An Eulerian-Eulerian CFD modeling analysis of gas-liquid flow under elevated pressure

In this study, a newly proposed pressure correction method coupled with DBS drag force model was used to simulate the gas-liquid flow in a bubble column, which combines Eulerian-Eulerian approach to simulate gas-liquid flow under elevated pressure conditions. This further verifies the accuracy of the model and expands its application range. The gas holdup and velocity distributions of different models were compared, and the performance of the new model was similar to that of Yan et al., while the model of Tran et al. was not sensitive to the influence of the pressure in this study. It was found that the drag model using the new pressure correction method showed the best consistency.


Introduction
Multiphase flow is the key and difficult point in the study of fluid mechanics.There are various multiphase flow processes in both daily life and industrial production.Among them, gas-liquid twophase flow plays a very important role in many industrial processes such as petroleum, chemical industry, energy, food, etc.For bubbles in liquids, their dynamic characteristics are Co-determination by a variety of factors.
In most cases, there is a strong interaction between the gas phase and the liquid phase.Since the gas-liquid two-phase process in many industrial processes needs to be carried out under pressurized conditions, it has caused great difficulties to accurately predict this special gas-liquid flow phenomenon.Due to the complexity of gas-liquid interaction, this is still a difficult problem so far.Some researchers have studied the characteristics of gas-liquid two-phase flow affected by pressure.Usually, the numerical modeling method for this kind of problem is to add the pressure correction factor to the drag force model.Krishna et al. [1] were the first to apply this idea.The gas phase density correction item ,0 ( ) was added to the drag force model to consider the influence of different pressures on gas-liquid flow, but it does not have very good applicability.Chen et al. [2] added a new gas-phase density correction item 0.25 ,0 ( ) to the drag force model based on Krishna et al.This method can well predict the gas-liquid flow in the pressure range of 0.1-1.0MPa.Tran et al [3,4] proposed two drag factors with pressure correction and coupled them with the Ishii-Zuber drag model.The applied pressure conditions were 0.1-3.5 MPa and up to 16 MPa.Yan et al. [5] recommended three modified models suitable for 0.5-2.0MPa,which provide a very useful reference for further numerical model research.
In this study, a new pressure-corrected drag model is improved, and the drag coefficient is

Governing equations
The Eulerian-Eulerian method is used in this work for the gas-liquid phases.The governing equations for mass and momentum can be given as: Continuum equation: Momentum equation: where the lower index i denotes the different phases, with α, ρ and ui the volume fraction, density and resolved velocity, respectively.P is the pressure and eff  is the effective viscosity.Fi,j is the total interphase forces.

Drag force mdoels 2.2.1. Yan drag model.
The pressure correction model recommended by Yan et al. is based on the EMMS method, which simplifies the structural parameters and ignores further differentiation of bubbles of different sizes, velocities, or movement tendencies [6,7].They fitted a curve formula of the concentrated parameter CD/db of the gas phase as a function of overall apparent gas velocity and density correction.Its expression is as follows: ( ) We have developed a new drag force correction method based on the DBS model, which correlates the effective drag coefficient to the pressure changes, making it applicable to a wider range of pressure variations.The obtained correction value increases with increasing pressure, reflecting the inhibitory effect of bubbles.We found that this inhibitory relationship undergoes a change at 1 MPa, showing a linear distribution below 1 MPa and a power-law distribution above 1 MPa.
0.05 0 0 0.45 9.9 1.0MPa 0.076 0.4 1.0MPa where P and g  are current operating pressure and gas density, P 0 and ,0 g  are the atmospheric pressure and the density of air at this pressure.

Numerical details
In present work, the simulations make use of the commercial CFD package Ansys Fluent for estimating the key parameters of air-water systems in column.The drag models were imported by user-defined functions.The column is 6.6m high and 0.3m in diameter.The velocity inlet is used as the inlet boundary condition which the gas volume fraction at the inlet is set to 1, and the superficial gas velocity at the inlet is 0.121m/s.The top adopts degassing boundary condition.The time stepping strategy was 0.001s at the beginning, and when the flow is stable, change it to 0.004s.The simulation time was set to 80 s, and the calculation data between 40 and 80 s were time averaged for data analysis.

Local gas holdup
Figure 1 compares the radial gas holdup distribution predicted by different drag models under lower pressure condition.It can be clearly seen that the present model predicts a good result at P = 0.5MPa.Among them, Yan et al.'s model predicts a better value at the center, but near the wall, the gas holdup is overestimated.On the contrary, the model of Tran et al. underestimated the gas holdup at the center, and the predicted value near the wall almost overlapped with the present model.

Bubble rise velocity
Figure 3 shows the radial distribution of the axial gas velocity at lower pressure condition (P = 0.5 MPa).It can be seen that the difference in the axial gas velocity predicted by the three models is not large, especially in the area of the column center.But near the wall, the differences among the three models are very obvious, among which the axial gas velocity estimated by Tran et al.'s model is the lowest, and the distribution gradient of the present model is the largest.
Figure 4 shows the radial distribution of the axial gas velocity predicted by the three models at higher pressure condition (P = 2.0 MPa).It can be seen that the prediction value of the present model is almost consistent with that of Yan

Conclusions
In this study, an improved novel drag model was proposed that is based on the DBS drag model.We investigated the influence of the new model, Yan et al's model and Tran et al's model on the gas holdup, axial gas velocity using the Eulerian-Eulerian approach at different operating pressures.This study revealed that the simulation results of the new model about parameters were satisfactory.
In these simulations, the average gas holdup predicted by various drag models increases with increasing pressure.Furthermore, the new drag model has good applicability under all tested pressure conditions.In the meantime, other drag models under the influence of pressure were evaluated

Figure 2
compares the radial gas holdup distribution predicted by different drag models under higher pressure condition.It can be seen that the present model and the Yan et al.'s model maintain good predictive ability in the center of the column, but in other regions, the present model is closer to the experimental value.But Tran et al.'s model predicts unsatisfactory results, the gas holdup is grossly underestimated in the entire flow region.

Figure 1 .
Figure 1.Radial gas holdup distributions at lower system pressure.

Figure 2 .
Figure 2. Radial gas holdup distributions at higher system pressure.
et al.According to the research of Jin et al., with the increase of pressure, the distribution of axial gas velocity should be reduced.The simulation results of the present model also verified this, but the model of Tran et al. did not show it well.

Figure 3 .
Figure 3. Axial gas velocity distributions at lower system pressure.

Figure 4 .
Figure 4. Axial gas velocity distributions at higher system pressure.
. The model of Yan et al. has a relatively similar performance, while the model of Tran et al. is insensitive to pressure changes.
[8]local gas holdup αg are specified values at a fixed superficial gas velocity according to the following formulas[8]: .2.3.Modified DBS drag model.In recent years, A mesoscale multiphase flow model called the DBS model is proposed, which has natural advantages in dealing with gas liquid flow systems.This model is obtained by curve fitting to obtain the dependence of the global parameter (CD/db).A kind of CD/db which based