# Inner Damping of Water in Conduit of Hydraulic Power Plant

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Continuity Equation

#### 2.2. Momentum Equation

#### 2.3. Second Viscosity

## 3. Measurement

## 4. Evaluation

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Symbols

a | wave speed |

b | bulk viscosity |

D | pipe diameter |

E | Young’s modulus |

e | pipe wall thickness |

f | frequency |

${g}_{p}$ | gravity acceleration projection to pipe axis |

i | imaginary unit |

${J}_{0}$ | Bessel function |

${J}_{1}$ | Bessel function |

n | normal vector oriented out of liquid |

P | pipe wall area |

p | pressure |

Q | flow rate |

S | pipe cross-section |

s | parameter of Laplace transformation |

T | transfer matrix |

t | time |

V | volume |

v | velocity |

x | longitudinal coordinate |

$\beta $ | pipe wall damping |

$\epsilon $ | deformation |

$\eta $ | dynamic viscosity |

$\lambda $ | friction coefficient |

$\xi $ | second viscosity |

$\rho $ | density |

$\sigma $ | stress |

$\omega $ | angular velocity |

Subscripts: | |

c | complex number |

0 | see Figure 1 |

1 | see Figure 1 |

Superscripts: | |

* | value divided by steady value |

$\overline{}$ | variable after Laplace transformation |

## References

- Alligne, S.; Nicolet, C.; Tsujimoto, Y.; Avellan, F. Cavitation surge modelling in Francis turbine draft tube. J. Hydraul. Res.
**2014**, 52, 399–411. [Google Scholar] [CrossRef] - Liu, H.; Ouyang, H.; Wu, Y.; Tian, J.; Du, Z. Investigation of unsteady flows and noise in rotor-stator interaction with adjustable lean vane. Eng. Appl. Comp. Fluid
**2014**, 8, 299–307. [Google Scholar] [CrossRef][Green Version] - Vítkovský, J.P.; Bergant, A.; Simpson, A.R.; Lambert, M.F. Systematic evaluation of one-dimensional unsteady friction models in simple pipelines. J. Hydraul. Eng.
**2006**, 132, 696–708. [Google Scholar] [CrossRef][Green Version] - Keramat, A.; Kolahi, A.G.; Ahmadi, A. Waterhammer modelling of vicoelastic pipes with a time-dependent Poissons’s ratio. J. Fluid. Struct.
**2013**, 43, 164–178. [Google Scholar] [CrossRef] - Pochylý, F.; Habán, V.; Fialová, S. Bulk viscosity—Constitutive equations. Int. Rev. Mech. Eng.
**2011**, 5, 1043–1051. [Google Scholar] - Cannizzaro, D.; Pezzinga, G. Energy dissipation in transient gaseous cavitation. J. Hydraul. Eng.
**2005**, 131, 724–732. [Google Scholar] [CrossRef] - Jablonská, J. Compressibility of the fluid. EPJ Web. Conf.
**2014**, 67, 322–327. [Google Scholar] [CrossRef][Green Version] - Graves, R.E.; Argrow, B.M. Bulk viscosity: Past to present. J. Thermophys. Heat Tr.
**1999**, 13, 337–342. [Google Scholar] [CrossRef] - Dellar, P.J. Bulk and shear viscosities in lattice Boltzmann equations. Phys. Rev. E
**2001**, 64, 11. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zuckerwar, A.J.; Ash, R.L. Volume viscosity in fluids with multiple dissipative processes. Phys. Fluids
**2009**, 21, 12. [Google Scholar] [CrossRef][Green Version] - Marner, F.; Scholle, M.; Herrmann, D.; Gaskell, P.H. Competing Lagrangians for incompressible and compressible viscous flow. Roy Soc. Open Sci.
**2019**, 6, 14. [Google Scholar] [CrossRef] [PubMed][Green Version] - Scholle, M. A discontinuous variational principle implying a non-equilibrium dispersion relation for damped acoustic waves. Wave Motion
**2020**, 98, 11. [Google Scholar] [CrossRef] - Karim, S.M. Second viscosity coefficient of liquid. J. Acoust. Soc. Am.
**1953**, 25, 997–1002. [Google Scholar] [CrossRef] - Dukhin, A.S.; Goetz, P.J. Bulk viscosity and compressibility measurement using acoustic spectroscopy. J. Chem. Phys.
**2009**, 130, 13. [Google Scholar] [CrossRef] [PubMed] - Holmes, M.J.; Parker, N.G.; Povey, M.J.W. Temperature dependence of bulk viscosity in water using acoustic spectroscopy. J. Phys. Conf. Ser.
**2011**, 269, 7. [Google Scholar] [CrossRef][Green Version] - He, X.; Wei, H.; Shi, J.; Liu, J.; Li, S.; Chen, W.; Mo, X. Experimental measurement of bulk viscosity of water based on stimulated Brillouin scattering. Opt. Commun.
**2012**, 285, 4120–4124. [Google Scholar] [CrossRef] - Himr, D.; Habán, V.; Fialová, S. Influence of second viscosity on pressure pulsation. Appl. Sci.
**2019**, 9, 5444. [Google Scholar] [CrossRef][Green Version] - Dörfler, P. Pressure wave propagation and damping in a long penstock. In Proceedings of the 4th International Meeting on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Belgrade, Serbia, 26–28 October 2011. [Google Scholar]
- Zielke, W. Frequency-dependent friction in transient pipe flow. J. Basic Eng.
**1968**, 90, 109–115. [Google Scholar] [CrossRef] - Zielke, W. Frequency Dependent Friction in Transient Pipe Flow. Ph.D. Thesis, The University of Michigan, Ann Arbor, MI, USA, 1966. [Google Scholar]
- Záruba, J. Water Hammer in Pipe-Line Systems, 2nd ed.; Academia: Prague, Czech Republic, 1993. [Google Scholar]

**Figure 5.**Pressure upstream of the ball valve during the regular stop after the ball valve fully closed.

Real Part ($\mathit{\alpha}$), Imaginary Part ($\mathit{\omega}$), Frequency (f) | Shape of Pressure Amplitudes along the Conduit |
---|---|

$\alpha $ = 0.0121522 rad/s $\omega $ = 1.403 rad/s f = 0.223 Hz | |

$\alpha $ = 0.01469911 rad/s
$\omega $ = 4.235 rad/s f = 0.674 Hz | |

$\alpha $ = 0.03323047 rad/s
$\omega $ = 6.991 rad/s f = 1.113 Hz | |

$\alpha $ = 0.03837075 rad/s
$\omega $ = 9.792 rad/s f = 1.558 Hz | |

$\alpha $ = 0.04869796 rad/s
$\omega $ = 12.610 rad/s f = 2.007 Hz | |

$\alpha $ = 0.08408768 rad/s
$\omega $ = 15.355 rad/s f = 2.444 Hz | |

$\alpha $ = 0.07548372 rad/s
$\omega $ = 18.156 rad/s f = 2.890 Hz |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Himr, D.; Habán, V.; Štefan, D. Inner Damping of Water in Conduit of Hydraulic Power Plant. *Sustainability* **2021**, *13*, 7125.
https://doi.org/10.3390/su13137125

**AMA Style**

Himr D, Habán V, Štefan D. Inner Damping of Water in Conduit of Hydraulic Power Plant. *Sustainability*. 2021; 13(13):7125.
https://doi.org/10.3390/su13137125

**Chicago/Turabian Style**

Himr, Daniel, Vladimír Habán, and David Štefan. 2021. "Inner Damping of Water in Conduit of Hydraulic Power Plant" *Sustainability* 13, no. 13: 7125.
https://doi.org/10.3390/su13137125