Hydrocarbon Processing - July 2022 - 24

Valves, Pumps and Turbomachinery
G2 =
⎡
⎣
⎢
⎢
−2 × ∫p0
p
2
vt
v × dp
⎤
⎦
⎥
⎥
where,
G = the mass flux through the nozzle, lb/sec-ft2
P = the pressure of the fluid, psia
v = the specific volume of the fluid, ft3
/lb
t = the throat of the nozzle.
The calculation precision associated with the HDI method
is determined by the size of the pressure intervals, with smaller
pressure intervals providing better results. Some cases may
require many data sets for acceptable sizing results. The HDI
method requires intensive computational efforts; however, the
numerical integral result can also be obtained accurately and
quickly if evaluated analytically with pressure-specific volume
(P-v) models or employed P-v models.
Reduced P-v models. The equation of state (EOS) typically
involves four variables. Simpson presented a total of six reduced
P-v models involving only two variables.2
model is obtained by a few constant entropy flash calculations,
for which the specific volume reflects changes in the compressibility
factor, pressure and temperature. The reduced P-v model
based on the homogeneous equilibrium model (HEM) well
represents the pressure expansion processes in the PRV.
This article evaluates two out of the six P-v models and a
conventional model in a simple vapor equation to select the
best P-v model for isentropic expansion flow. Eq. 4 is good
for ideal vapors, but not suitable for flashing liquids. Eq. 5, the
Omega method, is good for flashing liquids, but not so suitable
for vapors. The two-point Omega method is most widely used
for sizing two-phases. Eq. 6 is a combination of Eqs. 4 and 5, so
it should be suitable for both vapors and two-phases. Simpson
noted that Eq. 6 is the only reduced P-v model that gave consistently
outstanding fits of the data.
v = v0 ×
v
− 1 = ω ×
v0
v
v0
⎛
⎜⎝
p0
p
⎞
⎟⎠
1
n
⎡
⎢⎣
⎡
⎣
− 1 =α × ⎢
⎛
⎝
P0
P
P0
P
− 1
⎞
⎠
β
⎤
⎥⎦
⎤
⎦
−1⎥
where,
n,ω,α,β = the P-v model constants.
For subcooled liquids, the initial pressure (P0
(4)
(5)
(6)
The reduced P-v
G = 68.07 ×
⎡
Pec =
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
−2 × α × β × P0
⎣ ( 1−α) × p−p0
⎛
⎝
⎜
⎜
⎜
)+
(
( p0 −psc
α × p0
1−β
)
where,
ec-max = the critical pressure (Pc
equivalent pressure
), maximum
ec = the equivalent critical pressure at the calculated mass flux
sc = the subcooled liquid.
Eq. 10 has been developed to determine if a subcooled liquid
is compressible or incompressible. Eq. 10 is similar to the
API 520 Part 1 Equation (C.32), but easier to follow. If the subcooled
pressure is greater than the calculated Pec-max
, the fluid is
incompressible; the fluid is not considered a slightly subcooled
liquid because there is no change in density during the expansion
process. In this case, the fluid should be sized using a liquid
equation and the fluid chokes at the saturated pressure. If the
subcooled pressure is not greater than the calculated Pec-max
, the
fluid is compressible. In this case, the fluid should be sized as a
two-phase fluid:
) shall be at the
saturated pressure, ignoring subcooled conditions. Eq. 5, a twopoint
method, uses two data sets at initial pressure and typically
90% of the initial pressure. Eq. 6, a three-point method, uses
three data sets at initial pressure, a middle pressure and 50% of
the initial pressure (not greater than the critical pressure).
Upgrading the numerical integration method. A new
approach to make the rigorous HDI method more convenient
is discussed here. To reduce significant computational efforts,
24 JULY 2022 | HydrocarbonProcessing.com
Psc−max = p0 +
p0
2 × α × β
For subcritical mass flux, use Eqs. 8 and 9. The Pec
(10)
in Eq. 8 is
in
replaces
calculated using Eq. 9 at P (typically at backpressure) without
iteration. For critical mass flux, use Eqs. 7 and 9. The Pec-max
Eq. 7 is also calculated using Eq. 9 at P (initially at backpressure)
with iteration until Pec
is equal to P. The calculated Pec
P in the next trial. The maximum Pec value (Pec-max
when Pec
is equal to P.
) is obtained
β
pec
β+1
α × β × p0
β × v0
1
β× α ×
⎡
⎣
⎢
⎢
× p1−β− p0
1−β
(
⎛
⎜⎝
p0
p
⎞
⎟⎠
β
− α+ 1
) +
⎞
⎠
⎟
⎟
⎟
⎤
⎦
⎥
⎥
−2
×
⎤β+1
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Kim, et al. proposed Eqs. 7-9, which are based on Eq. 6.3
(3)
A compressible
fluid reaches a condition of maximum flow when the
velocity at the nozzle reaches the speed of sound. The isentropic
maximum mass flux for Eq. 2 is given by Eq. 7. Eqs. 7 and 8
are in an identical derivative form.
An identical and simple equation form for both critical mass
flux and subcritical mass flux is preferred to simplify the calculation
method. This allows for an analytical solution for mass flux
calculations to make the numerical integration easier. Eqs. 2 and
3 are in an identical integral form rather than derivative. The
integral form does not allow the analytical solution for mass flux
calculations. Eq. 9 is derived from Eqs. 3 and 8, and has been
upgraded here to handle slightly subcooled liquids. Eq. 9 simplifies
the numerical integration and provides rigorous results.
Both Eqs. 7 and 8 provide the results of P-v equation-based integration.
This new method is applicable to compressible fluids
(vapors and two-phases, including slightly subcooled liquids):
Gc =
⎛
⎜⎝
−∂P
∂v
⎞
⎟⎠s
= 68.07 ×
pec−max
β+1
α × β × p0
β × v0
(7)
(8)
(9)
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