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ABSTRACT
Time evolution of water surface oscillations in surge tanks is of importance in pipeline and hydropower engineering, and is often solved graphically or numerically because the associated nonlinear second order ordinary differential equation, with a quadratic damping term, cannot be solved exactly. This research then innovates an accurate approximation based on the Lambert function, Padé approximant, and elliptic integral. Specifically, (i) the characteristic length depends on the pipe-tank geometry, the system resistance, and an unsteady factor; and the characteristic time depends on the pipe-tank geometry, the gravity, and the unsteady factor, but independent of the system resistance; (ii) the surge transit squared velocity is approximated by a Padé approximant of order [2/1], resulting in a simple approximation for the surge tank water surface oscillations in terms of an elliptic integral of the second kind; and (iii) the approximate solution accurately reproduces numerical and laboratory data thereby being applicable in practice.
Acknowledgements
The authors wish to thank the anonymous reviewers, Associate Editor and Editor for their constructive comments, which significantly improved the accuracy, brevity, and clarity of the presentation.
Notation
| = | pipe cross-sectional area (m2) | |
| = | surge tank cross-sectional area (m2) | |
| C | = | integration constant (–) |
| c | = | wave celerity (m s−1) |
| = | pipe diameter (m) | |
| = | diameter of circular surge tank (m) | |
| = | elliptic integral of the second kind (–) | |
| = | bulk modulus of water (Pa) | |
| = | composite modulus of elasticity of water-pipe system (Pa) | |
| = | elasticity modulus of pipe (Pa) | |
| F | = | functional symbol (–) |
| f | = | pipe friction factor (–) |
| g | = | gravity acceleration (m s−2) |
| = | initial head loss (m) | |
| I | = | integral (–) |
| = | pipe resistance coefficient (–) | |
| = | throttle resistance coefficient or a lumped coefficient of both throttle and surge tank (–) | |
| k | = | constant related to pipeline anchoring (–) |
| L | = | pipe length (m) |
| = | characteristic length (m) | |
| = | Mach number (–) | |
| m | = | parameter of elliptic integral (–) |
| Q | = | discharge (m3 s−1) |
| = | Reynolds number (–) | |
| T | = | dimensionless t (–) |
| = | initial value of T (–) | |
| = | value of T at turning point k (–) | |
| t | = | time (s) |
| = | characteristic time (s) | |
| = | initial time (s) | |
| V | = | pipe velocity (m s−1) |
| = | surge tank velocity (m s−1) | |
| = | initial value of | |
| = | primary branch of Lambert function (–) | |
| = | secondary branch of Lambert function (–) | |
| X | = | interim variable (–) |
| x | = | variable (–) |
| Y | = | dimensionless y (–) |
| = | model parameter (–) | |
| = | initial condition (–) | |
| = | value of Y at turning points (–) | |
| = | maximum velocity position (–) | |
| y | = | distance above reservoir water surface (m) |
| = | initial value of y (m) | |
| = | interim parameter (–) | |
| β | = | unsteady factor (–) |
| δ | = | pipe wall thickness (m) |
| ε | = | pipe wall roughness (m) |
| λ | = | model parameter (–) |
| ρ | = | density of water (kg m−3) |
| = | wall shear stress (Pa) | |
| φ | = | argument of elliptic integral (–) |
| = | initial value of φ (–) | |
| ψ | = | interim variable (–) |
| ν | = | kinematic water viscosity (m s−2) |