# Leak from a container w/ a bellow and air bubble

snyper10x
Hey guys!
I'm doing a project and having a tough time deriving the equations to model it (Matlab (ODE/PDE) or Excel (cell/dv/dt)). I would love some insight/help etc!

To start off there's a sealed rigid container which has a bellow attached internally and a valve.
The inside of the bellow has its own constant air pressure source.
The bellow has a spring rate and can only extend in the x direction (diameter is constant).
The container is full of water at an initial pressure and also contains an air bubble.
The air bubble will always be a bubble (i.e. will never go into solution).
The bellow starts with some deflection (in -x, compressed) from the initial water pressure.
At t=0 the valve is opened so there is a leak out of the container. (m_dot_e)
The leak causes a pressure drop in the fluid and the bubble and bellow expand to equalize pressure.

I started this off as cell based modeling with just the container and a bubble, plotting pressure as a function of volume of water removed. That gave a characteristic isothermal p-v chart.
Then I did the container with just the bellows, again plotting pressure as a function of volume of water removed. That gave a linear decrease in pressure (as the spring rate is linear) (I ensured the bellows never got to free state).
Next I assumed it was differential between the volume of the bubble and volume of the bellows but wasn't able to derive them to where I could model it.
I then tried to use 1st Law to get an analytical solution but appear to be stuck.
Did start in the wrong place?!

Thanks in advance for the help guys!
Jason

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