Got it. I understand what you are saying now. I will have to go back and look at your math later.The intake is not energy absorbing but energy providing. You can harvest this energy with a piston that is pushed inside by the incoming air.
I do not talk about the expansing but about pushing the air outside away. This isobar process is not spontaneous and consumes energy. This energy is provided by the expansion and it is equal to the work of the isobar intake of air at the bottom.
EDIT: The thing that I have found confusing (and is once again confusing to me) is that the work you are describing already seems accounted for in an isothermal expansion formula. The expanding high pressure air in a cylinder expanding can generate energy. The energy it generates is during the process of moving aside a volume of low pressure air.
As I said, I am confused again, but I will look at it later.