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I am trying to calculate the final pressure and temperature in the pressure regulator and connecting line of a gas cylinder when the valve is open to pressurize the line.

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The type of process across the regulator is Joule Thompson or isenthalpic ($\Delta h=0$) expansion. An ideal gas has zero temperature change during this process, but any real gas will have such a change, determined by its Joule Thompson coefficient.

In addition, if it is a closed system you are also compressing and heating the gas already in the line, which often overwhelms any JT cooling effect.

It is not a trivial problem you are trying to solve. My advice is do a test with a temperature probe if you're able.

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Yes, mechanical work is done. This is similar to mechanical work done by a balloon when you untie the knot and let the balloon fly all over the place. Or similar to a water rocket. Except in a water rocket the ejected media is water, much denser than air but the concept is the same: gas pressure in a container ejecting some gas or liquid through some orifice.

The Bernoulli flow equation, Q is:

$$Q= C_AA\sqrt{2gh}$$

  • g is the gravity acceleration.
  • h is the difference between the tank's pressure and ambient pressure.

The momentary work is: $$dW=\dot Q P$$ $$P = P_{contaner}-P_{atm}$$

You need the Volume of the container and Ca and P container tp get the total work.

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