How many liters of water come out? ....Physics Fluid question.?

Question

A cylindrical steel pressure vessel with volume 1.47 m^3 is to be tested. The vessel is entirely filled with water, then a piston at one end of the cylinder is pushed in until the pressure inside the vessel has increased by 3000 kPa . Suddenly, a safety plug on the top bursts.

How many liters (L) of water come out?

Answer 1

Let's see. We may assume that the compression of the water is the dominant contributor to the internal pressure, and that the weight of the water contributes little.

Bulk modulus of water, Β = V*ΔP / ΔV

When water is initially poured in, we'll call that the equilibrium state. There will be stresses set up due to the weight of the water, however the top of the vessel (where the plug is) will experience no stress or pressure.

Then the water is compressed by the piston. Two things are going to happen. The piston is going to decrease the length of the cylinder, thus increasing the internal water pressure. This internal pressure is then going to cause the walls of the container to expand under stress. This is the only stress that the plug feels.

You haven't given any information on the dimensions of the container, so it's hard to take into account the expansion of the wall. So let's ignore that for now. We take the container as being rigid.

For the pressure to increase by 3e6 Pa, the water must be compressed by an amount

ΔV = V*ΔP / Β = 1.47*3e6 / 2.2e9

= 0.002 m^3 = 2 L

Now the pressure is 3 MPa. The plug bursts allowing water to escape until there are no more stresses at the top i.e. until only gravity effects exist once again. Once we are reduced to the initial gravity state, there is no longer any driving force for water to escape from the top.

This means that 2 L of water that must escape before the initial  equilibrium is restored.

The only thing that bothers me still is that the question specifically mentioned that the vessel is made of steel, which indicates that wall expansion may be an issue that affects this result. In order to consider the elasticity of the container, we need to know the value of R/t, the radius to thickness ratio. If we use R/t = 10 with properties of steel, we'll find that 2.44 L of water escape instead.

Here's a follow up question if you're interested.

http://answers.yahoo.com/question/index;...