Water hammer

However difficult these transients may be there is another very serious problem to contend with. It is separation and its audible effect, water hammer.

 

I am referring to the pipe system in which  where  is the velocity at the start of closure when the pressure is . In figure 17-4 I stressed that the diagram had negative going half waves that only just went below atmospheric pressure. I did not point out how easy it is for the pressure to attempt to drop below zero absolute.

 

We have seen that for typical velocities of flow e.g. 3 m/s the rise in pressure following sudden or rapid closure the value of  can be large.

 

For  . .

This means that almost any system fed from a tank will have  less than 428 m. (The typical height for a power station building is about 40 metres.)  It is very common indeed for  to exceed . This leads to separation and water hammer.

 

We need to understand what actually happens. We need to go back to figure 17-12 in which friction is ignored and the instant when the first returning wave front reaches the valve and the water in the pipe is flowing from the valve towards the tank at velocity  . The pressure now attempts to fall by hundreds of feet of head when it can only fall to zero absolute. So the pressure falls to zero absolute and the water continues to flow towards the tank at an initial velocity .

 

Two things now happen. The first is that the water separates from the valve and starts to create a void that can contain nothing other that water vapour and continues to flow towards the valve against a pressure difference of  and this causes the column of water of length  to be retarded uniformly to rest and then to be accelerated towards the valve to just reach velocity  when it returns to the valve. Whilst this phase is going on the void in the pipe between the valve and the surface of the flowing water increases to some maximum length and then collapses again as the water returns to the valve. The second thing is that a negative going wave of magnitude  sets off towards the tank and, in the time taken for the whole column to come to rest and then return to the valve, the wave front makes many passes from end to end of the column and may even be attenuated to nothing because of friction. In trying to assimilate this it should be recalled that the speed of the water through the pipe would be about 3 m/s where the speed of the wave would be 1400 m/s.

 

We must decide how we are to quantify this cycle. The new feature is that we have a wave front producing the basic square wave for the first half of the cycle and then a half cycle where the dominant feature is the behaviour of the a column of water of length  moving under the effect of a constant force.

 

So let me start pressure-time diagram for a point just upstream of the valve following sudden and complete closure from velocity  in the system depicted in figure 17-1 that I repeat here as figure 17-27.

 

In order to draw this diagram we need an expression for the interval of time between the start of separation and the return of the column to contact with the valve. The column separated from the valve with a velocity of  and it moves under a retarding force that equals the pressure difference multiplied by the area. The pressure difference is given by  and when this is multiplied by the area we get a force of  . The mass involved is given by  if the length of the void is small when compared with .

 

Then if we equate the impulse of force and the change in momentum per second we can write :-                     where  is the time taken for the void to form and collapse and  is the net change in velocity.

 

From this we get :-                  

 

When the column hits the closed valve there is a loud noise produced by the collision. Then the pressure rises by  as before and the cycle starts again but this time the rise in pressure is from zero absolute and not from .

I have gathered all this together in figure 17-28.

 

At one time I had a rig for student use to demonstrate water hammer. It comprised a water tank on a roof from which a steel pipe of 1² diameter and 50 metres long ran down to the laboratory in straights and swept bends to terminate in a plug cock. There were tappings for pressure transducers at the valve and at the mid length.

 

I made many tests on this rig and only one has survived in my care. It is for the pressure-time diagram at the valve. I have copied it to give figure 17-29. The length of 50 metres gave a value of  of about 0.075 seconds. The rig was of realistic proportions so this trace is not just of academic interest. Separation is real and potentially destructive