Energy contained in compressed air

[Ameen1985]

Hello all,

For a vessel with a constant volume of 8 m^3 with a 15 bar pressurized air in it. What is the contained energy in the vessel?? If the pressure changes from 15 bar to 7 bar or to 20 bar; how much energy does the vessel lose or gain ? Assuming isentropic processes with n or k = 1.4
I found in the literature different equations to solve such a problem. It I think it should be easier than that. Many thanks in advance.
As you can see three different equations with different answers.

 

[Snickster]

It looks like all three equations are the same but in different form. The equations are for obtaining the work (energy) for the very slow quasi static expansion of a gas from a high to low pressure (like in a piston cylinder arrangement). This is the maximum possible energy that can be extracted from the gas.

The first equation indicates that for air for an isothermal process the polytropic exponent "n" factor is 1.0 and for an isentropic process the same factor is 1.4. So when the gas is expanding, if heat is added in enough quanity while it is expanding the gas will remain at constant temperature during the expansion and the exponent will be 1.0 for a constant temperature expansion. If there is no heat input to the gas during expansion then the temperture will drop since the energy of the gas is going into the work to push against the piston (or surroundings) and no heat is coming in to keep the gas at constant temperature. If there is no heat input and there is no irreversible losses then the process is isentropic where the exponent "n" is 1.4 and the temperature drops during the expansion in accordance with the isentropic expansion temperature relationship equations. Note for an isentropic process the exponent is the same as the ratio of specfic heats, so for air Cp/Cv = 0.24/0.17 = 1.4 the isentropic expansion exponent. A polytropic process is one that is somewhere between an isothermal and isentropic process and the exponent "n" is called the polytropic exponent "n" rather than the isentropic exponent "k".

In a typical expansion the process is somewhere between isothermal and isentropic which puts the polytropic exponent somewhere between 1.0 and 1.4. For instance if you are expanding a gas in a piston-cylinder arrangement as you expand the gas it is picking up heat from the walls of the cylinder and depending on how hot the cylinder is will determine how much the expansion goes from isentropic to isothermal and therefore how much the expansion goes from isentropic 1.4 to isothermal 1.0 factor. In your case of a gas exploding in a vessel I think the process will be closer to isentropic as not much heat can be picked up from surroundings that fast.

The second equation looks like the same equation but in different terms and assuming an isothermal process with n=1.0. It even states that and isothermal process was considered in the write up.

The third equation is the same as the first with "gamma" replacing "n" and making other substitutions to just change the form a little.

 

[Ameen1985]

Thank you @Snickster for your reply.
However, regardless of the volume, in the third equation you can see P/(gamma - 1) while in the first equation it is P*gamma/(gamma - 1).

 

[Snickster]

The first equation gives energy in terms of available work per unit volume that can be extracted from a complete cyclical thermodynamic compression-expansion process. In thermodynamics textbooks this is the "area behind the curve" = -INTEGRAL(VdP). For a compression cycle it would be the net work input of the compressor to the fluid. In an expansion cyclical process it would be the net work output of the fluid. If you take the textbook equation for -INTEGRAL(VdP)and then divide both sides by V to get work in terms of work per unit volume then you get the first equation. The write-up for the first equation states that the work is per unit volume for a thermodynamic CYCLE.

The second equation is the total work for the ISOTHERMAL compression or expansion of a gas = INTEGRAL(PdV) This is the non-cyclical TOTAL work (not the work per pound but total work) of the single expansion of a gas from a high pressure to a lower pressure.

The third equation is also TOTAL work of a single expansion or compression (non-cyclical) of a gas from a high to lower pressure in an ISENTROPIC (or polytropic) process = INTEGRAL(PdV).

 

[shvet]

The stored energy of the equipment or piping system and TNT equivalent see ASME PCC-2 art. 5.1 app. 2

 

[pierreick]

Hi,
You may find pointers in the handbook attached.
Pierre

 

[Ameen1985]

Thank you @Snickster for your reply but I asked you about the missing gamma between the first and third equation. In the third equation you can see P/(gamma - 1) while in the first equation it is P*gamma/(gamma - 1).

 

[Ameen1985]

Thank you @shvet for your reply.
The ASME equation is the same as the third equation in my post. However, is it (1/(gamma - 1)) or (gamma/(gamma - 1)) ?

 

[shvet]

What does gamma stand for? Heat capacity ratio (coefficient of Laplace)?

 

[Ameen1985]

K or gamma or n is the ratio of specific heat of the fluid

 

[shvet]

What does "(1/(gamma - 1)) or (gamma/(gamma - 1))" stand for? Which piece of which equation is it?

 

[Ameen1985]

The first equation and third equation in my post.
n/(n-1) in the first equation and 1/(gamma -1) in the third and ASME equation.
n or gamma or K are the same = 1.4 the ratio of specific heat of the fluid. I hope my question is clear.

 

[shvet]

Question was

What is the contained energy in the vessel?

Answer was

ASME PCC-2 art. 5.1 app. 2

I hope my question is clear.

What is question now? Is ASME's equation correct? Yes, it is correct.

 

[Ameen1985]

Thank you Shvet

 

[Snickster]

Thank you @Snickster for your reply but I asked you about the missing gamma between the first and third equation. In the third equation you can see P/(gamma - 1) while in the first equation it is P*gamma/(gamma - 1).


My response in my second post correctly explains the three equations. The difference between the 1st and 3rd equations is that the 1st equation is -INTEGRAL(VdP) in energy per unit volume units, and the 3rd is INTEGRAL(PdV) in total energy units. 2nd equation is INTEGRAL(PdV)for a constant temperature process in total energy units.

 

[pierreick]

Hi,
Consider this book to support your query:

https://books.google.co.th/books?id...N.+Brown&cd=1&redir_esc=y#v=onepage&q&f=false

Pierre