The hydraulic performance of a high-speed submersible axial flow pump is investigated to reduce its energy consumption. A more efficient and stable optimization method that combines parametric design, computational fluid dynamics, and a computer algorithm is proposed. The main aim is to broaden the high-efficiency operating zone, so the average efficiency under multiple conditions is optimized while considering rotor–stator matching. The design-of-experiments method and a radial-basis-function neural network are combined to form the optimization platform, and automatic optimization of the pump design is realized through repeated execution of design and simulation. The flow loss mechanism inside the pump is studied in depth via the entropy generation rate, and regression analysis shows that the pump efficiency is influenced mainly by the blade angles. After optimization, the target efficiency is increased by 8.34%, and the flow field distribution shows that the channel vortex and hydraulic loss are controlled effectively. Finally, the results are validated by experiment. The proposed optimization approach has advantages in saving manpower and obtaining globally optimal solutions.

Extreme weather around the world is causing increasingly many serious natural disasters, among which the increasing frequency of urban flooding poses a serious threat to social security and national economies. In response to such situations, various new types of drainage equipment are emerging. In particular, to aid the rapid deployment of drainage pumps by firefighters, lightweight submersible axial flow pumps have been developed and have promising applications in emergency rescue. However, such pumps are powered by vehicle engines, thereby limiting their output power, and so an important way to ensure completion of rescue tasks is to improve the efficiency of such pumps.1 

As the most important part of energy conversion in a pump, designers usually focus on the impeller design.2 For example, Chen et al.3 corrected the circumferential velocity at the impeller outlet to improve the pump head. Using sequential quadratic programming, Shi et al.4 subjected the impeller of an axial flow pump to multi-condition optimization design. Using orthogonal design, Dai et al.5 assessed which geometric parameters had the greatest influence on the performance of an axial flow pump. Tong et al.6 proposed a novel hybrid optimization algorithm for optimizing the impeller of a centrifugal pump, thereby reducing the shaft power significantly. Zanetti et al.7 investigated how blade loading affects impeller performance, and they used the 3D inverse design method to optimize the impeller of a pump turbine.

However, despite the impeller being the most important hydraulic part, pump performance cannot be improved fully by impeller optimization alone.8 Using computational fluid dynamics (CFD) and experiments, Kang et al.9 analyzed how the guide-vane number affects the efficiency of axial flow pumps, finding that having a higher number creates a more uniform distribution of axial velocity at the impeller outlet. Shi et al.10 found that the guide-vane rotation angle can affect the performance of an axial flow pump under overload conditions, a choosing a suitable rotation angle widens the high-efficiency zone of the pump. To suppress unstable flow and internal loss, Wang et al.11 optimized 14 parameters of a pump diffuser based on modified heuristic algorithms, and the results showed that guide-vane optimization can further improve the performance of centrifugal pumps.

Now that it is so highly developed, CFD technology has largely replaced expensive prototype experiments.12 During optimization design, especially when there are many optimization parameters, many schemes must be tested. However, although using many design schemes can give more-comprehensive optimization results, it is obviously unrealistic to try all possible combinations from the point of view of saving time, and clearly using too few schemes will not yield satisfactory results. This contradiction can be resolved well by using the design of experiments (DOE) approach combined with an approximate model. Yang et al.13 investigated how the diffuser meridian affected the performance of an electrical submersible pump and optimized the meridional profile by using the Taguchi method. To achieve higher prediction precision, Pei et al.14 proposed a neural network with two hidden layers, and they subjected centrifugal pumps to multiparameter optimization based on a modified particle-swarm algorithm. Regarding approximate models, each has its own prediction accuracy, and previous research has shown that a radial-basis-function neural network (RBFNN) can provide higher prediction accuracy in the optimization of centrifugal pumps.15 Zhao et al.16 proposed a backpropagation neural network (BPNN) with six neurons to further improve the prediction precision in the optimization of multistage double suction pumps. Similarly, Wu et al.17 used the same model to improve the flow rate of a centrifugal pump. Tong et al.18 proposed a novel hybrid optimization algorithm and improved the efficiency of slanted axial flow pumps based on a BPNN. Zhao et al.19 applied a genetic-algorithm–BPNN method to the optimization of a multistage centrifugal pump, and the shapes of the impeller and channel were optimized to increase the efficiency at high flow rate.

In summary, the geometric parameters of the hydraulic parts of pumps are very complicated, and most studies usually focus on only some important parameters, with some studies even ignoring the effect of rotor–stator interaction.20 In the study presented herein, impeller–diffuser matching optimization is investigated in depth. An RBFNN algorithm is used to improve the fitting accuracy of the mathematical model, and batch processing is used to realize the repeated execution of software in the background. Finally, multi-island genetic algorithm (MIGA) optimization is used to search for the globally optimal solution. Flow loss visualization and evaluation are used to analyze the flow pattern and internal flow loss with the aim of clarifying the underlying mechanism for performance improvement.

As shown in Fig. 1, the investigated pump has a very compact structure. Its main parts are a front guide vane, an unshrouded axial impeller, and a spatial diffuser; not included in Fig. 1 is the electromotor located in front of the front guide vane. The operating conditions are described using two dimensionless parameters, i.e.,
ϕ=8QπD3ω,
(1)
ψ=gHr22ω2,
(2)
where Q is design flowrate, D is impeller diameter, H is design head, r2 equals to 0.5D, ω is angular velocity and the main hydraulic design parameters of the pump are given in Table I.
FIG. 1.

Structure of axial flow pump.

FIG. 1.

Structure of axial flow pump.

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TABLE I.

Design specifications of studied pump.

ParameterSymbolValue
Flow-rate coefficient ϕ 1.71 
Head coefficient ψ 0.11 
Impeller diameter D 244 mm 
Impeller hub diameter dh 72 mm 
Inlet angle of blade hub β1h 37.2° 
Inlet angle of blade shroud β1s 13.5° 
Outlet angle of blade hub β2h 43.3° 
Outlet angle of blade shroud β2s 20.2° 
ParameterSymbolValue
Flow-rate coefficient ϕ 1.71 
Head coefficient ψ 0.11 
Impeller diameter D 244 mm 
Impeller hub diameter dh 72 mm 
Inlet angle of blade hub β1h 37.2° 
Inlet angle of blade shroud β1s 13.5° 
Outlet angle of blade hub β2h 43.3° 
Outlet angle of blade shroud β2s 20.2° 
The flow in the pump is highly turbulent, and in engineering, using time averaging to deal with flow variables can greatly reduce the calculation time. Based on these principles, the basic equations describing the flow regime are as follows:21 
uj̄xj=0,
(3)
uīt+uj̄uīxj=fī1ρp̄xi+ν2uīxjxjuiuj̄xj.
(4)

A turbulence model is needed to resolve the Reynolds stress in Eq. (4), and the shear stress transport kω model is suitable for flows with large curvature or separation. Considering the complexity of the flow in the pump, this model was used to solve for the flow field in the pump. The automatic wall function was used to treat the near-wall flow, and the inlet and outlet boundaries were opening and mass flow, respectively. Frozen-rotor interface treatment was applied to the steady simulations, and the steady results were used as initial values for transient calculations in which the interface mode was transient rotor–stator. One time step represented a 2° rotation of the impeller, and the convergence condition was that the residuals were less than 10−4.

For a given number of nodes, using a hexahedral mesh results in significantly fewer cells. Also, a hexahedral mesh has better orthogonality and quality, which can accelerate the calculation and convergence. According to the geometric characteristics of each fluid domain, Turbo-Grid was used to generate a single channel grid of the front guide vane, the impeller, and the diffuser, and meshing tool ICEM was used to discretize the outlet pipe. Figure 2 shows the details of the hexahedral mesh. To meet the needs of the turbulence model,22 the dimensionless distance y+ was controlled to within 30 by setting a reasonable first layer mesh height. The expansion of the boundary layer was controlled by the ratio of 1.2.

FIG. 2.

Details of hexahedral mesh.

FIG. 2.

Details of hexahedral mesh.

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In theory, having more grid points should lead to higher computational accuracy, but it also consumes more computing resources and slows the rate of convergence. Therefore, for a particular case, it is necessary to find a suitable set of mesh numbers to balance calculation speed and accuracy. To achieve this, various grid numbers were tested, and Table II shows that when the grid number exceeded 3.23 × 106, the difference in pump head was negligible. Therefore, a grid number of 3.23 × 106 was deemed suitable for the present calculations.

TABLE II.

Mesh independent check.

Grid number (million)Head (m)Relative error (%)
1.37 9.97 ⋯ 
1.91 10.76 7.92 
2.48 11.15 3.62 
3.23 11.39 2.15 
4.19 11.43 0.35 
Grid number (million)Head (m)Relative error (%)
1.37 9.97 ⋯ 
1.91 10.76 7.92 
2.48 11.15 3.62 
3.23 11.39 2.15 
4.19 11.43 0.35 

To test the pump efficiency accurately, a hydraulic prototype with a straight inlet pipe and an elbow outlet pipe was manufactured, and a four-quadrant test rig was used for the performance experiment. A plan diagram of the test section is shown in Fig. 3. The tested pump was installed horizontally, and to reduce electromagnetic interference, a DC motor was used to adjust the rotational speed of the pump. A torque meter was installed between the motor and the pump shaft to transmit rotational-speed, torque, and power data, and the test accuracy was ±0.1%. The water tanks ensured the stability of the pressure in the inlet and outlet pipes. Because of the pipeline resistance of the test rig, it would have been difficult for a low-head axial flow pump to operate at high flow rate, so instead a high-head mixed flow pump was used to increase the pipe flow. An electromagnetic flowmeter with an accuracy of ±0.2% was used to collect the flow data in real time. The pump head was obtained from the data of a differential pressure meter whose error was better than ±0.1%. Before the start of the test, data under known conditions were collected repeatedly to ensure that the random error was within an acceptable range. The results showed that the comprehensive uncertainty of the test rig was lower than ±0.3%.

FIG. 3.

Plan diagram of test section.

FIG. 3.

Plan diagram of test section.

Close modal

Figure 4 shows the prediction accuracy of external characteristics. Excessive numerical errors would lead to untrustworthy optimization results, but the steady results agree well with the experimental data. The CFD-predicted efficiency is always higher than the test value, but this is because the test data were subject to mechanical losses such as bearing friction, which the CFD did not consider. The head prediction accuracy is very high, especially for the design conditions. In general, the numerical errors are deemed acceptable, and so the same CFD method was used for the subsequent optimization study.

FIG. 4.

Comparison of performance curves.

FIG. 4.

Comparison of performance curves.

Close modal
In fluid machinery, loss is an invisible physical quantity, but thermodynamic entropy can be used to locate the flow loss and quantify its magnitude. According to the second law of thermodynamics, the flow direction and energy loss can be expressed in mathematical equations, which is known as entropy generation theory. Herein, the heat transfer is neglected, and after time averaging of variables, the entropy generation equations are as follows:23,
ΦT̄=S=SD̄+SD,
(5)
SD̄=μT̄2ūx2+v̄y2+w̄z2+ūy+v̄x2+ūz+w̄x2+v̄z+w̄y2,
(6)
SD=μT̄2ux2̄+vy2̄+wz2̄+uy+vx2̄+uz+wx2̄+vz+wy2̄,
(7)
where ū, v̄, and w̄ are the Cartesian components of the averaged velocity, and u′, v′, and w′ are the respective velocity fluctuations. The term SD̄ is the dissipation term of entropy generation in the average flow field. The term SD is mainly affected by the gradients of the fluctuating velocity components, but we cannot calculate them directly from the numerical results. Fortunately, Herwig and Schmandt24 proposed a method for calculating SD from the current numerical results, i.e.,
SD=ρε̄T.
(8)
The total entropy production (Pv) is obtained via volume integration, i.e.,
Pv=SD̄dV+SDdV,
(9)
and all physical quantities are averaged in the last revolution of the impeller.

In this work, an intelligent algorithm and batch processing were used to realize automatic pump optimization. As shown in Fig. 5, this involves six main steps:25 (i) sample data are generated via DOE; (ii) hydraulic design and (iii) numerical simulation are executed in turn based on the sample space; (iv) the surrogate models are established according to the CFD results; (v) regression analysis is performed to determine whether the surrogate models meet the requirements; (vi) the optimization algorithm is used to solve the surrogate models and obtain the best solution.

FIG. 5.

Overall process of design optimization.

FIG. 5.

Overall process of design optimization.

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This study is dedicated to widening the efficient operating zone of pumps. The average efficiency under multiple operating conditions is specified as the objective, and the corresponding mathematical problem is as follows:
Max FXm=η1+η2+η33subject to 11mH13mXmLXmXmU,
(10)
where η is the hydraulic efficiency, the subscripts 1–3 represent the working conditions of 0.9Qd, 1.0Qd, and 1.1Qd, respectively, and X=x1,x2,,xm contains the optimization variables.

Rotor–stator matching was investigated, and Fig. 6 defines the relevant parameters. Based on the assumption of cylindrical layer irrelevance, five equidistant cylindrical surfaces were created from the hub to the shroud, and hydrofoils on the cylindrical surfaces were designed to generate a 3D blade. To obtain rational design solutions, some parameters were strictly constrained: β1, β2, l, α3, and φd were distributed linearly from hub to shroud. Finally, 14 parameters were selected for optimization, and their ranges are given in Table III.

FIG. 6.

Definition of geometric parameters: (a) meridional flow channel; (b) hydrofoil; (c) diffuser.

FIG. 6.

Definition of geometric parameters: (a) meridional flow channel; (b) hydrofoil; (c) diffuser.

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TABLE III.

Ranges of parameters.

ParameterLower boundBaselineUpper bound
Impeller diameter D (mm) 220 240 260 
Hub diameter dh (mm) 60 72 100 
Inlet angle of blade hub β1h (°) 25 37.2 45 
Inlet angle of blade shroud β1s (°) 10 13.5 25 
Outlet angle of blade hub β2h (°) 30 43.3 50 
Outlet angle of blade shroud β2s (°) 15 20.2 30 
Chord length of blade hub lh (mm) 60 79.4 110 
Chord length of blade shroud ls (mm) 120 137.6 150 
Inlet angle of vane hub α3h (°) 35 49.3 55 
Inlet angle of vane shroud α3s (°) 60 65.2 80 
Wrap angle of vane hub φdh (°) 35 45.1 50 
Wrap angle of vane shroud φds (°) 10 11.3 20 
Impeller blade number Z 
Diffuser vane number Zd 
ParameterLower boundBaselineUpper bound
Impeller diameter D (mm) 220 240 260 
Hub diameter dh (mm) 60 72 100 
Inlet angle of blade hub β1h (°) 25 37.2 45 
Inlet angle of blade shroud β1s (°) 10 13.5 25 
Outlet angle of blade hub β2h (°) 30 43.3 50 
Outlet angle of blade shroud β2s (°) 15 20.2 30 
Chord length of blade hub lh (mm) 60 79.4 110 
Chord length of blade shroud ls (mm) 120 137.6 150 
Inlet angle of vane hub α3h (°) 35 49.3 55 
Inlet angle of vane shroud α3s (°) 60 65.2 80 
Wrap angle of vane hub φdh (°) 35 45.1 50 
Wrap angle of vane shroud φds (°) 10 11.3 20 
Impeller blade number Z 
Diffuser vane number Zd 
DOE is a branch of applied statistics that helps in studying relationships between inputs and outputs, and it is used widely in many engineering fields because of its powerful data processing function. A popular DOE technique is optimal Latin hypercube (OLH), which is an improved version of Latin hypercube. By optimizing the order of two factor levels, a more uniform design space can be obtained.25 To build an accurate surrogate model, spatial sampling was performed using OLH. Regression analysis is a set of statistical methods used for estimating the relationships between a dependent variable and one or more independent variables. In statistics, regression analysis refers to a method of analysis that determines the quantitative interdependence of two or more variables. The coefficient of determination R2 is a parameter that reflects significance levels, and it is defined as follows:26 
R2=1i=1nyiȳ2i=1nŷyi2,
(11)
where yi is an actual value, ȳ is the average response, and ŷ is the estimated value. R2 takes values between 0 and 1, with stronger correlation indicated as R2 approaches 1. In this work, the sample size was controlled rationally with the help of regression analysis, and the threshold of R2 was 0.9.

Artificial neural networks mimic the processing power of biological nervous systems, and their main characteristics are their massively parallel processing architecture and fast self-learning capability. Neural networks have been used successfully to solve various problems, and the RBFNN used in the present study is an example of a feedforward neural network. Figure 7 shows the topology of the RBFNN,27 which contains three layers. Each input variable is associated with one independent neuron and is passed to the hidden layer through the function.

FIG. 7.

Architecture of radial-basis-function neural network (RBFNN).

FIG. 7.

Architecture of radial-basis-function neural network (RBFNN).

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Euclidean distance is generally used to describe radial basis functions, the most popular of which is the Gaussian kernel function27 given by
Rxm=expxmci2σi2.
(12)
Together, the center vector ci and width σi form the ith Gaussian function. The output layer y is expressed as27 
y=Ωiexpxmci2σi2,
(13)
where Ωi is the weight factor.

Genetic algorithms are search algorithms inspired by Darwin’s theory of evolution and were developed by Holland in the 1970s.28 They are randomized search algorithms, and by mimicking the processes of natural selection and reproduction, they can provide high-quality solutions to a wide range of problems involving search, optimization, and learning. At the same time, genetic algorithms resemble natural evolution and so can overcome some of the obstacles encountered by traditional search and optimization algorithms, especially for problems with many parameters and complex mathematical representations. However, genetic algorithms also have some drawbacks, and any improper selection of parameters such as mutation rate and crossover rate will cause convergence difficulties.

After years of development, Miki et al.29 proposed MIGA, a modified genetic algorithm whose main feature is that each population of individuals is divided into several sub-populations called “islands.” A migration operation was introduced to perform search actions on different islands, and the migration process is controlled by two parameters: migration interval and migration rate. Superior to other genetic algorithms, MIGA can avoid falling into local search space. Figure 8 shows the search behavior of MIGA,25 and the algorithm settings are given in Table IV.

FIG. 8.

Principle of multi-island genetic algorithm (MIGA).

FIG. 8.

Principle of multi-island genetic algorithm (MIGA).

Close modal
TABLE IV.

Selection of parameters.

ParameterValue
Number of generations 10 
Number of islands 10 
Sub-population size 20 
Migration rate 0.01 
Crossover rate 0.9 
Migration interval 
Mutation rate 0.01 
ParameterValue
Number of generations 10 
Number of islands 10 
Sub-population size 20 
Migration rate 0.01 
Crossover rate 0.9 
Migration interval 
Mutation rate 0.01 

By constantly adjusting the number of schemes, the accuracy of the approximate model finally reaches the requirement at 250 sample points. Figure 9 shows the numerical results for the sample points, and as can be seen, they differ greatly in the whole sampling space. This indicates that OLH samples are relatively uniform in space and the range of variables can reflect the difference of samples.

FIG. 9.

Sample distribution results: (a) head; (b) efficiency.

FIG. 9.

Sample distribution results: (a) head; (b) efficiency.

Close modal

The results of regression analysis are shown in Fig. 10. As can be seen, the R2 values for head and efficiency are 0.9616 and 0.9437, respectively, which exceed the threshold (0.9).

FIG. 10.

R2 values for surrogate models: (a) head; (b) efficiency.

FIG. 10.

R2 values for surrogate models: (a) head; (b) efficiency.

Close modal
Sensitivity analysis measures the closeness of correlation between two variable factors, and the main types of correlation analysis are Pearson and Spearman. In this work, Pearson correlation was used to calculate the correlation coefficient,25 i.e.,
rXY=XX̄YȲXX̄2YȲ2.
(14)

The sign of the correlation coefficient indicates positive or negative correlation, and its absolute value indicates the degree of association. For example, a correlation coefficient of 1 indicates a perfect positive correlation, whereas a correlation coefficient of 0 indicates that there is no linear relationship between the two variables. An absolute value of 0.4 is usually used as the threshold: when −0.4 < r < 0.4, it is considered that there is no significant association between the two variables.

The results of the correlation analysis are shown in Fig. 11, where positive and negative effects are marked with blue and red bars, respectively. Figure 11(a) shows that Z, β2h, and β1s greatly influence the head of the axial flow pump, and the blue bars in Fig. 11(a) indicate that increasing D, Z, and β2h can effectively improve the pump head; by contrast, the effects of φdh, φds, and α3h on the pump head are almost negligible. Figure 11(b) shows that the pump efficiency is affected mainly by the impeller blade angles and especially the blade outlet shroud angle β2s. Therefore, when designing axial flow pumps, more attention should be paid to the design of the impeller.

FIG. 11.

Correlation analysis: (a) effect on head; (b) effect on average efficiency.

FIG. 11.

Correlation analysis: (a) effect on head; (b) effect on average efficiency.

Close modal

The optimal solution is obtained by solving the RBFNN surrogate model. The design variables before and after optimization are given in Table V, and as can be seen, the hub diameter and the blade chord length have changed significantly. Figure 12 shows that after optimization, the blade surface becomes more distorted; for the diffuser, the curvature of the pressure side increases. The numerical results of the optimized scheme are compared with the predicted values in Table VI, and as can be seen, the forecasts by the surrogate models have very small errors (<2%). Compared with the CFD results, it is found that the efficiency of the optimized scheme at the design point has increased by 7.81%, the average efficiency has increased by 7.65%, and the high-efficiency operation area has been significantly improved.

TABLE V.

Changes in geometric parameters.

VariableInitialOptimized
D (mm) 240 244 
dh (mm) 72 96.2 
β1h (°) 37.2 39.1 
β1s (°) 13.5 18.6 
β2h (°) 43.3 46.2 
β2s (°) 20.2 23.3 
lh (mm) 79.4 97.3 
ls (mm) 137.6 152.7 
α3h (°) 49.3 48.2 
α3s (°) 65.2 68.4 
φdh (°) 45.1 41.1 
φds (°) 11.3 13.8 
Z 
Zd 
VariableInitialOptimized
D (mm) 240 244 
dh (mm) 72 96.2 
β1h (°) 37.2 39.1 
β1s (°) 13.5 18.6 
β2h (°) 43.3 46.2 
β2s (°) 20.2 23.3 
lh (mm) 79.4 97.3 
ls (mm) 137.6 152.7 
α3h (°) 49.3 48.2 
α3s (°) 65.2 68.4 
φdh (°) 45.1 41.1 
φds (°) 11.3 13.8 
Z 
Zd 
FIG. 12.

Comparison of 3D geometry: (a) initial scheme; (b) optimized scheme.

FIG. 12.

Comparison of 3D geometry: (a) initial scheme; (b) optimized scheme.

Close modal
TABLE VI.

Comparison of predicted and CFD results.

CaseMethodFlow rateHead (m)Efficiency (%)F (%)
Initial CFD 0.9Qd 12.93 78.72 77.85 
1.0Qd 11.39 79.26 
1.1Qd 9.64 75.56 
Optimized RBF 0.9Qd 13.35 84.06 86.19 
1.0Qd 11.49 87.57 
1.1Qd 8.91 86.94 
 CFD 0.9Qd 13.49 83.31 85.5 
 1.0Qd 11.54 87.07 
 1.1Qd 8.79 86.11 
CaseMethodFlow rateHead (m)Efficiency (%)F (%)
Initial CFD 0.9Qd 12.93 78.72 77.85 
1.0Qd 11.39 79.26 
1.1Qd 9.64 75.56 
Optimized RBF 0.9Qd 13.35 84.06 86.19 
1.0Qd 11.49 87.57 
1.1Qd 8.91 86.94 
 CFD 0.9Qd 13.49 83.31 85.5 
 1.0Qd 11.54 87.07 
 1.1Qd 8.79 86.11 

Figures 1315 compare the velocity streamlines at different spanwise locations under three studied conditions. When the original pump is operating under low flow rate and the design conditions, the inlet flow angle of the impeller is smaller than the blade inlet angle, and the positive incidence angle will cause serious impact loss. By adjusting the blade angle, the inflow of the optimized impeller is more uniform, and the incidence angle is almost zero at the blade leading edge. In addition, serious velocity slip occurs inside the original impeller at span =0.1. When the hub diameter and blade number are increased, the channel width decreases, and the influence of the limited blade number is weakened. The function of the diffuser is to achieve energy conversion, and the vortex structures in the diffuser will cause energy dissipation and reduce the pump efficiency. The unreasonable design of the guide-vane profile is prone to cause flow separation, especially at span =0.1 and 0.5, but the optimized guide vane can better control the diffusion of fluid. The separation vortex in the diffuser is effectively suppressed; in particular, no vortex structures exist within the diffuser at high flow rate.

FIG. 13.

Flow patterns at different spanwise locations for 0.9Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 13.

Flow patterns at different spanwise locations for 0.9Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
FIG. 14.

Flow patterns at different spanwise locations for 1.0Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 14.

Flow patterns at different spanwise locations for 1.0Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
FIG. 15.

Flow patterns at different spanwise locations for 1.1Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 15.

Flow patterns at different spanwise locations for 1.1Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
Pressure pulsations in the pump not only cause structural vibrations but also increase the energy dissipation therein. An important indicator of pressure fluctuations is the pressure pulsation intensity (PPI), which is expressed as
PPI =i=1mpip̄m,
(15)
where pi is the instantaneous pressure and p̄ is the average pressure.

Figures 1618 show the PPI at different spanwise locations. As can be seen, the PPI decreases gradually with increasing flow rate. Because of the inflow shock on the blade pressure side, significant pressure fluctuations can be observed in the impeller of the initial scheme at both 0.9Qd and 1.0Qd, and these propagate downstream. When the pressure fluctuations propagate to the suction sides of the diffuser vanes, the flow separation induces larger pressure fluctuations. The optimized blades suppress pressure pulsations inside impeller very well, thus reducing those in the diffuser.

FIG. 16.

Pressure pulsations at different spanwise locations for 0.9Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 16.

Pressure pulsations at different spanwise locations for 0.9Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
FIG. 17.

Pressure pulsations at different spanwise locations for 1.0Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 17.

Pressure pulsations at different spanwise locations for 1.0Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
FIG. 18.

Pressure pulsations at different spanwise locations for 1.1Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 18.

Pressure pulsations at different spanwise locations for 1.1Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal

At high flow rate, the optimization of the guide-vane profile is very effective in suppressing pressure pulsations. It is difficult to detect pressure pulsations inside the diffuser passages. Also, the rotor–stator interaction is weakened, and the pressure pulsations do not propagate along the guide vane to the outlet, thus the energy dissipation is decreased significantly.

Figure 19 compares the total entropy production (EPT). The total energy loss in both the impeller and diffuser of the optimized scheme has decreased considerably, which explains the considerable efficiency improvement compared with the original model from the perspective of energy conversion. Dramatically, the total entropy production of the diffuser was reduced by 75.87%, 83.61%, and 83% at 0.9Qd, 1.0Qd, and 1.1Qd, respectively. Therefore, matching the rotor and stator is a more reliable way to optimize axial flow pumps.

FIG. 19.

Comparison of total entropy production between original and optimized models: (a) impeller; (b) diffuser.

FIG. 19.

Comparison of total entropy production between original and optimized models: (a) impeller; (b) diffuser.

Close modal

To further understand the flow loss distribution, Figs. 2022 compare the distributions of volume entropy production rate (EPR) inside the impeller and diffuser. As can be seen, the hydrofoil near the hub of the initial scheme is not designed properly, and under partial load, the improper incidence angle of the initial scheme causes significant flow loss. By contrast, the optimized scheme shows a more reasonable profile design, and the huge flow loss between the impeller and diffuser has disappeared.

FIG. 20.

Volume entropy generation at different spanwise locations for 0.9Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 20.

Volume entropy generation at different spanwise locations for 0.9Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
FIG. 21.

Volume entropy generation at different spanwise locations for 1.0Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 21.

Volume entropy generation at different spanwise locations for 1.0Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal
FIG. 22.

Volume entropy generation at different spanwise locations for 1.1Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

FIG. 22.

Volume entropy generation at different spanwise locations for 1.1Qd: (a) span =0.1; (b) span =0.5; (c) span =0.9. Left: initial scheme; right: optimized scheme.

Close modal

When the pump is operating under nominal flow, the flow loss is found mainly at the impeller outlet and inside the diffuser. The rotor–stator interaction and vortex flow together affect the flow loss, and the optimized scheme significantly reduces the rotor–stator interaction and suppresses the flow separation in the diffuser.

At high flow rate, there is no obvious difference in the flow loss within the impeller. However, the loss in the diffuser always starts at the inlet and spreads downstream. This means that the diffuser inlet blockage and inflow angle will affect the flow loss. The optimized diffuser vane profile provides better inflow for the diffuser, which almost eliminates the flow loss in the diffuser.

To investigate the flow regime in the impeller, the streamlines on the blade surfaces are shown in Figs. 23 and 24. Theoretically, the flow on the blade surfaces of the axial flow pump should be only in the circumferential direction, but the interaction between the cylindrical layers means that radial flow may occur on the blade surfaces, and this will cause flow loss on the blade surfaces. For the initial scheme, radial flow can be observed on the pressure side for 0.9Qd and 1.0Qd, and it occurs near the hub and blade trailing edge. By contrast, the optimized blade suppresses the corner vortex flow for 0.9Qd. For 1.0Qd and 1.1Qd, there is little radial flow on the blade pressure side. Thus, the blades can transfer more energy to the fluid.

FIG. 23.

Flow patterns on blade pressure side: (a) initial scheme; (b) optimized scheme.

FIG. 23.

Flow patterns on blade pressure side: (a) initial scheme; (b) optimized scheme.

Close modal
FIG. 24.

Flow patterns on blade suction side: (a) initial scheme; (b) optimized scheme.

FIG. 24.

Flow patterns on blade suction side: (a) initial scheme; (b) optimized scheme.

Close modal

It is necessary to carry out experiments to enhance the reliability of optimization. To ensure the accuracy of the 3D surface, the impeller was processed using a numerical control machine, and performance tests were carried out on the test rig mentioned in Sec. II C. Figure 25 shows the machined impeller and the pump installation. Except for the impeller and diffuser, all other factors in the test were the same as in the initial scheme.

FIG. 25.

Setup for experimental measurements.

FIG. 25.

Setup for experimental measurements.

Close modal

Figure 26 compares the performance curves. The optimized solution results in a significant increase in the best efficiency of the pump, and the performance curves show that the efficiency improvement at high flow rate is impressive. However, the efficiency decreases slightly at low flow rate. Furthermore, the optimized solution broadens the high-efficiency operating range considerably. Comparing the head curves shows that the unstable point in the optimized case is brought forward, but this has less impact on the operation of the pump because it basically will not operate at too low a flow rate.

FIG. 26.

Comparison of experimental performance curves.

FIG. 26.

Comparison of experimental performance curves.

Close modal

In this work, an optimization strategy was proposed for axial flow pumps, and impeller–diffuser matching optimization was performed. From the results, the following conclusions are drawn.

  1. According to the regression analysis, the RBFNN has strong modeling capability, especially for complex nonlinear relationships.

  2. The results showed that D, Z, and β2h have significant effects on the pump head. The pump efficiency is affected mainly by the impeller blade angles and especially the blade outlet shroud angle β2s.

  3. According to the CFD results, the optimized scheme considerably improves the impeller inflow incidence and the flow separation in the diffuser. Meanwhile, the high-pressure pulsation intensity in the pump is effectively alleviated.

  4. The flow loss was visualized and estimated via entropy generation. After optimization, the flow loss in the impeller and diffuser is effectively reduced.

  5. The experimental verification showed that the pump efficiency is increased when the flow rate exceeds 0.82Qd, and the efficiency improvement at high flow rate is impressive.

The authors thank the China Scholarship Council for its assistance and encouragement.

The authors have no conflicts to disclose.

Lu Rong: Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Martin Böhle: Supervision (equal). Gu Yandong: Formal analysis (equal).

The authors confirm that the data supporting the findings of this study are available within the article.

1.
H.
Bozorgasareh
,
J.
Khalesi
,
M.
Jafari
, and
H. O.
Gazori
, “
Performance improvement of mixed-flow centrifugal pumps with new impeller shrouds: Numerical and experimental investigations
,”
Renewable Energy
163
,
635
648
(
2021
).
2.
Y. D.
Gu
,
J. X.
Li
,
P.
Wang
,
L.
Cheng
,
Y.
Qiu
,
C.
Wang
, and
Q. R.
Si
, “
An improved one-dimensional flow model for side chambers of centrifugal pumps considering the blade slip factor
,”
J. Fluids Eng.
144
(
9
),
091207
(
2022
).
3.
B.
Chen
,
Z. B.
Li
,
H.
Zhang
,
Y.
Ma
, and
Y. Z.
Gu
, “
Optimization design of axial flow pump based on circumferential velocity
,”
Int. J. Fluid Dyn.
06
(
04
),
124
133
(
2018
).
4.
L. J.
Shi
,
F. P.
Tang
,
C.
Liu
, and
R. S.
Xie
, and
W. P.
Zhang
, “
Optimal design of multi-conditions for axial flow pump
,”
IOP Conf. Ser., Earth Environ. Sci.
49
(
6
),
062028
(
2016
).
5.
Z. X.
Dai
,
L.
Tan
,
B. F.
Han
, and
S. Y.
Han
, “
Multi-parameter optimization design of axial-flow pump based on orthogonal method
,”
Energies
15
(
24
),
9379
(
2022
).
6.
Z.
Tong
,
J.
Xin
, and
C.
Ling
, “
Many-objective hybrid optimization method for impeller profile design of low specific speed centrifugal pump in district energy systems
,”
Sustainability
13
(
19
),
10537
(
2021
).
7.
G.
Zanetti
,
M.
Siviero
,
G.
Cavazzini
, and
A.
Santolin
, “
Application of the 3D inverse design method in reversible pump turbines and Francis turbines
,”
Water
15
(
12
),
2271
(
2023
).
8.
H. S.
Shim
and
K. Y.
Kim
, “
Design optimization of the impeller and volute of a centrifugal pump to improve the hydraulic performance and flow stability
,”
J. Fluids Eng.
142
(
10
),
101211
(
2020
).
9.
C.
Kang
,
X. J.
Yu
,
W. F.
Gong
,
C. J.
Li
, and
Q. F.
Huang
, “
Influence of stator vane number on performance of the axial-flow pump
,”
J. Mech. Sci. Technol.
29
(
5
),
2025
2034
(
2015
).
10.
L. J.
Shi
,
Y.
Chai
,
L.
Wang
,
T.
Xu
,
Y. H.
Jiang
,
J.
Xing
,
B. Y.
Yan
,
Y. Y.
Chen
, and
Y.
Han
, “
Numerical simulation and model test of the influence of guide vane angle on the performance of axial flow pump
,”
Phys. Fluids
35
(
1
),
015129
(
2023
).
11.
W. J.
Wang
,
Z. H.
Han
,
J.
Pei
,
G.
Pavesi
,
X. B.
Gong
, and
S. Q.
Yuan
, “
Energy efficiency optimization of water pump based on heuristic algorithm and computational fluid dynamics
,”
J. Comput. Des. Eng.
10
(
1
),
382
397
(
2023
).
12.
Y. D.
Gu
,
J. W.
Cheng
,
P.
Wang
,
L.
Cheng
,
Q. R.
Si
,
C.
Wang
, and
Y.
Shouqi
, “
A flow model for side chambers of centrifugal pumps considering radial wall shear stress
,”
Proc. Inst. Mech. Eng., Part C
236
(
13
),
7115
7126
(
2022
).
13.
Y.
Yang
,
L.
Zhou
,
J. W.
Hang
,
D. Y.
Du
,
W. D.
Shi
, and
Z. M.
He
, “
Energy characteristics and optimal design of diffuser meridian in an electrical submersible pump
,”
Renewable Energy
167
,
718
727
(
2021
).
14.
J.
Pei
,
W. J.
Wang
,
M. K.
Osman
, and
X. C.
Gan
, “
Multiparameter optimization for the nonlinear performance improvement of centrifugal pumps using a multilayer neural network
,”
J. Mech. Sci. Technol.
33
(
6
),
2681
2691
(
2019
).
15.
W. J.
Wang
,
J.
Pei
,
S. Q.
Yuan
,
J. F.
Zhang
,
J. P.
Yuan
, and
C. Z.
Xu
, “
Application of different surrogate models on the optimization of centrifugal pump
,”
J. Mech. Sci. Technol.
30
(
2
),
567
574
(
2016
).
16.
J. T.
Zhao
,
J.
Pei
,
J. P.
Yuan
, and
W. J.
Wang
, “
Energy-saving oriented optimization design of the impeller and volute of a multi-stage double-suction centrifugal pump using artificial neural network
,”
Eng Appl. Comput. Fluid Mech.
16
(
1
),
1974
2001
(
2022
).
17.
Y. Z.
Wu
,
D. H.
Wu
,
M. H.
Fei
,
H.
Sørensen
,
Y.
Ren
, and
J. G.
Mou
, “
Application of GA-BPNN on estimating the flow rate of a centrifugal pump
,”
Eng. Appl. Artif. Intell.
119
,
105738
(
2023
).
18.
Z. M.
Tong
,
Z. Q.
Yang
,
S. G.
Tong
,
Z. K.
Shu
, and
X. E.
Cao
, “
Investigating the hydraulic performance of slanted axial flow pumps using an enstrophy dissipation-based hybrid optimization approach
,”
Phys. Fluids
35
(
5
),
057117
(
2023
).
19.
J. T.
Zhao
,
J.
Pei
,
J. P.
Yuan
, and
W. J.
Wang
, “
Structural optimization of multistage centrifugal pump via computational fluid dynamics and machine learning method
,”
J. Comput. Des. Eng.
10
(
3
),
1204
1218
(
2023
).
20.
W. X.
Ye
,
C.
Geng
,
A.
Ikuta
,
S.
Hachinota
, and
K.
Miyagawa
, and
X.
Luo
, “
Investigation on the impeller-diffuser interaction on the unstable flow in a mixed-flow pump using a modified partially averaged Navier-Stokes model
,”
Ocean Eng.
238
,
109756
(
2021
).
21.
R.
Lu
,
J. P.
Yuan
,
L. Y.
Wang
,
Y. X.
Fu
,
F.
Hong
, and
W. J.
Wang
, “
Effect of volute tongue angle on the performance and flow unsteadiness of an automotive electronic cooling pump
,”
Proc. Inst. Mech. Eng., Part A
235
(
2
),
227
241
(
2020
).
22.
F.
Zhang
,
D.
Appiah
,
F.
Hong
,
J. F.
Zhang
,
S. Q.
Yuan
,
K. A.
Adu-Poku
, and
X. Y.
Wei
, “
Energy loss evaluation in a side channel pump under different wrapping angles using entropy production method
,”
Int. Commun. Heat Mass Transfer
113
,
104526
(
2020
).
23.
Y. D.
Gu
,
H.
Sun
,
C.
Wang
,
R.
Lu
,
B. Q.
Liu
, and
J.
Ge
, “
Effect of trimmed rear shroud on performance and axial thrust of multi-stage centrifugal pump with emphasis on visualizing flow losses
,”
J. Fluids Eng.
146
(
1
),
011204
(
2024
).
24.
H.
Herwig
and
B.
Schmandt
, “
How to determine losses in a flow field: A paradigm shift towards the second law analysis
,”
Entropy
16
(
6
),
2959
2989
(
2014
).
25.
R.
Lu
,
J. P.
Yuan
,
G. J.
Wei
,
Y.
Zhang
,
X. H.
Lei
, and
Q. R.
Si
, “
Optimization design of energy-saving mixed flow pump based on MIGA-RBF algorithm
,”
Machines
9
(
12
),
365
(
2021
).
26.
D. N.
Moriasi
,
J. G.
Arnold
,
M. W.
Van Liew
,
R. L.
Bingner
,
R. D.
Harmel
, and
T. L.
Veith
, “
Model evaluation guidelines for systematic quantification of accuracy in watershed simulations
,”
Trans. ASABE
50
(
3
),
885
900
(
2007
).
27.
T. S.
Li
,
S. K.
Duan
,
J.
Liu
, and
L. D.
Wang
, “
An improved design of RBF neural network control algorithm based on spintronic memristor crossbar array
,”
Neural Comput. Appl.
30
,
1939
1946
(
2018
).
28.
A.
Sinha
,
P.
Malo
, and
K.
Deb
, “
Evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping
,”
Eur. J. Oper. Res.
257
(
2
),
395
411
(
2017
).
29.
M.
Miki
,
T.
Hiroyasu
,
M.
Kaneko
, and
K.
Hatanaka
, “
A parallel genetic algorithm with distributed environment scheme
,” in
Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Tokyo, Japan, 12-15 October
(
IEEE
,
1999
), Vol.
1
, pp.
695
700
.