Modelling the number of times a gas tank can be filled

Question

I'm currently working on a project where I need to estimate how many times a large, high-pressure gas tank can refill a smaller tank. For this purpose, I'm using a pressure regulator to manage the flow between the two tanks and have assumed that the regulator operates perfectly without any inefficiencies.

To model the gas transfer, I started with the ideal gas law and applied the adiabatic expansion formula to derive the following relationship:

$$ P_1 \left(\frac{n_1}{n_2}\right)^\gamma = P_2 ​ $$

and then ran it itervely using a pyhon program to find out how many times it can fill the smaller tI implemented this in a Python program, running iterations to determine how many refills are possible before the pressure in the large tank falls below a specified threshold.

However, I'm questioning one of my underlying assumptions related to the adiabatic process(as there is a regulator in between) :

$$ P_1​V_1^γ​=P_2​V_2^γ​ $$

Is this correct? if not why is it incorrect and how would I be able to model this correctly?

Answer 1

Theoretically, this seems perfect but in real world applications, there are some ramifications that you'd be facing. One of these is that the pressure regulator introduces a source of non-adiabaticity which means that the regulator will tend to dissipate some energy. This is the most important thing that you should take care of before applying the gas laws. You can try to use the first law of thermodynamics.

Q=ΔU+W

Using this equation, you can model equations as apt for the work done and heat transfer during the aforementioned process taking the efficiency into account.