Matching Optimization of a Mixed Flow Pump Impeller and Diffuser Based on the Inverse Design Method

When considering the interaction between the impeller and diffuser, it is necessary to provide logical and systematic guidance for their matching optimization. In this study, the goal was to develop a comprehensive matching optimization strategy to optimize the impeller and diffuser of a mixed flow pump. Some useful tools and methods, such as the inverse design method, computational fluid dynamics (CFD), design of experiment, surrogate model, and optimization algorithm, were used. The matching optimization process was divided into two steps. In the first step, only the impeller was optimized. Thereafter, CFD analysis was performed on the optimized impeller to get the circulation and flow field distribution at the outlet of the impeller. In the second step of optimization, the flow field and circulation distribution at the inlet of the diffuser were set to be the same as the optimized impeller outlet. The results show that the matching optimization strategy proposed in this study is effective and can overcome the shortcomings of single-component optimization, thereby further improving the overall optimization effect. Compared with the baseline model, the pump efficiency of the optimized model at 1.2Qdes, 1.0Qdes, and 0.8Qdes is increased by 6.47%, 3.68%, and 0.82%, respectively.


Introduction
As one of the most important machines in modern civilization, the performance of rotating machinery will have a great influence on the development of wider society. Published data in the Annual Work Report of the Chinese Government in 2019 revealed that more than 98% of the country's electric power conversion is done by rotating machinery. Consequently, a marginal increase in the efficiency of rotating machinery will produce unimaginable economic benefits. For instance, in China, for every 1% increase in rotating machinery efficiency, about 70 billion kilowatts of electricity can be saved annually. As a kind of rotating machinery, mixed flow pumps play an important role in industrial production, agricultural irrigation, and urban drainage due to their moderate head, wide high efficiency range, and good anti-cavitation performance. Thus, it is of great significance to study the optimization design of mixed flow pumps.
In the field of rotating machinery optimization, computational fluid dynamics (CFD) prediction is a better method than experimental investigation, because the former is more convenient and cheaper [1]. More importantly, CFD prediction can provide flow details inside the rotating machinery, which can help designers better understand the reasons for performance changes and make targeted optimization [2]. However, CFD prediction cannot directly provide the optimal solution for the optimization of rotating machinery. Like most complex optimization problems, the optimization of rotating machinery usually entails multiple indicators, and each target affects the other. Therefore, considering the multi-objectivity of the rotating machinery optimization is inevitable [3]. Several attempts have been made to solve the above problem. Eventually, the method of combined optimization strategy consisting of CFD, design of experiment (DOE), surrogate models, and

Mixed Flow Pump Model
The mixed flow pump as shown in Figure 1 was selected as the baseline model, which consists of an outlet elbow, a seven-blade diffuser, a four-blade impeller and a straight inlet pipe. The design flow rate Q des is 0.4207 m 3 /s, the design head H des is 12.66 m, the rotational speed N of the impeller is 1450 r/min, and the specific speed n s can be calculated by Equation (1): The performance of the baseline model was tested by Tianjin experimental bench, China. On the test bench, an intelligent differential pressure transmitter and intelligent torque speed sensor are used to measure head and torque, respectively. These devices yielded measurement errors of <±0.1%. Also, an intelligent electromagnetic flowmeter is used to measure flow rate, and the measurement errors is <±0.2%. The random uncertainty and overall uncertainty of this test bench are less than 0.1% and 0.3%, respectively. The test results shown in Figure 2 show that the efficiency of the baseline model at the design point is 86.3%. In this figure, Q * = Q/Q des is the normalized flow rate, and H * = H/H des is the normalized head. (The same dimensionless method was used in other parts of this paper.) introduced. Finally, the optimization mechanism was clarified by a comparative analysis of the internal flow field of the two models.

Mixed Flow Pump Model
The mixed flow pump as shown in Figure 1 was selected as the baseline model, which consists of an outlet elbow, a seven-blade diffuser, a four-blade impeller and a straight inlet pipe. The design flow rate Qdes is 0.4207 m 3 /s, the design head is 12.66 m, the rotational speed of the impeller is 1450 r/min, and the specific speed can be calculated by Equation (1) The performance of the baseline model was tested by Tianjin experimental bench, China. On the test bench, an intelligent differential pressure transmitter and intelligent torque speed sensor are used to measure head and torque, respectively. These devices yielded measurement errors of <±0.1%. Also, an intelligent electromagnetic flowmeter is used to measure flow rate, and the measurement errors is <±0.2%. The random uncertainty and overall uncertainty of this test bench are less than 0.1% and 0.3%, respectively. The test results shown in Figure 2 show that the efficiency of the baseline model at the design point is 86.3%. In this figure, * = ⁄ is the normalized flow rate, and * = ⁄ is the normalized head. (The same dimensionless method was used in other parts of this paper.)   introduced. Finally, the optimization mechanism was clarified by a comparative analysis of the internal flow field of the two models.

Mixed Flow Pump Model
The mixed flow pump as shown in Figure 1 was selected as the baseline model, which consists of an outlet elbow, a seven-blade diffuser, a four-blade impeller and a straight inlet pipe. The design flow rate Qdes is 0.4207 m 3 /s, the design head is 12.66 m, the rotational speed of the impeller is 1450 r/min, and the specific speed can be calculated by Equation (1): The performance of the baseline model was tested by Tianjin experimental bench, China. On the test bench, an intelligent differential pressure transmitter and intelligent torque speed sensor are used to measure head and torque, respectively. These devices yielded measurement errors of <±0.1%. Also, an intelligent electromagnetic flowmeter is used to measure flow rate, and the measurement errors is <±0.2%. The random uncertainty and overall uncertainty of this test bench are less than 0.1% and 0.3%, respectively. The test results shown in Figure 2 show that the efficiency of the baseline model at the design point is 86.3%. In this figure, * = ⁄ is the normalized flow rate, and * = ⁄ is the normalized head. (The same dimensionless method was used in other parts of this paper.)

Optimization Strategy
As shown in Figure 3, the optimization system was built by combining the IDM, CFD, optimal Latin hypercube sampling (OLHS), RSM, Multi-island genetic algorithm (MIGA) and NSGA-II. The entire optimization process was divided into two steps. In the first step of optimization, only the impeller was optimized, and the objective function was set as the impeller weighted efficiency at 1.2Q des , 1.0Q des and 0.8Q des . Thereafter, CFD analysis was performed on the optimized impeller to get the circulation and flow field distribution at the outlet of the optimized impeller. In the second step of optimization, the diffuser was optimized, and the objective functions were the levels of pump efficiency at 1.2Q des , 1.0Q des and 0.8Q des . To improve the optimization effect of this step, the flow field and circulation distribution at the inlet of the diffuser were set to be the same as the optimized impeller outlet.

Optimization Strategy
As shown in Figure 3, the optimization system was built by combining the IDM, CFD, optimal Latin hypercube sampling (OLHS), RSM, Multi-island genetic algorithm (MIGA) and NSGA-Ⅱ. The entire optimization process was divided into two steps. In the first step of optimization, only the impeller was optimized, and the objective function was set as the impeller weighted efficiency at 1.2Qdes, 1.0Qdes and 0.8Qdes. Thereafter, CFD analysis was performed on the optimized impeller to get the circulation and flow field distribution at the outlet of the optimized impeller. In the second step of optimization, the diffuser was optimized, and the objective functions were the levels of pump efficiency at 1.2Qdes, 1.0Qdes and 0.8Qdes. To improve the optimization effect of this step, the flow field and circulation distribution at the inlet of the diffuser were set to be the same as the optimized impeller outlet.

3D Inverse Design Method
The parameterization of the impeller is done by the inverse design software TUR-BOdesign 6.4 developed by Advanced Design Technology. In this procedure, we assume that the fluid is steady, inviscid and uniform, so that the only vorticity is the bound vorticity on the blades, and its strength was determined by a specified distribution of circumferentially averaged swirl velocity (directly related to the bound circulation 2 ) [11,25]: Here, , , and are the circumferentially averaged velocity, blade numbers, and radius, respectively. When the meridional shape and the distribution of are given, the pressure field in the blade passage can be calculated by the meridional derivation of circulation ∂( ) ∂ ⁄ : Here, − , , , and are pressure difference across the blade, fluid density, normalized streamline on the meridional shape, and pitch-wise averaged relative velocity, respectively.

3D Inverse Design Method
The parameterization of the impeller is done by the inverse design software TURBOdesign 6.4 developed by Advanced Design Technology. In this procedure, we assume that the fluid is steady, inviscid and uniform, so that the only vorticity is the bound vorticity on the blades, and its strength was determined by a specified distribution of circumferentially averaged swirl velocity rV θ (directly related to the bound circulation 2πrV θ ) [11,25]: Here, rV θ , B, and r are the circumferentially averaged velocity, blade numbers, and radius, respectively. When the meridional shape and the distribution of rV θ are given, the pressure field in the blade passage can be calculated by the meridional derivation of circulation ∂ rV θ /∂m: Here, p + − p − , ρ, m, and W m are pressure difference across the blade, fluid density, normalized streamline on the meridional shape, and pitch-wise averaged relative velocity, respectively.
The blade shape can be calculated by the following equation: Here, v and V are the periodic velocity and circumferential average velocity, respectively, and subscripts r and z represent the radial and axial components of velocity, respectively. f is the blade wrap angle that is θ value at the blade between the leading edge and trailing edge.

CFD Analyses and Validation
In this study, CFD analyses have the following three functions: one is to calculate the objective functions, the other is to analyze the flow field distribution of the optimized impeller and provide inlet flow field information for diffuser optimization, and the third is to verify the final optimization results. Therefore, the accuracy of CFD analyses is critical to the reliability of this work.
Thus, 3D steady incompressible Reynolds-Averaged Navier-Stokes (RANS) equation was used in the full-passage simulation of the mixed flow pump. The RANS equation was solved by the shear stress transport k − ω turbulence model, because this model has the advantage of accurately calculating the internal flow pattern of the mixed flow pump [26]. A high-resolution scheme was selected to discretize the convective-diffusion terms [27]. The mass flow rate was set at the inlet, and the static pressure was set at the outlet. The frozen rotor frame of reference was adopted at the interface between the stationary and rotating domains. The convergence criteria were set to 5 × 10 −5 .
The discretization of the computational domain is the basis of CFD analysis. In this study, the discretization of the computational domain is completed by structured grids, which have the advantages of controllable quality and quantity compared to unstructured grids. Mesh refinement was performed to all walls to ensure a small Y+ near the wall, O-type grids were used near the blade surface, and H/C-type grids were used near the blade edge. The meshing of the entire computational domain was completed by hexahedral grids as shown in Figure 4. Table 1 shows the results of the mesh independence analysis by using the same boundary conditions and governing equations. When the total number of grids is greater than 4.71 million, the head and efficiency reveal a small difference with the increase in grid numbers. Meanwhile, the maximum Y+ on the blades is no more than 65.
To verify the accuracy of the CFD analysis, the baseline model was numerically calculated using the above grid division and calculation setting, and the results are shown in Figure 2. The maximum head difference does not exceed 4% and the maximum efficiency difference does not exceed 2.5%. Therefore, the numerical simulation method adopted in this study is reliable.

Optimization Process
The optimization process can be accelerated by constructing the approximate model between optimization objectives and design parameters. In this study, the second-order RSM was used to construct the approximate model: Here, , , and are the undetermined coefficients and can be obtained by the least square method from the optimization objectives and design parameters.
In DOE, the optimal Latin hypercube sampling (OLHS) method was employed to generate the random, equiprobable, and orthogonally distributed sample points [28]. The structure of the sample space is consistent with the design space, which helps to reduce the number of calculation times.
A genetic algorithm (GA) was used for global optimization in the entire design space. In GA, crossover and mutation were adopted to ensure that the final result is the global optimal solution. Generally, the two main strategies for solving multi-objective optimization problems are the aggregation approach [29] and Pareto front [30]. Compared with the Pareto front, the aggregation approach has lower complexity because it converts the multi-objective optimization problem into a single-objective optimization problem through the weighted average method. However, in Pareto front, the nature of the tradeoffs between optimization objectives can be more intuitively reflected.
The optimization process starts from the selection of the optimization objectives and design parameters. After determining the range of the design parameters, OLHS was used to generate different combinations of design parameters. Thereafter, IDM was used to perform 3D modeling for each parameter combination, and CFD analysis was used to calculate the model optimization objectives. Then, RSM was used to construct the approximate model between optimization objectives and design parameters. Finally, GA was used to determine the global optimal solution.

Design Parameters
In this study, no changes have been made to the meridional shape of the mixed flow pump; thus, the mixed flow pump can be parameterized by the parameterization of the

Optimization Process
The optimization process can be accelerated by constructing the approximate model between optimization objectives and design parameters. In this study, the second-order RSM was used to construct the approximate model: Here, α 0 , α i , α ii and α ij are the undetermined coefficients and can be obtained by the least square method from the optimization objectives and design parameters.
In DOE, the optimal Latin hypercube sampling (OLHS) method was employed to generate the random, equiprobable, and orthogonally distributed sample points [28]. The structure of the sample space is consistent with the design space, which helps to reduce the number of calculation times.
A genetic algorithm (GA) was used for global optimization in the entire design space. In GA, crossover and mutation were adopted to ensure that the final result is the global optimal solution. Generally, the two main strategies for solving multi-objective optimization problems are the aggregation approach [29] and Pareto front [30]. Compared with the Pareto front, the aggregation approach has lower complexity because it converts the multi-objective optimization problem into a single-objective optimization problem through the weighted average method. However, in Pareto front, the nature of the tradeoffs between optimization objectives can be more intuitively reflected.
The optimization process starts from the selection of the optimization objectives and design parameters. After determining the range of the design parameters, OLHS was used to generate different combinations of design parameters. Thereafter, IDM was used to perform 3D modeling for each parameter combination, and CFD analysis was used to calculate the model optimization objectives. Then, RSM was used to construct the approximate model between optimization objectives and design parameters. Finally, GA was used to determine the global optimal solution.

Design Parameters
In this study, no changes have been made to the meridional shape of the mixed flow pump; thus, the mixed flow pump can be parameterized by the parameterization of the blade. As described in Section 3.1, circulation, blade loading, and stacking have the greatest effect on blade shape in IDM. As a result, these parameters were selected as design parameters.
Generally, we assume that the circumferential distribution of the circulation is uniform. Therefore, the three-dimensional distribution of the circulation can be simplified to a twodimensional distribution along the spanwise. Wang et al. [31,32] and Chang et al. [33] pointed out that the non-linear circulation distribution has more advantages than the linear circulation distribution in the mixed flow pump optimization design. Therefore, the curve shown in Figure 5 was used to control the circulation distribution with controlled parameters of rV h and rV s . In this figure, r V θ = rV θ /ω 2 r shroud is the normalized circulation, and r = (r − r hub )/(r shroud − r hub ) is the normalized spanwise distance.
Processes 2021, 9, x FOR PEER REVIEW 7 of 16 blade. As described in Section 3.1, circulation, blade loading, and stacking have the greatest effect on blade shape in IDM. As a result, these parameters were selected as design parameters.
Generally, we assume that the circumferential distribution of the circulation is uniform. Therefore, the three-dimensional distribution of the circulation can be simplified to a two-dimensional distribution along the spanwise. Wang et al. [31,32] and Chang et al. [33] pointed out that the non-linear circulation distribution has more advantages than the linear circulation distribution in the mixed flow pump optimization design. Therefore, the curve shown in Figure 5 was used to control the circulation distribution with controlled parameters of and . In this figure, = / is the normalized circulation, and is the normalized spanwise distance. The blade loading distribution is usually controlled by two parabolas and a connecting straight line as shown in Figure 6, where = ⁄ is the normalized meridional distance. The control parameters are the loading at the leading edge, the locations and of the connection point, and the slope of the middle straight line. The stacking condition shown in Figure 7 is usually imposed linearly along the blade trailing edge. Zangeneh [13] pointed out that it plays an important role in suppressing the flow separation in mixed flow pump. Zhu [34] also reported that it has a greater influence on the pressure pulsation in the impeller. The blade loading distribution is usually controlled by two parabolas and a connecting straight line as shown in Figure 6, where m = m/m total is the normalized meridional distance. The control parameters are the loading DRVT at the leading edge, the locations NC and ND of the connection point, and the slope K of the middle straight line.
Processes 2021, 9, x FOR PEER REVIEW 7 of 16 blade. As described in Section 3.1, circulation, blade loading, and stacking have the greatest effect on blade shape in IDM. As a result, these parameters were selected as design parameters.
Generally, we assume that the circumferential distribution of the circulation is uniform. Therefore, the three-dimensional distribution of the circulation can be simplified to a two-dimensional distribution along the spanwise. Wang et al. [31,32] and Chang et al. [33] pointed out that the non-linear circulation distribution has more advantages than the linear circulation distribution in the mixed flow pump optimization design. Therefore, the curve shown in Figure 5 was used to control the circulation distribution with controlled parameters of and . In this figure, = / is the normalized circulation, and is the normalized spanwise distance. The blade loading distribution is usually controlled by two parabolas and a connecting straight line as shown in Figure 6, where = ⁄ is the normalized meridional distance. The control parameters are the loading at the leading edge, the locations and of the connection point, and the slope of the middle straight line. The stacking condition shown in Figure 7 is usually imposed linearly along the blade trailing edge. Zangeneh [13] pointed out that it plays an important role in suppressing the flow separation in mixed flow pump. Zhu [34] also reported that it has a greater influence on the pressure pulsation in the impeller. The stacking condition α shown in Figure 7 is usually imposed linearly along the blade trailing edge. Zangeneh [13] pointed out that it plays an important role in suppressing the flow separation in mixed flow pump. Zhu [34] also reported that it has a greater influence on the pressure pulsation in the impeller.

Optimization Setting
To determine the undetermined coefficients in Equation (5), the minimum number of sample points required is = ( + 1) × ( + 2) 2 ⁄ , where N is the number of design parameters. However, to improve the accuracy of the approximate model, the number of sample points used in this study is 2 . As shown in Figure 3, the impeller and diffuser of the mixed flow pump were optimized by the two-step optimization method. In Table 2, the design parameters (subscripts h for hub and s for shroud), constraints, and optimization objectives during the two-step optimization process were given.
In the first step of optimization, only the impeller was optimized. The design parameters are the two circulation distribution control parameters, eight blade loading distribution control parameters and stacking condition. The range of these eleven parameters is shown in Table 2. To achieve the two purposes of maximizing overall efficiency and reducing the complexity of multi-objective optimization at the same time, the aggregation approach was used in this step. The weighted efficiency of the impeller at 1.2Qdes, 1.0Qdes and 0.8Qdes was set as the optimization objective, with weights of 0.25, 0.5, and 0.25, respectively. To make the head difference between the optimized impeller and the baseline impeller fall within an acceptable range, the impeller head change at the design point of less than 3% was taken as the constraint condition. The impeller head ( ) and efficiency ( ) are calculated by Equations (6) and (7), respectively. out in where , , , , , and are the impeller outlet total pressure, impeller inlet total pressure, fluid density, acceleration due to gravity, rotational angular velocity of the impeller, and the torque of the impeller, respectively.

Optimization Setting
To determine the undetermined coefficients in Equation (5), the minimum number of sample points required is S min = (N + 1) × (N + 2)/2, where N is the number of design parameters. However, to improve the accuracy of the approximate model, the number of sample points used in this study is 2S min . As shown in Figure 3, the impeller and diffuser of the mixed flow pump were optimized by the two-step optimization method. In Table 2, the design parameters (subscripts h for hub and s for shroud), constraints, and optimization objectives during the two-step optimization process were given. In the first step of optimization, only the impeller was optimized. The design parameters are the two circulation distribution control parameters, eight blade loading distribution control parameters and stacking condition. The range of these eleven parameters is shown in Table 2. To achieve the two purposes of maximizing overall efficiency and reducing the complexity of multi-objective optimization at the same time, the aggregation approach was used in this step. The weighted efficiency of the impeller at 1.2Q des , 1.0Q des and 0.8Q des was set as the optimization objective, with weights of 0.25, 0.5, and 0.25, respectively. To make the head difference between the optimized impeller and the baseline impeller fall within an acceptable range, the impeller head change at the design point of less than Processes 2021, 9, 260 9 of 16 3% was taken as the constraint condition. The impeller head (H) and efficiency (η) are calculated by Equations (6) and (7), respectively.
where p out , p in , ρ, g, ω, and T are the impeller outlet total pressure, impeller inlet total pressure, fluid density, acceleration due to gravity, rotational angular velocity of the impeller, and the torque of the impeller, respectively. After the first step of optimization, the impeller with the best performance was selected. CFD analysis was performed on the optimized impeller to extract the axial and circumferential velocity distribution at the outlet. Due to the non-uniformity of velocity distribution at the impeller outlet, directly using it as the inlet condition of the diffuser will result in the divergence of the IDM calculation, and thus the shape of the diffuser cannot be obtained. Therefore, the optimized impeller outlet velocity distribution needs to be smoothed, and the smoothed velocity will be taken as the initial condition for the second step of optimization.
In the second step of optimization, the diffuser was optimized. To reduce the hydraulic loss at the inlet of the diffuser, the circulation and flow field distribution at the diffuser inlet was set to be consistent with the optimized impeller outlet. Therefore, only the blade loading and stacking were selected as design parameters in this step. To comprehend the nature of the trade-offs made in choosing the final solution, Pareto front was used in this step, and the pump efficiencies at 1.2Q des , 1.0Q des , and 0.8Q des were selected as the optimization objectives. To reduce the head change of the optimized mixed flow pump at the design point, the pump head change at the design point was restricted to less than 3%.
Therefore, in this study, to improve the accuracy of the RSM in the optimization process, the number of sample points used in the first and second steps is 156 and 110, respectively. The parameter settings for MIGA and NSGA-II are shown in Table 3, and the number of impellers and diffusers with different configurations generated in the first and second steps are both 12,000.

Optimization Result
The iteration history of the first step optimization is shown in Figure 8, and the best impeller A is obtained after 12,000 steps of calculation. The performance predicted by RSM and CFD of optimized impeller A is shown in Table 4, which indicates a good consistency between the two. The weighted efficiency of the optimized impeller A is 94.29%, which is 1.63% higher than the baseline impeller. In detail, the maximum improvement of the impeller efficiency occurred at 1.2Q des , which is 5.5%. At 1.0Q des , the efficiency of the optimized impeller is improved by 0.79%. However, at 0.8Q des , the efficiency of the optimized impeller is reduced by 0.56%. Moreover, the best efficiency point in the optimized impeller A is consistent with the design point, while in the baseline impeller, the best efficiency point appears at small flow conditions. Processes 2021, 9, x FOR PEER REVIEW 10 of 16 Figure 8. Optimization results of the first step. Figure 9 shows the axial and circumferential velocity distribution at the outlet of the impeller A, these values were extracted at 0.15 ̃0.85 ̃ after considering the influence of the wall on the flow field. As described in Section 4.2, the velocity distribution needs to be smoothed. In this study, the widely used linear distribution assumption was used, and the results of the smoothing treatment are shown in Figure 9.  The optimization result of the second step is shown in Figure 10. In this step, the Pareto front seems separated, which means there is a trade-off relationship between the pump efficiency at 0.8Qdes ( . ), 1.0Qdes ( . ) and 1.2Qdes ( . ). Figure 10a shows that . and . have a strong competitive relationship, while Figure 10b shows an interesting fact that . and . are positively correlated to some extent. After carefully considering the relationship between . , . and . , the optimized diffuser B was selected for further study. The performance predicted by RSM and CFD of the optimized mixed flow pump is shown in    Figure 9 shows the axial and circumferential velocity distribution at the outlet of the impeller A, these values were extracted at 0.15 r~0.85 r after considering the influence of the wall on the flow field. As described in Section 4.2, the velocity distribution needs to be smoothed. In this study, the widely used linear distribution assumption was used, and the results of the smoothing treatment are shown in Figure 9.  Figure 9 shows the axial and circumferential velocity distribution at the outlet of the impeller A, these values were extracted at 0.15 ̃0.85 ̃ after considering the influence of the wall on the flow field. As described in Section 4.2, the velocity distribution needs to be smoothed. In this study, the widely used linear distribution assumption was used, and the results of the smoothing treatment are shown in Figure 9.   The optimization result of the second step is shown in Figure 10. In this step, the Pareto front seems separated, which means there is a trade-off relationship between the pump efficiency at 0.8Q des (η 0.8 ), 1.0Q des (η 1.0 ) and 1.2Q des (η 1.2 ). Figure 10a shows that η 0.8 and η 1.2 have a strong competitive relationship, while Figure 10b shows an interesting fact that η 1.0 and η 1.2 are positively correlated to some extent. After carefully considering the relationship between η 0.8 , η 1.0 and η 1.2 , the optimized diffuser B was selected for further study. The performance predicted by RSM and CFD of the optimized mixed flow pump is shown in Table 4. It is observed that the RSM prediction results corroborate with the CFD calculation results with the maximum error not exceeding 1%. The pump efficiency of the optimized mixed flow pump at 1.2Q des , 1.0Q des is 0.8Q des is 80.31%, 88.89% and 81.30%, respectively, which is 6.47%, 3.68% and 0.82% higher than the baseline model. CFD calculation results with the maximum error not exceeding 1%. The pump efficiency of the optimized mixed flow pump at 1.2Qdes, 1.0Qdes is 0.8Qdes is 80.31%, 88.89% and 81.30%, respectively, which is 6.47%, 3.68% and 0.82% higher than the baseline model.  Figure 11 shows the blade loading and circulation distribution of the optimized impeller and diffuser. It can be seen that the blade loading distribution at the hub and shroud of the optimized impeller A is fore-loaded and mid-loaded, respectively, while the blade loading distribution at the hub and shroud of the optimized diffuser B is fore-loaded and aft-loaded, respectively. The circulation distribution at the optimized impeller A outlet and the optimized diffuser B inlet is a second-order parabola, and the value of the circulation at the mid-span is the smallest.  Figure 12 shows the performance comparison of the optimized mixed flow pump with the baseline model. When the flow rate is greater than 0.75Qdes, the pump efficiency of the optimized model is higher than the baseline model, and the location of the best efficiency point does not change. The head of the optimized model presented an interesting change compared to the baseline model. Under the design condition, the pump head of the optimized model is almost the same as the baseline model, which means that the matching optimization results meet the constraints. However, under small flow conditions, the pump head of the optimized model is lower than the baseline model, and the lower the flow rate, the greater the head difference. The decreased head and increased  Figure 11 shows the blade loading and circulation distribution of the optimized impeller and diffuser. It can be seen that the blade loading distribution at the hub and shroud of the optimized impeller A is fore-loaded and mid-loaded, respectively, while the blade loading distribution at the hub and shroud of the optimized diffuser B is fore-loaded and aft-loaded, respectively. The circulation distribution at the optimized impeller A outlet and the optimized diffuser B inlet is a second-order parabola, and the value of the circulation at the mid-span is the smallest. CFD calculation results with the maximum error not exceeding 1%. The pump efficiency of the optimized mixed flow pump at 1.2Qdes, 1.0Qdes is 0.8Qdes is 80.31%, 88.89% and 81.30%, respectively, which is 6.47%, 3.68% and 0.82% higher than the baseline model.  Figure 11 shows the blade loading and circulation distribution of the optimized impeller and diffuser. It can be seen that the blade loading distribution at the hub and shroud of the optimized impeller A is fore-loaded and mid-loaded, respectively, while the blade loading distribution at the hub and shroud of the optimized diffuser B is fore-loaded and aft-loaded, respectively. The circulation distribution at the optimized impeller A outlet and the optimized diffuser B inlet is a second-order parabola, and the value of the circulation at the mid-span is the smallest.  Figure 12 shows the performance comparison of the optimized mixed flow pump with the baseline model. When the flow rate is greater than 0.75Qdes, the pump efficiency of the optimized model is higher than the baseline model, and the location of the best efficiency point does not change. The head of the optimized model presented an interesting change compared to the baseline model. Under the design condition, the pump head of the optimized model is almost the same as the baseline model, which means that the matching optimization results meet the constraints. However, under small flow conditions, the pump head of the optimized model is lower than the baseline model, and the lower the flow rate, the greater the head difference. The decreased head and increased    Table 5 shows the comparison of hydraulic losses of each flow passage component between the optimized model and baseline model under different flow rates. Compared with the baseline model, the hydraulic loss of the optimized impeller under large flow conditions is effectively suppressed, and the hydraulic loss of the optimized diffuser under the design condition is significantly reduced. Moreover, the hydraulic loss of the inlet pipe is positively related to the flow rate and independent of the impeller. However, the hydraulic loss of the outlet pipe is related to both the diffuser and flow rate. Compared with the baseline model, the hydraulic loss of the optimized model outlet pipe is reduced under all flow conditions. To clarify the reasons for the change of hydraulic loss in detail, the internal flow field of the optimized model and the baseline model were analyzed and compared. The streamline contours and total pressure on the mid-span of the baseline model and optimized model are shown in Figure 13. At 0.8Qdes and 1.0Qdes, a large-scale flow separation occurs at the diffuser outlet of the baseline model, which not only increased the hydraulic losses at the diffuser but also at the outlet pipe. At 1.2Qdes, an obvious low-pressure region appeared on the working surface near the impeller inlet of the baseline model. Zhang [35] pointed out that this region has a great influence on the blade vibrations and pressure fluctuation. After the matching optimization, the flow separation in the optimized diffuser was effectively suppressed, especially at 1.0Qdes, and the low-pressure region at the impeller inlet at 1.2Qdes was also weakened. As Zangeneh [12,13,36] mentioned, in the optimization design of mixed flow pump, the flow separation can be effectively suppressed by fore-loading at the hub and aft-loading at the shroud. This paper verifies this point of view again.  Table 5 shows the comparison of hydraulic losses of each flow passage component between the optimized model and baseline model under different flow rates. Compared with the baseline model, the hydraulic loss of the optimized impeller under large flow conditions is effectively suppressed, and the hydraulic loss of the optimized diffuser under the design condition is significantly reduced. Moreover, the hydraulic loss of the inlet pipe is positively related to the flow rate and independent of the impeller. However, the hydraulic loss of the outlet pipe is related to both the diffuser and flow rate. Compared with the baseline model, the hydraulic loss of the optimized model outlet pipe is reduced under all flow conditions. To clarify the reasons for the change of hydraulic loss in detail, the internal flow field of the optimized model and the baseline model were analyzed and compared. The streamline contours and total pressure on the mid-span of the baseline model and optimized model are shown in Figure 13. At 0.8Q des and 1.0Q des , a large-scale flow separation occurs at the diffuser outlet of the baseline model, which not only increased the hydraulic losses at the diffuser but also at the outlet pipe. At 1.2Q des , an obvious low-pressure region appeared on the working surface near the impeller inlet of the baseline model. Zhang [35] pointed out that this region has a great influence on the blade vibrations and pressure fluctuation. After the matching optimization, the flow separation in the optimized diffuser was effectively suppressed, especially at 1.0Q des , and the low-pressure region at the impeller inlet at 1.2Q des was also weakened. As Zangeneh [12,13,36] mentioned, in the optimization design of mixed flow pump, the flow separation can be effectively suppressed by fore-loading at the hub and aft-loading at the shroud. This paper verifies this point of view again. sses 2021, 9, x FOR PEER REVIEW 13 of 16 Figure 13. Comparison of the internal flow field between the optimized model and baseline model.

Performance Comparison and Analysis
The comparison of the total pressure distribution along the streamline between the baseline model and the optimized model is shown in Figure 14. The horizontal axis is the standardized streamline distance , = 0 means at the impeller inlet and = 2 means at the diffuser outlet. It can be seen that the total pressure rises rapidly between 0.2 ~0.8 due to the work done by the impeller to the fluid. However, the total pressure drops rapidly between 0.8 ~1.2 , because the fluid in this interval has just left the blade zone of the impeller and has not yet entered the blade zone of the diffuser. Compared with the baseline model, the hydraulic loss of the optimized model in this interval was significantly reduced, which may be related to the setting of the diffuser inlet conditions during the matching optimization. The total pressure decreases slowly between 1.2 ~2 due to the rectification effect of the diffuser.  The comparison of the total pressure distribution along the streamline between the baseline model and the optimized model is shown in Figure 14. The horizontal axis is the standardized streamline distance S, S = 0 means at the impeller inlet and S = 2 means at the diffuser outlet. It can be seen that the total pressure rises rapidly between 0.2 S~0.8 S due to the work done by the impeller to the fluid. However, the total pressure drops rapidly between 0.8 S~1.2 S, because the fluid in this interval has just left the blade zone of the impeller and has not yet entered the blade zone of the diffuser. Compared with the baseline model, the hydraulic loss of the optimized model in this interval was significantly reduced, which may be related to the setting of the diffuser inlet conditions during the matching optimization. The total pressure decreases slowly between 1.2 S~2 S due to the rectification effect of the diffuser. The comparison of the total pressure distribution along the streamline between the baseline model and the optimized model is shown in Figure 14. The horizontal axis is the standardized streamline distance , = 0 means at the impeller inlet and = 2 means at the diffuser outlet. It can be seen that the total pressure rises rapidly between 0.2 ~0.8 due to the work done by the impeller to the fluid. However, the total pressure drops rapidly between 0.8 ~1.2 , because the fluid in this interval has just left the blade zone of the impeller and has not yet entered the blade zone of the diffuser. Compared with the baseline model, the hydraulic loss of the optimized model in this interval was significantly reduced, which may be related to the setting of the diffuser inlet conditions during the matching optimization. The total pressure decreases slowly between 1.2 ~2 due to the rectification effect of the diffuser.