Multiobjective Optimization Design and Performance Prediction of Centrifugal Pump Based on Orthogonal Test

In order to improve the hydraulic performance of the centrifugal pump, based on the original model, the optimization mathematical model with the four indexes head, efficiency, shaft power, and pump net positive suction head as objective function was established, and the multiobjective optimization design of the centrifugal pump was carried out by orthogonal test. Based on the L16(4) orthogonal table, 16 sets of orthogonal test schemes were made by selecting the four parameters impeller outlet width, blade inlet angle, blade outlet angle, and cape angle; the flow field numerical simulation was carried out by computational fluid dynamics technique; and the influence order of geometric parameters on optimization indexes was obtained by range analysis..e weight of each test factor on the optimization index was calculated by weight matrix, and a set of optimal schemes was obtained. Based on the external characteristic experimental bench of the IH 65-60-190 chemical centrifugal pump, the simulation values and test values of the prototype pump and the optimization pump were obtained under different working conditions. Under the rated flow, the head was reduced by 17.00%, the efficiency was increased by 9.14%, the shaft power was reduced by 21.50%, the pump net positive suction head was reduced by 16.69%, the curve humpwas eliminated, the performance of centrifugal pumpwas improved, and the feasibility of the weight matrix optimization method was verified. .e particle image velocimetry measurement system was used to measure the relative velocity of the internal media in the centrifugal pump. .e results showed that the optimization pump had no obvious “jet-wake” flow structure, its maximum velocity was less than the prototype pump, the area of low-speed zone was larger than the prototype pump, the efficiency of the centrifugal pump was improved, and the shaft power and pump net positive suction head were reduced. .e reason of the head decrease was analyzed from the internal flow situation, and the accuracy of the design optimization process was proved.


Introduction
Centrifugal pumps are widely used in various fields of national economy and play an important role in the fields of petroleum, chemical industry, national defense, and aerospace [1].For a long time, how to solve the low efficiency of the centrifugal pump, the head flow curve prone to hump, and poor anticavitation performance is an important index to meet the need of the centrifugal pump design [2].Centrifugal pumps are mainly based on the velocity coefficient method, extreme value loss method, criterion screening method, flow field analysis method, and other methods for hydraulic design and model conversion [3][4][5][6].With the development of computer technology, CFD (computational fluid dynamics) technology has become an important means to predict the performance of the pump and shorten the development cycle.Haifeng et al. [7] used the pressure correction method to simulate the turbulent flow in the impeller of two centrifugal pumps, and through comparative analysis of the calculation results and experimental data, provided valuable information for the comparison and selection of the impeller scheme.Yanshu et al. [8] used G-A (genetic algorithm) to predict the performance of the centrifugal pump; compared with the BP (back propagation) network prediction algorithm, the sample training time was reduced, which was more suitable for engineering application.Wang et al. [9] used neural network to improve the fuzzy rules of centrifugal pump control system and achieved better control effect.Tan Lei et al. [10] proposed a new hydraulic design method for 3D guide vane of centrifugal pump, which improved the hydraulic performance of the centrifugal pump under nondesigned working conditions.For the performance of composite impeller centrifugal pump prediction, Shouqi et al. [11] used the improved BP neural network, introduced Levenberg-Marquardt and other optimization algorithms, and discovered that the predicted value and experimental values had good consistency.Lei et al. [3] used the direct and inverse iterative design methods to design the centrifugal pump impeller, which improved the pump working efficiency compared with the traditional design method.Fu et al. [12] established a multiobjective mathematical optimization model with the highest efficiency of the lowspecific speed centrifugal pump, the minimum NPSH r (pump net positive suction head), and the maximum theoretical head, and the genetic algorithm was used to solve the multiobjective mathematical optimization model and achieved good optimization effect.
In this paper, the orthogonal test is used to optimize the structural parameters of the centrifugal pump; combined with CFD numerical simulation, the influence of the main structural parameters of the impeller on the pump head, efficiency, shaft power, and NPSH r is studied, and the optimal combination of parameters is calculated by weight matrix, in order to improve the hydraulic performance of the centrifugal pump.e hydraulic design of the main structural parameters of the impeller is carried out by the velocity coefficient method [13], and the initial design scheme of the impeller is obtained as shown in Table 1.

Centrifugal Pump Hydraulic Design
According to the initial results in Table 1, the threedimensional model of the impeller can be obtained by using CFturbo software, as shown in Figure 1.

Orthogonal Test Factors.
Orthogonal test is a design method of using the orthogonal table to arrange and analyze the multifactor test [14,15].It is from the whole level combination of test factors, the selection of some representative horizontal combination to carry on the test, through the analysis of this part of the test results, to understand the situation of comprehensive test, to find out the optimal level combination.Its basic characteristic is to use partial test instead of comprehensive test, and it is an efficient, fast, and economical experimental design method.
Considering the influence of impeller geometry parameters on the centrifugal pump head, efficiency, shaft power, and NPSH r , the impeller outlet width, inlet angle, outlet angle, and cape angle are selected as test factors, and four levels are set for each factor, as shown in Table 2.
In accordance with the L 16 (4 4 ) orthogonal table, 16 sets of orthogonal test schemes are established, as shown in Table 3.

Grid Division.
PumpLinx is a simulation software for hydraulic simulation of pump research, which can accurately predict pressure, power, cavitation, heat transfer, flow, and other conditions.e impeller three-dimensional model is imported into PumpLinx software, and the unstructured grid can be divided, as shown in Figure 2.
As shown in Figure 2, the grid number of the impeller is 44,935.As a professional fluid mechanical analysis software, PumpLinx software can automatically select the appropriate mesh density for meshing.In order to further verify the   influence of grid number on the numerical simulation performance of the centrifugal pump, the centrifugal pump outlet pressure is measured by changing the grid number, as shown in Figure 3.
As can be seen from Figure 3, with the increase in the grid number, although the outlet pressure of the centrifugal pump has changed, the change range is not large.It is visible that the grid density in the current situation does not have a greater impact on the numerical simulation results.

Boundary Conditions.
Before numerical simulation, the boundary conditions of the centrifugal pump are set.Centrifugal pump boundary conditions are pump rotational speed, pump outlet flow rate, inlet pressure, turbulence model, and basic properties of fluid media.Among them, the pump rotational speed is 2900 r/min, the pump outlet flow rate is 25 m 3 /h, the inlet pressure is 1 atmosphere, and the k − ε model is chosen for turbulence model.e basic properties of the fluid medium include density of the medium, reference temperature, kinematic viscosity, saturated vapor pressure, and volume elastic modulus.e basic parameters of fluid medium are set, as shown in Table 4.

Control Equation.
e control equations of the pump numerical calculation include the continuous equation, momentum equation (Navier-Stokes), and Reynolds equation.e numerical simulation medium of the centrifugal pump is water, which belongs to incompressible fluid, and its motion law is in accordance with the momentum equation.
e incompressible Navier-Stokes equation is the mathematical expression of momentum conservation law, and the tensor form of its differential equation is [16] In ( 1), u i is the velocity component parallel to the axis x i , t is the time, f i is the mass force on the unit volume fluid, ρ is the fluid density, p is the pressure, and v is the kinetic viscosity.

Factors
Corresponding parameters  Advances in Materials Science and Engineering

Orthogonal Test Results.
e 16 sets of experimental data in Table 3 are numerically simulated by PumpLinx software.
e pump head, efficiency, shaft power, and NPSH r are calculated by (2) [17].e orthogonal test results are shown in Table 5: where g is the gravity acceleration, P is the shaft power, ΔP is the inlet and outlet pressure difference, η is the efficiency, and n s is the specific speed.3) is the range analysis formula, E j is the value of a certain factor, K i is the sum of factor values of j levels, k i is the average of each factor, and s i is the range [18,19].In general, the greater the k i , the higher the indicators under this level.e larger the s i , the greater the influence of this factor on the performance of the centrifugal pump:

Range Analysis. Equation (
( Equation ( 3) is used to carry out the range analysis of orthogonal test results.
e analysis results are shown in Tables 6-9.
From the range analysis of Table 6, it can be concluded that the primary and secondary order of influence of each parameter on the pump head is ACDB.
e optimal combination of the pump head as the evaluation criterion is From the range analysis of Table 7, it can be concluded that the primary and secondary order of influence of each parameter on the pump efficiency is ABCD.e optimal combination of pump efficiency as the evaluation criterion is From the range analysis of Table 8, it can be concluded that the primary and secondary order of influence of each parameter on the pump shaft power is ACDB.
e optimal combination of pump shaft power as the evaluation criterion is From the range analysis of Table 9, it can be concluded that the primary and secondary order of influence of each parameter on the NPSH r is ACDB.e optimal combination of NPSH r as the evaluation criterion is

Multiobjective Weighted Matrix Optimization Design
According to the above orthogonal test, the optimal scheme of the four indexes head, efficiency, shaft power, and NPSH r  4 Advances in Materials Science and Engineering is solved, respectively.However, the optimal scheme obtained by this method is independent and does not take into account the optimization scheme of multiple objectives.In order to effectively solve the above problems, based on the orthogonal test results, the weight matrix is used to calculate the influence degree of each test factors on each index, through comparative weight to determine the primary and secondary order of factors, and the optimal scheme is obtained [20].
According to the orthogonal test results of each factor, the matrices are established as shown in the following equation: In ( 4), the greater the value of head and efficiency, the better the control effect, then K ij � k ij ; the smaller the value of shaft power and NPSH r , the better the control effect, then In (5), T i � 1/∑ m j�1 K ij .In (6), S i � s i /∑ l i�1 s i .us, the weight matrix that affects the test index value is obtained as According to the orthogonal test results, the weight matrix is calculated by calculating the weight matrix of each index, and the weight matrix ω avg can be obtained by calculating the arithmetic mean of four indexes; the calculation results are as follows:  Advances in Materials Science and Engineering From the weight matrix ω avg , it can be seen that the primary and secondary order of the influence of each factor on the comprehensive performance of orthogonal test is ACBD, and the weights of factors A 2 , B 2 , C 1 , and D 4 are the largest, so that the total optimal scheme can be determined as follows:  Based on the optimal scheme, the impeller is made of polyvinyl chloride (PVC), as shown in Figure 6 As shown in Figure 7, although there are errors in the simulation and experimental results, the simulation values and experimental values of each index before and after optimization are consistent with the trend of the flow variation.

Experimental Verification
e simulation values and the experimental values of each index parameter are in good agreement, which can accurately reflect the trend of the changes with the flow rate.e main reasons for the error of the simulation and experimental values are that the leakage of the mouth ring and the volume loss of the front and rear pump cavity are not considered in the numerical simulation, and the accuracy of the impeller model may also be different in the production process.e simulation values of head and efficiency are greater than the experimental values, the simulation value of shaft power is less than the experimental value, and the simulation value of NPSH r is greater than the experimental value.Before and after optimization of β 2 Z 0.773 are 119.85 and 79.90, respectively.As the blade outlet angle is reduced, the β 2 Z 0.773 < 90 is less than 90 and the curve hump is eliminated [11].
e experimental results before and after optimization of each index under different conditions are extracted from Figure 7, as shown in Table 10.
As shown in Table 10, at the rated flow, the pump head is reduced by 17.00%, the efficiency is increased by 9.14%, the shaft power is reduced by 21.50%, and the NPSH r is reduced by 16.69%.Compared with the prototype pump, the optimization pump reduces the width of the impeller outlet, improves the efficiency, and reduces the head.In addition, because the optimization model reduces the blade outlet angle, the shaft power is reduced as a whole.e cape angle of optimization pump is increased, which increases the flow channel area, reduces the blade blocking, improves the fluid flow efficiency, and improves the anticavitation performance of the centrifugal pump.In summary, although the optimization results of individual indicators is not ideal, the performance of other indicators is significantly improved; it can be concluded that the weight matrix optimization method is feasible, and the

PIV (Particle Image Velocimetry) Test on Internal Flow
Field. e changes of external characteristics of the centrifugal pump and NPSH r are related to the internal flow field, and the internal flow field change is the intrinsic factor of the above characteristics.In order to further analyze the changes of flow field in the centrifugal pump before and after optimization, the centrifugal pump flow field test platform is built based on PIV measurement system, as shown in Figure 8. e PIV system mainly includes pulse laser, CCD camera, image acquisition card, lens group, and encoder.
At fixed speed (2900 r/min), the flow field of centrifugal pump before and after optimization under rated flow condition is measured, and the relative velocity vector diagrams and cloud diagrams of the prototype pump and the optimization pump are shown in Figures 9 and 10, respectively.e fluid area between the two blades is shown in   Advances in Materials Science and Engineering  From Figures 9 and 10, the relative velocity of cross section in the impeller channel flows along the blade profile direction, the relative velocity of fluid from the center of the impeller to the edge of the impeller is gradually increased, the change of gradient is relatively flat, and there is no obvious "jet-wake" flow structure.e inlet of blade pressure surface of the prototype pump and the optimization pump has low velocity area, the relative velocity of working face is less than the back surface, and the maximum speed appears at the outlet of the back surface.e relevant data are extracted 5th page through Figures 9 and 10, the maximum velocity of the prototype pump is 30.8466m/s, the maximum velocity of the optimization pump is 28.7978 m/s, and the optimization pump velocity is 6.64% smaller than the prototype pump.Because the optimization pump adopts the smaller impeller outlet width and the blade outlet angle, the optimization pump speed zone is larger than the prototype pump, the medium is more easily controlled in the flow channel, and the efficiency of centrifugal pump is improved significantly.In addition, when the optimization pump low-speed zone area is large, there will be a certain hydraulic loss and reduction in the head, which is consistent with the above external characteristics experimental results.At the same time, when the optimization pump blade inlet angle is small, the flow of internal media in centrifugal pump is more stable, and to some extent, the anticavitation resistance of centrifugal pump is improved, and the cavitation phenomenon is reduced.

Conclusion
(1) e velocity coefficient method was used to design the hydraulic design of the centrifugal pump, the main structure parameters of the impeller were obtained, and the three-dimensional model of the impeller was established by CFturbo software.(2) e 16 sets of experimental schemes were designed by orthogonal test, the internal flow field of centrifugal pump was simulated by PumpLinx software, the weight matrix was used to optimize the head, efficiency, shaft power, and NPSH r , and the optimal scheme was obtained.(3) To build the external characteristics test of the centrifugal pump, the simulation values and experimental values before and after optimization under different working conditions were obtained.e simulation values and experimental values of each index are consistent with the law of flow variation, the relative error of simulation value and experimental value was relatively small, the optimization of pump indexes except head had significant improvement, the curve hump was eliminated, and the feasibility of the weight matrix optimization method was verified.(4) Based on the PIV test system, the experimental results show that there was no obvious "jet-wake" flow structure in the optimization pump, its maximum velocity was less than the prototype pump, the area of Advances in Materials Science and Engineering low-speed zone was larger than the prototype pump, the efficiency and anticavitation performance of centrifugal pump were improved, the head and shaft power were reduced, and the accuracy of the optimization design process was proved.(5) Taking the centrifugal pump as the research object, the hydraulic performance of the centrifugal pump was improved by orthogonal test, which laid a foundation for the research of hydraulic design and external characteristics of the centrifugal pump.However, due to the limitations of the research conditions, there were still some deficiencies that need to be improved.In this paper, the weight matrix method was used to optimize the orthogonal test results, the influence of the impeller geometric parameters on the comprehensive performance of centrifugal pump was obtained by calculating the weight, and a set of optimal scheme was obtained.In the experimental results, although the performance of other indicators was significantly improved, the head was greatly reduced, which had a greater impact on the efficiency of pump.erefore, in the followup work, the influence of the main geometric parameters of the impeller and shell on the hydraulic performance of the centrifugal pump will be comprehensively considered, in order to improve the overall performance of the centrifugal pump.

e
basic parameters of the centrifugal pump are flow rate Q � 25 m 3 /h, head H � 50 m, and rotational speed n � 2900 r/min.

5. 1 .
External Characteristics Test.In order to verify the feasibility of the optimization scheme, the external characteristics test bench is set up by the IH 65-60-190 chemical centrifugal pump, as shown in Figures 4 and 5. e IH 65-60-190 chemical centrifugal pump is a single-stage singlesuction cantilever centrifugal pump, which can transport corrosive liquid.e chemical pump mainly consists of the pump body, blade transmission line, sealing ring, impeller nut, pump cover, sealing components, intermediate bracket, shaft, suspension components, and so on.
. e external characteristics of the centrifugal pump are measured, respectively, under the 6 operating conditions 0.2Q d , 0.4Q d , 0.6Q d , 0.8Q d , 1.0Q d , and 1.2Q d (where the rated flow Q d � 25 m 3 /h), and the external characteristics curves of the simulated and experimental values of the centrifugal pump before and after optimization are shown in Figure 7.

Figure 10 :
Figure 10: Optimization pump relative speed velocity (a) vector diagram and (b) cloud diagram.

Table 1 :
e main structural parameters of the impeller.

Table 2 :
Level table of orthogonal test factors.

Table 4 :
Basic parameters of fluid media.

Table 8 :
Influence analysis of parameters on the shaft power.

Table 9 :
Influence analysis of parameters on the NSPH r .

Table 7 :
Influence analysis of parameters on the efficiency.

Table 10 :
Experimental results before and after optimization of each index.