Hydrocarbon Processing - August 2021 - 32

Valves, Pumps and Turbomachinery
tegration rule. A smaller step size decreases the truncation error,
resulting in a more accurate solution; however, more calculation
steps (numerous isentropic phase-equilibrium calculations) are
needed, so it is a time-consuming procedure. Using the higherorder
integration rule brings more complexities to the solution.
A few analytical solutions have been presented for an indiBased
on Eq. 1
and a pressure-specific volume correlation, for instance, a universal
mass flux equation (similar to the Omega method for critical
flow) and an equivalent critical pressure equation were applied
to predict mass flux for any fluid at any condition except flashing
flow of initially subcooled liquid.6
The pressure-specific volume
correlation had two parameters that can be calculated using one,
two or three data points. Although the models showed a good
accuracy for the mass flux and the critical pressure estimations,
they were limited to the fluid state and condition.
To avoid such issues related to numerical integration and to
present a universal model applicable to any fluid at any condition,
an alternative solution (analytical) is proposed to solve
Eq. 1. The main objectives of the presented work are to bring
all PSV sizing equations under one umbrella by using Eq. 1, to
reduce the number of isentropic process calculations to a few
data points, to solve Eq. 1 analytically by targeting the valve
throat pressure, and to calculate mass flux at the throat for sizing
a PSV. This requires that the density of the fluid with respect to
pressure variations can be expressed as a linear, a second-order
polynomial, or an exponential function with high accuracy. Additionally,
for each fluid state, a detailed example is provided to
show the applicability of the approach for any fluid and to provide
guidelines for PSV sizing.
Theory: An analytical approach. The approach provided is
easy to follow:
1. Determine the fluid state and properties, including
pressure, temperature and density at the relief
condition (Point 1).
2. Determine the total backpressure and the pressure
range (Pr
- Pb
).
3. Based on the authors' experiences, 3-6 data points
(including the relief condition) are enough to
recognize the trend of fluid density with respect to
pressure changes. Divide the pressure range into
approximately n equal increments (Eq. 4):
(Pr
- Pb)/n
(4)
where n would be 2, 3, 4 and 5, based on the number
of selected data points between 3 and 6. In this article,
three data points are selected to examine the proposed
model-in general, using more data points provides
a better trend line for the fluid density-pressure
relationship, leading to a more accurate model.
4. Decrease the relief pressure by one increment through
an isentropic process and find the fluid density at the
new state. In a proprietary simulatora
, for example,
use an expander model palette with 100% efficiency
to simulate an isentropic process and ensure that the
mass entropy remains constant for both initial and final
states. Repeat this step two times until the pressure
approaches the total backpressure.
32 AUGUST 2021 | HydrocarbonProcessing.com
Fr
1 =
5. Plot the three data points (density vs. pressure) and find
a proper trend for density as a function of pressure. Our
experiences show that the data can be fitted to a linear, a
second-order polynomial or an exponential function.
vidual fluid to overcome the numerical issues.1,6,8
6. The regression process can be calculated by many math
software tools (in Excel, for example, use a combination
of index and linest functions). Fit the data points to one
of the above functions and find all required constants,
as well as R2
. Although other statistical rules exist, R2
is sufficient to judge the validity of the curve fit.
7. Estimate the throat nozzle pressure from one of the
following equations, calculate corresponding mass flux,
and then size the PSV using Eq. 2.
< Pr. If Pb
8. In all cases 0 ≤ Pt
≤ Pt < Pr, then use Pt
as the throat pressure. Otherwise, total backpressure
should be used as the throat pressure.
A linear function. For liquids, the density as a function of
pressure usually can be expressed linearly (Eq. 5):
ρ = aP + b
G2 = −
( )
dPt
2,000
a
= 0
( aPt +b)2
⎛
⎝
ln⎜
aPt + b
aPr + b
⎞
⎠
⎟
To maximize the mass flux at the throat (Eq. 7):
d G2
(5)
where a and b are the equation constants. Substituting Eq. 5
into Eq. 1 and taking the integral of the function results in Eq. 6:
(6)
(7)
Applying Eq. 7 to Eq. 6 results in an explicit equation, which
relates the throat pressure to the relief pressure as (Eq. 8):
e−1/2
Pt =
(aPr + b) − b
a
Eq. 6 gives the maximum mass flux, Gmax
2
(8)
, at this pressure.
A second-order polynomial. If the fluid density-pressure
trend can be fitted to a second-order polynomial, then (Eq. 9):
ρ = aP2
+ bP + c
(9)
Substituting Eq. 9 into Eq. 1, taking the integration results in
three cases (depends on a, b and c coefficient values and signs)
to calculate the mass flux, and then applying Eq. 7 to the obtained
mass flux, the throat pressure can be targeted.
Case 1: 4ac - b2 < 0 (Eqs. 10, 11 and 12):
2 + bPt + c)2
G2 =
where
Ft
−2,000 aPt
(
b2 − 4ac
1 =
2aPt + b− b2 − 4ac
2aPt + b+ b2 − 4ac
2aPr + b− b2 − 4ac
2aPr + b+ b2 − 4ac
ln
⎛
⎝
⎜⎜
Ft
Fr
1
⎞
⎠
1 ⎟⎟
(10)
(11)
(12)
In this case, the throat pressure must satisfy the following
implicit equation (Eq. 13):
http://www.HydrocarbonProcessing.com
Hydrocarbon Processing - August 2021

Table of Contents for the Digital Edition of Hydrocarbon Processing - August 2021
