Hindawi Publishing Corporation Advances in Mechanical Engineering Article ID 936218
Research Article Investigation on Vibration Characteristics in a Centrifugal Pump with Special Slope Volute Ning Zhang, MinGuan Yang, Bo Gao, and Zhong Li School of Energy and Power Engineering, Jiangsu University, Zhenjiang 212013, China Correspondence should be addressed to Ning Zhang;
[email protected] Received 13 June 2014; Revised 2 September 2014; Accepted 7 September 2014 Academic Editor: Lu´ıs Godinho Copyright © Ning Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vibration is one of the main problems which should be considered seriously in designing and operating a centrifugal pump. In this paper, a special slope volute was proposed to reduce vibration level of centrifugal pump. The goal of this paper was to analyze vibration characteristics of slope volute pump and provide a new method to reduce vibration energy in centrifugal pump. Experiments were carried out to obtain vibration signals at different flow rates of slope volute pump, which were compared with conventional spiral volute pump. Results show that vibration level is closely related to flow structure interior pump. At low flow rates, especially, vibration energy increases rapidly with the onset of rotating stall. Rotor-stator interaction mechanism changes when adopting slope volute. In slope volute pump, the peaks in power spectra correspond to different harmonics of 𝑓BPF at three measuring directions; however, in spiral volute pump, the predominant components always correspond to 2𝑓BPF . From comparison of vibration level in different frequency bands, it is found that vibration level of slope volute pump is smaller than conventional spiral volute pump. So vibration energy of centrifugal pump can be reduced effectively using slope volute.
1. Introduction As general machinery, centrifugal pump is widely used in all fields. In the entire system, centrifugal pump is the highspeed rotating equipment, and periodic forces generated by rotating impeller may lead to fatigue failure of shaft, bearing, and other components. Meanwhile, vibration induced by centrifugal pump is crucial to stable operation of the system. However it is still a tough problem for us to reduce vibration energy in centrifugal pump in order to diminish its negative effect on the whole system. Many researches have been done to illustrate vibration characteristics of centrifugal pump. Flow field in centrifugal pump has a great effect on vibration energy. Geometric parameters of impeller, including blade leading edge, blade number, blade inlet angle, and diameter of impeller, affect performance and vibration of model pump evidently [1–3]. It is well known that flow structure at blade exit shows a typical jet-wake pattern, and the striking effect between jet-wake flow and volute tongue is the main factor causing unsteady rotor-stator interaction [4, 5]. When centrifugal pump works
at off-design conditions, separate flow structure may easily occur leading to vibration characteristics changing remarkably [6, 7]. Meanwhile, parameters of volute also have a huge impact on flow field discharged from impeller, and vibration magnitude increases rapidly when exit angle of the flow out of the impeller does not match the geometry of the volute [8]. It is well accepted that in centrifugal pump unsteady flow structures are the main reason inducing low frequency vibration signals, including flow separation on blade suction side, vortex shedding, and rotating stall. Many researches have been carried out to investigate unsteady flow in centrifugal pump. At off-design conditions, separate flow would develop on blade suction side leading to partial impeller channels being blocked [9]. At low flow rates, especially, the separate flow structure may transfer from one blade channel to the adjacent channel forming rotating stall phenomenon [10]. In centrifugal pump, flow out of impeller interacts with volute tongue periodically due to the limited blade number [11]. Rotor-stator interaction between impeller and volute tongue would cause pressure pulsating and nonuniform flow field distribution on periphery of impeller. Usually predominant
2 component in pressure spectrum corresponds to blade passing frequency 𝑓BPF . Pressure magnitude at 𝑓BPF varies under different flow rates, and it usually falls to minimum at nominal flow rate [12]. The clearance between impeller and volute tongue is also an important factor, which will have a great effect on pressure pulsation. Rotor-stator interaction becomes intensely with clearance decreasing, which will lead to pressure fluctuation increasing steeply [13]. Many researches have proved that rotor-stator interaction keeps the predominant role in causing low frequency vibration signals in centrifugal pump [14, 15]. So vibration energy of centrifugal pump would be reduced significantly by diminishing rotorstator interaction between impeller and volute tongue. However, in order to reduce rotor-stator interaction, most of the researches are focusing on parameters of impeller and the shape of volute has been rarely taken into consideration. A special slope volute was proposed to reduce rotorstator interaction. The diffusion section of the volute has an acute angle with impeller axis. According to our previous study [16], it has an effective impact on reducing pressure pulsation level by means of numerical method. In this paper, vibration characteristics of slope volute were investigated, which were compared with spiral volute pump. Also, flow structure interior model pump was analyzed using numerical simulation method to clarify correlation between vibration and unsteady flow structure. This paper attempts to illustrate vibration features of model pump at variable flow rates and explore a new method to reduce vibration level in centrifugal pump.
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Outflow
Outlet duct
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Figure 1: Computational domains.
2. Numerical Setup Two model pumps, the slope volute pump and the spiral volute pump, were first designed in this paper. To investigate unsteady flow structures inside the slope volute pump, numerical method was applied. During numerical calculation, the front and back chambers of the model pump were neglected for simplification. Finally, the computational domains included pump inlet, impeller, slope volute, and pump outlet as presented in Figure 1. Standard wall functions were employed to simulate flow structure near the wall boundary, and due to the rigid requirement of the first layer grid height [17], grid cells near the solid wall were encrypted in order to achieve a better numerical precision. On surface of the blades, especially, separate flow structure may easily develop at off-design conditions. Finally, the total size of structured grid applied is 1.15 × 106 after mesh independent check, and 𝑦+ values of the model pump domains meet the requirement of standard wall functions according to numerical results. Figure 2 presents structured grid of the partial computing domains. The commercial CFD code Fluent was used to solve the full 3D Reynolds-averaged Navier-Stokes equations for flow field in the model pump, and turbulence was simulated using SST turbulent model. For transient calculation, steady simulation results were set as the initial condition to obtain a fast convergence process. The boundary condition imposed at inlet domain was a constant velocity and pressure condition
Figure 2: Structured grid of the model pump.
at the outlet, and the static pressure was defined as 1.0 × 105 Pa. The time step was set to 1.15 × 10−5 s, and at least forty revolutions were achieved to guarantee numerical accuracy.
3. Experimental Setup Vibration experiments of the slope volute pump were conducted in a closed-loop test platform to guarantee measuring accuracy, as seen in Figure 3. And the detailed description of the experimental platform could be found in our preceding research [18]. To alleviate the intense rotor-stator interaction in the centrifugal pump, the special slope volute is investigated, which is presented in Figure 4. And the structure comparison with conventional spiral volute has already been analyzed in [18].
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The most widely used vibration measuring systems LMS were applied to attain the vibration signals of the model pump. To have a comprehensive understanding of vibration characteristics, seven accelerometers, including four triaxial sensors and three single direction sensors, were mounted on the slope volute casing covering the sensitive regions of the pump. The relative positions of the transducers on the spiral volute are in accordance with the slope volute. Figure 6 presents the detailed positions of the accelerometers along the slope volute, and both the measuring direction of each sensor and the measuring scheme are the same as described in [18]. Autopower spectrum algorithm is applied to analyze vibration spectra.
4. Experimental Results
Figure 4: Configuration of the slope volute.
Generally, it is known that vibration of centrifugal pump depends on not only excitation forces but also boundary conditions of the whole system. To ensure the vibration results to be comparable, some special measures have been applied. The tested two pumps used the same impeller, bracket and motor, which were kept unchanged during experiments. So mechanical noises were considered to be approximately the same, as illustrated in Figure 5. Meanwhile, flexible joints were fixed at the inlet and outlet of model pump to reduce the influence of interference signals from experimental system on vibration signals of model pump. The two different volutes were made with the same material and installed on the base by using bolts, so stiffness matrixes and damping matrixes of two pumps were regarded to be the same. Due to the different structures of two volutes, mass matrixes are different. However the tested model pumps are of small size, so the differences of mass matrixes of two pumps may not be significant. From the above analysis, it is inferred that two tested pumps are of approximately the same boundary conditions; thus the vibration levels of two pumps are considered to be comparable.
4.1. Performances of Two Model Pumps. Figure 7 presents performance characteristics of two model pumps. From Figure 7(a) it is found that the best efficiency points of slope volute and spiral pump are around 1.0B𝑁. At nominal flow rate, the efficiency of spiral pump is 2.5% higher than slope volute pump. From shutoff to 0.3B𝑁 conditions, the head and efficiency curves of two model pumps are almost unanimous. Meanwhile, positive slope phenomenon occurs in both head curves, which indicates that rotating stall develops interior blade channels [19]. With the increasing of flow rate, hydraulic loss of slope volute pump increases leading to performance descending. When operating at high flow rates, performance of slope volute pump drops significantly. At flow rate 1.1B𝑁, efficiency of slope volute pump is 3% lower than spiral volute pump, which reaches 10% at flow rate 1.4B𝑁. Shaft power of slope volute pump is slightly smaller than spiral volute pump at various working conditions. Figure 7(b) presents comparison of numerical and experimental results. At low flow rates, numerical results are in good agreement with experimental values. At nominal flow rate, calculation error of slope volute pump is 2.3%, and it is 3% for spiral volute pump. However, at high flow rates, numerical errors of both model pumps increase. From performances comparison of two pumps, it is concluded that efficiency of slope volute pump decreases significantly when model pump works at high flow rates. Numerical results agree well with experimental results at low and nominal flow rates.
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Advances in Mechanical Engineering Slope volute
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Figure 5: Installation conditions of model pumps.
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where 𝐺𝑥 (𝜔) is unilateral power spectrum. Vibration experiments of model pumps were conducted at different operating conditions. Vibration signals in 0–10 Hz were filtered, so acceleration signals in 10–8000 Hz band were analyzed in this paper. Frequency domain signals of two model pumps were first divided into three frequency bands: low frequency band (10–500 Hz), middle frequency band (10– 4000 Hz), and full frequency band (10–8000 Hz) to analyze vibration characteristics of two model pumps. Root mean square (RMS) method was used to deal with acceleration signals in each frequency band. And it is defined as
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Figure 6: Measuring position of the accelerometer.
4.2. Vibration Characteristics Comparision of Two Pumps. For any given time signal 𝑥(𝑡), the correlation function is defined as follows: 1 𝑇 𝑅𝑥 (𝜏) = lim ∫ 𝑥 (𝑡) 𝑥 (𝑡 ± 𝜏) 𝑑𝑡, 𝑇→0𝑇 0
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where 𝑇 represents observation time of 𝑥(𝑡). 𝑅𝑥 (𝜏) describes correlation between 𝑥(𝑡) and 𝑥(𝑡 + 𝜏). Correlation coefficient is shown in 𝑅 (𝜏) 𝜌𝑥 (𝜏) = 𝑥 2 , 𝜎𝑥
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where 𝜎𝑥 is standard deviation of time signal 𝑥(𝑡). Power spectral density function of 𝑥(𝑡) is faster Fourier transform (FFT) of correlation function, and it is defined as 𝑆𝑥 (𝜔) = ∫
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𝑆𝑥 (𝜔) is power spectral density function of 𝑥(𝑡), also called bilateral power spectrum.
2 1 𝑛 RMS = √ ∑ (𝐴 𝑛 − 𝐴) , 𝑛 𝑛=1
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For triaxial accelerometer, total vibration energy 𝐸 was used to evalute vibration energy at different measurinig directions. And it is defined as shown in 𝐸=√
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To investigate the influence of slope volute on vibration characteristic in centrifugal pump, vibration energy in each frequency band of slope volute pump was compared with spiral volute pump. Figure 8 presents vibration energy trends versus flow rate at sensor 5 and sensor 6. At measuring point
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sensor 5, vibration energy of slope volute pump is obviously smaller than spiral volute pump in each frequency band. At nominal flow rate, vibration energy of slope volute pump is 1.5 dB lower than spiral volute pump in low frequency band, 1.87 dB lower in middle frequency band, and 1.73 dB lower in full frequency band. At measuring point sensor 6, vibration energy of slope volute is much smaller than spiral volute in middle and full frequency band, while, in low frequency band, vibration energy of slope volute pump is larger than spiral volute pump at some operating conditions. At nominal flow rate, the difference between two model pumps is 1.1 dB in low frequency band, 2.5 dB in middle frequency band, and 1.8 dB in full frequency band. For two model pumps, total vibration energy values of each measuring point at variable flow rates are averaged. Figure 9 presents the averaged results. In low frequency band, vibration energy of slope volute pump is much smaller than spiral volute pump at each measuring point. At sensor 1, the difference even reaches 6.5 dB. In middle frequency band, vibration energy of slope volute pump is much lower at sensors 1, 4, 5, and 6, compared with spiral volute pump, which achieves maximum 6 dB at sensor 4. However, at sensors 2, 3, and 7, vibration energy of slope volute pump is larger than spiral volute pump. At sensors 2 and 3, the difference between two pumps is within 0.5 dB, but the difference rises to 2.2 dB at sensor 7. In full frequency band, vibration energy of slope volute pump is lower than spiral volute pump at every measuring point. From vibration energy comparison of two model pumps, it is concluded that the shape and relative position of volute tongue changes significantly when adopting slope volute leading to the fact that striking between impeller and volute tongue decreases rapidly, so intense rotorstator interaction will be reduced. On the whole, vibration level can be reduced effectively using slope volute.
Centrifugal pump is a complicated flow-structure interaction system, and unsteady phenomena of interior pump are the main factors that induced vibration [20, 21]. Many studies have shown that vibration signals caused by hydraulic factors mainly concentrate in low frequency band [22–24]. So vibration signals in low frequency band of two model pumps were investigated to analyze influence of slope volute on vibration characteristic in centrifugal pump. Rotating speed of model pump keeps a constant value 1450 r/min, so shaft rotating frequency 𝑓𝑅 is 24.2 Hz and blade passing frequency 𝑓BPF is 145 Hz. Figure 10(a) shows power spectra of slope volute pump at sensor 3 under different flow rates. It is found that power spectra characteristics of slope volute pump exhibit great diversities at different measuring directions. At 𝑥-direction, acceleration magnitudes at blade passing frequency 𝑓BPF and 𝑓𝑅 are quite small, and peak in spectra corresponds to higher harmonic 3𝑓BPF . At 𝑦-direction, the spectrum feature changes significantly, and maximum value no longer locates at 3𝑓BPF . Peak value of acceleration signal deviates to 5𝑓BPF . At other frequencies, acceleration amplitudes are rather small. At 𝑧-direction, frequency of peak value changes again, and the predominant component occurs at 2𝑓BPF . Vibration experiments of spiral volute pump were carried out under the same experimental conditions with slope volute pump. Figure 10(b) presents power spectra of spiral volute pump at different operating conditions. At different measuring directions, vibration acceleration magnitudes are really small at blade passing frequency 𝑓BPF and 𝑓𝑅 . And it is noted that the maximum amplitudes of vibration acceleration signals at three measuring directions always correspond to 2𝑓BPF . At 𝑥-direction, other peaks at higher harmonic of blade passing frequency (3𝑓BPF , 4𝑓BPF ) are also evident. But at 𝑦- and 𝑧-directions, vibration acceleration magnitudes at higher harmonic of 𝑓BPF are really small. From power spectra
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Figure 8: (a) Vibration energy in different frequency bands at sensor 5; (b) vibration energy at sensor 6.
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In slope volute pump, predominant components in vibration spectra are different at three measuring directions, while, in spiral volute pump, evident peaks in power spectrum always correspond to 2𝑓BPF .
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Figure 9: Averaged results of vibration energy at different sensors.
comparison of two model pumps, it is found that rotor-stator interaction mechanism changes by using slope volute, which has a significant effect on power spectra of vibration signals.
4.3. Vibration Characteristics of Slope Volute Pump. Vibration signals in low frequency band (10–500 Hz) of slope volute pump were studied to analyze vibration energy trend versus flow rate. Numerical simulation method was applied to attain typical flow structure interior model pump. Finally, the correlation between flow structure within centrifugal pump and vibration energy would be carried out. Flow rate was first divided into four regions: region I (0–0.3B𝑁), region II (0.3B𝑁–0.8B𝑁), region III (0.8B𝑁–1.1B𝑁), and region IV (1.1B𝑁–1.45B𝑁). Figure 11 shows vibration energy varying trends as a function of flow rate. At different working conditions, vibration energy at sensor 2 (slope volute outlet) is much larger than that at other sensors. But, at sensor 6, vibration energy achieves minimum value under different flow rates. It is also evident that, at low flow rates, vibration energy increases rapidly due to the separate flow structures. Figure 12 presents transient relative velocity streamlines within blade channels at different moments, when model pump works in region I. From Figure 12(a), it is noted that some patterns of flow structures exist in different blade channels at 0.2B𝑁. At 𝑇 moment, flow structures in channels 1, 2, 3, and 4 are similar, and two separate flow regions could be found near blade inlet and outlet. And the periphery of vortex structure almost reaches middle of blade outlet, which would lead to partial channel being blocked. But only one
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separate structure exists near blade inlet in blade channels 5 and 6. At 𝑇 + 14Δ𝑡 moment, flow structure in channel 4 changes, and the scale of vortex structure starts to shrink significantly. With impeller rotating, at 𝑇 + 30Δ𝑡 moment, vortex structure disappears completely at blade outlet. Then flow structure in adjacent channel 3 will experience the whole periodic process. But the separate structures near blade inlet in different blade channels almost keep unchanged and unaffected by rotating impeller at different moments. At flow rate 0.05B𝑁, scale of separate region near the impeller inlet expands, and the core of separate vortex structure moves to middle of the channel. Almost half of blade inlet region is occupied by separate structure. At 𝑇 moment, flow structures in channels 1, 2, 3, and 6 are almost unanimous. Two vortex structures occur, respectively, locating near inlet and outlet of impeller, which would lead
to blade channels being blocked severely. In channels 4 and 5, another pattern of vortex structure exists on suction side of blade outlet. It rotates along clockwise direction, and three vortex structures exist in blade channels at the same time. At 𝑇 + 14Δ𝑡 moment, in channel 5, vortex structure near pressure side of blade outlet splits into two separate vortex structures; meanwhile four individual separate vortexes exist in impeller channel. At 𝑇 + 44Δ𝑡 moment, two vortexes at blade outlet merge into single vortex structure. From analysis of structure in impeller, it is evident that, from 0.3B𝑁 to 0.05B𝑁, separate flow region in impeller channels expands causing partial channels to be blocked. At flow rate 0.05B𝑁, multiple vortexes exist in impeller channels and vortex structure at blade outlet develops with impeller rotating. Intensely unsteady flow structures in impeller lead to vibration energy increasing rapidly at every measuring
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point. From Figure 11 it is found that, from 0.3B𝑁 to 0.05B𝑁, vibration energy increases 1.5 dB at sensor 2, 1.3 dB at sensor 3, 1.0 dB at sensor 5, and 1.0 dB at sensor 6. When slope volute pump works in region II, Figure 13 presents relative velocity streamlines interior blade channels. At flow rate 0.6B𝑁, separate structure occurs on suction side of blade, but the scale of separate region is rather small. The separate structure on the blade almost remains unchanged at different moments. When flow rate decreases, flow structure in blade channel changes obviously at flow rate 0.4B𝑁. At 𝑇 moment, flow structures in channels 1, 2, 4, 5, and 6 are similar, and the separate region near blade inlet expands. In blade channel 3, the second vortex structure at pressure side of blade outlet develops. At 𝑇 + 14Δ𝑡 moment, the scale of vortex region at blade outlet shrinks. With the impeller rotating, at 𝑇 + 30Δ𝑡 moment, vortex structure at outlet of blade channel 3 completely disappears. In region II, from 0.8B𝑁 to 0.7B𝑁, separate flow structure starts to occur on blade, which leads to vibration energy increasing slightly. And it has about an increment of 0.4 dB at each measuring point. From 0.7B𝑁 to 0.3B𝑁, although the scale of separation region expands and unsteady vortex structure develops at blade outlet, vibration energy almost keeps constant. So it is concluded that flow structure within model pump has little effect on vibration energy from 0.7B𝑁 to 0.3B𝑁. When slope volute pump works in region III, flow field interior impeller is uniform, and separate flow structure no longer occurs on blade. But flow structure around slope volute tongue alters, as shown in Figure 14. At flow rate 1.1B𝑁, fluid flows into diffusion section of volute smoothly, and backflow phenomenon does not exist. However, at flow rate 0.8B𝑁, partial fluid flows from diffusion section back into volute, striking with volute tongue severely. The changing of flow structure around volute tongue has a great influence on vibration energy as shown in Figure 11. From 1.1B𝑁 to 0.8B𝑁,
vibration energy around nominal flow rate falls to a local minimum. When flow rate decreases, backflow phenomenon develops causing vibration energy to increase rapidly. At flow rate 0.95B𝑁, vibration energy achieves a local maximum, which has about an increment of 0.5 dB at different sensors. Then vibration energy of model pump starts to descend, and the whole process is like wave shape. When model pump operates in region IV, Figure 15 presents pressure pulsation amplitudes versus flow rate at sensor 3. The predominant components in spectra correspond to blade passing frequency 𝑓BPF , and spikes at 2𝑓BPF are also evident. At high flow rates, flow strikes on pressure side of blade leading to vortex shedding, which would cause pressure pulsation amplitude increasing. So vibration energy induced by pressure pulsation increases gradually. From 1.1B𝑁 to 1.45B𝑁, vibration energy increases 0.5 dB at sensors 2 and 5 and about 0.8 dB at sensors 3 and 6.
5. Conclusions In this paper, a special slope volute was proposed and vibration characteristics of model pump were measured, which were compared with conventional spiral volute pump. Meanwhile, flow field interior slope volute pump was investigated using numerical simulation method to clarify correlation between flow structure and vibration level of model pump. It is found that, at low flow rates, the head and efficiency curves of two different pumps almost have the same trend and values. Positive slope phenomenon occurs, which indicates that rotating stall develops in blade channels. But, at high flow rates, efficiency of slope volute pump is smaller than spiral volute pump. Flow structure interior model pump has a great impact on vibration characteristics. At low flow rates, separate flow structure may easily occur on blade suction side. From 0.3B𝑁 to 0.05B𝑁, several different types of vortexes exist. And vortex structure at blade exit changes at different moments, which leads to vibration level of model pump increasing rapidly. Also, flow field around volute tongue is closely related to vibration energy, and vibration level of model pump increases obviously when backflow phenomenon occurs. From power spectra comparison of two model pumps, it is found that rotor-stator interaction mechanism changes using slope volute. Peaks at different measuring directions in slope volute pump locate at different frequencies, while in spiral volute pump main peaks always correspond to 2𝑓BPF . From comparison of vibration energy of two pumps, it is observed that vibration energy of conventional spiral volute pump is much larger than slope volute pump at different working conditions. So vibration energy of centrifugal pump can be reduced significantly using slope volute. In the further study, detailed analysis of unsteady flow field and pressure pulsation features of slope volute pump will be carried out to validate fluid forces reduction effect of the proposed slope volute. And optimal design of slope volute will be conducted to obtain lowest vibration level. Finally,
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Figure 12: (a) Relative velocity streamlines in impeller at flow rate 0.2B𝑁 ; (b) relative velocity distribution at flow rate 0.05B𝑁 .
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Figure 13: (a) Relative velocity streamlines distribution at flow rate 0.6B𝑁 ; (b) relative velocity distribution at flow rate 0.4B𝑁 .
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Advances in Mechanical Engineering V 0.8 0.7 0.6 0.4 0.3 0.2 0.1 0.0 (a)
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Figure 14: (a) Absolute velocity distribution around volute tongue at 0.8B𝑁 ; (b) absolute velocity distribution at 1.1B𝑁 .
Nomenclature
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Figure 15: Spectra of pressure pulsation at high flow rates.
our paper would provide a new way to reduce rotor-stator interaction in centrifugal pump.
Appendix Validation of Pressure Pulsation Reduction Effect of Slope Volute To validate lower pressure pulsation characteristics of slope volute, pressure pulsation of a centrifugal pump with conventional spiral pump was measured for comparison. Figure 16 shows details of pressure transducers mounting on two different volutes. Figure 17 presents comparison of amplitude at 𝑓BPF between slope volute and spiral volute pump. As observed, the experimental results agree with our hypothesis very well. It is found that pressure amplitude of spiral volute pump is much larger than slope volute pump from shutoff to maximum flow rate conditions. According to the comparison results, it is demonstrated that the special slope volute has an evident impact on reducing pressure energy in centrifugal pump.
𝑄, 𝑄𝑁: Flow rate, nominal flow rate, 48 m3 /s 𝐻, 𝐻𝑁: Head of model pump, nominal head, 7.8 m B, B𝑁: 𝑄/(𝜔𝑑22 𝑏2 ) flow coefficient, nominal flow coefficient Ψ, Ψ𝑁: 𝑔𝐻/(𝜔2 𝑑22 ) head coefficient, nominal head coefficient Impeller outlet diameter, m 𝑑2 : Impeller outlet width, m 𝑏2 : 𝑍: Blade number 6 𝑛, 𝜔: Impeller rotating speed 1450 r/min, rad/s Blade rotating frequency, 24.2 Hz 𝑓𝑅 : 𝑓BPF : Blade passing frequency, 145 Hz 𝐸: Vibration total energy of different measuring direction, dB RMS: Root mean square of vibration signals 𝑤, 𝑊: Relative velocity, m/s, 2𝑤/𝜔𝑑2 relative velocity coefficient V, 𝑉: Absolute velocity, m/s, 2V/𝜔𝑑2 absolute velocity coefficient 𝑃, 𝜆: Shaft power, kW, 𝑃/𝑑52 𝑛3 shaft power coefficient Δ𝑡: Time step, s 𝑝: Static pressure, Pa 𝜌: Water density, kg/m3 𝑢2 : Tangential velocity at impeller periphery, m/s.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments The authors gratefully acknowledge the financial support of National Natural Science Foundation of China (51206063,
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Figure 16: Pressure transducers mounting on two vloutes.
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Figure 17: Comparison of amplitudes at 𝑓BPF between two pumps at variable flow rates.
51106066), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and the Research and Innovation Project for College Graduates of Jiangsu Province (KYLX 1036).
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