An equation-oriented approach for handling thermodynamics based on cubic equation of state in process optimization

This paper deals with handling of the cubic equation of state (CEOS) in equation-oriented (EO) process optimization. The roots of the CEOS are generally computed by subroutine-based calculations that involve logical if-else conditions. This approach can lead to nonsmoothness and convergence issues when used with gradient-based solvers in an EO framework. In this work, we propose a new general EO approach for selecting the appropriate root of the CEOS by incorporating derivative constraints specific to the desired (vapor or liquid) phase. We prove that these constraints are capable of isolating the liquid, middle and vapor root. The derivative constraints are tested using the EO phase-equilibrium formulation of Gopal and Biegler (1999) which is suitably extended for handling CEOS and relaxing the derivative constraints in the event of disappearance of phases. Using numerical examples involving simulation of flash vessels and optimization of distillation column, it is demonstrated that the proposed EO formulation always selects the appropriate roots, handles missing phases on distillation trays and flash units, and is fairly robust to different starting points in the phase envelopes.