Hydrualic pressure moving plug in pipe

[BobJones1234]

Homework Statement

A plug with a given mass is placed in the end of a horizontal pipe to hold water at a given pressure behind it. Is there any way to calculate how far this plug would move in the x-axis if it suddenly lost it's grip against the pipe wall, and it was situated a given height over the ground? Time of movement before it hits the ground is ok (no air resistance etc), but need to know if it is possible to find the initial speed in the x-direction (which also would be final speed).

Homework Equations

Force acting on plug (when still) = pressure x area
Time until plugs hits the ground = sqrt(2*height/g)

The Attempt at a Solution

 

[Nidum]

It all depends on what the water does once the plug starts to move .

If the pipe is connected at the bottom of say a very tall large volume tank so that any flow of water into the pipe does not change the pressure at the bottom of the tank significantly then the plug will be propelled by a constant value pressure force while it is still in the pipe and then once out of the pipe propelled in a less easily definable way by the stream or jet of water coming out of the now open pipe end . The plug would probably in most cases leave the water stream at some point and then fly in a free trajectory until it hit the ground .

If the pipe is connected to a more uncertain supply of water such that any flow of water into the pipe causes a large pressure loss then the plug may not be propelled very far at all .

The second case is of little interest but the first case would certainly be challenging to try to analyse !

 

[BobJones1234]

Thanks nidum. I did not mention this, but there is no inflow/supply of water. The pipe is just a sylinder which is closed in the other end. Water is static until the plug losses its grip against the pipe wall. The water pressure then drops to atmospheric (as on the outside of the pipe), and the plug should be propelled along the x axis.

 

[haruspex]

water at a given pressure

Unless you are going to treat it as compressible, the initial pressure is only that due to the weight of the water, i.e. zero at the top and 2rg at the bottom, where r is the radius. On that basis, atmospheric pressure would be plenty enough to hold it in.
To get around that, we have to allow an air leak around the plug, so adding atmospheric pressure to all of the above.

the initial speed in the x-direction (which also would be final speed).

I don't think you can assume that. Released from rest, the water will accelerate, pushing the plug in front of it.

 

[Nidum]

In any realistic version of this problem and with free admission of air as suggested by @haruspex the plug would probably just be sluiced out of the pipe and fall to the ground quite close to where it left the pipe .

 

[BobJones1234]

Thanks @haruspex, intial speed in x-direction was meant to be the speed of the plug after loosing grip against the pipe wall. What I meant was that it should not slow down in the x-direction, but accelrate with gravity towards the ground, which would determine the time until it hits the grount.
@Nidum @haruspex It was unclear from my side, water is pressurized behind the plug inside the pipe. Typical the pressure could be 300-400 bar behind the plug, and therefore we have seen that there is a danger for the plugs if they loose the grip against the pipewalls that the plugs could have a high velocity, and I am trying to generate a spreadsheet for safety distance.

 

[Nidum]

This problem keeps evolving .

Please tell us exactly what the actual set up is and supply some basic dimensions .

 

[BobJones1234]

@Nidum it would be different setups, but I want to just create a simplified spreadsheet giving an estimate with input pressure, pipe diameter, weight of plug and height above ground.

For example:

A plug with weight 1000kg is holding water at a pressure of 300 bar behind it in a horizontal pipe with 500mm diameter 1 meter above the ground.
Should then get an area of 0.39m2 and a force of 1.18e7 Newton acting on the plug. If for some reason the plug was released, this force should accelrate the plug up to a velocity in the horizontal direction, but the water would not be acting on the plug while it moves if we consider the water as incompressible.
Dropping 1 meter in y-direction should take 0.45 sec.

 

[haruspex]

water is pressurized behind the plug

I do not see how that can be without an inflow source. What maintains the pressure?

 

[BobJones1234]

@haruspex volume behind plug would be filled through a port in the pipe. Water is pressurized up to a given pressure, and then pressure should be static when target pressure is reached, there would be no more inflow.

 

[Nidum]

The problem described in this thread can't actually be 'solved' so as to get exact numerical answers . There are too many unknown quantities .

We can though look at the limit state case where all the stored energy in the pipe and water gets transferred to the plug at 100% efficiency .

So the plug leaves the pipe with a kinetic energy equal to all the stored energy in the pipe and water . After that we would just have a standard ballistic trajectory problem to work through so as to get the horizontal distance traveled by the plug before it hits the ground .

Finding the amount of stored energy in the pipe and water is not entirely straightforward but it can be done if all relevant factors are taken into account .

https://www.physicsforums.com/threads/pressure-drop-across-a-valve-in-tank.923867/#post-5829979