r"""Hydraulic transients: Joukowsky's water hammer.
Closing a valve stops the water next to it, and a pressure wave carries the
news upstream at the speed of sound in the pipe. Joukowsky (1898) measured
the jump :math:`\Delta p = \rho a \Delta v` in the Moscow water mains and
used Korteweg's formula (1878) for the wave speed in an elastic pipe. A
closure is "fast" when it takes less than the round trip :math:`2L/a`.
"""
from __future__ import annotations
import numpy as np
from engineeringkit._validation import finite as _finite
from engineeringkit._validation import nonnegative as _nonnegative
from engineeringkit._validation import positive as _positive
from engineeringkit._validation import validity as _validity
__all__ = ["water_hammer_pressure", "pipe_wave_speed", "critical_closure_time"]
[docs]
def water_hammer_pressure(rho, wave_speed, delta_v):
r"""Joukowsky pressure rise for a sudden velocity change, :math:`\Delta p = \rho a\,\Delta v`.
Parameters
----------
rho : float or array_like
Fluid density [kg/m^3], positive.
wave_speed : float or array_like
Pressure-wave speed in the pipe [m/s], positive (see :func:`pipe_wave_speed`).
delta_v : float or array_like
Velocity reduction [m/s]; negative for a sudden opening.
Returns
-------
float or ndarray
Pressure rise [Pa].
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/fluid_engineering`. Holds for closures shorter than
:math:`2L/a` and neglects friction.
References
----------
N. Joukowsky, "Über den hydraulischen Stoss in Wasserleitungsröhren,"
*Mém. Acad. Imp. Sci. St.-Pétersbourg*, 8th ser., 9(5) (1900) 1-71
(read 1898); English translation by O. Simin, *Proc. AWWA* 24 (1904)
341-424.
Examples
--------
>>> water_hammer_pressure(1000.0, 1200.0, 1.0)
1200000.0
"""
_finite(rho=rho, wave_speed=wave_speed, delta_v=delta_v)
_positive(rho=rho, wave_speed=wave_speed)
return rho * wave_speed * delta_v
[docs]
def pipe_wave_speed(bulk_modulus, rho, D, wall_thickness, E, c1=1.0):
r"""Korteweg's pressure-wave speed in a liquid-filled elastic pipe.
.. math::
a = \sqrt{\frac{K/\rho}{1 + c_1\,K D/(E e)}}
Parameters
----------
bulk_modulus : float or array_like
Bulk modulus of the liquid :math:`K` [Pa], positive (2.2 GPa for water).
rho : float or array_like
Liquid density [kg/m^3], positive.
D : float or array_like
Pipe inner diameter [m], positive.
wall_thickness : float or array_like
Wall thickness :math:`e` [m], positive.
E : float or array_like
Young's modulus of the wall [Pa], positive.
c1 : float or array_like, optional
Pipe-support factor [-], nonnegative: 1 for expansion joints
throughout, :math:`1 - \nu^2` for a pipe anchored against axial
movement, 0 for a rigid pipe.
Returns
-------
float or ndarray
Wave speed [m/s]; at most the speed of sound :math:`\sqrt{K/\rho}` in the liquid.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/fluid_engineering`. Thin-walled pipe, :math:`D/e \ge 25`;
a ``ValidityWarning`` is issued for thicker walls.
References
----------
D. J. Korteweg, "Ueber die Fortpflanzungsgeschwindigkeit des Schalles in
elastischen Röhren," *Ann. Phys.* 241 (1878) 525-542.
Examples
--------
>>> round(float(pipe_wave_speed(2.2e9, 1000.0, 0.5, 0.01, 200e9)))
1191
"""
_finite(bulk_modulus=bulk_modulus, rho=rho, D=D, wall_thickness=wall_thickness, E=E, c1=c1)
_positive(bulk_modulus=bulk_modulus, rho=rho, D=D, wall_thickness=wall_thickness, E=E)
_nonnegative(c1=c1)
_validity("pipe_wave_speed", "D/e", np.asarray(D, dtype=float) / wall_thickness, 25)
stiffness = np.asarray(E, dtype=float) * np.asarray(wall_thickness, dtype=float)
return np.sqrt((bulk_modulus / rho) / (1 + c1 * bulk_modulus * np.asarray(D, dtype=float) / stiffness))
[docs]
def critical_closure_time(length, wave_speed):
r"""Round-trip time :math:`2L/a` of a pressure wave along a pipe.
Parameters
----------
length : float or array_like
Pipe length from valve to reservoir [m], positive.
wave_speed : float or array_like
Pressure-wave speed [m/s], positive.
Returns
-------
float or ndarray
Time [s]; a valve closed faster than this sees the full Joukowsky rise.
Notes
-----
Inputs are raw SI numbers with no dimensional metadata; see
:doc:`/guides/fluid_engineering`.
Examples
--------
>>> critical_closure_time(600.0, 1200.0)
1.0
"""
_finite(length=length, wave_speed=wave_speed)
_positive(length=length, wave_speed=wave_speed)
return 2 * length / wave_speed
