CSTRs in series

The effluent coneeniration of reactant A from the first CSTR can be found using Equation (5-9) 

Solving for concentration exiting the second reactor, we obtain 

If both reactors are of equal size (t, = Tt = t) and operate at the same temperature (Jt, = kj = k), then 

If instead of two CSTRs in series we had n equal-sized CSTRs connected in series (t, = Tt = = = T =(V/V( )) operating at the same temperature 

Example S-2 Producing 200 Million Pounds per Year in a CSTR 

Suppose three CSTRs each with the same volume V are arranged in series, as shown in the following figme. The molar fl ow rate of reactant A into the first reactor is Fao. The fractional conversion of A in the effluent from the first reactor is xa,i, the conversion of A in the effluent from the second reactor is xa,2. and the conversion of A leaving the third reactor, and the system as a whole, is xa,3. 

It is very important to be consistent in defining the fractional conversions for a series of reactors. The easiest definition, which will be followed in this book, is to base the conversion on the molar flow rate to the first reactor, i.e., Fao- Let Fa,i be the molar flow rate of A out of the first reactor. Fa,2 be the molar flow rate out of the second reactor, and Fa,3 be the molar flow rate out of the third reactor. Then 

The conversion a a,i is the fractional conversion of A in the stream leaving the first reactor. This is the same definition that is used for a single reactor. The conversion xp is the overall conversion of A in the stream leaving the second reactor. In other words, jca,2 is the conversion for the first and second reactors combined. Finally, jca3 is the conversion of A in the stream leaving the third (last) reactor. It is the overall conversion for the series of three reactors. 

With this basis, the fractional conversion of reactant A in the stream leaving reactor N +lis always greater that the fractional conversion of A in the stream leaving the reactor immediately upstream, i.e., reactor N. Moreover, the fractional conversion of the stream entering reactor iV + 1 is the same as the conversion in the stream leaving reactor N. 

Let s carry out a material balance on the second reactor in the figure above. This balance will illustrate how to use the preferred approach to defining conversions and molar flow rates. At steady state, the material balance for A, using the whole second reactor as a control volume, is 

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. 

In the second reactor, the rate of disappearance of A, is evaluated at the conversion of the exit stream of reactor 2, Xj, A mole balance on the second reactor 

For the second CSTR recall that is evaluated at Xt and then use (X -X] to calculate Vi at X . 

In the examples that follow, we shall use the molar flow rate of A we cal 

For the two CSTRs in series, 40 i conversion i,s achieved in the first reactor. Wha is the volume of each of the two reactors necessary to achieve 80 overall conversion of the entering species A  

Let us write down the mole balance for reactant A for each reactor i  

Multiplying each of these equations for i = 1,2,. . . , n results in a correlation between the outlet concentration of the system, i.e. outlet concentration of the n-th reactor, and inlet concentration of the system C0, as well as an expression for the overall conversion  

Let us now apply the above equation in order to calculate the conversion of a varying number of CSTRs in series with total volume Vtot = 0.6 m3 and volumetric flow rate V = 0.1 m3 h 1 for a first-order reaction with reaction rate constant k = 0.25 h The results are presented in Table 1. 

It is obvious from the table that the conversion of the series of CSTRs increases with the number of CSTRs (total volume is constant), approaching an upper limit that is the value of the conversion of a PFR with the same volume as the total volume of the CSTR system. The PFR itself can be considered as an infinite number of differential homogeneous slices, each one of which behaves like a CSTR. The increase in conversion with the number of CSTRs is at first very rapid, levelling off later and hence, in practice, a number of CSTRs between 10 and 20 has an overall conversion similar to the equivalent total volume PFR. How 

Number of CSTRs Volume of one reactor (m3 ) Residence time (h) Conversion 

---

Simple combinations of reactor elements can be solved direc tly. Figure 23-8, for instance, shows two CSTRs in series and with recycle through a PFR. The material balances with an /i-order reaction / = /cC are... 

Figure 23-8 develops the overall transform of a process with a PFR in parallel with two CSTRs in series. C(t) is found from C(.s) by inversion of the output transform. 

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 weneedrecursive programming. There are threeprogrammingstylesin Mathematica Rule-Based,Functional,and Procedural.Wewill attackthisprobleminrecursionwith Rule-Based,Functional,and Procedural programming. WecanbeginbylookingattherM/e-tosed recursioncodesforCaandCbinanynCSTRs. 

Monod kinetics are considered in a CSTR with an organism growing with an initial substrate concentration of 50g-l 1 and kinetic parameters of Ks = 2g-l 1 and /Amax = 0.5lr. (a) What would be the maximum dilution rate for 100% yield of biomass with maximum rate (b) If the same dilution is used, what would be number of CSTRs in series ... 

Numerical calculations are the easiest way to determine the performance of CSTRs in series. Simply analyze them one at a time, beginning at the inlet. However, there is a neat analytical solution for the special case of first-order reactions. The outlet concentration from the nth reactor in the series of CSTRs is... 

Determine the minimum operating cost for the process of Example 6.2 when the reactor consists of two equal-volume CSTRs in series. The capital cost per reactor is the same as for a single reactor. 

Example 14.6 derives a rather remarkable result. Here is a way of gradually shutting down a CSTR while keeping a constant outlet composition. The derivation applies to an arbitrary SI a and can be extended to include multiple reactions and adiabatic reactions. It is been experimentally verified for a polymerization. It can be generalized to shut down a train of CSTRs in series. The reason it works is that the material in the tank always experiences the same mean residence time and residence time distribution as existed during the original steady state. Hence, it is called constant RTD control. It will cease to work in a real vessel when the liquid level drops below the agitator. 

Part (c) considers the mixing extremes possible with the physical arrangement of two tanks in series. The two reactors could be completely segregated so one limit remains 0.233 as calculated in part (b). The other limit corresponds to two CSTRs in series. The first reactor has half the total volume so that Uinkii = 2.5. Its output is 0.463. The second reactor has (ai )2ki2 = 1.16, and its output is 0.275. This is a tighter bound than calculated in part (b). The fraction unreacted must lie between 0.233 and 0.275. 

Part (c) in Example 15.15 illustrates an interesting point. It may not be possible to achieve maximum mixedness in a particular physical system. Two tanks in series—even though they are perfectly mixed individually—cannot achieve the maximum mixedness limit that is possible with the residence time distribution of two tanks in series. There exists a reactor (albeit semi-hypothetical) that has the same residence time distribution but that gives lower conversion for a second-order reaction than two perfectly mixed CSTRs in series. The next section describes such a reactor. When the physical configuration is known, as in part (c) above, it may provide a closer bound on conversion than provided by the maximum mixed reactor described in the next section. 

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. 

LDPE tabular reactor is divided into several reaction zon acoirding to fhe feed injection points. Here we apply mixing cell model for tobidar rcsictor which considea s the reactor axis as series of cells which is conceptually the same as CSTRs in series. In tiiis study 40 cells are used for each reactor spool of 10 m long. The mass balant equation of a single cell at steady state can be written as follows. 

Example 4.5 Derive the state space representation of two continuous flow stirred-tank reactors in series (CSTR-in-series). Chemical reaction is first order in both reactors. The reactor volumes are fixed, but the volumetric flow rate and inlet concentration are functions of time. 

Example 4.7 We ll illustrate the results in this section with a numerical version of Example 4.5. Consider again two CSTR-in-series, with V] = 1 m3, V2 = 2 m3, k] =1 min-1, k2 =2 min-1, and initially at steady state, x, = 0.25 min, x2 = 0.5 min, and inlet concentration cos = 1 kmol/m3. Derive the transfer functions and state transition matrix where both c0 and q are input functions. 

Example 4.7B Let us revisit the two CSTR-in-series problem in Example 4.7 (p. 4-5). Use the inlet concentration as the input variable and check that the system is controllable and observable. Find the state feedback gain such that the reactor system is very slightly underdamped with a damping ratio of 0.8, which is equivalent to about a 1.5% overshoot. 

The main conclusions to be drawn from this study are that the reactor design works well, and that steady state continuous flow operation requires excellent mixing of the gases and two liquid phases and high conversions. Improvements in the catalyst (ligand) are required to reduce leaching still further, but commercialisation will also require a different reactor design or more than one CSTR in series. 

Some multiple-vessel configurations and consequences for design and performance are discussed previously in Section 14.4 (CSTRs in series) and in Section 15.4 (PFRs in series and in parallel). Here, we consider some additional configurations, and the residence-time distribution (RTD) for multiple-vessel configurations. 

For CSTRs in parallel with the feed split as for optimal performance, the fact that two (or more) reactors behave the same as one CSTR of the same total volume means that the RTD is also the same in each case. Here, we consider the RTD for CSTRs in series, as in a multistage CSTR (Section 14.4). In the following example, the RTD is obtained for two tanks in series. The general case of N tanks in series is considered in Chapter... 

To develop E(B) for two CSTRs in series, we use a slightly different, but equivalent, method from that used for a single CSTR in Section 13.4.1.1. Thus, consider a small amount (moles) of tracer M, nMo = F,dt, where Ft is the total steady-state molar flow rate, added to the first vessel at time 0. The initial concentration of M is cMo = nMo/(V/2). We develop a material balance for M around each tank to determine the time-dependent outlet concentration of M from the second vessel, cM2(l). 

As indicated in the problem statement, E(t) is based upon two cases a single CSTR, and two CSTRs in series. The respective E(t) expressions are ... 

Table 20.2 gives the results obtained for the two cases, with E(t) based upon a single CSTR, and upon two CSTRs in series. 

Packed beds usually deviate substantially from plug flow behavior. The dispersion model and some combinations of PFRs and CSTRs or of multiple CSTRs in series may approximate their behavior. 

Erlang with time delay expC-t Vd + t2s/n)n The last item is of a PFR and an n-stage CSTR in series. More complex combinations are the subject of problems P5.01.33, P5.03.10, P5.03.02 and... 

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 ... 

Hofmann [34] has performed more extensive calculations, setting the bounds on possible reactant conversion for various numbers of CSTRs in series. For the particular case of a second-order reaction with = 10,... 

---