We propose in this paper a superstructure that considers feasible flowsheet alternatives for the all process configurations and model it as a Mixed-Integer Nonlinear Programming (MINLP) problem in order to identify optimum configurations for the crystallization process. This superstructure is a generalization of the flowsheets studied by Mendez et al. (2005). In this work we have embedded all the flowsheets studied by these authors and considered new connections between stages. The superstructure includes a sub-superstructure for the first crystallization stage and a new sub-superstructure for the set of centrifuges with wash, which allowed establishing new connections between the different process stages so as to consider flowsheets that have been proposed in US patents.
The proposed MINLP model includes several nonlinearities, mainly in the solubility correlations and heat balances, as well as discontinuities in the solubility correlations, giving rise to a nonlinear nonconvex MINLP problem of a large dimension. In order to cope with the complexity of this MINLP model and to avoid its direct solution, we propose a two-level decomposition consisting of the iterative solution of an aggregated and a detailed model. The two key ideas in the aggregated model are: a) merging the units in centrifuge blocks and slurry drums into single input-output blocks so that the aggregated model is defined in the space of interconnection of major blocks; b) elimination of the constraints that set an upper bound on the inlet flowrate of each centrifuge unit. In this way a large number of equations and variables are eliminated because groups of individual units are replaced by a single equivalent unit. However, in order to meet the same production targets, the constraints that impose operating ranges for the centrifuges are relaxed in the aggregated model. For the definition of the detailed model, the number of units in each aggregated block is calculated by the ceiling of the ratio between the calculated capacity of the block and the upper bound of the size for each unit. As an example, for the first stage of centrifuges the number of units is determined by the ceiling of the ratio between the individual feed flowrate of solid p-xylene in the aggregated block divided by the maximum individual feed flowrate of solid p-xylene treated for each centrifuge. In each iteration of the two-level decomposition an integer cut is added to the aggregated model to eliminate previous combinations of number of units. Therefore, this requires the introduction of binary variables in the aggregated model in order to calculate the number of units associated with each aggregated block. The solution of the aggregated model is used to initialize the detailed model as well as to define a reduced superstructure for the detailed model. While the proposed approach has the advantage of providing an effective solution method, its limitation is that convergence cannot be guaranteed in terms of lower and upper bounds. Therefore, the approach we take is to simply iterate over a fixed number of major iterations.
The results obtained show that the aggregated MINLP model is easier to initialize and and to solve than the MINLP model of the overall superstructure. Furthermore, the solution from the aggregated model provides a good initial point for the detailed model. The optimum flowsheet that was obtained yields a new design alternative compared to previous flowsheets studied by Mendez et al (2005). Comparing the configuration of the optimum flowsheet obtained by these authors with the one obtained in this work, the latter has three fewer slurry drums, one less melting stage, four fewer centrifuges, but one more crystallizer. This led to a reduction of 12% in the total annualized investment cost but at the expense of a small increase in the total annual operating cost, resulting in a modest net improvement in the total annual profit.
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