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UNSTEADY FLOW IN PIPES
When water is flowing through a pipe is suddenly stopped by the use of valve then we can see sudden rise in pressure inside the pipe and this pressure waves transmitted along the length of the pipe and it creates knocking like sound effects knows as 'water hammer'. In doing so, there are rapid pressure oscillation in the pipe, often accompanied by a hammering like sound, this phenomenon is called as 'Water hammer effect'. Sometimes the pressure may rise to the greater extent and may causes serious damage to the pipe. Therefore the 'water hammer' effects should be properly studied in the pipes.
- The magnitude of pressure rise depends on:
i) Speed at which the valve is closed
(ii) The velocity of flow (V)
(iii) The length of the pipe (L)
(iv) Elastic properties of pipe material (E) and
flowing fluid (K).
- Causes of water hammer (or hammer blow):
(i) Valve operation (i.e. closing or opening of
valve)
(ii) Mechanical failure of control devices like valve
(iii) Starting or shut down of pumps and hydro turbine
(v) Fluctuation in power demand in turbine
Effect of water hammer (or hammer blow):
(i) High pressure fluctuation in pipeline
(ii) Rupture of pipe or valve beyond safety limit
(iii) High pressure requirements for the design of
pipeline and pressure pipe such as penstocks
Theory of water hammer phenomenon
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| Fig: Simple Description of Water Hammer |
As the valve at the end of the pipe is closed suddenly, the
water in the pipe retards and hence there is a pressure increases
instantaneously. This pressure swings normal HGL to a position as indicated
by dotted line (i.e. upward) in the Figure.
Since, the pressure at the reservoir is atmospheric and
hence constant, the +ve sing results in the back flow from the pipe into the
reservoir. As the water flows back into the reservoir, it creates partial
vacuum conditions in the pipe and the pressure in the pipe swings in the -ve
direction (i.e. downward). This indicates the reservoir water to flow into the
pipe. But, the valve being partially closed, much of this water is again
retarded giving rise to a +ve swing again.
Consider a long pipe of length (L) and of cross-sectional
area (A), carrying liquid flowing with a velocity (V=Q/A). Due to the closing
of the valve, let the liquid be brought to rest in time (t) seconds.
So, the total mass of liquid contained in the pipe (m) = ρ x Volume = ρ (AL); (Since, ρ = Density )
However we assumed that the we adjusted rate of closure of valve such that we brought the liquid inside the pipe to the rest with uniform -ve acceleration from initial velocity 'V' to '0' in time 't' seconds.
Thus, rate of retardation (a) =V/t
Now, Force on valve (F) is= m*a=ρ(AL)*V/t i)
Again, due closure of valve the pressure will be increased.
So, increased pressure force (Fi) available for producing retardation is
:. Fi= Pi*A ii)
Where, Pi = Inertia (or Increased or Dynamic) pressure = γwHD
Now, equating equation (I) and (Il), we get
Pi*A= ρ (AL) (V/t)
:. Pi=ρ (L/t)V
:. Pi=ρcV
Which is the required expression to calculate the 'Increased (or Inertia or Dynamic) pressure'; where, 'C' = Velocity of wave (= L/t), 'HD' = Inertia (or Increased or Dynamic) head (=Pi/γ =ρ(L/t)V/γ =VL/gt) and 'Hs' = Static head (Ps/γ).
- Time to travel by pressure wave from 'Valve' to 'Reservoir' is
- Time to travel by pressure wave from 'Valve' to 'Reservoir' and back to 'Valve' is
- Time
of closure of 'Valve' = T
The valve may be closed either gradually or instantaneously
and according to the nature of the closure of the valve, the expressions for
calculating the pressure head due to water hammer may be developed as indicated
below.
Slow (or Gradual or Partial) Closure of Valve [Incompressible Theory]: (T > tc.)
Inertia (or Increased or Dynamic) pressure (Pi) = ρcv
Where, [Velocity of wave (C) = 2L/T for 1 < T/tc < 1.5 ] and
[Velocity of wave (C) = L/T for T/ tc > 1.5 ]
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Instantaneous (or Rapid or Fast) Closure of Valve [Elastic Theory]: (T < tc)
From the equation Pi = ρ(L/T)V ; if the time of closure (T) = t = 0, then, Pi = ∞ . But from the experimental observations, even for a very rapid closure of the valve, the pressure rise (Pi) is quite finite and measurable.
So, it may therefore be concluded that in the derivation of
equation (Ill) certain essential factors have been omitted. These factors are;
i)Compressibility of liquid (K)
and
(ii) Elasticity of pipe material (E).
Hence, both these factors make it impossible for the liquid
column to be stopped instantaneously even if it was possible to close the valve
instantaneously.
This may however be explained by means of a simple example as shown Figure below. If the head of loosely trucks is suddenly checked the following trucks still keep on moving until all the gap between them has been covered and only then the whole train will come to rest. This means that even after first truck is stopped but the last truck will move through the certain distance and it finally comes to the rest.
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| Fig: Stopping of a Loosely Coupled Train |
This similar process can be observed inside the pipe during the phenomenon of 'water hammer'.
Figure 4-3 (a) shows the normal flow condition of the liquid in the pipe with normal velocity (V), under normal pressure head (HS), at the moment just before the valve at end 'B' is instantaneously shut. Here friction head is quite small in comparison with inertia head and hence we can neglect it.
Figure 4-3 (b) shows the condition immediately after the
valve at 'B' is closed. Thus a wave of inertia pressure, as shown by the step
in the HGL, has begun to travel with velocity (C) in the U/S, which results in
compressing the water and also expanding the pipe. Meanwhile the liquid on the
left of the advancing wave continues to move on as nothing has happened to it.
Figure 4-3 (c) shows that the wave has advanced further.
Figure 4-3 (d) shows that the liquid
in the entire pipe is at rest under a pressure head (HS+HD); the whole of the
pipe having been distended under this pressure.
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| Fig: Transmission of Pressure Wave along a Pipe due to
Instantaneous Closure of Valve |
Thus, if 'dt' is the time required to stop the liquid column, then in this interval of time a transverse plane (AX) at the end of the column has been able to move through a distance 'dL' to a new position (AX'); in other words, the compression of the liquid column together with the increase in the pipe diameter have made space for an amount of liquid '(IQ' (shown by hatching) to enter the pipe after the valve has been shut.
Now we know;
Now,
- v/c
Inertia (increased or dynamic) Pressure (Pi)=ρcv
In this case, pipe is stressed in the circumferential direction.
So, circumferential stress (Q) is taken for the analysis. The longitudinal
stress (Q) is more or less compensated by support given to pipe. While taking
the circumferential stress (a), strain is also taken as circumferential.
so,
Which is the required expression of velocity of wave (C) for
elastic pipe.
(a) Rigid
pipe (or Rigid Water Column Theory; RWCT)
Inertia (or Increased or Dynamic) pressure (Pi) = pcv
While taking the pipe as rigid, then the stretching of pipe
along longitudinal and circumferential direction will not be occurred (i.e. 'E'
is not considered). So, additional stress due to increased pressure (Pi) can be
compensated by compressing the liquid in the pipe column.
Now we know,
(b) Elastic
pipe (or Elastic Water Column Theory; EWCT)
Inertia (or Increased or Dynamic) pressure (Pi) = pcv
So, from the equation (XII),
Progress of Water Hammer Pressure Wave
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Stage: 1
When the valve is closed, the layer of water close to the valve is brought to rest. The layer of water is compressed and consequently the pressure will rise by 'Pi/γ = HD+' and the pipe will get stretched. Successive layers are brought to rest in succession and the pressure rises for greater length of the pipe. The high pressure wave moves upstream with a velocity of 'C' reducing the incoming velocity to zero and reaches the reservoir at time, 'T=L/C'.
Stage: 2
Since, the velocity in the reservoir is approximately zero, there is no change in head. At the end of Stage: 1, the pressure in the pipe is equal to 'Hs + HD+' throughout and the velocity is zero. Since, the pressure in pipe is higher than that in the reservoir, water starts to flow back to the reservoir reducing the pressure in the pipe, an expansion wave of equal magnitude travels with a velocity 'C' towards the valve. The pressure of water falls back and the pipe walls resume their original size. Thus, at the time, ' T=2L/C', the pressure in the pipe in its entire length returns to the normal or original value and water has a velocity 'V' in the backward direction.
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Fig: One Cycle of Wave Motion in a Pipe
due to Instantaneous Closure of Valve |
Stage: 3
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Stage: 4
Since, reservoir pressure is higher than
that in the pipe, the water flows into the pipe attaining a velocity 'V'
towards the wave in the normal condition. At 'T = 4L/C', the process repeats to the initial conditions when valves are closed.
The stages mentioned above now repeat. The
pressure is dampened with friction against pipe and elasticity of water and
ultimately water comes to rest.
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