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Phase redistribution problem
As a three-phase benchmark, phase redistribution [9] represents a test to verify the numerical
convergence of multiphase codes. In this case, we consider a closed column of 2.5 m height and
0.1 m in diameter. The system is uniformly filled with a three-phase mixture composed of 0.2, 0.4
and 0.4 gas, oil and water volume fraction, respectively. We start from this an unphysical initial
condition and let the system reach the equilibrium condition and separate the light and heavy phases.
It is interesting to visualize the time-evolution of the phase fraction field and pressure field.
Figure 7: pressure distribution (left) and gas distribution profile (right)
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Figure 8: oil (left) and water (right) fraction profile
From figure 7 and 8, we can conclude that the algorithm correctly converges in 400 s to the solution
of perfect phase separation. This is visible not only analysing the phase distribution at the latest
instant but also considering the pressure distribution that progressively converges to the well known
static solution (figure 7a). Regarding the phase fraction distribution, we can notice that gas fraction,
due to the limited density compared to liquids, tend to separate immediately by forming a single-
phase domain within 10 s of simulation (figure 7b). For the remaining part of the simulation, the three-
phase settling reduces to a two-phase separation process between oil and water. This is quite visible
observing the specular nature of phase fraction profile for water and oil at t=250 s (figure 8). Finally,
at t=400 s, each phase forms the correct hold-up according to its own initial composition.
SEPARATOR TESTS
Finally, we present the results of the simulation for a geometry that reproduces more realistically the
separator layout. In this case, a mixture composed of gas, oil and water is provided from the inlet
nozzle located at the top of the separator (phase properties are reported in table 1).
Phase
Density [kg/m
3
]
Viscosity [Pa
⋅
s]
𝛼
𝑖𝑛𝑙𝑒𝑡
[-]
𝑣
𝑖𝑛𝑙𝑒𝑡
[m/s]
Air
1.2
1.84 ⋅ 10
−5
0.90
2
Oil
800
3.64 ⋅ 10
−2
0.05
2
Water
1000
3.94 ⋅ 10
−4
0.05
2
Table 1: Phase properties
Three outlets for three phases are located at different heights with prescribe boundary conditions for
the modified pressure in order simulate the control level techniques. The whole domain is initially
filled with the gas phase.
Since the gas separates immediately from the liquids stream, the focus is on the effectiveness of
separation of the mixture oil-water. We report the phase distribution of oil at different time-instant (in
figure 9):
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Figure 9: the first three figures represent the oil distribution at 50, 100, 200 sec. The last one represent all three-phase
completely separated at 500 s (alphas field) with water (red), blue (oil), grey (air)
The inlet mixture separates in the three main components with an efficiency of 99 % within 500 s
(around 8 min). Following the simple rule of thumb of API standard [10] (in this case, the oil gravity
is 45°API), the retention time is in the correct range for effective separation.
Figure 10: Sketch of the separator vessel to highlight the zone where liquids separate from gas
In figure 10, it is reported the zone where it is assumed that occurs most of liquids separation from
the gas stream. According to the industrial validation process, this area is sized using the Stokes law
that provides the terminal velocity of single droplet immersed in an infinite medium at rest:
𝑉
𝑡,𝑘𝑗
= √
4𝑔𝑑
𝑃
(𝜌
𝑘
− 𝜌
𝑗
)
3𝜌
𝑘
𝐶
𝑑,𝑘𝑗
Clearly, the assumptions behind the Stokes law are generally not satisfied inside the separator where
the convection processes represents a continuous obstacle to separation. Moreover, the terminal
velocity is generally not uniform along the vessel. For this reason, it is worth analysing the vertical
components of the two liquids
𝑣
𝑤,𝑥
and
𝑣
𝑜,𝑥
computed with a three-dimensional approach and
evaluate the deviation from their ideal values provided from the previous equation. To make a
significant comparison, we selected a zone of the separator sufficiently distant both from the inlet
section and the gas-liquid interface where local phenomena render the assumptions behind the
Stokes-law not valid. In figure 11, we report the comparison between the vertical components of
liquids velocity and the terminal velocity. We notice that the local values of the liquids velocity may
be significantly different from those provided using the simplified analytical model. Moreover, it is
possible to see the inherent variation of the local settling velocities of liquid phases with respect to
the gas stream. Nevertheless, we can observe that the average values of
𝑣
𝑤,𝑥
and
𝑣
𝑜,𝑥
are much
more consistent with the respective terminal velocities, confirming that this value represents a useful
average quantity for the sizing of oil and gas separators.
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Figure 11: water velocity vertical components (left), oil velocity vertical components (right)
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