Abstract
A dual-technique-based inline strategy was investigated in this study as a sustainment to conventional-technique skills in terms of limitation of wave oscillation period spread-out. Instead of the single polymeric short section employed by the latter technique, the former is based on replacing an up- and downstream short section of the primitive piping system using another couple made of polymeric pipe-wall material. Numerical computations used the method of characteristics for the discretization of unconventional water-hammer model based on the Vitkovsky and the Kelvin–Voigt formulations. The efficiency of the dual technique was considered for two operating conditions associated with up- and downsurge frames. Moreover, two pipe-wall material types were utilized for short-section pipe wall, namely the HDPE or LDPE materials. Additionally, the conventional technique was also addressed in this study, for comparison purposes. First, analyses of pressure-head, circumferential-stress and radial-strain wave patterns, along with wave oscillation periods examination, confirmed that the dual technique could improve the efficiency of the conventional one, providing acceptable trade-off between the attenuation of pressure-head and circumferential-stress peaks (or crests), and limitation of period spreading and radial-strain amplification. Second, a parametric study of the sensitivity of the wave damping to the employed short-section dimensions was performed in terms of short-section length and diameter. This parametric study helped estimate the near-optimal values of the short-section dimensions.
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Abbreviations
- A :
-
Cross-sectional area of the pipe (\(\hbox {m}^{2})\)
- \(a_0 \) :
-
Elastic wave speed (m/s)
- \(C_r \) :
-
Courant number (–)
- D :
-
Main-pipe diameter (m)
- \(d_{\mathrm{short}\hbox {-}\mathrm{section}} \) :
-
Polymeric (sub-) short-section diameter (m)
- \(E_0 \) :
-
Young’s modulus of elasticity of the pipe wall (Pa)
- e :
-
Pipe-wall thickness (m)
- f :
-
Darcy–Weisbach friction factor (–)
- g :
-
Gravity acceleration (m/\(\hbox {s}^{2})\)
- \(h_f \) :
-
Head loss per unit length (–)
- \(h=p/\gamma +z\) :
-
Pressure head (m)
- \(J_k =1/{E_k} \) :
-
Instantaneous or elastic creep compliance (\(\hbox {Pa}^{-1})\)
- K :
-
Bulk modulus of elasticity of the fluid (Pa)
- z :
-
Elevation or pipe axis elevation (m)
- \(k_v \) :
-
Vitkovsky’s unsteady decay coefficients (–)
- L :
-
Main steel-pipeline length (primitive system) (m)
- \(l_{\mathrm{(sub}\hbox {-}\mathrm{)short}\hbox {-}\mathrm{section}} \) :
-
Polymeric (sub-) short-section length (m)
- \(h^{*}\) :
-
Absolute pressure head (Pa)
- q :
-
Average flowrate (\(\hbox {m}^{3}\)/s)
- \(R=f/{2DA}\) :
-
Pipeline resistance coefficient (–)
- t :
-
Time (s)
- x :
-
Coordinate along the pipe axis (m)
- z :
-
Elevation (m)
- \(\alpha _0 \) :
-
Pressure-dependent volumetric ratio of gas in mixture (void fraction) (–)
- \({\alpha } '\) :
-
Dimensionless parameter (–)
- \(\Delta t\) :
-
Time-step increment (s)
- \(\Delta x\) :
-
Space-step increment (m)
- \(\rho \) :
-
Fluid density (kg/\(\hbox {m}^{3})\)
- \(\mu \) :
-
Viscosity of the Kelvin–Voigt dashpot (\(\hbox {m}^{2}\)/s)
- \(\tau =\mu /E\) :
-
Retardation time for Kelvin–Voigt model (s)
- \({\nu } '\) :
-
Kinematic fluid viscosity (\(\hbox {m}^{2}\)/s)
- \(\nu \) :
-
Poisson’s ratio (–)
- \(\psi \) :
-
Numerical weighting factor (–)
- \(\theta \) :
-
Relaxation coefficient for the local acceleration numerical scheme (–)
- \(\forall \) :
-
Volume of cavity (\(\hbox {m}^{3})\)
- 0:
-
Steady state
- i :
-
Section index
- k :
-
Kelvin–Voigt element index
- ns :
-
Number of sections
- \(n_{\mathrm{kv}} \) :
-
Number of Kelvin–Voigt elements
- g :
-
Gas
- j :
-
Pipe index
- k :
-
Index of Kelvin–Voigt element
- np :
-
Number of pipes
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Triki, A. Dual-technique-based inline design strategy for water-hammer control in pressurized pipe flow. Acta Mech 229, 2019–2039 (2018). https://doi.org/10.1007/s00707-017-2085-z
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DOI: https://doi.org/10.1007/s00707-017-2085-z