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Units in Pressure Relief Valve Calculator: barrel=42 U.S. gallons, cm=centimeter, cP=centiPoise, cSt=centiStoke, ft=foot, g=gram, gal=U.S. gallon, hr=hour, kg=kilogram, kPa=kiloPascal, lb=pound, m=meter, min=minute, mm=millimeter, N=Newton, psf=pound per square foot, psi=pound per square inch, s=second, SSU=Saybolt universal seconds.
Pressure Relief Valve Drawing (Spring) (with permission of Wikimedia Commons).
Topics: Pressure Relief Valve Equations Variables
Messages References
Introduction to Sizing Pressure Relief Valves for Liquids
Pressure relief valves are installed in nearly all refinery, chemical, and industrial facilities. Pressure relief valves are installed to protect equipment from failing due to overpressurization. The calculation on this page is specifically for sizing and flow rate determination for rupture disks, spring pressure relief valves, and pilot operated pressure relief valves, or valves that have a rupture disk in combination with a spring or pilot operated relief valve for the flow of liquids. Balanced bellows valves can be spring or pilot operated and are specified where backpressure is expected.
The equation methodology on this web page mostly follows the American Petroleum Institute standard 520 for pressure relief valve sizing for liquids in order to protect equipment having a maximum allowable working pressure of 15 psig (103 kPa gage) or greater (API, 2014, p. 1). For an orifice in a pipe (rather than in a relief valve), LMNO Engineering has written additional software based on ISO (International Organization for Standardization) methodologies. The calculations can be seen by clicking on the links in the yellow background on the right side of this page.
LMNO Engineering's pressure relief valve for liquids calculator can compute relief valve area or flow rate. If area is computed, the area will be shown for the inputs that you enter. After viewing the area, refer to a pressure relief valve manufacturer's catalog and select a relief valve having the area computed or, if such a pressure relief valve is not available, the next larger area. Then, rerun our calculator using the manufacturer's actual area to compute the liquid flow rate through the pressure relief valve.
Equations for Pressure Relief Valve Sizing for Liquids Back to calculator
The following methodology is used to compute relief valve area or flow rate.
Discharge coefficient, K_{d}
The following guidance is from API (2014, p. 76):
If there is a pressure relief valve with or without a rupture disk, enter K_{d} of 0.65.
If there is only a rupture disk, enter K_{d} of 0.62.
If there is a value provided by the valve manufacturer, enter the manufacturer's value for K_{d}.
Combination correction factor, K_{c}
Guidance from API (2014, p. 76):
If there is no rupture disk with the pressure relief valve, enter K_{c}=1.0.
If there is only a rupture disk (no relief valve), enter K_{c}=1.0.
If there is a rupture disk and a pressure relief valve, enter K_{c}=0.9.
If there is a value provided by the valve manufacturer, enter the manufacturer's value for K_{c}.
Upstream relieving pressure, P_{1}
The upstream relieving pressure (expressed as a gage pressure) is computed from the set pressure (expressed as a gage pressure) and overpressure percent (API, 2014, p. 76; Crowl & Tipler, 2013, p. 73):
P_{1} = P_{s} (1+ 0.01P_{o})
Percent gage back pressure, P_{g}
The percent gage back pressure is computed from the back pressure (expressed as gage) and set pressure (expressed as gage) (API, 2014, p. 49 Figure 31; Crowl & Louvar, 2011, Figure 104):
Back pressure correction factor, K_{w}
K_{w} is set automatically by the LMNO Engineering calculation based on your selection of "Spring and pilot operated valves" or "Balanced bellows valve." The following guidance is from API (2014, p. 76, Figure 31):
For spring and pilot operated relief valves, K_{w}=1.0 (set by LMNO Engineering calculation by selecting "Spring and pilot operated valves").
For balanced bellows relief valves, a graph for K_{w} is provided in API (2014, p. 49, Figure 31). Crowl & Louvar (2011, Figure 104) provide an equation fit to the API graph. The equation is:
K_{w} = 1.165  0.01P_{g}
The equation is valid in the range 16.5% ≤ P_{g} ≤ 50%. Thus, in LMNO Engineering's calculation above when a balanced bellows valve is selected, K_{w} is:
If P_{g} < 16.5%, K_{w}=1.0.
If 16.5% ≤ P_{g} ≤ 50%, the K_{w} equation is used.
If P_{g} > 50%, K_{w} is not defined.
Graphically, the equation for K_{w} for a balanced bellows pressure relief valve is:
Viscosity correction factor, K_{v}
K_{v} is a function of Reynolds number. Reynolds number in general is defined in terms of mass density, velocity, diameter, and dynamic viscosity (Munson et al., 1998, p. 462):
Velocity in terms of flow rate and effective discharge area, as well as area in terms of diameter:
In the pressure relief valve literature, Reynolds number is often written in terms of area. Thus, making the above substitutions:
API (2014, p. 76) provides an equation for K_{v} for 0.3 ≤ K_{v} ≤ 1:
A graphical representation of K_{v} developed from the equation is:
Thus for K_{v} in LMNO Engineering's pressure relief valve calculator:
If R < 32.662, K_{v} is not defined.
If 32.662 ≤ R ≤ 192,282.5, the equation above for K_{v} is used.
If R > 192,282.5, K_{v}=1.
Computation of pressure relief valve area and diameter
Equations for area and diameter of the pressure relief valve written in general with Kfactors (similar to Crowl & Tipler, 2013, p. 73; Crowl & Louvar, 2011, Eqn 102; API, 2014, p. 76):
Pressure relief valve flow rate equation:
Since either area or flow rate is to be computed, Reynolds number is not known directly. Thus, an iterative solution is used which converges when the percent change in area or flow rate (whichever is being computed) is less than 10^{10}.
Variables for Liquid Relief Valve Sizing and Flow Back
to calculation
Units shown in SI for use in equations: kg=kilogram, m=meter, N=Newton, s=second (the calculator allows a variety of units).
A = Effective discharge area, m^{2}.
D = Diameter, m.
g = Acceleration due to gravity = 9.8066 m/s^{2}.
K_{c} = Combination correction factor.
K_{d} = Effective discharge coefficient.
K_{v} = Viscosity correction factor.
K_{w} = Back pressure correction factor.
P_{g} = Percent gage backpressure, %.
P_{o} = Percent overpressure (gage), %.
P_{s} = Set pressure, N/m^{2} gage.
P_{1} = Upstream relieving pressure, N/m^{2} gage.
P_{2} = Total backpressure, N/m^{2} gage.
Q = Liquid flow rate, m^{3}/s.
R = Reynolds number.
V = Velocity, m/s.
π = 3.1415926...
μ = Dynamic viscosity, kg/ms (i.e. Ns/m^{2}).
ρ = Mass density, kg/m^{3}.
LMNO Engineering uses the viscosity unit conversion from SSU to m^{2}/s of 1 SSU = 2.2e7 m^{2}/s. However, if viscosity in SSU is less than 100 SSU, the relation between SSU and m^{2}/s is significantly nonlinear. Therefore, the unit conversion is not valid, and a different viscosity unit should be selected. Additional discussion of SSU can be found in Weast (1984, p. F36).
Pressure Reief Valve for Liquids Messages Back
to calculator
Initial checks:
"Need A > 0",
"Need D > 0",
"Need P_{s} > 0",
"Need P_{2} ≥ 0",
"Need 1e30 ≤ Q ≤ 1e30 m^{3}/s",
"Need 0 ≤ P_{o} ≤ 100",
"Need 1e9 ≤ Density ≤ 1e9 kg/m^{3}",
"Need 1e19 ≤ Viscosity ≤ 1e9 m^{2}/s",
"Need 0 < K_{c} ≤ 1",
"Need 0 < K_{d} ≤ 1".
Runtime messages:
"Invalid input: P_{1} will be ≤ P_{2}",
"P_{g} out of range",
"R out of range: Q too small",
"R out of range: A too small",
"Solution not found".
References for Pressure Relief Valve Calculation for Liquids Back
to calculation
American Petroleum Institute. (2014, July). Sizing, selection, and installation of pressurerelieving devices in refineries. Part I  Sizing and selection. API Recommended Practice 520. 9ed.
Crowl, D. A. & Louvar, J. F. (2011). Chemical process safety: Fundamentals with applications. 3ed. Prentice Hall.
Crowl, D. A. & Tipler, S. A. (2013, Oct.). Sizing pressurerelief devices. CEP. American Institute of Chemical Engineers. Retrieved from
https://www.aiche.org/sites/default/files/cep/20131068_r.pdf
Munson, B. R., Young, D. F. & Okiishi, T. H. (1998). Fundamentals of fluid mechanics. 3ed. John Wiley & Sons, Inc.
Pentair Valves and Controls. (2012). Pentair pressure relief valve engineering handbook. Retrieved from
https://www.iomosaic.com/diersweb/docs/pvcmc0296us_tcm10635825.pdf
Weast, R. C. (Ed.) (1984). Handbook of chemistry and physics (65ed.). Chemical Rubber Company (CRC), Inc.
Wikimedia Commons. Author: Milton Beychok (Mbeychok). Retrieved from https://commons.wikimedia.org/wiki/File:Relief_Valve.png
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