A Hyperbolic One-Dimensional Model for Two-Phase Flows in Converging-Diverging Nozzles

Partial-evaporation organic Rankine cycle power systems are a promising technology for power generation from low-temperature heat sources, such as waste heat and geothermal heat. A specific challenge in two-phase turbines is converging-diverging nozzle design and performance analysis. Existing one-dimensional methods for two-phase flows in nozzles typically rely on space-marching approaches, which are unsuitable for predicting shock waves and thus limit their application to adapted expansion conditions. To address the limitations of existing models, this paper presents a new one-dimensional two-phase flow model suitable for capturing shock waves in converging-diverging nozzles. The model employs a finite volume method to solve the balance equations in a conservative form, using time marching methods to reach the steady-state solution. The predictive performance of the proposed model is validated against experimental data from converging-diverging nozzles using various working fluids, including organic molecules and CO₂. The results indicate that the proposed model formulation is suitable for predicting the performance of two-phase nozzles in terms of pressure distribution, critical mass flow rate, and shock wave characteristics across a wide range of operating conditions. These findings suggest that the developed model can be a reliable tool for the preliminary design and analysis of converging-diverging nozzles in two-phase turbines.