Solutions in a CSTR

One enormously interesting problem is optimal control of CSTRs. The dynamics and control of this kind of reactor for the following reactions 

The control problems raised by CSTRs will be investigated in detail in Vol. 4 -Buzzi-Ferraris and Manenti (in press). 

In this volume, however, we are interested in the subproblem that must be tackled to solve reactor control. Using the parameters assigned by Sistu and Bequette (1995), the following system must be solved with respect to the variables Yv Yi Y w  

The system is hard to solve since it has three distinct solutions and the one of interest for process control is the intermediate one, which is unstable. 

Find the three solutions to the previous system using an object from the BzzNonLinearSystem class by imposing limitations on the variables. The program is 

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Techniques based on the implicit function theorem have been used to predict the existence of multiple solutions in a CSTR (Chang and Calo, 1979). An extension of catastrophe theory known as singularity theory has also been effectively used to determine the conditions for the existence of multiple solutions in a CSTR and a tubular reactor (Balakotaiah and Luss, 1981, 1982 Witmer et al., 1986). In this subsection, the technique of singularity to find the maximum number of solutions of a single mathematical equation and its application to analysis of the multiplicity of a CSTR are presented (Luss, 1986 Balakotaiah at al., 1985). The details of singularity theory can be found in Golubitsky and Schaeffer (1985). 

An intrinsically rapid reaction is carried out in a very viscous solution, in a CSTR with a volume of 10 1. An anchor stirrer is used with a speed of 1 s" The diameter of the feed tube is 1.5 cm. When the residence time is 10 minutes, the degree of conversion is 0.9. It is found that the degree of conversion can be correlated with v. One wants to scale-up this reactor to 1 m, with a similar iimeller, but for technical reasons the impeller speed cannot be more than 0.2 s The diameter of the feed tube is now 3 cm. The degree of conversion has to be 0.9. What mean residence time would be advisable, and what is than the scale-up factor ... 

The result is that the driving force at the wall must increase by a factor of about 10 when scaling with S = 512 and constant power per unit volume. This may be acceptable when the pilot unit operates with a AT of 2°C but becomes problematic when the pilot plant operates with a 20°C AT. There are many solutions. In a CSTR, use cold feed. Some processes for PMMA use a 40°C feed to control the reaction exotherm. Diluents and low per-pass conversions can also be used this approach is typical of solution polyolefin processes. Reflux boiling can be used it is common in styrenic polymerizations where the reflux solvent is normally returned as a liquid. In some polypropylene processes, the returning propylene is flashed into the first reaction vessel. Finally, the external loop shown in Figure... 

Another important reaction supporting nonlinear behaviour is the so-called FIS system, which involves a modification of the iodate-sulfite (Landolt) system by addition of ferrocyanide ion. The Landolt system alone supports bistability in a CSTR the addition of an extra feedback chaimel leads to an oscillatory system in a flow reactor. (This is a general and powerfiil technique, exploiting a feature known as the cross-shaped diagram , that has led to the design of the majority of known solution-phase oscillatory systems in flow... 

Compare these results with those of Equation (2.22) for the same reactions in a batch reactor. The CSTR solutions do not require special forms when some of the rate constants are equal. A plot of outlet concentrations versus t is qualitatively similar to the behavior shown in Figure 2.2, and i can be chosen to maximize bout or Cout- However, the best values for t are different in a CSTR than in a PFR. For the normal case of bi = 0, the t that maximizes bout is a root-mean, t ix = rather than the log-mean of... 

Solution With Z>, = 0, a reaction wiU never start in a PFR, but a steady-state reaction is possible in a CSTR if the reactor is initially spiked with component B. An anal5dical solution can be found for this problem and is requested in Problem 4.12, but a numerical solution is easier. The design equations in a form suitable for the method of false transients are... 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

The final step is a hydrolyzing step with sulfatase enzymes (E.C. number 3.1.6.1), such as limpet sulfatase, Aerobacter aerogenes sulfatase, Abalone entrail sulfatase, or Helixpomatia sulfatase. This step was suggested to be carried out in a CSTR or fluidized bed reactors, with counter-current flow between the aqueous and the oil phase. A more efficient removal of the sulfate into the aqueous stream is expected to occur in this cross-flow manner. A final separation of the reacting mixture was suggested to obtain sulfur-free product and aqueous enzyme solution for recycle. 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Figure 14.8 Illustration of solution of material and energy balances for an endothermic reaction in a CSTR (no multiple stationary-states possible)...

A liquid phase reaction, A = 2B, is carried out in a CSTR. Input rate is W lb/hr with concentration Ca0 = 1.0. The density of the solution depends on the concentrations,... 

A liquid phase reaction, 2A = 2B, is conducted in a CSTR with 20% recycle through a heater as shown. Fresh input is at 300 K and consists of 500 kg/hr of water and 20 kgmol/hr of substance A. The recycle is at 350 K. Heat capacity of the solute is 40 kcal/gmol-K, the reaction is endothermic with AHr = +2000 cal/gmol of A converted, reactor volume is 25,000 liters and the specific rate is... 

J. Hamer, T. Akramov, and W. Ray. The dynamic behavior of continuous polymerization reactors II, nonisothermal solution homopolymerization and copolymerization in a CSTR. Chem. Eng. Sci., 36 1897-1914, 1981. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

A reactant A in liquid will be converted to a product P by an irreversible first-order reaction in a CSTR or a PFR reactor with a reactor volume of 0.1 m . A feed solution containing 1.0 kmol m of A is fed at a flow rate of 0.01 m min", and the first-order reaction rate constant is 0.12 min . ... 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

Solution Polymerization in a CSTR. Although many polymerization reactors in use by industry have the residence time distribution of a CSTR, they may not, at first glance, have the appearance of a CSTR (cf. Figure 1). Nevertheless, CSTR models, perhaps with some allowance for imperfect micromixing, are successfully employed to describe these reactors. Thus the behavior of the CSTR is of great practical interest. 

Figure 2. Isothermal polymerization of methyl methacrylate in a CSTR (1 5). a. Predicted steady-state monomer conversion vs. reactor residence time for the solution polymerization of MMA in ethyl acetate at 86 °C. h. Steady-state and dynamic experiments for the isothermal solution polymerization of MMA in ethyl acetate (solvent fraction O.k) ( ) steady states,...

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