Speed of propagation of a wave front

We need to relate the rise in pressure, the initial velocity of the water, the frequency of the cycle and the physical properties of the water and the pipe. Clearly the frequency depends on the length of the pipe and on the speed of propagation of the wave front. So we must have a value for the speed of the wave. We also need to find an expression for . The analysis is unusual because we have a finite rise in pressure  producing changes in density and volume of the water that are both relatively very small. I shall use both  and  as appropriate.

 

Consider some instant at some time  after the valve is suddenly closed. The wave will be at some point in the pipe that will be  from the valve where  is the speed of propagation of the wave.

Figure 17-16 shows a section of a pipe that is adjacent to a plug valve. It shows the position of a wave front at time  after the valve has been closed suddenly. The water is flowing towards the wave front at velocity  and at pressure  and density . The wave front moves at velocity  where  is the speed of propagation of the wave front in stationary water. Between the wave front and the valve the water is at rest at pressure  and at . During time  the wave front moves  and the area of cross section of the pipe over this length changes to .

 

The mass of fluid that has come to rest in time  is made up of the mass that was in the pipe when the valve was closed plus that which flowed in during time .

 

So we can say that :-

                             

                                     

from which we get :-

                                       .

 

The density term in these equations is always a nuisance so we must find a way to eliminate it in favour of pressure.

 

Density is mass per unit volume and the volume of a liquid is related to the pressure by the elastic properties of the liquid. We use a bulk modulus that is defined by  :-

                                        where  is the change in a volume  of liquid resulting from a change in pressure of the liquid of .  is the constant of proportionality and is called the bulk modulus. Clearly  has the units of pressure and, for water it has a value of about 2.05 GN/m².

 

For the mass of water that is at rest between the valve and the wave front the change in density resulting from the rise in pressure will be from  to  where  is the volume before the wave front and  is the volume after the passing of the wave front. From this :-  and  because .

 

Rearranging this we get . Then we get the crucial switch

                                             

 

Now we can eliminate the density terms from

                                     

                                       

                                            

                                            

                                            

                                             

                                           

This looks very formidable but it will yield if we now apply momentum to the compression. The mass of water that has been brought to rest has given up momentum as a result of a force equal to  acting on it for a time . So :-

                                         or,

                                                   

This simple step has given us an expression for  in terms of  and measurable quantities. It is important.  Here we shall need to use it in the form .

This expression can be combined with  to give a very useful outcome but it is tedious.

                                           

 

Multiply top and bottom by  to give :-

                                            

                                            

                                            

Divide top and bottom by

                                            

Now we have to extract

               

Rearranging gives :-

       

       

and then                                   

 

Now  will always be small when compared with 1 and  will also be minute. Using this we get                     . It is important to realise that  and  are of the same order of magnitude so we have to decide whether to take the expansion of the pipe into account. We must put some figures to this equation to find some speeds.

 

We have for water  and . Then .

 

This figure for the speed of the wave front is enormous when compared with the normally encountered speeds of water in practical pipe systems. Effectively the speed of the wave front is 1,432 m/s regardless.

 

However we cannot just forget the expansion of the pipe under stress. We make use of the elementary theories of hoop stress. Figure 17-17 represents one half of a short length  of a pipe of diameter  and wall thickness . It is in equilibrium under three concentrated forces that are the resultants of three distributed forces. The downward force is due to the internal pressure  and the two upward forces are due to the stress acting in the wall of the pipe.

 

The pressure rise  produces a downward force of  acting on a total area . So the hoop stress = . This stress produces a hoop strain, that is the circumferential extension/the circumference =  where  is Young’s modulus for the material of the pipe.

 

We need the value of . The increase in length of the circumference is  so the increase in area of the pipe is the circumference times the increase in radius which is equal to and . Then .

 

We had  and we can now substitute for  to give :-

                          or .

 

In the final form this becomes  .

 

This expression does not contain  so the speed of the wave in not dependent on  but is dependent on the elastic properties of both the water and the metal of the pipe.

 

W e can quantify the effect of the elasticity of the pipe. Suppose that the pipe were to be of 32 mm diameter and of 1 mm wall thickness and made of steel. This would be a thin walled pipe and if there is any effect this would make it large. Given that E for steel is  the speed of the wave front for this pipe :-

 

So the velocity of the wave front is primarily dependent on the elasticity of the water and for regular pipes a figure of 1350 m/s can be used.

 

We are now in a position to predict the pressure-time diagram for the pressure in a pipe, at the valve, after sudden closure. We know the speed of the wave front. If we know the length of the pipe we can predict the time taken for the wave front to traverse the pipe and if we know the velocity of flow before the valve is closed we can predict the pressure rise following sudden closure.

Figure 17-18 is figure 17-4 with the new information added. The duration of a crest is the time for a wave front to go from end to end and back, that is,  and the pressure rise is .

 

What we now need to know is what magnitude the pressure rise will have for any initial value of

 

This can be the result of using . I have plotted it for water and a range of  from 0 to 10 m/s.

 

The commonly used figure for the speed for economical flow of water in pipes is 3 m/s. This corresponds to a pressure rise in a wave front generated by sudden closure of about 40 bar. This is the pressure at a depth of 400 metres in water.

 

It is a large pressure when compared with the normal pressures encountered in, say, buildings and it must be treated with respect.