A. Savant and S. CHITALE, Worley, Mumbai, India
Sizing relief valves in supercritical conditions is a complex process that requires careful consideration of fluid properties. Key challenges include no latent heat of vaporization, density changes with pressure and temperature, non-ideal gas behavior and choked flow.
Supercritical fluids contain characteristics of both liquid and gases, and do not adhere to the ideal gas law. As pressure increases, the fluid—being gas-like—changes to become liquid in its properties. Supercritical fluids create fluids with densities the same as liquids and viscosities the same as gases. They are created by increasing the temperature and pressure beyond the substance’s critical point. During a fire, pressure safety valves (PSVs) on vessels containing liquid hydrocarbons that may be blocked can relieve a supercritical fluid if the relieving pressure is higher than the critical point. Compressibility factor and the fluid temperature are not constant while the vessel is relieving. These deviations from both ideal gas and incompressible fluid behavior present distinct challenges for relief valve sizing.
This article focuses on relief valve sizing methods for supercritical fluids in a fire case scenario for a pressure vessel containing hydrocarbons. Three approaches are discussed, and respective results are analyzed for study purposes. The first approach is based on American Petroleum Institute (API) standards wherein relief load calculations and size are calculated, assuming physical properties of air and the ideal gas laws. This may lead to a deviation in cases of near critical and supercritical fluids.
The second approach focuses on a rigorous approach (an alternative method described in API 521)—the pressure-relieving rate is derived from an unsteady-state analysis. This approach performs isentropic flashes using appropriate packages at relieving pressure at various temperatures to find the fluid density at each point. The required relief volume between any two steps is then the increase in volume of the fluid. The relief rate is then found by dividing the increase in volume by the time it takes to provide the enthalpy between the steps.
The third approach is based on the choked pressure phenomenon—if the choked pressure is higher than the backpressure, sonic flow exists at the orifice. The orifice area is calculated with the help of the mass flux found from the velocity and specific volume at sonic flow.
Fire scenario. A fire scenario has been considered for two sets of fluids: n-heptane and a mixture of n-nonane and n-decane. The entire vessel exposed to an external fire is considered. The pressure in the vessel will rise until the PSV set pressure is reached. At the PSV set pressure, the PSV will start to open. The valve will be fully open at the relieving pressure, normally 21% above the set pressure for a fire case design. If the relieving pressure is less than the critical pressure of the fluid, boiling liquid will be discharged through the PSV.
However, if the relieving pressure is higher than the critical pressure of the fluid, the liquid will not boil, and the phase of the fluid becomes difficult to know. As fluid becomes supercritical, the phase properties lie somewhere between a liquid and a vapor. The fire continues to heat the fluid content in the vessel, increasing the pressure to the PSV’s set pressure. The system will then send relief vapor to the flare header until the pressure in the vessel reaches the acceptable limit. FIG. 1 represents the fluid behavior path from normal operating conditions to the supercritical path through critical operating parameters. Parabolic curves show the fluid equilibrium condition. A fluid’s boiling point increases as pressure on it increases. This fluid behavior can be easily traced using the ideal gas law. Once the fluid nears its critical point, property estimates using the ideal gas law become uncertain. As heat continues to heat the content of vessel, after reaching the relieving pressure, the pressure must remain the same to avoid over-pressuring the vessel. The temperature and required relieving rate will vary as the relieving process continues.
Vessel data for the load calculation are given below:
Vessel diameter = 3 m
Vessel tan-tan = 4 m
Area exposed (Ae) = 43.3 m2
Insulation factor (f) = 1
Relieving pressure = 4,100 kPa
Backpressure = 300 kPa
Kd = 0.975 (1)
Kb = 1 (1)
Kc = 1 (1)
q = 950 kW
The heat transfer rate from the fire was calculated based on q = 21,000 x f x Ae0.82, as per API RP 521.
In FIG. 1, at constant relieving pressure and constant specific volume, multiple relieving enthalpy data are requested, considering the variation in temperature. This set of data is required to estimate maximum relief load to size the orifice of the PSV. The orifice is sized by considering the wide operating range of temperatures at constant relief pressure. The mass flowrate, volumetric flowrate and maximum orifice area are not found at the same temperature. A vertical straight line from constant relieving pressure to maximum backpressure represents a choked condition at the outlet of the orifice. The velocity through the orifice is sonic for these conditions, and the relieving rate is calculated accordingly. Sonic flow is typical for supercritical relief. The remaining four points in FIG. 1, along the backpressure line, are only needed when the orifice flow is subsonic.
SUMMARY OF METHODS
API 521 method based on the ideal gas equation. The derivations of equations used for calculating relief rate and orifice size in API-521 are based on the physical properties of air and ideal gas laws. The derivations assume that the vessel is uninsulated, has no mass, the vessel wall temperature does not reach rupture stress temperature, and there is no change in fluid temperature. The orifice area is calculated by using Eq. 1; however, this equation does not take credit for insulation:
F' can be determined using Eq. 2. If calculated using Eq. 2 and the result is < 182 SI units (< 0.01 in USC units), then use a recommended minimum value of F' = 182 SI units (0.01 in USC units). If insufficient information is available to use Eq. 2, then use F' = 821 SI units (0.045 in USC units).
The minimum relief load recommended for sizing when F' < 182 SI units is calculated by setting F' = 182 in SI units (Eq. 3):
Note: p1 is the set pressure plus the allowable overpressure plus the atmospheric pressure.
The relief rate and orifice area are calculated by API method derivations for components n-heptane and a mixture of n-nonane and n-decane. The results are computed in TABLE 1.
Rigorous real gas isentropic coefficient calculation method as an alternative to the ideal gas specific heat ratio for sizing relief valves. The pressure-relieving rate is derived from an unsteady-state analysis. Starting at the initial operating pressure and operating temperature, assume that the pressure increases via a constant-volume process until the relieving pressure is reached. This will be the starting condition for determining the relief rate and relief area. Perform constant pressure expansion using Eqs. 5–7, with the starting condition and the heat input over the time increment. As this procedure is intended to model a constant volume process, each step in the iteration ends with the removal of the incremental volume (as it is vented through the relief device); therefore, the term Vn remains constant and is equal to the initial volume V0.
Eq. 5 is used for calculating volumetric flowrate:
Eq. 6 is used for calculating mass flux:
G = m1/2 × ρ (6)
Eq. 7 is used for calculating integral:
mn+1 = k × [Pn – Pn+1 ] × [1 / ρn + 1 / ρn+1 ] + mn (7)
Ensure that the physical property correlation used is valid through the transition into the thermodynamic critical region. Various data points of temperature and fluid density are created to find the relief load by a series of flashes at constant relieving pressure. The simulation may be stopped at the maximum temperature limit of the vessel wall, as the vessel would fail if the temperature were higher. To narrow the range of temperatures investigated, an optional step is to determine the initial relieving temperature by matching the density at initial conditions to the density at relieving pressure by varying temperature. The required relief volume between any two steps is then the increase in volume of the fluid. The relief rate is then found by dividing the increase in volume by the time it takes to provide the enthalpy between the steps.
Once the peak load is found, those initial conditions are used to fill out the PSV ideal mass flux table. Then, a property table/heat curve, with isentropic expansion from the peak load temperature and pressure, is set up. The peak relief area occurs between the peak mass flow and peak volumetric flow. The area calculations for both peaks and several intermediate points are checked to ensure that the peak area requirement has been determined. The heat absorbed by the supercritical fluid is assumed to be the same as a wetted surface, though this is conservative. This method is analyzed for the mixture containing n-nonane and n-decane:
Calculate critical temperature and critical pressure using a simulator for the given component.
For the given component, calculate molar density at an estimated critical temperature and critical pressure.
A relieving temperature of 395.6°C is estimated by matching the molar density of the stream, which consists of the critical temperature (333.7°C) and pressure (2,215 kPa) with a simulated stream at relieving pressure (4,100 kPa) (constant volume process).
Fluid density and enthalpy data are constant at the relieving pressure (40 barg), and varying temperature is calculated to locate the maximum mass and volume flows. Results for the same are tabulated below in TABLE 2.
Several temperature points from the maximum mass flow to the maximum volume flow are chosen to locate the temperature for the maximum relief area.
For each relief temperature check, an isentropic flash is run, and the pressure is varied from relieving pressure (40 barg) to a backpressure of 3 barg. The pressure and density data of isentropic flash is used to calculate mass flux (refer to Eq. 6), and maximum mass flux is then used with the associated relief rate to calculate the minimum required orifice area. Results for the same are tabulated in TABLES 3–5 for 387°C and 404°C. The relieving temperature is 395.6°C, and the estimated maximum volumetric flow is at 404°C. This considers supercritical fluid where properties estimation using the ideal gas law is challenging, so that the orifice size is calculated for a higher volumetric flow rate, which is at 404°C. This is a conservative approach.
Similarly, the same steps are used to compute data for n-heptane. This data is detailed in TABLES 6–9.
Rigorous enthalpy-based calculation at choked pressure. This procedure consists of tabulating property data obtained from a process simulator onto a spreadsheet that calculates relieving rates and required orifice size. This method is analyzed for the fluid component pure n-heptane and for a mixture of n-nonane and n-decane. The following procedural steps are for a mixture of n-nonane and n-decane:
From a simulator program at a given relieving pressure, the relieving temperature is calculated using mass density estimated at critical pressure and temperature. This first point is defined as 5n on the relieving pressure line at the same mass density obtained at critical pressure and temperature.
For point 5n, enthalpy, entropy and Cp/Cv (k) ratio are calculated in the simulator, and choke pressure at the orifice is calculated by using Eq. 8.
If the choked pressure is greater than the backpressure (the backpressure is assumed to be 300 kPa), then the flow is sonic; therefore, for the corresponding point of the choked pressure’s physical properties (e.g., temperature, density), the Cp/Cv (k) ratio is recorded at constant entropy.
The next point is estimated by increasing and decreasing the temperature by 10°C until critical temperature is reached. A higher accuracy can be achieved with smaller temperature increments. Then, the volumetric flowrate, mass flowrate, orifice velocity, mass flux and orifice size are calculated using Eqs. 9–13.
Eq. 8 is used for calculating choked pressure:
Pch = Pn [ 2 / (1 + kn )](kn / (kn –1)) (8)
Eq. 9 is used for calculating sonic velocity:
vson = √kson gc (R / M)T (9)
Eq. 10 is used for calculating mass flux:
G = vson / V (10)
Eq. 11 is used for calculating volumetric flowrate:
Q = q(Vin,i + 1 – Vin,i ) / (hin,i+1 – hin,i ) (11)
Eq. 12 is used for calculating mass flowrate,
W = Q × ρ (12)
Eq. 13 is used for calculating orifice size:
A = W / (GKb Kc Kd Kv ) (13)
5. When vendor data is unavailable, the PSV discharge coefficient is 0.975, as recommended in the API RP-521 standard for preliminary sizing. The backpressure correction is considered to be 1 for supercritical relief, as the relieving pressure is high. The combination factor should also be 1. Orifice size is calculated for all data points. There will be a range of values to choose from for the orifice area; the maximum value should be selected for the design. It has been shown that the mass relieving rate, volume relieving rate and maximum orifice area are not necessarily found at the same temperature.
6. The highest orifice area configuration should be selected.
Supercritical relief valve sizing examples for n-nonane, n-decane and n-heptane are detailed in TABLES 10 and 11.
Results and discussion. The API method is based on extrapolation of the ideal gas law to the supercritical region. The orifice size for n-heptane and for a mixture of n-nonane and n-decane are “F” type. Similarly, the orifice sizes for n-heptane and for a mixture of n-nonane and n-decane for direct integration method are “F” and “G” types, respectively. The direct integration method uses integral pressure from operating pressure to PSV relief pressure. This leads to an estimation of thermodynamic properties at each pressure increment and sets fluid behavior near realistic.
The choked pressure method is based on the estimation of thermodynamic properties at choked pressure. The estimated orifice sizes for n-heptane and for a mixture of n-nonane and n-decane are “E” type and “D” type, respectively. Furthermore, choked pressure is a function of the specific heat and specific volume ratio at a given pressure. This ratio is more sensitive to change in pressure increments. The use of the ideal gas law at choked pressure for a fluid in supercritical regions to generate thermodynamic properties results in a wide variation in properties, and prediction of realistic fluid behavior in supercritical regions becomes a challenge.
Based on summarized results for three calculation approaches, the direct integration method is more reliable, as it estimates thermodynamic properties at regular pressure increments and sets fluid behavior near realistic.
Takeaways. This article summarizes various calculation approaches adopted to find relief rate or orifice size for supercritical fluids. All calculation approaches use the ideal gas law, either directly or by stepwise integration. Once the fluid crosses its critical temperature and pressure, it becomes highly unstable, and the estimation of physical properties using the ideal gas law becomes challenging. In such a scenario, users must adopt smaller steps to estimate physical properties and may omit temperature and pressure data points where physical properties are unusual. From the presented analysis, it was concluded that, as a conversative approach to keep the given system under a safe environment, the user may use a rigorous real gas isentropic coefficient calculation method. In this approach, the PSV orifice size might be one larger side, and may lead to chattering of the PSV; however, considering unstable fluid behavior in super critical regions, it is acceptable. HP
REFERENCES
API, “API 520 Part 1, 10th Ed.: Sizing, selection and installation of pressure-relieving devices in refineries,” API, online: https://www.api.org/products-and-services/standards/important-standards-announcements/520parti
API, “API 521: Pressure-relieving and depressurizing systems,” June 2020, online: https://www.api.org/products-and-services/standards/important-standards-announcements/standard521
Odenkirk, R., “Rigorously size relief valves for supercritical fluids,” CEP, August 2002.
Doane, R. C., “Designing for pressure safety valves in supercritical service,” Hydrocarbon Processing, January 2010.
Amit Ramesh Savant is a Principle Process Engineer for Worley. He earned an M.Tech degree in chemical engineering from the Institute of Chemical Technology in Mumbai, India. He has > 19 yrs of experience in refining, petrochemicals and chemicals industries in the areas of basic engineering, front-end engineering design and detailed design. Previously, he worked with Toyo Engineering Ltd. and GS E& C, Mumbai Ltd.
Shivani Kaustubh Chitale is a Process Engineer for Worley. She earned a BE degree in chemical engineering from Mumbai University, India, and has 3 yrs of experience in the petrochemicals and chemicals industries in the areas of basic engineering and detailed design.