The discharge coefficient for supercritical cryogenic fluids through an orifice

Abstract

The increasing application of cryogenic fluids in advanced propulsion systems and fusion reactor cooling technologies requires a deeper understanding of fluid behaviour under cryogenic supercritical conditions. In particular, accurate prediction of mass flow through flow restrictions such as valves, injectors, and orifices is essential for the reliable design and operation of high-pressure cryogenic systems. This research investigates the influence of supercritical cryogenic flow conditions on the discharge coefficient of an orifice, using liquid nitrogen as the working fluid.

The study focused on the development of a one-dimensional numerical model to predict the thermodynamic and hydraulic behaviour of cryogenic flow through an orifice. Experimental data from the Von Karman Institute on two-phase cryogenic flows for pressure conditions in the range of 1-10 bar were analysed and used to create a 1D model. A cavitating two-phase flow model is based on the Omega method. This two-phase model was extended and adapted to predict supercritical cryogenic
flow behaviour at pressures ranging from approximately 40 to 100 bar.

The developed model accounts for the strong variations in thermophysical properties occurring near the pseudo-boiling region, where supercritical fluids exhibit boiling-like behaviour despite the absence of a phase transition. By extending concepts traditionally used for compressible and two-phase flows, the model provides a framework for analysing flow behaviour in the supercritical regime. Pressure, temperature, and mass flow rate data were evaluated to investigate the evolution of the flow characteristics,
their influence on the discharge coefficient, and the predictive capability of the proposed model.

The results indicate that the transition into the supercritical flow regime significantly affects the discharge coefficient, with reductions observed depending on the initial thermal state relative to the pseudo-boiling line and zone, the position within the pseudo-boiling region, and the applied pressure drop. To describe this behaviour, a new parameter, the supercritical quality (yₛ꜀), is introduced. This parameter represents the equivalent of vapour quality (x) in two-phase flows and provides a means of quantifying boiling-like effects in supercritical conditions.

Although further experimental validation is required due to the limited availability of cryogenic supercritical data, the results show promising agreement with reference supercritical CO2 data. Furthermore, this work contributes to the fundamental understanding of cryogenic supercritical flows and provides a foundation for improved modelling of high-pressure cryogenic systems.