Automatic optimization of centrifugal pump based on adaptive single-objective algorithm and computational fluid dynamics

It is important to reduce carbon emissions caused by the energy consumption of pumps. This study used a centrifugal pump with a specific speed of 89.6 as the research object to improve pump efficiency. The adaptive single-objective method was adopted as the automatic optimization tool with computational fluid dynamics, which includes optimal space-filling experimental design, Kriging response surface, and mixed-integer sequential quadratic programming. Eight geometric parameters from the meridian section and the plan of the impeller and diffuser were chosen as the design variables. The maximum efficiency under the design flow conditions was set as the optimization target. The Spearman correlation coefficient analysis results show that the sensitivity of each variable to efficiency and head. Compared with the original scheme, the optimal scheme showed a 2.75% increase in efficiency and a 1.17 m increase in head under the design flowrate. The internal flow field after optimization was also improved. An external characteristic experiment of the original and optimized pumps was performed to validate the numerical results. This automatic optimization method presents great potential to improve the hydraulic performance of centrifugal pumps at a lower cost.


Introduction
The pump is one of the most popular and important fluid-conveying equipment  CONTACT Weidong Shi wdshi@ujs.edu.cn; Ling Zhou lingzhou@ujs.edu.cn et al., 2015;Tong et al., 2020). As reported by the European Commission, the pump system account for 22% of the world's motor energy consumption (Shankar et al., 2016), and centrifugal pump contributes to about 80% of all pumps due to their high efficiency and wide range of specific speed number (Ji et al., 2021). Therefore, many methods are proposed to improve the efficiency of the pump system (Armintor & Connors, 1987;Kaya et al., 2008;Suh et al., 2015). The remarkable method is to use variable frequency drives, which have numerous potential for energy saving. However, the pump system efficiency is determined by the pump, motor, controller, and so on. Improving pump efficiency can be done simultaneously with other methods to improve pump system efficiency without increasing manufacturing costs (Mandhare et al., 2019). In addition, for manufacturers, pump efficiency is an important criterion, as well as product competitiveness. Therefore, as the significant energy-consuming equipment, it is becoming increasingly important to improve pump efficiency to reduce energy consumption and carbon emissions Zhou et al., 2022). Owing to its lower cost and shorter cycle than experimentation (Shim et al., 2018), computational fluid dynamics (CFD) is widely applied to predict the performance of the pump, explore the internal flow mechanism in pumps, and offer significant guidance in pump optimization (Jafarzadeh et al., 2011). To reduce the number of experiments and shorten the optimization time, an increasing number of scholars have used the design of experiment (DOE), surrogate model, optimization algorithm, and numerical calculation to optimize centrifugal pumps to improve their performance (Bellary et al., 2016;Guleren, 2018;Wang et al., 2016;Xie et al., 2021;Zhao et al., 2022).  proposed a multi-objective optimization that combined numerical calculation, Latin hypercube design (LHS), Kriging or Raidal basis function surrogate model, and Genetic Algorithms (GA) to optimize the meridional plane parameters of a double-suction pump impeller. Heo et al. (2016) used an LHS, three different surrogate models, and sequential quadratic programming (SQP) optimization methods to optimize the impeller of the centrifugal pump and improved its efficiency. Based on a 3D Reynoldsaveraged Navier-Stokes equations (RANS) analysis, Kim and Kim (2012) employed the radial basis neural network (RBNN) model and SQP optimization algorithm to optimize the diffuser of a mixed-flow pump. The efficiencies at both the designed and high flowrates were enhanced with little change in the head. Donno et al. (2019) used an ERCOFTAC centrifugal pump as the research object and optimized the impeller parameterized by the Bezier surface. With the highest efficiency as the optimization objective and the flow head as the constraint condition, OpenFOAM was used for the numerical calculation, and LHS was used for the experimental design. It was found that Kriging had higher accuracy through the comparison of Kriging and artificial neural networks. Based on the Kriging surrogate model, a single-objective genetic algorithm optimization strategy was used for the optimization. The optimized pump efficiency improved by 2.63%.
As can be seen from the above, the impeller (as the most important energy conversion component in the pump) has a significant impact on the performance of the pump, which attracts the most attention among optimization objects (Chen et al., 2021;Kim et al., 2015). Simultaneously, it was found that the diffuser (as an energy-consuming component) should reduce energy loss as much as possible, which also contributes to the performance of the pump (Gülich, 2020). The impeller and diffusers determine the pump performance. Optimization of impellers or diffusers alone does not necessarily lead to the best results. In addition, owing to the error between the surrogate model and the true value, the optimization results using the surrogate model need to be verified through numerical calculations. If the error is large, the optimization results may not be reliable, and the surrogate model needs to be re-selected, or even the DOE design needs to be re-conducted, which is a large workload with a low degree of automation.
Aiming to find the optimal efficiency in a wider range, the impeller and diffuser were optimized simultaneously in this paper. An adaptive single-objective (ASO) optimization algorithm was adopted so as to automatically verify the optimal values of the surrogate model and further refine the optimization to obtain the global optimal solution. The ASO optimization method that combines experimental design, response surface analysis, and an optimization algorithm, and is a combination algorithm of intelligent optimization. The DOE adopts optimal space-filling, the response surface adopts the Kriging method, and the optimization algorithm adopts the MISOP method. It is a gradient-based algorithm, which is based on the response surface. It can directly verify the previous response surface optimization results, automatically reduce the optimization space, add new refined sample points, and conduct the next round of optimization to obtain gradually refined global optimization results that are suitable for the optimization of single-objective and multi-constraint problems with continuous variables. Because the candidate points must be calculated directly, it is a direct optimization algorithm (ANSYS, 2021b). To avoid too many direct calculation points, a surrogate model was also adopted. It is an intelligent optimization method that combines direct calculations with the surrogate model.
In this paper, the meridian and plane geometric parameters of the impeller and diffusers were considered as design variables. The optimization objective was to maximize the efficiency at the designed flow rate, and the corresponding head was the constraint condition. It was based on numerical calculations using the ASO optimization algorithm and automatic optimization. The Spearman correlation coefficient analysis method was used to analyze the correlation and sensitivity between each variable and the optimization goal. The optimal scheme was compared with the original scheme by simulation, and the internal flow and external characteristics were analyzed and finally tested and verified. The optimization method is a guide for the optimal design of centrifugal pumps with diffusers.

Problem description
The centrifugal pump consisted of an inlet chamber with a filter, impeller, diffuser, and outlet chamber. The design parameters were Q = 14 m 3 /h, H = 14 m, and n = 2 850 rpm with a specific speed of n s = 3.65n·Q 0.5 /H 0.75 = 89.6. The main geometric parameters that remained fixed were: D 1 = 48 mm, D 2 = 116.6 mm, D 3 = 122 mm, D 4 = 158 mm, number of impeller blades z La = 6, and number of diffuser vanes z Le = 7. Both the impeller and diffuser vanes had cylindrical blades. As the impeller is the component that generates energy, it has the greatest impact on the performance of the pump , while the diffuser, which is an energy-consuming component that also has a vital impact on the performance of the pump matched to the impeller, works together to determine the performance of the pump. To maximize the performance of the pump, the geometrical parameters of the impeller and diffuser were optimized simultaneously. As shown in Figure 1, the shroud and hub of the impeller in the meridian were composed of a line + arc, whereas the shroud and hub of the diffuser in the meridian were made up of a single line. The optimization variables were the angle between the impeller shroud line and radial direction θ 1 , angle between the diffuser shroud line and radial direction θ 2 , impeller outlet width b 2 , diffuser inlet width b 3 , impeller blade inlet angle β 1 , impeller blade outlet angle β 2 , impeller blade wrapping angle 1 , diffuser blade wrapping angle 2 , and eight design variables. The geometry of the impeller and diffuser in the meridian were controlled by changing b 2 , θ 1 , b 3 , and θ 2, and the blade profile of the impeller and diffuser were controlled by changing 1 , 2 , β 1 , and β 2 . All the variables were continuous, and the design variables and ranges are listed in Table 1. Because the head of the variable-frequency centrifugal pump can be adjusted by an appropriate speed, the efficiency is the most important performance indicator. Therefore, the efficiency at the design flow rate is the optimization target, and the corresponding head is the constraint for automatic optimization:

Numerical calculation
The centrifugal pump inlet chamber was equipped with a filter to achieve filtration in the swimming pool. The previous test showed that the filter had very little effect on the head of the pump with an average deviation of 0.22% and a maximum deviation of 1.9%; consequently, to simplify the calculation, the numerical calculation was implemented without the filter. The calculation domain is shown in Figure 2 and includes five parts: the inlet chamber, impeller, diffuser, outlet chamber, and front and rear chambers. Numerical calculations of the full flow field were carried out by considering the influence of the friction loss of the disc on the impeller and the leakage from the wear ring.

Computational grid
In numerical computation, a high-quality mesh can speed up the computation process and contribute to higher convergence accuracy (Pei et al., 2019). Structured grids have better orthogonality than unstructured grids, and are more conducive to the transfer of data in the flow field (Yang et al., 2021). In this study, all the computational domains were meshed with structural grids, in which the impeller and diffuser were meshed with turbogrid, and the remaining components were meshed with ANSYS-ICEM, as shown in Figure 3. To obtains  the characteristics of the boundary layer and meet the requirements of the turbulence model for the near-wall region grids, the near-wall region grids were refined. The average y + of the entire fluid domain was 5.13, and the maximum y + < 45, satisfying the requirements of the shear stress transport (SST) k-ω turbulence model (Han et al., 2021;Li et al., 2018). The greater the number of grids, the more accurate is the calculation, but excessive grids will consume extensive computational resources and prolong the computation time. Typically, when the number of grids increases to a certain level, the number of grids has little effect on the results. A grid sensitivity verification was performed to achieve a suitable grid to minimize the number of grids while satisfying computational accuracy. Five sets of different grid numbers were used for the calculation; the comparison results are shown in Figure 4. It can be seen that the performance tends to stabilize when the number of meshes increased to 4.61 million. To further predict the dispersion error of the grid, three sets of grids were selected for error analysis of the head by employing the grid convergence index (GCI) based on Richardson extrapolation (RE) method , as shown in Table 2. N 1 = 8802319, N 2 = 4607658,  N 3 = 2324123, the exponent p denotes the order of convergence (Kwaśniewski, 2013) and was calculated to be p = 2.13, with head extrapolation value H ext 21 of 13.07 m, and 0.45% and 0.71% for GCI 21 and GCI 32 , respectively, which are quite small. Next, we checked whether the two GCI values satisfied GCI 32 = r p GCI 21 , verified again that the error caused by the grid met the requirements. Finally, a grid setting of 4.61 million grids was chosen for the following optimization calculations.

Calculating settings
To better capture the strong shear characteristics of the boundary layer flow and improve the accuracy of the simulation results, 3D Reynolds time-averaged Navier-Stokes (NS) equations were solved through the SST turbulence model in the commercial CFD software ANSYS CFX-2021R1 (ANSYS, 2021a). The boundary conditions with mass flow inlet and static pressure outlet were used, the impeller domain was set as the rotating domain, the rest of the domain was set as the stationary domain, the intersection between the stationary and rotating domains was set as frozen rotor, and the rest of the intersection was set as none. The rotating wall surfaces in the stationary domain was set as rotating wall surface, the solid wall surface was set as no-slip boundary, the wall surface function was automatic, the convection term was set as high resolution, the turbulence numeric was set as fist order, and the residual convergence criteria was set as 10 −5 .

Experimental setup
As shown in Figure 5, the test rig consists of a centrifugal pump, a variable frequency motor, an electromagnetic flowmeter, two pressure sensors, two valves, a water tank, and pipes. The pressure sensors were located at the suction and discharge pipes with a diameter of 50 mm to measure the head of the centrifugal pump. The centrifugal pump was driven by a variable frequency motor that was set to a constant speed of 2850 rpm in this experiment. The flow rate was regulated by the outlet valve and measured by the electromagnetic flowmeter. The information on the experimental instruments is shown in Table 3.

Optimization design
Using ANSYS Workbench as the platform, CFturbo was used for parametric 3D modeling of the impeller and  diffuser, UG was used for 3D modeling of other parts, and the optimization variable was set as 'Input Parameter'. Turbogrid was used for generating the structural grid of the impeller and diffuser, ICEM was used for generating the structural grid of the rest of the components, numerical calculation was performed in CFX, and the efficiency and head at design flow rate were calculated in CFX-POST by using expressions, which was set as 'Output Parameter'. Finally, the direct optimization module in DesignXplorer (DX) was called automatic optimization.
Because there was only one optimization objective and one constraint, and ASO could handle the optimization of single-objective multi-constraint problems compared with the adaptive multi-objective (AMO) algorithm; the ASO optimization algorithm could be used to decrease the number of directly calculated samples and increase the efficiency of optimization work. Therefore, the ASO optimization algorithm was used for the optimization. ASO is a mixture optimization method that combines optimal space-filling design, Kriging response surfaces, mixed-integer sequential quadratic programming (MISQP), and domain reduction, which is applicable to direct optimization. It automatically adjusts the reduced design space based on the previous optimization results and re-performs the experimental design, response surface creation, and optimization search, making it an automatic global optimization algorithm with progressive refinement.

Optimal space-filling design
Optimal space-filling is essentially a DOE method that evolved from LHS, which is an iterative computation based on LHS that maximizes the sample point distance to make the sample distribution in the design space more uniform and fill the design space uniformly with a minimum number of sample points to capture more information in the design space. Optimal space-filling is an efficient DOE method for cases with more design variables and complex design spaces.

Kriging
Kriging is based on a polynomial model combined with deviations of the form where y(x) represents the response function, f (x) represents a polynomial model, and Z(x) represents a normally distributed Gaussian random process having zero mean, σ 2 variance, and non-zero covariance. The f (x) term gives a 'global' model of the design space, analogous to the polynomial model in a response surface.
Although f (x) is approximated globally to the design space, Z(x) causes local deviations that allow the Kriging model to interpolate N sample data points. The covariance matrix Z(x) is calculated as follows: where R is the correlation matrix and r(x i ,x j ) is the spatial Gaussian correlation function between any two points x i and x j among the N sample. R is a symmetric positivedefinite matrix of order N with 1s along the diagonal. r(x i ,x j ) can be written as: The θ k are the unknown parameter for fitting the model, M is the number of design variables, x k i and x k j are the k th components of the sample points x i and x j (ANSYS, 2021b;Stein, 1999).

MISQP
MISQP is a mathematical optimization algorithm developed by Oliver, Thomas, Lehmann, and Schittkowski. It is an efficient algorithm based on gradient-seeking and single-objective multi-constraint optimization, which solves the Mixed-Integer Non-Linear programming (MINLP) problem using a modified SQP approach (Exler et al., 2012). Owing to the use of gradient information and the linear search method, MISQP has a higher accuracy and solution efficiency for continuous singleobjective problem optimization.
ASO solves nonlinear programming problems with constraints of the form: minimize: where: The goal is to intelligently and automatically refine and decrease the domain to deliver global extrema.
In the ASO algorithm, the minimum of the objective function is solved, and to maximize the efficiency, the optimization objective function takes the opposite number of efficiencies as (Kim et al., 2019) where p T is the total pressure, Q is the volume flowrate, T is the sum of the torques of the rotating parts, and ω is the rotation speed. The head is the constraint condition that can be written as To quantitatively analyzes the energy loss within each component in the pump, each domain of the pump is considered as the control volume, the effect of gravity is not considered, and the energy loss within this control volume is where T is the torque generated by the rotating wall of the control volume, the first term on the right is the power of the external work impacted on the control volume, A is the outer surface of the control volume, p T is the total pressure on the microelement surface, v n is the velocity component in the direction normal to the outer surface of the control volume, and the second term on the right is the increment in the energy of the control volume. The impeller efficiency is where p 2 is the total pressure at the impeller outlet and p 1 is the total pressure at the impeller inlet. The diffuser efficiency is defined as (Zhou et al., 2012) where p 4 is the total pressure at diffuser outlet, p 3 is the total pressure at diffuser inlet, and p in is the total pressure at pump inlet.

ASO optimization process
In the ASO optimization, the initial sample point was 47, total sample point was 155, and maximum candidate point was 3. As shown in Figure 6, this study adopted the Optimal space-filling experimental design method with an initial sample size of 47. After the numerical calculation, the objective function value was obtained, and the response surface between the design variables and the objective function was created using the Kriging approach. Subsequently, the MISQP optimization algorithm was used to find the candidate points on the response surface, and different candidate points may have been obtained owing to the simultaneous search from different starting points. According to the Kriging error prediction, if the candidate point data are questioned, the numerical calculation will be verified, and the response surface establishment of the Kriging method will be performed again as an additional point; then, MISQP will be restarted for the optimization search until all verification points are free of problems and subsequently, the optimization space will be reduced according to the candidate points. If the candidate points are at the same location, the boundaries of the design variables are reduced with the candidate points as the center; otherwise, the boundary containing the candidate points is used as the reduction boundary. After each reduction in the optimization space, new Optimal space-filling sample points are generated in the new space, and numerical calculations are performed to obtain the output values and generate a new Kriging response surface. The optimization process is completed when the stopping condition is satisfied. Because the design space is a multidimensional space consisting of eight variables, the sample distributions on b 2 and b 3 were selected for analysis for ease of presentation. The distribution of sample points on b 2 and b 3 generated by the Optimal space-filling method is shown in Figure 7, and all the sample point results were obtained by direct CFD calculation. Figure 7(a) shows the initial samples and efficiencies generated by Optimal spacefilling, and the dashed boxes are the boundaries of the variables. The samples cover the entire sample space uniformly. With an increase in b 3 , the overall efficiency tends to decrease, and the efficiency point is placed close to the position of the lower limit of b 3 , that is, in the range of 9-11 mm. While the effect of b 2 variation on efficiency is not as obvious as for b 3 , the high-efficiency point is in the middle of b 2 , that is, in the 7.5-8 mm range. In the first round of optimization, two validation points were generated using the Kriging response surface and MISQP algorithm. Next, the boundaries were reduced near the candidate and efficient points to generate the second round of Optimal space-filling sample points. Figure 7(b) (left) shows the distribution of all the sample points after the second round of optimization and Figure 7(b) (right) shows the local zoomed-in view of the second round of Optimal space-filling; the dashed box is the boundary of the second Optimal space-filling. It can be observed that the samples are evenly distributed throughout the new reduced boundary. The overall efficiency improved, with the average efficiency increasing from 68.15% in the first round to 69.95% in the second round, and three verification points were generated. Thereafter, the distribution of all the sample points after the third round of optimization is shown. Figure 7(c) (left) shows the distribution of all the sample points after the third round of optimization and Figure 7(c) (right) shows the local enlargement of the third round of Optimal space-filling; the dashed box is the boundary of the third Optimal space-filling. The samples are evenly distributed throughout the new reduced boundary, and the overall efficiency of this round is improved again, with an average efficiency of 70.4%. After four rounds of Optimal space-filling + Kiriging + MISQP optimization and direct CFD calculation of 155 sample points, the efficiency of the best candidate parameters reached 70.69%. Table 4 shows the values of the variables and CFD calculation results for the sample points. A parallel coordinate plot is a major method for visualizing highdimensional spaces (Wegman, 1990), and Figure 8 shows a parallel coordinate plot of sample points, showing the distribution of the input and output variables for all the sample points. Qualitative analysis showed that the higher the efficiency, the denser is the distribution of the sample points. This is because the optimization is efficiency-oriented, and to seek the highest efficiency, denser sample points are set in the efficient region by Optimal space-filling through domain reduction. The denser distribution of efficient points on θ 1 , θ 2 , and b 3 indicates that these parameters are more sensitive to the effect of efficiency, whereas the distribution of efficient  points on β 1 and β 2 are more dispersed, indicating that they are not sensitive to the effect of efficiency. In addition, the high-efficiency points are mainly concentrated on the top of θ 1 and θ 1 and the bottom of b 3 , which means that the efficiency is positively correlated with θ 1 and θ 2 and negatively correlated with b 3 .

Correlation and sensitivity analysis
Correlation and sensitivity analyses was performed to quantitatively analyze the relationship between the parameters and their influence on the objective function. The correlation coefficient is a statistical index that indicates the degree of correlation between variables. Its value range is [−1,1], which means no correlation when the value is 0, negative correlation when the value is [−1,0), and positive correlation when the value is (0,1]. The two most commonly used methods for correlation coefficients are the Pearson and Spearman methods. Because Spearman performs a Pearson correlation coefficient calculation on the rank of the original data, it is a non-parametric test method that is independent of the distribution, resulting in a wider range of applications and more accurate results. The Spearman rank correlation coefficient method was used to perform a correlation analysis on the column vectors of variables and target values to determine the relationship between the variables and the effect of each variable on the target value. For two sets of random variables X and Y, the number of elements of which are both N, x i and y i are the variables of X and Y (1 ≤ i ≤ N), ranking X and Y in order of magnitude; x i andy i are the positions of x i and y i in the original sorted list; x i and y i are called as the ranks of x i and y i , respectively; and d i = x i − y i is the difference of the rank of x i and y i · ρ s is the Spearman correlation coefficient between random variables X and Y (Meng et al., 2019;Stephanou & Varughese, 2021).
If there is no identical rank order, then ρ s can be calculated as If the same rank order exists, the Spearman correlation coefficient between the ranks should be calculated as The correlation coefficients between the variables and the objective function are shown in Table 5. The correlation coefficients between the variables θ 1 and b 3 , θ 1 and θ 2 , θ 1 and 2 , 1 and 2 , b 3 and θ 2 , θ 2 and 2 , and b 3 and 2 are between 0.2 and 0.4, which have weak correlation, and their correlation coefficients are −0.389, 0.379, 0.296, 0.263, −0.236, 0.232, and −0.206, respectively. The correlation coefficients of the other variables are less than 0.2, showing a very weak correlation. This is due to the fact that each round of Optimal spacefilling is uniformly distributed with no correlation among the variables, while the optimization takes efficiency as the optimization objective, generating a denser sample of points in the efficient zone, resulting in the entire sample space not being uniformly distributed and thus, the correlation among the variables is affected by the sensitivity of the variables to efficiency. Figure 9 shows a sensitivity graph indicating the global sensitivity of each variable to the objective function η and H. Positive values indicate that the objective function increases as the variable increases, and negative values indicate that the objective function decreases as the variable increases. Figure 9(a) shows the histogram of sensitivity analysis, which shows that the sensitivity coefficients of each variable to efficiency are in descending order of: where the sensitivities of θ 2 , θ 1 , 2 , 1 , b 2 are positive and those of 1 , b 3 , β 1 are negative; the sensitivity coefficients of each variable to the head are in descending order of: b 2 > θ 1 > β 2 > θ 2 > 1 > b 3 > β 1 > 2 , where the sensitivities of b 2 , θ 1 , β 2 , θ 2 , and 2 are positive, and the sensitivities of 1 , b 3 , and β 1 are negative. Figure 9(b) shows the pie chart of the sensitivity analysis, which demonstrates the percentage of the effect of each variable on η and H. θ 2 and θ 1 have the largest effect on efficiency, reaching 26.7% and 22.8%, respectively, while β 1 , b 2 , and β 2 have the smallest and comparable effects on efficiency, all less than 5%. The effect of b 2 on the head  Figure 9. Sensitivity analysis. is the largest at 29.8%, followed by θ 1 and β 2 at 13.3% and 13.1%, respectively, and the effect of 2 on the head is the smallest at 5.3%. The most sensitive factors to efficiency θ 1 and θ 2 with sensitivities of 0.528 and 0.618, respectively, and the least sensitive factors to efficiency β 1 and β 2 with sensitivities of −0.102 and −0.098, respectively, were selected for analysis. Figure 10(a) shows the distribution of samples on θ 1 and θ 2 , which shows that the samples with high efficiency points are densely distributed in the smaller area in the upper right corner of θ 1 and θ 2 , indicating that θ 1 and θ 2 are sensitive to the efficiency. As θ 1 and θ 2 increase, the efficiency tends to increase; therefore, a strong positive correlation exists between θ 1 and η, and θ 2 and η. As θ 1 increases, θ 2 has a slightly increasing trend overall; therefore, there is a positive weak correlation between θ 1 and θ 2 with sensitivities of 0.379. Figure 10(b) shows the sample distribution on β 1 and β 2 , and it can be seen that the sample efficient points are more scattered in the middle larger areas of β 1 and β 2 , indicating the insensitivity of β 1 and β 2 to the efficiency. β 1 and β 2 had a very weak correlation with η. As β 1 increases, no obvious trend in change of β 2 can be observed, so there is a very weak correlation between β 1 and β 2 with a sensitivity of −0.063, which is consistent with the conclusion of Spearman's correlation coefficient.

External characteristics comparison
After optimization, the 121th scheme was selected as the optimal candidates, and Table 6 shows a comparison of the variables and target values of the optimized scheme and the original scheme. The comparison of the blade shapes between the original scheme and the  optimized scheme is shown in Figure 11. The simulation results of the optimized and original schemes were analyzed and compared. The optimized efficiency at design flowrate increases from 67.94% to 70.69%, with an efficiency increase of 2.75%, and the head increases from 13.16 to 14.33 m, with a head increase of 1.17 m. Figure 12 shows the simulated hydraulic performance comparison between the optimized scheme and the original scheme, from 10 to 24 m 3 /h, the optimized head experiences an increase of 1.17-2.33 m compared with the original scheme, and the shaft power also increases, but the proportion of head increase is higher than the proportion of shaft power increase; therefore, the overall efficiency is improved, and the increase in efficiency is more obvious as the flowrate increases, and the efficiency of 10 m 3 /h increases by 1.6%, while the efficiency of 24 m 3 /h is increased by 11.3%. Figure 13 shows the velocity triangle. The relative flow angle β is the angle between the relative velocity w and the opposite direction of the circumferential velocity u, and the absolute flow angle α is the angle between the absolute velocity v and circumferential velocity u. When the meridional velocity v m > 0, that is, flowing from the center to the outside, β and α are taken as positive values, and when the meridional velocity v m < 0, that is, flowing from the outside to the center, β and α are negative values. Figure 14 shows a comparison of the 3D streamlines of the impeller at 1.0Q before and after optimization, and its color is expressed by the relative liquid flow angle β. It can be seen that the overall streamline of the original impeller is smoother, but there is local backflow at the blade inlet, which is due to the blockage at the blade inlet side and the mismatch between the relative liquid flow angle of the incoming flow and blade inlet angle, increasing the impact loss. The flow at the inlet side of the optimized scheme impeller is improved, and the range of β becomes smaller and the flow is more uniform after optimization, indicating that the blade inlet design is more in line with the flow law. Figure 15 shows a comparison of the 3D streamlines of the local flow channel of the diffuser at 1.0Q before and after optimization, and its color is expressed by the absolute liquid flow angle α. It can be seen that the original scheme has a strong backflow area near the hub of the diffusion section in the diffuser caused by the short diffusion section and large diffusion angle, which lead to excessive flow deceleration and flow separation from the wall (Gülich, 2020). In the optimized scheme, the flow separation is reduced, and the backflow is eliminated by increasing the number of diffuser blades and wrap angle, which reduces the diffusion angle and flow deceleration. The range of α after optimization was very small, indicating that the flow was more stable and that the optimized flow field was significantly improved.

Internal flow analysis
The turbulent kinetic energy is used to describe the intensity of turbulent dissipation (Wang et al., 2014). Figure 16 shows a comparison of the turbulent kinetic energy volume rendering of the impeller and diffuser at 1.0Q before and after optimization, which shows that the original scheme impeller blade inlet has a large turbulent kinetic energy, which is due to the inconsistency between the original scheme impeller inlet angle and incoming flow that increases the turbulent kinetic energy loss. The turbulent kinetic energy of the impeller inlet of the optimized scheme is significantly reduced, which is consistent with the conclusions of the streamline analysis. The turbulent energy in the middle and rear of the impeller channels of the optimized scheme is reduced compared to that of the original scheme, which indicates that the flow in the impeller channels of the optimized scheme is improved. However, there is still a large turbulent energy at the exit of some impeller channels, which is the same as the original scheme and is resulted from the rotor-stator interaction between the impeller and diffuser. The large turbulent kinetic energy in the diffusion section of the diffuser of the original scheme is due to the diffusion section being too short, and the control of   the fluid is insufficient, resulting in flow separation and vortex, causing large turbulent kinetic energy loss. The overlap of the diffuser blade of the optimized scheme is increased, which strengthens the control of the fluid and suppresses the flow separation, and the turbulent kinetic energy in the diffuser is significantly reduced after optimization. Figure 17 shows a comparison of the turbulence energy rendering of the diffuser at 1.0Q before and after optimization, showing the location of the maximum turbulent energy of the diffuser. It can be seen that the diffusion section near the hub of the original scheme has a large turbulence kinetic energy, and the corresponding location after optimization of the turbulence kinetic energy is significantly reduced, which is consistent with the conclusions of the flow analysis in Figure 15. Figure 18 shows the comparison of the energy loss of each domain at 1.0Q before and after optimization. From Figure 18(a), we can see that the energy consumption of the diffuser decreases the most after optimization, from the original 44.29 W to 34.97 W, which means that the optimization effect of the diffuser is the most significant, followed by that of the impeller, from 45.05 W to 40.96 W. This is because the impeller and diffuser were considered as the optimization objects in this study, which allowed them to achieve better optimization results. The inlet and outlet chambers were not modified, but they were affected by the front and back flows. Because the optimized flow field is better in line with the flow law, the energy consumption of the suction and outlet chambers is slightly reduced. The energy consumption of the shroud and hub chambers slightly increased after optimization, which may be because the optimized scheme has a higher head and circulation than the original one, and the higher circumferential velocity will increase the wall friction loss as well as the flow loss inside the chamber. Figure 18(b) shows the proportion of the energy consumption of each component to the  total energy consumption before and after the optimization, in descending order of energy consumption: shroud and hub chambers > impeller > diffuser > outlet chamber > inlet chamber. The energy consumption of the shroud and hub chambers includes the friction loss of the disc, leakage loss of the wear ring, friction loss of the wall, and flow loss inside the chamber, which accounts for the largest energy consumption of more than 40%. The second type is the impeller runner. As an energy conversion component, the impeller has a high relative and absolute velocity, resulting in large wall friction loss and turbulent dissipation. Next is the diffuser, which decelerates the flow and consequently converts kinetic energy into static pressure through the diffusion section. This easily leads to larger impact losses in the inlet area owing to the mismatch of the setting angle, and it is easy to form larger along-range losses or turbulent losses generated by flow separation in the diffusion section owing to the unreasonable diffusion angle. The lowest energy loss occurred in the inlet chamber, owing to its lowest velocity.

Energy characteristics analysis
The energy losses of the main components under different working conditions before and after optimization are analyzed below. Figure 19 shows the energy losses of the impeller and diffuser between the original and optimized schemes. It can be seen that, as the flowrate increased from 10 to 24 m 3 /h, the impeller energy loss before and after optimization first decreases and then increases, and the minimum value exists at 12 m 3 /h, similar to the concave parabola. This is because the flow angle at a low flowrate is less than the blade inlet angle, the impeller-inlet impact loss is larger, and the vortex and reflux in the flow channel are more significant, intensifying the instability of the flow and increasing the energy loss. An excessive flowrate will also lead to large impact losses, and an increase in the flowrate will lead to a dramatic increase in losses along the path. Except for the individual working conditions (Q = 24 m 3 /h), the optimized impeller energy losses were lower than those of the original scheme. The optimized diffuser energy loss has a minimum value of 14 m 3 /h, while the minimum value of the original scheme is at 10 m 3 /h or lower. Except  for the individual working conditions (Q = 10 m 3 /h), the energy loss of the optimized diffuser was lower than that of the original scheme.
The absolute magnitude of the energy loss before and after optimization is analyzed above, and the relative energy index, that is, the efficiency comparison before and after optimization, is analyzed below. Figure 20 shows the comparison of impeller and diffuser efficiency before and after optimization under different working conditions. The optimized impeller efficiency is 1.2-2.8% higher than the original scheme for each working condition. The optimized diffuser efficiency is comparable to the original scheme at 10 m 3 /h. As the flowrate increases, the efficiency improvement of the optimized scheme is more obvious, indicating that the optimized diffuser is more suitable for high flowrate.
The above simulation data analysis shows that after the optimization design of the impeller and diffuser, the performance of the impeller, diffuser and the whole pump have been greatly improved.

Comparison of experimental and simulation results
All parts of the original and optimized solutions were processed and manufactured. Figure 21 shows the impeller and diffuser parts before and after the optimization. To ensure accuracy, the main overflow parts were 3D printed, tested, and verified on a closed test rig. Figure 22 shows the comparison of simulation and experiment performance before and after optimization. Figure 22(a) shows the experiment and simulation Q-H curves, and that the head curves of experiment and simulation before and after optimization are in the same trend, i.e. the simulated head of the optimized scheme is improved and its experiment head is also improved. The simulated head is lower than the experiment value at low flowrate and higher than the experiment value at high flowrate, and the simulated and experiment heads are closest near the design point. The simulation and experiment deviation of the design point of the original and optimized schemes are 2.4 and 2.8% respectively. This is because, when deviating from the design point, the internal flow becomes more turbulent and the instability of the flow increases, leading to the increase in simulation calculation error. After optimization, the experimental head of design point is increased by 1.1 m, from 12.8 to 13.9 m. Figure 22(b) shows the experimental and simulated Q-η curves. The efficiency curves of the experiment and simulation before and after the optimization are in the same trend. The experimental efficiency of the scheme with improved simulation efficiency is also improved. After optimization, the experimental efficiency at the design point is increased by 3.4%, from 66.8 to 70.2%. The experimental and simulation efficiencies are relatively close at the design point, with large deviations at high flowrates. The position of the highest efficiency point of the experiment and simulation is shifted. This may be due to the fact that when deviating from the design point, the simulation error of head and power increases, which reduces the efficiency of low flowrate and increases the efficiency of high flowrate, and causes the maximum efficiency point of simulation to shift to a high flowrate compared with the experiment.

Conclusion
In this study, a combination of the ASO optimization algorithm and numerical calculation was applied to automatically optimize the impeller and diffuser of a centrifugal pump with eight geometrical parameters, namely θ 1 , θ 2 , b 2 , b 3 , β 1 , β 2 , 1 , and 2 as optimization variables, the highest efficiency at the design point as the optimization objective, and the head as the constraint. The correlation between the variables and the sensitivity to the objective function were analyzed, and the results of the original and optimized schemes were analyzed and compared with the external characteristics, internal flow, and energy characteristics. Experimental verification was carried out, and the following conclusions were drawn: (1) After four rounds of adaptive optimization and 155 sets of direct CFD calculations, the optimal candidate solution was obtained. Compared with the original scheme, the simulation efficiency of the design point was increased by 2.75%, and the head was increased by 1.17 m. (2) Using Spearman's correlation coefficient analysis, the sensitivity of each variable to efficiency was obtained in the following descending order: θ 2 > θ 1 > 2 > b 3 > 1 > β 1 > b 2 > β 2 , where θ 2 , θ 1 , 2 , 1 , and b 2 were positively correlated with efficiency, and b 3 , β 1 , and β 2 were negatively correlated with efficiency. The sensitivity of each variable to the head was obtained in the descending order of: b 2 > θ 1 > β 2 > θ 2 > 1 > b 3 > β 1 > 2 , where b 2 , θ 1 , β 2 , θ 2 , and 2 were positively correlated with the head, and 1 , b 3 , and β 1 were negatively correlated with the head. (3) By comparing the simulation data of the original and optimized schemes, the optimized head, power, and efficiency were improved from 10 to 24 m 3 /h. The head was increased by a range of 1.17 to 2.33 m, and the efficiency was increased by a range of 1.6 to 11.3%.
(4) Through internal flow and energy characteristics analysis, it was found that there is a large turbulence kinetic energy and backflow at the impeller inlet in the original scheme. After optimization, the turbulence kinetic energy at the impeller inlet and in the flow passage was reduced, and the flow was more stable and uniform with lower energy loss. The strongest vortices and backflows of the original diffuser lie in the hub of the diffusion section with high turbulence kinetic energy. After optimization, the flow field of the diffuser is clearly improved, the turbulent kinetic energy is reduced significantly, and the energy loss reduction is the largest of all the domains, which means that the optimization effect of the diffuser is the most significant. (5) As leakage at the wear ring and disc friction loss are considered in the shroud and hub chambers, the energy loss of the shroud and hub chambers accounts for the largest proportion (more than 40%), followed by the impeller, diffuser, and outlet chambers, and the energy loss is the smallest in the inlet chamber. (6) The comparison between the simulation and experiment performance shows that they have the same trend, and they are closest near the design point; the deviation increases further away from the design point. After experimental verification, the optimized design point efficiency increased by 3.4% and the head increased by 1.1 m, proving that the optimization method is feasible.
In this paper, only the efficiency under the design operating condition at constant speed is considered in the optimization process. As a variable frequency pump, the efficiency of different operating conditions at different speeds should be considered in the future. In addition, cavitation will also have an impact on the efficiency and reliability of the pump, so multi-speed and multioperating condition efficiency and cavitation should be taken as the optimization objectives in the future.

Disclosure statement
No potential conflict of interest was reported by the author(s).