Gas Flow into Vacuum Paradox ?

[arhanbezbora]

Gas Flow into Vacuum Paradox ??

Hi,

I've just been reading up on 1-D isentropic, compressible flow through a nozzle (constant cross-sectional area) and have hit a bit of a conceptual road block which I was hoping someone would help resolve.

I'm interested in the mass flow rate of the gas through the pipe which should be proportional to the difference in pressures between the two ends of the nozzle. Thus if we hold the front pressure constant and reduce the back pressure (or do vice versa), then presumably the mass flow would increase (proportionally with respect to the difference in pressure) until the advent of choked flow whereupon the mass flow would attain a constant value and be independent of further reductions in the back pressure. The critical pressure at which this occurs can be easily obtained from the stagnation pressure via the specific heat ratio of the gas.

Now what if start our experiment with the back pressure set to 0 (a vacuum) and have a gas in the reservoir at a pressure P that is just a tiny bit greater than 0, say 0.001 Pa. Based on my understanding of the equations, it seems that the flow will be choked no matter what the front pressure is, as long as it is nonzero. This seems hard to believe; the pressure gradient is tiny but we still have choked flow ?? Is this true at all ?

Does anyone know any resources that will help to understand the compressible flow of a gas into a vacuum ?

Thank you very much.

 

[maxpower2008]

This is an interesting question, but I think your paradox can be explained by a few small details.
First, choked flow is not the result of increasing flow rate, but rather the increase of the fluid velocity to the speed of sound, which is a direct result of the fact that the mass flow must be conserved. Likewise, the flow acceleration is governed by the continuity equation, which says that for a constant density fluid, the only method of acceleration is area decrease (for subsonic flows) or increase (for supersonic flows). The pressure ratio and area ratio of a nozzle are related and they have optimum values for a given nozzle, but the basic gist is that the lower your pressure drop, the greater your area ratio must be.
Additionally, the laws of thermodynamics don't allow you to expand a gas all the way to a complete vacuum as that requires 100% usage of the fluid's energy.
So if you start with 0.001 MPa, you would need to exhaust down to at most 0.0005 MPa at the nozzle exit to get sonic (and hence choked). That small expansion would take a huge area ration, and hence enormous nozzle to achieve, which is why rocket nozzles are all slightly underexpanded in the vacuum of space.
It's worth noting that the mathematics of isentropic flow also require the gas to have some small pressure at the exit, because to achieve absolute expansion, the area ratio from throat to exit goes to infinity.
I hope that helped.

-Max