(177c) Energy-Efficient Distillation Configurations: Novel Formulation, Relaxations and Discretizations

This work addresses optimal distillation column sequencing for non-azeotropic multicomponent mixtures: a task that plays a central role in the design of separation units for chemical and petrochemical industries. Identifying the optimal configuration is challenging because of the combinatorial explosion in the number of admissible distillation configurations with an increase in the number of components in the mixture, and because the governing equations (Underwood constraints) are nonconvex. To appreciate the challenge, consider a test set of 496 cases involving separation of five-component mixtures[1]. On this test set, the current best approach solves only 10% of the cases to 1%-optimality within five hours using state-of-the-art global solvers. Given the ubiquity of applications and significant benefits from optimal designs, there is a need to develop better models and solution procedures that screen the entire choice set and identify attractive configurations.

We propose a novel Mixed Integer Nonlinear Program (MINLP) to identify configurations that require the least vapor (heat) duty, a metric that acts as a proxy for energy consumption. To address the challenges with solving this model, we make various advances. First, inspired by physical insights, we model the space of admissible configurations in a lifted space by introducing new variables representing specific products of binary variables, where the latter capture the absence or presence of a mixture in a configuration. We show that the proposed model is contained in the convex hull of various important substructures, and it is strictly tighter than prior formulations for configuration choices. Second, we adapt classical Reformulation-Linearization Technique[2] (RLT) to obtain a family of cuts for fractional terms. These cuts are useful, because they exploit the mathematical structure of Underwood constraints. Third, we use simultaneous convexification techniques to construct convex hull of multiple nonconvex terms over polytopes (mass balances). The resulting simultaneous hull is tighter than the set obtained by relaxing nonconvex terms individually over a box. Fourth, the denominator of some fractions in Underwood constraints can approach arbitrarily close to zero. Prior formulations have imposed arbitrary lower bounds on the denominator of such fractions. However, we construct a provably valid relaxation using rigorous bounds that are inferred from cuts derived using our RLT variant. Fifth, we employ piecewise relaxation techniques by discretizing the domain of Underwood roots. Here, we adaptively partition the region by introducing partition points, where we suspect optimal Underwood roots lie, while ensuring exhaustiveness of the partitioning scheme, until we prove optimality. The proposed formulation, with all the above improvements, provides the first ever system to identify, for further exploration, a few attractive configurations reliably. In particular, the proposed method solves 72% of cases in the test set within 1200 s, and all 496 cases are solved within 1%-optimality in less than five hours.

[1] Nallasivam, U., Shah, V.H., Shenvi, A.A., Tawarmalani, M. and Agrawal, R., 2013. Global optimization of multicomponent distillation configurations: 1. Need for a reliable global optimization algorithm. AIChE Journal, 59(3), pp.971-981.

[2] Sherali, H.D. and Alameddine, A., 1992. A new reformulation-linearization technique for bilinear programming problems. Journal of Global optimization, 2(4), pp.379-410.