The concepts of rapid and slow closure

We know that the speed of a wave front  does not vary much from about 1400 m/s. It follows that, as a wave will take a time of  to travel from the valve to the inlet end of the pipe and back, this time is quite predictable. It follows that we can categorise the times taken to close a valve and the character of the subsequent events by whether the valve has closed fully before the first returning wave reaches the valve. We call a closure where the closure time <  a rapid closure

I know from electronic measurement of pressure transients that even small changes in valve setting produce wave fronts with measurable magnitudes of pressure. Provided that the time of closure is less than  we can say that the original velocity of flow  where  is a small change in  and we can say that the total rise in pressure at the vale . If we propose some relationship between velocity at the valve and time we can draw a diagram showing the change of pressure at the valve.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I have drawn the diagram for a closure in less than  in figure 17-25. It is probably impossible to predict the graph of pressure at the valve-time during closure but, for the sake of illustration, I have drawn a smooth curve in blue. I have divided the rise in pressure, that will equal  where  is the original velocity of flow, into seven equal increments of pressure . The diagram has been drawn on the supposition that each rise in pressure takes place at the instant when the pressure has risen by . The seven rises in pressure are shown as red up-going arrows. Once the valve is fully closed the pressure at the valve remains constant with time until the first returning wave front reaches the valve. Then the pressure drops by  shown by a red down-going arrow. Ultimately the seven returning wave fronts reduce the pressure at the valve by  to give pressure of . I have drawn a blue line through the mid points of the down-going red arrows to give pressure-time graph up to the instant when the pressure reaches  when this pressure prevails until the instant .

 

On this graph I have added the limiting case for this construction to be valid when the time of closure is . If time of closure exceeds  and the same construction is used to produce a pressure-time diagram at the valve the fact that wave fronts begin to return to the valve before the valve is closed adds a severe complication. I have started to draw a diagram for the case of time of closure greater than  and it is quickly evident that the complexity makes it unlikely that a working engineer would see it as a viable way to look at this closure.

 

I have used the same proportions for figure 17-26 as I used for figure 17-35. The graph of pressure versus time is now much less steep and before the valve is fully closed the first returning wave arrives to start reducing the pressure. A way of constructing the graph after the first wave arrives is to imagine that the pressure goes on rising as if the wave was absent and then subtract the appropriate pressure drop = . Then one must imagine the pressure to rise again as if the valve is still producing a pressure rise at the same time as the returning waves are reducing the pressure. This is shown as two steps one in blue and the other in red. This can be repeated until the valve is fully closed. Then the imaginary rise ceases and the blue line goes horizontal.

 

Figure 17-26 is for a closure time of about 1.5 times the value of  and it is complex. The value of the closure time could be in minutes not milliseconds. If the closure time is long compared with  we know that we can use  and we know that when the closure time is of the same order as  the pressure transients are complex and probably unpredictable. This whole range is called slow closure and it is all too complicated to be viable. The only conclusion that we can draw is that it is better to design to eliminate the effects of pressure transients than to try to try to predict the transients.