CFD Analysis and Optimization of Three Phase Oil Separator
Abhijith Joshy, Adithya B, Anfal M A, Anirudh Nambiar Dept. of Mechanical Engineering, GEC Thrissur
Abstract: The multiphase separators are usually the primary and largest technical devices in oil manufacturing platforms, furthermore this primary separation step is a key element in the oil and gas industries in the downstream equipment, together with compressors, are absolutely dependent on the efficient performance of those multiphase separators. This project is concerned with CFD simulation of three phase oil-separator consisting of all the internal parts. The VOF model was used to simulate the phase behaviour and fluid flow patterns. The project also evaluated conventional separation design methodologies using detailed CFD simulations, and adjustments in the design. This project studies various models of the separator and the best optimized model has been obtained. The CFD model consisted of the momentum, continuity and standard k-𝜀turbulence equations. Assumptions and effective model configurations were used to establish design criteria for new or modified separators. These criteria will be combined with an algorithmic design method used in industry to specify a realistic optimum separator design. This allows you to lay down an effective most suitable separator, a beneficial approach become settled for estimation of the droplet sizes used to calculate sensible separation velocities for numerous oil refinery conditions. In difference with classic design techniques, the conclusions of the CFD simulation implied enhancement in design. In this extra settling time is critical for droplets to penetrate through the interfaces. As a result, this research project does display the advantages that CFD analyses can provide in optimizing the design of new separators and fixing troubles with present design.
Key words: CFD; Oil Separator; Coalescer; Baffles; VOF INTRODUCTION
The simultaneous flow of oil, water, and gas is of practical significance for the oil and gas industry. Crude oil is actually produced with water in the form of a mixture of water droplets in oil and it is important to remove water from the oil for economic and operational reasons. Various physical treatment methods including pH adjustment, centrifugation, and gravitational settling have been employed (Zolfaghari et al., 2016) which can improve phase separation by helping, collision, migration, and subsequent coalescence of droplets (Xie et al., 2015). At the beginning of the separation process, the three-phase separator is used for the separation of oil, gas, and water in the petroleum industry. In order to determine the capacity of subsequent equipment such as heat exchangers, liquid extractors, and distillation columns, a proper sizing of the oil separator is crucial (Laleh et al., 2012c). The three- phase oil separator has mainly two parts: a gas-liquid section and an oil- water section. A large equipment is used in the gravity settling approach for separation, which is not convenient, practical, and economical for offshore conditions. The liquid level can vary from 0.20-0.27 m
up to 80-90% of the cross-sectional area of the vessel (Smith, 1987). The configuration of the internal devices is a factor that determines the oil removal efficiency which is related to the minimum required volume of the separators (Mostafaiyan et al., 2014). Therefore, in order to enhance the separation efficiency, gravity separators are equipped with different types of internals (Han et al., 2017; Kharoua et al., 2013, Simmons et al., 2004).
The phase separation process is always accomplished in three zones, in spite of the variety of design configurations proposed for multiphase separators. In the first zone, that is the primary separation zone, an inlet diverter is used so that an abrupt change in flow direction and velocity causes the largest liquid droplets to impinge on the diverter and then drop by gravity. The bulk of the liquid phase is separated from the gas phase, in this zone. The next zone is the secondary separation zone. In the secondary separation zone, the liquid droplets settle out of the gas stream due to gravity and gravity separation of fine droplets occurs as the vapor and liquid phases flow through the main section of the separator at relatively low velocities. Because of gravity and buoyancy, the liquid collection section in the bottom half of separator provides the retention time required for entrained gas bubbles or other liquid droplets to join their corresponding phases. While small accumulated liquid droplets are again separated by gravity, gas flows above the liquid phase. The final zone is the coalescing media zone. It is designed for mist elimination in which very fine droplets that could not be separated in the gravity settling zone are separated by passing the gas stream through a mist eliminator. Vanes, wire mesh pad, or coalescing plates may be used in this zone, to provide very fine droplets to coalesce and form larger droplets which can be separated out of gas stream by gravity. By changing the inlet device of the separator, the opening area and position of the perforated plate, the
performance of the separator can be improved
(Kharoua et al., 2012a; Kharoua et al., 2012b; Lee et al., 2008; Vilagines and Akhras, 2010).The single perforated plate installed in the separator was modeled as porous media of finite thickness with directional permeability, and the CFD results showed that the flow streamlines developed quickly downstream of the perforated plate were short circuiting (Frankiewicz and Lee, 2002). It was recommended that the fluids could choose the path of least resistance for travelling of least resistance for travelling and could anticipate the outlet zone. . As a result of this behaviour, the result was a significant loss in thethe effective liquid retention time. According to the CFD studies, a second perforated plate just upstream of the outlet nozzle was required to prevent this problem.
Practical considerations such as location, configuration of the inlet nozzle, position of the weir and water outlet nozzle, and the vessel dimensions were taken into account in determining the position of the perforated plate. It was emphasized that a fully symmetric setting of the internal baffles was not effective. A dual mist eliminator system composed of a vane-type demister at a lower level and a wire mesh pad at higher level with a gap of 0.15 to 0.30 m between them is usually used for separators suffering from liquid carryover while processing foamy crude oils, or glycols, amines, and similar materials (with high foaming tendency) (Lyons and Plisga, 2005). The GPSA Engineering Data Book (1998) states while horizontal separators are most efficient for high-capacity operations, where large amounts of solution of gas are in the liquid phase, vertical separators are usually used if the gas to liquid ratio is high or total gas volumes are low. To remove smaller droplets from the gas phase, mist eliminators are usually required (Smith, 1987).
The efficiency can be increased to around 100% by installing mist eliminators as 95% of droplets entrained in the gas stream can be separated in economically- sized separators without coalescing media (Walas, 1990;
Sinnott, 1997; Arnold and Stewart, 2008). To modify the design and to improve the operation of process equipment, Computational fluid dynamics (CFD) has been used (Laleh et al., 2012c; Pham et al., 2017). Two CFD methods have been used to simulate a three Phase separator. Eulerian and Volume of fluid (VOF) method.
The VOF method is based on the assumption that two phases do not interpenetrate (Ngo et al., 2018), while in the Eulerian method the two phases are treated as interpenetrating continua and also solving continuum and momentum equation for each phase (Kharoua et al., 2013; Laleh et al., 2012a; Lim, 2013; Pham et al., 2015b).
Liang et al. (2013) examined the effect of the flow rate, gas ratio, and water ratio on the separation efficiency using a VOF CFD model with a k-ε turbulence equation (Liang et al., 2013) for a horizontal three phase separator. Fewel and Kean (1992) emphasised that CFD analysis of separator internals is very similar to a physical test because laboratory tests performed on various arrangements usually match CFD results remarkably well. 9 different models with internals of different configurations have been considered in the is paper. Same boundary condition was used in carrying out the simulation on each model. The flow parameters of the separator internals were also observed. CFD simulation was conducted for inspecting separation efficiency of the separator with only baffles by Tathagata Acharya, Lucio Casimiro et al. (2019). Primary phase was oil with volume fraction of 0.652 in the crude oil mixture.
The model used by Tathagata Acharya, Lucio Casimiro was not sufficient to provide enough separation efficiency for validating the CFD simulation model. So, CFD simulation model for separator was developed using
curved end plate and 2 baffles. The developed CFD model using coalescer and demister was used to run Fluent analysis.
MODELLING
A three phase oil-water-gas separator has a primary separation zone, including a feed inlet for liquid-gas separation, a zone where water-oil separation takes place by gravitational separation with the help of coalescer, and a mist elimination zone along with a gas outlet as shown in Figure 1. The cylindrical gravity settling vessel is 2.1 m long, 0.6 m in diameter and is horizontal in geometry. There are three outlets and one inlet. The separator has various internals such as coalescer, weir plate, perforated plate, and demister. In the primary separation zone, the mixture with oil, water, and air was injected at the top of the separator, and the liquid and gas phases were separated. In order to remove the liquid droplets contained in the gas flow, the demister was placed at the gas outlet. The liquid phase drops down to the bottom. To maintain a good fluid flow distribution and to moderate liquid sloshing, the perforated plate was used. Oil-droplets were captured, coalesced, and rose to the surface (American Chemical Society [ACS], 2014) in the corrugated plate-type coalescer. The height and length of the semi cylindrical coalescer used was 0.4 m and 0.5 m respectively (PETROL 5181-Journal of Petroleum Science and Engineering). The weir plate used was 0.35 m in height. The water outlet was located between the weir plate and the coalescer. Oil flowing over the weir plate was discharged through the oil outlet. The pressure at which the separator was operated was,10
bar and at room temperature
.
Figure 1: Geometric model of three phase oil separator Table 1:Physical properties of air, water and oil
Compon ents
Phase Densi ty(kg/
𝑚3
)
Viscosity (Pa-s)
Surface Tension(N/
m)
Air Gas 1.1 1.8*10
–5-
Water Liquid 980 2.42*10
–373*10
–3Oil Liquid 831 7.21*10
–327.2*10
–3It was assumed that, the perforated plate, demister, and the coalescer were porous media of finite thickness and directional permeability (Frankiewicz
and Lee, 2002).
To capture mist,the demister was composed of fibro matrix. Because of high permeability, the porous resistance of the perforated plate was ignored, but due to high velocity, the inertial resistance (Cr.x) was considered in the x- direction. Loss coefficient was used to obtain the inertial resistance (Cr.x) (Weber et al., 2000). The inertial resistance (Cr.x) was assumed to be of large value in the y or z direction, which means that there is no flow in the two directions. In the CFD model, the porous and inertial resistances were used (see table 2(Le, T.T., Ngo, S.I., Lim, Y.-I., Park, C.-K., Lee, B.-D, Three-phase Eulerian computational fluid dynamics of air–water–oil separator under off-shore operation, Journal of Petroleum Science and Engineering (2018))).
The inertial resistance in the y- or z-directions was assumed to be a large value, which means that there is almost no flow in the two directions. The porous and inertial resistances were used in the CFD model (see Table 2).
Table 2: Material properties
Material
Coalescer
Demister
Perforated plate
Porosity (𝜀
P)0.954
0.980
0.25
Porous resistance (1/𝑚
2)(
1)
Kx
3.76*105
2.57*106
0
Porous
resistance(1/𝑚
2)(
1)
ky
0
2.57*106
0
Inertial
resistance (1/m)
(𝑐rx)0 0
1.03*104
Inertial resistance (1/m)
(𝑐ry)0
0
1*10
6 Because of high flow velocity, a high density mesh was applied to one inlet and the three outlets. A perforated plate with a thickness of 3 mm was used to concentrate the mesh on it. Owing to the porous media assumption, the fluid flows of the coalescer and the demister were not sensitive. At the interfaces of the gas and the liquid, a relatively high mesh density was used. The domain of the gas phase had a relatively low mesh density. To attain skewness of 0, element size was made to 0.005 m. With default size and angle with medium smoothing, capture curvature was applied.
Figure 2: Meshed Model GOVERNING EQUATIONS
Continuity equation
𝜕
𝜕𝑡 (𝜌𝛼) + 𝜕
𝜕𝑥 (𝛼𝜌𝑢) + 𝜕 𝜕𝑦 ⁄ (𝛼𝜌𝑣) = 0
Momentum equation
𝜕
𝜕𝑡(𝛼𝜌𝑢), +𝑢 𝜕
𝜕𝑥(𝑎𝜌𝑢) + 𝑣 𝜕
𝜕𝑦(𝛼𝜌𝑢)
= −𝛼𝜕𝑃
𝜕𝑥+ 𝛼𝜌𝑔𝑥+ 𝜕
𝜕𝑥(𝛼𝜇𝜕𝑢
𝜕𝑥) + 𝜕
𝜕𝑦(𝛼𝜇𝜕𝑢
𝜕𝑦)
𝜕
𝜕𝑡(𝛼𝜌𝑣), +𝑢 𝜕
𝜕𝑥(𝑎𝜌𝑣) + 𝑣 𝜕
𝜕𝑦(𝛼𝜌𝑣)
= −𝛼𝜕𝑃
𝜕𝑦+ 𝛼𝜌𝑔𝑦+ 𝜕
𝜕𝑥(𝛼𝜇𝜕𝑣
𝜕𝑥) + 𝜕
𝜕𝑦(𝛼𝜇𝜕𝑣
𝜕𝑦) For the three-phase mixture, the momentum and continuity equations were solved. The concentration fields of dispersed phase were obtained by tracking their volume fraction and relative motions of the dispersed phase were approximated by algebraic equations. It is considered that, the Reynolds number of the liquid phase at the inlet was up to 58,000. The flow in the separator
also involved in significant turbulence under the operating conditions. The k-ε model
was implemented to account for the turbulent effects. The turbulence model was applied to all the three phases.
The oil phase was set as the primary phase because it is the phase, with highest mass and volume fraction among the three phases at the inlet. The gas and water were the secondary phases ( or dispersed phases), which were assumed to form droplets or bubbles within the continuous phase. For the three-phase mixture, the momentum and continuity equations were solved. The concentration fields of the dispersed phase were obtained by tracking their volume fraction and the relative motions of the dispersed phase were approximated by algebraic equations. It is considered that, the Reynolds number of the liquid phase at the inlet was up to 58,000.
The flow in the separator also involved significant turbulence under the operating conditions. The k-ε model was implemented to account for the turbulent effects. The turbulence model was applied to all the three phases. The oil phase was set as the primary because it is the phase, with highest mass and volume fraction among the three phases at the inlet. The gas and water were the secondary phases (or dispersed phases), which were assumed to form
droplets or bubbles within the continuous phase. Using the standard k-ɛ turbulence equation and Reynolds- averaged multiphase VOF(Volume of fraction) CFD approach, the air–water–oil separator was modelled(Lim et al., 2017).VOF is used if there are more than two immersible fluids, where interface position is of importance and is actually a type of surface tracking technique. It also has a set of momentum equations, shared by all fluids and volume fraction of each of the fluid in each computational cell is tracked.
The standard k − ε (Launder and Spalding, 1972) model was selected for a wide range of turbulent flows in industrial flow simulations and also for its robustness, economy and accuracy. After importing the mesh file and making the modification to it, the necessary material properties for various phases were input. Mesh modification consisted of converting the highly skewed grids to polyhedral grids which resulted in a minor reduction in the number of cells.
Table 3: Standard k-ɛ turbulence model
Parameters Value or reference Turbulence intensity
of inlet velocities (%)
5 Turbulent Prandtl
number for k
1 Turbulent Prandtl
number for ε
1.3 Dispersion Prandtl
number
0.75 Turbulence energy
dissipation coefficients
1.44,1.92
Turbulent viscosity coefficient
0.09
The plate-type coalescer is assumed to be the porous medium having a uniform porosity instead of the real geometry. The porous media model does not represent the fundamental phenomena such as capturing, aggregation, and rising of droplets on the plate. Since the coalescer, demister, and the perforated plate are considered as porous media, the number of computational cells is reduced.
BOUNDARY CONDITIONS AND CFD SIMULATION MODEL
In table 4, the inlet and outlet boundary conditions for each phase is represented. The volume fraction (α) and the inlet mass flow rates (m) were given at P= 10 bar and T=20℃. In order to set the boundary conditions for inlet, the velocity and volume fractions of phases were set. The oil phase was set as the primary phase because it had the highest mass and volume fraction among the three phases at the inlet. The gas and water phases are the secondary phases or dispersed phases, which were assumed to form droplets within the continuous phase.
For the gas-outlet boundary, volume fractions and outlet pressures were set while for the liquid-outlet
boundaries, volume fractions and outlet velocities were set. 10 bar pressure was fixed as the outlet static pressure (or 0 bar in gauge pressure). To the wall, the no-slip condition was applied. At P=10 bar and T=20℃, the separator was initially filled with air, water, and oil where the initial liquid level of the water was 0.37 m. The gas phase had 100% air. When setting the flow regime in the inlet and outlet nozzles, the turbulence intensity and hydraulic diameter of the nozzles were determined.
The flow regime of the three-phase horizontal separator is dominantly stratified flow (Taitel et al., 1995) of diluted oil, released gas and it produced water in longitudinal direction, which was accompanied by vertical settling and floating motions of fluids which results in the separation of the phase. Therefore, actual flow is a combination of stratified and dispersed flow.
Table 4: Boundary conditions Inlet BC Phase Mass
flow rate(m,kg /s)
Volume fraction
(α)
Pressure(
P)
&Temper ature(T) (Secondary) Air 0.001829 0.1852 P=10 bar,
T=20℃
(Secondary) Water 2.1 0.1630 P=10 bar, T=20℃
(Primary) Oil 4.2 0.6518 P=10 bar, T=20℃
Outlet BC Air - - P(gauge)=
0KPa The transient VOF CFD model was solved using a commercial code, ANSYS Fluent 21.0 (ANSYS Inc., USA). As explained by Anderson (1995), the incompressible Navier Stokes equations cannot be solved explicitly because of some stability issues and solution techniques for the incompressible equations are usually different from those used for solution of the Navier-Stokes equations for compressible flow. To overcome this difficulty, the pressure correction approach has been proposed. With good success, this accepted and widely used approach has been applied to both compressible and incompressible flows (Anderson, 1995). The phase-coupled SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) method was selected for the pressure–velocity coupling. QUICK (Quadratic Upwind Interpolation for Convective Kinetics) schemes and second-order upwind were used for the spatial discretization of the momentum and volume fraction, respectively. For the transient simulation the time was taken as 0.001 sec and the maximum number of iterations was limited to 20 at each time step. Having set all the CFD parameters for the redesigned separator, approximately 10,000 iterations are required for continuous phase solution convergence. A personal-computer (PC) run time of approximately 36 hours is required per solution of the continuous phases and each iteration takes approximately 30 seconds. For simulation of interactions between the dispersed droplets and continuous phases, a further PC run time of approximately 4 hours is also required. To set the position of interface between
phases, the volume fractions of phases above and below the assumed interface planes were set to the reasonable values by the Patching tool of Fluent.
The iterations need to be stopped regularly again, and if necessary, by patching the volume fractions of phases, the position of interfaces should be checked and corrected.
RESULTS AND DISCUSSION Effect of internal elements
CFD study on different models and effect of internal elements such as baffles, coalescer, demister, deflector are studied through simulation.
Deflector shape
For studying the effect of deflector shape in separating efficiency, two models were selected. One model was designed using a curved plate deflector and the other using a flat one. Both models had weir plate designated in base geometry and consisted of single baffle. For both the models, CFD simulation was carried out applying the same boundary conditions and operating parameters.
Contours of volume fraction of oil, air, and turbulent kinetic energy were calculated
MODEL 1 (With flat deflector)
(a)
(b)
(c)
Figure 3. (a) Volume fraction of oil (b) Volume fraction of air (c) Turbulent kinetic energy
MODEL 2 (With curved deflector)
(a)
(b)
(c)
Figure 4. (a) Volume fraction of oil (b) Volume fraction of air (c) Turbulent kinetic energy
The conclusion that can be obtained from the plots is that significant turbulence occurs at the curved portion compared to the flat portion. Air-oil interface is higher in model 2 compared to model 1 which leading to significance of deflector in separator efficiency. Hence flat plate deflector can be preferred over curved one for greater efficiency.
Number of baffles
The distribution baffles generally improve the quality of liquid flow distribution in the vessel, break the large scale circulations into smaller ones, and reduce the short-circuiting flow streams. This was shown in the recent CFD-based study by Lu et al. (2007). Hence the flat plate deflector gives greater efficiency compared to the curved one.
MODEL 3 (Two baffles)
Figure 5: Volume fraction of oil
MODEL 4 (3 Baffles)
Figure 6: Volume fraction of oil
Results concluded that the efficiency of the separator can be increased using baffles. Traces of water can be seen at oil outlet of model 3, whereas 100% oil at oil outlet of model 4. It can be expected that the baffles will improve the quality of liquid flow distribution in the vessel and increase the separation efficiency as concluded by Lu et al. (2007) in their recent CFD- based study.
Effect of coalescer
The droplet coalescence and breakup were again modelled, and the simulation results confirmed that droplet coalescence at a rate of less than 1% was not a common phenomenon. Based on the simulation case study results, a general conclusion for droplet coalescence could not be reached. However, droplet breakdown was a common phenomenon and did show significant variations to operating conditions. Highest velocities usually intensified the number of droplet breakups in horizontal separators, based on the simulation results
Model 5 (With coalescer)
(a)
(b) Figure 7. (a)Volume fraction of oil (b)Velocity distribution
The Coalescer is a oil-water separator that uses coalescing material to remove up to 95% of all floating oil from the water's surface. The Coalescer is also very helpful in removing sludge, scum, and dirt particles which are often found floating on coolant or degrease solution surfaces. From volume fraction of oil plot, it is clear that efficiency is improved compared to previous model. Significant rise of oil & water level at respective
outlets in latter design conclude this fact. There is normal distribution of mixture at coalescing zone while comparing the velocity distribution. It ensures the smoother running of equipment and reduces maintenance requirements.
Optimized models
Based on conclusions from above, three optimized models were created
Model Internal elements
Model 6
One baffle, coalescer, demister, flat plate deflector
Model 7
Three baffle, coalescer, demister, flat plate deflector Model 8
Three baffle, coalescer, demister, flat plate deflector, additional outlet at oil outlet for water removal
MODEL 6
(a)
(b)
Figure.8 (a)Volume fraction of oil (b)Turbulent kinetic energy v/s position
Model 7
(a)
(b)
Figure 9 (a)Volume fraction of oil (b)Turbulent kinetic energy v/s position.
Model 8
(a)
(b)
(c)
Figure.10 (a)Volume fraction of oil (b)Volume fraction of water (c) Turbulent kinetic energy v/s position
Comparing the three optimized models, it can be concluded that turbulence distribution is fair in case of model 7 & 8. Turbulence kinetic energy (TKE) is one of the most crucial variables in micrometeorology. It has a direct bearing on the movement of momentum, heat, and moisture across the boundary layer .Higher turbulent energy creates mixed flow which causes vibration to walls resulting in crack. Oil and air interface is higher in 7&8 compared to model 6 which results in greater separation. Influence of baffles can be seen clearly from the plot of turbulent kinetic energy v/s position. More separation of water takes place in models 7 &8, although oil separation takes place in three models . Exclusively considering Model 8,it have two oil outlets-vertical &
horizontal where pure oil can be obtained from horizontal outlet. Since there is a chance of water being accumulated at the oil outlet, model 8 can be applied for mixtures containing significant amount of water.
CONCLUSION
The performance of the separator was enhanced when modifications as a result of the CFD simulation of the research project were made. The redesigned separator dealt satisfactorily with 1988 production conditions, in that the total separation efficiency was as high as 99.1%
(a bit higher than the original separator efficiency) with its components of 100% and 98.7% as separation efficiencies for oil and water droplets, respectively. There would be no droplet carry-over in the gas phase outlet, i.e. all the injected droplets exited in either the oil outlet or the water outlet as was the case with the original separator. The distribution of baffles had a minor effect on the separation efficiency. This was pointed out by the results generated for the original separator and its modified versions. They are helpful in improving flow distribution quality, however, poor designs cannot be overcome by installing distributing baffles. Only VOF approach was used in the CFD simulation of the three- phase separator, based on the obtained results for the CFD simulation for the three-phase separator. The developed model did provide high-quality details of fluid flow profiles, leading to a very realistic overall picture of phase separation in all zones of the separator compared to the original study of Hansen et al. (1993). An understanding of both the microscopic and macroscopic features of the three-phase separation phenomenon was obtained by the realistic CFD simulation of the three- phase separator. The water outlet as was the case with the original separator. The distribution of baffles had a minor effect on the separation efficiency. This was pointed out by the results generated for the original separator and its modified versions. They are helpful in improving flow distribution quality, however, poor designs cannot be overcome by installing distributing baffles. Only VOF approach was used in the CFD simulation of the three- phase separator, based on the obtained results for the CFD simulation for the three-phase separator. The developed model did provide high-quality details of fluid flow profiles, leading to a very realistic overall picture of phase separation in all zones of the separator compared to the original study of Hansen et al. (1993). An
understanding of both the microscopic and macroscopic features of the three-phase separation phenomenon was obtained by the realistic CFD simulation of the three- phase separator.
Nomenclature
Cr=inertial resistance factor (m-1) G= gravitational acceleration (m/𝑠2)) 𝐼t = turbulence intensity (%)
K= turbulence kinetic energy (J/kg or 𝑚2/𝑠2) K=
permeability (𝑚2) L =length of separator (m)
m =inlet mass flow rate (kg/s) P= pressure (Pa) 𝑃exit= exit pressure (Pa)
𝑃gage=gage pressure (Pa)
Re =Reynolds number t =flow time (s or min) T =temperature or period (℃)
𝑢=velocity in x direction 𝑣=velocity in y direction 𝑥=longitudinal direction (m) 𝑦=gravitational direction (m) 𝑧=transversal direction (m) Greek letters
α= volume fraction
ε= dissipation rate of k (𝑚2/𝑠3)
µ =viscosity (kg/m/s) ρ =density (kg/𝑚3) σ =surface tension (N/m)
α - Volume fraction of first phase in the two phase mixture β - Volume fraction of second phase in the two phase mixture
ρ - Density (kg/𝑚3)
µ - Dynamic viscosity, (Pa-s)
u - Fluid velocity in horizontal direction (m/s) ν - Fluid velocity in vertical direction (m/s) g - Acceleration due to gravity (m/𝑠2) P- Pressure (Pa)
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