Attainable Metastability — Volume Balancing Method

When a liquid is depressurised rapidly — for example the rarefaction wave that travels into a pipe after a full-bore rupture — phase change is delayed and the liquid becomes metastable (superheated) below its saturation pressure. The maximum attainable superheat (equivalently, the pressure undershoot) controls the driving force for the subsequent flashing two-phase flow and is an important input to depressurisation and pipeline-rupture safety studies.

NeqSim implements the volume balancing method of Log (2025) in AttainableMetastabilityVolumeBalance (package neqsim.util.nucleation).

Background

Classical nucleation theory (CNT, see ClassicalNucleationTheory) reproduces the attainable superheat near the critical point but strongly over-predicts it at low reduced temperature, where an empirical, temperature-dependent correction factor is normally required.

The volume balancing method removes that empirical factor by recognising that, away from the critical point, the limit is set by evaporation into pre-existing bubbles (gas pockets trapped in wall crevices) rather than by homogeneous nucleation. The metastability limit is the point on the depressurisation path where the rate of vapour volume created by the growth of those bubbles first balances the rate of liquid volume lost through the rarefaction outflow:

\[\left.\frac{dV}{dt}\right|_{\text{evaporation}} \;\ge\; \left.\frac{dV}{dt}\right|_{\text{outflow}} \quad\Longrightarrow\quad \text{metastability limit reached.}\]

Method

Volume loss (rarefaction outflow)

The depressurising liquid is treated as an isentropic rarefaction wave. Starting from the initial pressure $p_0$ at constant molar entropy, the outflow speed $u$ is integrated along the metastable isentrope using the Riemann invariant for a simple wave,

\[u_i = u_{i-1} - \frac{2\,(p_{i-1} - p_i)}{\rho_i c_i + \rho_{i-1} c_{i-1}}, \qquad \left.\frac{dV}{dt}\right|_{\text{outflow}} = -\,u_i\,A_{\text{pipe}},\]

where $c$ is the single-phase (metastable) liquid speed of sound and $A_{\text{pipe}} = \pi r_{\text{pipe}}^2$ the pipe cross-sectional area. The metastable liquid root is obtained with setForceSinglePhase(PhaseType.LIQUID).

Volume creation (bubble growth)

Each pre-existing bubble grows by heat-transfer-limited evaporation following the asymptotic Plesset & Zwick (1954) solution (Collier & Thome, Eq. 4.27). Evaluated at a short reference time $\Delta t$, the bubble radius and growth rate are

\[R = \frac{2\,\Delta T_{\text{sat}}\, k_l}{h_{lg}\,\rho_g}\sqrt{\frac{3\,\Delta t}{\pi\,\alpha_l}}, \qquad \frac{dR}{dt} = \frac{R}{2\,\Delta t}, \qquad \left.\frac{dV}{dt}\right|_{\text{evaporation}} = 4\pi R^2\,\frac{dR}{dt}\,n_{\text{bub}},\]

with liquid superheat $\Delta T_{\text{sat}} = T - T_{\text{sat}}(p)$, liquid thermal conductivity $k_l$, thermal diffusivity $\alpha_l = k_l / (\rho_l c_{p,l})$, latent heat $h_{lg}$, saturated vapour density $\rho_g$, and a single temperature-independent tuning parameter $n_{\text{bub}}$ (the number of pre-existing bubbles).

All thermodynamic and transport properties are taken from the supplied NeqSim SystemInterface, so any equation of state available in NeqSim (SRK, PR, GERG-2008, reference EOS, …) can be used. Unlike the original reference implementation, the liquid thermal conductivity is taken from NeqSim’s general transport-property models rather than a substance-specific correlation, so the method is not limited to a single fluid.

Parameters

Parameter Setter Default Meaning
Number of bubbles $n_{\text{bub}}$ setNumberOfBubbles 1.0e8 Single tuning parameter; more bubbles → faster evaporation → smaller attainable superheat
Pipe radius $r_{\text{pipe}}$ [m] setPipeRadius 0.02 Sets the outflow cross-sectional area
Minimum pressure [Pa] setMinimumPressure 3.0e5 Lower bound of the depressurisation search
Number of steps setNumberOfSteps 1000 Pressure steps used to integrate the isentrope
Reference time $\Delta t$ [s] setBubbleReferenceTime 1.0e-4 Time at which the asymptotic growth rate is evaluated

Note: $n_{\text{bub}}$ is a free parameter fitted to data; values in the range $10^8$–$10^9$ reproduce the CO₂ depressurisation experiments in Log (2025). It is the only temperature-independent tuning constant in the model.

Example — liquid CO₂ blowdown

import neqsim.thermo.system.SystemInterface;
import neqsim.thermo.system.SystemSrkEos;
import neqsim.util.nucleation.AttainableMetastabilityVolumeBalance;

// Pure liquid CO2 at 0 C and 120 bara
SystemInterface co2 = new SystemSrkEos(273.15, 120.0);
co2.addComponent("CO2", 1.0);
co2.setMixingRule("classic");

AttainableMetastabilityVolumeBalance vb = new AttainableMetastabilityVolumeBalance(co2);
vb.setNumberOfBubbles(1.0e8);
vb.setPipeRadius(0.02);
vb.setNumberOfSteps(600);

vb.calculateLimit(273.15, 120.0e5); // T0 [K], p0 [Pa]

if (vb.isLimitFound()) {
  double pLimitBar = vb.getLimitPressure() / 1.0e5; // metastability limit pressure [bara]
  double superheat = vb.getSuperheat();             // attainable superheat [K]
}

// Machine-readable summary (LinkedHashMap / JSON)
String json = vb.toJson();

toMap() / toJson() return the inputs and, when a limit is found, the limit pressure, temperature, saturation temperature, and superheat — suitable for direct inclusion in a study results.json.

This is a screening / research method for a pure (or pseudo-pure) liquid. It predicts the limit of metastability, not the subsequent two-phase flashing flow.

References

  1. Log, A.M. (2025). On the attainable metastability of liquids during depressurisation — effect of pre-existing bubbles. Journal of Fluid Mechanics 1018, R2. https://doi.org/10.1017/jfm.2025.10545
  2. Plesset, M.S. and Zwick, S.A. (1954). The growth of vapor bubbles in superheated liquids. Journal of Applied Physics 25, 493–500.
  3. Collier, J.G. and Thome, J.R. (1994). Convective Boiling and Condensation, 3rd ed. Oxford University Press, Eq. 4.27.