Abstract
In order to comprehensively optimize the axial force and hydraulic performance of the multistage pump, considering that there are relatively more secondary impeller stages and the blade profile has a greater impact on the axial force and hydraulic performance, Plackett–Burman test design method in this paper is adopted to conduct significance analysis and screening of the secondary impeller parameters. Based on the response surface methodology, a central composite test is designed for three control variables with strong sensitivity. The multiple regression model between the parameters of the secondary impeller and the hydraulic performance and axial force of the multistage pump is established. The optimal parameter combination which takes the performance and axial force into account is obtained. The accuracy of the optimization results is verified through tests. The results show that the blade exit angle, outlet diameter and blade wrap angle of the secondary impeller have the most significant influence on the axial force and hydraulic performance of the multistage pump. The results of variance analysis and coefficient test show that the regression model is highly significant and can reflect the objective relationship between the control parameters of the secondary impeller shape and the response objectives. A larger outlet diameter and blade wrap angle of the secondary impeller can improve the head of the multistage pump. A larger blade wrap angle and a smaller blade exit angle of the secondary impeller can reduce the axial force of the multistage pump. By solving the multiple regression equation, it is found that when the outlet diameter of the secondary impeller is 292 mm, the blade exit angle is 22°, and the blade wrap angle is 150°, the axial force of the multistage pump is the lowest and the hydraulic performance is slightly improved. It is verified by experiments that the head and efficiency of the optimized multistage pump increase by 0.95% and 1.71%, respectively, the temperature of the front and rear bearings decreases by 16.49% and 16.17%, respectively, and the vibration speed of the multistage pump along three directions is significantly reduced.
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1 Introduction
In the operation of multistage centrifugal pump, due to the asymmetry of the impeller structure and the impact of the medium on the impeller and other reasons, the liquid action of multistage impeller produces higher axial force, which will cause harm to the normal operation of multistage pump. Under high pressure, the axial force sometimes reaches hundreds of tons [1,2,3]. The problem caused by excessive axial force is more serious than efficiency, wear and other factors, and has become the decisive factor for the stable operation of multistage pump.
Many scholars have done a lot of research on the hydraulic performance optimization and axial force control of centrifugal pumps. In terms of hydraulic performance optimization, Liu et al. [4] found that the head of the centrifugal pump increases with the increase in the number of blades, and there is an optimal value of efficiency matching with the number of blades. Tan et al. [5] studied the influence of blade wrap angle on the performance of centrifugal pumps, and proposed that the higher the wrap angle is, the higher the efficiency and stability of the pump will be. Pan et al. [6] analyzed the influence of three different blade angles on pump performance, and found that the slip theory could accurately calculate the head at nominal flow rate. Bacharoudis et al. [7] comprehensively evaluated the performance of impellers with the same outlet diameter and different blade exit angles, and found that the performance curve of the pump became more flat with the increase in blade exit angle through experiments. Shi et al. [8] optimized the performance of the deep-well centrifugal pump by changing the outlet width of the impeller, and compared the numerical simulation with the experiment to verify the accuracy of the optimization method. Yang et al. [9] analyzed the influence of blade thickness and profile changes on centrifugal pump performance. Cui et al. [10] analyzed the influence of blade exit angle on the flow and performance of the centrifugal pump with low specific speed. Zhou et al. [11] studied the influence of pump rear cavity radius on hydraulic performance. Ding et al. [12] conducted numerical simulation and experiment on five impeller models with different blade exit angles. And it was found that when the flow rate gradually increased, the hydraulic loss of the impeller increased with the increase in blade exit angle, and the influence of blade exit angle on efficiency was relatively obvious. In terms of axial force control of the pump, Yasuyuki Nishi et al. [13] studied the influence of the blade exit angle changes on axial force of a single-blade closed centrifugal pump. And the results showed that the time average value of axial thrust of the front and rear pump cavities increases with the increase in blade exit angle under large flow rate. Kazakov et al. [14] conducted an experimental study on the axial force of submersible well pump, and the results showed that the axial force in the pump increased with the increase in sealing clearance. Lino et al. [15] conducted theoretical and experimental research on a single-stage centrifugal pump, and developed a computer program for axial force of multistage centrifugal pump. Guelich et al. [16] discussed the two prediction methods of axial force with multistage boiler feed pump as the research object. Baun and Flack [17, 18] carried out flow visualization research, measured the axial force in plexiglass pump directly, and analyzed the influence of the shape of volute and the number of blades on the axial force and hydraulic performance. Gantar et al. [19] studied the axial force in a multistage pump by using laser Doppler anemometer (LDA) and numerical flow analysis method. The results showed that the impeller side cavity may be affected and the hydraulic axial force may be reduced.
The above research shows that there are many studies on single-stage centrifugal pump impeller profile line optimization and axial force control. While the research on axial force control and performance optimization of multistage pump is relatively small. Due to the consideration of cavitation resistance performance of multistage pumps, the structure design of the first-stage impeller and the secondary impeller is mostly different. Among them, the series of the secondary impeller is relatively more, and the blade profile has a great influence on the axial force and hydraulic performance. In recent years, it is more and more popular to use statistical methods to study the matching relationship between impeller parameters and explore the optimal structure combination to improve the performance of fluid machinery. Dai et al. [20] selected the key parameters with significant influence on noise based on sensitivity. The response surface method was used to construct a response surface multiple regression model of significant variables and multiple objective functions, and the interaction between the parameters affecting hydraulic efficiency and noise was analyzed. Guo et al. [21] established a multiple regression model between the blade profile and pump performance of jet centrifugal pump based on the response surface method. The research results showed that the blade wrap angle, the blade exit angle and the blade inlet diameter had the most significant influence on the hydraulic and acoustic performance of the jet centrifugal pump. Duccio et al. [22] combined test design method, response surface method and multi-objective optimization algorithm to optimize the impeller of centrifugal compressor.
In this paper, the high-pressure double-volute 11-stage multistage pump used by an auxiliary plant of Lanzhou Company is taken as the research object, and the primary and secondary factors affecting pump efficiency, head, shaft power and axial force are analyzed by response surface methodology combined with CFD technology, and the multiple regression model between the objective function and the main control variables is obtained. The interaction between different control variables is analyzed, the mapping relationship between hydraulic performance and axial force of the multistage pump is clarified, and the blade type parameters of the secondary impeller that meet the optimal comprehensive performance are obtained. Finally, the accuracy of the optimization method is verified by the test method. The research results can provide a basis for impeller design and axial force research of multistage pump.
2 Model and numerical calculation method
2.1 Structure parameters of multistage pump
The main design parameters of the P101B segmenting multistage centrifugal pump selected in this paper are as follows: nominal flow rate Q = 128m3/h, stage = 11, single-stage head H = 106 m, nominal rotating speed n = 2986r/min. Table 1 shows the main parameters of the model pump. The sketch of the main parameters is shown in Fig. 1. D1 in the figure represents the impeller inlet diameter. And the corresponding relationship with Table 1 is that D11 is the inlet diameter of the first-stage impeller, D12 is the inlet diameter of the second-stage impeller, and other parameters are shown as above.
2.2 Computational domain and grid division
The calculation domain of P101B pump model is composed of suction chamber, first-stage impeller, secondary impeller, guide vane, final guide vane and pressure water chamber. The three-dimensional modeling software Pro/E 5.0 is used to carry out geometric modeling for the multistage pump, as shown in Fig. 2.
The structured grid is adopted for the computational domain generated by the ANSYS ICEM 16.0 software, and the near-wall grid is important for the result, it means that the height of the first layer grid should be carefully selected. The near-wall region y + (dimensionless height of the first layer of grid near the wall) is less than 5. Figure 3 shows the grids in flow components. The efficiency and axial force of the multistage pump are taken as criteria for grid independence verification. The relationship between pump efficiency and grid, axial force and grid at nominal flow rate is shown in Fig. 4. It can be seen from the figure that when the number of grids increases from 26.62 million to 31 million, the absolute increment of pump efficiency is less than 0.023%, and the increase in axial force is less than 0.35%. Therefore, the total mesh grid of the model pump is about 26.62 million.
2.3 Numerical calculation method
The shear stress transport (SST) turbulence model implements a gradual transition from a model inside the boundary layer to a model with a high Reynolds number outside the boundary layer. It has a great advantage in predicting near-wall flows or flows with adverse pressure gradient [23,24,25,26,27]. Therefore, the SST \(k - \omega\) model is selected in this paper. FLUENT commercial software is used to solve the internal flow field of multistage pump. The flow in the pump is set as incompressible constant flow. The inlet boundary condition is set as velocity inlet. It is assumed that the incoming flow direction is perpendicular to the inlet section, and the magnitude of the incoming flow velocity is given. The outlet boundary condition is set as free outflow, the flow is considered to be absolutely developed, the fixed wall boundary condition is selected as no slip wall, and the MRF rotation reference frame method is used.
3 Experimental verification of numerical calculation method
The test bed for the multistage pump P101B is shown in Fig. 5a. The multistage pump transfers the liquid in the raw material tank to the reactor for redox reaction. The schematic diagram of the test bed is shown in Fig. 5b. The motor used in the test bed is AMD400L2RBABM (ABB of Switzerland), which can meet the design speed of 2986r/min for pump operation. The pump’s inlet and outlet liquid pressure are measured by 7MF403 pressure sensor of Siemens company, Germany, with a measurement error of ± 0.075%. The pump outlet flow is measured by AE215 electromagnetic flowmeter of Yokogawa Company of China, and the measurement error is ± 0.5%. Speed and power are measured by a torque sensor installed between the pump and the motor. The MH6150 orifice flowmeter (class 0.5) of Shanghai Meng Hui Company is installed at the axial direction of the balance tube 153 mm far from the pump inlet to measure the flow rate, and the YB-150 standard manometer (class 0.4) of Shanghai Automation Instrument Factory No.4 at 239 mm to measure the pressure.
It can be seen from Fig. 6a that the numerical prediction performance curve is basically consistent with the experimental curve, but the numerical result is always higher than the experimental result. The main reason for the energy loss caused by mechanical seal and bearing friction is not taken into account in the numerical simulation, and the maximum errors of head and efficiency are 4.17 and 2.81%, respectively. For the high-pressure double-volute multistage pump, it is difficult to measure the pressure in the pump chamber and the balance drum clearance leakage directly. Therefore, this paper measured the pressure and flow in the balance tube and compared it with the numerical results, so as to verify the reliability of the calculation method adopted in this paper. From comparison, we can find that CFD predicts higher leakage through the balance tube, and the maximum error of the balance tube flow at nominal flow rate is 4.49%. This suggests larger internal recirculation in the flow passages of the impeller in CFD. Since the impeller flow rates are larger than experiments by the amount of internal recirculation, the Euler head is lowered in CFD and the total head generation may be reduced. Besides, the curve of calculated result of the pressure in the balance tube agrees well with the experimental result, with a maximum error of 2.5%, and the experimental error is within the allowable range, indicating that the calculation method selected in this paper can provide a reliable guarantee for this study.
4 Experimental design
4.1 Variable selection
In order to clarify the main blade profile control variables that have the most obvious impact on the performance of multistage pumps and save analysis cost, based on Minitable16 software, Plackett–Burman test design and significance analysis are carried out for the six blade profile control variables of the secondary impeller of the model pump, as shown in Table 2.
The analysis results show that the three most sensitive factors of the secondary impeller to the axial force of multistage pump are blade exit angle β22, outlet diameter D22, blade wrap angle φ2. Therefore, this study chooses these three variables for the next analysis.
4.2 Design scheme
By using Design-Expert software, three significant factors of the secondary impeller are designed by central composite design (CCD) and the other parameters of the multistage pump are original values. The factor coding comparison is shown in Table 3. A total of 20 tests are conducted to construct 3d models and grid division for each blade profile test point, and then the hydraulic performance and axial force values of each test point are solved according to fluent software. The calculation results are shown in Table 4.
4.3 Test results
The third-order polynomial is used to construct the response surface model, and the regression equation based on actual control variables for each response target is obtained by fitting, which is, respectively, expressed as:
Head regression equation
Shaft power regression equation
Efficiency regression equation
Axial force regression equation
In order to ensure the adaptability and accuracy of the fitted third-order polynomial regression equation, significance test should also be carried out on the regression equation. The analysis results of the fitted regression equation are shown in Table 5. The closer the results of the complex correlation coefficient R2 and the correction determination coefficient Radj2 in the table are to 1, the better the regression effect is. As can be seen from Table 5, the complex correlation coefficient R2 of the regression equation of head, axial power, efficiency and axial force is all greater than 0.9427, and the correction determination coefficient Radj2 is all greater than 0.8185, both of which are close to 1. Therefore, it can be considered that the fitted regression equations are significant. The value of Prob (P) > F is less than 0.0001, indicating that the model is highly significant. The above results show that the regression model can be used to predict multistage performance and axial force, and the test scheme is correct.
5 Results analysis
5.1 Response analysis of single factor and objective function
The change rule of the three control variables of the secondary impeller on head, shaft power, efficiency and axial force is analyzed, as shown in Fig. 7. It can be seen from Fig. 7a–b that the hydraulic performance has obvious monotonicity within the range of outlet diameter and blade wrap angle of the secondary impeller. With the increase in outlet diameter and blade wrap angle, the head and efficiency increase gradually, while the shaft power decreases gradually. The axial force performance of the secondary impeller shows a concave parabolic distribution law with the increase in the outlet diameter, and the axial force of the secondary impeller increases first and then decreases with the increase in the blade wrap angle. It can be seen from Fig. 7c that with the increase in the blade exit angle of the secondary impeller, the head and efficiency increase monotonically and the axial power decreases monotonically. The axial force of both reaches the minimum value near the blade exit angle of 19°. The above analysis shows that the influence law of impeller geometric parameters on axial force is more complicated than the hydraulic performance of multistage pump.
5.2 Interaction between secondary impeller parameters
Figure 8–11 show the interaction between the secondary impeller parameters affecting the performance of the multistage pump and axial force, respectively. The secondary impeller parameters are outlet diameter D22, blade exit angle β22 and blade wrap angle φ2.
It can be seen from Fig. 8a that the head variation caused by changes in outlet diameter and blade exit angle of the secondary impeller is large. When the blade exit angle is at a low level, the head decreases first and then increases with the decrease in the impeller outlet diameter. When the blade exit angle is at a high level, the head increases first and then decreases with the decrease in the impeller outlet diameter. The contour shows that the interaction between outlet diameter and blade exit angle is not significant. According to Fig. 8b, the outlet diameter and the blade wrap angle of surface slope are relatively steep. When the impeller outlet diameter is high and the blade wrap angle is low, the maximum value of the head appears. The change of contour density along the impeller outlet diameter axis is greater than that of the wrap angle axis, indicating that the influence of the blade outlet diameter on the head is more significant than that of the impeller outlet diameter. It can be seen from Fig. 8c that the influence of blade exit angle and blade wrap angle on head presents a concave surface distribution. With the increase in blade number, the head first decreases and then increases. The contour density variation along the blade exit angle axis is greater than that of the wrap angle axis, indicating that the blade exit angle has a more significant influence on the head than the blade number.
It can be seen from Fig. 9a that the influence of the secondary impeller outlet diameter and blade exit angle on the shaft power presents a concave distribution law. When the control code of the impeller outlet diameter and blade exit angle is around 0, the shaft power of the pump can be reduced. And the contour line shows an ellipse with a small curvature, indicating that the interaction between outlet diameter and blade exit angle is not significant. It can be seen from Fig. 9b that the shaft power of multistage pump decreases first and then increases with the increase in secondary impeller outlet diameter and blade wrap angle. When the control code of outlet diameter and blade wrap angle is in the range of 0 and 0.85, the shaft power is minimum. In addition, the contour curve shows an ellipse with a large curvature, indicating significant interaction between outlet diameter and blade wrap angle. Figure 9c shows that when blade exit angle is at a low level, the shaft power decreases gradually with the increase in blade wrap angle. And when the blade exit angle is at a high level, the shaft power increases gradually with the increase in blade wrap angle. The best combination of the two is located near the diagonal. The contour distribution shows that the interaction between blade exit angle and blade wrap angle is not obvious.
Figure 10a shows that the slope of the curved surface of the secondary impeller outlet diameter and the blade exit angle is relatively slow. When the blade exit angle is 0 and the outlet diameter is 1, the efficiency of multistage pump is the highest. With the decrease in impeller outlet diameter, the efficiency of pump decreases obviously. And the contour line shows an ellipse with a large curvature, indicating that the interaction between outlet diameter and blade exit angle is obvious. It can be seen from Fig. 10b that the efficiency of pump can be improved by larger impeller outlet diameter and smaller blade wrap angle. The contour line presents an ellipse with a large curvature, indicating that blade wrap angle and impeller outlet diameter have a significant interaction. And the contour density variation along the blade exit angle axis is greater than that along the blade wrap angle axis, indicating that the blade exit angle has a more significant impact on efficiency than that of blade wrap angle. It can be seen from Fig. 10c that the curvature of the curved surface of the blade exit angle and the blade wrap angle is relatively large. When the blade exit angle is at a low level and the blade wrap angle is at a high level, or the blade exit angle is at a high level and the blade wrap angle is at a low level, the pump efficiency can be improved. The best match between the two is located near the diagonal.
It can be seen from Fig. 11a that the axial force caused by the secondary impeller blade exit angle and outlet diameter changes greatly. When the control code of blade number is 0 and the impeller outlet diameter is at a low level, the axial force of the multistage pump is the smallest. The contour line shows an ellipse with a large curvature, indicating that the interaction between blade exit angle and outlet diameter is obvious. And the contour density variation along the outlet diameter axis is greater than that along the blade exit angle axis, indicating that the outlet diameter has a more significant impact on axial force than that of blade exit angle. It can be seen from Fig. 11b that when the outlet diameter and blade wrap angle are small, or the outlet diameter is near the zero level and the blade wrap angle is at a high level, the axial force can be reduced. The contour density variation along the outlet diameter axis is greater than that along the blade wrap angle axis, indicating that the outlet diameter has a more significant impact on axial force than that of blade wrap angle. It can be seen from Fig. 11c that the surface of blade exit angle and the blade wrap angle is concave. When the blade exit angle is near zero level and the blade wrap angle is at a low level or at a high level, the axial force of the multistage pump is small. The contour density variation along the blade exit angle axis is greater than that along the blade wrap angle axis, indicating that the blade exit angle has a more significant impact on axial force than that of blade wrap angle.
From the above analysis, it can be seen that for the secondary impeller, the larger impeller outlet diameter and the larger blade wrap angle can improve the head of the multistage pump. The axial force of multistage pump can be reduced by the larger blade wrap angle and the smaller blade exit angle. To optimize the hydraulic performance of multistage pump, the coding level of each control variable should be close to 0. When the blade exit angle and the outlet diameter of the impeller are close to zero level, and the blade wrap is at low level and high level, the axial force performance of the multistage pump is the best.
5.3 The mapping relationship of the objective function
The mapping relationship between the control code of the entire test design sample of the secondary impeller and the objective functions (efficiency, head and shaft power) is shown in Fig. 12. In order to unify the dimension, the relative values of each value and control variable code are taken as 0 level samples. It can be seen from the figure that, in the design variable space, the head increases linearly with the increase in efficiency, while the shaft power decreases linearly with the increase in efficiency. The mapping points of secondary impeller are relatively dispersed, and the variation ranges of efficiency, head and shaft power are [0.48, 1.05], [0.95, 1.15] and [0.90, 1.69], respectively. The performance results of the whole test sample show that the variation in secondary impeller parameters has a great influence on the hydraulic performance of multistage pumps.
5.4 Performance optimization results
Under the constraint condition, the regression Eqs. (1)–(4) of hydraulic efficiency and axial force are solved simultaneously. Three factor values that minimize the axial force under the condition that the multistage pump efficiency is not lower than the original efficiency are obtained. And the parameters are D22 = 292 mm, β22 = 22°, φ2 = 150°. Figure 13 shows the blade profile of secondary impeller before and after optimization.
The hydraulic performance and axial force performance of the multistage pump before and after the optimization of the secondary impeller parameters calculated by CFD are shown in Fig. 14. It can be seen from Fig. 14a that with the increase in flow rate, the head decreases gradually and the shaft power increases gradually, and the efficiency increases first and then decreases. The hydraulic performance of the optimized multistage pump is slightly improved under different working conditions. At nominal flow rate, the efficiency increases by 4.62%, the head increases by 0.89%, and the shaft power decreases by 3.56%. Figure 14b shows that the residual axial force of the multistage pump decreases first and then increases with the increase in flow rate. At nominal flow rate, the residual axial force is the minimum, indicating that the multistage pump is stable. After optimization, the residual axial force of multistage pump is significantly reduced, and the optimized post-axial force is reduced by 4.92% on average under different working conditions.
6 Test comparison before and after optimization
In order to verify the optimization effect, the hydraulic performance and axial force performance tests of the prototype pump and optimized pump are carried out. If the axial force measurement equipment such as force sensor was installed on the raw material pump of methyl-ethyl ketone reaction device, it will affect the delivery of raw materials and the safe operation of the pump. Once the leakage of raw materials occurs, it will cause serious consequences. The changes in axial force of multistage pump before and after optimization are compared under the condition of ensuring normal operation of production equipment. Considering that too much axial force of the pump will cause stress friction of the thrust bearing, which will increase the bearing temperature and cause the vibration of pump [28,29,30,31,32,33]. Therefore, the WSS-181 thermometer is installed on the front and rear bearing boxes of the test pump. The thermometer is produced by Tian Chang Hui Ning Electrical Appliance and Meter Factory, with a measuring range of 0 ~ 100℃, an accuracy of 1.5 and a tail length of 100 mm. The vibration of the front and rear bearing of the pump is measured by the VT-67 portable digital vibration meter produced by ACCPOM, Germany. The measuring range is 11.2 ~ − 45 mm/s, and the measurement error is 0 ~ 20 m. The axial force test device is shown in Fig. 15.
Figure 16 shows the hydraulic performance curve of the multistage pump before and after optimization. As can be seen from Fig. 14, compared with the original pump, the head and efficiency of the optimized multistage pump are improved, among which the head is increased by 0.95% and the efficiency by 1.71%. Figure 17 shows the temperature curve of multistage pump bearing before and after optimization. It can be seen from Fig. 15 that the temperature of the multistage pump bearing decreases significantly after optimization. Under different working conditions, the average temperature of the front bearing is decreased by 16.49%, and that of the rear bearing is decreased by 16.17%. Moreover, the temperature of the front bearing does not change significantly with the increase in flow rate, while the temperature of the rear bearing decreases first and then increases with the increase in flow rate, which is the same as that of the residual axial force of the multistage pump. This is because the thrust bearing bears the residual axial force. When the residual axial force is large, the friction of the thrust bearing increases and the temperature rises. When the residual axial force is small, the thrust bearing bears less stress and the temperature is lower. So the residual axial force can be judged by the temperature of the rear bearing.
Figure 18 shows the bearing vibration curves before and after optimization. It can be seen from the figure that the vibration velocity of the optimized bearing decreases in three directions. After optimization, the tangential, radial and axial vibration velocity of the front bearing decrease by 37.57, 62.38 and 3.07%, respectively, and the tangential, radial and axial vibration velocity of the rear bearing decrease by 32.9, 21 and 6.09%.
7 Conclusion
(1) The blade exit angle, outlet diameter and blade wrap angle of secondary impeller are the three most significant factors affecting the axial force and hydraulic performance of the multistage pump.
(2) Based on the response surface methodology, a central composite test is designed for experimental design (a total of 20 test programs). The multiple regression model between the hydraulic performance and axial force of the multistage pump is established. At the same time, the significant verification of the multiple regression equation of each response target is carried out. The results show that the regression model is highly significant, which can reflect the objective relationship between the control parameters of the secondary impeller profile and the hydraulic and axial force performance of the multistage pump.
(3) The secondary impeller blade profile has certain restrictions on the optimal hydraulic performance and the optimal axial force of the multistage pump. A larger secondary impeller outlet diameter and a larger blade wrap angle of secondary impeller can improve the head of multistage pump. And a larger blade wrap angle of secondary impeller and a smaller blade exit angle of secondary impeller can reduce the axial force of multistage pump.
(4) By solving the multiple regression equation, the optimal parameter combination with the lowest axial force and no lower hydraulic performance than the original model is obtained. And in the optiminal parameter combination, the secondary impeller outlet diameter is 292 mm, the secondary impeller blade exit angle is 22° and the secondary impeller blade wrap angle is 150°. It is verified by experiment that the head of the optimized multistage pump increases by 0.95%, and the efficiency increases by 1.71%. After optimization, the front and rear bearing temperature decrease by 16.49 and 16.17%, respectively, the vibration velocity of the multistage pump along three directions reduces significantly, and the vibration of the front bearing has a greater reduction.
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Qian, C., Luo, X., Yang, C. et al. Multistage pump axial force control and hydraulic performance optimization based on response surface methodology. J Braz. Soc. Mech. Sci. Eng. 43, 136 (2021). https://doi.org/10.1007/s40430-021-02849-1
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DOI: https://doi.org/10.1007/s40430-021-02849-1