Energy of a water tank with compressed air

[davidpele]

Homework Statement

Lets fill a 1m x 1m x 2m enclosed water tank with 1m3 of water. Lets place the tank at sea level resulting in the tank having 1 atm of pressure. Now lets pump air into the tank until it is at 2 atm. Has the tanks potential energy changed and by how much?

Relevant Equations

mgh

At sealevel, the water in the tank is just mgh so 1000 kg * g * 1m so about 10 kjoules.
Now if we pump air until it is 2 atm.

The potential energy of a compressed gas is given by the formula:
U = -nRTln(V2/V1)

But in this case it is not the gas that we care about but the water.

I can understand that the tank has higher potential energy because of the higher pressure inside it. But I can't figure out how to calculate how much it has changed by.

I can imagine increasing the air pressure is like lowering it in height. 10m deep in the sea is the equivalent of 2atm.

The other way I can think of this is like the compressed air is raising the height of the tank.
The energy required to compress 1 m³ of air at 1 atm and 25°C to 2 atm is approximately 2,436 J.
So that would mean we've supplied 2436 joules of potential energy to the water.

So the change in height would h = 2436/1000g around 25cm.

Am I thinking about this in reasonable way?

Thanks.

 

[Chestermiller]

Homework Statement: Lets fill a 1m x 1m x 2m enclosed water tank with 1m3 of water. Lets place the tank at sea level resulting in the tank having 1 atm of pressure. Now lets pump air into the tank until it is at 2 atm. Has the tanks potential energy changed and by how much?
Relevant Equations: mgh

At sealevel, the water in the tank is just mgh so 1000 kg * g * 1m so about 10 kjoules.
Now if we pump air until it is 2 atm.

The potential energy of a compressed gas is given by the formula:
U = -nRTln(V2/V1)

But in this case it is not the gas that we care about but the water.

I can understand that the tank has higher potential energy because of the higher pressure inside it. But I can't figure out how to calculate how much it has changed by.

I can imagine increasing the air pressure is like lowering it in height. 10m deep in the sea is the equivalent of 2atm.

The other way I can think of this is like the compressed air is raising the height of the tank.
The energy required to compress 1 m³ of air at 1 atm and 25°C to 2 atm is approximately 2,436 J.
So that would mean we've supplied 2436 joules of potential energy to the water.

So the change in height would h = 2436/1000g around 25cm.

Am I thinking about this in reasonable way?

Thanks.

No. Are you talking about gravitational potential energy or some other kind of potential energy? Can you please provide a schematic of the system you are describing?

 

[davidpele]

No. Are you talking about gravitational potential energy or some other kind of potential energy? Can you please provide a schematic of the system you are describing?

Thanks for your reply. I realized I missed one important contextual thing. The way to measure the "energy" present in that system would be to put the water through a turbine and measure the amount of energy generated. I guess that means gravitational potential energy plus the additional energy due to air in the tank being at 2 atm thus exerting additional pressure on the water and increasing the amount of energy it could generate through the turbine.

 

[Chestermiller]

Thanks for your reply. I realized I missed one important contextual thing. The way to measure the "energy" present in that system would be to put the water through a turbine and measure the amount of energy generated. I guess that means gravitational potential energy plus the additional energy due to air in the tank being at 2 atm thus exerting additional pressure on the water and increasing the amount of energy it could generate through the turbine.

Please show how you would set this up schematically in terms of piping and turbine. Why do you feel you need to put a number on this?

 

[davidpele]

I'm sorry, I don't think I understand what you mean by schematically. Could you give an example? The question is interesting to me and hopefully to others. Thanks for the discussion.

 

[davidpele]

I found this diagram of an airbattery. This looks like a good example.