Sudden opening of a valve

Let me try to use this expression. Suppose that in the simple system of figure 17-1 the free surface is 10 metres above the valve, that the pipe is of 100 metres in length, of diameter 50 mm and has a friction coefficient of 0.001. Now suppose that the valve is suddenly opened from being fully closed. The flow will obviously increase in some way until a steady condition is achieved when the head lost to friction in the pipe plus the kinetic energy head in the outflow is equal to 10 metres. We can find out how the flow in the pipe varies with time.

 

The steady state that will finally be reached will come when:-

                                       where  is the final steady velocity of the water.

 

If the valve is a gate valve or a plug valve it would be useful to ignore the loss in the valve and then the equation becomes :-  and we can evaluate  for the given figures to give  =4.67 m/s

 

We can set up a differential equation to relate the change of velocity and time using the energy equation and the expression for the head required to produce an acceleration. We get :-   where  is the instantaneous velocity of the water in the pipe at time  after closure.

 

This can be rearranged to give :-

 

This can be integrated repeatedly using Mathcad to give a velocity versus time graph. This is graph 17-1. In principle the water in the pipe will take an infinite time to reach its final steady state. The dotted red line gives the final velocity nd we can see that, after one minute there is still a few percent to go.

 

In engineering practice the system that I have proposed is most unlikely to exist, as the tank will have to be filled and then the water in it is not still. This will affect the flow and some equilibrium condition will quickly be established.

 

I have only once seen really steady flow. It was a fountain in an ornamental fishpond. The fountain was a nozzle set with its axis vertical and fed by a long pipe from a large lake. The jet was like a glass trumpet with no observable motion until the top edge broke up into separate parts held together by surface tension that fell back into the fishpond.