Thermodynamic analysis of formulations to discriminate multiple roots of cubic equations of state in process models

- Continuous criterion to separate roots of cubic equations of state proposed.
- Proposed criterion violates constraint qualification but numerical result unaffected.
- Proposed criterion correct when criterion from literature excludes valid solutions.
- Proposed criterion and criteria from literature only necessary for stability.
- KKT conditions of minimizing Gibbs free energy yield formulation for vanishing phase.

Several discrimination criteria have been proposed to select the correct root in the presence of multiple real roots to a cubic equation of state. Herein, a criterion which exploits the well-known method of Cardan to solve cubic equations is presented. The criterion is straightforward, continuous and covers different root scenarios. As a reference, minimization of Gibbs free energy is formulated for const. (P, T). From its KKT conditions, it follows that, in case a phase vanishes, relaxation of cubic equality is an alternative to relaxation of root discrimination as proposed by Kamath et al. (2010). The proposed criterion and those from literature satisfy at most a necessary condition, while minimization of Gibbs free energy obviously is a necessary and sufficient condition for thermodynamic stability of the selected root. In certain cases, formulations from literature are found to exclude valid solutions, whereas the proposed criterion overcomes this deficiency.