Introducing Non-Hierarchical RSM and MIGA for Performance Prediction and Optimization of a Centrifugal Pump under the Nominal Condition

: In order to improve the operation performance of the multi-stage double-suction centrifugal pump and reduce the internal energy loss of the pump, this paper proposes a single-objective optimization design method based on the non-hierarchical response surface methodology (RSM) and the multi-island genetic algorithm (MIGA). Nine parameters, such as the blade outlet width and blade wrap angle, were used as design variables, and the optimization objective was efﬁciency under design conditions. In total, 149 sets of valid data were obtained under the Latin hypercube sampling method (LHS), the corresponding thresholds were set for efﬁciency and head, and 99 sets of valid data were obtained. A cross-validation analysis of the sieved data was carried out based on non-hierarchical RSM, global optimization of the efﬁciency was carried out using MIGA, and numerical veriﬁcation was carried out via CFD. The research results show that compared with hierarchical RSM, non-hierarchical RSM can approximate the nonlinear relationship between the objective function and the design variables with higher accuracy, and the model ﬁtting R 2 value was 0.919. The efﬁciency was improved by 3.717% after optimization. The overall prewhirl of the impeller inlet after optimization decreased, the internal speed of the volute signiﬁcantly improved, the large-area vortex at the volute and the outlet pipe was eliminated, the impact loss at the volute separating tongue disappeared, and the overall hydraulic performance of the pump was improved. The total entropy output value of the optimized pump was reduced by 4.79 (W/K), mainly concentrated on the reduction in the entropy output value of the double volute, and the overall energy dissipation of the pump was reduced.


Introduction
As general mechanical equipment in the field of fluid machinery, pumps are widely used in production and in life for the purpose of conveying fluid media. For the multi-stage double-suction centrifugal pumps used in the fields of sewage treatment, water diversion irrigation, and industrial water supply, during the large-flow and high-head operation and due to the complexity of the structure, it is easy to cause an internal flow disorder, which results in the low overall efficiency of the pump [1,2].
However, current pump manufacturers and users have increasingly higher requirements for pump performance, and obtaining a high-efficiency pump type has become essential. In the field of hydraulic machinery, the use of numerical simulation methods to optimize the mechanical properties of pumps has been widely used [3]. Traditional pump design is accomplished via a combination of numerical calculations and experiments; the design process is very complicated, and the calculation process takes a long time. At present, with intelligent optimization algorithms being applied more and more widely, optimization design that combines numerical calculations and an intelligent optimization algorithm is also very common. The operational speed and accuracy of this combination a non-hierarchical polynomial [17]. Bao et al. [18] proposed an efficient stochastic update method based on statistical theory and developed an incomplete fourth-order polynomial RSM. Combining RSM with Monte Carlo Simulation (MCS) reduces computation and enables fast random sampling. Tanaka et al. [19] applied an interactive hierarchical RSM to the parameter optimization of photonic crystal nanocavities, and they demonstrated the effectiveness of this method for parameter optimization.
In summary, although there are many studies on the application of RSM at home and abroad, there are few applications for a complex high-order RSM. In this paper, the efficiency of a multi-stage double-suction centrifugal pump is optimized based on the improved fourth-order non-hierarchical RSM polynomial. The effects of different polynomial terms on the approximate accuracy of RSM are compared. In Section 2, the hydraulic model, mesh generation, and numerical calculations are presented. Then, in Section 3, the optimization objectives, optimization variables, variable ranges, agent model, and the algorithm in the optimization process are described. In Section 4, the sensitivity analysis of each geometric parameter is carried out, and the inner flow state and entropy generation performance of the pump before and after optimization are compared and analyzed. Finally, the conclusion is given in Section 5.

Hydraulic Model
The first-stage single-suction impeller and the secondary double-suction impeller of the multi-stage double-suction centrifugal pump use the same impeller hydraulic model. In order to better eliminate the radial force of the impeller when the pump is running, the volute of the flow passage adopts a double volute design. At the design operating point, the design performance parameters of the pump are: flowrate Q = 540 m 3 /h, design head H = 132 m, speed n = 1490 r/min, and specific speed n s = 64. The formula for calculating the specific speed is as follows: The overall 3D pump fluid domain was modeled in the model design software UG NX, as shown in Figure 1. After the water enters the suction chambers on both sides, it flows into the middle symmetrical flow channel perpendicular to the axis through the single-suction impellers on both sides. It then flows into the double-suction impeller from the flow channel and finally discharges through the middle-pressure water chamber. The specific details of the flow through the impeller are shown in Figure 2. The main design parameters of the multi-stage double-suction centrifugal pump are shown in Table 1.

Mesh Generation and Numerical Calculation
Due to the complexity of the double volute internal structure of the multi-stage double-suction centrifugal pump, ANSYS ICEM was used to generate unstructured meshes. The impeller, suction chamber, interstage runner, and other components were based on the commercial software TurboGrid, which has high precision and good convergence performance in its high-quality structural grid. In order to better satisfy the subsequent highprecision flow field analysis and more accurately characterize the complex flow phenomena around the solid wall, the mesh of the solid surface was refined. Part of the computational domain grid is shown in Figure 3.
The CFD in the commercial software ANSYS CFX was used to study and analyze the hydraulic characteristics of the pump. The turbulence model adopted was the shear stress transfer model (SST k-ω), which is widely used in multi-stage double suction centrifugal pumps and can predict the flow separation and pump performance with good accuracy [20][21][22]. In order to meet the requirements of the above turbulence model, the maximum y + used for the impeller blade was less than 10; Figure 4 shows the contour of y + .

Mesh Generation and Numerical Calculation
Due to the complexity of the double volute internal structure of the multi-stage double-suction centrifugal pump, ANSYS ICEM was used to generate unstructured meshes. The impeller, suction chamber, interstage runner, and other components were based on the commercial software TurboGrid, which has high precision and good convergence performance in its high-quality structural grid. In order to better satisfy the subsequent highprecision flow field analysis and more accurately characterize the complex flow phenomena around the solid wall, the mesh of the solid surface was refined. Part of the computational domain grid is shown in Figure 3.
The CFD in the commercial software ANSYS CFX was used to study and analyze the hydraulic characteristics of the pump. The turbulence model adopted was the shear stress transfer model (SST k-ω), which is widely used in multi-stage double suction centrifugal pumps and can predict the flow separation and pump performance with good accuracy [20][21][22]. In order to meet the requirements of the above turbulence model, the maximum y + used for the impeller blade was less than 10; Figure 4 shows the contour of y + .

Mesh Generation and Numerical Calculation
Due to the complexity of the double volute internal structure of the multi-stage double-suction centrifugal pump, ANSYS ICEM was used to generate unstructured meshes. The impeller, suction chamber, interstage runner, and other components were based on the commercial software TurboGrid, which has high precision and good convergence performance in its high-quality structural grid. In order to better satisfy the subsequent high-precision flow field analysis and more accurately characterize the complex flow phenomena around the solid wall, the mesh of the solid surface was refined. Part of the computational domain grid is shown in Figure 3.
The CFD in the commercial software ANSYS CFX was used to study and analyze the hydraulic characteristics of the pump. The turbulence model adopted was the shear stress transfer model (SST k-ω), which is widely used in multi-stage double suction centrifugal pumps and can predict the flow separation and pump performance with good accuracy [20][21][22]. In order to meet the requirements of the above turbulence model, the maximum y + used for the impeller blade was less than 10; Figure 4 shows the contour of y + .
In the independence analysis of the effect of the number of grid cells on the numerical calculation results, a total of five groups of independent grid numbers were generated; the calculation results of the corresponding lift and efficiency are shown in Table 2. After the grid-independence analysis, the grid size was finally determined. The final number of cells was 130.858 × 10 5 . The total pressure inlet and mass flow outlet were set as the boundary conditions of the pump. The computational domain generated a total of 11 networks, including the suction chamber, suction pipe, first-stage impeller, inter-stage flow channel, second-stage impeller, double volute, and outlet pipe grid. The number of grid cells for each computational domain is shown in Table 3.
In the independence analysis of the effect of the number of grid cells on the numerical calculation results, a total of five groups of independent grid numbers were generated; the calculation results of the corresponding lift and efficiency are shown in Table 2. After the grid-independence analysis, the grid size was finally determined. The final number of cells was 130.858 × 10 5 . The total pressure inlet and mass flow outlet were set as the boundary conditions of the pump. The computational domain generated a total of 11 networks, including the suction chamber, suction pipe, first-stage impeller, inter-stage flow channel, second-stage impeller, double volute, and outlet pipe grid. The number of grid cells for each computational domain is shown in Table 3.   In the independence analysis of the effect of the number of grid cells on the numerical calculation results, a total of five groups of independent grid numbers were generated; the calculation results of the corresponding lift and efficiency are shown in Table 2. After the grid-independence analysis, the grid size was finally determined. The final number of cells was 130.858 × 10 5 . The total pressure inlet and mass flow outlet were set as the boundary conditions of the pump. The computational domain generated a total of 11 networks, including the suction chamber, suction pipe, first-stage impeller, inter-stage flow channel, second-stage impeller, double volute, and outlet pipe grid. The number of grid cells for each computational domain is shown in Table 3.

Experimental Verification
A comparison between the test results and the numerical calculation results is shown in Figure 5. It can be seen from the figure that the trends of the test curve and the numerical calculation curve are almost the same. Since the energy loss generated by the pump itself was not fully considered during the test, the test results for the head and efficiency were generally lower than the numerical calculation results. At the design operating point, the numerical calculation result of the pump was 76.512%, the test result was 73.705%, and the absolute error of the two was 2.807%. Under non-design conditions, the error between the numerical calculation results and the experimental results did not increase greatly, therefore the numerical simulation method in this paper is reliable and can be used for subsequent optimization studies.

Experimental Verification
A comparison between the test results and the numerical calculation results is shown in Figure 5. It can be seen from the figure that the trends of the test curve and the numerica calculation curve are almost the same. Since the energy loss generated by the pump itsel was not fully considered during the test, the test results for the head and efficiency were generally lower than the numerical calculation results. At the design operating point, the numerical calculation result of the pump was 76.512%, the test result was 73.705%, and the absolute error of the two was 2.807%. Under non-design conditions, the error between the numerical calculation results and the experimental results did not increase greatly therefore the numerical simulation method in this paper is reliable and can be used for subsequent optimization studies.  Figure 6 presents the optimization flow chart for this paper. The efficiency under the design condition of the multi-stage double-suction centrifugal pump was selected as the optimization objective, the nine design parameters of the pump were used as the optimi zation variables, and respective boundary conditions were set for the nine variables. The Latin hypercube sampling (LHS) method was used to generate 149 groups of valid sample data, the performance of the original scheme was compared, the data were screened, the functional relationship between the objective function and the design variables was estab lished, and the objective function was fitted based on the improved response surface methodology (RSM) using the multi-island genetic algorithm (MIGA). This algorithm  Figure 6 presents the optimization flow chart for this paper. The efficiency under the design condition of the multi-stage double-suction centrifugal pump was selected as the optimization objective, the nine design parameters of the pump were used as the optimization variables, and respective boundary conditions were set for the nine variables. The Latin hypercube sampling (LHS) method was used to generate 149 groups of valid sample data, the performance of the original scheme was compared, the data were screened, the functional relationship between the objective function and the design variables was established, and the objective function was fitted based on the improved response surface methodology (RSM) using the multi-island genetic algorithm (MIGA). This algorithm finds the optimal efficiency point for CFD verification and finally obtains the optimal geometric parameter design of the volute and the impeller. finds the optimal efficiency point for CFD verification and finally obtains the optimal geometric parameter design of the volute and the impeller.

Optimization Objective
Due to the long-term and continuous operation of the pump and its frequent operation under low load or variable load, the operating point of the pump easily deviates from the high-efficiency area; the operating efficiency of the pump is then greatly reduced, and a large amount of energy is wasted. In order to save energy and reduce the internal energy loss so as to improve the operating efficiency of the two-stage split centrifugal pump, this paper takes the efficiency at the design operating point as the optimization goal. The efficiency equation is as follows: where Q is the flow rate at the design operating point (m 3 /h); p2t and p1t are the total pressure at the inlet and outlet, respectively (Pa); T is the torque of the impeller (N m); and ω is the rotational speed of the impeller (rad/s).

Design Variables and Parameter Ranges
Since this paper only addresses the design and optimization of the blade profile, in order to reduce the complexity and error of the overall calculation, the diameter of the impeller inlet and outlet and the thickness of the blade were kept unchanged. There were nine design variables to be optimized and controlled, and the range of each design variable is shown in Table 4. In the table, x1 represents the outlet width of the blade, which is used to control the variation range of the size of the impeller on the axial projection diagram. x2 and x3 represent the inlet placement angles of the rear and front cover plates of the blade, respectively, while x4 and x5 represent the outlet placement angles of the rear and front cover plates of the blade, respectively. The blade wrap angle was set as the

Optimization Objective
Due to the long-term and continuous operation of the pump and its frequent operation under low load or variable load, the operating point of the pump easily deviates from the high-efficiency area; the operating efficiency of the pump is then greatly reduced, and a large amount of energy is wasted. In order to save energy and reduce the internal energy loss so as to improve the operating efficiency of the two-stage split centrifugal pump, this paper takes the efficiency at the design operating point as the optimization goal. The efficiency equation is as follows: where Q is the flow rate at the design operating point (m 3 /h); p 2t and p 1t are the total pressure at the inlet and outlet, respectively (Pa); T is the torque of the impeller (N m); and ω is the rotational speed of the impeller (rad/s).

Design Variables and Parameter Ranges
Since this paper only addresses the design and optimization of the blade profile, in order to reduce the complexity and error of the overall calculation, the diameter of the impeller inlet and outlet and the thickness of the blade were kept unchanged. There were nine design variables to be optimized and controlled, and the range of each design variable is shown in Table 4. In the table, x 1 represents the outlet width of the blade, which is used to control the variation range of the size of the impeller on the axial projection diagram. x 2 and x 3 represent the inlet placement angles of the rear and front cover plates of the blade, respectively, while x 4 and x 5 represent the outlet placement angles of the rear and front cover plates of the blade, respectively. The blade wrap angle was set as the design variable x 6 ; variable x 7 is the Stepanoff number that controls the change of the cross-sectional area in the volute; x 8 and x 9 represent the volute inlet width and the starting position of the volute baffle, respectively.

Latin Hypercube Sampling Method
As an important step in the optimization process of an experimental design, it is necessary to choose an appropriate sampling technique. Since there are many variables in this optimization design, in order to obtain better space-filling randomness, accuracy, and robustness for the sample parameters, the LHS method was used to generate 149 sets of valid data for the defined nine variables and the range of the design variables. In order to further reduce the error of the sample and obtain more concentrated data sample points, thereby improving the convergence and fitting accuracy of the data, corresponding thresholds were set for the head and efficiency in the sample data. The threshold of the head was set to 135 m and the threshold of efficiency to 78%. Finally, three partial data sets, as shown in Figure 7, were screened out. Namely, zone 1 represents data that exceeds the efficiency threshold; zone 2 represents data within the set efficiency and head thresholds; zone 3 represents data that exceeds the head threshold. The 99 groups of valid data screened in the zone 2 were selected for subsequent model training and prediction. design variable x6; variable x7 is the Stepanoff number that controls the change of the crosssectional area in the volute; x8 and x9 represent the volute inlet width and the starting position of the volute baffle, respectively.

Latin Hypercube Sampling Method
As an important step in the optimization process of an experimental design, it is necessary to choose an appropriate sampling technique. Since there are many variables in this optimization design, in order to obtain better space-filling randomness, accuracy, and robustness for the sample parameters, the LHS method was used to generate 149 sets of valid data for the defined nine variables and the range of the design variables. In order to further reduce the error of the sample and obtain more concentrated data sample points, thereby improving the convergence and fitting accuracy of the data, corresponding thresholds were set for the head and efficiency in the sample data. The threshold of the head was set to 135 m and the threshold of efficiency to 78%. Finally, three partial data sets, as shown in Figure 7, were screened out. Namely, zone 1 represents data that exceeds the efficiency threshold; zone 2 represents data within the set efficiency and head thresholds; zone 3 represents data that exceeds the head threshold. The 99 groups of valid data screened in the zone 2 were selected for subsequent model training and prediction.

Non-Hierarchical Response Surface Methodology
As a common approximation model established between the objective function and the design variables, RSM has multiple selectable polynomial orders, such as first-(linear), second-, third-, and fourth-order polynomial functions. Based on the multi-parameter optimization design in this paper, in order to improve the accuracy of the model prediction

Non-Hierarchical Response Surface Methodology
As a common approximation model established between the objective function and the design variables, RSM has multiple selectable polynomial orders, such as first-(linear), second-, third-, and fourth-order polynomial functions. Based on the multi-parameter optimization design in this paper, in order to improve the accuracy of the model prediction results, the fourth-order RSM was selected for the fitting calculations. The fourth-order RSM polynomial function is expressed as follows: where x = (x 1 , x 2 , . . . , x n ), x i (i = 1, 2, . . . , n) are design variables, a 0 , b i , c ij , d i , e i , g i are the regression coefficients of each polynomial, and the number of hierarchical polynomials is 1 + 9 + (81 − 9)/2 + 9 + 9 + 9 = 73. The non-hierarchical RSM was selected to be able to use non-hierarchical polynomials in analyzing and verifying the accuracy of the model.

Optimization Algorithm
As an improved genetic algorithm based on the traditional genetic algorithm, MIGA is a pseudo-parallel genetic algorithm based on population grouping. The function of diversity and the prevention of premature maturity solve the problem experienced by traditional genetic algorithms, which are prone to falling into local optima [23,24].
Based on the 99 groups of sample data obtained by screening the original data, the above-mentioned fourth-order RSM polynomial function was used to establish the relationship between the optimization objective and the design variables; the MIGA was then used for optimization, and the performance of the impeller was finally verified. The parameter settings of the optimization algorithm are shown in Table 5.

Approximate Model Fit Accuracy
In order to verify the accuracy of the approximate model, this paper compares the model prediction accuracy of the third-order and fourth-order RSM polynomials in the hierarchical and non-hierarchical models. The R 2 value is used to represent the degree of agreement between the approximate model and the sample points. The closer the value is to 1, the higher the prediction accuracy of the approximate model. For the fourth-order RSM polynomial, the number of polynomials when layered is 73, and for the third-order RSM polynomial, the number of polynomials when layered is 64. Figure 8a,b present the corresponding R 2 values of the third-order and fourth-order polynomials under the hierarchy. It can be seen that the R 2 value is higher under the third-order hierarchical fitting, and the fitting effect is better. Figure 8c,d present the R 2 values corresponding to thirdorder and fourth-order non-hierarchical polynomials. In this optimization process, the cross-validation method was used for error analysis, and 50 groups of random data were selected for cross-validation error analysis. At the same time, automatic three-dimensional modeling and numerical simulation were performed on these 50 groups of data, and the corresponding calculation results were finally obtained. After many instances of repeated training, it can be seen that the fitting effect of the fourth-order non-hierarchical model is better than that of the third-order model, and the fitting accuracy of non-hierarchical model is higher than that of the hierarchical model.
In this model verification, when the fourth-order model is non-hierarchical, and when the number of polynomials selected is 40, a higher fitting accuracy can be obtained. Table 6 shows the design variable values before and after optimization. The efficiency of the optimal scheme is 80.939%, which is 4.427% higher than the 76.512% before optimization. The efficiency value verified by CFD is 80.229%, and the relative error is 0.88%. Therefore, the optimization model has good reliability and can be accurately used for pump performance prediction.
Processes 2022, 10, x FOR PEER REVIEW 10 of 16 corresponding calculation results were finally obtained. After many instances of repeated training, it can be seen that the fitting effect of the fourth-order non-hierarchical model is better than that of the third-order model, and the fitting accuracy of non-hierarchical model is higher than that of the hierarchical model. In this model verification, when the fourth-order model is non-hierarchical, and when the number of polynomials selected is 40, a higher fitting accuracy can be obtained. Table 6 shows the design variable values before and after optimization. The efficiency of the optimal scheme is 80.939%, which is 4.427% higher than the 76.512% before optimization. The efficiency value verified by CFD is 80.229%, and the relative error is 0.88%. Therefore, the optimization model has good reliability and can be accurately used for pump performance prediction.

Sensitivity Analysis
In order to verify the influence of the design variables on the performance and efficiency of the pump, a sensitivity analysis was carried out for the nine variables in the optimal design. Table 7 shows the corresponding coefficient values of each polynomial using the fourth-order non-hierarchical fortieth-degree polynomial. It can be seen from

Sensitivity Analysis
In order to verify the influence of the design variables on the performance and efficiency of the pump, a sensitivity analysis was carried out for the nine variables in the optimal design. Table 7 shows the corresponding coefficient values of each polynomial using the fourth-order non-hierarchical fortieth-degree polynomial. It can be seen from the table that the coefficients of x 1 , x 4 , x 6 , x 8 , and x 9 are negative numbers; that is, the blade outlet width b 2 , the blade front cover inlet placement angle β 1s , the blade wrap angle ϕ, the volute outlet width b 3 , and the double-volute starting position θ of the diaphragm have a negative effect on the overall efficiency of the pump. The blade wrap angle ϕ and the starting position θ of the diaphragm of the double volute have a significant impact on the hydraulic power of the pump. The blade outlet width, the blade front cover inlet placement angle, and the volute outlet width have little influence on the overall performance of the pump and can almost be ignored. Because the coefficients of x 3 and x 7 are positive values, the outlet placement angle β 2h of the rear cover plate of the blade and the Stepanoff number K s have a positive impact on the overall efficiency of the pump, with the Stepanoff number K s having a greater influence. The influence of the placement angle β 2h at the outlet of the rear cover plate of the blade is small.   Figure 9 presents a comparison of the impeller inlet peripheral speed before and after optimization. The inlet peripheral speed of the impeller has an important influence on the pump head. If the peripheral speed is too large, it easily forms a prewhirl at the inlet and affects the impeller head. In the steady calculation, due to the uneven distribution of the dual-inlet flow channels and the water suction chamber, the inter-stage flow channels of the second-stage impeller have a great influence on the flow distribution. Under the centrifugal force of the first-stage impeller, the overall increase in the velocity distribution of the second-stage impeller is higher than that of the first-stage impeller. Compared with that before optimization, the peripheral velocity distribution of the first-stage impeller inlet shows almost no great change. After optimization, the average circumferential speed at the inlet of the second-stage impeller is 4.45 (m s −1 ), and the average circumferential speed of the second-stage impeller inlet of the original scheme is 6.14 (m s −1 ). Thus, compared with the original model, the overall prewhirl of the impeller inlet is reduced, therefore significantly improving the hydraulic performance of the impeller and the optimized second-stage impeller.    Figure 12 presents a comparison of the distribution of the velocity streamlines of the double volute before and after optimization. It can be seen from the figure below that the velocity of the volute after optimization is significantly improved, the vortex at the volute and the outlet pipe is eliminated, and the impact loss at the volute tongue is eliminated. The overall velocity inside the volute is reduced, so the overall hydraulic performance of the volute is improved. Since the volute is a static water-passing component, we generally think that it has little effect on the pump's efficiency. However, as an energy-recovery component that converts kinetic energy into pressure energy, the volute has a considerable impact on the efficiency of the pump. Therefore, an improvement of the internal flow performance of the volute can effectively improve the overall operating efficiency of the pump [25].

Characteristic Analysis of the Entropy Field
Due to the phenomena of secondary flow, backflow, pressure pulsation, and flow separation that aggravate energy dissipation during pump operation, energy dissipation can be effectively evaluated by comparing the entropy production results before and after Figure 11. Comparison of the second-stage impeller velocity distribution before (a) and after (b) optimization. Figure 12 presents a comparison of the distribution of the velocity streamlines of the double volute before and after optimization. It can be seen from the figure below that the velocity of the volute after optimization is significantly improved, the vortex at the volute and the outlet pipe is eliminated, and the impact loss at the volute tongue is eliminated. The overall velocity inside the volute is reduced, so the overall hydraulic performance of the volute is improved. Since the volute is a static water-passing component, we generally think that it has little effect on the pump's efficiency. However, as an energy-recovery component that converts kinetic energy into pressure energy, the volute has a considerable impact on the efficiency of the pump. Therefore, an improvement of the internal flow performance of the volute can effectively improve the overall operating efficiency of the pump [25].  Figure 12 presents a comparison of the distribution of the velocity streamlines of the double volute before and after optimization. It can be seen from the figure below that the velocity of the volute after optimization is significantly improved, the vortex at the volute and the outlet pipe is eliminated, and the impact loss at the volute tongue is eliminated. The overall velocity inside the volute is reduced, so the overall hydraulic performance of the volute is improved. Since the volute is a static water-passing component, we generally think that it has little effect on the pump's efficiency. However, as an energy-recovery component that converts kinetic energy into pressure energy, the volute has a considerable impact on the efficiency of the pump. Therefore, an improvement of the internal flow performance of the volute can effectively improve the overall operating efficiency of the pump [25].

Characteristic Analysis of the Entropy Field
Due to the phenomena of secondary flow, backflow, pressure pulsation, and flow separation that aggravate energy dissipation during pump operation, energy dissipation can be effectively evaluated by comparing the entropy production results before and after

Characteristic Analysis of the Entropy Field
Due to the phenomena of secondary flow, backflow, pressure pulsation, and flow separation that aggravate energy dissipation during pump operation, energy dissipation can be effectively evaluated by comparing the entropy production results before and after optimization. For this analysis of entropy production, the inclusion of wall dissipation, turbulent dissipation, and direct dissipation is considered based on the Reynolds timeaveraged turbulent motion. Table 8 shows the entropy production values corresponding to the wall, turbulence, and direct dissipation of different components before and after optimization, where Imp 1 st 1 represents the first-stage impeller on the left side of Figure 1, Imp 1 st 2 is the first-stage impeller on the right side, Vol is the double volute, and Imp 2 nd is the second-stage impeller. It can be seen from the table that the entropy production of various dissipations of the optimized first-stage impeller is increased, and the wall dissipation of the second-stage impeller is reduced to a certain extent. The dissipation of the volute in all three parts is reduced, mainly concentrated in the dissipation of the wall, which is reduced by 9.07 (W/K) as compared with that before optimization. Figure 13 presents a comparison chart of the entropy production results of the firststage impeller, the second-stage impeller, and the double volute before and after optimization. As can be seen from the figure, due to the optimization of the structure of the double volute, the entropy production value after optimization is reduced by 9.64 (W/K), and the energy dissipation of the volute is significantly reduced. The other parts may have a small increase in entropy production due to the deterioration of the optimized flow state, but the overall entropy production of the pump is decreased by 4.79 (W/K). Hence, the overall energy loss of the pump is reduced, the performance is improved, and the optimization effect of the volute is better than that of the impeller. optimization. For this analysis of entropy production, the inclusion of wall dissipation, turbulent dissipation, and direct dissipation is considered based on the Reynolds timeaveraged turbulent motion. Table 8 shows the entropy production values corresponding to the wall, turbulence, and direct dissipation of different components before and after optimization, where Imp 1 st 1 represents the first-stage impeller on the left side of Figure 1, Imp 1 st 2 is the first-stage impeller on the right side, Vol is the double volute, and Imp 2 nd is the second-stage impeller. It can be seen from the table that the entropy production of various dissipations of the optimized first-stage impeller is increased, and the wall dissipation of the second-stage impeller is reduced to a certain extent. The dissipation of the volute in all three parts is reduced, mainly concentrated in the dissipation of the wall, which is reduced by 9.07 (W/K) as compared with that before optimization. Figure 13 presents a comparison chart of the entropy production results of the firststage impeller, the second-stage impeller, and the double volute before and after optimization. As can be seen from the figure, due to the optimization of the structure of the double volute, the entropy production value after optimization is reduced by 9.64 (W/K), and the energy dissipation of the volute is significantly reduced. The other parts may have a small increase in entropy production due to the deterioration of the optimized flow state, but the overall entropy production of the pump is decreased by 4.79 (W/K). Hence, the overall energy loss of the pump is reduced, the performance is improved, and the optimization effect of the volute is better than that of the impeller.

Conclusions
In this paper, the efficiency of a multi-stage double-suction pump under its design conditions was selected as the optimization target, and nine design parameters were used as the optimization variables. The LHS method was used to sample and screen the data

Conclusions
In this paper, the efficiency of a multi-stage double-suction pump under its design conditions was selected as the optimization target, and nine design parameters were used as the optimization variables. The LHS method was used to sample and screen the data based on the improved RSM in order to optimize the efficiency. Finally, the MIGA was used for global optimization, and the hydraulic performance of the pump before and after optimization was compared and analyzed.
(1) The non-hierarchical RSM selected in this paper, namely the fourth-order fortiethdegree non-hierarchical polynomial, can effectively approximate the nonlinear relationship between the optimization target efficiency and the design variables. The fitted R 2 value was 0.919, which was significantly improved compared with the fourth-order hierarchical polynomial and met the accuracy requirements. The efficiency under the design case after the final numerical verification was increased by 3.717%. (2) For the fourth-order fortieth-degree hierarchical polynomial selected in this paper, the degree of influence of each variable on the efficiency can be obtained through the coefficients of each polynomial, among which the blade front cover inlet placement angle β 1s , the baffle starting position θ, and the blade wrap angle ϕ were found to have a greater impact on the efficiency, while the other variables were found to have less impact. (3) The internal flow of the optimized double volute was well improved, eliminating the large-area vortex phenomenon in the low-pressure area at the outlet of the volute. The overall velocity inside the volute was reduced, therefore converting kinetic energy into pressure energy to a greater extent, and the energy loss was reduced. (4) By comparing and analyzing the entropy production value of each component before and after optimization, it can be concluded that the total entropy production of the pump is reduced by 4.79 (W/K) as compared with that before optimization, while the optimized double volute entropy production is reduced by 9.64 (W/K). This is mainly due to the reduction in the wall surface entropy generation as well as the dissipation value in the double volute, which effectively reduces the energy loss of the pump and improves the overall operating performance of the pump.