J. S. KIM, Covestro, Baytown, Texas (U.S.); and O. TSENG, Novetus Engineering, Houston, Texas (U.S.)
The accurate sizing of pressure relief valves (PRVs) for low vapor-quality and subcooled flashing liquids is critical due to significant non-equilibrium effects that are not adequately captured by conventional models. The homogeneous equilibrium model (HEM), commonly recommended by the American Petroleum Institute (API) 520 standard, tends to provide conservative estimates but may not reflect actual flow behavior in metastable, superheated liquid conditions.1
For high vapor-quality flows, the homogeneous non-equilibrium model (HNE-KH) developed by Kim, et al. offers improved accuracy.2 However, this model underpredicts mass flux for vapor qualities below 0.001. In such regimes, flow behavior resembles frozen flow—characterized by delayed boiling, minimal density variation and incompressibility. This article proposes a new non-equilibrium frozen model (NEF-KT) tailored for low vapor quality (vapor quality < 0.001) and subcooled flashing liquids. Unlike the HNE-KH model, the NEF-KT model incorporates frozen flow assumptions and applies the liquid discharge coefficient to better represent the metastable nature of the flow. Validation against experiment data from Bolle, et al. and Darby, et al. demonstrates strong agreement, indicating the NEF-KT model provides a more accurate method for PRV sizing in these challenging flow regimes.3,4
Review of experimental data. Bolle, et al. conducted a series of experiments using a Crosby PRV to study subcooled flashing water at various stagnation pressures ranging from 89 psia to 58.7 psia.3 Sixteen data points were recorded using the Crosby 1D2 JLT-JOS PRV, with the valve disc fixed at a 5-mm lift (certified lift: 3.835 mm). Darby, et al. also presented eight experiment data points for saturated and subcooled flashing water at stagnation pressures between 162.9 psia and 489.7 psia, using Crosby JLT-JBS and Crosby 900 valves.4 The specifications of the valves used in these experiments are summarized in TABLE 1.
Based on the experimental data, it is evident that steam/water mixtures with vapor qualities below 0.001 behave as frozen flow through PRVs. This frozen flow appears to choke in the valve; however, due to the nearly constant density of incompressible flow, choking is not associated with reaching critical velocity. The HEM significantly underestimates the mass flux in case of near-zero vapor quality. While several frozen non-equilibrium approaches have been proposed in the literature, none have demonstrated consistently satisfactory results.
Mass flux for non-equilibrium frozen flow. For low vapor quality and subcooled flashing liquids, which behave as frozen flow, the mass flux can be estimated using the incompressible Bernoulli equation. In this regime, the flow can be approximated as incompressible due to negligible density variation. Eq. 1 was developed to determine the mass flux using the actual liquid discharge coefficient, as flashing liquids tend to choke in the valve. The liquid discharge coefficient is appropriate for incompressible flow because it accounts for exit losses without assuming significant compressibility.
Eqs. 2 and 3 are empirical correlations derived from the experimental data of Bolle, et al. and Darby, et al., and are designed to provide good agreement with measured results.3,4 These correlations may be refined in the future as more experimental data becomes available. The actual liquid discharge coefficient (Kdl) used in the calculations is obtained by removing the 0.9 derating factor from the certified value. However, it is important to note that PRV design and sizing should always be based on the certified liquid discharge coefficient, as required by applicable codes and standards.
G = Kdl × √(2 × 4,633 × ρin × (pin – pc-NE )) (1)
where:
G = the mass flux through the nozzle, lb/ft2-sec
Kdl = the actual liquid discharge coefficient
ρin = the density of the fluid at the valve inlet, lb/ft3
pin = the stagnation pressure of the fluid at the valve inlet, psia
pc-NE = the non-equilibrium critical pressure of the fluid, psia.
Eq. 2 is used for low vapor quality flashing liquids (0 ≤ vapor quality < 0.001):
pc-NE = Pin – 2.5 × (Pin – Pc at x = 0 ) × (–520 × x + 1) × Z0.8794 (2)
Eq. 3 is used for subcooled flashing liquids:
pc-NE = Pin – (Pin + 1.5 × P0 – 2.5 × Pc at x = 0 ) × Z 0.8794 (3)
po = the saturation pressure of the fluid, psia
pc at x = 0 = the critical pressure of the HEM at x = 0 , psia
x = the vapor quality at po, mass fraction
Z = the vapor compressibility factor at po.
Use the backpressure value as pc-NE if the actual PRV backpressure exceeds the calculated pc-NE. The vapor compressibility factor is included to account for pressure effects on non-equilibrium behavior, consistent with the approach used in the HNE-KH model.2 According to Leung, higher pressures reduce the extent of non-equilibrium effects.5 Eq. 2 appears to provide a smooth transition in behavior as vapor quality deceases from 0.001 to zero. However, experimental data in this range is limited.
Critical pressure of homogeneous equilibrium flashing liquids. The following is a step-by-step procedure to determine the critical pressures of homogeneous equilibrium flashing liquids used in Eqs. 2 and 3.
Step 1—Generate a three-point data set (P0 , v0 ,P1 , v1 ,P2 , v2 ) using isentropic flash calculations. P0 is the saturated stagnation pressure at the PRV inlet. P1 and P2 are taken as 75% and 50% of P0, respectively.
Step 2—Solve for the two parameters in Eq. 4 using Eq. 5 to determine and Eq. 6 to determine . Eq. 4 is one of the most accurate pressure-specific volume correlation models, developed by Simpson.6
(v / v0 ) – 1 = α × [(P0 / P)β – 1] (4)
α = the parameter in the pressure-specific volume model
β = the parameter in the pressure-specific volume model
P = the pressure of the fluid, psia
v = the specific volume of the fluid, ft3/lb
P0 = the stagnation pressure at the PRV inlet, psia
v0 = the specific volume at the PRV inlet, ft3/lb.
Step 3—Calculate the critical pressure using Eq. 7 from Kim, et al.7 Use P2 as the initial trial value for P.
A = [(α × p0β ) / (1 – β)] × (p1–β – p01–β ) + (1 – α) × (p – p0 )
Pec = the equivalent critical pressure for the mass flux, psia.
If Pec > P, the flow is considered critical. Iteratively repeat the calculation of Pec until Pec is nearly equal to P. The calculated Pec from each iteration is used as P in the next iteration. Once Pec is almost equal to P, the Pec becomes the critical pressure, pc. Note that Pc and pc at x = 0 are identical if the three-point data set is for zero vapor quality.
Mass flux calculation procedure for low vapor quality flashing liquids. The following is a step-by-step procedure to determine the mass flux for non-equilibrium frozen flashing liquids with low vapor quality (0 ≤ vapor quality < 0.001).
Step 1—Generate a three-point data set (P0 , v0 ,P1 , v1 ,P2 , v2 ) at the saturated stagnation pressure P0 and zero vapor quality using isentropic flash calculations. P1 and P2 are taken as 75% and 50% of P0, respectively.
Step 2—Solve for the two parameters using the three-point data set: calculate β using Eq. 5 and α using Eq. 6.
Step 3—Calculate the critical pressure at zero vapor quality, pc at x = 0 using Eq. 7.
Step 4—Calculate the non-equilibrium critical pressure using Eq. 2.
Step 5—Calculate the mass flux using Eq. 1.
Mass flux calculation procedure for subcooled flashing liquids. The following is a step-by-step procedure to determine the mass flux for non-equilibrium frozen subcooled flashing liquids (vapor quality < 0). For subcooled flashing liquids, the vapor quality is defined as negative.8
Step 4—Calculate the non-equilibrium critical pressure using Eq. 3.
Validation of NEF-KT. The results of the NEF-KT model and the HEM for each data point are plotted with the experimental data from Bolle, et al. and Darby, et al.3,4 Unfilled orange color markers represent the experimental data from Bolle, et al., while solid-filled orange color markers represent from Darby, et al. The experimental data shows clear agreement with the NEF-KT model predictions, as illustrated in FIG. 1. The experimental data and the corresponding calculation results depicted in FIG. 1 are presented in TABLES 2–5. All calculations are based on the discharge coefficients of vapor and liquid listed in TABLE 1. The NEF-KT model uses the liquid discharge coefficient, assuming frozen flow is incompressible. On the other hand, the HEM uses the vapor discharge coefficient, as recommended by Darby, when the nozzle flow is choked.9
For the experimental data from Bolle, et al. in TABLE 2, the NEF-KT model clearly aligns well with the observations. However, the HEM underpredicted the mass flux at PC-NE / Pin values greater than 0.6. Red-colored values in TABLE 2 indicate deviations in mass flux greater than 30%. At PC-NE / Pin values lower than 0.5, the HEM showed good agreement with the measured mass flux, likely because Bolle, et al. conducted their experiments under high lift conditions.3 Under high lift, the estimated liquid discharge coefficient is approximately 0.956, minimizing the difference between the vapor and liquid discharge coefficients. This can lead to the mistaken assumption that the vapor discharge coefficient should be used. If the experiment were performed under certified lift conditions, the measured mass flux should be significantly lower.
For the experimental data from Darby, et al. in TABLES 3–5, the NEF-KT model again demonstrates good agreement. The model 9611, with its short nozzle length, did not exhibit any notable impact on the mass flux. Some instances of lower measured mass flux (highlighted in blue) in TABLE 3 at PC-NE / Pin lower than 0.5 were overpredicted by the NEF-KT model, likely due to high backpressure at the PSV outlet under high mass flux. The backpressure effect was not considered due to insufficient information for accurate evaluation. Conversely, the HEM significantly overpredicted the mass flux at PC-NE / Pin lower than 0.4. This contrasts with the Bolle, et al. case, underscoring the importance of using the correct discharge coefficient. The HEM overpredicted the mass flux by 30%–40%, likely due to the use of the vapor discharge coefficient. According to Kim, et al., exit losses contribute to differences between the liquid and vapor discharge coefficients (Kdl and Kdv).2 Therefore, exit loss in incompressible flow leads to differences between vapor and liquid discharge coefficients. Additionally, the HEM significantly underpredicted mass flux at PC-NE / Pin higher than 0.6. For saturated flashing liquid at x = 0 in TABLE 4, the measured mass flux was 180% higher than predicted by the HEM.
Takeaways. This article has introduced the NEF-KT model, a simple yet effective alternative specifically designed for low vapor qualities below 0.001 and subcooled flashing liquid conditions. By incorporating frozen flow characteristics and utilizing the liquid discharge coefficient, the NEF-KT model accounts for the incompressible nature and exit losses inherent in metastable liquid flows. Validation against experiment data from Bolle, et al. and Darby, et al. confirms the model’s ability to deliver accurate and consistent mass flux predictions.
The NEF-KT model offers a practical tool for engineers seeking reliable pressure relief valve sizing in challenging flashing flow conditions, improving both safety and design confidence in critical systems. Future work may focus on refining the model further with broader datasets for various fluid types. HP
LITERATURE CITED
American Petroleum Institute (API) Standard 520, Part I, “Sizing, selection, and installation of pressure-relieving devices, Part I-sizing and selection,” 10th Ed., October 2020.
Kim, J. S., et al., “Sizing PRVs for homogeneous nonequilibrium saturated flashing liquids,” Hydrocarbon Processing, April 2025.
Bolle, L., et al., “Experimental and theoretical analysis of flashing water flow through a safety valve,” Journal of Hazardous Materials, 1996.
Darby, R. and J. Stockton, “Evaluation of two-phase flow models for flashing flow in relief valves,” Design institute for Emergency Relief Systems (DIERS), October 1999.
Leung, J. C., “PRV discharge and non-equilibrium effects in two-phase flows – Part II,” Design institute for Emergency Relief Systems (DIERS), May 2006.
Simpson, L. L., “Navigating the two-phase maze,” International symposium on runaway reactions and pressure relief design, Boston, Massachusetts (U.S.), August 2–4,1995.
Kim, J. S., et al., “Use a simple vapor equation for sizing two-phase pressure relief valves,” Hydrocarbon Processing, July 2022.
Sozzi, G. L. and W.A. Sutherland, “Critical flows of saturated and subcritical water at high pressure,” NEDO-13418, General Electric Co., San Jose, California (U.S.), July 1975.
Darby, R., “On two-phase frozen and flashing flows in safety relief valves,” Journal of Loss Prevention in the Process Industries, 2004.
Jung Seob Kim is a Principal Pressure Safety Engineer at Covestro and has more than 40 yrs of experience in different roles within the petrochemical industry, including with ioMosaic, SK E&C USA, Bayer Technology Services, Samsung BP Chemicals and Samsung Engineering. He earned a BS degree in chemical engineering from the University of Seoul (Korea). He is a member of AIChE and is a registered Professional Engineer in the State of Texas (U.S.).
Olivia F. Tseng is a Process Safety Engineer at Novetus Engineering with 2 yrs of experience working various roles, including PSM, pressure safety, and process and structural engineering. Her primary interests include research and development and process design. She earned a BS degree in chemical engineering from the University of Houston (U.S.).