An analysis of the pressure in a runner channel of a low-specific speed Francis model runner during resonance is presented, which includes experiments and the development of a pressure model to estimate both the convective and acoustic pressure field from the measurements. The pressure was measured with four pressure sensors mounted in the runner hub along one runner channel. The mechanical excitation of the runner corresponded to the forced excitation from rotor-stator interaction. The rotational speed was used to control the excitation frequency. The measurements found a clear resonance peak in the pressure field excited by the second harmonic of the guide vane passing frequency. From the developed pressure model, the eigenfrequency and damping were estimated. The convective pressure field seems to diminish almost linearly from the inlet to outlet of the runner, while the acoustic pressure field had the highest amplitudes in the middle of the runner channel. At resonance, the acoustic pressure clearly dominated over the convective pressure. As the turbine geometry is available to the public, it provides an opportunity for the researchers to verify their codes at resonance conditions.
Power plants with recently installed Francis runner have experienced breakdown after few running hours [
The way of measuring frequency response found in the literature is with the use of accelerometers and strain gauges, and the excitation methods found is pressure field excitation, impact excitation, or excitation with various vibration mechanisms as electronic muscles and shakers. The surrounding structure of the runner has high impact on the natural frequencies and damping; hence, the analysis should preferably be carried out with the runner mounted in the housing [
The objective of this paper is to investigate the use of pressure sensors to find the resonance frequency of a runner. RSI is used to excite a model of Francis turbine runner with forced excitations, and the mechanical response is measured with the pressure sensors mounted in one runner channel and one accelerometer mounted above one runner channel close to the inlet. In addition, to allow numerical research to verify their calculations, a model which separates the convective pressure field from the incompressible flow field and the acoustic pressure field from the acoustic-mechanical eigenmodes, have been developed. It produces useful estimates for the pressure fields, damping, and eigenfrequency which can be evaluated individually against the numerical calculations. This is a significant advantage as the researcher can verify each step of the calculation process and not only the final resulting pressure field. The turbine geometry in the current study is openly available through the Francis99 project and provides a unique opportunity for numerical researchers to verify their codes at resonance conditions [
The guide vanes create lift to direct the flow, and as a result, a circumferential repetitive pattern around the runner with zones of higher and lower pressure is created [
The runner in the current study is equipped with
Instantaneous illustration of the DM2 excitation. A sinusoidal signal with 28 periods is divided into 30 equal segments. The intersection line for each segment represents the pressure in each of the 30 runner channels. By plotting a curve through the intersection points, the overall pressure with 2 diametral modes appears.
The second harmonic of the guide vane passing frequency,
Instantaneous illustration of the DM4 excitation. A sinusoidal signal with 2∗28 periods is divided into 30 equal segments. The intersection line for each segment represents the pressure in each of the 30 runner channels. By plotting a curve through the intersection points, the overall pressure with 4 diametral modes appears.
The deflection pattern of the runner is defined by the excitation from the pressure field. Each deflection pattern with diametral modes can have different deflection amplitudes for the blades, hub and shroud. The blade mode is a deflection pattern of the runner where the blades have the largest amplitude, while the disc mode is a deflection pattern of the runner where the hub and shroud have the largest amplitudes. Modes with high deflection of the hub and ring, disc modes, could have higher damping due to the surrounding water and structure in the housing compared to blade modes [
The Francis test rig available at the Waterpower laboratory, Norwegian University of Science and Technology, was used for the experimental studies as shown in Figure
The Francis test rig, including the high- and low-pressure vessel.
Three-dimensional view of the investigated Francis turbine.
Hill chart of the investigated runner.
Measurements involving moving fluids can be severely influenced by the mounting method of the sensor [
Figure
Onboard pressure sensors. The distance between the sensors is indicated.
Overview of the measurement chain.
All pressure values presented were calculated as percentage of specific hydraulic energy of the machine (E = gH) and denoted
Both the dynamic convective pressure (
The acoustic pressure was modelled as the response of a coupled acoustic mechanical 2DOF system to accommodate higher modes while still limiting the number of unknowns in the model. As the fluid at the wall acts with the effect of added mass, the pressure has to be proportional to the frequency squared. A frequency proportionality
Here
The model has a total of 28 unknowns, 8 of which are for the complex convective pressure field, another 16 are for the complex acoustic pressure field, two are for the eigenfrequencies, and two are for the scaling constants
The accelerometer was modelled without the frequency squared in the amplification
With the accelerometer included in the fit, the total number of unknowns was 34.
The frequencies for the forced excitation were controlled by changing the speed of the runner while changing the head to keep the speed factor (
Measurement summary.
Description | Flow |
|
|
Efficiency | Head |
|
Speed |
---|---|---|---|---|---|---|---|
(m3/s) | (–) | (–) | (–) | (m) | (°) | (rpm) | |
BEP1 | 0.107 | 0.185 | 0.152 | 0.916 | 3.45 | 10.0 | 185.1 |
BEP2 | 0.134 | 0.179 | 0.154 | 0.916 | 5.2 | 10.0 | 219.8 |
BEP3 | 0.160 | 0.176 | 0.156 | 0.920 | 7.2 | 10.0 | 254.3 |
BEP4 | 0.183 | 0.178 | 0.154 | 0.919 | 9.6 | 10.0 | 297.8 |
BEP5 | 0.209 | 0.178 | 0.155 | 0.920 | 12.6 | 10.0 | 340.5 |
BEP6 | 0.232 | 0.180 | 0.154 | 0.920 | 15.55 | 10.0 | 381.7 |
The uncertainty of the measurements was calculated from calibrations, vibration sensitivity, and repeatability of the measurements. Static calibration of the pressure sensors was initially done in an estimated pressure range for the measurements with a GE P3000 Series pneumatic deadweight tester as the primary reference. As the evaluation of the pressure amplitudes was a dynamic quantity, dynamic uncertainty was addressed. All components in the current pressure measurement chain, from the sensors to the data acquisition, were stated to have resonance frequencies above 10 kHz; hence, the dynamic uncertainty was assumed to be neglectable, and only repeatability and hysteresis from static calibration remained in the uncertainty evaluation [
A vibration test with the runner surrounded by air was conducted to analyze the pressure sensors vibration sensitivity. An unbalanced mass shaker was used to excite the runner from 0 to 850 Hz. The response was measured with the accelerometer and the pressure sensors to evaluate the vibration sensitivity of the pressure sensors. Four test points were selected from the frequency response with different acceleration amplitudes and repeated with constant frequency. The results are shown in Figure
Vibration sensitivity test of the pressure sensors. The runner was excited in air with a frequency sweep with an unbalanced mass shaker. Four test points were repeated with constant frequency. The response was measured with the accelerometer and the pressure sensors.
The highest vibration amplitude measured by the accelerometer in the vibration test was 17 ms−2. The highest measured vibration amplitude for the second harmonic of the guide vane passing frequency in the measurements BEP1 to BEP6 was 0.05 ms−2 in BEP6. From the results shown in Figure
To analyze the variation of the blade-passing amplitude for each sensor, a short-time fast Fourier transform (STFFT) was used. The analysis was performed with a window length equal to 50 periods of the RSI signal with each window starting at the same relative position to the signal period. The amplitudes were found to be normally distributed, and a 95% confidence interval was calculated. The uncertainty budget for the RSI amplitudes is presented in Table
Uncertainty budget for the fundamental RSI amplitudes, BEP5.
Location | Amplitude RMS of fundamental frequency RSI | Calibration repeatability, |
Vibration sensitivity, |
Repeatability of the measurements, |
Total relative uncertainty, |
---|---|---|---|---|---|
R1 | 0.94 kPa | 0.01 kPa | 0.003 kPa | 0.015 kPa | 1.9 |
R2 | 0.70 kPa | 0.01 kPa | 0.003 kPa | 0.011 kPa | 2.2 |
R3 | 0.53 kPa | 0.006 kPa | 0.003 kPa | 0.010 kPa | 2.3 |
R4 | 0.30 kPa | 0.006 kPa | 0.003 kPa | 0.012 kPa | 4.6 |
A | 0.028 ms−2 | 1% | — | 0.002 m/s−2 | 7.1 |
The vibration sensitivity was calculated as 0.13 times the acceleration amplitude.
Antialiasing filters according to the Nyquist–Shannon sampling theorem were used on all measurement channels. To reduce the noise sensitivity, amplification of the measured signals was done close to the sensors inside the runner hub.
For the accelerometer measurements, no normalization recommendations could be found in the literature. The goal of the normalization was to compare the acceleration as if the frequencies and the driving force were the same for each step in rotational speed and head. A speed ratio coefficient
The acceleration is
Rearranged to achieve constant acceleration amplitude independent of speed and head,
The amplitudes of the fundamental guide vane passing frequency are shown in Figure
The RSI fundamental frequency. The error bars represent the uncertainty from calibration and the experimental repeatability of the amplitudes.
The second harmonic of the RSI guide vane passing frequency had an increasing trend towards the measurement at 280 Hz as shown in Figure
The RSI second harmonic frequency. Pressure amplitudes are normalized to the potential energy. The error bars represent the uncertainty from calibration and the experimental repeatability of the amplitudes.
The following steps were performed for the fitting of the measured data: An appropriate mathematical model was selected as described in the pressure model section A merit function was defined as the sum of square error (equation ( The parameters were adjusted for the best fit by minimizing the merit function with a constrained nonlinear minimizing routine The goodness of fit was evaluated with the coefficient of determination, The accuracy of the best fit parameters was estimated with Monte Carlo simulation
The number of data points from the measurements was 60. Six measurements with 5 sensors were with amplitude and phase information. The number of model parameters was required to be less than the number of measurement to limit the degree of freedom in the fitting and avoid overfitting. The measurements were fitted to a model with convective pressure and acoustic pressure with two acoustic modes, giving 34 model parameters. The merit function was calculated as the sum of the weighted square errors as follows:
Goodness of fit
Amplitude | Phase | |
---|---|---|
R1 | 0.903 | 0.933 |
R2 | 0.986 | 0.961 |
R3 | 0.994 | 0.990 |
R4 | 0.996 | 0.984 |
A | 0.991 | 0.894 |
The
Residuals for the calculated model with two degrees of freedom in the acoustic pressure.
The uncertainty of the fitting coefficients was calculated from Monte Carlo Method (MCM) simulation. The calibration uncertainties in each measurement point were not independent as required in the MCM simulation. The input to the calculation was therefore generated from normally distributed random numbers in the uncertainty interval for the calibration, combined with a normally distributed random number from the measurement uncertainty as shown in equation (
The first 1000 inputs to the MCM simulation of the amplitude data from pressure sensor R1. Each line is plotted with light color, and darker color represents coinciding lines. Measurement data are indicated.
The distribution of the coefficients from the MCM simulation was analyzed, and the 95% confidence interval was found with the empirical cumulative distribution function. Two example calculations are shown in Figure
Two examples of the histogram, the distribution and the cumulative probability of the coefficients
Figure
Least square fitting of the pressure measurements. (a) R1. (b) R2. (c) R3. (d) R4.
The accelerometer fit.
Fitting coefficients.
Description | Value and 95% interval |
---|---|
|
|
|
0.0905 (−0.0105, 0.0109) |
|
0.0644 (−0.0093, 0.0098) |
|
0.0397 (−0.0051, 0.0055) |
|
0.0194 (−0.0047, 0.0052) |
|
0.0002 (−0.0001, 0.0001) |
|
0.0058 (−0.0033, 0.0033) |
|
0.0024 (−0.0046, 0.0048) |
|
0.0001 (−0.0020, 0.0022) |
|
0.0002 (−0.0025, 0.0025) |
|
−0.0012 (−0.0002, 0.0002) |
|
|
|
0.0057 (−0.0020, 0.0019) |
|
0.0061 (−0.0022, 0.0021) |
|
0.0111 (−0.0021, 0.0019) |
|
0.0123 (−0.0021, 0.0019) |
|
0.0003 (−0.0001, 0.0000) |
|
0.0099 (−0.0014, 0.0014) |
|
0.0150 (−0.0018, 0.0018) |
|
0.0146 (−0.0015, 0.0015) |
|
0.0102 (−0.0013, 0.0014) |
|
0.0003 (0.0000, 0.0000) |
|
|
|
0.0013 (−0.0010, 0.0011) |
|
0.0053 (−0.0016, 0.0020) |
|
0.0052 (−0.0013, 0.0018) |
|
0.0013 (−0.0011, 0.0013) |
|
−0.0005 (−0.0001, 0.0001) |
|
−0.0020 (−0.0014, 0.0011) |
|
−0.0006 (−0.0014, 0.0014) |
|
−0.0015 (−0.0011, 0.0009) |
|
−0.0041 (−0.0017, 0.0012) |
|
0.0003 (−0.0001, 0.0001) |
|
|
|
272.4895 (−0.9329, 0.8634) |
|
0.0510 (−0.0051, 0.0045) |
|
325.5277 (−2.6368, 2.7486) |
|
0.0813 (−0.0125, 0.0153) |
The calculated amplitudes from the fitting coefficients in Table
With the use of equation (6) and
The first measurement point (BEP1) had the highest uncertainty, and the proposed pressure model did not fit the phase in any of the pressure measurements, while the accelerometer model was within the range of the uncertainty. The amplitudes in the BEP1 were low, and also the test rig was operated at a very low head, giving the high uncertainty. For the higher frequencies, the fit was very good for all sensors giving reasons to believe the proposed model is valid. Questions can be raised about the number of measurements and the frequency step, and the accuracy would likely be better. The analysis method can separate experimental pressure data into convective and acoustic pressure. The convective pressure represents the pressure field not influenced by the vibrating structure, while the acoustic pressure represents the pressure from the fluid structure interaction.
The pressure measurements were able to find a resonance, and the proposed pressure model was able to calculate the convective and acoustic parts of the pressure for the investigated runner. By separating the measured pressure amplitudes into an acoustic and a convective pressure field, their individual shapes together with the eigenfrequency and the damping were estimated. The convective pressure field diminishes almost linearly from the inlet to the outlet, while the acoustic pressure field had the highest amplitudes in the middle of the runner channel. At resonance, the acoustic pressure clearly dominates over the convective. For the acoustic1 mode, the estimated eigenfrequency was 272 Hz while the damping was in the range of 2.5% to 5.1% depending on the runner speed. The acoustic2 mode was estimated to 326 Hz with the damping in the range of 4% to 8%. It is shown that the analysis method can separate experimental data corresponding to the different results of numerical analyses. The convective pressure represents the output from computational fluid dynamics (CFD), and the acoustic pressure, resonance frequency, and damping represent the output from simulations as modal and flutter analysis.
Two degrees of freedom
Amplifier
Best efficiency point
Computation fluid dynamics
Data acquisition
International Electrotechnical Commission
Integrated Electronics Piezo-Electric
Monte Carlo method
Diametral mode
Rotor-stator interaction
Short-Time fast Fourier transform
Volt direct current.
Accelerometer mode
Guide vane opening
Fluctuating acceleration
95% repeatability from calibration
95% vibration sensitivity
95% repeatability of the measurements
95% total uncertainty
Damping coefficient
Standard deviation
Standard deviation of the calibration
Standard deviation of the measurement
Acoustic pressure mode
Frequency
Frequency proportionality
Head of machine
Damping proportionality constant
Dimensionless speed factor
Fluctuating pressure
Dimensionless discharge.
Acoustic pressure
Convective pressure
Specific hydraulic energy of machine
Sensor identification
Imaginary part
Real part
Total pressure.
The measurement data and geometry used to support the findings of this study are available at the Francis-99 website [
The authors declare that they have no conflicts of interest.
The experiments were conducted under the HiFrancis research project. The authors are grateful for all the support from the technical staff at the Waterpower laboratory to make the measurements possible. This work was funded by Energy Norway, Norwegian Research Council, and the Norwegian Hydropower Center.