April 2025
Valves, Pumps, Compressors and Turbomachinery
Sizing PRVs for homogeneous non-equilibrium saturated flashing liquids
The sizing method of pressure relief valves (PRVs) for homogeneous equilibrium saturated flashing liquids is well-established. The homogeneous equilibrium method (HEM) is generally used in the American Petroleum Institute (API) 520 standard because the result is conservative. The HEM is based on the assumptions that the two phases in a mixture are well-mixed, have the same temperature and pressure in the liquid and vapor phases, and allow the mixture to be considered as a single component.
Although the non-equilibrium effects have been addressed by the Design Institute of Emergency Relief Systems (DIERS), there is presently no sizing method for the homogeneous non-equilibrium saturated flashing liquids in API 520. However, a homogeneous non-equilibrium (HNE) model is a standard sizing method in the International Organization for Standardization/Draft International Standard 4126-10 (ISO/DIS-4126-10) known as the HNE-DS (homogeneous non-equilibrium model developed by Diener and Schmidt).1,2 The HNE-DS extended the Omega method developed by Leung for the HEM to include the degree of thermodynamic non-equilibrium.
This article proposes a simple HNE model (HNE-KH) for saturated flashing liquids (vapor quality ≥ 0.001). In the case of saturated flashing liquids, the flashing liquids are considered as compressible due to the reduced boiling—though boiling delay is often used in industry. However, the boiling delay for subcooled flashing liquids leads to a metastable state, where the liquids are superheated and considered as incompressible. The HNE-KH is developed to consider the non-equilibrium effects based on the experiments of Friedel, et al.3,4 The HNE-KH is in good agreement with the experimental measurements. The results of the HNE-KH have also been compared with the ISO/DIS 4126-10 model for experimental data, methanol, propane, benzene, n-hexane and 1,000-psia water. In addition, this article offers important insights for determining the discharge coefficient of two-phase flow.
Review of experimental data. Friedel, et al. conducted a set of experiments with a Leser DN25/40 PRV for saturated water at four different stagnation pressures: 78.3 psia (5.4 bar), 98.6 psia (6.8 bar), 116 psia (8 bar) and 153.7 psia (10.6 bar). The measured discharge coefficient of vapor without derating given by the valve manufacturer is 0.77. The measured discharge coefficient of liquid without derating is 0.55, which is slightly different from the value reported in NB-18, Pressure Relief Device Certification. The experiments by Friedel, et al. measured the mass flux through the PRV for the inlet stagnation vapor qualities of the saturated water from 0.001 up to 0.05.3,4
Based on the experimental data, in fluids with less than a stagnation vapor quality of about 0.05, the HEM underestimates the mass flux. Although a few experimental data points indicate some uncertainty, the results demonstrate that the effects of non-equilibrium lead to the underestimation. The non-equilibrium model typically results in a significantly higher mass flux than an equilibrium model at low stagnation vapor qualities. The higher mass flux could be due to a reduction in vaporization. However, the mass flux approaches the value of the homogeneous equilibrium model at a stagnation vapor quality of > 0.05. According to Leung, higher pressures reduce the magnitude of non-equilibrium effects.3
Non-equilibrium factor N. The HNE-KH is a simple modification to the HEM with a three-point method.5 After analysis of the experimental results of Friedel, et al., Eq. 1 for the non-equilibrium factor (NN) was developed to consider the non-equilibrium effects as a function of stagnation vapor quality and stagnation vapor compressibility factor.
where: Xo = stagnation vapor quality, vapor weight fraction
Z0 = stagnation vapor compressibility factor.
If the calculated N value from Eq. 1 is negative, the N value should be zero.
Eq. 2 is one of the best pressure-specific volume models developed by Simpson.6 The three-point data sets (P0, v0, P1, v0, P1, v1, P2, v2) are determined by isentropic flash calculations. P0 is stagnation pressure at the PRV inlet. P1 and P2 are 75% of P0 and 50% of P0, respectively.
where:
α𝛼 = the parameter in the pressure-specific volume model
β𝛽 = the parameter in the pressure-specific volume model
PP = the pressure of the fluid, psia
vv = the specific volume of the fluid, ft3/lb
P0P0 = the stagnation pressure at the PRV inlet, psia
v0v0 = the specific volume at the PRV inlet, ft3/lb.
However, the v1 and v2 should be corrected for an HNE state by incorporating the non-equilibrium factor, N, calculated from Eq. 1. The HNE-KH uses the three-point data corrected (P0,v0, P1, v-NE, P2, v2-NE) for an HNE state. The three-point data for HNE-KH is obtained using Eq. 3:
FIG. 1 shows the non-equilibrium factors for the experiments of Friedel, et al. As shown in FIG. 1, the non-equilibrium factor from Eq. 1 does not show a substantial deviation from the actual non-equilibrium factor.
FIG. 1. Non-equilibrium factor for saturated flashing liquids.
Mass flux calculation. The following is a step-by-step procedure to determine the mass flux for HNE saturated flashing liquids (vapor quality ≥ 0.001).
- Step 1: Generate the three-point data sets (P0, v0, P1, v1, P2, v2) for the HEM using isentropic flash calculations.
- Step 2: Determine the three-point data sets corrected (P0, v0, P1, v1-NE, P2, v2-NE) for the HNE-KH by incorporating the non-equilibrium factor from Eq. 1.
- Step 3: Solve for two parameters for the three-point data sets corrected using Eq. 4 for B and Eq. 5 for a.
- Step 4: Calculate the equivalent critical pressure using Eqs. 6 and 7. The initial trial value for P is the maximum expected backpressure of P2, whichever is greater.
where:
G = the mass flux (mass flow per unit area) through the PRIV, lb/ft2-sec
Kd = the discharge coefficient of vapor
Pec = the equivalent critical pressure for the mass flux for subcritical flow, psia
Pc = the critical pressure for the mass flux for critical flow, psia.
Validation of HNE-KH. FIGS 2–5 show the results of the HNE-KH are in good agreement with the measurements. However, the ISO/DIS 4126-10 model underestimates the mass flux for low stagnation vapor qualities. The ISO/DIS 4126-10 model uses the weighted discharge coefficient from Eq. 10. Using the weighted discharge coefficient is one of the causes for the lower prediction of the mass flux.
FIG. 2. Calculated and measured mass flux at 78.3 psia.
FIG. 3. Calculated and measured mass flux at 98.6 psia.
FIG. 4. Calculated and measured mass flux at 116 psia.
FIG. 5. Calculated and measured mass flux at 153.7 psia.
According to Kim, et al., an exit loss makes the difference between Kdl and Kdv.7 There is no exit loss for compressible flow. The HNE-KH uses Kdv for saturated flashing liquids since the flow is compressible. Generally, the maximum available Kdl is 0.7071 (0.5 = 0.7071) because an exit loss limits the use of only 50% of the total available pressure if the exit loss is based on the narrowest flow cross section. Only the exit loss explains the significant pressure losses. The non-derated for major U.S. manufacturer valves is slightly higher than 0.7071. However, the disk lift significantly affects Bolle, et al. performed experiments with different lifts.8 The higher disk lift than the certified lift affects Kdl. Conversely, the higher disk lift than the certified lift does not affect Kdv.
Extension of HNE-KH. This comparison is to evaluate if the HNE-KH can be extended to chemicals and high-pressure water: methanol, propane, benzene, n-hexane and 1,000-psia water. In FIGS. 6–10, the calculated mass flux values are plotted against stagnation vapor qualities: 0, 0.001, 0.01, 0.05 and 0.1, respectively. Note that the results at the vapor quality of 0 are plotted for reference only and should not be considered valid.
FIG. 6. Calculated mass flux for saturated flashing methanol.
FIG. 7. Calculated mass flux for saturated flashing propane.
FIG. 8. Calculated mass flux for saturated flashing benzene.
FIG. 9. Calculated mass flux for saturated flashing n-hexane.
FIG. 10. Calculated mass flux for saturated flashing 1,000-psia water.
Calculations were performed on two different stagnation pressures for chemicals and on one stagnation pressure for water. Due to the lack of experimental data, the results of HNE-KH are compared with the results of the ISO/DIS 4126-10 model. The PRV with double-certified trim has a vapor discharge coefficient of 0.9667 (non-derated) and a liquid discharge coefficient of 0.7289 (non-derated). In general, both methods consistently show a similar trend in the saturated water-steam experiments of Friedel, et al. The ISO/DIS 4126-10 model underestimates the mass flux at low-vapor qualities (X0 < 0.05). On the other hand, the ISO/DIS 4126-10 model overestimates the mass flux at high vapor qualities (X0 ≥ 0.05).
FIG. 10 shows the comparison of the HNE-KH with the ISO/DIS 4126-10 model for saturated 1,000-psia water. The results are also consistent, as expected. The ISO/DIS 4126-10 model under-predicts the mass flux over less than the inlet vapor qualities of 0.05 but over-predicts over high inlet vapor qualities. In an Electric Power Research Institute Report, the mass flux is about 10,000 lb/ft2-sec for saturated water at 1,000 psia.9 The report provides the most extensive data sets for steam/water at inlet pressures of 600 psia–1,000 psia in various nozzles.
FIG. 10. Calculated mass flux for saturated flashing 1,000-psia water.
TABLE 1 is prepared to figure out why the mass flux of the ISO/DIS 4126-10 model for 300-psia n-hexane significantly deviates, as shown in FIG. 9.
The one-point Omega method used in the ISO/DIS 4126-10 model is generally not valid if the fluid is close to its thermodynamic critical point (Pr ≥ 0.5).10 A prevailing thought is that near the thermodynamic critical conditions with one-point Omega method many contribute to more deviation than the other cases.
Takeaways. The most rigorous HEM with a three-point method is extended by employing a non-equilibrium factor to size the PRVs for HNE saturated flashing liquids (vapor quality ≥ 0.001) because there is presently no sizing method in API 520. The non-equilibrium factor is based on the experimental data of Friedel, et al. on a PRV to account for reduced boiling. The results of the proposed HNE-KH are satisfactory. The proposed HNE-KH is a simple method to determine the accurate mass flux for homogeneous non-equilibrium saturated flashing liquids. The accurate mass flux can be used for PRV stability predictions to be reasonably assessed, as well as PRV sizing.
The proposed HNE-KH with the discharge coefficient of vapor is better agreement with the experimental measurements than the ISO/DIS 4126-10 model with the weighted discharge coefficient. According to Kim, et al., the discharge coefficient of vapor is generally greater than the discharge coefficient of liquid because an exit loss is not required for the compressible flow. The exit loss makes the difference between the discharge coefficient of liquid and the discharge coefficient of vapor. Therefore, using the discharge coefficient of vapor for saturated flashing liquids is a more appropriate choice.
LITERATURE CITED
1 International Organization for Standardization (ISO)/DIS 4126-10, “Safety devices for protection against excessive pressure—Sizing of safety valves and connected inlet and outlet lines for gas/liquid two-phase flow,” 1st Ed., 2010.
2 Diener, R. and J. Schmidt, “Sizing of throttling device for gas/liquid two-phase flow Part 1: safety valves,” Process Safety Progress, Vol. 23, No.4, 2004.
3 Leung, J. C., “PRV discharge and non-equilibrium effects in two-phase flows—Part II,” Design Institute for Emergency Relief Systems (DIERS), May 2006.
4 T. Lenzing, et al., “Prediction of the maximum full lift safety valve two-phase flow capacity,” Journal of Loss Prevention in the Process Industries, Vol. 11, Iss. 5, September 1998.
5 Kim, J. S., et al., “Use a simple vapor equation for sizing two-phase pressure relief valves,” Hydrocarbon Processing, July 2022.
6 Simpson, L. L., “Navigating the two-phase maze,” International symposium on runaway reactions and pressure relief design, Boston, Massachusetts, August 2–4, 1995.
7 Kim, J.S., et al., “Proper use of conventional PRV discharge coefficients,” Chemical Engineering, May 2017.
8 L. Bolle, et al., “Experimental and theoretical analysis of flashing water flow through a safety valve,” Journal of Hazardous Materials, 1996.
9 Sozzi, G. L. and W. A. Sutherland, “Critical flows of saturated and subcritical water at high pressure,” NEDO-13418, General Electric Co., San Jose, CA, July 1975.
10 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.
The Authors
Jung Seob Kim is a Principal Pressure Safety Engineer at Covestro. He has more than 35 yr 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 holds a BS degree in chemical engineering from the University of Seoul. He is a member of AIChE, and is a registered professional engineer in Texas.
Ryan J. Haines is a Process Safety Engineer at Covestro. He has 5 yr of experience in different roles, primarily focusing on reliability and process engineering. He earned a BS degree in chemical engineering from Texas A&M University.
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