277375 Evaluating Trade-Offs in the Operation of a Semi-Batch Polymerization Reactor
Evaluating trade-offs in the operation
a semi-batch polymerization reactor
F. Assassaa,c, F. Logistb, J. Van Impeb, W. Marquardta
a AVT – Process Systems Engineering, Aachener Verfahrenstechnik, RWTH Aachen University, Turmstraße 46, D-52056 Aachen, Germany
& OPTEC, Department of Chemical Engineering, Katholieke Universiteit
W. de Croylaan 46, B-3001 Leuven, Belgium
c German Research School for Simulation Sciences GmbH, 52425 Jülich, Germany
Abstract. Optimizing the design and operation of polymerization reactors often involves multiple and conflicting objectives. In the current study, the trade-off between economic profit and energy consumption is evaluated for the operation of a semi-batch reactor, which produces a co-polymer of styrene and butyl-acrylate. A set of trade-off or so-called Pareto optimal solutions is generated based on a dynamic reactor model. Consequently, we have to solve a multi-objective dynamic optimization problem. A scalarization method (normal boundary intersection) is employed to tackle the multi-objective aspect. This method converts the original multi-objective optimal control problem into a series of single-objective optimal control problems, which are each solved with direct single shooting (adaptive control vector parameterization). For this series of optimization problems, it can be assumed that neighboring values of the scalarization parameter result in neighboring solutions. We exploit this feature by building an accurate approximation for the solution of the first single-objective optimization by grid adaptation and structure detection. The detected structure is repeatedly updated along the Pareto set. The resulting Pareto set as well as the resulting profiles for controls and states are discussed.
Process. The case study deals with a semi-batch reactor, which produces a co-polymer of styrene and butyl-acrylate in solution with pentyl acetate. Di-cumyl peroxide is chosen as the initiator. Both monomers and the diluted initiator are fed separately. Their feed flow rates as well as the coolant inlet temperature are time-varying control variables.
Model. Hartwich  developed a model of the reactor from kinetic models of [2, 3] including reaction rates, polymer distribution moment equations and mass balances. The energy balance including the cooling jacket has been added. This gives rise to a model with 199 algebraic and 23 differential states.
Decision variables. The feed flow rates of the monomers and the diluted initiator as well as the coolant inlet temperature are decision variables of the optimal control problem. The optimization algorithm tries to minimize the objective function by varying these control variables.
Constraints. Feasible and safe batch operation is ensured by specifying bounds along the batch duration for the reactor content, the reactor temperature set-point, the reactor temperature and the accumulation of monomer. To guarantee production, lower bounds are imposed on the reactor content, the conversion, the amount of polymer and the reactor temperature at the end of the batch. Moreover, pre-specified product quality bounds on, e.g., the weight average molecular weight and the polydispersity, have to be fulfilled at the end time of the batch. The batch duration time is fixed.
Objectives. Two conflicting objectives are evaluated: (i) the profit, i.e., the return due to the product minus the cost for monomers and initiator, and (ii) the energy consumption, i.e., the energy needed to heat up the reactor.
The approach for computing the Pareto set of the multi-objective dynamic optimization problem is similar to  and combines (i) the normal boundary intersection (NBI) method  for scalarization of the objectives with (ii) control grid adaptation and structure detection techniques [6-8] to efficiently solve the resulting dynamic optimization problem. The approach was implemented in the software package DyOS.
Normal boundary intersection (NBI). Das and Dennis  have proposed NBI, a geometrically intuitive approach, to mitigate the drawbacks of the weighted sum approach for scalarization of multiple objectives. NBI first builds a plane in the objective space which contains all convex combinations of the individual minima, i.e., the convex hull of individual minima (CHIM), and then constructs (quasi-)normal lines to this plane. The rationale is that the intersection between the (quasi-)normal from any point on the CHIM, and the boundary of the feasible objective space closest to the utopia point (i.e., the point which contains the minima of the individual objectives) is expected to be Pareto optimal. Hereto, the multi-objective optimization problem is reformulated as to maximize the distance from a point on the CHIM along the quasi-normal through this point, without violating the original constraints.
Grid adaptation and structure detection. The scalarized multi-objective dynamic optimization problem is converted into a nonlinear programming problem (NLP) by adaptive control vector parameterization developed by Schlegel and Marquardt . Here, the control profiles are successively refined employing a wavelet-based analysis of the optimal solution obtained in the preceding optimization step. Furthermore, a control structure detection algorithm [7-8] was used to improve computational performance of the optimal solution.
Results have been obtained for some points along the Pareto set. Figure 1 shows an exemplarily Pareto set for (i) the negative value of the scaled profit vs. (ii) the scaled energy consumption with the scalarization vector as w = [1-A, A]T in which A varies from 0 to 1 in steps of 0.1. In general, the process is profitable but requires energy supplied from outside. More profit can be made at the expense of consuming more energy. Hence, an engineer can use this information to select a suitable operation according to the current economic situation. Grid adaptation techniques, structure detection and multi-stage reformulations allow a lean control parameterization, which is updated going along the Pareto set.
|Figure 1: Sampled Pareto set of -profit vs. energy consumption with 11 points. For increasing A-values in w = [1-A,A]T optimizations proceed along the set from left to right.|
Acknowledgements. Work supported in part by Projects OPTEC (Center-of-Excellence Optimization in Engineering) PFV/10/002 and SCORES4CHEM KP/09/005 of the KULeuven. FL acknowledges the FWO travel grant. JVI holds the chair Safety Engineering sponsored by the Belgian chemistry and life sciences federation essenscia.
This work has been partially funded by the German Research School for Simulation Sciences.
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