Behaviour of water in a pipe following sudden closure of a valve

Figure 17-2 shows a facility to test a horizontal pipe. The test pipe is U shaped for convenience with a large radius sweep. In effect the roof tank and the small closed tank, joined together by a pipe, replace a tank of very small cross-section but very tall.

 

At the outlet end of the test pipe there is a plug cock that can be turned very quickly to give what amounts to sudden closure. A pressure transducer is fitted close to the plug cock.[1]

 

At this point we cannot analyse just any sudden closure. We must start with the case of a system with a high value of the static head and a low flow that would, of course, be the result of having the valve partially open. With some care it is possible to get a pressure-time trace from the pressure transducer near to the valve for the events following sudden closure of this partly open valve. It would look like figure 17-3.

 

This graph is a trace of the pressure against time as recorded by the transducer, processed to give readings above and below the pressure corresponding to atmospheric pressure, against time. Before closure the loss of head to friction would be small compared with the static head

 

The most obvious features are the cyclic nature of the trace and that the periodic time is constant. The ramps on the tops to each half wave are a little puzzling but can be explained. If it were not for these ramps this trace would look like a square wave. In fact the real trace is a square wave combined with the effects of friction loss to make it into a decaying wave.

 

When one seeks an explanation it makes sense to start by ignoring friction during the events that produce the square wave and then take friction into account later. Such a decision leads us to the position where there is no mechanism for attenuation of the wave but does let us get to grips with the physical explanation of the cycle of events.

Figure 17-4 shows the square wave. It takes equal half waves that are alternately positive and negative relative to the pressure head at the instant of closure. I have shown the negative half waves as going below atmospheric pressure just to introduce the idea of pressures that can be less than atmospheric and, indeed, go down to the vapour pressure of the water.

 

In order to explain this trace we must look at events in the pipe adjacent to the valve just after the instant of closure. In figure 17-5 I am endeavouring to represent a wave that is moving through a pipe full of water that is moving with one-dimensional flow at uniform speed  and pressure . I cannot usefully draw a closed plug cock but we have to imagine it to be just to the right of the arbitrary section so that between the closed plug cock and the wave there is water that is at rest at pressure . I use  to indicate “a finite increase in” as distinct from  to mean “a small increase in”.

So figure 17-5 represents a moving wave front going from right to left advancing into water moving at  from left to right and bringing the water to rest as it advances. At the wave front kinetic energy is given up in exchange for pressure energy and that energy is stored as strain energy in the water. The wave front moves at constant speed and it can never be in equilibrium.

 

Now we must look to see what happens to this wave front.

 

We have a system to analyse and we need to know the starting condition. Figure 17-6 shows both the system and the starting condition. The system is a horizontal pipe connected to a supply tank. At the inlet end the pipe is connected to a supply tank and, at the outlet end, there is a plug valve. Now we must suppose that water flows in this pipe at a steady velocity  with uniform pressure . This defines the system and the steady flow condition before the valve is suddenly closed.

 

We know that a pressure wave will be propagated at some uniform velocity towards the inlet end and that the pressure will rise by  at the wave front. We can represent the activity in the pipe at some instant after closure. It is figure 17-7. The wave is part way along the pipe and moving right to left. Between the valve and the wave the water will be at rest at pressure . Between the inlet and the wave the water continues to flow at the same velocity  at the original pressure . This can go on until the pressure wave reaches the open end.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 17-9 shows the state of the water in the pipe at the instant that the pressure wave reaches the open end. All the water is at rest at pressure . However, at the wave front there is only a pressure  from left to right and a pressure  from right to left. This condition is unstable and we must decide what happens.

 

The first thing to recognise is that the wave front can only occur in an elastic medium and in this case the elastic medium is the water. So, during the passage of the wave over the length of the pipe, the kinetic energy once possessed by the water that is now at rest, is stored as strain energy in the water. (The pipe is also elastic and will store strain energy but we will take that into account later.)

At the instant that the wave front reaches the open end of the pipe it is in the condition shown in figure 17-10 where the water in the tank is not moving yet there is a large difference in pressure between the two sides of the wave. This cannot persist and the prevailing process of exchanging kinetic energy for strain energy is reversed so that the water flows out of the pipe from right to left and a wave front in which the pressure drops by  to  moves from left to right. In the process the strain energy changes back to kinetic energy but now in the opposite direction and the water flows out of the pipe into the supply tank.

 

Figure 17-11 shows the wave front on its way towards the valve leaving water to the left moving at  towards the open end. The water to the right of the wave front is still stationary at a pressure of .

 

The next interesting point is when this wave front reaches the valve when the pipe is full of water all moving away from a closed valve!

This instant is shown in figure 17-12 and the pressure is again  throughout. We have to decide what now happens at the valve because the situation shown in figure 17-12 cannot be in equilibrium. In fact a new wave front is generated at the valve and in this wave front the pressure drops by  and the water between the valve and the wave front comes to rest again. We can see the result in figure 17-13.

The pressure at the valve drops by  and I have shown the resulting pressure in the pipe as being less than atmospheric but greater than zero absolute. We shall see that typical conditions in a pipe is more likely to bring the pressure to zero absolute when this diagram would not be correct.

 

The next changeover point will be when the wave reaches the open end. Then the water in the pipe will be at rest from end to end at pressure . This is again not in equilibrium and yet another wave front is propagated towards the valve causing the velocity of the water to change back to the original value  and the pressure to rise to . This is shown in figure 17-14. When that wave reaches the valve the cycle is complete and will start again.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This gives an explanation of the square wave that we postulated for the events following sudden closure of a valve at the end of a pipe. We can use it to draw the diagrams for the pressure variations with time at, say, the mid point of the pipe and at the open end.

 

I have drawn the pressure-time traces for this square wave for pressures measured at these three points. The trace for the mid point is self-explanatory and the trace for the open end merely notes that there will be pressure spikes as the wave front reaches the open end and reverses.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In order to make this explanation of the events in a pipe after sudden closure we ignored friction as it suited the explanation. We still need an explanation of the ramps on the top of real traces but I will leave that until later in this text.

 

All this makes sense but it lacks any physical reality because we cannot put values to the magnitudes of the pressure changes or to the speed of the waves. This can be done using our normal physics.