A simple logical problem regarding pressure and force

Question

Suppose a 2 meter long pipe, with area of cross section is $1 m^2$. The pipe is vertically placed, with the bottom end closed. Inside the pipe, at the closed bottom end, there is $1 m^3$ of air, closed with a disc. This disc is movable, like a piston in a cylinder (the cylinder is the pipe).

Now, I'm pressing on the disc downwards and compressing the air to 5 bar of pressure (the volume of the air decreased).

Now I'm pouring water on to the disc, $1000$ $kg$ of water.

Now in the pipe,there is the compressed air at the very bottom, then the piston on it, and on the piston, filled with water of $1 m$ height ($1 m^3$ i.e., $1$ ton of weight).

If I release the pressure of the compressed air at the bottom very suddenly, UP TO WHAT HEIGHT will the water column will throw off(like a bullet)?

Consider the atmospheric pressure also, which is acting downwards,when it is shooting up..

Answer 1

The way I see it, the air will expand, probably adiabatically ("suddenly", so that there's no time for heat transfer out of the piston), until the pressure in the air volume is 1 + 0.1 bar (i.e. atmospheric pressure + hydrostatic pressure from the water). Mind that the pressure in the air volume was already 1bar.

Also, "atmospheric pressure also, which is acting downwards" is imo incorrect - the pipe being in the atmosphere, the atmospheric pressure is acting from all sides, not only downwards.

I won't do the calculations for you, as this is homework, but you should have everything you need to apply a couple formulas from your textbook.

Some more considerations/assumptions: The piston must be assumed massless, otherwise its weight would already compress the air volume initially. The air volume after compression has been given time to cool down after the compression to 5 bar. Regarding the "sudden"ness - this needs further thought in reality - the timescale of expansion (i.e. how long will it take - you have to consider you have to accelerate 1ton of water, and the acceleration won't be constant) must be small compared to the timescale of heat transfer (conduction/convection/radiation) out of/into the volume in question. this makes it a bit more complicated to justify the assumption of adiabaticity.

Answer 2

(Assumes that air occupies 1m^3 before it is compressed to 5 barr) To evaluate this problem we may use the ideal gas equation:

$PV = nRT$,

where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is a constant and $T$ is temperature. If we assume that temperature doesn't change, we may write this equation as simply:

$PV = k$, where $k$ is some constant.

Now we know the following:
Initial pressure on air = $P$ = 1 barr (from atmosphere)
Initial volume of air = $V$ = 1m^3

Therefore, we have

$PV = 1\times1 = 1 units = k$

Now once we compress the water to 5 barr, we have:
$PV = 5*V = 1$, therefore $V = 0.2 m^3$.

We then add water. The total downward pressure (excluding external force) is thus: $P_{total} = P_{water} + P_{air} = 0.1 + 1 = 1.1 barr$ Therefore,
$1.1V = 1$, therefore: $V = \frac{1}{1.1}m^3 = 0.91m^3$

Therefore the water will rise to a height of $0.91$m. Since the water occupies $1m$, there will be no spillage, which is consistent with our calculations.