The test rig is part of the hydraulics laboratory of Eafit University and consists of a closed circuit of pipes connected to a tank and a channel. The energy to move the water within the hydraulic system is supplied by recirculating pumps. The direction in which the flow is addressed inside the loop makes it possible to simulate the conditions of the hydraulic head, either to operate a turbomachine as a pump or as a turbine. A simplified sketch of the test rig is shown in figure 1.

Figure 1. Figure 1. Test rig sketch Source: Own elaboration

Pump under study was an ITT-Goulds Pump of 1.5 HP and manufactured in 316 stainless steel. Its main technical specifications are displayed in table 1. The pump was operated in a frequency range from 380 to 1751 rpm in pump mode and from 351 to 1902 rpm in turbine mode. Different rotation speeds were imposed with a frequency driver.

Instrumentation used to perform the tests was as follows:

[1] Wika A-10 piezoresistive pressure transducers in the high- and low-pressure zones.

From now on, the sensors mounted on the volute will be called as follows: The Dytran 2005V as PES (Piezo Electric Sensor), the Futek PMP300 as PRS (Piezo Resistive Sensor) and the Kistler 8704B50M1 as accelerometer. Figure 2 shows the location of the measuring instruments and figure 3 the polar location of PRS and PES.

Table 1. Table 1. ITT-Goulds Pump technical specifications Source: Own elaboration

Electrical signals transmitted by the sensors were received and processed by a National Instruments® CompactRio® 9076 Data Acquisition Equipment (DAE) and a software called Turbologger, developed in a LabView® environment by researchers from EAFIT University. The DAE was configured with four acquisition modules: Two NI-9232, one NI-9215 and one NI-9203. General technical specifications of the modules are shown in [18].

All signals were logged simultaneously and composed of 655360 data for each one of the variables at each operation point (OP). Operation points (OPs) were obtained by varying the impeller rotational speed and the flow rate. Between consecutive OPs the speed varied among 1 and 20 rpm and the flow rate among 0.1 and 0.2 l/s. Sampling rate was 20480 Hz. Acquired signals were used to observe harmonic phenomena within the alias-free band ranging from 0.03 Hz (fundamental Fourier frequency) and 10240 Hz (Nyquist frequency). Current and voltage signals were converted to engineering units of the International System of Units (SI).

Figure 2. Figure 2. Location of the instruments used in the measurements Source: Own elaboration

Figure 3. Figure 3. Polar location of PRS and PES Source: Adapted from ITT Goulds Pumps [19]

A convenient way to characterize all operation modes of a centrifugal pump is through the four-quadrant curve proposed by Knapp in 1937, which relates rotation speed and discharge [20]. The four-quadrant curve used in this investigation differs from Knapp’s in the use of dimensionless numbers, velocity and discharge factors, n_ED and Q_ED respectively, defined as established by IEC-60193 [13]. In addition to this, characteristic curves of velocity (n_ED) vs. torque (T_ED) factors, velocity (n_ED) vs. power (P_ED) factors and energy (E_nD) vs. flow (Q_nD) coefficients were made. Dimensionless factors and coefficients were also estimated according to the same standard. Factors and coefficients definitions involved in the characteristic curves are presented in equations (1)–(6).

(1)

(2)

(3)

(4)

(5)

(6)

Where, n is rotational velocity (s^-1), D is reference diameter (m), Q is discharge (m³/s), E is specific energy (Nm/kg), T is torque (Nm), ρ is density (kg/m³) and P stands for the power (Nm/s).

The characteristic curves were built with 335 points located in all possible modes of operation of the turbomachine under study.

Spectral analysis was performed on the signals from pressure sensors installed in the volute (PES and PRS) and for all OPs considered in the characteristic curves. The pressure signals were converted to their corresponding pressure fluctuation coefficients, , defined by equation (7), according to IEC-60193 [13].

(7)

Where, p is pressure (N/m²) and is the average pressure (N/m²)

Frequencies were also represented as dimensionless coefficient, f_n, defined by equation (8), agreeing with the standard IEC-60193 [13].

(8)

Spectral analysis was based on the Fast Fourier Transform of pressure signals. Frequency band from 0 to 2.5 times the rotation speed of the machine was regarded; i.e., 0 ≤ f_n ≤ 2.5. The spectral components clearly identified by the two sensors in the volute at the same frequency were taken under consideration. From this information, spectral behavior patterns were determined for each quadrant in the Q_ED-n_ED characteristic. Procedure details carried out can be found in [21].

The wave number, k, is estimated from the phase analysis. k is useful for determining the number of rotating cells and rotation speed in the case of Rotating Stall [10]. However, this estimate might be ambiguous since the frequency of passage of the phenomenon (f_pg) perceived by a sensor is either the effect of the passage of one cell rotating at high speed or the effect of several cells rotating at low speeds [22]. This ambiguity can be resolved with the graph of the cross-correlation of the pressure signals, previously filtered, as proposed by Best, LaFlamme and Moffatt [22], and Fortin and Moffat [23]. The proposal of these authors consists of identifying pairs of consecutive peaks with the greatest correlations, which represent the period mark of the hydrodynamic phenomenon. Then, counting the oscillations between these peaks corresponds to k. Nevertheless, the application of the method is not always suitable because the result of cross-correlation of filtered signals does not clearly show the peaks representing the period mark of the phenomenon and counting fluctuations between them becomes a subjective task. In order to overcome this problem, the following relationship was proposed between the phase angle estimated by the Fourier coefficients on the original signals, i.e. unfiltered signals, and the polar angle separating the two sensors on the volute. This is referred as the k-Fourier-coefficients method.

(9)

Where,

k_θ: Wave number estimated by the k-Fourier-coefficients method,

Deduction of equation (9) is shown in [21]. It was assumed that cells rotate at a constant speed and if several cells are present, they are equidistant.

To validate the k-Fourier-coefficients method, k was also estimated by filtering the signals and then calculating the cross-correlation of the signals. In this validation the pressure signals were filtered with digital bandpass filters that were chosen in such a way that they did not significantly affect the response of the signals in phase or amplitude. The filters regarded were Elliptic, Butterworth, Chebyshev type 1 and type 2 and Kaiser. The validation can be seen in [21].

For the analysis of rotating cells, k was adjusted according to the following hypotheses: First, the wave number is an integer number of stall cells that occupies the circular domain. Second, the passage frequency perceived by the sensors (f_pg ) is equal to the wave number by the frequency of the phenomenon (f), w.r.t. an inertial frame of reference [22]. Third, in order to fulfill the above hypothesis, it is necessary to correct the value of the frequency of the phenomenon (f) to guarantee that:

(10)

In terms of frequency coefficients, equation (10) is written as:

(11)

Phase analysis was performed for some components identified in the spectral analysis at operating points such as best efficiency, minimum torque, change of slope sign on the Q_ED-n_ED characteristic and those representing significant variations in pressure fluctuation.

Rotating Stall: It can be defined as a flow disturbance in a turbomachine resulting from backflows that partially or totally block the channels between the rotor blades. The backflows are associated with flow shedding at the solid-fluid interphase due to alterations in the angle of attack of the flow when it enters the channels between the blades. In turbines it is common to find RS at frequencies from 0.5 to 0.7 times the rotation frequency of the rotor [24]. In pumps, RS can be found at frequencies between 0.05 and 0.10 times the rotation frequency of the rotor [8, 28].

Figure 4 depicts characteristic curves Q_ED-n_ED (figure 4a), T_ED-n_ED (figure 4b) and P_ED-n_ED (figure 4c). Table 2 presents the values of speed factor, flow factor, torque factor and power factor for the Best Efficiency Point (BEP) in pump mode, the BEP in turbine mode and the runaway.

Figure 4. Figure 4. Characteristic curves of the pump under study

Source: Own elaboration

Q_ED-n_ED characteristic is properly called the four-quadrant curve, which are pointed out in figure 4a. The first quadrant corresponds to a pump, the second to a brake, the third to a turbine and the fourth to a reverse pump. The runaway point splits the third quadrant into two regions: turbine and brake turbine. [13]. In the case of the T_ED-n_ED and P_ED-n_ED curves, their quadrants do not correspond directly to the quadrants of the Q_ED-n_ED characteristic (see figures 4b and 4c).

Table 2. Table 2. Speed, flow, torque and power factors of the relevant points of the characteristic curves Source: Own elaboration

Figure 5 illustrates spectral patterns as a function of frequency and flow coefficients for the PES signals in the band 0 ≤ f_n ≤ 2.5. The spectra of the BEP in pump mode, BEP in turbine mode and runaway were highlighted. In quadrants 1, 2 and 3 a 2D representations Q_ED vs f_n are added too. Since only matching spectra were taken from the PES and PRS sensors, it is not necessary to display the PRS graph. Spectral components of the PES and PRS signals only differ in amplitude. The amplitude of the spectral components obtained by PRS signals was greater than that of the PES signals in most cases.

Figure 5. Figure 5. Spectral patterns in the QED-nED characteristic curve obtained from PES signals Source: Own elaboration

Spectral components 1X and 2X are present in all operation modes, albeit they are not evenly distributed along the whole range of flow coefficients. Component 1X represents the shaft frequency and 2X emerges from non-sinusoidal excitations at the shaft frequency [24]. Description of the other spectral patterns is shown in table 3.

Table 3. Table 3. Description of the spectral patterns Source: Own elaboration

Phase analysis was focused on the calculation of k-number by the k-Fourier-coefficients method (k_θ ) on some spectral components identified in spectral analysis. Most of these components are part of the spectral patterns depicted in figure 5.

Twelve OPs were selected from the Q_ED-n_ED characteristic curve for the phase analysis. The cross-correlation of the PES and PRS signals was computed to identify the most highly correlated spectral components for the selected OPs. Figure 6 shows the cross spectrum of the two signals with the spectral components considered in the phase analysis.

Figure 6. Figure 6. Cross-spectra of PES and PRS signals Source: Own elaboration

A chart depicting the spectra of 12 representative OPs on the Q_ED-n_ED characteristic curve is given in figure 7. It highlights the matching spectral components of PES (black spectra) and PRS (red spectra) and it shows each component of interest with its frequency detected by the sensors (f_n-pg), its k-number and the frequency of the hydrodynamic phenomenon (f_n). Spectral components f_n = 1 and f_n = 2 are not shown in the graph. Figures 8a and 8b display the location of the same 12 OP in the T_ED-n_ED and P_ED-n_ED characteristics, respectively.

The discrepancies between the spectra of the two sensors mounted on the volute (see figure 7) are given by the type of sensor. The PRS is a sensor used mainly for industrial process control and measures total pressure values, while the PES is a highly sensitive sensor used to measure pressure variations with respect to the average (pressure fluctuations). While the PRS can measure phenomena with frequencies up to 250 Hz, the PES can measure dynamic phenomena up to 40 KHz. Although the two sensors are very different, it is important to mention that the frequency range considered in this investigation was from 1 to 79.25 Hz, which is within the span of the two sensors.

Figure 7. Figure 7. QED-nED characteristic curve and some representative operating points with their corresponding amplitude spectra Source: Own elaboration

Figure 8. Figure 8. TED-nED and PED-nED characteristics with 12 operating points highlighted

Source: Own elaboration

It is important to notice that in the time domain the relationship between f_n and k-number can be seen using correlation of the filtered signals [22], [23]. Figure 9 shows an example of the correlation between the PES and PRS signals for a spectral component f_n = 1.85 in OP244 (BEP Turbine) with n = 18.3452 Hz. This graphical representation will be referred from now on BB-Plot (Bolaños-Botero Plot) since the contribution from the authors. The strips represent the period of the phenomenon (T) and the number of peaks or valleys into each strip corresponds well to k. The time differential between consecutive peaks corresponds to the passage period (T_pg), i.e., the period perceived by the sensor. The results shown in equations (12) and (13) can be estimated from figure 9.

(12)

(13)

Each strip in figure 9 contains 6 peaks or valleys, which means that k = 6. Hence the frequency of the phenomenon can be computed by dividing equation (13) by k, yielding what is presented in equation (14).

(14)

As expected, the passage frequency computed from the passage period estimated by the BB-Plot corresponds to the frequency identified in the spectrum of the sensors mounted in the volute (see figures 6 y 7).

The power of this approach lies in the ability to feature the period of the phenomenon that, as can be seen, is not evident in the bare graph of the cross correlation. This is a significant improvement to the graphical method proposed by Best, LaFlamme and Moffatt [22], and Fortin and Moffat [23], which is inappropriate for this case.

Figure 9. Figure 9. BB-Plot Source: Own elaboration

Pressure fluctuations and acceleration may be indicative of hydrodynamic phenomena [27]. In order to identify possible areas of operation of the turbomachine under study with the presence of this type of phenomena, the results of the spectral patterns (figure 5), the components of interest (figure 6) and the location of the OPs highlighted in the characteristic curves (figures 7 and 8) were related with the RMS curves of the PES and accelerometer versus Q_nD. Figure 10 shows the PES and accelerometer RMS curves and the possible operating zones (OZ) with hydrodynamic phenomena.

[1] Quadrant four did not display spectral components of interest other than f_n = 1 and f_n = 2 (see figure 5). However, it is important to notice that the high-pressure pulses and high vibrations are experienced in this quadrant. Figure 10b was deliberately trimmed on the vertical scale to maintain the aspect ratio with figure 10a. However, from Q_nD = 0, and below, the pressure pulses increase vertiginously. In the case of the RMS of a maximum value of 3.37 was reached, and for the RMS of the acceleration a value of 9.45 m/s².