Research on the influence of piping system on the fluid-dynamic noise of centrifugal pump

During the experimental research of the fluid-dynamic noise generated by a centrifugal pump, the accurate measurement of its acoustic characteristics can only be achieved when the pump is connected to a pipeline system. If the noise at a specific position of the centrifugal pump or its suction and discharge pipelines is used to directly describe the internal sound source characteristics of the pump during the test, there may be a broad difference in the experimental data depending on the specific system and the measuring position. Therefore, in order to verify the influence of different piping systems on the acoustic characteristics at a certain position of the inlet and outlet pipeline of the centrifugal pump, three different discharge pipeline schemes of the piping system were set up in this study. Meanwhile, this study solved the scattering matrix of a centrifugal pump using experimental methods. This method can be used independently of the testing system to describe the transmission and reflection characteristics at the ports of a centrifugal pump, and accurately solve the acoustic characteristics of the pump itself in future research.


Introduction
As the core equipment for fluid medium transportation in hydraulic systems, centrifugal pumps have extremely important applications in various fields such as urban municipal, biotechnology pharmaceuticals, petrochemicals, energy and power [1][2][3].In recent years, with the development of centrifugal pumps towards high power, high speed, compact structure and high reliability in various fields, the issue of vibrationand noise reduction of pumps and their piping systems, as the most important source of noise in hydraulic systems, have significant practical significance [4][5][6].
In the numerical simulation of centrifugal pumps, it is difficult to evaluate the sound pressure amplitude of a pump in the hydraulic system if the resonance condition is taken into account.This is because numerically solving the flow-induced noise of a pump requires the establishment of monopole sources, dipole sources, and quadrupole sources following Lighthill's acoustic analogy theory, and accurate settings are required for the selection of appropriate boundary conditions, turbulence models, and pipeline material characteristics [7,8].Considering the influence of the entire hydraulic system, this numerical simulation process is very complex.In addition, the working medium is usually considered as the homogeneous and incompressible ideal fluid, so the transmission characteristics and coupling effects of the sound pressure are not considered [9].
Therefore, more exhaustive and precise test data in hydraulic pipeline system are needed to support and validate the investigation of the internal acoustic characteristics of blade pumps.

Methodology
Due to the transmission of the medium in a centrifugal pump, the mutual effect of sound field parameters among different pipelines of a pump objectively exists in the hydraulic circulatory system [10,11].Therefore, a scattering matrix model is established to analyze this coupling relationship and to identify the acoustic characteristics of a test pump.In this study, it is hypothesized that the sound pressure waves propagating along the pipelines can be treated as plane waves.This hypothesis is based on the fact that, within the frequency range of up to 280 Hz, the diameter of the pipeline in the pump-pipeline system is significantly smaller (at least 40 times) compared to the wavelength of the sound pressure waves.Meanwhile, two different approaches are depicted to calculate the acoustic model of a pump at its ports as follows.

Pressure and velocity as port variables
Figure 1 shows a schematic representation of an acoustic pipeline section with two state variables (pulsating pressure   and volume velocity   ) at each monitoring point.In a pipeline, an ideal acoustic emission device radiates plane-wave along a uniform pipe of circular cross-section.The state variables between the upstream and downstream of the pipeline are linearly related by a transfer matrix τ: Based on acoustic wave equation ( ), plane-wave assumption and boundary conditions of detection areas (| =0 = 1 , | =0 = 1 , | = = 2 , | = = 2 ), the elements of the transfer matrix τ for a pipe section can be observed by means of algebraic manipulation, as following: where j represents the imaginary unit, k denotes the wave number, ρ•c represents the specific acoustic impedance of the plane wave, l and   are the distance of the monitoring points and the corresponding cross-sectional area of the reference pipeline, respectively.Figure 2 proposes an acoustic system of the composition based on pulsating pressure and volume velocity to analyze the acoustic properties of a centrifugal pump in a piping system.In terms of the transfer matrix T for the pump is indicated by the following equation: In this acoustic system, the measurable variables are pulsating pressure at monitoring points  1 ,  2 ,  5 and  6 .To identify the elements of the transfer matrix T of a pump, the corresponding relationships between the measurable variables of the pipeline and the port variables of the pump ( 3 ,  3 ,  4 ,  4 ) need to be established, showed as: where A and α are transfer matrices of pipeline section 1-2 and 2-3, respectively, which can be determined from the equations ( 1) and (2).Furthermore, in equation ( 4),  1 can be represented by the measurable variables  1 and  2 , that is  1 =( 2 −  11 •  1 )  12 ⁄ , then this value is substituted into the equation ( 5), obtained as: Similarly, we can calculate another set of port variables:  As a result, the elements of the transfer matrix T can be solved by substituting the equation ( 6)-( 9) into equation (3).To estimate the transfer matrix T of the test pump at different frequencies, the above calculation process is to be extended to complex quantities.

Plane pressure waves as port variables
The sound pressure signal decomposition technique is shown in Figure 3. On one end of the pipeline, there is an ideal sound source that emits plane pressure waves, and on the other end, the pipeline is equipped with the corresponding acoustic impedance  R .Two piezoelectric pressure sensors are flushinstalled at positions  1 and  2 , separated by a distance of l, to monitor the transient pressure fluctuations emitted by the ideal sound source.The values measured by these sensors are P1(x,t) and P2(x,t).The formulas for calculating the amplitudes  + ,  − and phases  + ,  − of the plane pressure waves propagating in opposite directions inside the pipeline at each frequency are as follows: where ω is an angular frequency.The presence of an external sound source and the acoustic impedance  R on the other end of the pipeline together result in the generation of pressure waves propagating in opposite directions.These pressure waves are a result of the acoustic interaction between the sound source and the pipeline system.It is important to note that when the acoustic impedance ZR is modified while keeping the sound source unchanged, the characteristics of these pressure waves also change accordingly.In Figure 4, the centrifugal pump is described as a linear dual-port acoustic component in the pipeline system.The suction/discharge pipelines of a pump in this system each contain two state variables: the pressure waves entering the pump (  + ,   + ) and the pressure waves exiting the pump (  − ,   − ).If the sound source originates from an external source, the relationship between these port variables can be depicted using the scattering matrix S, which is defined by the following equation: where the subscripts "i" and "o" represent the plane pressure waves propagating along the inlet and outlet pipelines of a pump,  11 and  22 ( 12 and S 21 ) denote the reflection coefficients (transmission coefficients) at the inlet port and outlet port, respectively.
To ascertain the scattering matrix, two pressure sensors were strategically positioned at the suction and discharge pipelines of a pump, separated by distances l1 and l2 respectively.By applying equations (10,11) to process signals acquired at different frequencies and pipeline configurations, a comprehensive dataset of experimental data on traveling pressure waves under various acoustic loads was generated.For each specific frequency, based on the acoustic characteristics of a pump itself, the scattering matrix elements will keep unchanged.This means that every individual set of pressure wave result (   + ,   + and   − ,   − ) mentioned above should validate equation (12).Consequently, by systematically applying equation ( 12) to all N data groups, two extensive systems of N equations are obtained.Every system involves only two unknown parameters, which correspond to the two elements of either the first or the second row of the scattering matrix.The overdetermined equations consisting of N equations mentioned above can be solved using the least-square error method.This method minimizes the error  1 and  2 defined in equations (13)(14): The advantages of this method are that it does not require determining the positions of the inlet and outlet ports at a pump, as well as the transfer matrices of the suction and discharge pipelines.Additionally, the computational solving process for this method is straightforward and convenient.

Experimental facility
In this study, a test rig for studying flow excitation noise was set up in which pressure waves generated by an external sound source were propagated through the connected pipeline towards the test pump.This test rig includes three interconnected systems: a hydraulic circulatory system, an operational control system, and a signal acquisition system.The diagram in Figure 5 illustrates the interdependence and interaction among these systems.The schematic diagram in Figure 6 shows the hydraulic circulation test system and experimental instruments.The test pump is a single-suction single-volute centrifugal pump, and its impeller contains 7 blades.The previous study has provided a detailed description of the main geometric parameters and hydraulic performance data of the test pump [12].The auxiliary pump, serving as an external sound source in the hydraulic circulation system, has a similar structure to the test pump, with 7 blades as well.
In this system, the test pump and auxiliary pump are connected through piping network.By adjusting the opening and closing of different valves, the test pump and auxiliary pump can be arranged in either series or parallel connection.The auxiliary pump operates at a very low flow rate as a sound source, which allows it to generate significant pressure pulsations.In Figure 6, the hydraulic circulation test system provides three options to adjust the magnitude and frequency of pressure fluctuations at various monitoring points along the inlet and outlet pipelines of the test pump: 1. Adjusting the valve opening, which controls the water flow rate in the pipeline system (change the amplitude of pressure pulsations at specific frequencies at monitoring points).
2. Modulating the variable-frequency drive of the auxiliary pump, i.e., adjusting the rotational speed of the auxiliary pump (alter the measurement frequency of monitor points and the amplitude of pressure pulsations at different frequencies).
3. Utilizing the transmission or reflection effects of sound at each component of the system, by changing the opening and closing of different valves in the hydraulic circulation system, resulting in different pipeline configurations (modify the amplitude of pressure pulsations at specific frequencies at monitor points).

Pressure wave monitoring under different pipeline configurations
Figure 7 illustrates the frequency domain plots of pressure pulsations captured by four pressure sensors (as shown in Figure 6) at various positions of the test pump.The monitor points P1 and P2 are arranged on the suction pipeline of the test pump, and monitor points P3 and P4 are mounted on the discharge pipeline.During the experiment, valve 3 and 5 remain closed while the rest of the valves were open (in Figure 6).The auxiliary pump, serving as the sound source, was running at lower flow rate (n=1200 r/min, fB=140 Hz) and is capable of generating larger pressure pulsation signals within the pipeline.To avoid the occurrence of any noise resulting from fluid dynamic excitation within the pump, the test pump was kept in a stationary state throughout the entire duration of the experiment.In Figure 7, pressure pulsation signals (at fS=20 Hz and fB=140 Hz) were obtained at the different monitor points along both the upstream/downstream pipelines of the pump.This results show that the upstream/downstream pipelines of the pump are able to receive energetic sound pressure waves transmitted from the sound source.In addition, it is noticeable from the figure that there are certain differences in the amplitude of pressure pulsations (especially at monitor points P1 and P2) at discrete frequency points with larger pressure pulsation amplitudes, such as at fS and fB.This indicates that the sound pressure waves from the sound source to the suction and discharge pipelines of the test pump are influenced by the acoustic impedance of the components in the system, and the data at each monitor points are the result of the superposition of sound pressure waves.To investigate the influence of pipeline system on the propagation of entering and exiting sound pressure waves within the upstream/downstream pipelines of a pump, four pipeline configuration schemes (Conf.#1 ~ Conf.#4) were designed as shown in Figure 8.By changing the opening or closing of different valves, the test pump and the pipelines formed different closed-loop circuits.In this experimental setup, the auxiliary pump served as an external sound source for a pump-pipeline system, and its operating conditions remained constant (defined fB=140 Hz).In Figure 9, SP and DP represent the suction and discharge pipelines of the test pump, respectively. + and  − indicate the amplitudes of the sound pressure waves entering and exiting the test pump, while  + and  − represent their corresponding phase values (calculated using equations (10a) ~ ( 11)).From the figure, it can be observed that the experimental results align with the expected results of the experimental design, indicating that the pipeline configuration has a significant impact on the state variables at both ports of the test pump.In comparison between Conf.#1 and Conf.#3, the amplitudes and phases of the pressure waves entering/exiting the upstream/downstream pipelines of a pump are similar at fB, while deviations occur at 2fB.This could be attributed to the fact that the valves located at the suction and discharge pipelines of the test pump have the same open/closed states.Similar phenomena also occur in Conf.#2 and Conf.#4.Meanwhile, comparing Conf.#1 and Conf.#2 (and Conf.#4), there are significant differences in sound pressure wave amplitudes and phases, especially in DP.The observed phenomenon can be attributed to the open state of the valve in the discharge pipeline of the test pump, which provides an explanation for it.Additionally, the observed phenomenon in Figure 9 indirectly reflects that it is not appropriate to use directly measured pressure pulsation spectra in the centrifugal pump pipelines to reflect the pump noise.This is because in certain cases monitoring results can potentially be misinterpreted due to interactions of sound pressure waves during sound propagation process within the pipelines.10 shows how the modulus and phase of each element in the scattering matrix of the test pump various with different frequencies.S11 and S22 (S12 and S21) represent the reflection (transmission) coefficients of the inlet and outlet ports of a pump, respectively.Experimental data for each element are obtained by solving equations ( 12) to (14).From the figure, it can be observed that although the test results are some dispersion, they exhibit a certain trend of variation with frequency.Based on the consideration of energy conservation, the modulus of elements S11 and S22 should not exceed unity at any frequency.When the modulus approaches unity, it indicates that most of the sound pressure waves are reflected back at the ports of the pump.In addition, the modulus of element S21 is always less than unity, while the modulus of element S12 is relatively high compared to S12.This phenomenon also satisfies the energy conservation.This experiment data show some similarities with the measurement data from the reference [13], including the data dispersion.Furthermore, as mentioned in reference [13], an alternative method for obtaining the pump scattering matrix involves solving the pump transfer matrix (refer to Section 2.1 for a detailed explanation of the procedure).In other words, there exists a direct relationship between the scattering and transfer matrices.Once one of them is solved, the other can be calculated correspondingly.

Conclusion
Directly measuring pressure pulsation in a centrifugal pump or at a certain point on its inlet or outlet pipeline cannot accurately depict the fluid-induced noise characteristics generated by the pump.In certain cases, monitoring results can potentially be misinterpreted due to interactions of sound pressure waves during sound propagation process within the pipelines.
This study presents a method that utilizes opposite direction plane pressure waves as port variables to solve the scattering matrix parameters of a centrifugal pump.This method can describe the transmission and reflection characteristics at the ports of the centrifugal pump independently of the testing system (including the pipeline system, valves, etc.), and accurately solve the acoustic characteristics of the pump itself in future research.

Figure 1 .
Figure 1.Schematic representation of an acoustic pipeline.

Figure 2 .
Figure 2. Acoustic system based on the transfer matrices.

𝐸 2 =Figure 4 .
Figure 4.A linear dual-port acoustic component in the pipeline system.

Figure 5 .
Figure 5. Schematic view of the interdependence and interaction among systems.

Figure 6 .
Figure 6.Schematic diagram of the hydraulic circulation test system and experimental instruments.

Figure 7 .
Figure 7. Frequency domain plots of pressure pulsations captured by four pressure sensors.

Figure 9 .
Figure 9. Pressure waves inside the suction and discharge pipelines under different pipeline configuration schemes (A represents the auxiliary pump, T represents the test pump).

Figure 10 .
Figure 10.Experimental values of each element of the pump scattering matrix. 11