How to determine when compressible flow is choked?

[MysticDream]

TL;DR Summary

Seeking clarification on the behavior of gas flow through venturi tubes and converging diverging nozzles and ducts.

In a compressible flow there is a critical pressure ratio between the upstream and downstream stagnation pressures in order for the flow to become choked, which always happens at the minimum cross-sectional area, or throat. In a duct (or nozzle) that only converges and exits to a stagnation environment, the downstream pressure, for air, would have to be .528 times the upstream pressure as I understand.

However, it is said that with a diverging section of duct (or nozzle) after the throat, the upstream to downstream pressure ratio can be much lower and still result in a choked condition.

This seems counterintuitive as this would mean the velocity and flow rate for a given converging nozzle with a subsonic flow rate could be increased by simple adding a diverging section. How does this work? Does the diverging section kind of redirect the gas in the direction of the flow and thus "pull" more on the throat section? If so, then the angle and size of the diverging section would affect the flow, it seems. If the diverging section had a sharp angle and large diameter, at what point is it no different than not having a diverging section at all? Is there a limiting angle that would prevent the diverging section from having an effect on the flow? Any insight would be appreciated.

 

[jack action]

the upstream to downstream pressure ratio can be much lower

The pressure is much lower in the divergent part but the throat pressure ratio is still 0.528.

This seems counterintuitive as this would mean the velocity and flow rate for a given converging nozzle with a subsonic flow rate could be increased by simple adding a diverging section.

The velocity increases in the divergent part but not the [mass] flow rate as it is still choked at the throat.

How does this work?

At one point, a normal shock will appear to reequilibrate everything (d, e, or f in the next image).

The location where the normal shock appears is a function of the area ratio. Even if you have no divergent part per se, you still have a transition when changing suddenly areas in a duct. The normal shock will thus always appear at the apparent sudden change. The shock might be "oblique" or as an expansion fan at the sudden area change, depending on the larger area pressure.

(source)​

 

[MysticDream]

Thanks for the detailed explanation. There's one thing I'm still trying to wrap my head around. Say the nozzle ended at point 2 and there is no divergent part (the exit is point 2). The only way for the velocity to reach mach 1 at point 2 is for the external pressure to be .528 * p1. Now lets add the divergent part back on and the exit is at the receiver. Now all of a sudden the external pressure can be much higher (higher than critical), yet the velocity at point 2 can still reach mach 1. This is shown in curve "c" on the graph. If we maintained this higher external pressure and again took away the divergent part, the velocity at point 2 has now reduced because the external pressure is higher than .528 * p1. If this is all correct, then indeed, simply adding the divergent part of the nozzle increased the velocity at point 2 and the overall mass flow rate. Correct me if I'm wrong.

Now assuming this description is correct, and we're maintaining this higher external (exit) pressure, flowing according to curve "c", what would then happen if the diameter at point 3 increased significantly in steps and at each step the flow rate was recorded? It seems at some point that the diameter and angle of the divergent section would increase so much that the velocity at point 2 would decrease because you're approaching having no divergent section at all, in which case, again the external (exit) pressure would have to be critical for the throat to reach mach 1. So there seems to be a limit here, and I'm trying to figure out what that is and how to calculate it.

To calculate the overall flow rate, it seems to me that first the geometry of the exit (or receiver) area has to be taken into account. Then the exit pressure and initial upstream (stagnation) pressure and temperature. Then, knowing the throat area, you should be able to calculate the flow rate.

 

[jack action]

Sorry I did not answer this thread sooner, PF did not send me a notification for a new reply. I'm guessing the discussion continued - without me realizing it - on https://www.physicsforums.com/threa...for-choked-flow-through-a-c-d-nozzle.1063800/ .

 

[MysticDream]

Sorry I did not answer this thread sooner, PF did not send me a notification for a new reply. I'm guessing the discussion continued...

Yes sir, and I appreciate you sharing your knowledge. No problem.