Three CSTRs in Series

The equations describing the series of three isothermal CSTRs were developed in Sec. 3.2. 

DISTURBANCE IS A STEP CHANGE IN FEED CONCENTRATION AT TIME EQUAL ZERO PROM O.S TO 1.8 KG-MOLES OF A/CURIC METER. 

10 CA1=CA1+CA1D0T DELTA CAa=CA2+CA2DOT DELTA CA3=CA3+CA3D0T+DELTA TIME=TIME+DELTA IF(TIME.LE.3.)G0 to 100 STOP END 

The right-hand sides of the DDEs [Eqs. (5.9) to 5.11)] are the functions/J, discussed in Sec. 4.4.1. Let us caJl these derivatives CAIDOT, CA2DOT, and CA3DOT. At the nth step in time, 

Then to step to the next point in time, using Euler integration with a step size DELTA, 

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P4-33 A CSTR with two impellers is modeled as three CSTRs in series. [3rd Ed. p4-2... 

P4-29. Apply the unsteady state mole balances to three CSTRs in series. Use POLYMATH to solve three coupled ODEs. 

These numbers show that the optimal economic steady-state design is the multiple-stage reactor system. The higher the conversion, the larger the economic incentive to have multiple stages. At 95 percent conversion, the capital cost of a three-CSTR process is 67 percent that of a one-CSTR process. At 99 percent conversion, the cost is only 38 percent. Thus, if only steady-state economics is considered, the design of choice in these numerical cases is a process with two or three CSTRs in series. 

TABLE 1-3 Restricted Residence-Time Optimization for Three CSTRs in Series Operating at the Same Temperature"... 

Two or three CSTRs in series or a CSTR-PFR array can be used in mass processes. In the most common two CSTR or CSTR/PFR arrays, the PB solution is added to the first CSTR that operates between 20% and 40% solids at steady state. The second CSTR operates at solid contents between 50% and 80% depending on reactor volume and design. In the three-CSTR array, the first reactor operates at low conversions, <20%, and the second CSTR operates... 

Consider the following reactor structure involving three CSTRs in series. 

Note the reduction in the volume requirement when three CSTRs in series are employed. The reader is left the exercise of showing that the total volume requirement for three CSTRs of equal volume is... 

Case 1 Suppose that you are asked to calculate the performance of the three CSTRs in series. The volume of each reactor, Vi, is given, as is the inlet molar flow rate to the first reactor, Fao- The rate equation for the disappearance of A is also specified. The convo ion of A leaving the third (last) reactor is to be calculated. 

Albanis TA, Pomonis PJ, Sdoukos AT. 1988a. Describing movement of three pesticides in soil using a CSTR in series model. Water Air Soil Pollut 39 293-302. 

Consider the flow mixing through three identical ideal CSTRs in series. Each tank has a space time, r, or mean residence time, Mj, of 2 min. An idealised impulse of tracer is made in the inlet to the first tank what tracer response will be observed from the third tank ... 

This result is, of course, identical to that found previously. Responses of a number of CSTRs to other forcing functions may be found with equal ease. Thus, for example, the step response of the three tanks in series considered above is given by... 

The design equation for three equal-volume CSTRs in series is... 

Let us return to the comparison we made in Chapter 2 of multiple CSTRs in series. In Section 2.6.1 the steady-state designs of 1-, 2-, and 3-CSTR processes were developed, and in Section 2.9.1 the capital costs of the three alternative processes were determined. The steady-state calculations indicate control problems in the first reactor of the 2-CSTR and 3-CSTR processes because the jacket temperatures are getting closer to the supply cooling water temperature. 

CSTR Substrate and nutrients including oxygen are continuously fed so as to achieve the desired steady state. The beer containing ethanol, biomass and unconsumed nutrients is removed continuously. The productivity can reach 6g/l h, about three times of a batch reactor. The yield of ethanol is limited by the inhibition effect Total productivity is also limited by low biomass concentration (10-12g/l). A sugar concentration of 10% in the feed gives the highest productivity. Two or more CSTR in series can be used. 

We will now use the value of FiS,o calculated in Example 2-1 together with Figure 2-1 to size each of the reactors for the three reactor schemes. The first scheme to be considered is the two CSTRs in series shown in Figure 2-3, For the first reactor in which the rate of disappearance of A is -jai at conversion Xi, the volume necessary to achieve the conversion Xj is... 

The RTD will be analyzed from a tracer pulse injected into the first reactor of three equally sized CSTRs in series. Using the definition of the RTD presented in Section 13.2. the fraction of material leaving the system of three reactors (i.e., leaving the third reactor) that has been in the system between time t and / + Ar is... 

Figure 14-2 illustrates the RTDs of various numbers of CSTRs in series in a two-dimensional plot (a) and in a three-dimensional plot (b). As the number becomes very large, the behavior of the system approaches that of a plug-flow reactor. 

The reader is left the exercise of calculating the volume requirements for three equal-volume CSTRs in series. For this condition, Vj = 4.77 m with Vi — V2 = Vj = 1.59m. This is left as an exercise only for those with a strong analytical mathematics background. 

To demonstrate these ideas, let us consider three different schemes of reactors in series two CSTRs, two PFRs, and then a combination of PFRs and CSTRs in series. To size these reactors, we shall use laboratory data that gives the reaction rate at different conversions. 

We see once again that the overall conversion is higher in the PFR than in the CSTR and that the fractions of D and E in the products are therefore larger than in the CSTR. The efficiency of the PFR with positive order kinetics versus the perfectly mixed CSTR raises an interesting question—if one were to divide the volume of one CSTR into two CSTRs in series of equal volume, would there be any change in efficiency What if there were three or even more CSTRs in series with equal volumes that all summed up to that of the original one— would there be a significant difference We address this in the next section. 

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. 

Case ni Three Identical CSTRs in Series. In this case there are two intermediate unspecified reactant concentrations instead of just the single intermediate concentration... 

Figure 9.3 Instantaneous yield versus reactant concentration curves and their relation to overall product concentration changes (a) plug flow or batch reactor, (b) single CSTR (c) cascade of three arbitrary-sized CSTRs in series. (Adapted from O. Levenspiel...

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