**3.1 QED-nED characteristic curve**

**Figure 2** shows the PAT operating modes in quadrants 3 and 4 of the QED-nED characteristic. The OPs corresponding to each operating mode are distinguished by colors. According to Greitzer [40], a hydraulic system can exhibit two forms of instability. The first, static instability, occurs when a small change in the flow rate at one OP can cause an increase in pressure forces and deviate the system to another OP farther away. The second, dynamic instability, occurs when a disturbance oscillates continuously increasing in amplitude. These forms of instability are associated with

*Periodic Instabilities in a Specific Low-Speed Pump Working as a Turbine DOI: http://dx.doi.org/10.5772/intechopen.109210*

hydraulic transients and clues can be found by analyzing the QED-nED and TED-nED characteristic curves [41–43]. The occurrence of static instability is a necessary but not sufficient condition for the occurrence of dynamic instability [43].

A practical method to identify static instability in the QED-nED curve is by the slope of a segment tangent to the runaway. If the slope is negative, the system is considered stable; if the segment is vertical, it is critical; and if the slope is positive, the system is unstable [41, 43]. The characteristic curve of an unstable system acquires the form "s," which means that for the same value of nED, several values of QED can occur. In some cases, the QED can reach values in all operating modes (turbine, turbine brake, and reverse pump). According to this criterion, the system under study is stable (see **Figure 2**) and, therefore, the system does not exhibit transient phenomena. This suggests that the instabilities occurring in the system are stationary.

## **3.2 Spectral analysis**

### *3.2.1 Identification of periodic or quasi-periodic components*

This analysis was based on the waterfall spectra of the pressure fluctuation signals from DYT1 and DYT2 sensors (see **Figures 3** and **4**). In **Figures 3** and **4**, the spacing between the spectra (OP axis) is defined by the Euclidean distance between the OPs in the QED-nED plane. Comparing **Figures 3** and **4**, it is observed that there is a frequency correspondence in the components with higher spectral power, suggesting the existence of periodic or semi-periodic hydrodynamic phenomena detected by the two sensors. From this inspection, three spectral groups of interest were defined. Group 1 (G1) is composed of OPs from 32 to 48 in the band 0.76 ≤ fn ≤ 0.86, group 2 (G2) by OPs from 38 to 48 in the band 1.61 ≤ fn ≤ 1.71, and group 3 (G3) by all OPs in the band 5.95 ≤ fn ≤ 6.05. **Figures 3** and **4** also emphasize the groups of interest with shading.

G1 can be divided into two subgroups, with the following characteristics:

• Subgroup 1: It includes OPs 32–41 and describes a phenomenon that appears at OP 32, reaches its maximum instability at OP 38, and then decays up to OP 41. Its frequency coefficients vary between 0.7995 and 0.8378.

**Figure 3.** *Waterfall power spectra for the DYT1.*

**Figure 4.** *Waterfall power spectra for the DYT2.*

• Subgroup 2: It is constituted by OPs 42–48. It describes a phenomenon that initiates at OP 42, reaches its maximum instability at PO 44, and decays until it reaches OP 48. Its frequency coefficients vary between 0.7798 and 0.8381.

G2 contains OPs 38–48, where the way the spectral power is distributed suggests the same hydrodynamic phenomenon. The phenomenon begins at OP 38, increases in power up to OP 41, and from there decreases to OP 48. It occurs in the lower part of the turbine-brake mode, and its frequency coefficients range between 1.6438 and 1.6834.

G3 comprises all operating points and corresponds to the blade passage with fn = 6. It is distributed throughout all operating modes in quadrants 3 and 4 and is most noticeable in turbine-brake and reverse pump modes. Near the runaway, the spectral power increases gradually until it reaches the last operating point in the reverse pump mode. In the high part of the turbine operating mode (OPs 3–9), some spectral powers are observed that outstand with respect to those of the other OPs in this part.

Another finding from comparing **Figures 2** and **3** is the spectral power of the components of interest in the DYT2 sensor signal spectra is higher than that of the DYT1 sensor. On average, the spectral powers in DYT2 are 2.6 times those of DYT1.

#### *3.2.2 Characterization of periodic or quasi-periodic components*

The characterization was performed for the groups of interest under the hypothesis that the components are the representation of hydrodynamic phenomena detected by the two sensors. For this purpose, power spectra were used, and three criteria were applied for the selection of periodic components. First, components with sharp, clearly distinguishable, and isolated peaks [36], that is, without contiguous spectral components of similar spectral power. Second, corresponding components in the spectra of both sensors in terms of frequency factor, and third, coherence between them greater than or equal to 90%.

The characterization of G1 and G2 was performed by phase analysis. In G1, OPs 32, 33, 34, 36, 37, 38, 39, 44, 45, and 46 satisfy the periodicity criteria. These subsynchronous components have frequency coefficients between 0.8018 and 0.8378 and signal coherence between 90.3 and 99.7%. In OP 34, 37, 38, 39, and 45 the signal from DYT1 is ahead of DYT2, and they show phase shifts between �150.6 and �135.2°, approaching 135° when considering the separation of the sensors in the opposite direction of flow from DYT1. For OPs 32, 33, 36, 36, 36, 44, and 46, the phase shifts are between 206.3 and 236.9°, which are approximately 225°, the physical separation of the sensors in the opposite direction of flow but starting at DYT2. These components were subjected to an additional test, the wave number determination. This test serves to identify phenomena that sometimes decompose in rotating cells, such as rotating stall [44, 45]. For these phenomena, the wave number is equal to the number of rotating cells. The wave number was estimated as follows:

$$K = \frac{f\_{n\_{\text{fl}}}}{f\_n} \tag{9}$$

where *K* is the wave number, *f npg* is the passage frequency, and *f <sup>n</sup>* is the frequency identified in the PS. The passage frequency is the frequency perceived by the sensors and is estimated from the correlation between signals in a band containing the component of interest. Details of the method can be found in Refs. [44, 45]. For the periodic components of G1, it was found the passage frequency is approximately equal to the PS frequency, so the wave number is equal to one. The evidence collected suggests the existence of a one-cell subsynchronous hydrodynamic phenomenon (K = 1), moving around the volute in the opposite direction of flow. Full characterization of the phenomenon requires further studies that are beyond the scope of this investigation.

In G2, OP 39–47 fulfill the periodicity criteria. These components present frequency coefficients between 1.6438 and 1.6742, and coherence between 92.00 and 99.51%. OPs 39, 41, 43, 44, 46, and 47 present offsets between 108.1 and 161.8°, and OPs 40, 42, and 45 between �228.3 and �204.8°. Based on the same arguments used in the characterization of the periodic components of G1, it can be concluded the components of this group correspond to a one-cell phenomenon (K = 1) moving in the direction of the flow.

The G3 components correspond to instabilities due to rotor-stator interaction (RSI) blade excitation. In this case, the excitation is probably caused by the asymmetry of the volute [46] since the PAT has no guide vanes. The effect of the rotor blades on the volute produces a periodic disturbance or force whose frequency, expressed in terms of frequency coefficient, is given by:

$$f\_{\boldsymbol{\pi}\_{\mathcal{S},k}} = Z\_b k, \quad (k = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots) \tag{10}$$

where *f nS*,*<sup>k</sup>* is the frequency coefficient of the disturbance due to the blades, *Zb* is the number of blades, and *k* is harmonic. In our case, *f nS*,1 ¼ 6, is the blade passage.

Notice in **Figures 3** and **4** that OPs 20 (runaway) and 21 (next to runaway) do not show significant perturbations compared to the interest group components. On an appropriate scale, it can be seen that OPs 20 and 21 have a component that matches the periodicity criteria, with frequency coefficients of 0.776 and 0.758, respectively, coherence of 99.64% and 98.21%, and phase shifts of 0.1 and �3.3°, suggesting an inphase phenomenon. This phenomenon is probably a surge, which is characterized by pressure and flow fluctuations that affect all parts of the hydraulic system simultaneously [33]. **Figures 5** and **6** show the PS of these points, where the highlighted spectral components are clearly distinguishable.

**Figure 5.** *PS of OP 20 (runaway).*

**Figure 6.** *PS of OP 21 (next to runaway).*

## **3.3 Measurement of flow instabilities due to low-frequency phenomena**

The instability measure of the prominent spectral components (G1, G2, and G3) at each OP was estimated based on the filtered signals at 2 Hz bandwidth. The bandwidth was equally distributed with respect to the spectral component of interest. In other words, the center of the filtering band corresponds to this component. Notice a distinction in the terms "interest band" and "filtering band." The former refers to the band defined by frequency coefficients (fn) in which the spectral components of interest are found, and the latter refers to the frequency band in which the filtering was done around the prominent component identified in the interest band.

The filtered signals were used to estimate the instability due to periodic or quasi-periodic phenomena in the time domain and the frequency domain. In the time domain, we considered the filtered signals and computed the variance. In the frequency domain, we computed the PSD of the filtered signal and then estimated the area under the spectrum. **Figures 7** and **8** depict the three-dimensional representation of the instability estimates for the DYT1 and DYT2 sensor signals, respectively, based on the QED-nED characteristic curve and classified according to interest bands.

Comparing the estimates of instabilities in the time domain with the estimates in the frequency domain (**Figure 7a** vs. **b** and **Figure 8a** vs. **b**), it is observed they are not equal, but quite similar. Regarding the sum of variances, we obtain a mean error of 4.9% and a standard deviation of the errors of 13.5% for DYT1, and a mean error of 4.0% and a standard deviation of the errors of 11.7% for DYT2.

Looking at **Figures 7a** vs. **8a** (time domain instabilities estimation) and **Figures 7b** vs. **8b** (frequency domain instabilities estimation), it is suggested that the distribution of the pressure pulses is not homogeneous in the volute, since the magnitude of the instabilities in DYT1 is smaller than their corresponding ones in DYT2. If the most unstable operating zone is considered (OPs 32–48), the magnitudes of the instabilities in DYT2 are larger than their corresponding ones in DYT1 by 1.74–3.32 times.

In terms of the contribution of periodic or quasi-periodic phenomena to the total instability, DYT2 is analyzed, since the magnitude of instabilities perceived by this sensor are higher than those perceived by DYT1. In this case, the sum of the instabilities estimated with the filtered signals is compared with those of the original unfiltered signals, which were used to estimate the total instability. **Figure 9** shows a 3D representation of the sum of the periodic or quasi-periodic instabilities with respect to the total instability. Total instability estimations in the time and frequency domain yield a mean error of 0.3% with a standard deviation of the errors of 4.8%.

Given the contribution of periodic or semi-periodic instabilities to total instability (see **Figure 9**), three zones of operation can be defined. **Table 4** describes these zones.
