Abstract
In order to study the unsteady performance of a twostage selfpriming centrifugal pump, the unsteady numerical calculation in a twostage selfpriming centrifugal pump was performed and energy characteristics experiments and selfpriming experiments were carried out. The pressure pulsation and radial force in the pump were then analyzed. The results show that numerical calculation values are close to the experiment values. Head deviation of the pump is less than 3 %, and efficiency deviation of the pump is less than 2 percentage points. Compared with monitoring point ${P}_{1}$, the pressure fluctuation coefficient of monitoring point ${P}_{3}$ at the design flow rate is reduced by 61 %. Compared with monitoring point ${P}_{8}$, the pressure fluctuation coefficient of monitoring point ${P}_{5}$ is reduced by 70 %. The radial force on the radial guidevane is obviously smaller than that on the volute. Under the same flow rate, radial force on the volute of secondstage pump is almost 20 times larger than that on the radial guidevan of firststage pump.
1. Introduction
Due to its small size, easy maintenance and simple installation, selfpriming centrifugal pumps are widely used in agricultural irrigation, urban environmental protection, fire control, chemical industry and water supply field. With the development of technology and applications of selfpriming centrifugal pumps, multistage selfpriming centrifugal pumps are made. Because multistage selfpriming centrifugal pump combines the advantages of selfpriming centrifugal pump and multistage centrifugal pump. It can be used in some occasions where either the head of pump is required to be high, the pump and motor installation location is limited, or equipment needs to be started and stopped frequently.
At present the researches about selfpriming centrifugal pump are mainly aimed at the singlestage selfpriming pump [18]. Sato et al. [2] analyzed the influence of impeller blade angles of centrifugal pump on air/water twophase flow performance by using of the numerical simulation and experiment, and found that the larger inlet angle and outlet angle of the impeller are disadvantageous to the air/water twophase flow in the selfpriming pump. John [6] introduced the basic working principle of the selfpriming pump and compared the influence of the volutetype water chamber and the guidevanetype water chamber on the selfpriming performance. Henke [8] installed inducer in the inlet of the selfpriming pump, which reduces the energy consumption and decreases the noise generated during the operation process of the pump.
However, researches about multistage selfpriming centrifugal pump [9, 10] are relatively few, especially unsteady performance and selfpriming experiment of multistage selfpriming centrifugal pump. As to multistage centrifugal pumps, some scholars have always made some research achievements. Yutaka et al. [11] measured the dynamic characteristics of multistage centrifugal pump and researched the unsteady flow in multistage centrifugal pump, especially pressure fluctuation in the flow passage of guide vane and volute. Plutcki et al. [12] studied the single stage of multistage centrifugal pump by numerical simulation and analyzed the flow law of fluid in guide vane with ballshape surface. Cukurel et al. [13] studied the flow law of fluid in circumferential and axial direction in radial guide vane as well as the variation in load.
Therefore, the main motive of this paper is to analyze the unsteady characteristics of pressure fluctuation and radial force in a twostage selfpriming centrifugal pump with numerical simulation and experimental study.
2. Calculation model and experimental device
2.1. Calculation model
The selfpriming centrifugal pump is required to be installed in the sprinkler. The design parameters of the selfpriming centrifugal pump are as followed. Flow rate ${Q}_{d}$ is 60 m^{3}/h, head $H$ is 105 m, rotation speed $n$ is 3540 r/min, and efficiency $\eta $ is 60 %. When the selfpriming height is 4 m, the selfpriming time of the pump ${T}_{s}$ is less than 180 s.
The selfpriming centrifugal pump used in the sprinkler is designed to be a twostage selfpriming centrifugal pump with a gear case. The head of firststage pump is ${H}_{1}=$ 50 m, and the head of secondstage pump is ${H}_{2}=$ 55 m. The structure of the pump is shown in Fig. 1. Main design parameters of the pump are shown in Table 1.
Fig. 1Structure of the twostage selfpriming centrifugal pump: 1. Firststage impeller, 2. Radial guidevane, 3. Pump body, 4. Secondstage impeller, 5. Mechanical seal, 6. Overdrive gear, 7. Gear shaft, 8. Pump shaft, 9. Gear case
Table 1Main design parameters of the pump
Parameter  Inlet diameter of pump  Inlet diameter of firstimpeller  Outlet diameter of firstimpeller 
Value  75 mm  75 mm  170 mm 
Parameter  Outlet blade width of firstimpeller  Blade outlet angle of firstimpeller  Blade wrap angle of firstimpeller 
Value  14 mm  25 °  110 ° 
Parameter  Blade number of firstimpeller  Inlet diameter of secondimpeller  Outlet diameter of secondimpeller 
Value  6  75 mm  175 mm 
Parameter  Outlet blade width of secondimpeller  Blade outlet angle of secondimpeller  Blade wrap angle of secondimpeller 
Value  13 mm  25 °  110 ° 
Parameter  Blade number of secondimpeller  Inlet diameter of guide vane  Outlet width of guide vane 
Value  6  174mm  17.5 mm 
Parameter  Blade number of guide vane  Outlet diameter of guide vane  Inlet diameter of volute 
Value  7  237 mm  180 mm 
Parameter  Inlet width of volute  Tongue angle of volute  Outlet diameter of pump 
Value  22 mm  29 °  70 mm 
Fig. 2 shows the calculation area of the whole flow field, including inlet extension part, firststage impeller, radial guidevane, firststage pump chamber, secondstage impeller, volute, secondstage pump chamber, gasliquid separation chamber and outlet extension part.
Fig. 2Calculation areas
a) Explosion chart
b) Profile chart
2.2. Experimental device
2.2.1. Energy characteristic test bench
Energy characteristic test device of the twostage selfpriming centrifugal pump is shown in Fig. 3.
2.2.2. Selfpriming test bench
Selfpriming test bench of the twostage selfpriming centrifugal pump is shown in Fig. 4. In order to measure selfpriming time of the twostage selfpriming centrifugal pump, the twostage selfpriming centrifugal pump is placed at a platform whose height is 4 m from the grand. A pipe upward bending is connected at the inlet of the pump, and the pipe is about 300 mm higher than the pump shaft.
Fig. 3Sketch of energy characteristic test bench: 1. Electric motor, 2. Coupler, 3. Speed changing case, 4. Coupler, 5. Pump, 6. Pressure sensor in the outlet, 7. Pressure displayer, 8. Pressure sensor in the inlet, 9. Electromagnetic flowmeter, 10. Flow meter displayer, 11. Sluice valve, 12. Water tank
Fig. 4Sketch of selfpriming test bench: 1. Electric motor, 2. Coupler, 3. Speed changing case, 4. Coupler, 5. Pump, 6. Pressure sensor in the outlet, 7. Pressure displayer, 8. Pressure sensor in the inlet, 9. Sluice valve 10. Water tank
3. Numerical method
3.1. Grid generation
In this paper, calculation area of the whole flow field was divided with the ICEM code. It is one of the key factors which affects the accuracy of numerical simulation and the calculation time, concerning whether grid generation is reasonable or not. Due to the complexity of the pump structure, the tetrahedral grid with strong adaptability is used during grid generation. Too few meshes will affect the accuracy of the calculation results. Too many meshes will increase the workload and consume too much calculation time. In order to ensure the accuracy of the calculation and save time, the grid independence check is performed. CFX software was used to simulate the centrifugal pump numerically, and the results are shown in Table 2.
Table 2Grid independence analysis
No.  1  2  3  4  5  6 
Grid number  3845623  4581564  6862158  7912235  8516352  10013496 
Head /m  105.89  106.08  106.38  106.98  107.04  106.56 
As seen from Table 2, although the difference in the total grid numbers is large, the deviation of head is within 0.5 %. As a result, the simulation results are stable. Considering the accuracy and time of the simulation, the total number of 6862158 is selected to do the next numerical simulation.
3.2. Boundary conditions
The inlet boundary condition was set as a standard atmospheric pressure (1 atm), which assumed that the flow velocity in the inlet section is uniformly distributed. The outlet boundary condition is set as the mass flow rate. All physical surfaces were set as noslip wall and the nearwall regions were disposed with standard wall functions method. That is, the component of time indicated that velocity and the pulse velocity in all directions were zero.
3.3. Arrangement of monitoring point
According to the rotation speed of the impeller, the time step is set to 4.7081×10^{5}^{}s. That is, 1 degree which the impeller rotated was chosen as a time step. In order to ensure the accuracy of analysis, the impeller was set to rotate 6 cycles, and the sixth period of the calculation results was selected to be analyzed.
In order to study the pressure fluctuation in the flow channel of radial guidevane and volute under different flow rate, the monitoring points ${P}_{1}$, ${P}_{2}$, ${P}_{3}$, ${P}_{4}$, ${P}_{5}$, ${P}_{6}$, ${P}_{7}$, ${P}_{8}$ and ${P}_{9}$ are shown in Fig. 5.
Fig. 5Distribution of monitoring point
a) Radial guidevane
b) Volute
4. Results and analysis
4.1. Comparison with numerical simulation and experiment
Under the design flow rate, the experiment value of head of the twostage selfpriming centrifugal pump is 108.65 m, and the efficiency of the pump is 60.58 %. Standard $k$$\epsilon $ model, RNG $k$$\epsilon $ model, Standard $k$$\omega $ model and SST $k$$\omega $ model is used to simulate the internal flow of the twostage selfpriming centrifugal pump. Numerical results are shown in Numerical results are displayed in Table 3.
Table 3Comparison of different turbulence models
Turbulence model  Head  Efficiency /%  
Calculation value / m  Relative error / %  Calculation value  Absolute error  
Standard $k$$\epsilon $  106.29  –2.17  61.27  0.69 
RNG $k$$\epsilon $  106.46  –2.01  61.15  0.57 
Standard $k$$w$  107.11  –1.42  61.08  0.50 
SST $k$$w$  107.65  –1.03  60.99  0.41 
Which shows the slightly smaller or larger calculation of head values with four turbulence models in comparison with experiment head value and calculation efficiency value. When comparing experiment data, calculation error with SST $k$$\omega $ model is minimum. In accordance, SST $k$$\omega $ model is selected to predict energy characteristics and analyze transient performance of the twostage selfpriming centrifugal pump.
The numerical simulation and experiment of the twostage selfpriming centrifugal pump were carried out, and the results are shown in Fig. 6. As can be seen from Fig. 6, the numerical results are close to the experiment values. The head deviation is less than 3 %, and the efficiency deviation is less than 2 percentage points. The results show that the numerical simulation method is reliable and can accurately predict the energy characteristics of the twostage selfpriming centrifugal pump. It can be seen from Fig. 6 that the high efficiency area of the pump is relatively wide. From the flow rate of 52 m^{3}/h to 85 m^{3}/h, efficiency of the pump is higher than 58 %.
In order to improve the accuracy and repeatability of the data, selfpriming time of the pump was measured three times. When the selfpriming height is 4 m, the selfpriming time was 154 s, 155 s and 153 s. So, the selfpriming time of the twostage selfpriming centrifugal pump is 154 s, which meets the requirements of selfpriming.
Fig. 6Comparison with numerical simulation and experiment
4.2. Velocity distribution
Fig. 7 shows the relative velocity distributions of middle section in the firststage impeller under 0.8${Q}_{d}$, 1.0${Q}_{d}$ and 1.2${Q}_{d}$. The relative velocities near pressure surfaces of the blades are obviously lower than that near suction surfaces of the blades. With the increase of flow rate, the relative velocity in flow channel of the firststage impeller gradually increases. Under the 0.8${Q}_{d}$,_{}vortices appear near the inlet of firststage impeller and the relative velocity distribution in each channel of the impeller outlet is uneven. With the increase of the flow rate, the vortices decrease gradually and the relative velocity in each channel of the impeller outlet becomes welldistributed.
Fig. 7The relative velocity distributions of middle section in the firststage impeller
a) 0.8${Q}_{d}$
b) 1.0${Q}_{d}$
c) 1.2${Q}_{d}$
Fig. 8 shows the absolute velocity distributions of middle section in firststage impeller and guidevane under 0.8${Q}_{d}$, 1.0${Q}_{d}$ and 1.2${Q}_{d}$. The absolute velocities in the inlet of firststage impeller are smaller and the velocity distributions are more uniform. Under the 0.8${Q}_{d}$, the absolute velocity in the channel of the firststage impeller is larger than that in the positive of radial guidevane. And vortices appear in the radial guidevane. With the increase of the flow rate, the vortices in the radial guidevane decrease gradually.
Fig. 9 shows the relative velocity distributions of the secondstage impeller under 0.8${Q}_{d}$, 1.0${Q}_{d}$ and 1.2${Q}_{d}$. The relative velocity distributions in flow channel of firststage and secondstage impeller is basically the same. Compared with the firststage impeller, the relative velocities in the secondstage impeller are lager. The vortices near the impeller inlet are obviously reduced. At the same time, the uneven distribution of velocity at the impeller outlet is relieved.
Fig. 10 shows the absolute velocity distributions in the volute and gasliquid separation chamber under 0.8${Q}_{d}$, 1.0${Q}_{d}$ and 1.2${Q}_{d}$. It can be found in Fig. 10 that the absolute velocity distributions in flow channel of secondstage and firststage impeller are basically the same.
Fig. 8Absolute velocity distributions of middle section in firststage impeller and guidevane
a) 0.8${Q}_{d}$
b) 1.0${Q}_{d}$
c) 1.2${Q}_{d}$
Fig. 9The relative velocity distributions of middle section of the secondstage impeller
a) 0.8${Q}_{d}$
b) 1.0${Q}_{d}$
c) 1.2${Q}_{d}$
Fig. 10Absolute velocity distributions of middle section of volute and gasliquid separation chamber
a) 0.8${Q}_{d}$
b) 1.0${Q}_{d}$
c) 1.2${Q}_{d}$
Due to the influence of volute tongue, there are differences in each channel of impeller, and the absolute velocity in the channel close to the volute tongue is lager. Because the pump is a selfpriming pump, the fluid flowing out of the volute outlet enters the gasliquid separation chamber. Due to the structure restriction of gasliquid separation chamber, the liquid impacts the wall which generates vortices. The gasliquid separation chamber is essential in selfpriming pump, so the loss of efficiency is inevitable, which is one of the reasons for the relatively lower efficiency of selfpriming pump.
4.3. Pressure fluctuation analysis
In order to further analyze pressure pulsation and radial force of the twostage selfpriming centrifugal pump, the unsteady numerical simulation of internal flow in the pump was carried out.
Due to the highspeed rotation of the impeller, periodic rotorstator interference can be caused at the boundary condition between the outlet of firststage impeller and inlet of radial guidevane, which resulted in pressure pulsation in the flow field of radial guidevane.
4.3.1. Analysis of time domain of pressure fluctuation
Fig. 11 shows time domain diagram of pressure fluctuation in 9 monitoring points. The horizontal coordinate $\theta $ represents angle which the impeller rotates 1 cycle and the longitudinal coordinate $P$ is the static pressure of the monitoring points.
Fig. 11Time domain diagrams of pressure fluctuation at monitoring points
a) 0.8${Q}_{d}$
b) 1.0${Q}_{d}$
c) 1.2${Q}_{d}$
For monitoring point ${P}_{4}$_{}in the outlet of radial guidevane, the average value of pressure fluctuation at ${P}_{4}$ under the 0.8${Q}_{d}$ is larger. With the increase of the flow rate, the pressure fluctuation at the outlet of the radial guidevane gradually reduces. This is consistent with the change of pressure in the flow channel of positive guide vane. It as well suggested that the pressure pulsation of the monitoring point ${P}_{4}$ does not have obvious periodicity, which is because the monitoring point ${P}_{4}$ is far from firststage impeller outlet and the influence of the rotorstator interference on the monitoring point ${P}_{4}$ is small.
For monitoring point ${P}_{5}$, ${P}_{6}$, ${P}_{7}$, ${P}_{8}$ and ${P}_{9}$ in volute, pressure fluctuation in volute under different flow rate presents obvious periodicity. Six peaks and six troughs of pressure fluctuation have appeared in volute, which are the same with the blade number of secondstage impeller. This is consistent with the pressure pulsation regularity in the radial guidevane of firststage pump. Compared with the pressure fluctuation in the radial guidevane, the pressure pulsation in volute is significantly larger than that in radial guidevane. Under the same flow rate, the pressure increases gradually from the volute tongue to the diffuser section. The pressure at the monitoring point ${P}_{9}$ in the diffuser section of volute is obviously higher than that at other monitoring points in volute. With the increase of the flow rate, the pressure in volute gradually reduces, and the difference value of the pressure between the three monitoring points is gradually reduced. It can clearly be seen that under the larger flow rate, when impeller blade moves through volute tongue, it causes great pressure fluctuations in the monitoring point ${P}_{5}$. Under the small flow rate, this phenomenon is not obvious.
4.3.2. Analysis of frequency domain of pressure fluctuation
In order to quantitatively analyze the pressure fluctuation in the positive guide vane flow channel of radial guide vane, the pressure fluctuation coefficient is defined as followed:
where $\mathrm{\Delta}p$ is the difference between instantaneous pressure and mean value, $\rho $ is fluid density, ${u}_{2}$_{}is circumferential velocity of firststage impeller outlet.
The pressure fluctuation coefficients were calculated with the Eq. (1). And the frequency domain diagram of the pressure fluctuation coefficients at monitoring points were obtained with fast Fourier transform, as shown in Fig. 12.
As it can clearly be seen from the Fig. 12(a), 12(b) and 12(c), pressure fluctuations at the monitoring points ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$ are not obvious in the highfrequency region, while there was a significant pressure pulsation in the lowfrequency region. The maximum value of pressure fluctuation coefficient corresponds to the frequency of about 354 Hz, which is the passing frequency of firststage impeller blades. And with the decrease of the flow rate, the pressure fluctuation coefficient in the low frequency region at the same monitoring point gradually becomes larger. For example, the pressure fluctuation coefficient at the monitoring point ${P}_{1}$ increases from the 1.2 ${Q}_{d}$ of 0.041 to the 0.8 ${Q}_{d}$ of 0.050, which causes the pressure fluctuation coefficient to increase by 21 %. Under the same flow rate, the pressure fluctuation coefficient at the monitoring point ${P}_{1}$ is the maximum, and the pressure fluctuation coefficient at the monitoring point ${P}_{3}$ is the minimum. For example, under the design flow rate, the pressure fluctuation coefficient at the monitoring point ${P}_{1}$ is 0.044, and the pressure fluctuation coefficient at the monitoring point ${P}_{3}$ is reduced to 0.017, which is equivalent to a reduction of 61 %. This is because the monitoring point ${P}_{1}$ is located in inlet of stationary radial guidevane close to the outlet of rotating firststage impeller. Thus, the effect of the rotorstator interference at the monitoring point ${P}_{1}$ is the biggest. With the monitoring location moving away, the effect of the rotorstator interference gradually becomes smaller.
It can be seen from Fig. 12(d) to 12(h) that pressure pulsation tendency in the volute and are basically the same in the positive guidevane of radial guidevane. The pressure fluctuation coefficient decreases with the increase of flow rate. At the monitoring point ${P}_{5}$, pressure fluctuation coefficient is reduced from the 0.8${Q}_{d}$ of 0.058 to the 1.2${Q}_{d}$ of 0.028. And the frequency corresponding to the maximum value of the pressure fluctuation coefficient at monitoring points in the volute is all about 354 Hz, which is the pass frequency of the secondstage impeller blade. Under the same flow rate, the pressure fluctuation at the monitoring points in the spiral flow channel of volute is gradually reduced with moving away from the volute tongue. Under the design flow rate, the pressure fluctuation coefficient is reduced from the monitoring point ${P}_{5}$ of 0.039 to the monitoring point ${P}_{8}$ of 0.011, which is reduced by about 70 %. After fluid moving into the diffusion section of the volute, the pressure and the pressure fluctuation coefficient increase rapidly.
Fig. 12Frequency domain diagrams of pressure fluctuation at monitoring points
a)${P}_{1}$
b)${P}_{2}$
c)${P}_{3}$
d)${P}_{5}$
e)${P}_{6}$
f)${P}_{7}$
g)${P}_{8}$
h)${P}_{9}$
4.4. Radial force analysis
Time domain diagrams of radial force on the wall of radial guidevane at the 0.8${Q}_{d}$, 1.0${Q}_{d}$, 1.2${Q}_{d}$ are shown in Fig. 13. It can be seen from Fig. 13 that the radial force on the wall of the radial guidevane is obviously periodic. With the increase of flow rate, the radial force on the wall of the guidevane gradually reduces. Under different flow rate, the radial forces on the wall of the radial guidevane are all small. It can only be 125 N under the small flow rate. This is because the radial guidevane is a central symmetrical hydraulic component. The radial force on the wall of radial guidevane generated by the fluid is mostly offset.
Fig. 13Time domain diagrams of radial force on the wall of radial guidevane
Time domain diagrams of radial force on the wall of volute under different flow rate are shown in Fig. 14. It can be seen from Fig. 14 that compared with the wall of the radial guidevane, radial force on the wall of volute is larger. This is because the volute is a noncentral symmetrical hydraulic component. Under the 0.8${Q}_{d}$, the average radial force on the wall of volute is about 2240 N. And with the increase of flow rate, the radial force on the wall of volute is gradually getting smaller and smaller, which is only about 1990 N under the 1.2${Q}_{d}$. At the same time, the fluctuation of the radial force also decreases with the increase of flow rate.
It can also be seen from Fig. 13 and Fig. 14 that radial force on the wall of volute is nearly 20 times larger than that on the wall of the radial guidevane. So, the central symmetrical hydraulic component can offset the radial force generated by the fluid flow.
Fig. 14Time domain diagrams of radial force on the wall of volute
5. Conclusions
Through analyses on numerical results and experiment data in a twostage selfpriming centrifugal pump, the following conclusions can be obtained.
1) Under the design flow rate, experiment head of the pump is 108.65 m and experiment efficiency of the pump is 60.58 %. High efficiency area of the pump is relatively wide. Efficiencies of the pump are all higher than 58 % in the flow range of 52 m^{3}/h to 85 m^{3}/h. When the selfpriming height is 4 m, the average selfpriming time of the pump is 154 s, which meets the design requirements of the selfpriming centrifugal pump.
2) The numerical simulation method in this paper is reliable and can accurately predict the energy characteristics of the twostage selfpriming centrifugal pump. The numerical calculation results with SST $k$$w$ turbulence model is close to the experiment values, head deviation is less than 3 %, and efficiency deviation is less than 2 percentage points.
3) With the monitoring point moving away from inlet of the radial guidevane, pressure fluctuation coefficients in the guide vane decrease. Compared with monitoring point ${P}_{1}$, the pressure fluctuation coefficient at monitoring point ${P}_{3}$ under the design flow rate is reduced by 61 %. The pressure fluctuation coefficient in the spiral flow channel of volute gradually reduces with the monitoring point moving away from the volute tongue. Compared with monitoring point ${P}_{8}$, the pressure fluctuation coefficient at the monitoring point ${P}_{5}$ under the design flow rate is reduced by 70 %.
4) The radial force on the radial guidevane is obviously smaller than that on the volute. Under the same flow rate, radial force on the volute of secondstage pump is almost 20 times larger than that on the radial guidevan of firststage pump.
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About this article
The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant No. 51509109), the IndustryUniversityResearch Cooperative Innovation Fund of Jiangsu Province of China (Grant No. BY201412309), the Science and Technology Support Program of Jiangsu Province of China (Grant No. BE2014116), China Postdoctoral Science Foundation(Grant No. 2016M600370), Postdoctoral Scientific Research Project of Zhejiang Province of China, and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
Wang Kai designed the experiment schemes and numerical schemes of the twostage selfpriming centrifugal pump. Liu Houlin designed the twostage selfpriming centrifugal pump. Jiang Linglin made the unsteady numerical calculation of the pump and analyzed unsteady performance of the pump. Zhang Zixu and Li Yu made energy characteristics experiments and selfpriming experiments of the pump.