Iterated linear programming strategies for nonsmooth simulation : a penalty based method for vapor-liquid equilibrium applications

We extend our iterated linear programming (LP) approach (Bullard and Biegler, 1991) to two-phase vapor-liquid equilibrium problems, which are characterized by regions of continuous operation with nonsmooth boundaries. Here we show that a simple reformulation allows us to handle the disappearance or reappearance of phases and thus allows us to solve a wider class of process problems. The proposed strategy uses a penalty function approach, called Penalty Simulation of Nonsmooth Algebraic Terms and Attributes (P-SONATA), to accommodate the nonsmooth nature of the system. To solve the vapor-liquid equilibrium problem, we also extend the theoretical results of the approach of Bullard and Biegler (1991) to characterize descent and convergence properties for PSONATA. The performance of this formulation is demonstrated for process models involving phase equilibrium, such as transitions from one and two phases in flash and distillation problems, where mass and energy balances must be satisfied but the phase equilibrium expression can be relaxed. Isothermal flash problems with ideal and nonideal phase equilibrium relations are considered as well as a case which exhibits retrograde condensation behavior near the critical point Finally, we examine limiting distillation cases including columns operating below the minimum reflux ratio (resulting in dry trays) and below the minimum reboiler heat duty (resulting in vaporless trays). Finally, we develop convergence properties for P-SONATA and discuss additional classes for nonsmooth problems. The results demonstrate that this approach is straightforward to implement, captures a wider range of phase equilibrium behavior, and otherwise performs competitively with conventional Newton-based approaches.