(inequality constraint) (model equations, equality constraints) (initial conditions, equality constraints) (inequality constraints)
where C is the conversion of the limiting reactant, xD(tf) is the composition of distillate at the end of the operation (tf), r(i) is the reflux ratio as a function of time (t). Solution method for such optimisation problem has been discussed earlier.
Note that other types of dynamic optimisation problems such as minimum time, maximum profit, etc. could also be formulated and solved using the algorithms mentioned in this book. However, Mujtaba and Macchietto (1997) used the results of optimisation problem PI to develop a computationally efficient technique to solve maximum profit problem which will be discussed in the following sections.
9.6. Example: Dynamic Optimisation
Mujtaba and Macchietto (1997) considered the esterification of ethanol and acetic acid to study dynamic optimisation of BREAD. The reaction products are ethyl acetate and water, with ethyl acetate being the main product. The reversible reaction scheme together with the boiling temperatures of the components are shown below:
Acetic Acid + Ethanol <=> Ethyl Acetate + Water Component (1) (2) (3) (4)
Boiling Points, K 391.1 351.5 350.3 373.2
Ethyl acetate has the lowest boiling temperature in the mixture and therefore has the highest volatility. Controlled removal of ethyl acetate by distillation will improve the conversion of the reactants by shifting the chemical equilibrium to further right. This will also increase the yield proportionately.
The number of plates (defining the column configuration), feed, feed composition, column holdup, etc. for the problem are given in Table 4.9 (Chapter 4). The vapour-liquid equilibrium data and the kinetic data are taken from Simandl and Svrcek (1991) and Bogacki et al. (1989) respectively and are shown in Table 4.10 (Chapter 4). The vapour and liquid enthalpies are calculated using the data from Reid et al. (1977). As mentioned in Chapter 4, these data do not account for detailed VLE calculations and for any azeotropes formed.
Mujtaba and Macchietto (1997) solved a series of optimisation problems (PI) for different but fixed batch time tj (between 5 and 30 hrs) and for two given product purities, x*D - 0.70 and x*D - 0.80. Reflux ratio level was optimised over the batch time of operation. Figure 9.2 shows the typical plots of accumulated distillate and reboiler composition profiles for t f= 15 hrs and x D = 0.80, with the reflux ratio being optimised.
The maximum conversion, the corresponding amount of product, optimal constant reflux ratio and heat load profiles for different batch times are shown in Figures 9.3-9.6. The maximum conversion profile achieved under total reflux operation (where no product is withdrawn) is also shown in Figure 9.3. The latter approximates the conversion which would be achieved in the absence of distillation. Note that if there is a large column holdup, the conversion under total reflux will not approximate the conversion achieved in the absence of distillation.
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