Application of the Modified Inverse Design Method in the Optimization of the Runner Blade of a MixedFlow Pump
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Abstract
To improve the design speed and reduce the design cost for the previous blade design method, a modified inverse design method is presented. In the new method, after a series of physical and mathematical simplifications, a saillike constrained area is proposed, which can be used to configure different runner blade shapes. Then, the new method is applied to redesign and optimize the runner blade of the scale core component of the 1400MW canned nuclear coolant pump in an established multioptimization system compromising the Computational Fluid Dynamics (CFD) analysis, the Response Surface Methodology (RSM) and the Nondominated Sorting Genetic AlgorithmII (NSGAII). After the execution of the optimization procedure, three optimal samples were ultimately obtained. Then, through comparative analysis using the target runner blade, it was found that the maximum efficiency improvement reached 1.6%, while the head improvement was about 10%. Overall, a promising runner blade inverse design method which will benefit the hydraulic design of the mixedflow pump has been proposed.
Keywords
Optimization Mixedflow pump Inverse design method Runner blade Nuclear coolant1 Introduction
Mixedflow pumps have the characteristics of both centrifugal pumps and axial pumps. They are widely used in various areas, such as industrial chemistry, industrial water applications, nuclear industry. As the heart of the power equipment in these areas, mixedflow pumps can provide stable energy for the circulation of fluid medium. The need for designing a higher efficiency pump with lower energy consumption is increasing with the current advocacy for green power, nowadays [1]. Accordingly, an effective design method for configuring the runner blade of the pump for high efficiency and reasonable high head would be valuable.
To design and optimize the blade shape, the distribution of the structural parameters, for instance the blade angles, were directly adjusted in previous studies [2, 3, 4]. Unfortunately, this approach is highly dependent on the designers’ experiences. Additionally, a fair number of design parameters would need to be selected to be optimized in order to get a new runner blade that is quite different from the target. Instead of geometrically optimizing the blade with blade angles, an important inverse design method was proposed by Borges [5] to design a blade under incompressible conditions, which was subsequently extended to compressible conditions by Zangeneh [6, 7]. In this inverse design approach, the blade shape was represented as sheets of blade loading with the approximate distribution controlled by specific variables at the hub and shroud sides, such a design method has already been adopted in the TURBODesign software. Using this sequential design method in the software, several optimal pumps have been successfully obtained [8, 9]. However, some problems still remain waiting to be solved. First, the blade shape is simply controlled by two cross sections, namely, the hub section and the shroud section, and this socalled 2D blade profile can only configure a blade with straight leading edge instead of a complex bend leading edge. Nevertheless, the leading edge affects the cavitation and secondary flows [10], thus a design method which can present a complex leading edge would offer extra choices for improving the flow characteristics. Moreover, the ranges for the design variables are not clearly defined. Taking the design method in the TURBODesign software as an example, seven independent variables [8, 9], namely, NCs, NCh, NDs, NDh, SLOPEh, SLOPEs, LEh and LEs, are adopted, and without obvious definition for the range of these variables, a large number of samples would have to be generated before optimization, which would require a considerable amount of time and computing resource.
The shortcomings of the previous design methods, thus indicates that the more cross sections controlling the blade shape (for configuring the complex blade) and the fewer design variables (for quickly designing blade), the better the design method would be. However, the number of cross sections and the number of variables have an inverse proportional relationship. In some traditional design methods [11], even four or five cross sections are adopted, but they can just configure a few blade shapes due to the excessive number of variables located on the cross sections. To increase the control sections and decrease the number of variables, a modified design method based on the previous research is proposed here. The scholars associated with such research proposed the idea of using swirl velocity distribution to design the runner blade and have carried out relevant works [12, 13]. However, hardly any optimization work has adopted this method before. Besides, Bing [14] tried to specify the constrained variables to control the swirl velocity distribution, but the design variables have not been effectively limited. As for the modified design method in this paper, a saillike constrained area controlling the design variables is deduced by some physical and mathematical simplifications based on Bing’s work, using just three points located in the constrained area are used to configure the blade by controlling the cross sections. Ultimately, the modified inverse design method is effectively applied in a multioptimization system to verify its effectiveness.
2 Modified Inverse Design Method
The Quasithreedimensional (Q3D) method which takes limited time and has successfully been applied to design the blade in previous studies [9, 12]. As for the Q3D theory, instead of taking the iteration computation of the S_{1} surface (bladetoblade) and S_{2} (hubtoshroud) surface, it takes the representative mean S_{2} surface (S_{2m}) to provide variables for the calculation of the S_{1} surface in the iterative process. To get the flow variables on the S_{2m} surface, the preliminary calculation was performed by the given meridional shape, the blocking factor (determined by the blade numbers, blade thickness, etc.), operating conditions (including mass flow, temperature, pressure, etc.) according to the 2D streamline curvature method [15, 16] or Computational Fluid Dynamics (CFD) [17]. Compared with the 2D streamline curvature method, the CFD method is much more accurate [17], thus, it was adopted here to calculate the averagewised variables on S_{2m} surface.
2.1 Basic Runner Blade Design Theory
Regarding Zangenh’s theory, after settling down the meridian shape, blade thickness and blade number, the preliminary averaged meridional velocity can be calculated at first. And with the preliminary meridional velocity and a given distribution of swirl velocity, the loading pressure can then be calculated to configure the blade shape according to Eq. (1). Afterwards, with the newly configured blade, the meridional velocity can be recalculated. Additionally, the configured blade can be subjected to minor modifications according to the new meridional velocity and the established swirl velocity before. With the iterative modification above, the blade shape can be finally established until the variables remain stable.
As a result, the averaged meridional velocity is the intermediate variable and becomes gradually stable during the design process above, so that the swirl velocity becomes the unique variable determining the loading pressure as well as the blade shape. Regarding the factor (V), the swirl velocity can also take the place of the loading pressure distribution in determining the runner blade shape [17].
After the calculation of the wrap angles, the results are then imported into a threedimensional modelling software among with the other established variables, such as blade thickness, meridional shape and blade number. Accordingly, it can be concluded that the distribution of swirl velocity is the key factor to design the runner blade shape, and its distribution law will be discussed next.
2.2 Derivation of the Constrained Area
where Eq. (10) is from Ref. [6]; Eq. (11) is from Ref. [17]; Eq. (12) is from Ref. [14].
2.3 Application of the Constrained Area in the Design Theory
The design variables to control the blade shape are given in Figure 3(b). The variables are selected from the saillike constrained area, and they are then used to establish the swirl velocity with Eq. (16).
3 Multioptimization System
 Step 1.

Design targets that refer to the performances of the design impeller should catch up with the target pump at the set 0.8Q_{d}, 1.0Q_{d} and 1.2Q_{d} mass flow conditions are set.
 Step 2.

The related design variables obtained from the target pump are input.
 Step 3.

A series of control design variables from the constrained area are selected to configure the blade shape by using the stratified sampling method and CFD analysis method.
 Step 4.

The mathematical relationships between the design variables and the objectives are established by using the RSM method.
 Step 5.

The right variables configuring the blade shape are chosen by applying the multiobjective algorithm NSGAII.
3.1 Computational Fluid Dynamics (CFD) Analysis
3.1.1 Mesh Generation
After being configured by the inverse design method, the geometry was then imported into NUMECA/Autogrid5. Taking into account the symmetry property of all blade passages in the pump, the flow characteristics of each blade passage can be considered to be the same. To reduce the grid number, it is necessary to select a single blade passage to simulate. The single blade passage is divided into several H blocks and I blocks with structured grids. To guarantee the quality, the meshed grids must have the right orthogonality (> 15°), expansion factor (< 5) and aspect ratio (< 2000). Furthermore, to reach maximal y+ around 10 in the computational domains, the minimum height of the cells is controlled to be 0.005 mm at the walls.
3.1.2 Numerical Simulation
The meshed grids are then imported into NUMECA/Fine. To simulate the threedimensional viscous compressible turbulent flow in the passage, the continuity equation, energy conservation equation and Reynolds Navier–Stokes equation processed by the turbulent model are combined. On account of the properties with good stability, small calculation and high accuracy, the Spalart–Allmaras turbulent model is chosen in the simulation process [21]. Moreover, with addition of a second and forth order artificial dissipation, the secondorder central scheme is adopted. The time marching is performed in a fourstage Runge–Kutta scheme, coupled with local time stepping and implicit residual smoothing technologies for convergence acceleration.
3.2 Optimization Process
3.2.1 Sampling Method
 Step 1.

As shown in Section 2.3, the constrained area at each cross section is divided into four equally small areas based on the equal area size. According to the data shown in this figure, it can be identified from the bottom to the top as: the small areas at the hub side, which are denoted as \( A_{1} ,A_{2} ,A_{3} ,A_{4} \) one by one; the small areas at the middle, which are identified as \( B_{1} ,B_{2} ,B_{3} ,B_{4} \); and the areas at shroud side, which are designated as \( C_{1} ,C_{2} ,C_{3} ,C_{4}. \)
 Step 2.

Continually, from the saillike area, the design variables \( A(P_{h} ,a_{h} ) \), \( B(P_{m} ,a_{m} ) \), \( C(P_{s} ,a_{s} ) \) are randomly sampled from the split areas \( A_{i} (i = 1,2,3,4) \), \( B_{j} (j = 1,2,3,4) \), and \( C_{k} (k = 1,2,3,4) \) respectively. Based on the rule of permutation and combination, 4^{3} samples were obtained at the end, and the sample numbers are listed in the first row of the table found in the Appendix.
 Step 3.

After getting the sampled results of the design variables above, geometries for the runner blade were then configured with the inverse design method described in Section 2. Then, applying the CFD analysis method described in Section 3.1, the partial simulated performances of these samples were obtained and are also listed in the Appendix.
3.2.2 RSM Method
Generally, the prediction accuracy is highly dependent on the sample scale in the design space. As for the RSM model here, the least number of sample points is \( S_{\text{min} } = (N + 1)(N + 2)/2 \). And \( N \) is 6 in this study, therefore, \( S_{\text{min} } \) is 28. Since the number of samples in the database is 64 and exceeds 28, the sample scale satisfies the RSM prediction.
3.2.3 NSGAII Method
 Step 1.

Initialize the random parent population of size n, and predict the optimized targets with the RSM method.
 Step 2.

The parent individuals are classified by the nondominated rank and crowding distance.
 Step 3.

Individuals with higher crowding distance and lower rank are preferred in the mating pool for generating the next generation.
 Step 4.

Simulated binary crossover and polynomial mutation are then applied with the crossover probability of 0.9 and mutation probability of 1/N (N is the number of variables, which is 6 here).
 Step 5.

After crossover and mutation, the mating pool generates a new generation of size n.
 Step 6.

Along with the preliminary population, the whole population (size 2n) is reduced to size n according to their rank and crowding distance.
 Step 7.

Return to Step 2, and repeat the process until the fixed generation is reached.
3.3 Optimization Targets
4 Optimization of the Runner Blade
4.1 Introduction to the Target Pump and Mesh Scheme
4.1.1 Target Pump
Geometrical variables of the runner blade
Parameters  Values 

Inlet diameter D_{1} (mm)  168 
Outlet diameter D_{2} (mm)  320 
Wrap angle at hub θ_{h} (°)  0‒130 
Wrap angle at middle θ_{m} (°)  0‒110 
Wrap angle at shroud θ_{s} (°)  0‒92 
4.1.2 Mesh Scheme and Its Validation
The target mixedflow pump is set as the simulation model, and to exclude the effects of the grids and prove the stability of the simulation method, its validation will be performed first. Under the operating conditions stated above, five kinds of grids ranging from 633548 to 1433940 are chosen to be simulated under the design condition. In addition, these grids satisfy the quality requirements described in Section 3.1, and the solver items are set as described in the same Part. The computational convergence is set below 10^{−6}. The calculations were conducted on a Dell Workstation with an Intel Core I56500 CPU.
The final simulation results of these grids are shown in Figure 6(b). According to the graph, when the number of grids exceeds 1261236, the simulation result would remain stable. Thus, this kind of grid would be the appropriate grid scheme in the multioptimization system. According to the numerical simulation results, the pump’s efficiency is \( \eta_{\text{d}} = 9 0. 4 {\% } \), and its head is H_{d} = 22.1 m. Taking the simulating performance of the target pump as the reference, the further optimization would be completed then.
4.2 Results for the Optimization Process
Basic setting of the NSGAII
Parameters  Values 

Number of generations  50 
Population size  20 
Crossover probability  0.9 
Crossover distribution index  10 
Mutation distribution index  20 
Initialization mode  Random 
Max failed runs  5 
Performances for the effective samples with relative high head
\( A(P_{h} ,a_{h} ) \)  \( B(P_{m} ,a_{m} ) \)  \( C(P_{s} ,a_{s} ) \)  θ_{h} (°)  θ_{m} (°)  θ_{s} (°)  Items  η_{0.8} (%)  η_{1.0} (%)  η_{1.2} (%)  H_{0.8} (m)  Thrust T (kN)  

Target  –  –  –  0–130  0–110  0–92  CFD  79.7  90.4  88.1  22.1  6.1 
S1 (Opt1)  (2.03, 0.57)  (1.08, 3.75)  (1.71, 5.73)  0–142  0–101  0–84  RSM  80.9  91.3  89.9  25.3  – 
CFD  81.5  91.6  88.9  23.9  5.9  
S2 (Opt2)  (1.96, 1.05)  (1.75, 1.78)  (1.13, 0.36)  0–132  0–115  0–91  RSM  80.7  91.8  90.8  25.3  – 
CFD  81.2  92.0  89.3  24.7  6.2  
S3 (Opt3)  (1.85, 1.04)  (1.75, 1.65)  (1.22, − 0.51)  0–129  0–116  0–93  RSM  80.2  91.1  90.9  25.2  – 
CFD  80.3  90.5  89.1  24.4  6.0  
S4  (1.66, − 0.07)  (1.67, 1.78)  (0.34, 3.26)  0–141  0–113  0–82  RSM  80.2  90.9  90.6  25.1  – 
CFD  78.9  91.1  89.4  24.2  6.2  
S5  (1.70, 0.40)  (1.63, 0.64)  (0.94, 0.19)  0–133  0–118  0–91  RSM  80.1  91.9  89.6  25.0  – 
CFD  80.3  89.9  88.5  23.9  6.2 
Considering the predicted errors and to guarantee the improvement of the head on the offdesign operating conditions near the design point, the optimal samples are to be chosen from the effective samples of a relatively high predicted design head (25.0–25.5 m). The predicted performances for these effective samples with relative high heads were verified by CFD analysis, and the results are listed in Table 3. Regarding the five samples with relatively high head shown in Figure 9(a), Table 3 shows their performances predicted by the RSM and simulated by CFD. From the results in Table 3, it is established that the efficiencies of three samples at the monitor mass flow points exceed those of the target mixedflow pump. Therefore, these three samples are chosen as the optimal samples, which are named as Opt1, Opt2 and Opt3. Most importantly, in Table 3, the axial thrust of the optimal results are very close to the target, which implies that they would not lead to any safety problems in the new design. The dimensionless swirl velocity distribution of the optimal samples is shown in Figure 9(b). Their characteristics will be further discussed along with the target pump.
4.3 Discussion of the Results
4.3.1 Experimental Validation for the Simulation
Sample performances in the optimization process above are mainly got by CFD analysis, though Section 4.1 has already excluded out the effect of grids, its accuracy should be further discussed. Taking the target impeller as the test model, the experimental verification was conducted on the test rig in Shenyang Blower Works.
4.3.2 Comparison of the Characteristics
The averaged meridional velocities on the S_{2m} surface are shown in Figure 11(b). These results indicate that that if the meridional shape, blade thickness, blade number, etc. from the target pump are settled, the average meridional velocities of the newly designed blades continue exhibiting a similar distribution compared with the target. Accordingly, meridional velocity of the target could be employed for the newly design blades as the initial before iterative design, which has already been discussed in Section 2.1.
The streamline distribution on the blade surface at the design point is shown in Figure 11(c). According to the results in this figure, there are four vortexes affecting the target pump’s blade surface, however, the vortexes disappear in terms of the Opt1 blade, and the number of vortexes is decreased to three for the Opt2 and Opt3 blades. Most importantly, the vortex size is greatly decreased compared with the target. Since a vortex consists of a multitude of rotating flows, it really poses a threat to the stable inner flows and can also lead to unsteady forces on the runner blade [25, 26]. Therefore, as for the optimal samples, the reduction of the vortexes would lead to a performance improvement of the inner flows relative to the target pump.
To verify the head improvement in Figure 11(a), the results in Figure 11(d) show the static pressure distribution of the impeller and vane at 50% blade height. According to the results in this figure, the static pressure of the optimal samples increases orderly from the inlet of the impeller to the outlet of the vane. Furthermore, regarding the optimal samples, the outlet static pressure is higher than the target pump (the color is much redder), which indicates that the optimal samples have a higher work capacity.
Apart from the discussion for the static pressure, Figure 11(e) shows the distribution of the quantitative total pressure along the streamline under the design condition. In this figure, at the outlet location, the total pressures of the optimal samples are larger than those of the target pump, which is consistent with the head improvement shown in Figure 11(a). Moreover, the pressure load of the optimal samples increased steadily along the streamline without any fluctuations, as shown in Figure 11(f), which means that the runner blades work on the flow medium continually without any distortions from the inlet to the outlet.
5 Conclusions
 (1)
For the modified inverse design method, a saillike constrained area generating the design variables is derived after a series of mathematical simplifications. Using a small scale of variables from the constrained area, the blades are effectively configured.
 (2)
Setting the scale core component of a 1400MW canned nuclear coolant pump as the target, a multioptimization system is established on the modified inverse design method. The effectiveness of the optimization system is demonstrated after experimental verification and analysis of variance.
 (3)
The optimal samples are ultimately obtained after optimization. Through simulation and inner flow analysis, the performances are improved when compared to the target from 0.8Qd to 1.2Qd mass flow. At the design mass flow point, the maximum efficiency improvement is as much as 1.6%, and the design head is improved by about 10%.
Notes
Authors’ Contributions
FMZ and YML were in charge of the whole trial; YML wrote the manuscript; YML, XFW and WW assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authors’ Information
YeMing Lu, born in 1991, is currently a PhD candidate at School of Energy and Power Engineering, Dalian University of Technology, China. He received his master degree from Tianjin University, China, in 2016, and his bachelor degree from Northwestern Polytechnical University, China, in 2013. His research interests include pump design and inner flow analysis of the unshroud centrifugal impeller
XiaoFang Wang, born in 1961, is currently a professor at Dalian University of Technology, China. Her research interests include design and structural reliability analysis of turbomachinery.
Wei Wang, born in 1967, is currently an associate professor at Dalian University of Technology, China. Her research interests include the cavitation study and inner flow analysis of fluid machinery.
FangMing Zhou, born in 1981, is currently a PhD candidate at School of Energy and Power Engineering, China. His research interests include the experimental investigation of the pump.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Basic Research Program of China (973 Program, Grant No. 2015CB057301), Research and Innovation in Science and Technology Major Project of Liaoning Province, China (Grant No. 201410001), and Collaborative Innovation Center of Major Machine Manufacturing in Liaoning Province, China.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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