The tanks-in-series model

This model can be used whenever the dispersion model is used and for not too large a deviation from plug flow both models give identical results, for all practical purposes. Which model you use depends on your mood and taste. 

The dispersion model has the advantage in that all correlations for flow in real reactors invariably use that model. On the other hand the tanks-in-series model is simple, can be used with any kinetics, and it can be extended without too much difficulty to any arrangement of compartments, with or without recycle. 

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 

The model is seen to be a series sequence of N equal sized CSTRs which have a total volume V and through which there is a constant flowrate Q. From the physical standpoint, it is natural to restrict N, the number of tanks, to integer values but, mathematically, this need not be the case. When N is considered as a continuous variable which lies between one and infinity, a model results which can be used to interpolate continuously between the bounds of mixing associated with the CSTR and PFR. For N less than unity, the model represents systems with partial bypassing [41]. For integral values of N eqn. (43) may be inverted directly (see Table 9, Appendix 1) to give 

Equation (44) represents a family of RTDs all with a mean residence time of V/Q. As mentioned in Sect. 3.1, it is frequently more convenient to present RTDs in terms of dimensionless time, 6, rather than real time t. 

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)] 

These dimensionless residence time aistributions E(0) can be obtained by inverting the fully normalised form of the system transfer function G(s). This is given by eqn. (45), which emphasises more clearly than eqn. (43) that the model in question contains only one flexible parameter, N. 

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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. 

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 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 ... 

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... 

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. 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

Figure 14.3 Properties of the RTD curve for the tanks-in-series model.

Chapter 14 The Tanks-in-Series Model 6-input signal... 

Since + V2 add up to 1, there is no dead volume, so at this point our model reduces to Fig. E14.3c. Now relax the plug flow assumption and adopt the tanks-in-series model. From Fig. 14.3... 

Given Qj, and as well as the location and spread of these tracer curves, as shown in Fig. E14.4a estimate the vessel E curve. We suspect that the tanks-in-series model reasonably represents the flow in the vessel. 

Equation 3 represents the tanks-in-series model and gives... 

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