Prediction and Control of Carbon Dioxide Frost Points in Cryogenic Systems Using Peng-Robinson Equation of State with Lagrange’s Cubic Solver

Carbon dioxide freezing in cryogenic systems presents significant operational challenges in natural gas processing, petroleum refining, and carbon capture applications. Accurate prediction of CO₂ frost points is critical for process design, operational safety, and economic efficiency. This paper presents a comprehensive thermodynamic framework for predicting CO₂ frost formation through both liquid/solid equilibria (LSE) and vapor/solid equilibria (VSE) mechanisms using the Peng-Robinson equation of state. A novel aspect of this work is the implementation of Lagrange’s analytical method for solving the cubic equation of state, which overcomes numerical instabilities encountered with traditional Cardano’s formula, particularly in the low-temperature region. The Peng-Robinson model with Lagrange’s solver demonstrates superior accuracy compared to commercial simulators, with maximum deviations of 6.4°F for binary methane-CO₂ systems and average deviations of 3°F for vapor/solid equilibria. Validation against high-quality experimental data demonstrates that this methodology provides a computationally efficient, numerically stable, and thermodynamically consistent approach suitable for industrial applications across a wide range of operating conditions.