In the modelling section I developed a model for a gas surge vessel:
This model is clearly non-linear and needs to be linearised prior to analytical solution. Have a go at linearising the model assuming that Pin is variable (Pout is constant) and the following nominal conditions:
My solution is given below:
The equations look horrendous, but that's just because the numbers are untidy - as you can see the final solution ends up as a simple first order equation. If the inlet pressure changes by a step it will take the surge vessel around 2.5 seconds to come to its new steady-state and the pressure change in the vessel will be around 75% of the change in the inlet pressure.
The numerical solution can be found here.