Investigations on affinity law under gas–liquid conditions in multistage radial and mixed-flow multiphase pumps Open Access

Pumps use moving mechanical parts such as blades, screws, and pistons to convert the mechanical energy of a prime mover into kinetic energy, pressure energy, and gravitational potential energy of a fluid. They are widely used in oil/gas development, chemical processing, aerospace engineering, agriculture, and other fields. Under abnormal working conditions, pumps will deliver gas–liquid two-phase fluid, as for example, in the case of a centrifugal pump starting with air in its volute (Huang , 2014; Sulzer, 2010). When a loss-of-coolant accident occurs in a nuclear reactor, uncondensed steam and water enter the nuclear main pump (Poullikkas, 2003; Lu , 2017). In a liquid oxygen/kerosene rocket engine, uncondensed oxygen may flow into the turbine pump together with liquid oxygen (Li , 2011). Equipment damage and serious accidents may happen, when a pump has to work under such gas–liquid conditions. With an improved blade profile, multiphase pumps can effectively pressurize gas–liquid–solid fluids within a certain range of gas volume fraction (Takacs, 2017). As a reliable pressurization method, multiphase pumps are widely used in important industrial processes, such as deep-sea long-distance oil/gas transportation and downhole oil–gas artificial lifts (Li , 2020; Zhu , 2018). The use of multiphase pumps in separation equipment can lead to substantial cost savings.

Owing to the complex geometry of a pump, the turbulent flow inside it cannot be accurately described by simple analytical methods. In practice, in guiding the design of new structures and for performance prediction based on test results from model pumps, the description of flow in pumps generally relies on the use of dimensionless numbers such as the Reynolds and Euler numbers (Zhu , 2018; Pei , 2021). On the basis of extensive research studies and practical experience, pump affinity laws based on the similarity theory of fluid mechanics have been established. Generalized pump designs can be developed on the basis of highly efficient model pumps using affinity laws. Affinity laws can be used to guide the cutting of impeller outer diameters to adapt to particular demands with regard to pump head. The application of affinity laws can effectively shorten the development cycle and reduce test costs by reducing the size of large pumps and to facilitate simulations of conveying high-temperature or corrosive fluids (Dhanasekaran and Kumaraswamy, 2020; Gülich, 2008; Zhu and Zhang, 2017, 2018).

In oil–gas transportation, the media are commonly gas–liquid two-phase and even multiphase fluids. It is of great value to study the applicability of affinity laws to multiphase pumps delivering gas–liquid two-phase fluids. Matsushita  2007 and 2009 experimentally studied the applicability of affinity laws under air–water conditions in the case of a single-stage impeller from three aspects: impeller outer diameter, blade height, and rotational speed. Two kinds of impeller with diameters of 190 and 235 mm were found to conform to an affinity law with rotational speeds of 1000–1600 rpm and gas–liquid flow rate ratios of 0–0.6. However, for blade heights of 13 and 20 mm, the heads predicted by the affinity law were found to be different from the experimental values. Si  (2018) studied the applicability of an affinity law to a single-stage volute centrifugal pump at different speeds (1800–2910 rpm) by experiment and numerical simulation. When the gas volume fraction was larger than 0.03, the gas distribution in the impeller changed, as a consequence of which the applicability of the affinity law becoming poorer. Morrison  (2018) focused on the application of an affinity law to a centrifugal pump delivering fluids with different viscosities. The affinity law for a pure water working fluid was modified for other fluids by the introduction of a flow rate coefficient Φ and a rotational Reynolds number Re_w. The experimental data on the head coefficient for different viscosities were normalized following the same Ψ–Φ(Re_w)^−a curve. Patil and Morrison (2019) predicted the head of a centrifugal pump for different viscosities based on the modified affinity law, and the predicted values were in agreement with experimental results. Current research to date on affinity laws has mainly focused on single-stage pumps and gas compressibility has been ignored. The application of affinity laws under gas–liquid conditions in pumps with higher stage numbers, high rotational speeds, and other blade structures needs further study.

Similarity theory indicates that two similar flow states need to satisfy geometric similarity, kinematic similarity, and dynamic similarity. For different pumps, geometric similarity and kinematic similarity can be determined by impeller structure and flow conditions. However, the complex structure of pumps and the large density difference between gas and liquid lead to an uneven spatial distribution of two-phase fluids (Ali , 2021). Experimental data indicates that the gas–liquid pressurization of centrifugal multiphase pumps is subject to a three-stage deterioration process as the inlet gas volume fraction increases (Zhu , 2017). Monte Verde  (2017); He  (2021); and Cubas  (2020) observed and classified four gas–liquid flow patterns inside a rotating impeller. The mechanism by which gas–liquid pressurization deteriorated in a centrifugal pump was revealed to be induced by a transition of the gas–liquid flow pattern in the impeller. Relations for predicting the critical gas volume fractions at which severe head degradation will occur were proposed by Gamboa and Prado (2011); Zhu  (2019); and Chang  (2022). Zhu  (2022) developed kinetic models for different gas–liquid flow patterns to predict the gas–liquid pressurization performance of a multiphase pump. Further study is needed to choose an appropriate parameter to judge dynamic similarity under gas–liquid conditions in multiphase pumps. In addition, in contrast to single-phase conditions, as gas and liquid flow rates are two-parameter variables, two independent variables are needed to determine the flow condition uniquely. Specific expressions for affinity laws under gas–liquid conditions in multiphase pumps and methods to quantitatively assess the applicability of affinity laws need to be further investigated.

This paper describes the construction of a high-pressure gas–liquid experimental platform and an experimental study of the applicability of an affinity law under gas–liquid conditions for three-stage and 25-stage radial multiphase pumps and a 15-stage mixed-flow multiphase pump at different rotational speeds. Based on those under single-phase conditions, expressions for the gas–liquid hydraulic parameters and corresponding dimensionless coefficients are proposed for multistage multiphase pumps, taking account of gas compressibility. The gas–liquid pressurization performances of radial and mixed-flow pumps at different rotational speeds are studied. The inlet gas volume fraction of the pumps is used to judge dynamic similarity. The goodness of fit is used to quantitatively evaluate the confidence that two-phase head and power coefficients conform to an affinity law. On the basis of experimental data, the applicability of the affinity law is verified for multiphase pumps with different stage numbers and blade structures under gas–liquid two-phase conditions.

The gas–liquid experiment on multistage radial and mixed-flow pumps is conducted on a high-pressure oil–gas–water experimental platform of industrial level developed by the State Key Laboratory of Multiphase Flow in Power Engineering. The maximum pressure of this platform is 30 MPa. A schematic of the experimental system is shown in Fig. 1. It consists of four main parts: a gas pipeline, a liquid pipeline, a multiphase pump test section, and a data acquisition and control system. Air and water are used as working fluids. The blue, green, and red sections in Fig. 1 represent the water pipe, air pipe, and air–water pipe, respectively.

Water is transported from the water tank by two high-pressure plunger pumps of the same type. The total maximum liquid flow rate is 29 m³ h⁻¹ and the maximum pressure is 30 MPa. The water flow rate is measured by a Coriolis mass flowmeter (RHM30FET20), with a range of 0–600 kg min⁻¹ and an accuracy of 0.15%. Air is supplied by an air compressor with a maximum flow rate of 1020 m³ min⁻¹ under standard conditions. An RHM015L flowmeter is used for low-flow-rate conditions, with a range of 0–0.6 kg min⁻¹ and an accuracy of 0.5%. An RHM06GET2 flowmeter is used for large-flow-rate conditions, with a range of 0–20 kg min⁻¹ and an accuracy of 0.2%. After flow rate measurement, the two-phase fluid flows into a straight pipe of thickness 7 mm and inner diameter 46 mm. The gas in the pipeline will move upward under the action of buoyancy. To enhance the mixing of gas and liquid at the pump inlet, a static gas–liquid mixer is installed upstream from the inlet with a horizontal length of 1.6 m. For flow rates ranging from 20 to 65 m³ h⁻¹, the inner diameter of the pipeline is 46 mm, and the flow time is 0.99–1.9 s from the mixer outlet to the pump inlet. This flow time is short, and there is not enough time for significant gas–liquid separation at the pump inlet. The stage numbers of the radial pump and mixed-flow pump studied are 25 and 15, respectively. The high-speed rotating impeller produces a strong shearing effect, which promotes the mixing of gas and liquid. The influence of inlet gas–liquid distribution on two-phase pressurization can be ignored. The gas–liquid fluid is pressurized and enters the gas–liquid separator to complete the separation of air and water. Pneumatic regulating valves are installed in the connecting pipelines of the gas–liquid separator to control the inlet pressure of the multiphase pump. In this experiment, the inlet pressure of multistage pump is controlled at a constant value of 0.5 MPa.

Two T-type thermocouples (Omega) are installed at the inlet and outlet of the multiphase pump to measure changes in temperature. The measurement deviation is less than 0.4% in the range of 0–200 °C. A variable-frequency motor is used to adjust the rotational speed of the multiphase pump. A speed and torque sensor is installed between the motor and the bearing box. The speed range is 0–6000 rpm, with an accuracy of 0.2%. The torque range is −1000 to 1000 N m, with an accuracy of 0.2%. The 4–20 mA standard current signals output by the sensors are collected by an NI-9253 acquisition module. The data are processed and stored by LabVIEW 2018 software. The sampling frequency is 100 Hz, with a sampling time of 1960s under stable conditions. The ranges of the experimental parameters are shown in Table I.

The hydraulic performances of three-stage and 25-stage radial pumps and a 15-stage mixed-flow multiphase pump under gas–liquid condition are studied. The hydraulic design parameters of the two multistage pumps are shown in Table II. The structures of the impeller and diffuser in the 25-stage radial pump are the same as those of the three-stage pump studied in our previous work (Chang , 2022). The flow rate at the best efficiency point of the radial pump is 28.5 m³ h⁻¹, and the head is 23.5 m. The designed rotational speed of the mixed-flow pump is 3500 rpm, with a designed flow rate of 75 m³ h⁻¹ and head of 17.5 m.

The inter-stage pressure inside the multistage pump increases stage by stage along the flow direction. The measurement points of the inter-stage pressure and differential pressure are carefully arranged as shown in Fig. 3. The pressure measuring holes are 8 mm in diameter in the middle part of the diffuser cavity, where the pressure is stable. O-ring seals are mounted between the diffusers and the pump casing to prevent any influence of inter-stage leakage on the differential pressure measurements. A large number of sensors are needed for the stage-by-stage arrangement of pressure measuring points in the 25-stage radial multiphase pump. To enable comparison with the pressurization performance of the three-stage pump, pressure and differential pressure measurement points are arranged every three stages in the 25-stage pump. In the 15-stage mixed-flow pump, differential pressure sensors are arranged in each stage. The pressure and differential pressure transmitters (Rosemount 3051S) used in the experiment have an accuracy of 0.075% in the range of −0.5 to 1 MPa.

In this study, the experimental errors mainly arise in the measurements of liquid/gas flow rate, pressure, temperature, rotational speed, and torque. The errors in the direct measurement parameters such as rotational speed and torque can be acquired by the measurement devices. The errors in the indirect measurement parameters such as gas volume flow rate and gas volume fraction can be calculated by error propagation according to Moffat (1988). The maximum relative error in the gas volume flow rate δQ_a/Q_a is 4.0% in the experiment, and the maximum absolute error in the inlet gas volume fraction δλ of the multiphase pumps is 3.4%. The detailed calculation is presented in the  Appendix.

To extend the application of the affinity law for a pump from water conditions to gas–liquid two-phase conditions, it is necessary to define the hydraulic parameters of a multistage multiphase pump under gas–liquid conditions. The main parameters are the two-phase flow rate, two-phase head, and the two-phase power and efficiency, as well as the corresponding two-phase dimensionless hydraulic coefficients. In particular, gas compressibility needs to be considered in defining and calculating the hydraulic coefficients, owing to the significant compression on the gas in multistage pumps.

This is consistent with the formula proposed by Mansour  (2018) under the hypothesis of isentropic compression.

The performances in terms of head, power, and efficiency of multistage radial and mixed-flow pumps are shown in Fig. 5 for different liquid flow rates. The head and power in the figure are average values for the whole multistage pump. In the flow rate range of 8–26 m³ h⁻¹, the head performances of the two pumps for one stage are close. With increasing flow rate, the heads of the two pumps gradually decrease and the hydraulic efficiencies gradually increase. In contrast to the shaft power of the radial pump, that of the mixed-flow pump decreases with increasing flow rate. The mixed-flow pump has a higher specific speed with a higher design flow rate at the best efficiency point. In the flow rate range of 8–26 m³ h⁻¹, the mixed-flow pump has higher shaft power and lower efficiency. At a flow rate of Q_w = 26 m³ h⁻¹, the average heads of the two pumps are the same.

According to the definitions of the dimensionless hydraulic coefficients in Sec. III D, the variations of the head coefficient Ψ_w and power coefficient Π_w with flow rate coefficient Φ_w at different rotational speeds (2500–3500 rpm) are shown in Fig. 6.

For the 25-stage radial pump, the Ψ_w–Φ_w data points at different rotational speeds are basically distributed on the same curve given by a quadratic function. Similarly, the Π_w–Φ_w data points at different rotational speeds are distributed on the same curve given by a cubic function. For the 15-stage mixed-flow pump, the experimental data points at different speeds are also distributed on the same curve. The power and head coefficients decrease approximately linearly with increasing flow rate coefficient. Therefore, under water conditions, the head and power performance of both the 25-stage radial pump and 15-stage mixed-flow pump satisfy an affinity law.

To judge the condition of dynamic similarity in multiphase pumps under gas–liquid conditions, the two-phase pressurization performances of radial and mixed-flow pumps at different rotational speeds are analyzed. The influence of rotational speed on the two-phase pressurization performance of the three-stage radial pump at different liquid flow rates Q_w = 10–22 m³ h⁻¹, is shown in Fig. 7. The single-phase boosting pressure ΔP_w at different rotational speeds is used to nondimensionalize the two-phase boosting pressure ΔP_m. With increasing inlet gas volume fraction, the dimensionless two-phase boosting pressure ΔP_m/ΔP_w decreases slowly at first, then rapidly, and finally slowly exhibits three obvious deteriorating trends. Two critical gas volume fractions λ_c1 and λ_c2 correspond to inflection points of the performance curve. The deteriorating trends of the two-phase boosting pressure can be divided into three stages separated by two critical gas volume fractions.

According to the experimental visualization results of Monte Verde  (2017) and Perissinotto  (2021), the transition of the gas–liquid flow pattern in the impeller is the main reason for the deterioration of the two-phase boosting pressure. The three stages of deterioration correspond to three flow patterns in the impeller: (I) bubble flow, (II) gas pocket flow, and (III) stratified flow. When the rotational speed is higher than the design speed, the data points of the normalized gas–liquid boosting pressure follow approximately the same distribution. In the speed range of 3500–4500 rpm, an increase in speed has no significant effect on delaying the occurrence of severe head degradation. The two-phase boosting pressure at different rotational speeds has a similar trend of variation with inlet gas volume fraction Si  (2018).

Figure 8 shows the variation of the inter-stage gas–liquid boosting pressure with inlet gas volume fraction in multistage radial pumps at different rotational speeds. In contrast to Fig. 7, with increasing stage number, when severe gas–liquid head degradation occurs, the gas–liquid pressurization performance curves at different speeds gradually separate. The critical gas volume fractions for head degradation increase clearly with stage number. At a rotational speed of 3500 rpm, the first and second critical gas volume fractions are 10.7% and 20.0%, respectively, for 4–6 booster stages. For 22–24 booster stages, the first and second critical gas volume fractions are 27.1% and 34.8%, respectively. After pressurization of the upstream booster stages, the inter-stage gas volume fraction decreases significantly along the flow direction, resulting in head degradation occurring at a higher inlet gas volume fraction in the downstream booster stages.

An increase in rotational speed can significantly increase the gas–liquid boosting pressure and reduce the gas volume fraction between stages. Therefore, at higher rotational speeds, the head degradation in the downstream stages is significantly delayed, which occurs at a higher pump inlet gas volume fraction. The gas compressibility results in the occurrence of different gas–liquid flow patterns in the booster stage at different positions under the same pump inlet conditions. When the rotational speed is 3000 rpm and the inlet gas volume fraction is 20%, there is stratified flow for 4–6 booster stages, where the pressurization ability is basically lost. For 10–12 booster stages, the performance curve indicates gas pocket flow in the booster stages, with obvious head degradation occurring. For 16–18 and 22–24 stages, the pressurization capacity is close to that under water conditions, indicating a bubble flow pattern in the booster stages.

Visualization of gas–liquid flow in a centrifugal impeller shows that increasing the rotational speed is conducive to reducing gas accumulation and delaying the occurrence of head degradation, by increasing the turbulent kinetic energy in impeller channels (Mansour , 2022). Figure 9 shows the variation of gas–liquid booting pressure in three-stage and 15-stage mixed-flow pumps at different rotational speeds and inlet gas volume fractions. With increasing rotational speed, the two-phase boosting pressures of three-stage and 15-stage mixed-flow pumps increase significantly. At a low gas volume fraction (λ = 10%), the two-phase boosting pressure is close to that in the case of a single phase, and increasing the rotational speed has a significant effect in improving the two-phase boosting pressure. At high gas volume fractions (λ = 30% and 40%), severe head degradation occurs in the mixed-flow pump, and the influence of rotational speed in improving the two-phase boosting pressure is very weak. There are similarities in pressurization curves between single-phase and two-phase conditions for the three-stage and 15-stage mixed-flow pumps, both of which conform to a quadratic function with increasing rotational speed.

The presence of gas always causes a decrease in pressurization capacity to different degrees. Thus, the boosting pressure under single-phase conditions is greater than that under two-phase conditions. The single-phase pressurization capacities of the two types of pumps are the same at a liquid flow rate of Q_w = 26 m³ h⁻¹. Therefore, Fig. 10 shows the two-phase pressurization performance curves for two pumps with three and 15 stages at a liquid flow rate of 26 m³ h⁻¹. For a low stage number (3), with increasing inlet gas volume fraction, gas–liquid head deterioration occurs in both pumps, corresponding to the three deterioration processes due to the transition of the flow pattern in the pump. The mixed-flow impeller has shorter and wider flow channels, which weaken the gas–liquid separation at higher gas volume fractions. Thus, the first and second critical gas volume fractions of the three-stage mixed-flow pump are much larger than those of the radial pump. The two critical gas volume fractions of the mixed-flow pump are 13.3% and 28.1%, respectively. The critical gas volume fractions of the radial pump are 4.8% and 9.8%.

In addition, among pumps with a higher stage number (15), a mixed-flow pump can still maintain high two-phase pressurization performance at high inlet gas volume fractions. Compared with a radial pump, two-phase pressurization of a mixed-flow pump changes more gently with inlet gas volume fraction. For the radial and mixed-flow pumps with three and 15 stages, the two-phase boosting pressures of the pumps are the same when inlet gas volume fractions are less than 5.2% and 7.2%, respectively. With increasing inlet gas volume fraction (λ > λ_c1), severe head degradation occurs in the radial pump. At a high gas volume fraction, there is a significant improvement in the two-phase pressurization of the mixed-flow pump compared with the radial pump.

According to the pump affinity law, when an experimental pump and model pump satisfy geometric similarity, kinematic similarity, and dynamic similarity, their hydraulic performances also have similar features. In the following, the applicability of an affinity law is analyzed for different rotational speeds in terms of the performances of the three-stage and 25-stage radial pumps and the 15-stage mixed-flow pump under gas–liquid conditions. Under single-phase conditions, at a constant rotational speed, the unique working condition of a pump is determined solely by the liquid flow rate. However, under gas–liquid two-phase conditions, at a constant rotational speed and pump inlet pressure, the gas flow rate and liquid flow rate are two-parameter variables, and at least two independent parameters are needed to determine the unique working condition.

The expression for the two-phase flow rate of multistage multiphase pump has been defined by analogy with that of a water pump, taking account of gas compressibility. In this paper, two parameters, namely, the two-phase flow rate Q_m and the inlet gas volume fraction λ, are used to determine the unique working condition. Moreover, the variation of the gas–liquid pressurization of a multiphase pump exhibits three typical deterioration processes with inlet gas volume fraction, corresponding to three gas–liquid flow patterns with different dynamic mechanisms. Therefore, the inlet gas volume fraction is used to judge dynamic similarity.

In Fig. 6, dimensionless hydraulic coefficients have been used to verify the applicability of the affinity law under single-phase conditions. The experimental data points at different rotational speeds are well distributed on the same curve, which conforms to the affinity law. In the following, the same method is used to study the affinity law for the three-stage radial pump under air–water conditions. At three rotational speeds n = 3500, 4000, and 4500 rpm, according to the critical gas volume fractions obtained from the pressurization curves in Fig. 7, inlet gas volume fraction of 5%, 10%, and 20% correspond to three gas–liquid flow patterns, namely, bubble flow, gas pocket flow, and stratified flow, respectively. The relationships between two-phase head and flow rate coefficients and between two-phase power and flow rate coefficients at different rotational speeds are shown in Figs. 11 and 12, respectively.

A quadratic function and a cubic function are used to fit the Ψ_m–Φ_m and Π_m–Φ_m data points by a least-square fitting approach. Similar to the single-phase condition, the Ψ_m–Φ_m and Π_m–Φ_m data points at the same inlet gas volume fraction conform to the same distribution under gas–liquid conditions. Therefore, the similarity law is applicable to the three-stage radial pump delivering gas–liquid fluids.

The R² values of the Ψ_m–Φ_m and Π_m–Φ_m fitting curves for the two-phase head and power coefficients and the two-phase flow rate coefficient of the three-stage radial pump are shown in Table IV. The inlet gas volume fractions are 5%, 10%, 15%, 20%, and 25%, respectively. R² is larger than 0.9 for different rotational speeds and flow patterns. Therefore, the affinity law has good applicability to the three-stage radial pump under two-phase conditions.

In multistage multiphase pumps, the inter-stage pressure increases significantly along the flow direction, and gas compressibility cannot be ignored. The applicability of the affinity law to radial pumps with higher stages is analyzed on the basis of the defined two-phase flow rate, head and power coefficients. Figures 13 and 14 respectively show the performances of the two-phase head coefficient and power coefficients for six-stage, 12-stage, 18-stage, and 24-stage radial pumps at different rotational speeds. In Figs. 13 and 14, the inlet gas volume fractions of 5%, 10%, 20%, and 30% cover three typical flow patterns of the multistage radial pump. Similar to the three-stage radial pump, a quadratic function is used to fit the experimental data points for the two-phase head and flow rate coefficients. The experimental data points for the two-phase power and flow rate coefficients are fitted by a cubic function. At low inlet gas volume fractions (λ = 5% and 10%) and a high inlet gas volume fraction (λ = 30%), the data points at different rotational speeds follow the same curve. The main reason for this is that most booster stages in multistage pumps are in the same flow pattern satisfying dynamic similarity. As shown in Fig. 8, most booster stages are in the bubble flow region at low inlet gas volume fractions (λ = 5% and 10%) before severe head degradation occurs. At a high inlet gas volume fraction (λ = 30%), most stages are in a stratified flow pattern, where severe head degradation has occurred.

At a moderate gas volume fraction (λ = 20%), severe head degradation occurs in only part of the upstream booster stages, and gas–liquid pressurizations are in flow patterns of gas pocket flow or stratified flow. However, head degradation does not occur in downstream stages, which are still in a bubble flow pattern. The gas–liquid pressurization performance of the pump is in an intermediate stage between severe degradation and no degradation. Therefore, data points at different speeds basically follow the same distribution at low stage number (six stages). However, for pumps with a higher stage number (12, 18, and 24 stages), the gas–liquid pressurization is in an intermediate stage. Dynamic similarity cannot be satisfied, and data points are more scattered. Thus, the applicability of the affinity law is poor in radial pumps with higher stages.

The goodness of fit R² is again used to quantitatively describe the confidence of the two-phase head and power conforming to the affinity law in multistage radial pumps at different rotational speeds. Table V shows R² of the Ψ_m–Φ_m and Π_m–Φ_m fitting curves for radial pumps with different stage numbers. For an inlet gas volume fraction of 20%, severe head degradation does not occur simultaneously in the upstream and downstream booster stages. Thus, the internal booster stages at different rotational speeds do not satisfy dynamic similarity, resulting in a low R² of the fitting curve. For other gas volume fractions, the multistage radial pumps do follow the affinity law well.

In the following, the applicability of the affinity law under gas–liquid conditions for three-stage and 15-stage mixed-flow pumps at different rotational speeds is studied. Figures 15(a) and 15(b) respectively show the variations of the two-phase head and power coefficients with the flow rate coefficient of the three-stage mixed-flow pump at different rotational speeds. At three inlet gas volume fractions (λ = 10%, 20%, and 30%), the experimental data points for four rotational speeds (2500–4500 rpm) follow the same distribution. Moreover, as can be seen from in Figs. 15(c) and 15(d), the affinity law shows good applicability to mixed-flow pumps with higher stage numbers (15 stages). The above results further confirm the rationality of the definitions of the two-phase head and power, taking account of gas compressibility.

To extend the applicability of the affinity law to different types of multiphase pumps, two-phase experimental data for a mixed-flow pump from Gamboa (2009) are analyzed. Gamboa (2009) experimentally studied the air–water pressurization characteristics of a Centrilift GC-6100 mixed-flow pump. The outer diameter of the impeller is 108 mm, with a specific speed n_s = 238. The flow rate at the best efficiency point is 42.9 m³ h⁻¹, the head is 10.9 m, and the design speed is 3600 rpm. Figure 16 shows the gas–liquid pressurization performance for one booster stage at inlet pressures of 0.69–1.72 MPa. In Fig. 16, as the inlet gas volume fraction increases, the gas–liquid pressurization performance curves also reveal three obvious trends of deterioration, corresponding to three different flow patterns.

At inlet gas volume fractions of 5% and 15%, the inner flow patterns of the mixed-flow pump are bubble flow and stratified flow, respectively. When the inlet gas volume fraction is 10%, dynamic similarity cannot be satisfied, owing to the flow pattern transition in the mixed-flow pump at different rotational speeds. As can be seen from Fig. 17, the experimental data points of the head coefficient and flow rate coefficient also follow the same distribution at rotational speeds ranging from 1500 to 3000 rpm. Therefore, the applicability of the affinity law to a multiphase pump under gas–liquid two-phase conditions is also confirmed by the experimental data of Gamboa (2009).

In this paper, a high-pressure (30 MPa) gas–liquid experimental platform has been constructed, and three-stage and 25-stage radial multiphase pumps and a 15-stage mixed-flow multiphase pump have been studied. The two-phase hydraulic parameters and corresponding dimensionless hydraulic coefficients of multistage multiphase pumps have been defined, taking account of gas compressibility. The pressurization performances of radial and mixed-flow pumps have been studied under single-phase and two-phase conditions. The applicability of an affinity law has been analyzed and summarized under gas–liquid conditions for multiphase pumps with different stage numbers and blade structures. The main conclusions are as follows.

1. Under water-only conditions, the experimental data points of the head coefficient and power coefficient at different rotational speeds follow the same distribution. The head and power performances of the 25-stage radial and 15-stage mixed-flow pumps conform to an affinity law.

2. The decline in gas–liquid pressurization performance of the three-stage radial pump can be divided into three processes with different dynamic mechanisms, corresponding to three flow patterns. The inlet gas volume fraction of pump can be used to judge dynamic similarity. At the same inlet gas volume fraction (λ₁ = λ₂), when the two-phase flows in two pumps are in the same flow pattern (λ₁ < λ_c1 or λ_c1 < λ₁ < λ_c2 or λ₁ > λ_c2), dynamic similarity will be satisfied.

3. Gas compressibility results in different gas–liquid flow patterns in the booster stages from upstream to downstream in a multistage pump under the same pump inlet condition. There are similarities in pressurization curves between single-phase and two-phase condition of mixed-flow pumps, both of which conform to a quadratic function with increasing rotational speed. Compared with a radial pump, a mixed-flow pump can maintain higher pressurization performance at high gas volume fractions.

4. Under gas–liquid conditions, when two multiphase pumps meet similarity conditions, for the same two-phase flow rate coefficients Φ_m1 = Φ_m2, the two-phase head coefficients and power coefficients of two pumps are respectively the same: Ψ_m1 = Ψ_m2 and Π_m1 = Π_m2. At different rotational speeds, the goodness of fit R² of the Ψ_m–Φ_m and Π_m–Φ_m fitting curves is higher than 0.9, indicating good applicability of the affinity law to the three-stage radial multiphase pump.

5. Finally, it is confirmed that the affinity law has good applicability to multiphase pumps with different stage numbers and blade structures (n_s = 107, 216, and 238) under gas–liquid conditions.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51888103 and 52076175).

The authors have no conflicts to disclose.

Liang Chang: Conceptualization (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Chenyu Yang: Data curation (equal); Methodology (equal). Xiaobin Su: Data curation (equal); Investigation (equal); Methodology (equal). Xiaoyu Dai: Data curation (equal); Investigation (equal); Methodology (equal). Qiang Xu: Conceptualization (equal); Resources (equal); Writing – review & editing (equal). Liejin Guo: Conceptualization (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data are available from the corresponding author on reasonable request.

In this study, the experimental errors arise mainly in measuring the liquid/gas flow rate, pressure and temperature. Measurement instruments and accuracies are listed in Table VI. The measurement accuracies for liquid flow rate, pressure and temperature are around 0.15%, 0.075%, and 0.4%, respectively. According the error propagation theory of Moffat (1988), the errors in the indirect measurement parameters can be calculated from the direct measurement parameters.

The calculated result for the maximum absolute error in the inlet gas volume fraction δλ of the centrifugal pump is 3.4% in the experiment.