#
Experimental Investigation on the Acoustic Scattering Matrix for a Centrifugal Pump^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

^{+}) and negative (P

^{−}) directions respectively, by means of the two-microphone procedure. This paper shows the elements of the scattering matrix determined for the test pump as a function of frequency. These results represent a reference for subsequent theoretical research on the acoustic scattering matrix of centrifugal pumps.

## 1. Introduction

## 2. Hydraulic System

## 3. Methodology

_{R}. The pressure sensors at positions X

_{2}and X

_{1}, separated by a distance S, measure the pressure fluctuations P

_{1}(x,t) and P

_{2}(x,t) respectively. The amplitudes ${P}^{+},{P}^{-}$ and phases ${\phi}_{+},{\phi}_{-}$ of each plane pressure wave must satisfy the following relationships for each frequency:

_{1}and L

_{2}were installed at the suction and discharge pipes of test pump respectively. The signals obtained at different frequencies and for different circuit configurations were processed by using Equations (1)~(3) resulting in a large set of experimental data on travelling pressure waves under different acoustic loads. However, the elements of the scattering matrix are expected to remain constant for a given frequency, i.e., each single set of experimental data should verify Equation (4). Therefore, the systematic application of Equation (4) to all the N data groups results in two large systems of N equations, each with just two unknowns: the two elements of either the first or the second line of S. These two overdetermined systems of N equations can be solved by means of a least square errror procedure, so that the errors ${\mathrm{E}}_{1}$ and ${\mathrm{E}}_{2}$ as defined by Equations (5) and (6) become minima:

## 4. Pump Scattering Matrix

_{11}and S

_{22}should not be higher than 1, and this is what happens at all frequencies. In particular, S

_{22}takes values close to 1, which means that most of the on-coming sound to the discharge port becomes reflected back. Besides, S

_{21}is always lower than unity but S

_{12}becomes higher. The latter is just due to the lower cross section of the discharge pipe relative to the suction side, and so the energy conservation is still verified, as expected. In general, the results obtained compare very well to those measured by Bardeleben et al. [6], data dispersion included, who tested a pump with a specific speed of 0.312. The most significant difference corresponds to the reflection coefficient S

_{11}, but this discrepancy can be explained by the different specific speed between both pumps (50% higher for the present pump) and the corresponding difference in suction geometry.

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Choi, J.S.; McLaughlin, D.K.; Thompson, D.E. Experiments on the unsteady flow field and noise generation in a centrifugal pump impeller. J. Sound Vib.
**2003**, 263, 493–514. [Google Scholar] [CrossRef] - Chu, S; Dong, R; Katz, J. Relationship between unsteady flow, pressure fluctuations, and noise in a centrifugal pump—Part A: Use of PDV data to compute the pressure field. J Fluid Eng.
**1995**, 117, 24–29. [Google Scholar] [CrossRef] - Yamamoto, K.; Müller, A.; Ashida, T.; Yonezawa, T.; Avellan, F.; Tsujimoto, Y. Experimental method for the evaluation of the dynamic transfer matrix using pressure transducers. J. Hydraul. Res.
**2015**, 53, 466–477. [Google Scholar] [CrossRef] - Munjal, M.L. Acoustics of Ducts and Mufflers; John Wiley & Sons: Chichester, UK, 2014. [Google Scholar]
- Parrondo, J.; Pérez, J.; Barrio, R.; Gonzalez, J. A simple acoustic model to characterize the internal low frequency sound field in centrifugal pumps. Appl. Acoust.
**2011**, 72, 59–64. [Google Scholar] [CrossRef] - Bardeleben, M.J.R.; Weaver, D.S. Estimation of the acoustic scattering matrix for a centrifugal pump. In Proceedings of the ASME 2002 International Mechanical Engineering Congress and Exposition, New Orleans, LA, USA, 17–22 November 2002. [Google Scholar]

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**MDPI and ACS Style**

Li, G.; Parrondo, J.; Wang, Y.
Experimental Investigation on the Acoustic Scattering Matrix for a Centrifugal Pump. *Proceedings* **2018**, *2*, 1489.
https://doi.org/10.3390/proceedings2231489

**AMA Style**

Li G, Parrondo J, Wang Y.
Experimental Investigation on the Acoustic Scattering Matrix for a Centrifugal Pump. *Proceedings*. 2018; 2(23):1489.
https://doi.org/10.3390/proceedings2231489

**Chicago/Turabian Style**

Li, Guidong, Jorge Parrondo, and Yang Wang.
2018. "Experimental Investigation on the Acoustic Scattering Matrix for a Centrifugal Pump" *Proceedings* 2, no. 23: 1489.
https://doi.org/10.3390/proceedings2231489