**2.2 QED-nED characteristic**

The PAT characteristic curve was made based on dimensionless factors of velocity (nED) and discharge (QED), according to IEC 60193 [35], as defined below:

$$m\_{ED} = \frac{nD}{E^{0.5}}\tag{1}$$

$$Q\_{ED} = \frac{Q}{D^2 E^{0.5}}\tag{2}$$

where *n* is the rotation frequency (s�<sup>1</sup> ), *D* is the reference diameter of the impeller (m), *E* is the specific energy (N.m/kg), and *Q* is the discharge (m<sup>3</sup> /s). The characteristic curve was calculated with 60 operating points (OP), which had flow rates between �0.0024 and 0.0061 m<sup>3</sup> /s, rotation frequencies between 5.04 and 33.85 s�1, and a hydraulic head around 8 m. The characteristic curve corresponds to quadrants 3 and 4 defined by IEC 60193 [35], with the operating modes given in **Table 3**.

#### **2.3 Spectral analysis**

Spectral analysis is performed to detect periodic or quasi-periodic components in a signal; therefore, it is important to differentiate these components from narrowband random contributions [36]. In practice, a simple way to identify periodic or quasiperiodic components is by means of a power density spectrum (PSD) or also known as an auto spectrum. In the PSD, the periodic or quasi-periodic component is identified as a sharp and clearly distinguishable peak. Other spectral representations derived from the PSD, such as the power spectrum (PS), linear spectrum (LS), and linear density spectrum (LSD) [37], can also be used to identify this type of component. In this work, PS was used to identify periodic components and PSD was used to measure the instability associated with hydrodynamic phenomena.

For the spectral analysis, we used a time series of 2,621,440 records (25.6 s) divided into 2.5 parts and overlapped by 50%. In this way, four segments were obtained, and the spectrum was estimated for each segment and then averaged. In addition, the Hanning window was used to fix possible discontinuities in the data series. The frequency resolution was 0.09765625 Hz.

#### *2.3.1 Fourier analysis*

In turbomachines, Fourier analysis assumes that the fluctuations are periodic and linear [34]. With this assumption, instability can be considered as the sum of instabilities from different types or sources. In other words,


**Table 3.** *Operation modes in quadrants 3 and 4.* *Periodic Instabilities in a Specific Low-Speed Pump Working as a Turbine DOI: http://dx.doi.org/10.5772/intechopen.109210*

$$
\tilde{\mathbf{x}} = \tilde{\mathbf{x}}\_1 + \tilde{\mathbf{x}}\_2 + \dots + \tilde{\mathbf{x}}\_n = \sum\_{i=1}^n \tilde{\mathbf{x}}\_i \tag{3}
$$

where *x*~ is the total instability of the variable *x* and *x*~1, *x*~2, … , *x*~*<sup>n</sup>* are the instabilities by type or source. Hence, when passing from time domain to frequency domain, the spectral representation of the instability of the interest variable contains the spectral components of the instabilities by type or source. Thus,

$$
\tilde{\mathbf{x}} \stackrel{F}{\to} \tilde{\mathbf{X}} \tag{4}
$$

$$
\tilde{X} = \tilde{X}\_1 + \tilde{X}\_2 + \dots + \tilde{X}\_n = \sum\_{i=1}^n \tilde{X}\_i \tag{5}
$$

where Eq. (4) denotes the shift from time domain to frequency domain by means of the Fourier transform. *X*~ is the frequency domain representation of the total instability of variable *x* and *X*~1, *X*~2, … , *X*~ *<sup>n</sup>* are the frequency domain contributions of instabilities by type or source. The Fourier transform estimation was performed by means of the Matlab® fft function.

### *2.3.2 Dimensionless representation of frequency and pressure fluctuations*

Frequency and pressure fluctuations were represented dimensionless. The frequency of the spectra was defined in terms of the frequency coefficient (fn), according to IEC 60193 [35]:

$$f\_n = \frac{f}{n} \tag{6}$$

where *f* is the frequency of the spectral component (s�<sup>1</sup> ) and *n* is the rotation frequency of the rotor or impeller (s�<sup>1</sup> ). The pressure fluctuation signals were represented by pressure fluctuation factor (*P*~*E*), which is defined by IEC 60193 [35] as:

$$
\tilde{P}\_E = \frac{\tilde{p}}{\rho E} \tag{7}
$$

where *p*~ is the pressure fluctuation (N/m<sup>2</sup> ), *ρ* is the density (kg/m<sup>3</sup> ), and *E* is the specific energy (N.m/kg).

### *2.3.3 Phase analysis*

The phase shift angle between corresponding spectral components was computed using the coefficients of the Fourier analysis. Phase analysis is useful to determine whether hydrodynamic phenomena are moving within the volute or whether they manifest as pressure pulses that simultaneously affect all parts of the system. In the first case, the phase difference is expected to be approximately equal to the angular separation between the sensors, and in the second case, approximately equal to zero. A negative phase lag means that the first signal is ahead of the second, and a positive phase lag is the opposite.

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

The spectral analysis yields the spectral components or bands associated with hydrodynamic phenomena. To determine the flow instabilities, the signals are filtered into frequency bands containing the components of interest, and their instability in the time domain and then in the frequency domain is estimated on the filtered signals. The signals were filtered in the interest bands with digital FIR passband filters, designed with the Matlab® *designfilt* function. Filtering was performed on the signals from the pressure fluctuation sensors to determine the contribution of low-frequency phenomena to the total instability at each OP.

According to Hasmatuchi et al. [38], one measure of instability is the standard deviation. However, this statistic has the disadvantage that it does not have linear characteristics. That is, the standard deviation of the sum of the filtered signals is not equal to the sum of the standard deviations of each of the filtered signals. To overcome this limitation, the variance (Var) can be used as an indicator of the instability, since this statistic is linear in nature, and the following can be verified:

$$\operatorname{Var}\left(\sum\_{i=1}^{n}\ddot{\mathbf{x}}\_{i}\right) = \operatorname{Var}(\ddot{\mathbf{x}}\_{1}) + \operatorname{Var}(\ddot{\mathbf{x}}\_{2}) + \dots + \operatorname{Var}(\ddot{\mathbf{x}}\_{n})\tag{8}$$

where *x*~1,*x*~2, … , *x*~*<sup>n</sup>* are the filtered signals in the frequency bands that were identified in the spectral analysis.

Considering that the area under the power density spectrum (PSD) is equal to the mean squared value of the signal [39], the variance of a zero-mean series can be used to measure the instability. This relationship between the area under the PSD and the variance of a zero-mean series allows for comparing the estimated instabilities in the frequency domain and in the time domain.

For estimation in the frequency domain, it would be enough to determine the area under the PSD in a frequency band associated with a component of interest, and this would be its approximate measure of instability. And as seen in Eq. (5), the linear nature of Fourier analysis allows the summing of the spectral power associated with the different phenomena to estimate the total instability.

To compare the estimated instabilities in the time and frequency domains, it is necessary to use the same units of measurement of the signals in the different domains. For this purpose, the technique proposed by Heinzel et al. [37] was used to convert the power spectral units into engineering units. In this research, the power spectrum is used since its representation in engineering units coincides with the units of variance.
