#
Numerical and Experimental Investigations on the Acoustic Characteristics of a Single-Stage Centrifugal Pump^{ †}

^{1}

^{2}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Approaches

#### 2.1. Determination of Acoustic Pump Parameters

- ${\underline{\widehat{f}}}_{s},{\underline{\widehat{g}}}_{s}\phantom{\rule{0.277778em}{0ex}}$ and $\phantom{\rule{0.277778em}{0ex}}{\underline{\widehat{f}}}_{d},{\underline{\widehat{g}}}_{d}$: up- and downstream-propagating pressure waves, with $\underline{\widehat{f}}$ and $\underline{\widehat{g}}$ quantifying the acoustic pressure fields on the suction (s) and discharge (d) side of the pump,
- ${\underline{\underline{S}}}_{fg}$: the pump’s scattering matrix, which contains the four transmission parameters ${\underline{\widehat{r}}}_{s}$, ${\underline{\widehat{t}}}_{sd}$, ${\underline{\widehat{r}}}_{d}$, and ${\underline{\widehat{t}}}_{ds}$,
- ${\underline{\widehat{f}}}_{src.}\phantom{\rule{0.277778em}{0ex}}$ and $\phantom{\rule{0.277778em}{0ex}}{\underline{\widehat{g}}}_{src.}$: the pump’s source (src.) parameters in the form of up- and downstream-propagating pressure waves.

- $\zeta <1$: the part different to one is dissipated (or transformed) within the four-pole;
- $\zeta =1$: the four-pole is a purely passive acoustic element;
- $\zeta >1$: the part different to one is generated (or transformed) within the four-pole.

#### 2.2. Experimental Approach

#### 2.3. 3D CFD Approach

- (i)
- the pump is left at standstill and treated with sound from external sources (src.) to evaluate the transmission parameters;
- (ii)
- several operation points are adjusted to calculate the monopole and dipole amplitudes from the CFD simulations.

#### 2.4. 1D Modelling

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## 5. Nomenclature and Abbreviations

a | Speed of sound | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

A | Cross-sectional area | ${\mathrm{m}}^{2}$ |

c | Mean velocity | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

${c}_{p}$ | Specific heat at constant pressure | $\mathrm{J}\phantom{\rule{0.277778em}{0ex}}\mathrm{k}{\mathrm{g}}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{K}}^{-1}$ |

$\underline{\widehat{c}}$ | Sound particle velocity (complex amplitude) | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

d | Diameter | m |

E | Young’s modulus | $\mathrm{P}\mathrm{a}$ |

f | Frequency | $\mathrm{H}\mathrm{z}$ |

$\underline{\widehat{f}}$ | Downstream pressure wave (complex amplitude) | $\mathrm{P}\mathrm{a}$ |

g | Gravitational acceleration | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}$ |

$\underline{\widehat{g}}$ | Upstream pressure wave (complex amplitude) | $\mathrm{P}\mathrm{a}$ |

h | Discharge head | m |

$He$ | Helmholtz number | − |

k | Wave number | ${\mathrm{m}}^{-1}$ |

l | Length | m |

L | Perimeter | m |

n | Rotational speed | $\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}$ |

p | Mean pressure | $\mathrm{P}\mathrm{a}$ |

$\underline{\widehat{p}}$ | Sound pressure (complex amplitude) | $\mathrm{P}\mathrm{a}$ |

${q}_{rel}$ | Relative flow rate | − |

$\underline{\widehat{q}}$ | Sound flux (complex amplitude) | ${\mathrm{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

$\dot{Q}$ | Flow rate | ${\mathrm{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}$ |

$\underline{\widehat{r}}$ | Sound reflection parameter (complex amplitude) | − |

s | Wall thickness | m |

$\underline{\underline{S}}$ | Scattering matrix | |

t | Time | s |

$\underline{\widehat{t}}$ | Sound transmission parameter (complex amplitude) | − |

T | Temperature | K |

$\underline{\underline{T}}$ | Transfer matrix | |

u | Circumferential speed | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

V | Fluid volume | ${\mathrm{m}}^{3}$ |

x | Local coordinate relative to the pump’s flanges | m |

${y}^{+}$ | Dimensionless wall distance | − |

z | Specific acoustic impedance | $\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

Z | Acoustic impedance | $\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-4}\phantom{\rule{0.166667em}{0ex}}{s}^{-1}$ |

$\alpha $ | Attenuation coefficient | ${\mathrm{m}}^{-1}$ |

$\gamma $ | Ratio of specific heats | − |

$\u03f5$ | Relative error | % |

$\zeta $ | Acoustic power ratio | − |

$\kappa $ | Thermal conductivity | $\mathrm{W}\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{K}}^{-1}$ |

$\lambda $ | Acoustic wave length | m |

$\mu $ | Dynamic viscosity | $\mathrm{k}\mathrm{g}\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{s}}^{-1}$ |

$\nu $ | Poisson’s ratio | − |

$\rho $ | Mean density | $\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

Amp. | Amplitude |

BC | Boundary condition |

BP | Blade-passing |

CFD | Computational fluid dynamics |

CH | Channel |

compl. | Compliant |

CSJ | Cross-section jump |

DFT | Discrete Fourier transformation |

DP | Discharge port |

EOS | Equation of state |

FEM | Finite element method |

FD | Finite difference |

FSI | Fluid–structure interaction |

FV | Finite volume |

IF | Interface |

MOC | Method of characteristics |

no. | Number of |

NRBC | Non-reflective boundary condition |

opt. | Optional |

PE | Piezoelectric |

URANS | Unsteady Reynolds-averaged Navier–Stokes |

RSI | Rotor–stator interaction |

SIMPLE | Semi-implicit method for pressure linked equations |

SP | Suction port |

src. | Source |

SST | Shear stress transport |

TMM | Transfer matrix method |

1D | One-dimensional |

3D | Three-dimensional |

ac | Acoustic |

d | Discharge |

CH | Chamber |

DP | Discharge port |

eff | Effective |

el. | Element |

f | Fluid |

n | Normalised |

N | Nominal |

p | Pump |

s | Suction |

SP | Suction port |

rel | Relative |

src. | Source |

sys | (piping) System (connected to the pump) |

t | Tube |

0 | Undeformed |

Δp | Deformed |

## Appendix A

## Appendix B

**Table A1.**Quantities used to determine $\alpha $ at ambient conditions ($T=20\xb0$, atmospheric pressure), taken from [25].

Parameter | Unit | Air | Water |
---|---|---|---|

$\mu $ | $\mathrm{k}\mathrm{g}\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{s}}^{-1}$ | $18.2\times {10}^{-6}$ | $1.002\times {10}^{-3}$ |

$\gamma $ | − | 1.4 | 1 |

$\kappa $ | $\mathrm{W}\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{K}}^{-1}$ | 0.0626 | 0.597 |

${c}_{p}$ | $\mathrm{J}\phantom{\rule{0.277778em}{0ex}}\mathrm{k}{\mathrm{g}}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{K}}^{-1}$ | $1.005\times {10}^{3}$ | $4.18\times {10}^{3}$ |

**Figure A1.**Amplitude and phase of transmission parameters ${\underline{\widehat{r}}}_{d}$ and ${\underline{\widehat{t}}}_{ds}$ evaluated from experiments (water and air), CFD, and 1D simulations as functions of Helmholtz number. Corresponding frequencies f for water are displayed in brackets below the abscissa. (Data: Supplementary Files S5–S12).

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**Figure 2.**(

**a**) Closed−circuit centrifugal pump test rig and (

**b**) schematic representation with measurement positions and named components.

**Figure 3.**Structure of the simulation area with position information on used monitor points (CSJ = cross-section jump, NR = non-reflective boundary condition, IF = interface, src. = source, opt. = optional).

**Figure 4.**FEM model setup and subdivision of fluid volumes in the pump.

^{1}Not considered for the simulation.

**Figure 5.**Total displacements of the pump’s casing calculated by means of static mechanical FEM simulations (the colour code corresponds to the original displacements whereas the moving patterns in the picture are scaled by $1300\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}$“true scale”).

**Figure 6.**Pump’s characteristic head curves evaluated from experiments and 3D CFD simulations for two different rotational speeds. (Data: Supplementary Files S1–S4).

**Figure 7.**Amplitude and phase of transmission parameters ${\underline{\widehat{r}}}_{s}$ and ${\underline{\widehat{t}}}_{sd}$ evaluated from experiments (water and air), 3D CFD, and 1D simulations as functions of Helmholtz number. Corresponding frequencies f for water are displayed in brackets below the abscissa. (Data: Supplementary Files S5–S12).

**Figure 8.**Acoustic power ratio $\zeta $ evaluated from transmission parameters for experiments, 3D CFD, and 1D simulations as functions of Helmholtz number $He$. (Data: Supplementary Files S13–S18).

**Figure 9.**Amplitudes of normalised source term parameters for monopole and dipole at blade-passing frequency (${f}_{BP}\approx [111;169]$ Hz) as functions of the relative flow rate evaluated from experiments and 3D CFD simulations for two different rotational speeds. (Data: Supplementary Files S19–S24).

**Table 1.**Specifications of the pump under investigation (${n}_{s}$ = specific speed (according to ${n}_{s}={n}_{N}\frac{\sqrt{{\dot{Q}}_{N}}}{{{h}_{N}}^{0.75}}$ with ${n}_{N}$ in $\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}$, ${\dot{Q}}_{N}$ in ${\mathrm{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, and ${h}_{N}$ in m), ${d}_{2}$ = impeller outlet diameter; abbreviations: N = nominal, no. = number of).

${\mathit{n}}_{\mathit{s}}$$\left({\mathbf{min}}^{-1}\right)$ | ${\mathit{d}}_{2}$$\left(\mathbf{mm}\right)$ | $\frac{{\mathit{d}}_{2}-{\mathit{d}}_{\mathbf{tongue}}}{2}$$\left(\mathbf{mm}\right)$ | No. Blades $(-)$ | ${\mathit{n}}_{\mathit{N}}$$\left({\mathbf{min}}^{-1}\right)$ | ${\mathit{h}}_{\mathit{N}}$$\left(\mathbf{m}\right)$ | ${\dot{\mathit{Q}}}_{\mathit{N}}$$\left({\mathbf{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\mathbf{h}}^{-1}\right)$ | ${\mathit{N}\mathit{P}\mathit{S}\mathit{H}}_{3\%}$$\left(\mathbf{m}\right)$ |
---|---|---|---|---|---|---|---|

12 | 222 | 7.5 | 7 | 1450 | 15 | 15.5 | 1.57 |

Sensor | ${\mathit{p}}_{\mathit{s}1}$ | ${\mathit{p}}_{\mathit{s}2}$ | ${\mathit{p}}_{\mathit{s}3}$ | ${\mathit{p}}_{\mathit{s}4}$ | ${\mathit{p}}_{\mathit{s}}$ | ${\mathit{p}}_{\mathit{d}}$ | ${\mathit{p}}_{\mathit{d}1}$ | ${\mathit{p}}_{\mathit{d}2}$ | ${\mathit{p}}_{\mathit{d}3}$ | ${\mathit{p}}_{\mathit{d}4}$ |
---|---|---|---|---|---|---|---|---|---|---|

Distance $\Delta x$ (m) | −1.16 | −0.89 | −0.57 | −0.26 | −0.145 | 0.175 | 0.26 | 0.57 | 0.89 | 1.16 |

Parameter | Unit | Volume 1 | Volume 2 | Volume 3 |
---|---|---|---|---|

$\Delta p$ | $\left(\mathrm{Pa}\right)$ | $1\times {10}^{5}$ | $1\times {10}^{5}$ | $1\times {10}^{5}$ |

${V}_{0}$ | $\left(\mathrm{m}{\mathrm{m}}^{3}\right)$ | 114,648.02 | 2,287,892.89 * | 40,854.596 |

${V}_{\Delta p}$ | $\left(\mathrm{m}{\mathrm{m}}^{3}\right)$ | 114,651.32 | 2,288,409.65 * | 40,854.540 |

$\Delta V\phantom{\rule{3.33333pt}{0ex}}$^{†} | $\left(\mathrm{m}{\mathrm{m}}^{3}\right)$ | 3.3 | 516.76 | −0.056 |

${a}_{eff}\phantom{\rule{3.33333pt}{0ex}}$^{‡} | $\left(\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\right)$ | 1158 | 607 | 1500 |

^{†}Determined by $\Delta V={V}_{\Delta p}-{V}_{0}$.

^{‡}Determined using Equation (14).

Parameter | Unit | Pipe (s) | SP | CH | DP | Pipe (d) |
---|---|---|---|---|---|---|

d | $\left(\mathrm{mm}\right)$ | 54.5 | 56 | 86 | 26 | 37.2 |

l | $\left(\mathrm{mm}\right)$ | 1 | 49 | 388 | 70 | 1 |

${a}_{eff}$ | $\left(\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\right)$ | 1350 | 1158 | 607 | 1500 | 1350 |

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## Share and Cite

**MDPI and ACS Style**

Lehr, C.; Munsch, P.; Skoda, R.; Brümmer, A.
Numerical and Experimental Investigations on the Acoustic Characteristics of a Single-Stage Centrifugal Pump. *Int. J. Turbomach. Propuls. Power* **2024**, *9*, 8.
https://doi.org/10.3390/ijtpp9010008

**AMA Style**

Lehr C, Munsch P, Skoda R, Brümmer A.
Numerical and Experimental Investigations on the Acoustic Characteristics of a Single-Stage Centrifugal Pump. *International Journal of Turbomachinery, Propulsion and Power*. 2024; 9(1):8.
https://doi.org/10.3390/ijtpp9010008

**Chicago/Turabian Style**

Lehr, Christian, Pascal Munsch, Romuald Skoda, and Andreas Brümmer.
2024. "Numerical and Experimental Investigations on the Acoustic Characteristics of a Single-Stage Centrifugal Pump" *International Journal of Turbomachinery, Propulsion and Power* 9, no. 1: 8.
https://doi.org/10.3390/ijtpp9010008