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

 

[Achy47]

Hello Jason,

I can see that you have made some progress in your project by modeling the container with just the bellows and the container with just the bubble. However, to accurately model the system, you will need to consider the interactions between the two components.

Firstly, it is important to understand the physical principles at play in this system. The pressure inside the container is determined by the volume of water and air present, as well as the volume of the bellow. When the valve is opened, water will start to leak out, causing a decrease in pressure and a corresponding expansion of the bellow and bubble to equalize the pressure. This process will continue until the pressure inside the container is equal to the pressure outside.

To model this system, you will need to use a combination of ODEs and PDEs. The ODEs will describe the behavior of the bellow and the bubble, while the PDEs will describe the behavior of the water inside the container. You will also need to consider the boundary conditions at the interface between the bellow and the water, as well as the valve.

One approach could be to use a mass balance equation to track the volume of water in the container, and then use the ideal gas law to calculate the pressure inside the container. The behavior of the bellow and bubble can be described using Hooke's law and the ideal gas law, respectively.

I suggest breaking down the problem into smaller, more manageable parts and then gradually incorporating them into a larger model. You may also find it helpful to consult with a colleague or mentor who has experience in modeling similar systems. Good luck with your project!