(346a) Process Synthesis without Integer Variables: Using Complementarity Conditions for Thermodynamic and Distillation Models

Authors: 
Dowling, A. W., Carnegie Mellon University
Biegler, L. T., Carnegie Mellon University

Discrete decisions are essential to process synthesis; they allow for units to be enabled and disabled in an optimization formulation. Unfortunately, the resulting mixed integer optimization problems may be difficult to solve, especially with highly nonlinear, non-convex detailed process equipment models (e.g. non-ideal thermodynamics). Typically researchers are forced to simplify models (e.g. linearization to MILP, simplifying assumptions regarding physics, etc.), develop complex decomposition strategies using generalized disjunctive programming and/or reduce the scope of the problem to be computationally tractable. All three of these options are not ideal, as they may produce unrealistic results, require substantial effort to implement and may not answer the actual design question.

As an alternative to mixed integer programming, researchers are exploring the application of complementarity conditions (and related reformulations) that allow (some) process synthesis problems to be reformulated as nonlinear programs (optimization problems). This approach is adventitious, as it enables optimization of the detailed, typically non-convex, process models along with some discrete decisions using generic, large-scale NLP algorithms. In this paper, we focus on two applications of complementarity conditions in process synthesis problems: (1) embedded cubic equation of state (e.g. Peng-Robinson) models and (2) optimization of distillation cascade size with tray-by-tray models.
Regarding embedded thermodynamic models, Kamath et al (2010) proposed an equation-based method for distinguish between liquid, vapor and spurious roots in a cubic equations of state. Phase disappearance is accommodated using complementarity conditions. We will summarize this approach, and propose an extension to prevent spurious vapor-liquid equilibrium solutions that occur when K = 1 (see Gundersen, 1982).

Furthermore, we will present a novel model for distillation optimization that replaces integer variables in the Viswanathan-Grossmann (1990) formulation with tray bypasses. This is essentially a relaxation of the integer problem, except that thermodynamics drives the continuous bypass fractions/efficiency to binary values, because mixing is inefficient. Care is taken in the model formulation to avoid degeneracies when distillation trays are bypassed. The efficacy of this approach and the tendency towards integer solutions will be demonstrated using a cryogenic air separation unit synthesis case study.

References:
Kamath, R. S., Biegler, L. T., & Grossmann, I. E. (2010). An equation-oriented approach for handling thermodynamics based on cubic equation of state in process optimization. Computers & Chemical Engineering, 34(12), 2085–2096.

Gundersen, T. (1982). Numerical aspects of the implementation of cubic equations of state in flash calculation routines. Computers & Chemical Engineering, 6(3), 245–255.

Viswanathan, J., & Grossmann, I. E. (1990). A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Computers & Chemical Engineering, 14(7), 769–782.