Hydrocarbon Processing - July 2022 - 25

Valves, Pumps and Turbomachinery
Critical pressure can also be obtained from the following
approximations (Eqs. 11 and 12) developed based on the
Omega method.
pc = nc
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
nc =
⎡
⎣
⎢
⎢
β−0.727144
×p0 + 1−nc
(
1−2 × α × β+
1−
⎡
⎣
⎢ 1− 1−
⎡
⎣
⎢1−
psc
p0
×
⎛
⎜⎝
β−0.727144
p0
psc
× 1 + 2 × α × β) × ⎞
) × p0 ×
(
2 × α × β−1
2 × α × β
(
⎤
⎦
2 × α × β⎥
⎞
⎟⎠
⎤
⎦
⎥
1−2 × α × β+ 1 + 2 × α × β)×
1
⎝
1 + 1.0446−0.0093431 × α0.5) ×
α−0.56261
(
All values of nozzle pressure and Pec
⎤
⎦
⎥
⎥
(
−0.70356 +
0.014685 × ln α
)
(12)
, along with mass flux,
are plotted in FIG. 1 to tangibly illustrate the relations among
nozzle pressure, Pec
, Pc and mass flux for API 520 Part 1, Example
B.3.3. The green line shows the subcritical mass flux
calculations for a known backpressure (at P, greater than the
critical pressure). In subcritical flow, there is no iteration. Because
Eq.9 provides a result of equivalent critical pressure at
the integrated mass flux from initial pressure to the backpressure,
the calculated Pec
lation. The calculated Pec
is directly used for the mass flux calcuvalue
shall be less than the known
backpressure (P).
The red line shows the critical mass flux calculations for a
known backpressure (at initial P, the backpressure is not greater
than the critical pressure). In critical flow, the calculations
are repeated until the calculated Pec
flux and Pec are maximized at the critical pressure (Pc
critical flow region, the maximum mass flux is taken at Pc
is equal to P. The mass
). In the
because
a further decrease in nozzle pressure does not affect the
mass flux. The calculated Pec
also decreases as the P is further
decreased below the critical pressure.
The HDI method uses only one blue graph. The upgraded
integration method uses two graphs (blue graph and gold
graph). The two graphs intersect at Pc
. Using the two graphs
significantly improves obtaining the critical pressure point
quickly and accurately. Eq. 9 is used to calculate both the Pec
for subcritical mass flux and the Pec-max
(Pc
) for critical mass
flux. The upgraded integration method quickly determines if
the flow is critical or subcritical. However, the HDI method
should complement the Eq. 2 calculations to determine if the
flow is critical or subcritical.
Isentropic expansion coefficient. Estimating the isentropic
expansion coefficient during the expansion process is
not simple because it is a function of pressure, volume and
the ratio of real gas-specific heats, as shown in Eq. 13. It also
varies throughout the expansion process, during which condensation
may occur. The isentropic expansion coefficient for
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
1
C = 520 ×
×
α × β
⎛
⎜⎝
pc
p0
⎞
⎟⎠
β+1
(15)
Kim proposed Eqs. 15-17 at the API SCPRC Meeting.4
Eq. 15 is derived from Eqs. 1 and 7.
The polynomial Eqs. 16 and 17 are used to estimate the
constant isentropic expansion coefficient for two different C
ranges. The C value is a result of the most rigorous calculation
method. This may be an outstanding solution for constant isentropic
expansion coefficients. This approach allows a simple
vapor equation to be extended to any compressible fluid,
including slightly subcooled liquids because Eq. 1 is based on
Eq. 4. Therefore, a simple vapor equation can be applied to
any compressible fluid, not only to ideal vapors.
However, n should be a constant isentropic expansion coefficient
obtained from Eq. 14 to be compatible with a simple
vapor equation. The constant isentropic expansion coefficient
reflects the assumption of ideal vapor relationship in Eq. 14
since it is simplified for ideal vapors with the ratio of the ideal
gas-specific heats; therefore, the constant isentropic expansion
coefficient is different from the average isentropic expansion
coefficient. For a non-ideal vapor, an average isentropic expansion
coefficient can result in undersized PRVs. For an ideal vapor,
an ideal gas-specific heat ratio at the relief inlet conditions
can be used as a constant isentropic expansion coefficient.
(11)
non-ideal vapors at the relief conditions is often used because
it is readily available. A constant isentropic expansion coefficient
can be easily back-calculated using Eq. 14 if the C value
is known. Eq. 14 can be expressed in terms of two-parameters
in a P-v model, critical pressure and initial relief pressure, as
shown in Eq. 15. The C value can be calculated by other methods
if the mass flux result is validated. Eq. 15 is recommended
as the calculation method of choice as a result of the reduced
P-v model evaluation.
n = −
v
P
×
⎛
⎜⎝
∂p
∂v
C = 520 × n ×
⎞
⎟⎠T
⎛
⎝
×
2
n+1
Cp
Cv
⎞
⎠
n+1
n−1
(13)
(14)
FIG. 1. Plot of nozzle pressure, Pec
and mass flux.
Hydrocarbon Processing | JULY 2022 25
Hydrocarbon Processing - July 2022

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