The multi-size group (MUSIG) model is employed in this paper to simulate the gas–liquid two-phase flow in pump as turbine (PAT) since the traditional Eulerian–Eulerian two-fluid model is unable to take into account the phenomena of breakup and coalescence of bubbles. First, the simulation of gas–liquid two-phase flow in a square column is compared with the experiment to verify the accuracy of the MUSIG model. Then, the results of gas–liquid two-phase flow in PAT simulated by the MUSIG model are compared with those by the conventional uniform bubble (UB) model and find that the MUSIG model is more favorable to capture the flow pattern at high gas content compared to the UB model. Based on the MUSIG model, the internal flow characteristics, pressure fluctuation, and bubble size distribution of the PAT are analyzed. The rotation of the blades breaks a part of big bubbles into small bubbles in the volute, resulting in a smaller diameter of the bubbles entering the impeller. As the gas content increases, the number and size of vortices in the impeller flow channel increase. The vortex is formed at locations where the gas phase distribution in the impeller flow channel is concentrated. The outlet of the impeller is more prone to bubble consolidation under high gas content conditions. In conclusion, the MUSIG model can well predict the complex flow characteristics of gas–liquid two-phase inside the PAT and identify the key influencing factors of energy acquisition, which can provide support for improving the performance of the PAT design.

## I. INTRODUCTION

Pump as turbine (PAT) is a centrifugal pump that operates in reverse under the action of high-pressure fluid.^{1–3} It is a kind of machine with high-pressure fluid as the working medium for energy conversion that converts the pressure energy of high-pressure fluid into the rotating mechanical energy of a hydraulic turbine impeller. There are two main uses of pump reversal: one is as a low-cost turbine for power generation^{4–6} and the other is for energy recovery in pipelines.^{7} With the development of industry, the fluid medium of hydraulic turbine is becoming more and more diversified. The medium can be single-phase water, liquefied natural gas, liquid hydrogen, liquid nitrogen, liquid oxygen, and so on, or gas–liquid two-phase or even gas–liquid-solid three-phase flow.^{8–10}

At present, many scholars have studied the flow of hydraulic turbines under gas–liquid two-phase conditions.^{11,12} Shi *et al.*^{13,14} simulated and analyzed the performance of the centrifugal pump as a turbine and the change law of the internal flow under different gas contents. However, in the simulation, the bubble was assumed to be a uniform diameter, and the bubble deformation and bubble collision were not considered. When the bubble moves in the impeller channel, it is affected by the rotation of the impeller and the shear of the wall surface. Collision, breakage, and coalescence will occur, resulting in flow separation. Therefore, some scholars have extended the two-fluid model and added the coalescence, breakup, and deformation of bubbles in the calculation, and the population balance mode-computational fluid dynamics (PBM-CFD) coupling model,^{15–19} and the multi-size group (MUSIG) model were proposed. For the MUSIG model, some scholars have made a preliminary exploration. First, the applicability of the MUSIG model in gas–liquid two-phase flow is studied. Yuan *et al.*^{20} used the MUSIG model to numerically simulate the air-nanofluid bubbly flow in a vertical tube and found that the MUSIG model is in good agreement with the experimental data of air–water bubbly flow; however, there is a big difference with the experimental data of air-nanofluid bubbly flow. Based on the MUSIG model, Zhou *et al.*^{21} conducted a numerical simulation study on subcooled flow boiling. It was found that the simulation results were qualitatively in good agreement with the experimental data, which laid a good foundation for the application of the simulation method in the engine cooling system. Afterward, some scholars used the advantages of the model to expand the application scope of the MUSIG model. Zhang *et al.*^{22} used the MUSIG model to study the three-dimensional complex flow in the side channel pump with the change of gas content and found that, under different gas contents, the suction side is more likely to accumulate bubbles, especially near the inner diameter of the impeller, while there is almost no gas at the outer diameter of the impeller. The MUSIG model was used by Lee *et al.*^{23,24} to numerically analyze the performance of the induced air flotation machine. The MUSIG model can predict relatively accurate bubble motion, coalescence, and rupture characteristics to select the rotor model with optimal performance. Some scholars have also modified the MUSIG model to improve the scope of application of the model. Liao^{25} optimized the discrete method of population balance equation in ANSYS CFX, discussed the inconsistency of bubble breakup, and updated the model.

In summary, the MUSIG model has good accuracy in the study of gas–liquid two-phase flows. Therefore, the MUISG model is introduced into the study of PAT. Based on the MUSIG model, the characteristics of the gas–liquid two-phase flow field in PAT can be revealed more accurately by considering the coalescence and breakup of bubbles.

## II. GOVERNING EQUATIONS AND NUMERICAL METHODS

### A. Geometry and mesh

As shown in Fig. 1(a), the calculation domain model of PAT is divided into six parts: inlet, volute, impeller, front cavity, back cavity, and outlet. In order to ensure the full development of the fluid flow, the length of the inlet section and the outlet section is appropriately extended. The structural mesh generation of the model was completed in ANSYS-ICEM, and the boundary layer of the structured mesh of the impeller single channel was refined, as shown in Fig. 1(b).

Taking the turbine efficiency under the design condition as the criterion, six groups of different numbers of grids are selected to verify the independence of the grid. As shown in Fig. 2, when the number of grids is greater than 5.2 × 10^{6}, the hydraulic efficiency of the PAT remains stable. Therefore, the final computational domain mesh number 5.2 × 10^{6} was selected. The mesh number of each hydraulic component in PAT is illustrated in Table I.

### B. Mathematical model

#### 1. Two-fluid model

*k*represents the

*k*th phase,

*ρ*is the density (kg/m

^{3}),

*α*is the volume fraction,

*u*is the relative velocity (m/s),

*μ*is the dynamic viscosity (Pa s), and

*F*is the momentum exchange between phases (

*N*).

*F*is mainly composed of drag, added mass force, lift, and turbulent dispersion force. The formula is as follows:

*U*

_{g}and

*U*

_{l}are the velocities of the gas and the liquid, respectively.

*F*

_{D},

*F*

_{A},

*F*

_{L}, and

*F*

_{T}are the drag, added mass force, lift, and turbulent dispersion force, respectively.

*D*

_{b}is the diameter of bubbles.

*K*is the turbulence kinetic energy.

*C*

_{A},

*C*

_{L}, and

*C*

_{T}are the coefficients of the added mass force, lift, and turbulent dispersion force, respectively. Furthermore,

*C*

_{A}= 0.5,

*C*

_{L}= 0.5, and

*C*

_{T}= 0.1.

^{26}

#### 2. MUSIG model

^{27}Regarding the breakup behavior in a turbulent field, the bubble coalescence model is from the work of Prince and Blanch,

^{28}

When the bubbles are divided into *N* groups, it means that there are *N* + 1 groups of equations to be solved; thus, the overall computational effort of the model will be greatly increased, and computational economy must be considered in the calculations.

#### 3. Boundary conditions and solving strategy

The ANSYS-CFX is used for the numerical simulation of the PAT in this investigation. Based on the simulation of PAT with pure water as a medium, research of PAT with gas–liquid two-phase flow is carried out. The boundary conditions in the inlet and outlet sections are mass flow rate and pressure. The turbulence model for the liquid phase is the *k*–*ω* SST model, and that for the gas phase is the zero-equation theoretical model. The liquid is set as incompressible continuous phase, and the gas is set as a dispersed phase. No-slip boundary conditions are applied to all stationary and rotating walls. In the case of using the Uniform Bubble (UB) model, the gas phase at the inlet entered the turbine uniformly with spherical bubbles with a diameter of 0.01 mm. In the case of MUSIG, the dispersed bubble diameters are set to five groups of 0.01, 0.05, 0.1, 0.15, and 0.2 mm, respectively. The inlet gas content is defined as the gas volume fraction (GVF), which equals 5%, 15%, 30%, and 45%, respectively.

## III. MODEL VALIDATION

### A. Experimental apparatus

In this investigation, a single-stage cantilever centrifugal pump with a specific speed *n*_{s} of 90 is reversed as a turbine, as shown in Fig. 3. The rotational speed *n* = 2900 rpm and the design flow rate is 81 m^{3}/h. The main geometric parameters of the impeller are illustrated in Table II.

### B. Validation under pure liquid conditions

*H*, efficiency

*η*, and shaft power

*P*are as follows:

*p*

_{1}and

*p*

_{2}are the inlet and outlet static pressures of PAT,

*T*represents the torque of the impeller, and

*n*represents the rotational speed.

*ρ*

_{m}represents the density of gas–liquid mixture, and it can be defined as

*α*represents the gas volume fraction,

*ρ*

_{l}represents the density of liquid, and

*ρ*

_{g}represents the density of gas.

The values of head, shaft power, and hydraulic efficiency of the simulated PAT under different flow rates with pure liquid conditions are presented in Fig. 4. It can be seen that the PAT efficiency increases rapidly with the increase in the flow rate at a small flow rate and decreases slowly after reaching the maximum value. The flow rate of the best efficiency of the experiment and numerical simulation appears at 81 kg/m^{3}. The efficiency of the numerical simulation is 2%–3% higher than that of the experiment. The reason can be that mechanical loss and partial volume loss are not considered in the numerical simulation. The values of head and shaft power initially increase slowly with the increase in the flow rate and then increase with a steep slope. Under the design conditions, the errors of efficiency, head, and shaft power of PAT are 2.8%, 0.8%, and 0.4%, respectively.

### C. MUSIG model validation

To verify the validity of the MUSIG model in the gas–liquid two-phase flow calculation method, the experimental results of gas–liquid two-phase flow in a three-dimensional square column by Deen *et al.*^{29} were used. The unsteady gas–liquid two-phase flow in a 3D square column was simulated using the UB model and the MUSIG model, and the numerical simulation results were compared with Deen’s experimental data.

The geometry of the 3D square column computational domain is shown in Fig. 5. The height of the water level in the square column is 450 mm. The gas enters through a 37.5 mm square area at the bottom of the column at a speed of 0.0784 mm/s. A straight line is set up at a height of 0.25 m in the center of the column. Deen experimentally measured the axial liquid and gas velocity to obtain the axial liquid and gas distribution by Particle Image Velocimetry (PIV) at the same height. In this paper, the results of the liquid-phase and gas-phase axial velocity distributions at this line will also be compared with Deen’s experimental results.

The boundary conditions of the square column were defined as the gas velocity inlet and pressure outlet, and the wall surface of the square column was set as non-slip wall. *k*–*ω* SST turbulence model was adopted for the liquid-phase and the zero-equation model was used for the gas phase. The diameter of the bubble is fixed at 1 mm for the UB model. In the MUSIG model, the bubbles are categorized into five groups based on the diameter, namely, 0.1, 0.5, 1.0, 1.5, and 2.0 mm, where the proportion of each group is 20%. For the unsteady simulation, the flow was simulated for totally 150 s with the time step set as 0.01 s. After the gas–liquid two-phase flow in the square column was computationally stable and showed periodic oscillations, the results of the last 50 s were selected for further analysis.

Figure 6 shows the axial velocity distribution of the liquid and gas phases at the monitoring line. The numerical simulation results are compared with Deen’s experimental data, and the numerical simulation results for the UB and MUSIG models follow the same trend as the experimental measurements. The maximum velocity of the liquid- and gas-phase axial velocity distributions in the square column is located in the center region of the column. The numerical simulation results of the MUISG model are closer to the experimental results than the UB model. The dispersion statistics of the errors between the two models shows that the standard deviation of the MUSIG model in the liquid-phase velocity is 0.032, which is significantly smaller than the standard deviation value of the UB model of 0.049. The results in the gas-phase velocity also show that the standard deviation value of the MUSIG model is smaller. Therefore, in this paper, the MUSIG model is chosen to analyze the internal flow characteristics of PAT under gas–liquid two-phase conditions.

## IV. RESULTS AND DISCUSSION

### A. Comparison between UB and MUSIG models

The comparison between the numerical simulation results of the external characteristics obtained by the MUSIG model and the UB model under the design flow rate is shown in Fig. 7. In general, the trends of the PAT external characteristic curves at different gas contents are basically the same when using the UB model and the MUSIG model. With the increasing gas content, the PAT efficiency decreases, and the higher the gas content, the faster the decline. The variation trend of the head curve is opposite to the efficiency. Under the same gas content, the efficiency of the MUSIG model is slightly lower than that of the UB model, and the head is slightly higher than that of the UB model. The reason may be that the MUSIG model considers the breakup and coalescence of bubbles, which increases the energy loss during the internal flow of the turbine. In addition, with the increase in the gas content, the difference between the two models increases, indicating that the MUSIG model has a greater influence on the numerical simulation results at high gas content, which is also consistent with the results of Qiaorui *et al.* in the multistage electrical submersible pump.^{30}

As shown in Fig. 8, the impeller is divided into six flow channels, starting from the closest cut-water channel, named C1–C6, respectively. In order to examine the cause of the external characteristics of the MUSIG model and the UB model are slightly different precisely, the gas phase distribution contours are shown in Fig. 10. Based on the gas-phase distribution, it can be found that there is almost no difference between the two models at low gas content [Fig. 9(a)], and significant differences are observed at high gas content [Figs. 9(b)–9(d)]. It can be clearly seen from Fig. 9(b) that there is a significant difference in the gas phase distribution between the two models at 15% gas content. When the MUSIG model is used, the phenomenon of gas–liquid separation in the impeller and volute is intensified. On the whole, the air volume fraction on the left side is significantly higher than that on the right side. Especially in the volute, the gas content gradually decreases with the flow direction. In the UB model, the gas-phase distribution in the volute flow channel is relatively uniform, and the whole flow channel is almost occupied by 15% gas content. Compared with the MUSIG model, the gas content of each flow channel of the impeller under the UB model is more evenly distributed. The gas in each flow channel of the impeller is distributed on the suction surface. The gas content in the left flow channel is higher than that in the right flow channel, which is determined by the asymmetry of the volute. At 30% and 45% gas content, the gas phase distribution of each impeller channel using the UB model is consistent, while under the MUSIG model, the gas phase distribution in different impeller channels is significantly different. The gas phase is concentrated in the C2–C5 channel on the left side, with the least distribution in the C1 channel. The reason may be that the MUSIG model considers the breakup and coalescence of bubbles, and small bubbles merge into large bubbles. Due to the large flow resistance of large bubbles, the velocity difference between gas and liquid is large, and the gas chamber is formed.

### B. Effect of gas content on external characteristics of PAT under different flow rates

The operating conditions of the hydraulic turbine are generally in the range of 0.7*Q*_{d}–1.3*Q*_{d};^{31,32} thus, the external characteristic curves of different gas contents in the flow range of 0.63*Q*_{d}–1.2*Q*_{d} are shown in Figs. 10 and 11. It can be seen from the efficiency curve that the design flow rate under pure water conditions is 81 kg/m^{3}. When the gas content is 5%, the best efficiency point (BEP) is 74 kg/m^{3}. When the gas content is 15% and 30%, the BEP is 66 kg/m^{3}. It is worth noting that the efficiency is higher under small flow conditions at 45% gas content, and the efficiency decreases rapidly with the increase in the flow rate. As the gas content increases, the optimal efficiency point gradually shifts toward smaller flow rates. The head curve has the same change trend as the shaft power curve. Under the same flow rate, as the gas content increases, both the head and the shaft power increase, and the higher the gas content, the greater the increase. At a flow rate of 81 kg/m^{3}, from 0% to 15% gas content, the head and shaft power increase by 40.8% and 25%, respectively. From 30% to 45% gas content, the head and shaft power increased by 69.6% and 69.6%, respectively. In addition, the greater the flow rate, the faster the growth rate of the head and shaft power value.

### C. Effect of gas content on internal flow characteristics of PAT

With the purpose of determining the internal flow characteristics in the impeller at the design flow rate, the location of the cascade plane and the velocity streamline at span 0.5 of the impeller cascade are shown in Figs. 12 and 13. According to Fig. 13, the backflow phenomenon occurs on the pressure side of the inlet. A velocity minimum is observed at the leading edge of the blade on the suction side, and a vortex is formed in the low-speed region. The maximum velocity is observed at the trailing edge of the suction side blade and the pressure side close to the vortex. As the gas content increases, the number and size of vortices in the impeller flow channel increase. The vortex changes in the C1 flow channel closest to the cut-water of volute are most obvious. From 5% gas content to 15% gas content, a vortex appears in the C1 flow channel. At 30% gas content, the vortex in the C1 flow channel increases and occupies about 1/2 of the flow channel. When the gas content is 45%, the vortex in the C1 flow channel is larger, occupying about 2/3 of the flow channel. In addition, the vortex is formed at locations where the gas phase distribution in the impeller flow channel is concentrated. It is concluded that due to the inconsistent speed of the gas and liquid phases, gas and liquid separate inside the impeller; thus, vortexes are generated at the low-speed gas phase position.

According to Fig. 14, the pressure distribution at 0.5 of the impeller cascades spans, reflecting the pattern of gas content on the pressure distribution within the impeller. Higher pressures appear near the region of interaction between the impeller and the volute. The value of pressure gradually decreases with the flow direction. As the gas content increases, the maximum pressure gradually increases. It is worth mentioning that compared with 5%–30% gas content, the inhomogeneity of pressure distribution at 45% gas content is significantly increased, and a small range of the low-pressure region is formed at the trailing edge of the blade.

Due to the increase in the gas content, the internal flow characteristics of the impeller will be significantly changed, and the higher the gas content, the more disordered the flow (see Figs. 12 and 13). Therefore, the pressure fluctuation in the flow channel of the 45% gas content impeller is analyzed to explore the evolution of pressure fluctuation with time under high gas content. Moreover, the pressure fluctuation near the impeller inlet is usually large because of the rotor–stator interaction.^{33,34} Therefore, we set up six monitoring points on the suction surface and the pressure surface in the flow channel C1. The distribution of monitoring points on the pressure surface and suction surface of the blade is shown in Fig. 8, which are located at the trailing edge of the blade (P_{1}, P_{4}), the middle of the blade (P_{2}, P_{5}), and the leading edge of the blade (P_{3}, P_{6}), respectively.

*C*

_{p}by dimensionless treatment of the pressure. The pressure fluctuation coefficient

*C*

_{p}is expressed as follows:

*P*represents the static pressure (Pa), $P\u0304$ represents the average value of the measured static pressure (Pa), and

*u*

_{2}represents the circumferential velocity of the impeller outlet (m/s). In addition, the pressure fluctuation spectrum analysis of different monitoring points is carried out by the fast Fourier transform (FFT) method.

Figure 15 shows the time-domain diagram and frequency-domain diagram of the pressure fluctuation on the suction surface and the pressure surface of the blade at 45% gas content. Regarding the time domain characteristic [see Figs. 15(a) and 15(b)], it is found that the pressure fluctuation is periodically distributed, and the impeller rotates one circle as a period (T). In the flow channel of the impeller, the pressure fluctuation near the impeller inlet is larger than that of the outlet, which is consistent with the study of Posa and Zheng.^{33,34} On the pressure surface, the amplitude of P_{6} is about two times that of P_{5} and four times that of P_{4}. The value of pressure fluctuation gradually decreases with the flow direction, and the amplitude of pressure fluctuation at the impeller outlet is the smallest, indicating that the flow gradually tends to be stable. In addition, the overall pressure fluctuation amplitude of the suction surface is smaller than that of the pressure surface. It is worth noting that the pressure fluctuation amplitude of P_{3} increases greatly at the last 1T/3. The reason may be that the blade rotates to the cut-water of volute position at this moment, resulting in more disordered flow. The frequency-domain characteristics of pressure fluctuation at 45% gas content during one-cycle rotation of the impeller are illustrated in Fig. 17(c). Points P_{4}–P_{6} on the suction surface are dominated by rotational frequency, whereas points P_{2} and P_{3} on the pressure surface show multiplicative rotational frequency, and there is a clear frequency-domain feature near the blade frequency, which may be related to the evolution of the gas–liquid two-phase flow pattern in the volute and impeller.

### D. Bubbles size distribution with different gas contents

The MUSIG model divides the bubbles into several groups for numerical simulation, and the breakup and coalescence of bubbles are considered so that the bubble diameter distribution in PAT can be well displayed. The bubble size distribution in the impeller and volute of PAT is shown in Fig. 16. As shown in the figure, in the impeller and volute, at each gas content, the bubble distribution with a diameter of 0.01 mm accounts for the highest proportion. The small bubbles with a diameter of 0.01 mm in the volute account for about 40% the large bubbles with a diameter of 0.2 mm account for about 20%–30%, and the remaining medium-sized bubbles (d_{2}, d_{3}, d_{4}) account for a very small proportion. This is most likely because some of the medium-sized bubbles in the flow path break up into smaller bubbles while others merge into larger bubbles. In the impeller of PAT, small bubbles with a diameter of 0.01 mm account for about 65%, followed by small bubbles with a diameter of 0.02 mm, accounting for about 20%. Small bubbles dominate in the impeller, which can be explained by the fact that due to the rotation of the impeller blades, the large bubbles flowing from the volute to the impeller are crushed into small bubbles by the high-speed rotating blades. The distribution of bubble diameter under different gas contents is analogous. It is worth mentioning that in the impeller and volute, the proportion of large bubbles with a diameter of 0.2 mm increases with the increase in the gas content. In the impeller and volute, at 5% gas content, the proportion of large bubbles with a diameter of 0.2 mm is 2.64% and 21.5%, respectively. At 45% gas content, the ratio of large bubbles with a diameter of 0.2 mm increases to 11.6% and 38.2%, respectively. It is indicated that the coalescence of bubbles will be more significant at high gas content than at low gas content. In addition, in the impeller and volute of PAT, the proportion of small bubbles with a diameter of 0.01 mm decreases with decreasing gas content. In the impeller and volute, at 5% gas content, the proportion of small bubbles with a diameter of 0.01 mm is 67% and 42.8%, respectively. At 45% gas content, the proportion of small bubbles with a diameter of 0.01 mm is reduced to 54.75% and 40%, respectively. It can be concluded that bubbles in PAT are more likely to break up at low gas content.

To deeply understand the effect of gas content on the bubble size distribution, the distribution of bubble diameter in the impeller and volute flow channels of PAT under different gas contents is shown in Figs. 17 and 18, respectively. As illustrated in Fig. 17, the gas with a large size distribution is mainly concentrated closer to the volute inlet. When the gas content is 5%, the distribution of large and small bubbles in the entrance area of the volute is not uniform and more dispersed. When the gas content exceeds 15%, the inlet area of the volute is occupied by large bubbles with a diameter of nearly 0.2 mm. It is worth noting that at 5% and 15% gas content, the distribution of small-diameter bubbles near the blade in the volute can be seen clearly. The reason is that a part of the bubbles is broken up by the rotation of the impeller in the volute flow channel. Along the contraction direction of the volute, due to the reduction in the volute section, the shear effect of the blade is enhanced, resulting in a gradual decrease in the bubble diameter along the contraction direction of the volute. The distribution of the bubble diameter at 15%, 30%, and 45% gas contents is roughly the same in the middle section of the volute, whereas the distribution of bubble diameters in the tail section of the volute shows that large diameter bubbles are more likely to aggregate when then the gas content above 30%. With the increase in the gas content, the distribution area of large-size bubbles gradually increases.

The bubble diameter in the impeller does not exceed 0.1 mm at 5% air content, as shown in Fig. 18. With the increase in the gas content, large bubbles gradually appear. The large bubbles in the flow channel are distributed on the pressure side of the leading edge of the blade. In addition, larger bubbles are distributed closer to the impeller center. Since the fluid converges at the center of the impeller, the gas phase distribution is concentrated, resulting in more obvious bubble aggregation (Fig. 19).

In order to reveal the time evolution of the bubble diameter distribution in the impeller, the distribution of the bubble diameter on the cascade plane at span 0.5 within the time of the impeller rotating one circle is provided in Fig. 20. It is obvious that in the impeller channel, most of the large-diameter bubbles are concentrated on the pressure side of the impeller outlet and the impeller inlet, while the other small parts are concentrated on the trailing edge of the blade.

Taking the change in the bubble diameter distribution in a complete cycle as an example, at 1/6T, position 1 is the region of the channel closest to the cut-water. It can be seen from the velocity streamline in Fig. 13 that a large vortex is formed on the suction surface of the impeller. The vortex causes the small bubbles generated by the shear breakage of the blade to converge into large-diameter bubbles. At 2/6T, position 1 rotates to position 2, where the relative size of the vortex decreases, the number increases from one to two, and the large-diameter bubbles decrease. Moreover, compared with position 1, the phenomenon of gas–liquid separation is intensified. When the impeller rotates to position 3, the bubble convergence area is more dispersed, and the average diameter is much smaller than the strong vortex area. When the blade rotates to position 4, the streamline becomes relatively flat, and the bubbles are evenly distributed throughout the impeller flow channel. Then, the impeller rotates to 1T again, and a large bubble gathering area similar to position 1 appears here. This is because as the blade passage approaches the cut-water of volute, the flow disorder intensifies with strong shear, resulting in the formation of vortices. Finally, as the fluid flows out of the impeller, large-diameter bubbles converge at the impeller outlet.

## V. CONCLUSIONS

In this work, the MUSIG model is used to investigate the flow characteristics and energy acquisition mechanism of PAT when the gas content is 5%, 15%, 30%, and 45%, respectively. The superiority of the MUSIG model compared to the traditional uniform bubble model is demonstrated. The major conclusions can be summarized as follows:

Compared with the traditional UB model, it can be concluded that the MUSIG model has better accuracy in predicting gas–liquid two-phase flow when compared with the experimental results. At the high gas content, the MUSIG model has a greater influence on the internal flow characteristics of PAT.

As the gas content increases, the number and size of vortices in the impeller flow channel increase. The vortex is formed at locations where the gas-phase distribution in the impeller flow channel is concentrated. Rotating the blade to the cut-water of the volute position will lead to more turbulent flow in the blade channel. The flow at the impeller outlet will be relatively stable and the flow close to cut-water will be disorder.

In the impeller and volute of PAT, the bubble distribution with a diameter of 0.01 mm accounts for the highest proportion of each gas content. The proportion of large bubbles with a diameter of 0.2 mm increases with the increase in the gas content. When the gas–liquid two-phase fluid enters the impeller from the volute, the small bubbles with a diameter of 0.01 mm increase by 24.2%, and the large bubbles with a diameter of 0.2 mm decrease by 18.86% under low gas content conditions. While under high gas content conditions, small bubbles with a diameter of 0.01 mm increased by 14.75%, and large bubbles with a diameter of 0.2 mm decreased by 26.6%.

The rotation of the impeller blade acts on the bubbles in the volute channel, resulting in a smaller diameter of the bubbles entering the impeller. The flow pattern in the flow channel changes from the air bag to the bubble flow with the rotation direction of the impeller. As the blade passage approaches the cut-water of volute, the flow disorder intensifies, resulting in the formation of vortices, and the broken small bubbles converge to form large bubbles. The outlet of the impeller is more prone to bubble consolidation under high gas content conditions.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52376036, 51906223, 52006197, and U2006221), the Science and Technology Key Plan Project of Xinjiang Production and Construction Corps (Grant No. 2021AA002), and the Science Foundation and Fundamental Research Funds of Zhejiang Sci-Tech University (ZSTU) (Grant No. 23242147-Y and 24242113-Y).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Hui Yang**: Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (equal). **Junhui Ying**: Writing – original draft (equal). **Tianyu Lu**: Investigation (equal). **Linmin Li**: Conceptualization (equal); Software (equal). **Xiaojun Li**: Supervision (equal); Validation (equal). **Yikun Wei**: Formal analysis (equal); Investigation (equal). **Zuchao Zhu**: Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

_{2}capture process based on Rectisol