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Leak from a container w/ a bellow and air bubble

  1. Sep 16, 2014 #1
    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
     

    Attached Files:

  2. jcsd
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