Pressurerelief valves
are reliable and effective pressurerelief devices that protect
personnel from the dangers of overpressurizing equipment,
prevent damage to equipment, and minimize release of hazardous
materials. Sizing relief valves involves determination of the
rate of material release through the relief valve during the
identified worstcase contingency. Relief valves are designed
to relieve liquids, vapors or two phases from protected
pressure vessels before excessive pressures are developed. A
mistake in the reliefvalve sizing can result in catastrophic
failures because relief valves are usually the last defense to
the process equipment against instrument failures, process
upsets and operator errors.
This article focuses on sizing the pressurerelief valves
for critical flow of gases or vapors. Twophase sizing
methods^{1,2} are not well established, but it is
generally understood that sizing methods for vapor or gas are
well established and the results are relatively accurate.
Reliefsystem designers favor simple sizing equations, and use
the conventional vaporsizing equation in American Petroleum
Institute (API) Standard 520, using the ideal gas specific heat
ratio.^{3} Although rigorous calculations using
isentropic flash calculations give the most accurate results,
the simple API reliefvalve sizing equation is still preferred
because of its simplicity.
However, sometimes the real gas specific heat ratio is more
readily available from a process simulator than the ideal gas
specific heat ratio. Thus, a simple sizing equation using the
real gas specific heat ratio was developed. Emerging from that
development is an improved sizing equation using the real gas
specific heat ratio. The results are compared with the
conventional API sizing equation. Furthermore, a rigorous
criticalflow equation with the isentropic flash is introduced
as a recommended estimation tool for gas or vapor critical flow
when the gas or vapor is known to deviate significantly from
ideal conditions.
Conventional API flow equation.
Eq. 1 is the conventional API reliefvalve sizing equation
for critical vapor flow. The conventional API sizing equation
requires five fluid property data: absolute pressure,
P; kPa, absolute temperature, T; K, molecular
weight, M, compressibility factor, Z, and
ideal gas specific heat ratio, k_{*}, at inlet
conditions. The equation provides satisfactory sizing results
over a wide range of process conditions. However, the sizing
equation is only valid for 0.8 < Z <
1.1.^{3} This means that the sizing results may not be
satisfactory for very high pressure conditions or
criticalpoint regions.
where: (1)
(2)
where:
A = Relief valve orifice area, mm^{2
}W = Mass flow rate, kg/h
C_{k*} = A function of the ratio of ideal gas
specific heat at inlet conditions
K_{d} = The coefficient of discharge
K_{b} = The capacity correction factor due to
backpressure
K_{c} = The combination correction factor for
installation with a rupture disk upstream
Subscripts
_{1} = Fluid conditions at the inlet of the relief
valve, where velocity is equal to zero
_{choke} = Choked (critical) conditions.
Real gas specific heat ratio.
The new equation uses the real gas specific heat ratio in
the sizing equation instead of the ideal gas specific heat
ratio. In order to figure out what assumptions are needed to
use the gas specific heat ratio as an isentropic expansion coefficient, it is
required to check with Eq. 3. This equation is one of the
widely used methods for calculating the isentropic expansion
coefficient where rigorous reliefvalve sizing is deemed
necessary. The equation is based on the assumption that the
isentropic expansion coefficient is constant. Although the
isentropic expansion coefficient is actually not constant
during the expansion process, the sizing results with the
isentropic expansion coefficient are relatively good. The
derivatives in Eq. 3 for the RedlichKwong and PengRobinson
equations are readily available in the
literature.^{4,5}
The PengRobinson equation of state appears to be the most
favorable with the SRK equation. Eq. 4 shows the derivative of
pressure with respect to specific volume, v;
m^{3}/kg, at constant temperature and constant
compressibility factor. The gas specific heat ratio becomes the
isentropic expansion coefficient (n) when the
compressibility factor is constant. Of course, the real gas
specific heat ratio, k, will be the ideal gas specific
heat ratio, k_{*}, if the compressibility
factor is 1. In conclusion, it is required to assume the
constant compressibility factor when developing a sizing
equation with the real gas specific heat ratio.
(3)
(4)
where:
Pv = ZRT
where R is a universal gas constant, 8.314
kPam^{3}/kgmoleK.
Eq. 5 is easily found in textbooks.^{6} When ideal
gases are expanded, they follow Eq. 5. However, Eq. 6 is for
real gases provided that the compressibility factor is
constant. Eq. 6 explains why the inlet compressibility factor
is not to be included in Eq. 7. The real gas specific heat
ratio already accounts for the value of the compressibility
factor and nonideality at high pressure conditions. Therefore,
when the real gas specific heat ratio is used in the sizing
equation, the compressibility factor is not necessary. If one
uses the real gas specific heat ratio in Eq. 1, the
compressibility factor will be accounted for twice. This may
result in inadequate relief valves, as addressed in
API520.^{7} A statistical analysis shows that about 7%
of the equipment in the oil, gas and chemical industries had
pressurerelief devices undersized.^{8}
Pv^{k}^{*} =
constant
(5)
(6)
The authors developed Eq. 7 for reliefvalve sizing for
critical vapor flow with the real gas specific heat ratio. The
simple equation follows fundamental thermodynamic rules.
where:
(7)
(8)
where C_{k} is a function of the ratio of real
gas specific heat at inlet conditions.
Although Eq. 7 may be satisfactory for criticalflow
reliefvalve sizing, the critical pressure prediction is not
sufficiently accurate. In order to predict the accurate
critical pressure, Eq. 9 is derived based on Eq. 7 being equal
to Eq. 10. Instead of solving Eq. 9 for the isentropic
expansion coefficient with the real gas specific heat ratio and
compressibility factor, Eq. 12 fits well the correlation among
the compressibility factor, real gas specific heat ratio and
isentropic expansion coefficient over the range of 0 <
n < 2.5.
(9)
Finally, Eq. 10 can be obtained as a criticalflow sizing
equation with the compressibility factor and isentropic
expansion coefficient. Eq. 7 and Eq. 10 are identical, and the
two equations give the same sizing results. However, Eq. 11
predicts better choked pressures than Eq. 8.
where:
(10)
(11)
(12)
(13)
a = 4.8422E5 b = 1.98366 c = 1.73684 d = 0.174274 e =
1.48802
where C_{n} is a function of the isentropic
expansion coefficient at inlet conditions.
Isentropic flash.
Eq. 14, which requires a few iterations in an isentropic
flash routine, is the most accurate sizing equation. The
rigorous method uses the best predictions of the actual fluid
properties since the calculated isentropic expansion
coefficient is constant between two data points. The first
trial of isentropic flash can start with an initial estimate of
choked pressure at 55% of the inlet pressure. The choked
condition is usually attained when the downstream pressure is
about 45% to 65% of the inlet pressure. Repeat the isentropic
flash with the new estimate until it stops changing. In case of
liquid formation during the isentropic flash, the overall
specific volume of the fluid has to be used in Eq. 15.
Otherwise, it is better to use the temperature and
compressibility factor in Eq. 15 to indicate that there is no
condensation during expansion. The calculation details
will be illustrated in the “example calculations”
section. However, the equivalent results can be obtained using
numerical integration with numerous flash
calculations. The intensive numerical integration method is presented in
API Standard 520.

where: (14)


(15)
where C_{r} is a function of the rigorous
isentropic expansion coefficient, and Subscript 2 refers to
fluid conditions at the outlet of the relief valve (at the
nozzle throat).
Comparison.
The predictions of the simple flow equations have been
compared with the most accurate estimates for the following two
cases at six different pressures. The two cases include high
pressures and criticalpoint regions to evaluate the
limitations of the new simple method. The PengRobinson
equation of state was used for the estimation of necessary
fluid properties.
Case 1: Air at 250 K. The first case
considers the discharge of air at 250 K that is relieving at
six different pressure levels of the inlet reduced pressure
from 0.5 to 8. The inlet compressibility factors are in the
range of 0.9 < Z < 1.1. Fig. 1 shows the mass
flux (W/A) deviations of Eqs. 1 and 7 from Eq. 14.
Here, the compressibility factor decreases during expansion.
Therefore, both of the simple sizing equations appear to
oversize the relief valves. The difference between the
compressibility factors increases with increasing the inlet
reduced pressure. Fig. 1 shows that both simple equations give
conservative estimates as expected. The results of both
equations are satisfactory at low pressures. However, if the
vapor or gas is at high pressure and low temperature, one
should use them with caution. Generally, the vapor or gas tends
to behave ideally at high temperatures.

Fig.
1. Calculation results for highpressure
air
at 250 K (Case 1). 
Case 2. Saturated nhexane vapor. The
second case considers the discharge of a pure
componentsaturated vapor (nhexane) that is relieving at six
different pressure levels of the inlet reduced pressure from
0.15 to 0.9. The inlet compressibility factors are in the range
of 0.4 < Z < 0.9. Here, the compressibility
factor increases during expansion. The difference between the
compressibility factors increases with increasing the inlet
reduced pressure. Therefore, the simple sizing equations appear
to undersize the relief valves. Fig. 2 shows that the
conventional simple API equation gives profoundly
unconservative estimates as expected. Unlike the conventional
simple equation, the results of Eq. 7 or Eq. 10 are
satisfactory up to the inlet reduced pressure of 0.75. However,
if the inlet reduced pressure is greater than approximately
0.7, one should use it with caution.

Fig.
2. Calculation results for saturated
nhexane vapor under critical conditions (Case
2). 
Examples.
A relief valve should release 10,000 kg/h of saturated
nhexane vapor (M = 86.18) at a relief pressure of
1,807.38 kPa (inlet reduced pressure of 0.6) and a relief
temperature of 474 K. The compressibility factor at the
conditions is 0.6279. Calculate the required actual orifice
area based on K_{d} = 0.877,
K_{b} = 1 and K_{c} = 1. The
authors used a process simulator with a selection of the
PengRobinson equation of state to obtain the necessary fluid
properties.
Calculations with Eq. 14. The results of
isentropic flashes for Eq. 14 that were obtained from a process
simulator are summarized in Table 1. Iterations were stopped
after four trials, as the new choked pressure was close enough
to the old one. The best estimate of required orifice area is
556 mm^{2}.
Calculations with Eq. 1. Using the ideal
gas specific heat ratio along with the inlet compressibility
factor gives the required orifice area of 482 mm^{2},
which is much smaller than the best estimate of 556
mm^{2}.
Calculations with Eq. 7. Using the real gas
specific heat ratio gives the required orifice area of 567
mm^{2}.
Calculations with Eq. 10. Eq. 10 is
technically identical to Eq. 7. This method will give a better
estimate of the choked pressure than the Eq. 7 method.
All calculation results are summarized in Table 2 for
comparison. The required orifice area estimated by the API
sizing equation is not satisfactory as expected, and this may
result in an undersized relief valve. On the other hand, the
new method by Eq. 7 or Eq. 10 produces satisfactory
results.
Recommended usage.
Proper sizing of a relief valve requires not only using an
accurate criticalflow equation, but also using accurate fluid
properties. The simple sizing equation for critical vapor flow
using the real gas specific heat ratio has been tested on two
cases: air at higher pressures and nhexane under critical
conditions. Although in both cases the conventional API
equation is difficult to properly size pressure relief devices,
the new approach results in a significant improvement. It is
also important to note that the compressibility factor should
be removed in the simple sizing equation where the real gas
specific heat ratio is used as an isentropic expansion coefficient. However,
reliefsystem designers should be careful when sizing the
pressure relief devices for vapor or gas at critical regions or
high pressures where the gas or vapor deviates significantly
from the ideal conditions.
Any simple equation involves assumptions that generally
introduce errors. As demonstrated in the example calculations,
the rigorous flow equation with isentropic flash is not so
complicated and difficult to use. The rigorous flow equation
with no assumptions uses the most accurate fluid properties.
Therefore, using the rigorous flow equation with isentropic
flash is recommended when the gas or vapor is known to behave
significantly nonideally. HP
LITERATURE
CITED
^{1} Darby, R., “Evaluation of twophase flow
models for flashing flow in nozzles,” Process Safety
Progress, Vol. 19, No. 1, pp. 32–39, 2000.
^{2} Diener, R. and J. Schmidt, “Sizing of
throttling device for gas/liquid twophase flow, Part 1: Safety
valves,” Process Safety Progress, Vol. 23, No.
4,
pp. 335–344, 2004.
^{3} “Sizing, selection and installation of
pressurerelieving devices in refineries,” API Standard
520, Part I—Sizing and selection, December 2008.
^{4} “Guidelines for pressurerelief and
effluenthandling systems,” Center for Chemical Process
Safety (CCPS), AIChE, New York, 1998.
^{5} Pratt, R. M., “Thermodynamic properties
involving derivatives using the PengRobinson equation of
state,” Chemical Engineering Education,
pp. 112–115, Spring 2001.
^{6} Crowl, D. A. and J. F. Louvar, Chemical
Process Safety: Fundamentals with Applications,
PrenticeHall, Englewood Cliffs, New Jersey, 1990.
^{7} Shackelford, A., “Using the ideal gas
specific heat ratio for reliefvalve sizing,” Chemical
Engineering, pp. 54–59, November 2003.
^{8} Patrick, C., R. A. Kreder and W. Lee,
“Analysis identifies deficiencies in existing
pressurerelief systems,” Process Safety
Progress, Vol. 19, No. 3,
pp. 166–172, 2000.