Hello, we are exploring the possibility of energy recovery of pressurised gas in a sealed gas adsorption vessel by using a scroll pump and letting it run in reverse, mechanically coupling its shaft to another pump. To see if it is viable I wanted to calculate the isentropic work that could be recovered from the pressure vessel but I am failing at the math and maybe I have wrong assumptions. I have put together a short latex document with the process description and the math, I believe my mistake will be somewhere in the assumptions on page 4 or page 5. everything after the 0= on page 5 everything else is garbage... It's been a while since I did some work like this and I don't have anybody to ask at work. I've spent days on the issue but without knowing if even my basic assumptions are correct I feel like I could be wasting my time... If somebody could have a look I would be immensely grateful