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Pipe Transients
Transients Maths

These few pages provide an overview of how transient pressure waves are generated in pipe networks; some of basic maths used to describe them and finally some videos from the 'Contaminant Ingress into Water Distributions Rig' at The University of Sheffield to show some of the effects that transients in a system might have.

What are Transients?

Water Distribution Systems are very rarely in a completely steady state, there are always changes in the pressures and the flow rates around the network.  These can be due to varying customer demands, maintainence works, pump and valve operations or changing the input flow to the system.  However, due to the limitation of the speed of sound in the pipes, the changes in the flow conditions cannot be transmitted around the system instantaneously.  For example if you close a valve at one end of a pipe it takes a certain amount of time (the length of the pipe divided by the speed of sound in the pipe) for the information that you changed the system to be transmitted to the other end.  The information and the effect of changing the system is transmitted as a wave along the length of the pipe.

Transient Waves
  • Transfer information about changes in the system
  • Water is not entirely incompressible and the walls are slightly elastic boundaries
  • Travel at a finite speed
  • Involve the conversion of kinetic to pressure energy
  • Quicker changes in velocity create bigger pressure waves

Considering again the flow of water in a straight section of pipe with reservoir at the upstream end and a valve at the downstream end.  If we ignore friction, then the entire pipe is at the pressure in the reservoir (p1) and if the valve is initially open then flow is travelling at a velocity V.

Pipeline with a valve at the end
A straight pipe with an open valve at the end.  Not considering friction the entire pipe is at a pressure P = p1 and  velocity V.

If the valve is closed rapidly the water just adjacent to the valve is stopped instantaneously, however the water further upstream continues to flow along the pipe at the same velocity.  The still moving water piles into the static fluid slightly compressing it, raising its pressure and expanding the walls of the pipes (again very slightly). The increased pressure stops that moving fluid, but the water further upstream is still flowing and the process repeats along the length of the pipe, generating a pressure wave.

Animation of a Valve Closure

Frictionless Transient Animation

When the high pressure wave reaches the end of the pipe the entire length of pipe is now at a pressure p2, which is greater than the pressure in the reservoir (p1), and with no velocity.  As flow will always travel from high to low pressures the flow in the pipe reverses, drawing the water out of the pipe (again at velocity V) relieving the pressure.  Again this effect starts with the water adjacent to the reservoir and moves back down the pipe as a wave. 

When this pressure relieving wave returns to the valve all the fluid in the pipe is at a pressure, P = p1, but now the velocity of the flow is in the oposite direction.  The flow is now attempting to suck the water away from the valve but as it is still closed the flow velocity is again reduced to zero.  This causes the fluid next to the valve to decrease in pressure, expand slightly and cause a slight shrinkage in of the pipe in the radial direction.  This low pressure wave then travels along the pipe to the reservoir end, at which point the entire length of the pipe is at P = p3 < p1, so flow again enters the pipe from the reservoir, increasing the pressure back to p1 again. 

Finally this pressure wave travels the length of the pipe to the valve, at which point the pressure in the system is again at p1 and the velocity is V, and the process of the reflecting waves starts over again.  The reflection of the waves will continue infinetly if we do not consider friction in the system, including friction means that each sucessive wave will be smaller thatn the last until the oscillations damp out and the system reaches it's new state.

Transient Schematic with Friction
Transient Schematic with friction, shows the damped pressure oscillations

Transients Maths

Maintained and updated by Richard Collins, June 2010