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 |
gravity acceleration projection to pipe axis | |
i | imaginary unit |
Bessel function | |
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 |
pipe wall damping | |
deformation | |
dynamic viscosity | |
friction coefficient | |
second viscosity | |
density | |
stress | |
angular velocity | |
Subscripts: | |
c | complex number |
0 | see Figure 1 |
1 | see Figure 1 |
Superscripts: | |
* | value divided by steady value |
variable after Laplace transformation |
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Real Part (), Imaginary Part (), Frequency (f) | Shape of Pressure Amplitudes along the Conduit |
---|---|
= 0.0121522 rad/s = 1.403 rad/s f = 0.223 Hz | |
= 0.01469911 rad/s
= 4.235 rad/s f = 0.674 Hz | |
= 0.03323047 rad/s
= 6.991 rad/s f = 1.113 Hz | |
= 0.03837075 rad/s
= 9.792 rad/s f = 1.558 Hz | |
= 0.04869796 rad/s
= 12.610 rad/s f = 2.007 Hz | |
= 0.08408768 rad/s
= 15.355 rad/s f = 2.444 Hz | |
= 0.07548372 rad/s
= 18.156 rad/s f = 2.890 Hz | |
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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
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 StyleHimr, 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
APA StyleHimr, D., Habán, V., & Štefan, D. (2021). Inner Damping of Water in Conduit of Hydraulic Power Plant. Sustainability, 13(13), 7125. https://doi.org/10.3390/su13137125