Decompression of dry air in relation to temperature

[CyanPhysics]

I have a problem that i cannot solve despite my best efforts. I do realize not all information here is relevant, but i still included it so you could get the full picture.

Imagining we have a cylinder that is filled with dry air (normal air without water vapor) that has a fixed volume of 8,5 liters. The initial pressure is 300 bar (30000 kPa). After use the final pressure is 50 bar (5000 kPa), therefore the change in pressure is 250 bar (25000 kPa). This happens in a span between 20 to 30 minutes. The cylinder is depressurizing into a tube thru a valve that reduces the maximum pressure inside that valve to 50 bar (5000 kPa) and then another reduction valve, furthermore reducing to 5 bar (500 kPa) and then again to 1.25 bar (125 kPa) the final chamber is of a volume approximately 1 liter, the sizes of the chambers in between are unknown. The final chamber is a mask and the air flows out with every exhale. From observation we have seen that there is some cooling effect, that is not measured.

My question is what is the theoretically maximum possible change in temperature at the reduction valve 300 bar -> 50 bar, if we disregard other factors from the inside. I will conduct an experiment on this and record the results, but i am still interested in the theory behind it as it will help me in the future.

Gas is normal air without water vapor.

P₁ = 500 bar
P₂ = 50 bar
ΔP = 250 bar
V = 8.5 Liters
t ≈ 1500 second
^{____________________________________________}
ΔT = ? Kelvin

 

[Chestermiller]

If heat transfer between the air within the cylinder and the cylinder itself is neglected, then the air remaining inside the cylinder at any point during the process has experienced an adiabatic reversible expansion in doing work to force the gas ahead of it out toward the exit valve. This adiabatic expansion is accompanied by cooling of the gas within the cylinder. This can be quantified.

 

[Anand Sivaram]

In an adiabatic process, there is no transfer of heat, then time does not matter. It should be easy looking at the classical equations of Adiabatic process and Ideal gas law.

 

[Chestermiller]

In an adiabatic process, there is no transfer of heat, then time does not matter. It should be easy looking at the classical equations of Adiabatic process and Ideal gas law.

The ideal gas law is a poor approximation to the behavior of air in the 50 - 300 bar pressure range. Non-ideal gas effects need to be included. So it's not as easy as you suggest.