Excess Gibbs free energy models, like the nonrandom two-liquid (NRTL) model , are widely used in process simulation. However, the models only yield predictions about stable phase splits if the global minimum of the Gibbs free energy is considered. Enforcing only necessary conditions can result in spurious phase splits inside the process simulations [2,3,4]. Optimization-based design problems subject to these equilibrium conditions result in bilevel-programs (BLPs), where the upper-level program maximizes an economic objective and lower-level program (LLP) minimizes the Gibbs free energy. Due to the nonconvexity of the LLP, these programs are commonly formulated as nonlinear programs by replacing the LLPs with their respective Karush-Kuhn-Tucker conditions [5,6]. Since solutions other than the global minimum of Gibbs free energy may be found, this can lead to erroneous results. In order to get reliable results in process optimization involving phase equilibrium, the original BLP must be solved. Gumus and Ciric  use a sequential approximation algorithm and Sahin and Ciric , a dual temperature simulated annealing approach to solve the BLP design problem with embedded phase equilibrium. Both do not consider solving systems with phase instabilities.
Herein, we rigorously solve design problems with deterministic global methods in the sense that the necessary and sufficient criteria of phase stability are satisfied and present a numerical study and a best practice to solve such programs efficiently. We reformulate the BLP as a semi-infinite program, which we then solve using the method of Blankenship and Falk . The problems are implemented in C++ using libALE  and the performance of the computations using different subsolvers, including MAiNGO  and BARON , are discussed on a set of binary and ternary flash optimization case studies with phase instabilities. Finally, the phase splits resulting from the optimizations are checked with AspenPlus flash models. We prove that the phase splits resulting from our approach are stable, as expected from theory, and show that AspenPlus models fail to provide the correct phase split at the optimal temperatures for the chosen case studies.
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