optimization on the impeller of a low-specific-speed

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 29,aNo. 5,a2016

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DOI: 10.3901/CJME.2016.0519.069, available online at www.springerlink.com; www.cjmenet.com

Optimization on the Impeller of a Low-specific-speed Centrifugal Pump for Hydraulic Performance Improvement

PEI Ji*, WANG Wenjie, YUAN Shouqi, and ZHANG Jinfeng

National Research Center of Pumps and Pumping System, Engineering and Technology, Jiangsu University, Zhenjiang 212013, China

Received September 29, 2015; revised May 18, 2016; accepted May 19, 2016

Abstract: In order to widen the high-efficiency operating range of a low-specific-speed centrifugal pump , an optimization process for

considering efficiencies under 1.0Qd and 1.4Qd is proposed. Three parameters, namely, the blade outlet width b2, blade outlet angle β2,

and blade wrap angle φ, are selected as design variables. Impellers are generated using the optimal Latin hypercube sampling method.

The pump efficiencies are calculated using the software CFX 14.5 at two operating points selected as objectives. Surrogate models are

also constructed to analyze the relationship between the objectives and the design variables. Finally, the particle swarm optimization

algorithm is applied to calculate the surrogate model to determine the best combination of the impeller parameters. The results show that

the performance curve predicted by numerical simulation has a good agreement with the experimental results. Compared with the

efficiencies of the original impeller, the hydraulic efficiencies of the optimized impeller are increased by 4.18% and 0.62% under 1.0Qd

and 1.4Qd, respectively. The comparison of inner flow between the original pump and optimized one illustrates the improvement of

performance. The optimization process can provide a useful reference on performance improvement of other pumps, even on reduction

of pressure fluctuations.

Keywords: low-specific-speed centrifugal pump, optimization, optimal Latin hypercube sampling, surrogate model,

particle swarm optimization algorithm, numerical simulation

1 Introduction

The low-specific-speed centrifugal pump with a specific speed between 30 and 80 is widely used in the agricultural irrigation, petroleum, and chemical industry. According to the traditional design method, the impeller passage becomes very narrow and long to meet the low flow rate and high head requirement. Dealing with operating problems, namely, low efficiency and overpower likelihood, is difficult. The low efficiency of the low-specific-speed centrifugal pump is mainly caused by the large disk friction loss. The pumps are required to efficiently work under different operating points to reduce energy loss because of the limited area and frequent adjustment for pump operations. Therefore, the development of a pump that can meet performance requirements under several operating points has received increasing attention in the market.

An optimization technique has been rapidly applied to

* Corresponding author. E-mail: [email protected] Supported by Jiangsu Provincical Natural Science Foundation of

China(Grant No. BK20140554), National Natural Science Foundation of China(Grant No. 51409123), China Postdoctoral Science Foundation (Grant No. 2015T80507), Innovation Project for Postgraduates of Jiangsu Province, China(Grant No. KYLX15_1066), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China (PAPD).

© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2016

improve the turbomachinery performance with numerical simulation in recent years. To some extent, turbomachinery optimization can reduce design time and experimental cost. First, the optimization design is to focus on investigating the effect of a single variable on pump performance. TAN, et al[1], numerically and experimentally analyzed different blade wrap angles on the pump performance. SHI, et al[2], investigated the influence of the impeller outlet width on the performance of a deep-well centrifugal pump. The second optimization design is the design of experiment (DOE). This optimization is used to find the importance level of the design parameters with respect to the optimization target and obtain the best combination of design variables. KIM, et al[3–4], applied the factorial design to optimize the pump impeller. They also analyzed the effect of the impeller’s parameters on efficiency using the response surface method(RSM). KIM, et al[5], further optimized the efficiency and cavitation of a mixed-flow pump by combining the DOE with a regression analysis. NATARAJ, et al[6]

, applied Taguchi’s method to design impellers and experimentally analyze the effect of the impeller’s parameters on pump performance. YUAN[7] applied the orthogonal experimental design method to solve the problems of the low-specific-speed centrifugal pumps with low efficiency and overload. ZHOU, et al[8], conducted an orthogonal experiment to improve the

CHINESE JOURNAL OF MECHANICAL ENGINEERING

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efficiency of a centrifugal pump by optimizing the impeller. They also analyzed the importance order of the parameters on performance.

The third optimization method is the combination of the DOE and optimization algorithms, which is widely applied to find the best combination of the parameters and make the best compromise between several controversial objectives. TANI, et al[9], proposed the multi-objective optimization of the low-specific-speed impeller by modifying the inlet and outlet blade angles. DERAKHSHAN, et al[10], applied a global optimization method based on the artificial neural networks(ANNs) and the artificial bee colony algorithm to optimize the impeller geometry of a Berkeh 32-160 pump and improve its performance. GOEL, et al[11], improved the performance of a diffuser in a turbopump using a surrogate model to select the best vane shape defined using Bezier curves and circular arcs. KIM, et al[12], conducted an optimization process based on the factorial design, RSM, and numerical simulation to improve the performances of the impeller and the diffuser of a mixed-flow pump. YANG, et al[13], applied inverse design method, response surface model, and multi-objective evolutionary algorithm to find the best blade shape for the high head and efficiency requirement of the pump–turbine impeller. WANG, et al[14], coupled the multi-objective genetic algorithm with the back propagation neural network to optimize a transonic axial compressor. CRAVERO, et al[15], also applied an optimization design approach combining DOE, RSM, and ANNs in a four-stage axial flow turbine. BONAIUTI, et al[16], proposed a multi-point optimization on a compressor by combining 3D inverse design, RSM, and numerical simulation. This method can be applied to the multi-discipline optimization. TOAL, et al[17], optimized a RAE-2822 airfoil under two operating points using the Cokriging model and CFD. JU, et al[18], applied the DOE, ANN, and genitive algorithm and simulation to optimize an industrial axial compressor cascade. The total pressure loss coefficient decreased by 1.54%, 23.4%, and 7.87% at three operating points. NEMEC, et al[19], presented the optimization of the B-spline based parameterizations of an airfoil considering multi-operating conditions.

This paper proposes an optimization procedure combining the DOE, surrogate models, and particle swarm optimization algorithm(PSOA) with numerical simulation to improve the efficiencies of a low-specific-speed centrifugal pump under 1.0Qd and 1.4Qd. The experimental performances of the original and optimized pumps are compared in the open test rig. The internal flow characteristic is then deeply analyzed to illustrate the performance improvement of the low-specific-speed centrifugal pump.

2 Pump Parameters and CFD Simulation 2.1 Pump parameters

The model pump is a typical IS centrifugal pump with specific speed ns (ns = 3.65nQ0.5/H0.75), which is 46.5.

The required performance parameters are as follows: design flow rate Qd is 6.3 m3/h, head H is 8 m, and rotating speed n is 1450 r/min. Table 1 presents the main geometrical parameters of the impeller.

Table 1. Main geometrical parameters of impeller

Parameter Value

Inlet diameter Dj /mm 50

Outlet diameter D2 /mm 160

Blade outlet width b2 /mm 6

Blade inlet angel β1 /(°) 24

Blade outlet angel β2 /(°) 30

Blade wrap angle φ /(°) 150

Optimization only aims to improve hydraulic efficiencies

at two operating points. Therefore, leakage loss is not considered in the computational domain of the low- specific-speed centrifugal pump(Fig. 1). The suction length is extended to five times as long as the impeller’s inlet diameter, and the pipe with a length four times longer than the volute’s outlet diameter is added to the volute to avoid backflow from occurring and extending to the whole inlet pipe and the recirculation at the outlet pipe.

Fig. 1. Computational domain of the centrifugal pump

2.2 Mesh and CFD settings

The quality and number of the mesh are very important to consider before conducting the numerical simulation, especially for the pump optimization process. Compared with the unstructured mesh, the structured mesh can increase the convergence speed. The computational domains are generated with the structured grids using the ICEM software(Fig. 2). The grids are refined in the near-wall flow regions. The mesh independence is checked using several numbers of grids[20]. The total number of grids is about 3.1 million.

Considering a 3D and viscous flow in the centrifugal pump, the 3D Reynolds-averaged Navier-Stokes equations are solved with the shear stress transport(SST) k–ω turbulence model with the commercial code ANSYS CFX 14.5. The multireference frame technique is applied in the steady simulation. The inlet and outlet boundary conditions are set as the total pressure and mass flow rate, respectively. The interface between the rotor and the stator is defined as

PEI Ji, et al: Optimization on the Impeller of a Low-specific-speed Centrifugal Pump for Hydraulic Performance Improvement

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the “frozen rotor.” The physical time scale is set to 0.1/ω, where ω is the impeller angular velocity. The root mean square residual values of the momentum and mass are set as less than 10−5. The performance results are obtained after approximately 500 iterations. The boundary conditions in the transient simulation are similar to that in the steady simulation. The interface between the rotor and the stator is changed to “transient rotor stator.” The accuracy of discretization in space is set to the second order. The second-order backward Euler scheme is used for time discretization. The total calculating time for the pump is 12 revolutions. Each timestep is 1.149 ´ 10−4 s, which corresponds to the rotating angle of 1°. The results of the three final revolutions are used to analyze the pressure fluctuation distributions in the pump. The steady simulation is used to calculate the pump performance, while the transient simulation is conducted to compare with the inner flow characteristics of the original pump and the optimized one. The numerical simulations are conducted on a Dell Workstation with an Intel Xeon CPU E5-1650(12 CPUs) and a clock speed of 3.2 GHz. 3 Experimental Validation

The performance predicted by the numerical simulation is necessarily validated by the experiment before the

optimization process is conducted. An open test rig is built to test the performance of the low-specific-speed pump (Fig. 3). The impeller is made of ABS materials, which is a good method of reducing the manufacturing time. The volute is made of polymethyl methacrylate(Fig. 4). The pressures at the inlet and outlet are measured by the WT200 pressure transducers with a precision of 0.1%. The flow rates are measured by the IFC 300 electromagnetic flowmeter with a precision of 0.3%. Fig. 5 shows a comparison of the simulated and experimental heads. The trend of the head predicted is similar to that of its experimental counterpart. The prediction deviations of the head are 3.2% and 4.3% under 1.0 Qd and 1.4 Qd, respectively. Therefore, the numerical simulation used in the optimization process is reliable.

Fig. 2. Structured mesh of the centrifugal pump

Fig. 3. Diagram of the open test rig

Fig. 4. Hydraulic components Fig. 5. Validation of the head

4 Optimization Process Fig. 6 illustrates the optimization process combining

optimal Latin hypercube sampling(OLHS) method, RSM, Kriging model(KRG), PSO, and numerical simulation. First, the design variables with a significant influence on the pump efficiencies are determined according to the design


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targets. Second, the DOE is used to generate different impeller schemes. The impellers are built according to the design space. The mesh generation and steady simulation are then executed to obtain the hydraulic efficiencies of each case. The databases are generated using the design variables and the objectives that contain the hydraulic efficiencies. The surrogate model is then constructed based on the database. The effect of the design variables on the

efficiencies is analyzed by regression analysis according to the response surface function. The best parameter combination is obtained by using the PSO to solve the KRG. The advantage of the optimization technique is that it can improve the efficiencies at two operating points, which significantly reduces the time of finding the best combination and the computational resource.

Fig. 6. Optimization flow chart

4.1 Optimization objective

The optimization aims to maximize the sum of efficiencies at two operating points defined as follows:

1.4 1.41 1

1 1.4

max ,gQ HgQ H

P P

æ ö÷ç ÷= +ç ÷ç ÷çè ø (1)

where ρ and g represent the density and acceleration of gravity, respectively. Q, H and P are calculated by the numerical simulation. Subscripts 1 and 1.4 represent the design point 1.0Qd and the overload condition 1.4Qd. 4.2 Design variables

According to research concerning the effect of various parameters on the performance of the low-specific-speed centrifugal pump[5], the geometric parameters including blade outlet width b2, blade outlet angle β2, and wrap angle φ were selected as the input variables, normalizing with respect to the reference value in Table 2.

Table 2. Range of design variables

Design variable Lower Reference Upper

b2 0.833 1.0 1.333

β2 0.9 1.0 1.1 φ 0.9 1.0 1.1

4.3 Optimal Latin hypercube sampling(OLHS)

Ensuring that the design space by the DOE is feasible and uniform is necessary in constructing the mathematical function approximation with high accuracy. The OLHS

method, which is improved from the Latin hypercube sampling, has great uniformity and space-filling ability. For example, the design points of two variables are randomly generated using the Latin hypercube sampling method without uniformity, whereas the optimal Latin hypercube sampling is more uniform(Fig. 7). Table 3 shows the objective values in the optimization process calculated by CFX at 20 design points sampled by the OLHS. The sum of the efficiencies of the sixth and ninth cases is relatively higher. 4.4 Surrogate models and optimization algorithm

The present study aims to analyze the effect of three design variables(i.e., outlet blade width, wrap angle, and blade outlet angle) on the efficiencies under two operating conditions. This research also aims to obtain the best variable combination using the optimization process. Therefore, two surrogate models, namely, RSM and KRG, are applied. The RSM is applied to investigate the importance level of the variables on the objectives, while the KRG is utilized to find the best impeller parameters. The surrogate model is a mathematic and statistical approach to build the mathematic relationship between the design variables and objectives. The advantage of the method is the reduction of the numerical simulation’s source and time.

The RSM can be written as first-order, second-order, and third-order functions. Increasing the order of the function leads to an increase in the size of the database required. The type of response surface is determined by the number of design variables and design points generated by the DOE. This study designs 20 impeller cases so that a second-order


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polynomial function is used to model the mathematical expression[21]:

20

1 1

,n n

i i j i ij i ji j i j

f x x x x = = ¹

= + + +å å å (2)

where n is the number of design variables, and x denotes the design variables. β0, βi, βj, and βij are the coefficients calculated using the least squares method.

Fig. 7 Schematic diagram of the Latin hypercube sampling method

Table 3. Designed schemes and objective values

Design scheme b2 β2 φ ηh(1.0Qd) ηh(1.4Qd) Sum of ηh

1 1.307 0.942 0.974 0.714 0.779 1.493 2 1.148 1.100 0.932 0.708 0.796 1.504 3 1.070 0.995 1.047 0.753 0.803 1.556 4 1.018 1.079 1.005 0.753 0.805 1.558 5 1.228 1.089 1.016 0.719 0.787 1.506 6 0.860 0.963 1.068 0.796 0.810 1.606 7 0.912 0.984 0.984 0.782 0.810 1.592 8 1.202 1.037 1.100 0.731 0.794 1.525 9 0.833 1.047 1.037 0.797 0.812 1.609 10 0.965 0.974 0.900 0.770 0.806 1.576 11 1.280 1.005 1.026 0.718 0.782 1.5 12 1.175 0.953 0.911 0.735 0.792 1.527 13 1.097 0.932 0.995 0.747 0.800 1.547 14 0.887 0.900 0.953 0.789 0.808 1.597 15 1.333 1.026 0.942 0.707 0.777 1.484 16 1.255 0.921 1.058 0.728 0.786 1.514 17 0.938 1.058 0.921 0.773 0.809 1.582 18 0.992 1.068 1.089 0.760 0.807 1.567 19 1.043 0.911 1.079 0.764 0.801 1.565 20 1.123 1.016 0.963 0.726 0.798 1.524

The KRG calculates the value of the objective function

as a sum of the regression function ( )i ib f x and random

function z(x)[22]:

( ) ( ).i if b f x z x= +å (3)

The random function z(x) is supposed to have mean zero

and covariance. The Gaussian function is used in the KRG:

[ ] 22( ), ( ) exp( ).E z x z y x y = - - (4)

The PSOA is a new approach for global

optimization[23–24]. The algorithm imitates the social behaviors exhibited by swarms of animals (e.g., birds, ants, etc.). A point in the search space is called a particle in the

PSOA. The terms “particle” and “swarm” are similar to “individual” and “population” used in common evolutionary algorithms that contain selection, crossover, and mutation operators. However, the algorithm is simpler and only has one operator comprising several components and moving the particle through the search space. Therefore, the PSOA is utilized to calculate the surrogated model and obtain the objectives of this study. The application of the surrogated models and PSOA is executed on a software platform called Isight. 5 Results and Discussions 5.1 Accuracy of surrogate model

Eqs. (5) and (6) illustrate two second-order response surface functions modeled for the efficiencies under 1.0Qd


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and 1.4Qd according to the numerical results in Table 3.

1 2 3

2 2 21 2 3

1 2

1.0

1 3 2 3

1.417 0.709 0.496 0.846

0.185 0.378 0.207

0.078 0.208 0.259 ,

x x x

x x x

x x x x x x

- + - +

- + -

+

=

+

(5)

1 2 3

2 2 21 2 3

1 2

1.4

1 3 2 3

0.647 0.153 0.256 0.059

0.106 0.124 0.0003

0.026 0.034 0.030 ,

x x x

x x x

x x x x x x

+ + - -

- - -

+

=

+

(6)

where x1, x2, and x3 represent b2, β2, and φ, respectively.

Assessing the accuracy of the response surface functions is very necessary before the optimization with PSOA. Therefore, the R-square error is used to analyze the difference between the efficiencies calculated by the numerical simulation and those predicted by the response surface functions. The R-square value is greater than 0.9, which indicates that the response surface functions can have faithful prediction accuracy. Fig. 8 shows the analysis results of the R-square after the approximation models are constructed using the RSM and KRG. The prediction accuracy of the KRG is higher than that of the RSM. Therefore, the KRG can be applied to obtain a fine mathematical relationship between the objectives and the design parameters combined with the genetic algorithm.

Fig. 8. Analysis of R-square error by the RSM and KRG

5.2 Regression analysis

The regression analysis is employed to investigate the parameters’ importance level on the performance based on the response surface model. The green bar in Fig. 9 illustrates the positive effect on efficiency, while the white bar reflects the negative effect on efficiency. Fig. 9(a) shows the effect of each variable on efficiency under 1.0Qd. The Pareto value of x1 is −52%, which indicates that the blade outlet width has a negative effect on efficiency. However, x12 has a positive effect on efficiency. The Pareto value of x2 is −10%. The blade outlet angle also exhibits a negative effect on efficiency. The Pareto value of x3 is 6%, and the wrap angle exhibits a positive effect on efficiency.

Moreover, the cross-effect of the variables on efficiency is relatively small. Fig. 9(b) illustrates the effect of each variable on efficiency under 1.4Qd. The blade outlet width still has a negative influence on efficiency, while the blade outlet and wrap angles have a positive effect. 5.3 Comparison of optimal parameters

The KRG is used to execute the optimization by comparing the prediction accuracy of two surrogate models. Table 4 shows the optimal design parameters for the centrifugal pump impeller predicted by the KRG. The blade width is optimized to be smaller, while the wrap angle is a little larger. The blade outlet angle is almost the same with the original one. The hydraulic efficiencies under 1.0Qd and


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1.4Qd of the original case obtained by the CFD calculation are 0.766 and 0.807, respectively. The hydraulic efficiencies under 1.0Qd and 1.4Qd of the optimized design are predicted to be 0.799 and 0.812, respectively. The impeller is simulated, and the efficiencies are calculated to be 0.798 and 0.812 according to the optimal impeller parameters, which shows an increase of 4.18% and 0.62%, respectively, compared to the original impeller. Moreover, the two efficiencies obtained by the KRG have a good agreement with those calculated by CFD. The relative variances are 0.01% and 0%, respectively.

Fig. 9. Pareto value of each coefficient on efficiencies under two operating points

Table 4. Result of optimization design

Design Design variable

b2 β2 φ

Original 1 1 1 Optimal 0.833 1.006 1.024

Objective

Design Prediction ηh(1.0Qd)

Prediction ηh(1.4Qd)

CFD ηh (1.0Qd)

CFD ηh (1.4Qd)

Original – – 0.766 0.807 Optimal 0.799 0.812 0.798 0.812

5.4 Comparison of performance

Fig. 10 shows a comparison of the performance curves of the hydraulic efficiencies of the original and optimized pumps, which are calculated by CFD. Considerable improvements in the hydraulic efficiency curves through

optimization are observed when the flow coefficients are less than approximately 0.019. Meanwhile, the hydraulic efficiency values for the optimized pump are only a little higher than those of the original pump under large flow coefficients. The optimization process significantly improves the hydraulic efficiency in the low flow rate region.

Fig. 10. Comparison of efficiency between the original pump and optimized impeller

5.5 Comparison of steady flow The inner flow characteristics of the original pump and

the optimized one are compared to find reasonable factors for the performance improvement of the centrifugal pump with low specific speed(Figs. 11–14). Figs. 11 and 12 show the total pressure distributions under 1.0Qd and 1.4Qd, respectively. The pressure reaches its lowest value at the suction side near the leading edge, where cavitation occurs. The pressure increases along with the passage and reaches the maximum at the trailing edge. The pressure distribution at the trailing edge of the original impeller is less uniform than that in the optimal one. Moreover, the maximum pressure area is larger. The total pressure distributions are improved under the two operation points.

Fig. 11. Total pressure distribution under 1.0Qd


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Fig. 12. Total pressure distribution under 1.4Qd

Fig. 13. Velocity distribution under 1.0Qd

Fig. 14. Velocity distribution under 1.4Qd

The relative velocity distributions under the design point

between the original and optimized pumps are compared (Fig. 13). The low velocity is mainly observed at the two impeller passages near the volute tongue. The flow separation happens on the pressure side, which reduces the pump efficiency. The low velocity area in the original impeller is smaller than that in the optimal one. Therefore, the flow in the optimal impeller can cause less hydraulic loss. Furthermore, the velocity distribution in the throat area of the volute is affected by the impeller. A larger low-velocity area occurs in the throat area of the optimal

pump. The impeller and volute matching should be considered when optimizing the pump performance. Fig. 14 shows the relative velocity distributions under 1.4Qd. The velocity flow fields have improved compared with those under the design flow rate. Consequently, the best efficiency point is moved forward to the large flow rate. The optimization reduces the low velocity area existing on the pressure side of the impeller. 5.6 Comparison of unsteady flow

Fig. 15 shows the frequency spectra of the efficiency fluctuation obtained by the FFT(Fast Fourier Transform). These spectra can directly reflect the operating stability of the low-specific-speed centrifugal pump. Fig. 15(a) presents the frequency spectrum of the efficiency fluctuation under the design point. The frequency components of the efficiency fluctuation principally comprise BPF(Blade Passing Frequency) and its higher harmonics. The peak at the BPF in the original pump is 5.04. This value decreases to 3.23 in the optimal pump. Fig. 15(b) shows that the peak at the BPF is lower than that under the design point. The optimization also increases the peak at the BPF from 3.79 to 2.93. The optimization reduces the periodic unsteadiness under the design point and increases it under 140% of flow rate.

Fig. 15. Frequency spectra of efficiency fluctuation


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The unsteady pressure fluctuation intensity distribution is directly and comprehensively assessed by adopting the statistical method to analyze the pressure fluctuation intensity on each grid node in the fluid domains. The pressure standard deviation is calculated to indicate the pressure fluctuation intensity[25]. A lower pressure standard deviation value indicates smaller pressure fluctuation intensity. The pressure on each grid node can be given as pi = p(x, y, z, t). The pressure fluctuation intensity psdv is defined as follows:

1

1, , ,

360

N

in

np p x y z T

N =

æ ö÷ç= ÷ç ÷çè øå , (7)

2

1

1( , , , )

1 360

N

sdvn

np p x y z T p

N =

æ ö÷ç= - ÷ç ÷çè ø- å , (8)

where T is the period of one impeller revolution, and N = 1080.

Fig. 16 shows a comparison of the pressure fluctuation intensity distribution between the original and optimized impellers. The maximum pressure fluctuation intensity exists at the pressure side near the trailing edge. The area of large pressure fluctuation intensity decreases in the optimal impeller under the design point. However, the pressure fluctuation intensity is larger at the pressure side near the trailing edge of the optimal impeller than that in the original one (Fig. 17).

Figs. 18 and 19 show comparisons of the pressure fluctuation intensity distributions in the volute inlet under the design flow rate and under 140% of the design flow rate, respectively. The rotor-stator interaction causes a large pressure fluctuation in the first volute section. The area of the large pressure fluctuation intensity decreases in the volute, with the optimal impeller evidently under the design point and slightly increasing under overload condition. The hydraulic efficiency improvement can also enhance the periodic flow stability.

Fig. 16 Pressure fluctuation intensity distribution under 1.0Qd

Fig. 17. Pressure fluctuation intensity distribution under 1.4Qd

Fig. 18. Pressure fluctuation intensity distribution at inlet of volute under 1.0Qd


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Fig. 19. Pressure fluctuation intensity distribution at inlet of volute under 1.4Qd

6 Conclusions

(1) The KRG can obtain a fine relationship between the objectives and the design parameters. The KRG can also be used to predict efficiencies and power. The optimization improves hydraulic efficiencies at increases of 4.18% and 0.62% under the two operating points.

(2) The pressure distribution at the trailing edge of the optimal impeller is more uniform than that in the original one under two operating points. The velocity flow fields also improve. The unsteady pressure fluctuation intensity in the optimal scheme decreases under the design flow rate. It slightly increases at the pressure side near the trailing edge of the impeller under 140% of the design flow rate. The stability of the periodic unsteady flow illustrates the performance improvement of the centrifugal pump.

(3) The optimization of the low-specific-speed centrifugal pump is feasible, and can provide suggestions in improving the performance of other pumps. Acknowledgement

The author wishes to thank to the 4CPump research group.

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Biographical notes PEI Ji is currently an associate professor at National Research Center of Pumps and Pumping System, Engineering and Technology, Jiangsu University, China. His research interests include unsteady flow, flow-induced vibration and fluid-structure interaction in turbomachinery. He received his PhD degree from Jiangsu University, China in 2013. Tel: +86-13776474939; E-mail: [email protected] WANG Wenjie is currently a PhD candidate at National Research Center of Pumps and Pumping System, Engineering and Technology, Jiangsu University, China. His research interests include the optimization design and analysis of unsteady flow of centrifugal pump. He received his BS degree from Jiangsu University, China in 2012. Tel: +86-15050850433; E-mail: [email protected] YUAN Shouqi is currently a professor at National Research Center of Pumps and Pumping System, Engineering and Technology, Jiangsu University, China. His research interests include the theory, optimization and design of fluid machinery. He received his PhD degree from Jiangsu University, China in 1994. Tel: +86-13705289318; E-mail: [email protected]

optimization on the impeller of a low-specific-speed

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