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Most pumping machineries have a problem of obtaining a higher efficiency over a wide range of operating conditions. To solve that problem, an optimization strategy has been designed to widen the high-efficiency range of the double-suction centrifugal pump at design (_{d}) and nondesign flow conditions. An orthogonal experimental scheme is therefore designed with the impeller hub and shroud angles as the decision variables. Then, the “efficiency-house” theory is introduced to convert the multiple objectives into a single optimization target. A two-layer feedforward artificial neural network (ANN) and the Kriging model were combine based on a hybrid approximate model and solved with swarm intelligence for global best parameters that would maximize the pump efficiency. The pump performance is predicted using three-dimensional Reynolds-averaged Navier–Stokes equations which is validated by the experimental test. With ANN, Kriging, and a hybrid approximate model, an optimization strategy is built to widen the high-efficiency range of the double-suction centrifugal pump at overload conditions by 1.63%, 1.95%, and 4.94% for flow conditions 0.8_{d}, 1.0_{d}, and 1.2_{d}, respectively. A higher fitting accuracy is achieved for the hybrid approximation model compared with the single approximation model. A complete optimization platform based on efficiency-house and the hybrid approximation model is built to optimize the model double-suction centrifugal pump, and the results are satisfactory.

There have been persistent efforts to reduce global fossil fuel consumption due to growing concerns of carbon emissions. This has made energy efficient centrifugal pumps a necessity since pumping power consumes about 10% of the global power share [

The theory of pump design has over the years advanced from the one-dimensional Euler equation and empirical correction theory to a three-dimensional design theory such as direct design with numerical simulation. During this period, several optimization strategies have been applied by various researchers. Early design methods applied theoretical formulas to solve the mathematical expressions of pump performance for the design parameters. Though these were simple and straightforward, it was not so accurate. For example, theoretical models of net positive suction head (NPSH) and efficiency have been used for single-objective optimization in centrifugal and mixed-flow pumps [

Recently, there has been a shift from the single-objective optimization methods to the application of genetic algorithms (GAs) and surrogate models to find optimum global geometrical parameter combinations that can solve multiobjective optimization problems. Practically, the single-objective optimization cannot meet the design requirements for multiple objectives. Thus, compared with theoretical and design of experiment method, the surrogate model-based optimization methods are much more accurate and can control more design variables [

Whereas all these studies have focused on improving the pump performance at the nominal flow condition, the problem of obtaining a higher efficiency over a wide range of operating conditions has been rarely researched. To solve that problem, an optimization strategy has been designed to widen the efficiency range of the double-suction centrifugal pump over multiple working conditions. To improve the optimization accuracy and obtain the global optimal variables, an optimization method based on “efficiency-house” theory was developed to convert all the multiple objective functions into a single optimization objective. A hybrid approximate model was then proposed to improve the prediction accuracy. The fitting principles and accuracy test methods of a single approximation model and the hybrid approximation model were introduced and optimized at 0.8_{d}, 1.0_{d}, and 1.2_{d} based on ANN and Kriging models. This paper introduces an “efficiency-house” theory into a hybrid approximation model for optimization of the high-efficient range of the double-suction centrifugal pump.

The computational research domain is a 250GS40 double-suction centrifugal pump. The three-dimensional model of the pump was built with Siemens NX. This is shown in Figure

Computational model for model pump. (a) Complete model, (b) suction, and (c) impeller.

Design specifications of model pump.

Design parameters | Value |
---|---|

Flow rate, _{d} (m^{3}/h) | 500 |

Head, | 40 |

Rotational speed, | 1480 |

Suction diameter, _{s} (mm) | 250 |

Impeller inlet diameter, _{1} (mm) | 192 |

Impeller outlet diameter, _{2} (mm) | 365 |

Delivery diameter, _{d} (mm) | 200 |

Efficiency, | 84 |

From the continuity equation, the time-dependent Reynolds-averaged Navier–Stokes (RANS) equation [_{t}, and

A test of grid sensitivity was carried out to determine the total mesh elements suitable for the numerical simulation. This was to save computation time while maintaining accuracy of the simulation. First ANSYS ICEM was used to build structural hexahedral mesh for the whole computational domain. Further grid refinement was carried out and concentrated at the walls to attain higher precision and boundary motion features. This has been presented in Figure

Mesh of calculation domain: (a) volute tongue, (b) impeller, and (c) suction tongue.

Arrangement of monitor points.

Grid cells of the selected mesh.

Item | Mesh I | Mesh II | Mesh III | Mesh IV | Mesh V | |
---|---|---|---|---|---|---|

Total mesh | 2878243 | 3679342 | 4266423 | 4958168 | 5847757 | |

GCI (%) | 6.021 | 4.324 | 1.962 | 1.921 | 1.835 | |

Ratio | _{1} | 1 | 1.156 | 1.221 | 1.220 | 1.221 |

_{1} | 1 | 1.132 | 1.304 | 1.304 | 1.304 | |

_{v1}_{v1,1} | 1 | 1.152 | 1.311 | 1.311 | 1.311 | |

_{v2}_{v2,1} | 1 | 1.156 | 1.282 | 1.282 | 1.282 |

Blade surface

The ANSYS software package was used for the numerical investigations. The working fluid for the flow domain was water at 25°C. The reference pressure was set to 0 atm, and an isothermal heat transfer rate was chosen to put the system into thermal equilibrium. The pressure gradient is to be maintained accurately irrespective of the distance to the wall, and therefore the SST (^{−5} and iterations were periodic stable. Performance indicators were calculated with reference to Gülich [

In order to validate the numerical model, experiments were conducted using the original pump model in an open test rig system. Figures

Schematics of the test setup.

Test pump (a) and data acquisition device (b).

Since the experimentally measured total pump efficiency includes hydraulic efficiency, volumetric efficiency, and mechanical friction loss efficiency, the efficiency of numerical simulation is further processed. In the numerical simulation, the efficiency calculated is the hydraulic efficiency only. The experimental efficiency (overall efficiency) of the pump comprises the hydraulic efficiency

The volumetric efficiency is expressed as

The relationship between the efficiencies is expressed as

This has been done in the previous studies [_{d}, representing about 2.49%, whereas for part load conditions, the highest deviation between the test and numerical results was 1.67%. In summary, the deviation between test and numerical values of both the head and efficiency for each flow condition are less than 3%, which is within a reasonable range, and therefore the numerical simulation method adopted in this paper is reliable.

Validation with test results.

Figure

Flow chart of optimization strategy.

For this study, the optimization objective was to maximize the pump efficiency for flow conditions 0.8_{d}, 1.0_{d}, and 1.2_{d} concurrently. This was obtained by numerical model calculation. The optimization objective can be expressed as_{d}, _{d}.

The expression for calculating efficiency is^{3}) is the density, ^{3}/h) is the flow discharge, and

The structure of the double-suction impeller is compact and complex, making space constraints a limitation due the structure of the suction and volute casing. Therefore, the shape of the impeller was maintained by holding constant the inlet diameter _{1}, the impeller outlet diameter _{2}, the hub diameter _{h}, and the blade width at outlet _{2}. The optimization therefore focused on the impeller hub and shroud angle effects on the blade profile, with all other parameters held constant. All eight geometry parameters were selected as decision variables for optimization. These are as follows: _{1_h} and _{2_h}, the blade angles at hub inlet and exit, respectively; _{3_s} and _{4_s}, the blade angle at the shroud inlet and exit; _{1_h} and _{2_s}, position of the leading edge at the hub and shroud; whereas Δ_{1_h} and Δ_{2_s}. are the leading edge wrap angle at the hub and shroud. The key to the parametric design of the impeller is to control of the blade profile. Figure

Impeller parameterization and meridional section.

The upper and lower limits for the decision variables were selected based on previous works [_{32} (8^{4}) orthogonal scheme was designed for parametrization and simulated. Through numerical simulation, the orthogonal test results for the objective functions

Range of decision variables.

Level | A | B | C | D | E | F | G | H |
---|---|---|---|---|---|---|---|---|

_{1_h}/° | _{2_h}/° | _{3_s}/° | Β_{4_s}/° | _{1_h}/° | _{2_s}/° | Δ_{1_h}/° | Δ_{2_s}/° | |

Original | 17 | 29.43 | 15 | 29.43 | 0 | 0 | 143 | 143 |

1 | 15 | 26 | 13 | 26 | −5 | −5 | 139 | 139 |

2 | 17 | 28 | 15 | 28 | −2.5 | −2.5 | 143 | 143 |

3 | 19 | 30 | 17 | 30 | 2.5 | 2.5 | 145 | 145 |

4 | 21 | 32 | 19 | 32 | 5 | 5 | 148 | 148 |

Orthogonal test results.

Level | |||
---|---|---|---|

1 | 84.70 | 88.03 | 86.87 |

2 | 84.14 | 88.69 | 87.51 |

3 | 84.86 | 89.21 | 87.60 |

4 | 83.31 | 88.16 | 87.59 |

5 | 82.25 | 87.41 | 86.45 |

6 | 83.86 | 87.26 | 84.61 |

7 | 83.17 | 87.20 | 86.04 |

8 | 84.58 | 88.43 | 87.25 |

9 | 83.72 | 87.88 | 87.43 |

10 | 83.68 | 87.83 | 86.70 |

11 | 85.07 | 88.44 | 85.61 |

12 | 83.99 | 88.70 | 87.80 |

13 | 84.25 | 88.62 | 88.35 |

14 | 84.87 | 87.68 | 85.07 |

15 | 83.74 | 88.02 | 87.40 |

16 | 85.00 | 88.77 | 86.77 |

17 | 84.96 | 88.66 | 86.28 |

18 | 83.79 | 87.51 | 85.31 |

19 | 83.99 | 87.88 | 85.27 |

20 | 83.92 | 87.85 | 85.66 |

21 | 84.70 | 89.40 | 88.82 |

22 | 84.44 | 88.98 | 86.74 |

23 | 83.29 | 87.83 | 87.37 |

24 | 83.91 | 88.52 | 87.54 |

25 | 83.85 | 88.36 | 86.94 |

26 | 84.17 | 88.90 | 87.88 |

27 | 84.40 | 88.93 | 87.99 |

28 | 83.77 | 87.79 | 85.69 |

29 | 84.15 | 87.87 | 87.72 |

30 | 83.86 | 88.43 | 85.41 |

31 | 84.89 | 88.18 | 87.09 |

32 | 83.01 | 87.17 | 84.72 |

It is necessary to conduct a range analysis to understand the extent to which each design variable affects the optimization target. To determine the variance _{i} is the sum of the values of the levels for each factor, _{i} is the average values, _{i,j} is the performance value for the factor, and

Table _{d}, _{2_h} and _{4_s} had greater impact on efficiency. _{2_h} and _{1_h} had the maximum influence on the optimization objective at the design point, whereas _{2_s} and Δ_{2}__{s} had the best influence at 1.2_{d}. Hence, three best parameter combinations according to Table _{d}, 1.0_{d}, and1.2_{d}, respectively.

Range analysis.

B | C | D | E | F | G | H | |||
---|---|---|---|---|---|---|---|---|---|

0.8Qd | 0.435 | 1.031 | 0.198 | 0.784 | 0.358 | 0.467 | 0.436 | 0.334 | |

1.0Qd | 0.753 | 0.832 | 0.444 | 0.550 | 0.227 | 0.409 | 0.227 | 0.509 | |

1.2Qd | 0.959 | 0.936 | 0.862 | 1.018 | 0.547 | 1.126 | 0.616 | 1.575 |

From Table _{d}, 0.8_{d}, 1.0_{d}, and 1.2_{d}) to their flow coefficients. The corresponding efficiency was then used as a sample for the polynomial fitting in the following equation:

Figure

Sketch map of “efficiency-house” model.

The three optimization models (model A, model B, and model C) obtained from the 32 sets of orthogonal experiments and range analysis were modelled according to the above steps, and the results are shown in Table _{2_h} (B), _{4_s} (D), _{1_h} (E), and _{2_s} (F) had the most influence on the single optimization target based on efficiency-house. Therefore, there four geometry parameters were set as decision variables for the optimization with the hybrid approximate model.

“Efficiency-house” of orthogonal tests.

Model | |
---|---|

1 | 9.0108 |

2 | 8.8285 |

3 | 8.8981 |

4 | 8.7494 |

5 | 8.5743 |

6 | 8.8071 |

7 | 8.7726 |

8 | 8.9455 |

9 | 8.8705 |

10 | 8.8197 |

11 | 8.9320 |

12 | 8.8157 |

13 | 8.9214 |

14 | 8.9795 |

15 | 8.8492 |

16 | 8.9430 |

17 | 8.9189 |

18 | 8.7994 |

19 | 8.7767 |

20 | 8.7942 |

21 | 8.9178 |

22 | 8.7892 |

23 | 8.7850 |

24 | 8.8110 |

25 | 8.7835 |

26 | 8.8254 |

27 | 8.8760 |

28 | 8.7750 |

29 | 8.9819 |

30 | 8.6676 |

31 | 9.0407 |

32 | 8.6526 |

A | 9.0885 |

B | 8.9772 |

C | 8.9057 |

Level of influence of decision variable on S.

The orthogonal experiment design only optimizes each parameter under the condition of four levels, so a more accurate optimization method is required. However, from the results of the range analysis for the single optimization objective, the decision variables were reduced to four geometry parameters of the double-suction impeller, namely, _{2_h}, _{4_s}, _{1_h}, and _{2_s}. Therefore, based on the reduced decision variables, the design variable range has been redefined in Table

LHS range of design parameters.

Input | ||||
---|---|---|---|---|

Lower bound | 264 24 | 265 24 | 266 145 | 267 145 |

Upper bounds | 269 30 | 270 30 | 271 155 | 272 155 |

The Latin hypercube test design method was used to randomly generate design points for the construction of the surrogate model. This is a method of approximately random sampling from a multivariate parameter distribution and belongs to the stratified sampling technique [_{d}, 1.0_{d}, and 1.2_{d} flow conditions as shown in Table

Cases designed by LHS.

Level | ||||
---|---|---|---|---|

1 | 26.8 | 25.7 | 146.0 | 147.1 |

2 | 29.3 | 27.9 | 149.7 | 152.3 |

3 | 26.4 | 29.2 | 148.7 | 153.4 |

4 | 27.5 | 29.3 | 148.8 | 152.2 |

5 | 29.6 | 28.4 | 146.4 | 148.9 |

… | … | … | … | … |

16 | 25.7 | 29.7 | 153.9 | 146.1 |

17 | 29.0 | 26.6 | 153.1 | 151.1 |

18 | 25.0 | 27.7 | 154.6 | 152.9 |

19 | 26.0 | 28.8 | 146.8 | 149.6 |

20 | 26.1 | 27.1 | 150.6 | 151.3 |

21 | 25.1 | 25.9 | 149.9 | 152.5 |

22 | 29.5 | 29.9 | 147.3 | 146.3 |

23 | 28.5 | 24.4 | 151.9 | 150.9 |

24 | 28.7 | 26.1 | 152.3 | 145.7 |

25 | 28.0 | 29.7 | 147.6 | 150.1 |

… | … | … | … | … |

36 | 28.2 | 28.2 | 150.8 | 147.9 |

37 | 26.5 | 24.1 | 149.5 | 148.7 |

38 | 28.9 | 26.9 | 152.9 | 149.1 |

39 | 27.4 | 24.3 | 151.3 | 148.4 |

40 | 27.0 | 28.7 | 153.6 | 151.9 |

Efficiency under three conditions.

Level | |||
---|---|---|---|

1 | 85.67 | 89.28 | 88.61 |

2 | 85.30 | 88.80 | 88.83 |

3 | 85.61 | 89.17 | 88.53 |

4 | 85.22 | 89.04 | 88.68 |

5 | 84.92 | 88.54 | 88.25 |

… | … | … | … |

… | … | … | … |

15 | 85.73 | 89.42 | 88.69 |

16 | 84.89 | 89.00 | 88.46 |

17 | 85.55 | 88.88 | 88.65 |

18 | 86.03 | 89.15 | 86.35 |

19 | 85.72 | 89.34 | 88.97 |

20 | 85.87 | 89.29 | 88.79 |

21 | 86.21 | 89.80 | 89.16 |

22 | 84.60 | 88.23 | 88.10 |

23 | 85.65 | 89.34 | 88.99 |

24 | 84.98 | 89.21 | 88.77 |

25 | 85.18 | 88.88 | 88.52 |

… | … | … | … |

… | … | … | … |

35 | 85.45 | 89.04 | 88.53 |

36 | 84.82 | 88.88 | 88.40 |

37 | 86.26 | 89.65 | 89.17 |

38 | 85.05 | 88.89 | 88.57 |

39 | 84.38 | 88.45 | 88.11 |

40 | 85.61 | 88.88 | 88.49 |

Efficiency-house of design.

Level | |
---|---|

1 | 9.1254 |

2 | 9.1436 |

3 | 9.1257 |

4 | 9.0765 |

5 | 9.0675 |

6 | 9.1459 |

7 | 9.1593 |

8 | 9.2032 |

9 | 9.1163 |

10 | 9.1646 |

11 | 9.0951 |

12 | 9.0851 |

13 | 9.1125 |

14 | 8.9159 |

15 | 9.1201 |

16 | 8.9991 |

17 | 9.1699 |

18 | 9.0651 |

19 | 9.1509 |

20 | 9.1779 |

21 | 9.1895 |

22 | 9.0421 |

23 | 9.1377 |

24 | 9.0044 |

25 | 9.0837 |

26 | 9.1818 |

27 | 9.0551 |

28 | 9.1167 |

29 | 9.0867 |

30 | 9.1925 |

31 | 9.147 |

32 | 9.0587 |

33 | 9.0867 |

34 | 8.962 |

35 | 9.114 |

36 | 9.0004 |

37 | 9.2256 |

38 | 9.0585 |

39 | 8.9602 |

40 | 9.1713 |

The approximate model theory is mainly used for complex engineering problems that are difficult to describe with mathematical expressions and is generally called “black box problem.” At present, the most widely used approximate models in engineering design are artificial neural network (ANN), Kriging, polynomial response surface (RSM) models. Jin [

For this study, a two-layer feedforward artificial neural network with sigmoid hidden neurons and linear output neurons was adopted. The Levenberg–Marquardt algorithm was adopted since it is the fastest of all traditional or improved networks [

The Kriging model is widely used for interpolation of nonlinear problems, and a series of methods suitable for various situations can be derived. For example, Ordinary Kriging is a univariate local linear optimal unbiased estimation method, while Ordinary Co-Kriging can be extended to two or more variables. Its mathematical expression is written as

In equation (

PSO is a widely used swarm intelligence algorithms because it has excellent global search capabilities. Its origin is from the process of birds searching for food using acoustics of echolocation. The algorithm imitates the cluster behavior and sets a series of behavior rules for each individual, so that the group can complete complex tasks. The basic mathematical model of particle swarm optimization is

This paper combines ANN and the Kriging model and uses the approximate model to fit the correlation coefficient ^{2} to solve the mixed model coefficients. The specific construction process is shown in equations (^{2} is the correlation coefficient,

Fitting process of hybrid approximate model.

The 40 groups of data generated by the above LHS were divided into two groups: 70% of the samples (28 groups) were used to train artificial neural networks and Kriging and 30% of the samples (12 groups) were used to verify the accuracy of the fit. After the approximate model was successfully established, the accuracy of the approximate model was evaluated using the ^{2} values of ANN was 0.8, whereas Kriging fitting accuracy had ^{2} value is about 0.88. However, both ^{2} values were less than 0.9. For the composite model, the fitting accuracy of the hybrid approximate model reached 0.95167, as shown in Figure

Fitting accuracy of composite model.

The ANN, Kriging, and hybrid approximation models established were used for optimization calculations. After convergence, the optimal design parameter combination and the optimal

Optimization results of different surrogate model.

ANN | 25.0235 | 28.8943 | 150.5623 | 152.8956 | 9.2268 |

Kriging | 24.8156 | 29.2201 | 149.7032 | 154.2331 | 9.2257 |

Hybrid | 24.8707 | 29.2325 | 148.8955 | 154.0112 | 9.2263 |

Error analysis of optimization results.

Error (%) | |||
---|---|---|---|

ANN | 9.2268 | 9.0243 | 2.24 |

Kriging | 9.2257 | 9.1296 | 1.05 |

Hybrid | 9.2263 | 9.2260 | 0.003 |

The numerical simulation results from hybrid approximate optimization model for each flow condition are compared with those of the original model in Table _{d}, 1.0_{d}, and 1.2_{d}, respectively. Although the head was slightly reduced under each working condition, the reduction in head lies within the optimization objective of no more than 5%.

Comparison of performance.

Deviation (%) | Head (m) | Deviation (%) | ||||
---|---|---|---|---|---|---|

Original | Optimized | Original | Optimized | |||

0.8 _{d} | 84.92 | 86.3 | 1.63 | 41.84 | 41.65 | 0.45 |

1.0 _{d} | 87.99 | 89.71 | 1.95 | 40.53 | 39.32 | 2.99 |

1.2 _{d} | 85.00 | 89.2 | 4.94 | 36.51 | 36.18 | 0.90 |

In order to study the flow loss inside different components, the head distribution analysis was performed on the original model and the optimized model. The head distribution in different flow channels before and after optimization is shown in Table

Hydraulic head distribution of optimized and original cases.

Flowrate | Model | Suction (m) | Impeller (m) | Volute (m) | Power (kw) |
---|---|---|---|---|---|

0.8_{d} | Original | −1.307 | 48.620 | −4.342 | 53.291 |

Optimized | −0.258 | 45.920 | −3.725 | 52.957 | |

1.0_{d} | Original | −0.167 | 44.196 | −3.141 | 62.735 |

Optimized | −0.150 | 42.372 | −2.673 | 59.877 | |

1.2_{d} | Original | −0.200 | 41.243 | −4.187 | 70.206 |

Optimized | −0.210 | 39.196 | −2.582 | 66.554 |

To clearly see and compare the changes in the high-efficiency zone before and after optimization, the efficiency curve for optimal model A in the orthogonal test and the efficiency from the hybrid approximation is compared with that of the original model in Figure

Comparison of high-efficiency zone before and after optimization.

To illustrate the pump performance improvement before and after optimization, the internal flow characteristics of the pump is compared. A comparison of the internal flow characteristics of the original model and the final optimized model after hybrid approximation is compared in Figures

Comparison of pressure distribution at 0.8 _{d}.

Comparison of pressure distribution 1.0 _{d}.

Comparison of pressure distribution at 1.2 _{d}.

Figures

Velocity and streamline distribution at 0.8 _{d}.

Velocity and streamline distribution at 1.0 _{d}.

Velocity and streamline distribution at 1.2 _{d}.

In this paper, an optimization strategy was designed to improve the high-efficiency range of the double-suction centrifugal pump. First the “efficiency-house” theory was introduced to convert the multiple objectives into a single optimization target. The fitting principles and accuracy test methods of single approximation model and hybrid approximation model were introduced, and multiple working conditions of the double-suction pump were optimized based on ANN, Kriging, and hybrid approximation models. The following conclusions were drawn:

The geometry parameters _{2_h}, _{4_s}, _{1_h}, and _{2_s} had the most influence on the single optimization target based on efficiency-house

The fitting accuracy of the composite approximate model is higher than that of the single approximate model, especially under the condition of a smaller sample size

The accuracy of the hybrid approximation model is higher than that of the single approximation model, ensuring the reliability of the hybrid approximation model

With ANN and Kriging models, a hybrid approximate model an optimization strategy was built to widen the high-efficiency range of the double-suction centrifugal pump at overload conditions by 1.63%, 1.95%, and 4.94% under for flow conditions 0.8_{d},1.0_{d}, and 1.2_{d}, respectively

The TKE distribution was reduced, and the secondary flow phenomena towards the exit of the volute during overload conditions improved significantly after optimization.

_{2}:

Blade width, mm

Impeller diameter, mm

Head, m

Kinetic energy of turbulence, m^{2}/s^{2}

Rotational speed, r/min

Pressure, Pa

_{S}:

Shaft power, kW

Flow rate, m^{3}/h

Area of efficiency-house

Velocity, m/s

Number of blades.

_{1_h}:

Blade angle at hub inlet, °

_{2_h}:

Blade angle at hub exit, °

_{3_s}:

Blade angle at shroud inlet, °

_{4_s}:

Blade angle at shroud exit, °

_{1_h}:

Leading edge position at hub, °

_{2_s}:

Leading edge position at shroud, °

_{1_h}:

Leading edge wrap angle at hub, °

_{2_s}:

Leading edge wrap angle at shroud, °

Turbulence dissipation rate, m^{2}/s^{3}

Efficiency, %

Density, kg/m^{3}

Dynamic viscosity, Pa.s

_{t}:

Turbulent viscosity, m^{2}/s

Specific dissipation of turbulent kinetic energy, s^{−1}

Flow coefficient.

Artificial neural network

Design of experiment

Grid convergence index

Latin hypercube sampling

Multiobjective generic algorithm

Net positive suction head, m

Nondominated sorting genetic algorithm

Particle swarm optimization

Reynolds-averaged Navier–Stokes

Radial bias neural network

Shear stress transport.

The data belongs to National Research Center of Pumps, Jiangsu University, China, and therefore cannot be made freely available. Requests for access to these data should be made to the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this article.

WW and MKO conceived and designed the study and analyzed the most of results. JP and SY refined the ideas and carried out additional analyses. MKO wrote the manuscript, and JC carried out the performance measurement in the lab. FKO proofread and edited of the final draft. The final manuscript was read and approved by all authors.

This work was supported by the Natural Science Foundation of Jiangsu Province (Grant no. BK20190851), Primary Research & Development Plan of Shandong Province (Grant no. 2019TSLH0304), Natural Science Foundation of China (Grant no. 51879121), Primary Research & Development Plan of Jiangsu Province (Grant no. BE2019009-1), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJB470005), and China Postdoctoral Science Foundation (No. 2018M640462).