Studies on Flow Characteristics of Gas–Liquid Multiphase Pumps Applied in Petroleum Transportation Engineering—A Review

Abstract

Flow and phase separation in gas–liquid multiphase pumps is easy to occur, which deteriorates their performance and mixed transportability. Many research achievements have been made in the experiment, CFD simulation and performance improvement of multiphase pumps. However, there are many challenges for the test technology, accurate numerical model development and gas–liquid flow control. This paper is mainly aimed at critically reviewing various technologies for experimental observation, flow calculation and analysis, and the optimization design of gas–liquid multiphase pumps. In this regard, the experimental results including the energy performance, flow pattern and bubble movement in the multiphase pump are presented in detail. Discussions on the turbulence model, multiphase flow model and bubble balance model are carried out for the flow prediction in such pumps. Various numerical results are presented, including energy performance, bubble distribution, vorticity, phase interaction and pressure fluctuation. Moreover, the flow control and optimization strategy are briefly introduced. Having carried out an extensive literature review of flow characteristics in multiphase pumps, the deficiencies of relevant fields and suggestions for future research direction are given.

1. Introduction

Oil and gas are important non-renewable resources, which support the development of the world’s economy. The “BP Statistical Review of World Energy 2021” reports that the global oil discovered reserves increased from 2012 to 2020, but global oil production declined in general [1].

In the process of oil resource development, a mixture of oil, gas, water and impurities directly extracted from oil wells need to be transported. The transportation model includes traditional separated transportation and mixed transportation. The former transports the extracted media separately through the separation system. The latter uses a multiphase pump to replace the oil pump and compressor, which saves the complicated separation equipment. Moreover, there is no need to lay two separate sets of gas and liquid transmission pipelines because of the shared oil and gas transportation pipeline. Therefore, compared with traditional separated transportation, the mixed transportation mode could improve transportation efficiency and reduces investment costs [2,3,4].

The gas–liquid multiphase pump is the key equipment in the multiphase transport of oilfield resource development [5,6]. The pump types mainly include the positive displacement pump and the vane pump [7]. The former is represented by the screw pump [8,9,10], which transports media through the volume variation between the mutually engaged screws. The latter is represented by the helical axial vane pump. Its working mode makes the medium obtain kinetic energy through the high-speed rotation of the impeller and convert it into pressure energy when passing through the guide vane [11,12]. Compared with the positive displacement pump, the vane pump has some advantages, such as large delivery flow, small volume, low manufacturing accuracy requirement and low sensitivity to solid particles [11,13,14].

Due to the existence of the gas phase, the flow in the multiphase pump is very complicated, accompanied by obvious flow separation and phase separation. Therefore, the energy performance of multiphase pumps is seriously worse than that of the single-phase flow pump [15,16]. The experiment is an effective means to obtain the energy performance and flow characteristics of multiphase pumps. The external characteristics of the pump under different conditions of rotating speeds, inlet gas void fractions (IGVFs) and flow rates can be obtained through performance testing. The flow pattern and flow parameters in the pump can be obtained using high-speed photography, PIV technology, LDV technology and high-precision sensors [17,18,19,20]. Due to the complex interface effect and relative velocity, the monitoring of relevant parameters for the two-phase flow is more difficult than that of the single-phase flow. Therefore, how to accurately measure gas–liquid flow parameters has become the major challenge in the experiment of multiphase pumps. Meanwhile, the pressure balance and uniform mixing between two phases should be ensured in the experiment, but in fact, it is difficult to achieve, especially for high IGVFs. Therefore, how to ensure the stability of test conditions is another important problem [21].

CFD simulation is widely adopted to predict the energy performance and internal flow of multiphase pumps due to its advantages of a short calculation period and low cost [22,23,24]. The stability of boundary conditions can be guaranteed in the CFD simulation. The energy performance, pressure fluctuation and gas–liquid flow characteristics could be obtained using CFD methods [25]. Moreover, the gas–liquid phase interaction information could be obtained with CFD methods, but it is difficult to achieve in the experiment [26,27,28]. Nevertheless, the CFD simulations of the gas–liquid multiphase pump are not flawless. The reliability and robustness of the turbulence model and phase interaction model need to be improved. The existing numerical model can only meet the requirements of CFD simulation for the multiphase pump with low IGVFs, but not for high IGVFs. As the IGVF is high, the gas–liquid phase separation and flow separation in the pump are more serious, resulting in the emergence of composite flow patterns with different mechanisms of formation, division and transition [27,29].

Due to the complexity of gas–liquid flow, the design of multiphase pumps has more requirements than single-phase flow pumps. It not only needs to avoid obvious phase separation and flow separation, but also needs to ensure strong pressurization capacity and high efficiency under conditions of high IGVFs. Unfortunately, limited by the design theory and calculation ability, the existing design methods cannot achieve the design of the multiphase pump with high energy performance and good transportability [13]. Therefore, the combination of CFD simulation and the optimization algorithm is often adopted to optimize the original multiphase pump. In the optimization design, the widely used optimization algorithms include design of experiment (DOE) [30], response surface methodology (RSM) [31,32] and multi-objective genetic optimization algorithm (MOGOA) [33]. The energy performance indicators of the pump are regarded as the optimization objectives [32], rather than the complex gas–liquid flow parameters [34]. The gas–liquid flow is not well controlled directly in the process of optimization design; thus, the energy performance of the multiphase pump has not been improved significantly.

To date, a substantial amount of research has been carried out on multiphase pumps. Some achievements have been made in the experiment, numerical model development, CFD simulation and analysis and optimization design. However, there is an absence of a single article that comprehensively summarizes the research results achieved so far. This is also the motivation for the current work. This work presents recent studies on the flow characteristics of multiphase pumps for obtaining an effective overview of the field. It is dedicated to disseminating the achievements of experimental measurement, CFD simulation and flow control of the gas–liquid multiphase pump. Firstly, the experimental results including the energy performance, flow pattern and bubble movement are introduced in detail. Secondly, a critical review of CFD simulations is carried out. It includes the turbulence model, multiphase flow model, bubble balance model, energy performance, flow characteristics, phase interaction and flow improvement strategy. Finally, a summary of the investigated literature, important findings and future directions is provided.

2. Multiphase Pump Type

The multiphase pump can be divided into a positive displacement pump and a vane pump. Table 1 shows the various types of multiphase pumps applied in the United States, Britain, Germany, Japan, China and Russia [35]. Table 2 shows the merit and defect of all types of multiphase pumps [35].

Table 1. Types of oil and gas multiphase pumps chosen by countries adapted from Ref. [35].

United States	Britain	Germany	Japan	China	Russia
Screw pump	Screw pump	Screw pump	Centrifugal pump	Screw pump	Screw pump
Liquid-ring pump	Axial pump	Vortex pump	Vortex pump	Liquid-ring pump	Liquid-ring pump
Centrifugal pump	/	/	Screw pump	/	/

Table 1. Types of oil and gas multiphase pumps chosen by countries adapted from Ref. [35].

United States	Britain	Germany	Japan	China	Russia
Screw pump	Screw pump	Screw pump	Centrifugal pump	Screw pump	Screw pump
Liquid-ring pump	Axial pump	Vortex pump	Vortex pump	Liquid-ring pump	Liquid-ring pump
Centrifugal pump	/	/	Screw pump	/	/

Table 2. Merit and defect of all types of the multiphase pump adapted from Ref. [35].

Types	Multiphase Pump
	Merit	Defect
Screw pump	High pressure (up to (5~12) × 105 Pa) High efficiency (20~50%) Large discharge capacity Simple structure	Small oil–gas ratio (up to 20) Easy temperature rise Sensitive to solid particles High manufacturing requirements High price
Liquid-ring pump	Large oil–gas ratio (up to 10~1000) Insensitive to solid particles Low price	Low pressure Low efficiency Small discharge capacity (120~600 m3/h)
Axial pump	Large discharge capacity (1500 m3/h) High efficiency Simple structure High pressure with a few units	Small oil–gas ratio (up to 10) performance deteriorates with the increased oil–gas ratio
Centrifugal pump	High pressure Large discharge capacity Simple structure Easy manufacturing Low price High efficiency	Small gas–liquid ratio
Vortex pump	Large discharge capacity	Small gas–liquid ratio (up to 10) Low efficiency Complex structure

Table 2. Merit and defect of all types of the multiphase pump adapted from Ref. [35].

Types	Multiphase Pump
	Merit	Defect
Screw pump	High pressure (up to (5~12) × 105 Pa) High efficiency (20~50%) Large discharge capacity Simple structure	Small oil–gas ratio (up to 20) Easy temperature rise Sensitive to solid particles High manufacturing requirements High price
Liquid-ring pump	Large oil–gas ratio (up to 10~1000) Insensitive to solid particles Low price	Low pressure Low efficiency Small discharge capacity (120~600 m3/h)
Axial pump	Large discharge capacity (1500 m3/h) High efficiency Simple structure High pressure with a few units	Small oil–gas ratio (up to 10) performance deteriorates with the increased oil–gas ratio
Centrifugal pump	High pressure Large discharge capacity Simple structure Easy manufacturing Low price High efficiency	Small gas–liquid ratio
Vortex pump	Large discharge capacity	Small gas–liquid ratio (up to 10) Low efficiency Complex structure

The positive displacement pump is represented by the screw pump, which is the main pressure-boosting equipment for mixed transportation in all major oilfields [36]. It can be mainly classified into the single-screw pump and twin-screw pump based on the number of mutually engaged screws [37]. The vane pump is represented by the helical axial multiphase pump, which mainly includes the supercharging unit, axial force balancing device, cooling system, sealing system and bearing support structure. Figure 1 and Figure 2 show the structure of the twin-screw pump [9] and helical axial multiphase pump [38], respectively.

3. Experimental Studies of Multiphase Pumps

The experiment is an effective means to obtain the energy performance and flow characteristics of the multiphase pump. It can provide basic data for the validation of CFD numerical methods.

3.1. Energy Performance Testing

The efficiency and head of the multiphase pump are important indicators to evaluate the level of hydraulic design. Through performance testing, the efficiency and head of the pump can be acquired under different conditions of IGVFs, rotating speeds and flow rates [17]. Xu et al. [39] conducted an experiment for a helical axial multiphase pump under different IGVFs. They found that the variation trend of the pump’s performance is similar to that at pure water conditions when the IGVF is lower than 15%. Moreover, the pump can operate stably at IGVF = 50% and be sustained momentarily at IGVF = 100%. When the IGVF becomes higher, the efficiency and head of the pump decrease. The hydraulic performance of the pump with different IGVFs is shown in Figure 3.

The variation in energy performance is closely correlated with the internal flow in the multiphase pump [18,40,41,42]. Bratu [41] conducted an experiment on the efficiency, operating stability and safety margin of Poseidon pump P300. The pump efficiency reduces significantly when the GVF oscillates and the slug flow pattern occurs. Zhao et al. [40] obtained the connection between flow pattern and energy performance through an experiment of a centrifugal pump at gas–liquid flow conditions, as shown in Figure 4. The pressurization and efficiency of the pump are almost unchanged by the entrained bubbles at the bubbly flow pattern. However, when the flow pattern is converted to agglomerated bubble flow and gas pocket flow, they decrease with the increase in IGVFs. Their downward trend becomes significant at the segregated flow pattern and a surge will occur in the pump.

3.2. Flow Patterns

One purpose of the experiment for the multiphase pump is to investigate the internal flow pattern. The flow pattern occurring in the pump depends on various factors such as the flow rate and physical properties of each phase, operating conditions and geometric structure [43,44]. The variation in flow pattern will make a great difference in the energy performance and transportability of the pump. Therefore, it is very important to explore the formation and transition mechanism of the flow pattern in the multiphase pump [45].

High-speed photography is usually applied to investigate various flow patterns in the gas–liquid multiphase pump. The isolated bubble flow, bubble flow, gas pocket flow and separated gas flow have been obtained using experimental observation and their characteristics have been summarized in detail [17,46,47]. Figure 5 shows four flow patterns observed in the first stage of a multiphase rotodynamic pump [17].

The flow patterns may be different in the impeller and guide vane of the multiphase pump. Cubas et al. [48] observed the flow characteristics in a radial centrifugal pump. They found that various flow patterns appear in the impeller, such as bubble flow, agglomerated bubble flow, gas pocket flow and annular flow, as shown in Figure 6. However, there is only one flow pattern in the guide vane for the whole range of tested operating conditions.

3.3. Bubble Motion

The motion and distribution of bubbles in the multiphase pump are complex due to the high-speed rotation of the impeller and the rotor–stator interaction [12,25,49]. Through the experimental technology, not only the flow pattern, but also the law of bubble movement and distribution in the multiphase pump could be obtained.

The bubble size is an important factor that affects the energy performance of the multiphase pump. As the IGVF increases and the rotating speed decreases, the bubble size usually becomes large, as shown in Figure 7 [50]. In addition, Zhang et al. [50] obtained a normal distribution of bubble size at the pump inlet by post-processing the high-speed photographic images of internal flow, as shown in Figure 8.

Figure 9 shows the bubble distribution in a multiphase rotodynamic pump at different IGVFs, which is obtained by high-speed technology [51]. Regardless of IGVFs (3%, 9%, 15%, 21%), the gas mainly gathers close to the impeller outlet and the blade suction surface. As the IGVF increases, the degree of gas accumulation in the impeller and guide vane becomes higher, as marked by yellow and red circles in Figure 9d. The law of bubble motion is similar at different IGVFs, as shown in Figure 10. The bubbles in the impeller move from the blade pressure surface to the blade suction surface along a similar path, and their sizes become smaller as they hit the blade wall.

4. Numerical Methods on Flow Mechanism

As computer technology and numerical algorithms advance, CFD technology becomes popular for flow calculation in the multiphase pump. It can eliminate the limitations of test conditions and test technology and provide an efficient means to comprehensively obtain the internal flow characteristics in the multiphase pump.

4.1. Turbulence Model

The turbulence model is the key factor that affects the accuracy of CFD simulation for multiphase pumps, but there is no unified model [52,53]. At present, the two-equation models are usually adopted to predict the gas–liquid flow in the pump, which includes k-ε [25,54,55,56], RNG k-ε [57,58,59] and SST k-ω [27,60,61,62,63] models. The information on these three turbulence models and their application to the flow prediction in the multiphase pump are summarized in Table 3 [64].

The k-ε model was proposed by Launder and Spalding in 1972 [65]. This model has a good prediction for the overall flow in the pump, but not for the large curvature and separated flow due to the neglect of turbulence anisotropy. According to renormalization group theory, Yakhot and Orszag established the RNG k-ε model in 1986 that could better solve the flow with a large curvature, strong rotation and high strain rate [66]. The SST k-ω model was developed by Menter, which combines the Wilcox k-ω model with the standard k-ε model [67]. The SST k-ω model has both the advantages of the Wilcox k-ω model in the near-wall region and the k-ε model in free shear layers.

Table 3. Turbulence models applied in multiphase pumps reprinted with permission from Ref. [64].

Turbulence Model	Author	Object	Main Conclusion
k-ε	Zhu et al. (2017) [68]	Three-stage centrifugal ESP	The relationship between bubble size and GVF is determined.
	He et al. (2020) [57]	Centrifugal pump	There is a great scattering of the void fraction at high-rotation speeds.
	Caridad et al. (2008) [55]	Centrifugal pump	The gas pocket increases hydraulic losses.
	Pineda et al. (2016) [56]	Centrifugal ESP	The head decreases at higher gas flow rates and lower intake pressures.
RNG k-ε	Ma et al. (2020) [58]	Reciprocating multiphase pump	The instantaneous two-phase distribution and fluctuation characteristics at high GVFs are determined.
	Liu et al. (2020) [59]	Three-stage helico-axial multiphase pump	Increases in viscosity and blade height as well as decreases in flow rate increase turbulent kinetic energy.
SST k-ω	Kim et al. (2015) [60]	Helico-axial multiphase pump	Hydrodynamic performance is improved by an optimization design method.
	Yu et al. (2015) [27]	Axial multiphase pump	Drag force is dominant and turbulent dispersion force can be neglected.
	Stel et al. (2015) [61]	Mixed-type ESP	Pressure difference and flow pattern are obtained at different flow rates.
	Pervaiz et al. (2022) [62]	Centrifugal pump	Turbulence increases with the increase in GVF in the impeller.
	Shu et al. (2022) [63]	Helico-axial multiphase pump	The relationship between vorticity and enstrophy dissipation rate is obtained.

Table 3. Turbulence models applied in multiphase pumps reprinted with permission from Ref. [64].

Turbulence Model	Author	Object	Main Conclusion
k-ε	Zhu et al. (2017) [68]	Three-stage centrifugal ESP	The relationship between bubble size and GVF is determined.
	He et al. (2020) [57]	Centrifugal pump	There is a great scattering of the void fraction at high-rotation speeds.
	Caridad et al. (2008) [55]	Centrifugal pump	The gas pocket increases hydraulic losses.
	Pineda et al. (2016) [56]	Centrifugal ESP	The head decreases at higher gas flow rates and lower intake pressures.
RNG k-ε	Ma et al. (2020) [58]	Reciprocating multiphase pump	The instantaneous two-phase distribution and fluctuation characteristics at high GVFs are determined.
	Liu et al. (2020) [59]	Three-stage helico-axial multiphase pump	Increases in viscosity and blade height as well as decreases in flow rate increase turbulent kinetic energy.
SST k-ω	Kim et al. (2015) [60]	Helico-axial multiphase pump	Hydrodynamic performance is improved by an optimization design method.
	Yu et al. (2015) [27]	Axial multiphase pump	Drag force is dominant and turbulent dispersion force can be neglected.
	Stel et al. (2015) [61]	Mixed-type ESP	Pressure difference and flow pattern are obtained at different flow rates.
	Pervaiz et al. (2022) [62]	Centrifugal pump	Turbulence increases with the increase in GVF in the impeller.
	Shu et al. (2022) [63]	Helico-axial multiphase pump	The relationship between vorticity and enstrophy dissipation rate is obtained.

4.2. Turbulence Model Two-Phase Flow Model

The two-phase flow model is also important in determining the numerical accuracy of the flow prediction in the multiphase pump [69]. It is mainly divided into the Euler–Lagrange model and the Euler–Euler model. The former is represented by the discrete phase model [70], which can more easily obtain the velocity and motion trajectory of the discrete phase. The latter has mainly experienced the development of the homogeneous model, diffusion model and two-fluid model [71]. In the homogeneous model, the two media are equivalent to a pure substance. The precondition is that the kinetic and thermodynamic properties of the two uniformly mixed media are similar or their velocity difference is small [72]. The diffusion model is used to describe the motion of a binary mixture and the variation in flow parameters, which combines the average motion and diffusion motion of the mixture [71]. The two-fluid model is widely applied in the flow prediction of multiphase pumps because of its relatively high calculation accuracy [27,73,74]. In this model, each medium has its conservation equations, which are coupled together through phase interactions [75]. The detailed comparison between the Euler–Lagrange model and the Euler–Euler model is shown in Table 4.

Table 4. Numerical methods for gas–liquid two-phase flow.

Numerical Methods	Two-Phase Flow Model	Feature	Disadvantage
Euler–Lagrange method	Discrete phase model [70]	The velocity and motion trajectory of the discrete phase are easily obtained.	Suitable for IGVF of less than 10% and very low computational efficiency.
Euler–Euler method	Homogeneous model [72]	The two media are equivalent to a pure substance.	Ignored the phase interaction and large calculation error.
	Diffusion model [71]	The average motion and diffusion motion are considered.	Not applicable for large diffusion speed.
	Two-fluid model [27,73,74]	Each medium has its conservation equations, and phase interactions are considered.	Low calculation efficiency and difficulty in convergence.

Table 4. Numerical methods for gas–liquid two-phase flow.

Numerical Methods	Two-Phase Flow Model	Feature	Disadvantage
Euler–Lagrange method	Discrete phase model [70]	The velocity and motion trajectory of the discrete phase are easily obtained.	Suitable for IGVF of less than 10% and very low computational efficiency.
Euler–Euler method	Homogeneous model [72]	The two media are equivalent to a pure substance.	Ignored the phase interaction and large calculation error.
	Diffusion model [71]	The average motion and diffusion motion are considered.	Not applicable for large diffusion speed.
	Two-fluid model [27,73,74]	Each medium has its conservation equations, and phase interactions are considered.	Low calculation efficiency and difficulty in convergence.

The interactions between phases in multiphase flow are calculated by the phase interaction model. The interaction model of two-phase flow mainly includes drag, added mass force, turbulence dispersion force, lift, Bassett force and Magnus force. Besides the interphase forces, the inertia force and pressure difference force are acting on the bubbles. Therefore, the force balance equation acting on bubbles can be expressed as Equation (1).

$$F_D{+F}_L{+F}_A{+F}_T{+F}_B{+F}_M{+F}_S{+F}_i{+F}_p=0$$

(1)

Table 5. Information on the forces acting on bubbles.

Items	Symbol	Expression
Drag	FD	$F_D=\frac{3}{4}{C}_D\frac{{\rho }_l}{d}{a}_g\left|V_g-V_l\right|\left(V_g-V_l\right)$
Lift	FL	$F_L=C_L{\alpha }_g{\rho }_l\left(V_g-V_l\right)\times \left(\stackrel{\sim }{N}\times V_l\right)$
Added mass force	FA	$F_A{=-\rho }_l{C}_A{\alpha }_g\left(\frac{{DV}_g}{Dt}-\frac{{DV}_l}{Dt}\right)$
Turbulent dispersion force	FT	$F_T{=-C}_T{\rho }_{l}k\stackrel{\sim }{N}{\alpha }_l$
Basset force	FB	$F_B=\frac{3}{2}{d}^2{\rho }_c\sqrt{\pi \nu }{\displaystyle {\int }_{t_0}^t\frac{\xi \left(t^{\prime }\right)}{\sqrt{{t-t}^{\prime }}}}{dt}^{\prime }$
Magnus force	FM	$F_M=\frac{1}{8}{\pi d}^3{\rho }_c\omega \left(u_c{-u}_p\right)$
Staffman force	FS	$F_S=1{.62d}^2\sqrt{{\rho }_c\mu }\left(u_c{-u}_p\right)\sqrt{\left|\frac{{du}_c}{dy}\right|}$
Inertia force	Fi	$F_i=-\frac{1}{6}{\pi d}^3{\rho }_p\frac{{du}_p}{dt}$
Pressure difference force	Fp	$F_p=-\frac{1}{6}{\pi d}^3\frac{dp}{dx}$

Table 5. Information on the forces acting on bubbles.

Items	Symbol	Expression
Drag	FD	$F_D=\frac{3}{4}{C}_D\frac{{\rho }_l}{d}{a}_g\left|V_g-V_l\right|\left(V_g-V_l\right)$
Lift	FL	$F_L=C_L{\alpha }_g{\rho }_l\left(V_g-V_l\right)\times \left(\stackrel{\sim }{N}\times V_l\right)$
Added mass force	FA	$F_A{=-\rho }_l{C}_A{\alpha }_g\left(\frac{{DV}_g}{Dt}-\frac{{DV}_l}{Dt}\right)$
Turbulent dispersion force	FT	$F_T{=-C}_T{\rho }_{l}k\stackrel{\sim }{N}{\alpha }_l$
Basset force	FB	$F_B=\frac{3}{2}{d}^2{\rho }_c\sqrt{\pi \nu }{\displaystyle {\int }_{t_0}^t\frac{\xi \left(t^{\prime }\right)}{\sqrt{{t-t}^{\prime }}}}{dt}^{\prime }$
Magnus force	FM	$F_M=\frac{1}{8}{\pi d}^3{\rho }_c\omega \left(u_c{-u}_p\right)$
Staffman force	FS	$F_S=1{.62d}^2\sqrt{{\rho }_c\mu }\left(u_c{-u}_p\right)\sqrt{\left|\frac{{du}_c}{dy}\right|}$
Inertia force	Fi	$F_i=-\frac{1}{6}{\pi d}^3{\rho }_p\frac{{du}_p}{dt}$
Pressure difference force	Fp	$F_p=-\frac{1}{6}{\pi d}^3\frac{dp}{dx}$

The information on each force is shown in Table 5. These forces are not equally important and are not necessary to be calculated completely. Therefore, it is of great significance to determine the important force through the magnitude analysis of forces.

4.3. Bubble Balance Model

Due to the high-rotational speed of impeller and rotor–stator interaction, bubbles in the pump will merge and split. Murakami and Minemura [76] adopted an empirical formula to correlate the observed bubble sizes in centrifugal pumps. Barrios [68] and Gamboa [77] developed a bubble-size prediction model from the experimental results in ESPs. Zhu et al. [25] developed a mechanistic model according to the maximum stable bubble for correlating the CFD simulated bubble sizes. The expression of the maximum bubble size is shown in Equation (2). Moreover, the accuracy of the bubble size model is verified by comparison with other models, as shown in Figure 11.

$$d_{\mathit{max}}{=C}^{\ast }{\lambda }_G{\left(\frac{{We}_{\mathit{crit}}^{\prime }}{2}\right)}^{3/5}{\left(\frac{\sigma }{{\rho }_c}\right)}^{3/5}{\left(\frac{\Delta Pq}{{\rho }_{c}V}\right)}^{-2/5}{\left(\frac{{\rho }_c}{{\rho }_d}\right)}^{1/5}$$

(2)

Our team has carried out some research on the CFD prediction of bubble size in the multiphase rotodynamic pump. According to the research results of the gas–liquid stirred tank obtained by Lane et al. [78], a conservation equation considering bubbles’ coalescence and break-up has been developed for the flow prediction in the multiphase rotodynamic pump [79]. Its accuracy is verified by comparison with the experimental data under different IGVFs, as shown in Figure 12.

$$\frac{\partial \mathit{n}\left(\mathit{x},\mathit{t}\right)}{\partial \mathit{t}}+\nabla ·\left({\mathit{nU}}_2\right){=\psi }_{\mathit{br}}{-\psi }_{\mathit{co}}$$

(3)

5. CFD Analysis of Energy Performance

The energy performance of the multiphase pump is crucial for evaluating its comprehensive characteristics. Due to the existence of gas, the energy performance of the multiphase pump is worse than the single-phase flow pump. Moreover, the gas content at the pump inlet is not constant during actual operation. This is the reason why the energy performance of a multiphase pump under different IGVFs is of great interest. Figure 13 shows the variation trend of the pump head with different IGVFs [79]. Regardless of the experiment and CFD simulation, the pump head decreases gradually as the IGVF increases. Especially, the pump head at IGVF = 21% is only about 91% of that at IGVF = 3%.

In the actual operation process, the multiphase pump is usually composed of different unit stages to meet the requirement of pressurization. This is the reason why the energy performance of the multistage multiphase pump is widely concerned [80]. After investigating the energy performance of a multiphase pump with three stages, Zhang et al. [73] obtained that the average static pressure decreases slightly at the rotor–stator region, but overall increases with the increased stage numbers, as shown in Figure 14.

6. CFD Analysis of Flow Characteristic

The gas–liquid phase separation is prone to occur in the multiphase pump. In fact, an air blockage may appear in the pump, which aggravates the flow separation and forms large-scale vortices [81,82]. Therefore, the energy performance and the operation stability of the pump will decrease.

6.1. Gas Distribution Law

Due to the density difference between phases, the centrifugal force of the liquid with a large density is greater than that of the gas phase, leading to the liquid and gas moving towards the shroud and hub, respectively [51]. CFD simulation is one effective method and is widely used in the flow prediction of the multiphase pump, which has the advantages of short calculation time and low cost [83].

Based on the Eulerian–Eulerian model and population balance model, He et al. [57] carried out CFD simulations for the gas–liquid flow in a centrifugal pump with the inlet bubbly flow pattern. They found that the air pocket is not completely filled with the impeller passage and some liquid exists between the rear cover and gas pocket, as shown in Figure 15. Figure 16 displays the relationship between the gas–liquid flow characteristics and the pump performance. When the IGVF is small (e.g., IGVF = 1~3%), the head and efficiency of the pump are slightly decreased. As the degree of gas accumulation increases, the pump performance decreases significantly.

6.2. Vorticity Characteristics

The flow separation and phase separation often occur in the gas–liquid multiphase pump, which accelerates the formation and development of the vortex [81,82,84]. Zhang et al. [81] conducted a CFD simulation for the gas–liquid flow in a multiphase rotodynamic pump with two unit stages. They found that the gas strongly accumulates in the first unit stages, leading to a complex internal flow, as shown in Figure 17. Moreover, obvious flow separation and large-scale vortices occur in the guide vane.

To deeply investigate the gas–liquid flow characteristics, the Q criterion is widely used to investigate the flow structure in the multiphase pump. The expression of the Q criterion is as follows [85,86]:

$$\mathit{Q}=\frac{1}{2}{(\Omega }_{\mathit{ij}}{\Omega }_{\mathit{ij}}{-\mathit{S}}_{\mathit{ij}}{\mathit{S}}_{\mathit{ij}})$$

(4)

S and Ω are respectively the symmetric and anti-symmetric components of the velocity tensor matrix. They can be calculated as follows:

$$\mathit{S}=\left(\nabla \mathit{V}+{\left(\nabla \mathit{V}\right)}^{\mathit{t}}\right)/2$$

(5)

$$\Omega =\left(\nabla \mathit{V}-{\left(\nabla \mathit{V}\right)}^{\mathit{t}}\right)/2$$

(6)

As the IGVF increases, the vortices in the pump are usually more obvious and the vortex structure becomes more complicated. Parikh et al. [24] investigated the characteristics of the vortex and wake in the impeller under different IGVFs by combining the Q criterion, as shown in Figure 18. The flow separation is obvious near the suction surface at the blade leading edge, while the weak flow separation occurs near the pressure surface at the blade trailing edge, as the marked separation vortices 1 and 2, respectively. Meanwhile, the vortex structure (black iso-surface) in the impeller becomes more obvious and more complicated with the increase in IGVFs.

6.3. Phase Interaction Characteristics

Phase interactions are the research focus and are difficult in the field of multiphase flows. Based on current detection technology, it is difficult to obtain detailed information on the multiphase flow interface parameters using experimental methods. With the progress and development of computing technology, CFD simulation is becoming an effective way to investigate the phase interaction of multiphase flows.

The complex phase interaction mode is the fundamental reason for the complex flow in the multiphase pump. Magnus and Saffman forces are generally considered to be negligible in the multiphase rotodynamic pump [72]. The drag, added mass force, lift and turbulent dispersion force in the pump have been explored in recent years. Their magnitude and variation laws are revealed gradually. Figure 19 is the numerical results of interphase forces in the impeller of a multiphase rotodynamic pump obtained by Yu et al. [27]. It shows that the drag is most important, followed by the lift and added mass forces, while the turbulent dispersion force is the smallest and its magnitude can be ignored.

The magnitude and variation laws of interphase forces are not constant, which varies with the pump’s operation conditions and the medium characteristics. Zhang et al. [80] analyzed the interphase forces in a gas–liquid multiphase pump with two unit stages. Figure 20 shows the variation trend of interphase forces with air–water and air–oil combinations. For the air–water combination, the ratios of lift/drag (F_L/F_D) and added mass force/drag (F_A/F_D) in the rotor–stator region are larger than 1, which indicates that drag is not the dominant force herein. For the air–crude combination, the ratios of lift/drag (F_L/F_D) and added mass force/drag (F_A/F_D) are generally less than 1, which illustrates that drag plays a leading role.

6.4. Pressure Fluctuation Characteristics

The pressure fluctuation in various types of single-phase pumps, such as centrifugal [87,88,89], axial flow [90,91,92] and mixed flow [93,94,95], has been investigated thoroughly. The relevant research results show that the rotor–stator interaction is the key factor to produce pressure fluctuation in the single-phase flow pump.

The pressure fluctuation in the gas–liquid multiphase pump is complex due to the coalescence and break-up of bubbles, the pulsation of phase content and the separation and mixing of gas–liquid phases [96]. Zhang et al. [97] analyzed the pressure fluctuation in a multiphase pump under different IGVFs. They obtained results showing that, regardless of IGVFs, the pressure fluctuation of point S1C at the guide vane inlet is the largest, as shown in Figure 21. Meanwhile, an interesting phenomenon emerges, showing that the weak pressure fluctuation of point R4C near the impeller outlet is obtained. This is attributed to the complex gas–liquid flow, therein weakening the rotor–stator interaction. Moreover, the inlet bubble diameter is another important factor that affects the pressure fluctuation in the multiphase pump. The pressure fluctuation in the pump usually increases with the increased inlet bubble diameter.

7. Flow Improvement Strategy of Multiphase Pump

At present, the mechanism of flow separation and phase separation in the gas–liquid multiphase pump is not clear enough, which is the reason why the complex gas–liquid flow cannot be controlled. Although relevant research on flow control in the multiphase pump has been widely carried out, a unified technical control strategy is absent. Generally, the current flow improvement strategies in the multiphase pump are divided two ways, i.e., indirect flow control strategy and direct flow control strategy.

7.1. Indirect Control Strategy of Gas–Liquid Flow

Due to the limitation of design theory and calculation ability, the three-dimensional inverse problem design method commonly used for the design of fluid machinery fails to directly obtain the multiphase pump with high performance. The indirect control for the gas–liquid flow in the pump is conducted by optimizing its hydraulic performance.

The combination of optimization design and CFD simulation is widely adopted to indirectly improve the gas–liquid flow in the pump. By combining Latin-hypercube sampling and response surface approximation, Suh et al. [13] carried out an optimization design on a multiphase rotodynamic pump with the objective of pump efficiency. The optimization result in Figure 22 shows that the maximum efficiency of the impeller is 77.01%.

Besides single-objective optimization, multi-objective optimization is also often widely adopted. Using the pressurization and efficiency of the pump as the optimization objectives, Zhang et al. [98] optimized a helico-axial multiphase pump by adopting an artificial neural network. After optimization, the pressurization and efficiency of the impeller are significantly improved, as shown in Figure 23. Moreover, Figure 24 shows that the gas in the impeller distributes more evenly.

7.2. Direct Control Strategy of Gas–Liquid Flow

Recently, the direct control strategy of gas–liquid flow has been applied in multiphase pumps. The pressure fluctuation, phase separation and flow separation are controlled during the optimization design of such pumps.

Liu et al. [23] optimized a multistage pump handling gas–liquid flow. Based on the Oseen vortex, a theoretical model that calculates the velocity moment downstream of the guide vane was developed. Therefore, the inlet blade angle of the later-stage impeller can be optimized using this model. Figure 25 shows that the length of the high Reynolds dissipation area at the blade leading edge is decreased significantly after optimization, which is attributed to the more matched inlet blade angle determined by the developed optimization method.

Our team has conducted some work on the direct control of gas–liquid flow in the pump [34,99]. The optimization design of an impeller was conducted with its geometry and blade-loading parameters as variables, and the pump efficiency and gas–liquid uniformity as objectives. A pump with high efficiency, good transport capacity and small pressure fluctuation was developed by directly controlling the gas–liquid flow in the optimization design.

Figure 26 shows that the low-velocity region, large-scale vortex range and turbulent kinetic energy in the optimized impeller are reduced significantly. This illustrates that the gas–liquid flow state is significantly improved by using the direct control flow strategy. Moreover, the optimization results in Figure 27 show that large loading at the leading edge, a large slope for the middle straight line and a large negative high-pressure edge angle are helpful to improve the pump efficiency [99].

8. Summary of the Investigated Literature, Important Findings and Future Directions

This paper investigates plenty of literature published in recent decades on the experiment, numerical model development, CFD simulation and analysis and optimization design in multiphase pumps. During this period, the flow measurement technology and numerical method of such pumps are developed greatly, thus the gas–liquid flow characteristics are revealed deeply. Here, the investigated literature is summarized to have a better visual and specialization of the current work. Moreover, the important findings and future directions are introduced in this section as follows:

• In this article, about 67.7% of the surveyed literature belongs to 2011–2023, 19.2% belongs to 2001–2010 and 13.1% belongs to the year before 2000. The annual distribution of the literature is shown in Figure 28;
• To have a comprehensive understanding of the current research on multiphase pumps, the types of the investigated literature are summarized in Figure 29, which includes experiment, numerical model development, CFD simulation and analysis and optimization design;
• Various results of the multiphase pump including performance, bubble distribution, vorticity, phase interaction and pressure fluctuation have been obtained. However, there is no single review article that discusses a comprehensive research overview of the gas–liquid multiphase pump.

Experimental observation and CFD simulation are two means to obtain the gas–liquid flow characteristics of the pump. In the experiment, high-speed photography technology is widely applied. The distribution and movement laws of the gas–liquid phases can be obtained by post-processing the flow field images. However, due to the limitations in obtaining comprehensive gas–liquid flow information using experimental technology, CFD simulation has been becoming more popular in the past decade. The information on gas–liquid phase interaction, pressure fluctuation and gas void fraction distribution is easily obtained using CFD methods. The improvement of the CFD prediction ability for the multiphase pump, especially at high IGVF conditions, is one of the major tasks.

Although the existing optimization design technology with the combination of CFD methods can improve the performance of multiphase pumps to a certain extent, the efficiency and head of such pumps have not reached a satisfactory level so far. The direct control strategy of gas—liquid flow is the development direction to improve the performance and operation stability of the multiphase pump.

9. Conclusions

Experimental observation is an important means to obtain the flow pattern in the gas–liquid multiphase pump. The flow patterns could be divided into isolated bubble flow, bubble flow, gas pocket flow, separated gas flow, etc. Due to the high-speed rotation of the impeller, the bubble in the pump will merge and split. Moreover, the aggregation degree of the bubbles increases with the increase in IGVFs.

The turbulence model and multiphase flow model determine the prediction accuracy of the gas–liquid flow in pumps, but they have not yet been unified. The two-equation turbulence model and two-fluid model are used as the numerical model of multiphase flow. The research focus on multiphase pumps is to develop superior numerical models for high IGVFs, including the phase interaction model and bubble balance model.

CFD simulation has become a popular and efficient means to investigate the energy performance and flow characteristics of multiphase pumps. The energy performance such as the efficiency and head of the pump, as well as the complex gas–liquid flow characteristics such as gas distribution, vortex, pressure fluctuation and phase interaction, could be obtained conveniently using CFD methods. Direct and indirect flow control strategies have been applied to improve the comprehensive performance of multiphase pumps, but the effect is still unsatisfactory so far.