An analysis and implementation of a fast single-root cardano method for cubic equations of state in process and compositional simulation

Cubic equations of state are widely used in process-flowsheet and compositional-reservoir simulators, where repeated TP-flash calculations require frequent evaluation of liquid and vapor roots. Conventional analytic implementations of Cardano's method often compute and sort all three real roots, although only one phase-relevant root is normally required in a given phase calculation. This work presents a phase-targeted Cardano procedure that directly returns the required liquid or vapor root in the three-real-root regime. The method uses the trigonometric structure of the cubic equation to avoid computing unused roots and does not require explicit discriminant evaluation during runtime. Numerical safeguards are limited to branch-consistent clipping of the inverse-cosine argument to avoid round-off violations. The procedure was evaluated through a controlled one-billion-case cubic-root benchmark and through a TP-flash profiling framework that separates cubic inversion from mixing-rule assembly, fugacity-coefficient algebra, and flash-iteration logic. In the cubic-root benchmark, the proposed procedure reduced cubic-inversion CPU time by 62.6% relative to a full Cardano solver that evaluates and sorts all three real roots. For equimolar n-alkane mixtures modeled with the Peng–Robinson equation of state, from 2 to 50 components, the measured cubic share of TP-flash time decreased from about 2.9% to 0.38%. These results show that the proposed method provides a deterministic acceleration of a fundamental cubic-EOS inversion kernel. Its end-to-end effect in full TP-flash calculations is naturally bounded by the fraction of time spent in cubic inversion, but the method improves the reliability and efficiency of a repeated phase-root selection step used in process-simulation and thermodynamic-property workflows.