Water flows inside infinitely long rigid pipe of diameter 10cm with speed 10m/s from left to right?

Question

There is a valve
http://upload.wikimedia.org/wikipedia/en/4/4f/USPatent6518683.png
installed in the 'middle' of the pipe.

What will be the force acting on the valve, if we intantly close it, and block water flow?

rigid means rigid

Answer 1

The water itself is slightly compressible so the force wouldn't be infinite. It's more like throwing a non-massless spring at a hard stop. The compression wave due to the stop will propagate upstream at the speed of sound in water. I'm sure calculus would provide a solution but it's getting late where I am.

Answer 2

That would be dependent on the elasticity of the pipe. The diametric expansion of the pipe would damp the inertial shock. One of the HDPE or plastic pipes with high elasticity would exhibit lower peek pressures.

Many contractor find that polymer pipe does not require dampers as metal plumbing did.

If you are talking about a hypothetical infinitely rigid pipe than the compressibility of water is the issue. The total mass of and infinite column of water would be infinite.

The force would be zero, the valve is broken.

Verner66 is wrong there would be a pressure curve that would increase based on compression and elasticity. The pipe or valve would fail at some point. The curve can be calculated but we are not given the parameters needed.

Answer 3

In closing the valve you reduce the momentum of the flow to zero.

Initially, volume flow rate = A*v = πd^2/4 * v
mass flow rate = ρπd^2/4 * v
momentum flow = mass flow rate * velocity
= ρπd^2/4 * v^2 kg m/s^2
= ρπ/4 Newtons

To kill the momentum of the fluid, this force must be acted on it by the valve, hence the reaction force on the valve
= ρπ/4 = 785 N

Answer 4

F=MA
IF YOU HAVE AN INFINITE MASS AND TAKE THE ACCELERATION TO ZERO INSTANTANEOUSLY
YOU WOULD END UP WITH AN INFINITE FORCE,
PIPE WOULD EXPLODE. ALL THIS IS JUST A THOUGHT EXPERIMENT ANYWAY. IT IS LIKE ASKING WHAT WOULD HAPPEN IF AN IMMOVABLE OBJECT WAS
HIT BY AN IRRESISTIBLE FORCE. THE ANSWER TO THIS IS THAT THE EXISTENCE OF EITHER ONE DENIES THE EXISTENCE OF THE OTHER.