The Design of an Optimal CSTR

Aha This is very interesting and instructive. Here we see that K1 is so small that at any of the holding times (even the highest ones) the rate is so small that the conversion is effectively zero throughout the range. This means that the catalyst just lacks the adsorption forces necessary to make the reaction run fast. In other words, for the reaction to take place efficiently, the reactant A must be adsorbed. If it is not adsorbed to an appreciable extent, then the rate is always going to be small unless the rate constant for the surface reaction is very high. 

Before we go on to the next section we should do some housekeeping. The command Names is shown below with its Mathematica explanation  

Names[ string ] gives a list of the names of symbols which match the string. Names[ string , 

SpellingCorrection - True] includes names which match after spelling correction. 

As we have worked through this session, or any other session, we have generated many new Names for functions. These show up in the Global context. If we ask for them we will get a list of those that we have created and used so far. (We show this in the following, but we have suppressed the output.) 

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Chapter 21. Chapter 7 in Shinskey [Ref. 3] is again an excellent reference for the practical considerations guiding the design of feedforward and ratio control systems. It also discusses the use of feedforward schemes for optimizing control of processing systems. Good tutorial references are the books by Smith [Ref. 2], Murrill [Ref. 8], and Luyben [Ref. 9]. The last one has a simple but instructive example on the nonlinear feedforward control of a CSTR. 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

The data are from Ref. 31. The objective for optimization is the maximization of the effluent concentration of component B. The performance limit of the system is identihed with each stochastic run requiring an average of only 120 CPU sec on an HP 9000-C100 workstation. Numerous designs are obtained from the stochastic search that perform close to the performance target, mostly variations of series arrangements of PFRs and CSTRs. A detailed discussion of this and other studies is given in Ref. 31. 

At this point the remark made in Section 4.1.3.1 about an optimized start-up strategy for the cooled CSTR shall be explained. The safety technical assessment procedure for the cooled isoperibolic SBR has demonstrated that in the case of correct design a prediction of the maximum reaction temperature is easily possible. This can be utilized for the optimization of the start up of the CSTR. The later steady state operating temperature of the CSTR is defined as the set value for the maximum SBR process temperature. In a next step one of the two reactants of the CSTR process is charged initially. Then the reactor is started as a semibatch process by feeding the second reactant. When the maximum temperature is reached, the feed of the initially charged reactant is started, and the feed streams are adjusted in such a way that the Stanton number of the CSTR is established. This way the initial oscillations are elegantly avoided. 

An equation relating temperature T and conversion Xa is required to design the non-isothermal reactors. This relationship between temperature T and conversion is obtained by setting up a heat balance equation around the reactor (Section 3.1.5.3). In certain cases, reactor temperature T is deliberately varied with conversion by regulating the heat supply to the reactor or heat removal from the reactor. One such case is the non-isothermal reactor in which a reversible exothermic reaction is carried out. In the case of a reversible exothermic reaction, there is an optimum temperature T for every value of conversion x at which the rate is maximum. A specified conversion Xaj will be achieved in a CSTR or a PFR with the smallest volume or in a batch reactor in the shortest reaction time if the temperature in the reaction vessel is maintained at the optimum level. This optimal temperature policy in which temperature is varied as a function of conversion x,i is known as the optimal progression of temperature presented in the following section. 

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