Effects of different vortex designs on optimization results of mixed-flow pump

Turbomachinery optimization based on the inverse design method (IDM) has been investigated in several previous studies, however, due to head constraints, most of these studies have adopted the constant impeller outlet angular momentum (IOAM) distribution, namely the free vortex design, which in turn reduces the optimization effect. To overcome these drawbacks, forced and compound vortex designs are proposed here by parameterizing the IOAM using a parabola, and the effects of different vortex designs on the optimization results of a mixed-flow pump impeller are investigated. First, a baseline mixed-flow pump is simulated and experimentally verified. Second, based on the IDM, the impeller is parameterized for the three vortex designs. Finally, it is optimized by artificial neural network and genetic algorithm to maximize the weighted efficiency at 0.8Q des, 1.0Q des and 1.2Q des, and the results are analyzed. The results show that the weighted efficiency of the forced and compound vortex design is improved by 1.33% and 1.69% respectively compared to free vortex design. The internal flow analysis reveals that the improved efficiency of the compound vortex design can be attributed to the improved impeller-outlet flow regime. Finally, the energy characteristics of the preferred and baseline models are compared using the entropy production method.


Introduction
Mixed-flow pumps are widely used in urban drainage, agricultural irrigation, and industrial circulating water systems because of their high flow rates and moderate heads . However, such pumps function based on a special principle -i.e. the impeller acts on the fluid with both axial thrust and centrifugal force. Consequently, the internal flow regime in these pumps is more complex compared to that observed in the axial-flow and centrifugal counterparts . This constitutes a significant challenge concerning the hydraulic design of mixed-flow pumps. Nevertheless, with the development of computational fluid dynamics (CFD), CFD-based optimization design has become the preferred method to solve this problem.
CFD is an interdisciplinary application involving computer science and fluid mechanics, which uses numerical calculations to simulate real flow situations, thereby obtaining detailed flow-field information. The effectiveness of CFD has been extensively demonstrated in investigations of the clocking effect, cavitation, hydraulic losses, and pressure fluctuations within several pump designs (Dong et al., 2019;Feng et al., 2016;Yang et al., 2019;Zhang et al., 2021), as well as in studies of combustion, heat-mass exchange simulation, and microscopic flow (Abadi et al., 2020;Ghalandari et al., 2019;Karimmaslak et al., 2021). More importantly, CFD-based optimized design is more cost-effective and easier to implement, as compared to the experiment-based optimized design (Lu et al., 2021;Wang et al., 2020b).
Among the many in-depth studies in this field, effects of the blade angle and meridional shape on the performance of a mixed-flow pump were studied by Suh et al. (2016) using a central composite design and response surface model. Their results showed that an increased incidence angle effectively improves the impeller cavitation performance, while reducing the efficiency only slightly. Further, Kim et al. (2017) explored the optimization of a mixed-flow pump by varying the hub ratio and reported that large-scale changes can be made to the impeller shaft diameters at a specific speed, with no loss of efficiency. In another study, Kim et al. (2018) studied the coupling optimization of a mixed-flow pump impeller and diffuser to solve the over-design problem of the impeller head. They found that by simultaneously adjusting the impeller-outlet blade angle and diffuserinlet blade angle, the impeller head could be reduced, and the efficiency improved; hence, the impeller shaft power was reduced. In a similar coupling optimization study, Zhu et al. (2019) found that the improved diffuser flow field is the fundamental reason for the increased efficiency of the mixed-flow pump achieved through such optimization. The effect of the diffuser geometry on the performance of a mixed-flow pump was investigated by Heo et al. (2016). Based on a sensitivity analysis of eight geometrical parameters of the diffuser, they reported that the diffuser inlet angle had the greatest effect on the performance. Further, Wu et al. (2015) studied the effect of the blade trailing edge on the performance. They found that by adjusting the trailing-edge geometrical parameters, the flow field distribution in the entire impeller flow channel was significantly affected, especially under high-flow conditions. The results of many similar studies indicate that the hydraulic performance of mixed-flow pumps can be improved by modifying the blade shape. Thus, blade parameterization has become a key focus for optimization design.
As a suitable approach to blade parameterization, Zangeneh (1991) proposed the 3D inverse design method (IDM), which involves hydrodynamic parameters, including the streamline loading and impelleroutlet angular momentum (IOAM), instead of geometric parameters for blade parameterization. There are several benefits to using hydrodynamic over geometric parameters, such as the ability to parameterize blades with fewer parameters and a closer connection with the hydraulic performance. More importantly, in the IDM, the blade angles can be calculated from the given hydrodynamic parameters, yielding more creative solutions than those obtained via the conformal mapping performed during geometric parameterization (Yin & Wang, 2014). The superiority of the IDM for the optimization design of turbomachinery has been extensively demonstrated, and some useful conclusions have been drawn. For example, using the IDM, Zangeneh et al. (1996) successfully reduced the hydraulic losses of a mixed-flow pump impeller by adjusting the streamline loading at the hub and shroud. A flow field analysis revealed that the main reason for the increased impeller efficiency is the suppressed flow separation at the blade suction surface. Zangeneh et al.'s work was experimentally verified by Goto et al. (1996), who noted that the improved uniformity of the impeller-outlet flow field is also an important factor responsible for the increased efficiency. Huang et al. (2015) performed optimization design of a mixed-flow pump by considering streamline loading and stacking as design parameters. They concluded that aft-loading is better than pre-loading as it smoothens the velocity gradient on the blade suction surface and eliminates flow separation at the impeller inlet. Ashihara and Goto (2002) also found that aft-loading increases the pressure at the front half of the blade, potentially improving the cavitation performance of the pump. Maillard and Zangeneh (2014) investigated the effects of streamline loading on the performance of an axial fan inducer using a combination of simulations and experiments and showed that an appropriate leading-edge loading contributes to advanced break-up of the tip leakage vortices. Further, Zhu et al. (2015) employed the IDM to improve the efficiency and cavitation performance of a pump-as-turbine. They noted that the separation vortex position could be controlled by changing the pressure gradient in the blade channel. In a subsequent study, Zhu et al. (2017) found that the pressure fluctuation amplitude could also be effectively controlled by changing the streamline loading distribution.
A clear similarity among the above studies is that a constant IOAM distribution from hub to shroud was set during the optimization design; this approach constitutes the free vortex design. This is mainly due to the assumption of equal meridional velocity used by the designer to facilitate control of the impeller Euler work. However, based on a theoretical derivation, Lang and Lang (1990) reported that a non-linear IOAM distribution, which represents a compound vortex design, offers more advantages compared to those of the free vortex design. Zhang et al. (2013) later verified this viewpoint through experimental measurements of several hydraulic models. Further, Bonaiuti and Zangeneh (2009) explored the effects of a linear IOAM distribution, i.e. a forced vortex design, on the internal flow field of a compressor impeller. They found that an increased angular momentum at the hub reduces the adverse pressure gradient at the suctionside trailing edge, limits the boundary-layer growth, and reduces the flow deviation at the impeller outlet. In a subsequent work, Bonaiuti et al. (2010) found that adjusting the IOAM distribution could also significantly affect the uniformity of the diffuser-outlet flow field. While maintaining a fixed distribution of the streamline loading, Chang et al. (2014) investigated the effects of three different forced vortex designs on the efficiency and cavitation performance of a mixed-flow pump. Referring to the previous work, the Wang et al. (2020b) of this paper studied the effects of 17 different compound vortex designs on the performance of a mixed-flow pump and found that the compound vortex design is more advantageous for multi-condition and multi-objective optimization design. Overall, as the IOAM distribution greatly affects the impeller performance, in-depth and systematic research into the effects of different vortex designs on the optimization results is imperative.
In this work, a parabola is used to control the distribution of IOAM, combined with the continuity equation, energy equation, and simplified radial equilibrium equation, a mixed-flow pump is parameterized under three typical vortex designs: free, forced, and compound. It was then parametrically optimized by using artificial neural networks and genetic algorithms. A detailed comparative analysis of the optimization results for free and compound vortex designs is carried out to determine the effect of different vortex designs on the flow field. Based on the entropy production method, an in-depth analysis of energy characteristics of preferred and baseline model is carried out to elucidate the optimization mechanism. The results show that the improved impeller outlet flow regime is the fundamental reason for the improved optimization effect. This study can therefore provide a reference for the optimization design of various turbomachinery.
The remainder of this paper is organized as follows. Section 2 describes the baseline model and presents the details regarding the calculations and experiments performed in this study. Section 3 presents the proposed IDM as well as explains the definition and selection of the design parameters. Section 4 explains the selection of the optimization objective and algorithm. Section 5 presents the analysis of the optimization results and describes the optimization mechanism. Finally, Section 6 summarizes the major findings of this study.

Description of baseline model
A widely used mixed-flow pump with a specific speed of 140 was used as the baseline model. Its performance was tested using the Jiangsu University test bench, shown in Figure 1. The main technical parameters of the baseline model and test bench are listed in Tables 1 and 2, respectively.

Establishment of calculation domain
In accordance with the experimental conditions, the calculation domain shown in Figure 2 was established. The   inlet and outlet pipes were extended to a certain length to avoid backflow in the inlet and outlet boundaries, respectively, thereby improving the simulation accuracy.
To facilitate the subsequent grid generation and calculation setup, the entire calculation domain was divided into four regions. As the grid quality and quantity significantly influence the accuracy and time of calculation results, a hexahedral structured grid was employed to mesh all regions. The grids of the outlet and inlet pipes were divided using ANSYS ICEM with O-type topology, while those of the diffuser and impeller were divided using ANSYS TurboGrid with H/C-type and O-type topologies. Mesh refinement was performed to ensure that the near-wall y + value remained below 50.
To eliminate the influence of the grid number on the calculation results, a grid independence analysis of the baseline model was performed. Table 3 details the relationship between the pump section performance and grid number under the same calculation conditions. The efficiency first increased with the number of grids until 4.6 million and then remained as such with further changes in grid numbers. A similar trend was observed for the head. Following the comprehensive consideration of the calculation accuracy and computing resource consumption, a grid number of 4.6 million was selected for the grid generation.

Calculation settings
To render the calculation conditions consistent with the experimental conditions and to achieve good convergence, the following settings of ANSYS CFX were used. 'Mass Flow Rate' and 'Static pressure' were set as the inlet and outlet boundary conditions, respectively, and a medium turbulence intensity (5%) was used. Smooth and no-slip boundary conditions were applied to all the walls. The 'Frozen Rotor' setting was selected for data transmission between the rotational and stationary domains (inlet pipe to impeller and impeller to diffuser), whereas the 'None' setting was selected between the two stationary domains of the diffuser to the outlet pipe. The convergence criterion was set to 5 × 10 −−5 to achieve a balance between the calculation accuracy and time. Finally, the shear stress transport (SST) k-ω turbulence model was selected to solve the Reynolds-Averaged Navier-Stokes equation as this model can effectively predict the flow regime of mixed-flow pumps (Menter, 1994). The ' Automatic wall function' was used to solve the flow field near the wall; it allows a consistent y + insensitive mesh refinement from coarse meshes, which do not resolve the viscous sublayer, to fine meshes placing mesh points inside the viscous sublayer (Menter et al., 2003).

Calculation accuracy verification
The hydraulic performance of the baseline model was calculated using the above-mentioned calculation settings and grid division, and the calculated performance was compared with the test performance whose results  are shown in Figure 3. In this figure, the head H and efficiency η are calculated using Equations (1) and (2), respectively, as follows: where P in and P out are the total pressure at the inlet and outlet, respectively, ρ is the density of water, g is the acceleration of gravity, M is the torque, and ω is the angular velocity. The efficiency η first increased and then decreased rapidly with the increasing flow rate, with the maximum values occurring at the design point. The head H was observed to decrease monotonically with increasing flow rate. The calculated values demonstrated strong agreement with the test data over the entire flow range, and the maximum error equaled 3%. Thus, the numerical results were considered credible.

Governing equations
The IDM was first proposed by Hawthorne et al. (1984) and Tan et al. (1984) to solve the two-dimensional cascade design problem. Borges (1990) and Zangeneh (1994) further developed the IDM based on potential flow theory and successfully applied it to threedimensional turbomachinery design. In this work, the Turbodesign 6.4.0 code developed by Zangeneh was used for blade parameterization.
The iterative IDM calculation process is shown in Figure 4. The detailed theoretical derivation of IDM can be found in the original literature (Zangeneh, 1991(Zangeneh, , 1994; thus, it will not be repeated here, and only the key points are presented. In this method, the fluid is assumed to be inviscid, steady, and incompressible, and the velocity can be decomposed into periodic and circumferentialaveraged velocities, which satisfy the potential function equation (Equation (3)) and stream function equation (Equation (4)), respectively: where r is the radial coordinate, z is the axial coordinate, B is the blade number, V θ is the circumferential-averaged tangential velocity, m is the number of Fourier expansion terms, B f is the blocking factor, i is the imaginary unit. m is the potential function, is the stream function, and f is the wrap angle, i.e. the θ value between the leading and trailing edges, which can be solved by numerical integration.
The blade shape is calculated according to the inviscid slip condition on the blade wall using where v zbl , v rbl , and v θ bl are the axial, radial, and circumferential velocity components of the periodic velocity, respectively; V z , V r , and V θ are those of the circumferential-averaged velocity, respectively; and ω is the angular velocity.

Design parameter definition
The governing equations reveal that the most important IDM design parameter is rV θ , i.e. the angular momentum. Thus, the key aspect of the IDM is the specification of the angular momentum distribution on the meridional shape.
To satisfy the assumption of no fluid pre-rotation at the impeller inlet, the impeller inlet angular momentum was set to zero. Then, to investigate the effects of different vortex designs on the optimization results, the IOAM distribution was parameterized using the equation proposed by Zhang et al. (2012): As shown in Figure 5, three different equations can be deduced by assigning different values to the variables a, b, and c in Equation (6). That is, for the free, forced, and compound vortex designs, a = b = 0 and c = 0, a = 0 and b = 0, and a = 0, respectively. Correspondingly, rV h , rV m , and rV s have different values in the three vortex designs (the sub-indexes h, m, and s refer to hub, mid-span and shroud, respectively). For the free vortex design, rV h = rV m = rV s = c. For the forced vortex design, rV h is selected as the design parameter, whereas rV m and rV s are determined automatically by the Turbodesign code based on the continuity equation, energy equation, and simplified radial equilibrium equation. Finally, for the compound vortex design, rV h and rV s are selected as the design parameters, and rV m is determined automatically by the Turbodesign code. The continuity equation, energy equation, and simplified radial equilibrium equation can be expressed as follows: where V z is the meridional velocity, V θ is the tangential velocity, H des and Q des are the head and flow rate under design condition, respectively. For convenience of description, the free, forced, and combined vortex designs are referred to as Cases 1-3, respectively, in the following sections. According to the incompressible potential flow theory, the derivative of the angular momentum along the streamline (i.e. the blade loading) is related to the pressure distribution. That is, where p is the pressure difference between the blade pressure surface and suction surface, W m is the relative velocity, and s is the streamline. Therefore, to control the pressure distribution directly in this study, the typical 'three-segment' loading distribution shown in Figure 6 was used to control the angular momentum distribution along the streamline. For each streamline, four control parameters were used: LE, NC, ND, and K; LE is the loading at the blade leading edge, NC and ND are the connection point locations of the middle straight line, and K is the slope of the middle straight line. The blade loading distributions were specified along the streamline at the shroud and hub, and the blade loading between the shroud and hub was determined using interpolation.

Design parameter selection
The design parameters are listed in Table 4, which actually represent the design parameters of this study. Hence, Cases 1-3 had 8, 9, and 10 design parameters, respectively.

Optimization objective and constraint
To provide the optimized mixed-flow pump with a higher efficiency and a broader high-efficiency area compared to those of the baseline model, as well as a similar specific speed, a head variation of less than 3% at 1.0Q des was selected as the constraint, where Q des is the design flow rate, and the weighted efficiencies at 0.8Q des , 1.0Q des , and 1.2Q des were selected as the optimization objective. The constraints and optimization objective could be expressed as: Constraints: Objective: where η 1 , η 2 , and η 3 denotes the pump efficiency at 0.8Q des , 1.0Q des , and 1.2Q des , respectively. ω j denotes the weight coefficient of each objective, and η ω denotes the weighted efficiency.
To eliminate the adverse effects of the subjective selection of the weight coefficient on the optimization results, the entropy weight method  was employed. This approach can objectively determine the weight coefficient. According to the information theory, entropy is used to measure the degree of uncertainty of an event. The greater the information volume, the smaller is the uncertainty, and the lower is the entropy value. According to this property of entropy, an objective with a small entropy has a large volume of information and plays a more important role in the comprehensive evaluation; thus, the weight coefficient of this objective is greater.
An optimization problem with m sample points and n objective numbers can be expressed by the matrix X X = (x ij ) m×n i = 1, 2, 3 . . . .m; j = 1, 2, 3 . . . .n. (13) The data in X are standardized to obtain the normalized matrix Y = (y ij ) m×n , where The entropy values E j of each objective can be calculated from the relation Finally, the weight coefficient of each objective can be calculated as shown below.

Optimization process
The optimization system used in this study is depicted in Figure 7 and comprises three parts: numerical simulation, IDM, and optimization algorithm. The optimization process can be divided into four steps. First, the optimization objectives and design parameters are determined. Second, design of experiment (DOE) is used to construct sample points, and IDM is employed to perform 3D modeling for all sample points. Then ANSYS-CFX is used to solve the objective functions of each sample point, and the approximate model is used to construct the mapping relationship between the design parameters and the optimization objectives. Finally, an optimization algorithm is used to find the optimum solution on the approximate model, and the accuracy and internal flow field of the optimal results are verified and analysed using CFD simulation. This subsection focuses on the algorithm selection.

Design of experiment
Compared with the commonly used Latin hypercube design (LHS) method, the sample points generated by the optimal LHS (OLHS) method are more evenly distributed in the design space and have a stronger spacefilling capability (Wang et al., 2020c). Hence, the sample points were generated using OLHS in this study. In each case, 110 sample points were generated in the design space. As the IOAM distribution has a greater effect on the head, monitoring the head variation of the sample-point at the design condition was necessary. If the head varied significantly for a sample point under the design condition (a difference of more than 10% from the baseline-model design head), that sample point was excluded and a new corresponding sample point was added. A comparison of the sample point heads under the design condition for the three cases is shown in Figure 8. As can be seen, the number of sample points in Cases 1-3 that do not meet the head requirements are 1, 0, and 3, respectively. Therefore, after excluding these sample points, adding additional 1, 0 and 3 corresponding sample points to Cases 1-3, respectively, was necessary. Hence, it is feasible to use different vortex designs for the optimization design of mixed-flow pumps.

Approximate model
The mapping relationship between the design parameters and optimization objective is also known as the approximation model, which helps to reduce the number of numerical simulation calls during optimization. The artificial neural network (ANN) proposed by McCulloch and Pitts (1943) has been widely used in engineering optimization recently because of its excellent nonlinear approximation ability. However, the ANN is also disadvantageous due to its slow convergence rate and tendency to fall into local optima. To overcome these shortcomings, the ANN improved by the genetic algorithm (ANN-GA) developed by Whitley et al. (1990) was employed in this work to construct the approximate model between the design parameters and optimization objective. The hidden-layer neurons, population size, generation numbers, crossover, and mutation probability are the most important parameters in ANN-GA, which were determined here through trial-and-error; the results are listed in Table 5. The prediction accuracy of these approximate models was measured using the correlation coefficient R 2 where y, y, and y are the actual, average, and predicted values of the sample point, respectively. To demonstrate the prediction accuracy of the approximate model, Figure 9 presents a comparison of the testdata predicted and calculated values. It can be seen that the approximate models of Cases 1-3 had sufficient prediction accuracies to be used for subsequent optimization.

Optimization algorithm
To find the optimal solution in the design space, the multi-island genetic algorithm (MIGA) in the MATLAB optimization toolbox was employed to solve the approximate model. Compared with the GA, MIGA avoids the disadvantage of premature convergence by introducing the 'island' concept for dividing the entire population into several sub-populations . The same MIGA settings were used for all cases: a population size of 200, a generation size of 20, an island number of 10, a crossover probability of 0.65, and a mutation probability of 0.1. To find the optimal solution, 40,000 impeller configurations were considered in each case. The entire optimization process was carried out on a Dell workstation with two Intel(R) Xeon(R) Gold 6154 CPUs, 128G of operating memory and 4 T of solid-state drives. The calculations were carried out in parallel model and took a total of approximately 300 h.
The optimization results of Cases 1-3 are compared in Table 6. The difference between the approximate model predictions and numerical simulations is less than 1% for all cases. Compared with the baseline model, the weighted efficiencies for Cases 1-3 increased but by different extents.

Comparison of angular momentum and loading distribution
In this study, clear differences in blades shape in Cases 1-3 were obtained, as shown in Figure 10; this demonstrates the usefulness of considering the IOAM distribution in the optimization design. Clear differences in the impeller-outlet angular momentum distribution for Cases 1-3 were also found. In Case 1, the angular momentum was uniformly distributed from hub to shroud, whereas in Case 2, it monotonically increased from hub to shroud. However, in Case 3, the angular momentum distribution first decreased and then increased from hub to shroud, reaching a minimum at the mid-span. The streamline loading distribution also varied among Cases 1-3. In Cases 1 and 2, those at the hub and shroud were fore-loaded and aft-loaded, respectively; this helped suppress the secondary-flow generation in the impeller (Zangeneh et al., 1997). However, in Case 3, the streamline loading distributions at the hub and shroud were both fore-loaded.

Performance comparison
A performance comparison of the optimization results for Cases 1-3 is shown in Figure 11. Compared with Case 1, Cases 2 and 3 exhibited negligible changes in head H but large increases in efficiency η, especially at high flow rates. Taking Cases 1 and 3 as examples, the efficiencies of Case 1 at 0.8Q des , 1.0Q des , and 1.2Q des equaled 80.86%, 87.52%, and 76.74%, respectively, whereas those of Case 3 were 81.6%, 88.86%, and 80.53%, respectively. Therefore, a compound vortex design in the mixed-flow pump optimization design is conducive to not only improving the design condition efficiency but also expanding the high-efficiency zone. A detailed internal flow analysis was performed for Cases 1 and 3 to elucidate the effects of these different vortex designs on the optimization results.
As shown in the boxes formed by the dashed red lines in Figure 12(a), compared to those in Cases 1, a greater axial velocity near the hub and shroud and a lower axial velocity at the mid-span were observed in Case 3. This trend is consistent with the impeller-outlet angular momentum distribution in Figure 10(e) and indicates that the impeller-outlet axial velocity distribution can be controlled by controlling the IOAM distribution. A Figure 11. Differences in head and efficiency of optimization results for Cases 1-3. similar phenomenon was observed for the total pressure distribution. Notably, the spanwise total pressure distribution was more uniform in Case 3 than in Case 1.
As depicted in Figure 13, the flow structures in Case 1 and Case 3 were essentially identical on the blade suction surface, as were the static pressure distributions. A key difference was the presence of a distinct backflow zone near the hub at the trailing edge of the Case 1 blade. The interaction between the backflow and mainstream near the hub forced the latter to flow in the mid-span direction. In Case 3, the backflow was effectively suppressed. This is because one of the main causes of such backflow is the accumulation of low-momentum fluids; due to the higher axial velocity near the hub in Case 3, more low-momentum fluid was removed, thereby preventing its accumulation. Adverse pressure gradients also contribute to this backflow (Chen et al., 2021). In Case 1, the pressure distribution was significantly distorted at the trailing edge of the blade near the hub, whereas in Case 3, it was uniform, and the pressure gradient was in the same direction as the mainstream.
The pressure difference between the shroud and hub is the main cause of hub-to-shroud (H-S)-type secondary flow (Wang et al., 2020a). In Case 1, the pressure distribution near the hub at the impeller outlet was distorted, resulting in a rapid increase of the pressure difference between the shroud and hub on the suction surface P s (Figure 14). This exacerbated the   secondary-flow formation. In Case 3, P s was reduced due to the increased pressure near the hub.
During the mixed-flow pump operation, a large amount of mechanical energy is irreversibly converted into thermal internal energy because of the Reynolds stress and fluid viscosity. Hence, it is feasible to use the entropy production theory to investigate the energy dissipation within the mixed-flow pump. During the pump operation, the temperature of the water can be approximated as constant. Therefore, the time-averaging entropy production by dissipation under incompressible turbulent flow can be calculated using Equation (19) where u, v, and w are velocities in the Cartesian coordinate system in the x, y, and z directions, respectively; μ is the dynamic viscosity, T is the temperature. The first term on the right side of Equation (19) is called direct entropy production rateṠ PRO,D , which can be obtained directly by solving the RANS equation. The second term is called indirect entropy production rateṠ PRO,D , which Herwig and Kock (2007) indicates can be calculating using Equation (20): where ε is the turbulent dissipation rate. Therefore, in incompressible turbulent flow, the entropy production rate can be calculated according to Equations (19) and (20). To clarify the effect of the impeller-outlet flow regime on the diffuser energy characteristics, the distribution of entropy produced in the diffuser was analysed. As shown in Figure 15, because of the wall-fluid interaction, high entropy production values were observed near the hub, shroud, and blade surface (Lee et al., 2021). However, the entropy produced near the hub was evidently reduced in Case 3, as compared with that in Case 1; this was mainly related to the improved flow regime near the hub at the impeller outlet ( Figure 13).

Comparison with baseline model
Case 3 was selected as the preferred model because of its good overall performance. To clarify the optimization mechanism, the performances of the baseline and preferred model were compared. Figure 16 shows that the preferred-model efficiencies at 0.8Q des , 1.0Q des , and 1.2Q des which were 1.12%, 3.66%, and 6.68% higher than those of the baseline model, respectively. Among the three flow rates, the efficiency at 1.2Q des exhibited the greatest increase because of its greater weight coefficient. In addition, the highest efficiency point of the baseline model was at 1.05Q des , whereas that of the preferred model coincided with the design point. The head of the preferred model also exhibited an interesting difference compared to that of the baseline model. At the design condition, both models had the same head; however, the head of the preferred model was higher at high flow rates, and the head difference increased with increased flow rate. A similar variation was observed at low flow rates.  Because of the interaction between the fluid and wall, higher entropy production values were observed near the blade and shroud, especially at the corner of the blade pressure surface near the shroud, as apparent for regions A, B, and C in Figure 17(a). After optimization, the entropy production in these regions (A', B', and C' in Figure 17(b)) reduced slightly.
The same phenomenon was observed in the impeller blade-to-blade channel (Figure 18). For the preferredmodel impeller, the entropy production value at the blade pressure surface near the shroud was markedly reduced compared to that of the baseline model, as apparent for the E region. At the mid-span, the entropy production distributions of both models were essentially the same, and higher entropy production values were observed in a small region near the impeller-outlet trailing edge. The entropy production distributions near the hub and the mid-span were essentially identical. However, because of the fluid mixing on the suction and pressure surfaces at the blade trailing edge, a higher energy loss was observed at the impeller outlet (Wang et al., 2020b), which implies the deterioration of the flow regime in this region. In region F of Figure 19(a), which depicts the spanwise axial-velocity distribution at the impeller outlet, the fluid axial velocity near the hub assumes a negative value, thereby indicating reverse flow. After optimization, the energy loss in region D reduced considerably, as described in region D' (Figure 18(b)). This was mainly related to the larger axial velocity near the hub in the preferred-model impeller. As previously discussed, a large axial velocity near the hub can effectively prevent the accumulation of low-momentum fluids, thereby preventing backflow.
As shown in Figure 19(b), the low-pressure region G near the hub was also eliminated after optimization. In addition, the spanwise axial velocity and pressure distributions observed in the preferred model were more uniform compared to those corresponding to the baseline model, indicating an improved flow regime at the impeller outlet of the former. This improvement affected the performance of the components located downstream of the impeller, thereby expanding the advantage of the preferred model over the baseline model. Table 7 lists the ratios of the energy loss to the total input energy for different components. In the baseline model, the impeller efficiency was already very high at 95.04%. After optimization, the impeller energy loss was reduced by only 0.47%, but the hydraulic loss in the impeller downstream components was reduced by 3.18%. Therefore, the improved flow regime at the impeller outlet, which reduces the hydraulic loss in the impeller downstream components, is the main reason for the increased efficiency of the preferred model.
To clarify the effect of the impeller outlet-flow regime on the performance of its downstream components, the entropy production distribution in the diffuser was analysed. As shown in Figure 20(a), a high entropy was produced for both regions I and J near the diffuser inlet. As the flow developed, the extent of region I was reduced, whereas that of region J was further increased, and a   new high entropy production region K was formed due to mixing of the two in the middle of the diffuser. The size of region K increased further at the diffuser outlet; however, the corresponding entropy-production value was reduced, and a new high entropy production region L was formed. The region L spanned the entire blade suction surface and occupied half the flow-channel volume. After optimization, region J was eliminated from the preferred model, but region I was increased to form region I'. Through the elimination of region J, the extent and entropy production values of regions K' and L' in the preferred model were significantly reduced compared to the baseline model. The same phenomenon was observed in the diffuser blade-to-blade channel, as shown in Figure 21(a). In the baseline model, high entropy production values were present throughout the flow channel near the hub, as well as in the second half of the flow channel at the mid-span. After optimization, high entropy production occurred in a small region near the blade suction surface in the first half of the channel.  Figures 20 and 21 reveals that the entropy production distributions in regions I and J decisively impact the entropy production distribution in the diffuser. To clarify the underlying factors behind the generation of regions I and J, a single blade channel (region H in Figure 20(a)) was selected for the flow field analysis. As depicted in Figure 22, the formation of region J is related to the vortex and pressure-surface to suction-surface (P-S)-type backflow near the hub, and that of region I to the included angle α between the blade inlet angle and inflow angle of the diffuser, which is also called the attack angle. After optimization, the backflow near the hub in the preferred model was suppressed; thus, region J was eliminated. However, the decrease and increase in the axial and circumferential velocities near the shroud resulted in α increasing to α , thereby increasing the size of region I. Figure 23 intuitively reflects the difference in the massflow average entropy production for the two models at different streamline positions. The preferred model had a lower entropy production over the entire streamline, especially in the diffuser. In the impeller, as the flow developed, the entropy production increased rapidly until s = 0.4. It then slowly decreased until s = 0.8 and finally rapidly decreased up to the impeller outlet.   The high entropy-production values observed between s = 0.2 and s = 0.8 could be attributed to the intense blade-fluid energy transformations. An interesting phenomenon is that the entropy production values at the impeller outlet were almost identical for both models, but the difference in the flow fields shown in Figure 19 induced an extremely large difference in the entropy production within the diffusers of the two models. In the preferred model, the entropy production in the diffuser gradually decreased with flow development, indicating a gradual improvement in the flow regime in the diffuser. However, such phenomena were not observed in the diffuser of the baseline model.

Conclusion
In this work, a mixed-flow pump with a specific speed of 140 was numerically simulated, and its performance was experimentally verified. Then, the impeller was parameterized using the IDM for three different cases: free, forced, and compound vortex designs and optimized using a combined optimization system composed of OLHS, ANN-GA, and MIGA. Through comparative analyses of the internal flow fields produced by the optimization results, the following conclusions were drawn.
(1) Compared with the commonly used free vortex design, no significant changes in head are obtained at the sample points under the design condition when forced and compound vortex designs were used for blade parameterization. However, the weighted efficiency of the compound vortex design is improved by 1.69% and 0.36% compared to those of the free and forced vortex designs, respectively. Hence, for further optimization, it is feasible and necessary to use the compound vortex design in the optimization design.
(2) The impeller-outlet axial-velocity and total-pressure distributions can be effectively controlled using different vortex designs. Compared with the free vortex design, the axial-velocity near the impelleroutlet hub is significantly increased with the compound vortex design. Hence, more low-momentum fluid is removed, which prevents backflow generation in this area. Further, the pressure difference between the blade suction surface shroud and hub near the impeller outlet is also reduced for the compound vortex design, which helps prevent the generation of the H-S-type secondary flow.
(3) The weighting efficiency of the compound vortex design is improved by 5.12% compared to that of the baseline model. Entropy production analyses showed that the increased efficiency of the compound vortex design is mainly due to two factors. First, the energy dissipation near the impeller blade pressure surface and outlet is reduced. Second, the backflow near the hub at the impeller outlet is eliminated, thereby suppressing the generation of the large-scale vortex near the diffuser hub. This resulted in a reduction of the hydraulic losses occurring downstream of the impeller.
This study provides ready reference to facilitate the optimized design of different turbomachines. However, due to the limitation of computational resources, only hydrodynamic parameters including IOAM and streamline loading are considered in this paper, while the influence of geometric parameters such as meridional channel on the optimization results is ignored. In future study, it is recommended that both hydrodynamic and geometrical parameters be considered in the optimized design to further improve the optimization effect, where computational resources allow.