3.2. Time Domain and Frequency Domain Analysis
Five main operating conditions from 0.8
QN to 1.2
QN are selected to analyze the pressure pulsation characteristics. In a centrifugal pump, the high−energy pressure pulsation caused by the RSI between the impeller and the tongue is the primary source of flow−induced vibration. Therefore, this paper analyzes the distribution characteristics of pressure pulsation near the tongue.
Figure 5 shows the time domain distribution of pressure pulsation under different working conditions at three pressure pulsation measuring points (P1, P2, P3) near the tongue of different schemes. Studies have shown that the unsteady flow at point P2 (near the tongue) is complex, and the pressure pulsation signals are composed of many components [
3,
24]. The time domain result shown in
Figure 5 is the instantaneous data collected after a period of stable operation of the pump. The sampling time is greater than one impeller rotation cycle, and the data is dimensionless. It can be found that there are apparent differences in the distribution of the time domain signal characteristics of the three measuring points from the test results. In the original impeller scheme, the pressure pulsation of each measuring point is the most stable under the design flow condition. The pressure pulsation has a certain regularity under the off−design flow condition. However, in the 0.8
QN and 1.2
QN working flow conditions, the pulsation peaks and troughs amplitude values are obviously inconsistent. In the staggered impeller scheme, the pressure pulsation amplitudes under the given conditions are smaller than those of the original impeller scheme, and the pulsation peaks and troughs are dense. In the same sampling time, the number of peaks and troughs increases in the staggered impeller scheme.
In order to further analyze the frequency domain characteristics of pressure pulsation under different working flow conditions of three monitoring points, FFT transformation is performed on the pressure pulsation signal.
Figure 6 shows pressure pulsation’s frequency domain distribution characteristics under different flow working conditions at three measuring points of different schemes. The number of blades in the original impeller model pump is six. In the frequency domain, the peak value is six times the axial frequency, that is, the main frequency is the impeller passing frequency. After the blades are staggered, the number of impeller tongue interferences is doubled. The twelve blades in the staggered impeller interfere with the tongue, and the main frequency becomes twelve times the axial frequency, that is, twice times the blade frequency. It can be seen from
Figure 6 that the amplitudes of the characteristic frequency of each measuring point under different flow working conditions are pretty different, and the primary characteristic frequency is the blade passing frequency (
fbpf = 145 Hz) and its higher harmonics. Especially at P1, the amplitude of the 2
fbpf has the most significant variation with the working flow conditions. It can be found that the primary characteristic frequency of the staggered scheme is 2
fbpf under different flow working conditions at the same measuring point, and the
fbpf amplitude is almost completely suppressed, especially at P3. The characteristic frequency amplitude of point P3 sharply increases under the development of RSI, while the characteristic frequency amplitude is almost completely suppressed in the staggered impeller scheme. This remarkable effect is brought about by the phase difference between the front and rear impellers, and the tongue separation after the blades are staggered. The vibration noise caused by hydraulic excitation in the pump system mainly comes from RSI. After the blades are staggered, the outflow of the two layers of impellers interferes with the tongue respectively, resulting in a phase difference in the vibration waveform. After superposition, the vibration energy is greatly reduced. Therefore, we can see the waveform with small amplitude and dense amplitude in the time domain diagram, and the amplitude energy of the characteristic frequency is also greatly reduced. This is also an effective means to suppress the source of pump vibration and noise.
The variation law of the primary characteristic frequency of each monitoring point of the two−impeller scheme is similar. Moreover, the amplitude of the characteristic frequency decreases first and then increases with the increase in the working flow condition, and the amplitude of the characteristic frequency is the smallest at the design condition. The shaft frequency signal fr = 24.16 and its higher harmonics with similar distribution characteristics can be captured under each working flow condition. Under different working flow conditions, fr, 2fr and 3fr are captured by both impeller schemes. The reason is that the rotor is not centered during the assembly process. The three monitoring points in the staggered impeller scheme captured the higher harmonics signals 4fr and 5fr.
In order to analyze the influence on the overall pressure pulsation characteristics of the staggered impellers, the amplitude values of
fbpf at 20 measuring points mounted on the volute are extracted in
Figure 7. The figure shows that the amplitude of
fbpf in the original scheme presents six peaks and troughs, while the magnitude of the amplitude of
fbpf is significantly reduced after the impeller is staggered, so that its value is too small to show the number of regular peaks. The amplitudes of the two schemes at
fbpf are the highest at point P3, which is downstream of the tongue. The amplitude of
fbpf in the staggered scheme is significantly reduced, and all the measured values of the 20 measuring points are smaller than in the original scheme. The distribution characteristics of the amplitude of each measuring point under different working flow conditions are similar. As the flow rate increases, the amplitude of
fbpf first decreases and then increases, and it is the lowest at the design flow condition.
Table 3 shows the average value and variation of
fbpf amplitude at 20 monitoring points under 0.8
QN–1.2
QN. At the same time, the suppressing effect of the staggered impeller scheme on the
fbpf amplitude is further quantified. The calculation method of the amplitude average value is listed in Equation (2). The expression of the
fbpf reduction amplitude is shown in Equation (3). It can be seen from the table that the staggered impeller at
fbpf has a significant inhibitory effect on each working flow condition. Moreover, the drop rate is greater than 80%.
AP−θ represents the pressure pulsation amplitude at fbpf with different angles, and Δ represents the reduction values of the fbpf amplitude of the staggered scheme compared with the original scheme.
As seen from
Figure 6 above, the primary excitation frequency of the staggered impeller scheme will move to the 2
fbpf after staggered. Moreover, the 2
fbpf of the staggered impeller scheme is dominant in the frequency spectrum near the tongue. Therefore, the amplitude distribution of 20 measuring points on the volute at 2
fbpf is further extracted to explore the main frequency characteristics of the staggered impeller, as shown in
Figure 8. It can be found that the amplitude of the original scheme is smaller at 1.0
QN at each measuring point and more prominent during 0.8
QN–0.9
QN. Under the staggered scheme, the amplitude increases at 2
fbpf with the flow rate increase at each measuring point.
Table 4 shows the average value and variation of the 2
fbpf amplitude. At 0.8
QN, the staggered impeller scheme has a certain inhibitory effect on the amplitude at 2
fbpf. However, the 2
fbpf amplitude of the staggered impeller scheme at 1.0
QN and 1.2
QN has a larger increase than the original scheme.
3.3. RMS Analysis
The Root Mean Square (
RMS) is used to judge the overall energy value of the pressure pulsation spectrum, and the
RMS processing is performed on the spectrum signal. The leakage of spectral energy during the test could be considered. Thus, the specific calculation method of the
RMS value is as shown in Equation (4) [
25]:
where
A0 and
An are the pressure pulsation amplitudes at the beginning and end of the sampling frequency band, respectively, and
An−1 are the amplitudes at different frequencies in the sampling frequency band.
The previous analysis shows that the high-frequency band has almost no high−amplitude excitation frequency. Therefore, the signal in the frequency band of 4
fbpf = 580 Hz is selected for the
RMS analysis of pressure pulsation energy. In the frequency spectrum of the two impeller schemes,
fr and its high harmonic frequency signal are captured. Therefore,
Figure 9 firstly shows the pressure pulsation energy at each monitoring point in the 0–
fbpf frequency band under 0.8
QN–1.2
QN. This frequency band includes multiple low−frequency signals, which are
nfr (
n = 1, 2, 3, 4, 5). In the original impeller scheme (
Figure 9a), the pressure pulsation energy under each working flow condition is stable in the 0–
fr frequency band, and the pressure pulsation energy of the P1−P20 measuring points shows a trend of first increasing and then decreasing. The pressure pulsation energy of 0.8
QN–0.9
QN is obviously higher than that of 1.0
QN–1.2
QN. It can be proved that the shaft frequency pulsation energy is larger under the small flow rate. It can be seen from
Figure 9b that, compared with the original impeller scheme, the monitoring points at the upstream and downstream of the tongue in the staggered impeller scheme have a larger difference in pressure pulsation energy under the partial working conditions. The RSI between the staggered blades and the tongue strengthens the amplitude energy at 0–
fbpf, so the pressure pulsation energy with the staggered impeller scheme is higher in this frequency band.
In order to analyze the energy characteristics as a whole,
Figure 10 shows the pressure pulsation energy distribution of each measuring point of the two schemes in the frequency band of 0–4
fbpf, which contains multiple characteristic frequency signals (
nfr,
nfbpf). In the original impeller scheme, the overall energy of the pressure pulsation at the monitoring point near the tongue is relatively large, and the pressure pulsation energy shows a gradually weakening trend at the measuring points around the volute. However, from 0.8
QN to 1.2
QN, the staggered impeller scheme dramatically reduces the pressure pulsation energy, and the overall energy amplitude is the lowest at 1.0
QN.
The suppression effect of the staggered impeller scheme on the pressure pulsation energy in different frequency bands is further quantified in detail.
Table 5 shows the two impeller schemes in different frequency bands (0–5
fr, 0–
fbpf, 0–2
fbpf, 0–4
fbpf, 0–6
fbpf)
RMS average value and the change relative to the original scheme under 1.0
QN, where the variation values calculation method of variation values is consistent with the Equation (4). The analysis found that the
RMS energy average values of the staggered scheme are significantly larger than that of the original scheme in the 0–5
fr frequency band. In addition to the influence of shaft frequency (including some shaft misalignment and installation problems), the reason is that the RSI effect of the staggered impeller and tongue is stronger than the original scheme, which increases the low−frequency excitation signal. However, in the frequency band including
nfbpf (
n = 1–6), the staggered impeller scheme has a significant suppression effect. Moreover, the suppression effect gradually weakens with the increase in high blade passing frequency, which decreases from 42.98% to 37.33%. The variation values Δ in the 0–4
fbpf and the 0–6
fbpf frequency band are almost the same, which indicates that the 4
fbpf–6
fbpf amplitude is small and has little effect on the pressure pulsation energy. It can be seen from the above analysis that the staggered impeller has a strong ability to suppress pressure pulsation energy and can be applied to the design of low vibration and noise pumps.
Figure 11 compares
RMS average values in different frequency bands under the two schemes from 0.8
QN to 1.2
QN, further showing more details of the pressure pulsation energy distribution under different working conditions. It can be seen from
Figure 11a that in the frequency band of 0–5
fr, the pressure pulsation energy of the staggered impeller scheme is higher than that of the original impeller scheme under 0.8
QN–1.2
QN. At the same time, the
nfr signal excitation source has been strengthened under 1.1
QN–1.2
QN. It can be seen from
Figure 11b–d that the
nfbpf signal is included in the other frequency bands, and the pressure pulsation energy of the staggered impeller scheme is smaller than that of the original impeller scheme. With the expansion of the frequency band, the energy suppression effect of the staggered schemes under five working conditions is gradually stable. In general, the staggered impeller scheme significantly inhibits the pressure pulsation energy in the pump.