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) 

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. 

Computationally, the solution of the dynamic optimisation problem is time consuming and expensive. Mujtaba and Macchietto (1997) reported that the number of Function and Gradient Evaluations for each maximum conversion problem is between 7-9. A fresh solution would require approximately 600 cpu sec in a SPARC-1 Workstation. However, subsequent solutions for different but close values of tf could take advantage of the good initialisation values available from the previous solutions. 

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Mujtaba and Macchietto (1994) presented an industrial case study in which dynamic optimisation method of Mujtaba and Macchietto (1993) is utilised for the development of the optimal operation of an entire batch distillation campaign where 100 batches of fresh charge have to be processed with secondary reprocessing of intermediate off-cuts. The process involved a complex separation of a five-component mixture of industrial interest, described using non-ideal thermodynamic models. In addition, the operation of the whole production campaign was subject to a number of resource constraints, for example -... 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

Another important aspect that we can learn fi-om this example is the optimisation of a recycle system. If only the steady state operation is studied, the optimal plant consists fi-om a small reactor coupled with a tall column, because usually the reactor is more expensive. If the cost of transient operation and off-spec products is taken into account, the steady state optimisation is no longer valid. Other sub-optimal designs might offer better dynamic performances, as for example somewhat larger reactor and shorter distillation column. The solution can be found by dynamic optimisation (Luyben, 1999). 

The theoretical models proposed in Chapters 2-4 for the description of equilibrium and dynamics of individual and mixed solutions are by part rather complicated. The application of these models to experimental data, with the final aim to reveal the molecular mechanism of the adsorption process, to determine the adsorption characteristics of the individual surfactant or non-additive contributions in case of mixtures, requires the development of a problem-oriented software. In Chapter 7 four programs are presented, which deal with the equilibrium adsorption from individual solutions, mixtures of non-ionic surfactants, mixtures of ionic surfactants and adsorption kinetics. Here the mathematics used in solving the problems is presented for particular models, along with the principles of the optimisation of model parameters, and input/output data conventions. For each program, examples are given based on experimental data for systems considered in the previous chapters. This Chapter ean be regarded as an introduction into the problem software which is supplied with the book an a CD. 

A general overview in Sect. 5.2 about the simulation of adaptronic (mechanical) systems is followed by a discussion of steps to be taken towards a mathematical model of an adaptronic structure in Sect. 5.3. Once a mathematical model of the adaptronic system has been derived and implemented numerically, analysis and simulations have to be carried out to characterise its dynamic behaviour. A survey of related methods and algorithms is given in Sect. 5.4. Simulation goals such as stability, performance and robustness are discussed, especially for the case of actively controlled structures. The modelling and simulation process is also demonstrated by a practical example in Sect. 5.5, while Sect. 5.6 gives an outlook on adaptronic system optimisation techniques. 

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