Towards global optimization of combined distillation-crystallization processes for the separation of closely boiling mixtures M. Ballersteinb , A. Kienlea,d , C. Kundea , D. Michaelsc , R. Weismantelc a Otto-von-Guericke-Universit¨ at Magdeburg, Lehrstuhl f¨ ur Automatisierungstechnik/Modellbildung, Universit¨ atsplatz 2, 39106 Magdeburg, Germany b Otto-von-Guericke-Universit¨ at Magdeburg, Institut f¨ ur Mathematische Optimierung, Universit¨ atsplatz 2, 39106 Magdeburg, Germany c Eidgen¨ ossische Technische Hochschule Z¨ urich, Institut f¨ ur Operations Research, R¨ amistrasse 101, 8092 Z¨ urich, Switzerland d Max-Planck-Institut f¨ ur Dynamik komplexer technischer Systeme, Sandtorstraße 1, 39106 Magdeburg, Germany

Abstract Separation of closely boiling mixtures is a challenging problem for process synthesis and process design. A typical example is the separation of mixtures of isomers like n/iso-aldehyde mixtures arising from oxo-synthesis. Standard distillation is often not favorable due to high process costs. A more energy and cost efficient separation process for such closely boiling mixtures is desirable and may be obtained by an optimal combination of distillation and melt crystallization, thus exploiting the advantages of both processes. The optimal design of such chemical processes with structural and operational degrees of freedom leads to mixed-integer nonlinear programs (MINLP). Due to nonconvexity and the integrality of some variables, MINLP problems are usually difficult to solve. Typical approaches for this purpose are either gradient based local optimization methods or stochastic optimization methods like simulated annealing and genetic algorithms. However, both methods cannot guarantee to find the global optimum. A general approach to determine the global optimum is to construct a convex relaxation of the original nonconvex problem. The convex relaxation can be solved for global optimality and its solution provides a global bound for the optimal solution of the original problem. This bound approaches the true global optimum as the convex relaxation is more and more refined [HGK+ 06]. Although this method is well known, many applications cannot be solved by general stateof-the-art global algorithms like BARON [TS04] within reasonable time. In this contribution we present first results towards a global optimization of combined distillation-crystallization processes for the separation of closely boiling mixtures. To improve global optimization algorithms the problem specific structure is taken into account. In the present work, first a suitable superstructure is proposed including simple model equations based on mass balances and simple thermodynamics. A simplified cost function is formulated that includes investment and operating costs. In particular, we show computational results for a benchmark problem whose process structure alternatives cannot be evaluated a priori. The existence of 1

multiple local cost minima is conjectured from physical insight and is validated in a first step by local optimization calculations from different starting points. For global optimization, improved relaxation strategies for mass balance equations are developed, which are stronger than the ones used in standard relaxation techniques. More precisely, we introduce novel convex relaxations for mass balance equations that are motivated by recent results [TS01, JMW08]. For this, we directly incorporate the phase equilibria into the mass balance equations and derive the best possible convex relaxations for the resulting terms. Additionally, a domain reduction strategy is applied which enables us to exclude several process configurations very fast. We show that such an approach allows us to reduce the number of mass balance equations, considerably, and still yields a good bound on the original problem. Key words: global optimization, local optimization, convex relaxations, domain reduction, closely boiling mixtures, superstructure, distillation, crystallization

References [HGK+ 06] U.-U. Haus, J. Gangadwala, A. Kienle, D. Michaels, A. SeidelMorgenstern, and R. Weismantel, Global bounds on optimal solutions in chemical process design, 16th European Symposium on Computer Aided Process Engineering - ESCAPE 16 (Amsterdam) (W. Marquardt and C. Pantelides, eds.), Elsevier, 2006, pp. 155–160. [JMW08] Matthias Jach, Dennis Michaels, and Robert Weismantel, The convex envelope of (n–1)-convex functions, SIAM Journal on Optimization 19 (2008), no. 3, 1451–1466. [TS01] Mohit Tawarmalani and Nikolaos V. Sahinidis, Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques, Journal of Global Optimization 20 (2001), 137–158. [TS04]

, Global optimization of mixed-integer nonlinear programs: a theoretical and computational study, Mathematical Programming 99 (2004), no. 3, Ser. A, 563–591.

July 16, 2010