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Wazza
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- TL;DR Summary
- Trying to find the right equations which described the energy transfer when compressing a gas.
Hi there,
I am working on a home project which involves compressing a gas and then retrieving the stored energy at a later date. Obviously in a real world scenario the recovered energy will be less than the energy it takes to compress the gas, due to system losses. However as a starting point I am trying to model the idealised situation, assuming zero losses, and am having trouble making the equations stack up. I was initially misled by isothermal equations and have now learned the difference between adiabatic and isothermal. Still, when I perform the seemingly trivial calculations, energy seems to be disappearing.
It may be simplest if I describe my question using an example.
My starting point is a belief in the first law of thermodynamics : energy cannot be created nor destroyed, in a closed system it is simply transferred from one state to another.
Suppose I have chamber which is perfectly insulated and fully sealed except for an opening for a piston which seals perfectly and is frictionless.
The chamber is full of air which starts off at atmospheric pressure.
Work is done to push the piston down which compresses the air inside the chamber.
As far as I know; Work = (the transfer of) Energy.
So the increase in energy in the gas in the chamber should equal the work done to compress it. Correct me if I'm wrong.
Let's say
Chamber is 2000 litres = 2 m^3
Piston plate is 1 m^2
Piston plate moves by 500 mm = 0.5m (down, i.e. compression)
Air inside is air pressure to start with = 101325 kPa
Temperature is 25C
Air can be thought of as an ideal diatomic gas with gamma = 1.4Energy in the chamber (to start with) is pressure x volume
= 101,325 Pa x 2 m^3
= 202,650 Joules constant = pressure x volume^gamma
= 101,325 x 2^1.4
= 267,398.3Final volume
= initial volume - (piston plate area x piston plate movement)
= 1.5 m^3 Final pressure = constant / volume_end^gamma
= 267,398.3 / (1.5^1.4)
= 151,576.14Energy in the chamber (to end with) is pressure x volume
= 151,576.14 Pa x 1.5 m^3
= 227,364.2 Joules Energy increase in the gas =
227,364 - 202,650 = 24,714work done pushing the piston = average pressure x change in volume
= 1/2(P2 + P1)(V2 - V1)
= 1/2 (151,576.14 + 101,325) (2.0 - 1.5)
= 126,450.6 x 0.5
= 63,225.3 Joules
So, the work done (63,225 Joules) is much greater than the energy increase in the gas (24,714 Joules).
Where has the energy gone ??
I note that gamma of 1.4 is an empirical value for a diatomic gas. If I arbitrarily fiddle this value to be 2.044 then I can make the energy equation balance, but arbitrarily fiddling values is not good physics. What am I missing ?
Please note that this question is not about what happens in the the real world. I am trying to start by modeling an idealised situation so that I can understand the basic physics. Once I have that, I can then start introducing losses and real world considerations.
Can anybody please tell which of my calculations are incorrect and what the right calculations should be ?
I am working on a home project which involves compressing a gas and then retrieving the stored energy at a later date. Obviously in a real world scenario the recovered energy will be less than the energy it takes to compress the gas, due to system losses. However as a starting point I am trying to model the idealised situation, assuming zero losses, and am having trouble making the equations stack up. I was initially misled by isothermal equations and have now learned the difference between adiabatic and isothermal. Still, when I perform the seemingly trivial calculations, energy seems to be disappearing.
It may be simplest if I describe my question using an example.
My starting point is a belief in the first law of thermodynamics : energy cannot be created nor destroyed, in a closed system it is simply transferred from one state to another.
Suppose I have chamber which is perfectly insulated and fully sealed except for an opening for a piston which seals perfectly and is frictionless.
The chamber is full of air which starts off at atmospheric pressure.
Work is done to push the piston down which compresses the air inside the chamber.
As far as I know; Work = (the transfer of) Energy.
So the increase in energy in the gas in the chamber should equal the work done to compress it. Correct me if I'm wrong.
Let's say
Chamber is 2000 litres = 2 m^3
Piston plate is 1 m^2
Piston plate moves by 500 mm = 0.5m (down, i.e. compression)
Air inside is air pressure to start with = 101325 kPa
Temperature is 25C
Air can be thought of as an ideal diatomic gas with gamma = 1.4Energy in the chamber (to start with) is pressure x volume
= 101,325 Pa x 2 m^3
= 202,650 Joules constant = pressure x volume^gamma
= 101,325 x 2^1.4
= 267,398.3Final volume
= initial volume - (piston plate area x piston plate movement)
= 1.5 m^3 Final pressure = constant / volume_end^gamma
= 267,398.3 / (1.5^1.4)
= 151,576.14Energy in the chamber (to end with) is pressure x volume
= 151,576.14 Pa x 1.5 m^3
= 227,364.2 Joules Energy increase in the gas =
227,364 - 202,650 = 24,714work done pushing the piston = average pressure x change in volume
= 1/2(P2 + P1)(V2 - V1)
= 1/2 (151,576.14 + 101,325) (2.0 - 1.5)
= 126,450.6 x 0.5
= 63,225.3 Joules
So, the work done (63,225 Joules) is much greater than the energy increase in the gas (24,714 Joules).
Where has the energy gone ??
I note that gamma of 1.4 is an empirical value for a diatomic gas. If I arbitrarily fiddle this value to be 2.044 then I can make the energy equation balance, but arbitrarily fiddling values is not good physics. What am I missing ?
Please note that this question is not about what happens in the the real world. I am trying to start by modeling an idealised situation so that I can understand the basic physics. Once I have that, I can then start introducing losses and real world considerations.
Can anybody please tell which of my calculations are incorrect and what the right calculations should be ?
