I've been puzzling over a question / class of questions I'd like to solve via rough estimate before turning to CFD analysis.
Say I have a pressure vessel with one inlet and one outlet. The pressure vessel is at some initial pressure $P_0$. The outlet is at some fixed pressure $P_{out}$. The inlet has some mass flow rate, $\dot{m}$ (assume it's attached to an infinite upstream reservoir and a mass flow controller).
With a compressible fluid, a known mass flow rate, and a set volume in the vessel, what mechanism would you use to approximate the final steady state (volume-average, we'll say) pressure in the vessel? Assume that the outlet pressure is such that the flow will choke. Assume no heat transfer to the walls and that the gas is ideal.
One solution I have thought of is to perform a dumb iterative solution:
- Pick some time step, $dt$
- Add $\dot{m} * dt$ mass to the tank
- Compute the new pressure in the tank via the ideal gas law
- Use choked pipe flow equation to determine $\dot{m}$ at exit (knowing exit area)
- Subtract $\dot{m}_{exit} * dt$ mass from the tank
- Repeat until "convergence"
Possibly you could do some Bernoulli-streamline-style analysis, but I'd like to also apply this sort of approximation to a multi-inlet / multi-outlet problem. This feels like some form of the Hardy Cross method might be applicable as well. Interested in anybody's suggestions.
Graphs of what the above technique gets me for a 1 kg/s flow rate into a 5 m^3 volume vessel, initial conditions T = 290K & P = 1 Pa in the vessel:
Edit: Added a few figures from my "estimation."