Optimization method investigation for centrifugal pump performance based on 3D inviscid inverse design

Centrifugal pump is widely used in many important fields such as large hydropower station, urban central heating, aerospace engineering, etc. It has high requirements for cavitation performance and hydraulic performance in actual use. This study centers around on a pipeline pump with a specific speed of 360, reconstructs the impeller in TURBOdesign, takes important parameters of the impeller as the input conditions for optimization, aims at enhancing the blade’s minimum pressure and curtailing the flow losses in the blade channel and progressively optimize the meridional-blade loading based on the inviscid flow which was carried out by using genetic algorithm. The optimization leverages genetic algorithm, progressively optimizing the impeller using meridional and blade loading. By analyzing the effects of meridional and blade loading parameters on cavitation and hydraulic performance, it is found that the optimized models’ NPSH are much better than the original one. It’s approved that this optimization method is valid.


Introduction
Optimization is a method to select the best scheme among various schemes.It is based on mathematical optimization theory and uses a computer to search for the best design strategy under a variety of restrictions in order to achieve the design's performance goals.With the rapid development of mathematical theory and computer technology, Optimization has emerged as a distinct autonomous engineering discipline and has been extensively applied in production practice.
TURBOdesign is a distinctive 3D aerodynamic and hydrodynamic design software based on the inverse problem design methodology.This paper mainly uses two modules, TurboDesign-1 and TURBOdesign Optima.TURBOdesign-1 is a 3D inverse problem design tool for inviscid flow.It can be applied to the design of blades for any rotating machine, including pumps, turbines, axial, centrifugal, or mixed-flow machines, rotating or fixed machines, and machines for compressible and incompressible flows.The 3D inverse problem design of rotating machinery originated in 1981.Hawthorne and Tan proposed a 3D inverse problem blade design method based on Clebsch formula [1][2].Subsequently, Borges extended this method to radial turbines with incompressible flows [3].Finally, Zangeneh further developed the theory and applied the 3D inverse problem design to the design of The reconstructed design parameters are consistent with the original model.Rated flow Qnonimal is 1380m 3 /h, rated head Hnominal is 19m, speed n is 1450r/min, blade number Z is 5, blade thickness t is 5mm.These five parameters are also the design parameters that must be defined in TURBOdesign-1.
rVt is an important parameter used in TURBOdesign-1 to define the impeller geometry [5].r represents the radius of the impeller, Vt represents the circumferential velocity of the liquid, and Cu is used in the general design method of the pump impeller.As shown in the impeller velocity triangle in Figure 2, C2u represents the circumferential component of the absolute velocity of the liquid at the exit of the impeller.
In the formula,  2 has been determined by the reconstructed impeller diameter and speed. 2 can be expressed as C2m is defined by the design flow and meridian plane outlet B2, so rVt is mainly used to define the outlet setting Angle of the blade  2 in this study.The reconstructed rVt parameters are shown in Table 1.Hub represents the rear cover, Shroud represents the rear and front cover, LE represents the inlet edge of the impeller, and TE represents the outlet edge of the impeller.LE is set to 0, indicating that the Angle of attack at the inlet of the impeller at the design point is 0 degrees.Blade loading is a crucial input variable that TURBOdesign-1 uses to determine the blade shape [6].Generally, it consists of three sections, shown in Figure 3.The first section is a parabola from 0 to NC, the second section is a straight line from NC to ND, and the third section is a parabola from ND to 1. Where, NC and ND are dimensionless quantities along the flow direction of the impeller, ranging from 0 to 1. Slope indicates the slope of the intermediate straight segment.LE loading refers to the inlet side load.When LE loading is positive, the inlet Angle of impact is positive.When LE loading is negative, the inlet impact Angle is negative.Generally, positive LE loading is conducive to the improvement of impeller efficiency, while negative LE loading is conducive to the improvement of cavitation performance, but it will affect the efficiency [7][8].2. For the shroud, the first half loading is big, but small in the second.It's called before loading.And for the hub, the first half loading is small, but bigin the second.It's called after loading.This type of Blade loading is mainly to reduce blade flow loss and improve impeller efficiency [9][10][11][12].Through the solver for the above design parameters, rVt and Blade loading, the performance of the reconstructed impeller is obtained, as shown in Table 3. Profile Loss_t includes pressure surface loss and suction surface loss, which is an important index to measure impeller efficiency.The minimum blade pressure Pmin represents the minimum pressure on the blade.For the design point, Pmin generally appears in the suction surface of the blade near the front cover plate.Mainly due to the effect of centrifugal force, the speed at the front cover plate is higher, so the pressure in this area is lower.NPSHr is the required cavitation allowance of the pump, which refers to the cavitation level of the pump itself.Figure 4 shows the pressure cloud map of the reconstructed impeller, from which it can be seen that Pmin is -40.042kPa,appearing on the upper edge of the suction surface.It is consistent with theoretical analysis.Due to the large number of geometric parameters and Blade loading parameters, in order to ensure the robustness of subsequent optimization, this paper adopts a progressive optimization method to optimize the meridional plane and blade successively.

Optimize settings
In this paper, process 1 in TD Optima is selected for optimization.This method has the benefits of a straightforward user interface, strong robustness and a quick resolution.ADT-NSGA II genetic algorithm is the optimization technique.Genetic algorithms are a mathematical approach to problem resolution that uses computer simulation to mimic the biological evolution of chromosome gene crossing and variation.When solving complex combinatorial optimization problems, better optimization results can be obtained relatively quickly compared with some conventional optimization algorithms.There are two important parameters in genetic algorithm: Population Size and Number of Generations.The term "population size" represents the entire population of given generation.The larger the population, the more probable it is that a global solution will be found, but the running time will also be rather long.In this paper, the Population size is set to 30.Genetic algebra represents the number of iterations of a genetic algorithm.Similar to population size, the likelihood of finding a universal solution increases with the number of iterations.In this paper, the genetic algebra is set to 30.The total number of solutions is 900.
Figure 6 shows the input parameters for the meridional optimization.M2_INLETSHRRIN represents inlet diameter D1, M2_OUTLETW2 represents meridional axial length Z2_Shr and blade outlet width B2, M2_XSHR and M2_YSHR represent meridional front cover curvature radius R2_Shr.The four meridional plane geometric dimensions and blade number Z are the key input parameters for this optimization.The blade number Z ranges from 5 to 7 and is an integer.M2_INLETSHRRIN, M2_OUTLETW2, M2_XSHR, and M2_YSHR have values in the range recommended by Optima, and the extreme values of the range are pre-debuggable to generate geometry.

Analysis of optimization results
Figure 8 shows the comparison between all the convergence results optimized by meridional genetic algorithm and the baseline design.ProfileLoss_t is the abscissa, and Pmin is the ordinate.The triangle in the figure shows the baseline design point, where ProfileLoss_t is 2.28 and Pmin is -40.042kPa.Along the Pareto frontier curve of all the results (the outermost curve in the upper left corner), it is found that no result can fulfill both ProfileLoss_t less than 2.28 and Pmin greater than -40.042kPa.It demonstrates that the demand for optimization cannot be satisfied by merely optimizing the meridianal.Therefore, in order to continue with optimization, No.44 are chosen along the Pareto frontier curve.4 shows the comparison of key parameters of different meridionals.Through comparative analysis, it is found that No. 10 meridional inlet diameter is the largest, whose value D1 is 272.7mm, which is 20.7mm larger than the baseline inlet diameter.The inlet diameter of No. 38 meridional is slightly larger than the baseline inlet diameter, and its value D1 is 257.7mm.No. 44 meridional inlet diameter is the smallest, D1 is 248.7mm.The axial length of the meridional of all three is longer than baseline, among which the longest length of No.38 and No.44 is 64mm, 13mm larger than that of the baseline.Additionally, The output width of the three impellers is less than the normal.Combined with Figure 7, it can be seen that a strong positive correlation between the inlet diameter of the impeller D1 and cavitation performance.In other words, the larger the inlet diameter, the better the cavitation performance; the inlet diameter of impeller D1 is adversely connected with pump efficiency.That is, the efficiency decreases with increasing inlet diameter.

Blade loading optimization
Based on the No.44 meridional, this research takes eight Blade loading parameters as input conditions, and uses the genetic algorithm again to carry out a new round of multi-objective optimization of cavitation and efficiency of the impeller.Figure 9 shows the input variables and their ranges for this optimization.The rest of the settings are the same as before.Comparative study revealed that the blade flow loss and minimum pressure at the majority of sites were better than Baseline 44 at the same time, and that the trend was comparable to the results from the two meridian planes mentioned above.An average of three segments were intercepted along the Pareto frontier with Baseline Pmin of No. 44 as the starting point and maximum Pmin as the ending point.From left to right are 785, 737, 881 and 844.In terms of efficiency alone, 785 is the optimal solution; 844 is optimal solution in terms of cavitation alone; Taking into account both efficiency and cavitation, and focusing on cavitation, the result of No. 881 is the optimal solution.The optimization results show that the optimization method is feasible for high specific speed pump and optimization space for Blade loading is large, and the majority of the convergence outcomes are better than their Baseline.

Conclusions
1.The genetic algorithm was used to gradually optimize the meridional-Blade loading based on inviscid flow.The inlet diameter of the impeller D1 has a significant impact on the cavitation of the pump and is positively associated.But the inlet diameter of the impeller D1 is negatively connected with pump efficiency, which means that the greater the inlet diameter, the larger the Pmin, the better the cavitation performance.and the efficiency decreases with increasing inlet diameter.2.Using the Blade loading parameter as an input condition, a genetic algorithm is employed to perform a fresh round of multi-objective optimization for cavitation and impeller efficiency.The results demonstrated that the optimized solution was better than the Baseline, which proves that the optimization space of Blade loading is larger.Comparative study reveals that the candidate models outperform the reconstructed base model in terms of cavitation performance.Combined with the meridional optimization results, the feasibility of this optimization design method is proved.

Figure 1 .
Figure 1.Reconstructed meridional and its parameters.The reconstructed design parameters are consistent with the original model.Rated flow Qnonimal is 1380m 3 /h, rated head Hnominal is 19m, speed n is 1450r/min, blade number Z is 5, blade thickness t is 5mm.These five parameters are also the design parameters that must be defined in TURBOdesign-1.rVt is an important parameter used in TURBOdesign-1 to define the impeller geometry[5].r represents the radius of the impeller, Vt represents the circumferential velocity of the liquid, and Cu is used in the general design method of the pump impeller.As shown in the impeller velocity triangle in Figure2, C2u represents the circumferential component of the absolute velocity of the liquid at the exit of the impeller.

Figure 2 .
Figure 2. Impeller velocity triangle.The formula of rVt can be expressed as

Figure 3 .
Figure 3. Blade loading definition.Blade loading parameters of the reconstructed impeller are shown in Table2.For the shroud, the first half loading is big, but small in the second.It's called before loading.And for the hub, the first half loading is small, but bigin the second.It's called after loading.This type of Blade loading is mainly to reduce blade flow loss and improve impeller efficiency[9][10][11][12].Table2.Blade loading parameters of reconstructed impeller.

Figure 4 .
Figure 4. Pressure cloud map of reconstructed impeller.And in order to make sure that the NPSHr and Pmin prediction based on 3D Inviscid Inverse Design is valid, this paper did a comparison between CFD and Inverse Design.The CFD cavitation calculations were conducted within the framework of ANSYS CFX19.0.In cavitation calculation, the homogeneous multiphase model and Rayleigh-Plesset model were used.So, in this paper thermal effects are not in the consideration, as the cavitation process is typically too rapid for the assumption of thermal equilibrium at the interface to be correct.The basic assumption of the model is that all phases share the same velocity and a mixture equation is solved for the conservation of momentum.Figure5shows the NPSHr difference between CFD and Inverse Design.CFD NPSHr at Best Efficiency Point flow is 11.56m, and Inverse Design NPSHr is 12.07m, which is close to each other.It means that the prediction based on Inverse Design is valid to be used as the results to continue the optimization study.

Figure 5
shows the NPSHr difference between CFD and Inverse Design.CFD NPSHr at Best Efficiency Point flow is 11.56m, and Inverse Design NPSHr is 12.07m, which is close to each other.It means that the prediction based on Inverse Design is valid to be used as the results to continue the optimization study.

Figure 6 .
Figure 6.Input parameters based on the optimal design method.This optimization is a multi-objective optimization to balance pump efficiency and cavitation.The optimization objectives are minimized ProfileLoss_t and maximized Pmin respectively, as shown in Figure7.

Figure 8 .
Figure 8. Optimization results of meridional genetic algorithm optimization.Table4shows the comparison of key parameters of different meridionals.Through comparative analysis, it is found that No. 10 meridional inlet diameter is the largest, whose value D1 is 272.7mm, which is 20.7mm larger than the baseline inlet diameter.The inlet diameter of No. 38 meridional is slightly larger than the baseline inlet diameter, and its value D1 is 257.7mm.No. 44 meridional inlet diameter is the smallest, D1 is 248.7mm.The axial length of the meridional of all three is longer than baseline, among which the longest length of No.38 and No.44 is 64mm, 13mm larger than that of the baseline.Additionally, The output width of the three impellers is less than the normal.Combined with Figure7, it can be seen that a strong positive correlation between the inlet diameter of the impeller D1 and cavitation performance.In other words, the larger the inlet diameter, the better the cavitation performance; the inlet diameter of impeller D1 is adversely connected with pump efficiency.That is, the efficiency decreases with increasing inlet diameter.Table 4. Comparison of key parameters of different meridional planes.

Figure 9 .
Figure 9. Input parameters optimized by Blade loading and their ranges.Figure 10 shows the optimization results of No.44 meridional.The abscissa is ProfileLoss_t and the ordinate is Pmin.The blue hollow point is the convergence result of all No. 44 meridionals, the red dotted line is the Pareto frontier curve of this optimization result, and the triangle is No. 44 Baseline performance.Comparative study revealed that the blade flow loss and minimum pressure at the majority of sites were better than Baseline 44 at the same time, and that the trend was comparable to the results from the two meridian planes mentioned above.An average of three segments were intercepted along the Pareto frontier with Baseline Pmin of No. 44 as the starting point and maximum Pmin as the ending point.From left to right are 785, 737, 881 and 844.In terms of efficiency alone, 785 is the optimal solution; 844 is optimal solution in terms of cavitation alone; Taking into account both efficiency and cavitation, and focusing on cavitation, the result of No. 881 is the optimal solution.The optimization results show that the optimization method is feasible for high specific speed pump and optimization space for Blade loading is large, and the majority of the convergence outcomes are better than their Baseline.
Figure 9. Input parameters optimized by Blade loading and their ranges.Figure 10 shows the optimization results of No.44 meridional.The abscissa is ProfileLoss_t and the ordinate is Pmin.The blue hollow point is the convergence result of all No. 44 meridionals, the red dotted line is the Pareto frontier curve of this optimization result, and the triangle is No. 44 Baseline performance.Comparative study revealed that the blade flow loss and minimum pressure at the majority of sites were better than Baseline 44 at the same time, and that the trend was comparable to the results from the two meridian planes mentioned above.An average of three segments were intercepted along the Pareto frontier with Baseline Pmin of No. 44 as the starting point and maximum Pmin as the ending point.From left to right are 785, 737, 881 and 844.In terms of efficiency alone, 785 is the optimal solution; 844 is optimal solution in terms of cavitation alone; Taking into account both efficiency and cavitation, and focusing on cavitation, the result of No. 881 is the optimal solution.The optimization results show that the optimization method is feasible for high specific speed pump and optimization space for Blade loading is large, and the majority of the convergence outcomes are better than their Baseline.

Table 2 .
Blade loading parameters of reconstructed impeller.

Table 3 .
Non-viscous energy parameters of reconstructed impeller

Table 4 .
Comparison of key parameters of different meridional planes.