The starting point for the new work is a Bernoulli-based description of the discharge along a representative streamline that runs from the gas bulk inside the vessel, through the outlet pipe, and out into the undisturbed ambient atmosphere. Because the vessel volume is far larger than the outlet cross section, the gas velocity in the bulk is negligible, and the unsteady acceleration along the streamline can be treated as quasi-stationary. What the formulation does keep, and what distinguishes it from many purely thermodynamic blowdown models, is the explicit treatment of irreversible pressure losses: local inertial losses at the vessel-to-pipe transition and at the discharge into ambient air, plus distributed pipe friction described by the Darcy-Weisbach equation with a friction factor obtained iteratively from the Colebrook relation. These losses are folded directly into an expression for the area-averaged outlet velocity, which in turn feeds a mass balance that yields the time-resolved pressure decay inside the vessel.

The first major extension concerns choked flow. When the ratio of ambient pressure to vessel pressure falls below the critical pressure ratio, which depends on the heat capacity ratio of the gas, the flow at the narrowest section reaches sonic conditions and the mass flow rate can no longer increase no matter how much the downstream pressure drops. The team uses the classical ideal-gas critical-pressure-ratio criterion as a regime-switching rule: at every time step of the numerical integration, the instantaneous vessel pressure is compared with the critical pressure, and the choked-flow pressure-decay equation is applied as long as the vessel pressure remains above that threshold. Because friction and inertia reduce the effective mass flow, the model introduces a loss-corrected equivalent outlet velocity, obtained by dividing the local speed of sound by the effective loss coefficient. The authors are careful to stress that this is an engineering approximation: the model does not resolve the spatially varying compressible flow field and does not claim to locate the sonic section precisely, which in a real vessel-pipe combination may sit near a vena contracta just downstream of the inlet or at the pipe exit depending on geometry and friction.

The second extension brings thermal physics into the picture. As gas expands during discharge, it cools, and in applications such as hydrogen storage that cooling can push temperatures below limits tolerated by polymer liners and seals. The Dortmund team couples the discharge model to an energy balance in which heat flows from the vessel wall to the gas by convection, with the heat-transfer coefficient derived from empirical Nusselt correlations based on the Rayleigh number for natural convection. Natural convection is the appropriate regime here because numerical simulations of the same geometry show that gas velocities inside the vessel are very low except near the outlet, so buoyancy driven by temperature-induced density gradients dominates the internal heat exchange. A conservative sensitivity analysis confirmed that the stainless-steel vessel behaves as a thermal reservoir on the timescale of the experiments: even under deliberately pessimistic assumptions, the predicted wall-temperature change during a single discharge remained close to one kelvin, justifying a constant wall-temperature boundary condition.

Numerically, the friction-and-heat-transfer formulation is implemented as a sequential two-pass procedure rather than a fully implicit coupled solve. In the first pass, a preliminary pressure history is computed without pipe friction, and at each time step the Rayleigh number, Nusselt number, and heat-transfer coefficient are evaluated to determine the heat transferred from the wall to the gas. From the ratio of diabatic to isentropic enthalpy changes, a global heat-transfer polytropic exponent is extracted. In the second pass, the pressure history is recalculated with the full frictional losses, using the global thermal exponent in place of the isentropic one, while the Darcy friction factor is iterated to convergence at every step. A local effective polytropic exponent is then matched to the loss-corrected outlet velocity for post-processing the gas temperature. The whole scheme runs with an explicit fourth-order Runge-Kutta method at a time step of one ten-thousandth of a second, and a Richardson extrapolation showed the numerical error to be negligible compared with the experimental deviations.

Validation rested on 108 discharge experiments: air, carbon dioxide, and helium released from a 2.085 cubic meter stainless-steel vessel through a 25 millimeter outlet pipe, at twelve initial overpressures from 100 to 1500 millibar, with three repetitions per condition. A quick-opening valve, reproducibly actuated within 34 milliseconds, released the gas, while a phase Doppler anemometer measured outlet velocities on the jet axis one pipe diameter downstream. Fog droplets of ethylene glycol and water served as tracers, and only particle-size classes up to 11.5 micrometers were retained, because exit Stokes numbers for those classes were far below unity, confirming that the particles faithfully followed the rapidly accelerating gas. Statistical analysis with linear mixed models, including adjusted intraclass correlation coefficients and conditional coefficients of determination, showed that repetition effects were negligible and justified combining the three repetitions at each condition.

The agreement between model and experiment is striking for a reduced-order formulation. Across all gases and pressures, the mean relative deviation in discharge duration was just 3.60 percent, with experimental scatter between repetitions ranging from 0.1 to 2.6 percent. The measured discharge durations fell between the idealized isentropic and isothermal limits, exactly as the polytropic formulation with friction and heat transfer predicts. Maximum outlet velocities agreed even more closely for the heavier gases: deviations ranged from minus 1.7 to plus 0.9 percent for air and from minus 1.0 to plus 1.5 percent for carbon dioxide, comparable to the sub-one-percent nominal uncertainty of the optical technique itself. Notably, the maximum velocity occurs at the very start of the discharge, before wall heat transfer has had time to influence the gas state, which explains why varying the empirical Nusselt correlation parameters by plus or minus ten percent barely moved the predictions.

Helium exposed both the power and the limits of the approach. Its low molar mass means the vessel empties in seconds rather than tens of seconds, and its high speed of sound drives outlet velocities beyond the 433 meters per second ceiling of the phase Doppler anemometer at almost all tested pressures. Direct velocity validation was therefore confined to the lowest overpressure, where the model deviated by minus 6.5 percent. Helium discharge durations, however, were predicted over the entire pressure range, with the model tending to slightly underpredict, a trend the authors attribute to fixed-timescale effects such as valve opening and sensor response becoming relatively more important in short discharges, and to the sensitivity of light-gas discharge to the effective loss coefficient and the choked-to-subcritical transition. The team is transparent that helium velocities at higher pressures remain model predictions rather than validated measurements.

The practical payoff is a tool that delivers, in seconds on an ordinary computer, what would otherwise demand computationally expensive three-dimensional simulations: time-resolved vessel pressure, discharge duration, mass flow rate, and maximum outlet velocity, with choking, friction, and thermal interaction all included. That makes the model well suited to sizing pressure relief and venting systems, assessing mechanical loads and jet momentum, running parameter studies over vent-line geometries, and supplying boundary conditions for downstream gas dispersion and risk analyses. It is particularly timely for alternative energy carriers such as hydrogen, where low molar mass produces exactly the high-velocity, choked conditions the model now handles. The authors caution that for safety-critical final designs the model should be paired with appropriate safety factors and, where necessary, detailed simulation, and they identify improved natural-convection correlations for enclosures and spatially resolved temperature measurements as the key next steps for extending the thermal validation to larger vessels, longer discharge times, and different wall materials.

Subject of Research: Transient gas discharge modeling from pressurized vessels including choked flow and heat transfer

Article Title: Modeling and validation of transient gas outflows from pressurized vessels under choked and thermal effects

Article References: Fischer, M.-D., Ryll, F., Baier, S., Boettcher, K. E., & Dieguez-Alonso, A. (2026). Modeling and validation of transient gas outflows from pressurized vessels under choked and thermal effects. Results in Engineering, 32, Article 113271. https://doi.org/10.1016/j.rineng.2026.113271

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113271

Keywords: pressurized vessels, gas discharge, choked flow, heat transfer, blowdown modeling, phase Doppler anemometry, hydrogen safety, pressure relief, reduced-order model, natural convection, fluid dynamics, process safety