1. Chemical Reaction Engineering II Note 4
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2. 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 to use depends on your 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.
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3. Pulse response experiment and the RTD
The system being considered is in figure 1. defining terms;
Figure 1: Tanks-in-series model
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4. At any particular time from Note 1;
For the first tank; consider a steady flow v (m3/s) of fluid into and out of the first ideal mixed flow units of volume V1. At time t = 0 inject a pulse of tracer into the vessel which when evenly distributed in the vessel has a concentration C0.
At any time t after the tracer is introduced, make a material balance;
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5. In symbols this expression becomes;
Where C1 is the concentration of tracer in tank “1”. Separating and integrating; Or
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6. Here C1 enters and C2 leaves, a material balance gives;
For the first tank;
For the second tank
Here C1 enters and C2 leaves, a material balance gives;
Separating give a first-order differential equation which on integration gives;
(1)
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7. (2)
For the nth-order. Integration for 2nd, 3rd, 4th …nth tank becomes more complicated so it is simpler to do all of this by Laplace transforms.
The RTDs, means and variances both in time and dimensionless time were first derived by MacMullin and Weber (1935), summarized by eqn. (3). Graphically the eqns. are displayed in Fig. 2. The properties are sketched in Fig. 3.
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8. (3)
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9. Figure 2: RTD curves for the tank-in-series
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10. Figure 3: Properties of RTD Curves for tank-in-series model
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11. Comments and extension
1)- If M tanks are connected to N more tanks (all same size) then the individual means and variances (time units) are additive; Because of this property, we can join incoming streams with recycle streams. This model is therefore useful for treating recirculating streams.
(4)
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12. One-shot tracer input
If one-shot tracer input is introduced into N tanks as in figure 4, then from eqn (3) and (4) we can write;
Because of the independence of stages it is easy to evaluate what happens to the C curve when tanks are added or subtracted. This model becomes useful
in treating recycle flow and closed recirculation systems. Let us briefly look at these applications.
(5)
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13. Figure 4: For any one-shot input eqn (4) relates input, output and number of tanks.
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14. Closed recirculation system
If we introduce a  signal into an N stage system as shown in Fig. 5, the recorder will measure tracer as it flows by the first time, the second time, and so on. In other words it measures tracer which has passed through N tanks, 2N tanks, and so on. It measures the superposition of all these signals.
To obtain the output signal for these systems simply sum up the contributions from the first, second, and succeeding passes. If m is the number of passes we shall have;
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15. Figure 5 shows the resulting C curve
We have for the 5 tanks the following equations (7).
(6a)
(6b)
(6c)
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16. Figure 5: Tracer signal in a recalculating system.
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17. Where the terms in brackets represent tracer signal from the first, second and successive passes. Recirculation systems can be represented as well by the dispersion model.
(7a)
(7b)
(7c)
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18. Recirculating with through flow
For relatively rapid recirculation compared to through-flow, the system as a whole acts as one large stirred tank hence, the observed tracer signal is simply the superposition of the recirculation pattern and the exponential decay of an ideal stirred tank.
This is shown in Fig. 6 where C, is the concentration of tracer if it is evenly distributed in the system.
This form of curve is encountered in closed recirculation systems in which tracer is broken down and removed by a first-order process, or in systems using radioactive tracers.
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19. Figure 6: Recirculation with slow through-flow
Drug injection on living organisms give this sort of superimposition because the drug is constantly being eliminated by the organism.
Figure 6: Recirculation with slow through-flow
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20. Step response experiment and the F curve
The output F curve from a series of N ideal stirred tanks is given in eqn (8) in its various forms.
(8)
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21. In Graphical form; Figure 7: F curves for the tank-in-series model.
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22. Chemical Conversion First Order Reaction
First order reaction in one tank, the eqn. is;
For N tanks in series;
For small deviation from plug flow (large N) comparison with plug flow gives;
(9)
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23. These equations apply for both micro and macro fluids.
Second order reaction of micro-fluid
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24. For a micro-fluid flowing thru N tanks in series equation is;
All other reaction kinetics of micro-fluids
Either solve the mixed flow equation for tank after tank
Or use graphical procedure
(10)
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25. Shown in figure 8
Figure 8: Graphical procedure of evaluating the performance of N tanks in series for any kinetics
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26. Chemical Conversion of Micro-fluids
It is rarely used for homogeneous system but if you need it, mostly for G/L heterogeneous systems
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27. Example 1: Modification to a winery
A small diameter pipe 32 m long runs from the fermentation room of a winery to the bottle filling cellar. Sometimes red wine is pumped through the pipe, sometimes white and whenever the switch is made from one to the other, a small amount of “house blend” rose is produced (8 bottles). Because of some construction in the winery the pipeline length will have to be increased to 50 m.
For the same flow rate of wine, how many bottles of rose may now be expected each time the flow is switched?
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28. Solution Let the number of bottles (the spread) be related to , so;
For small deviations from plug flow;
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