel of 2 volume under conditions when the flow rate is constant at 605 dm min and the reaction rate coefficient is 2.723 min the conversion of species A is 98%. Verify that this performance lies between that expected from either a PFR or a CSTR. Tracer impulse response tests are conducted on the reactor and the data in Table 6 recorded. Fit the tanks-in-series model to this data by (A) matching the moments, and (B) evaluating N from the time at which the maximum tracer response is observed. Give conversion predictions from the tanks-in-series model in each case. 

Now let us consider two possible ways of deciding what parameters should be chosen in the tanks-in-series model in order to describe this system well. 

Fig. 12. General design chart for the tanks-in-series model described by eqn. (43), first-order reaction A -r r with no change in volume (e = 0). Ordinate gives the total volume of all the tanks in series divided by the volume of an ideal PFR which achieves the same conversion.-------, Constant kr -------, constant N.

Imagine a first-order reaction taking place in such a system under conditions where rk, i.e. VkjQ, is 10 and R is 5. Using the technique previously adopted in Sect. 5.1 and outlined in Appendix 2, we can readily calculate that this system would achieve 96.3% conversion of reactant. Under these conditions, the recycle reactor volume turns out to be 3.03 times that of an ideal PFR required for the same duty. This type of calculation allows Fig. 14 to be constructed this is similar in form to Fig. 12, but lines of constant for the tanks-in-series model have been replaced by lines of constant recycle ratio for the recycle model. From a size consideration alone, the choice of a PFR recycle reactor is not particularly... 

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. 

If Fig. 12 and Fig. 14 were laid on top of each other, then conditions of equivalence could be determined under which the performance of the tanks-in-series model with specified N would be the same as that of the recycle model, that is the value of R could be found which would result in the same conversion and V/Vpp ratio. Levenspiel [17] gives these values for a variety of conditions for both first- and second-order reactions. His data are reproduced in Table 8. 

Recycle ratio for equivalent performance of the recycle and tanks-in-series models at specified conversions for both first- and second-order reactions [17]. 

Number of units in tanks-in-series model Required recycle ratio ... 

Consider the fully normalised tanks-in-series model (see Sect. 5.1) given by the equation... 

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