Is there a pressure ratio limit for choked flow through a c-d nozzle?

[MysticDream]

TL;DR Summary

Seeking clarification on how convergent-divergent nozzles behave in sonic/subsonic compressible flows.

Could line "b" become choked if the diameter at point 3 was increased? If so, what is the limit for the diameter and angle of the divergent section of the nozzle? Is there a point at which the angle is too great, and the diverging part is no longer effective, and the flow goes back to behaving like only having a converging section? It seems there must be a limit otherwise choked flow could be achieved with only a very slight pressure difference.

 

[jack action]

Could line "b" become choked if the diameter at point 3 was increased?

No. The choked flow at the throat depends solely on the conditions before the throat, namely the stagnation properties of the flow. If no energy is added to the flow (isentropic flow), they shall remain the same everywhere in the convergent part nozzle and before a normal shock in the divergent part. The stagnation temperature (i.e. a representation of the total enthalpy - or equivalent internal energy when at rest - of the flow) remains the same everywhere in the nozzle.

 

[MysticDream]

No. The choked flow at the throat depends solely on the conditions before the throat, namely the stagnation properties of the flow.

Then do you agree with this question and solution?

 

[jack action]

Yes, given the exit pressure is an initial condition, the exit Mach number is obtained by the exit pressure via the stagnation pressure definition. So you can choke the flow.

If so, what is the limit for the diameter and angle of the divergent section of the nozzle? Is there a point at which the angle is too great, and the diverging part is no longer effective, and the flow goes back to behaving like only having a converging section?

When too abrupt, any type of area change becomes a converging-diverging nozzle by the effect of vena contracta.

 

[MysticDream]

When too abrupt, any type of area change becomes a converging-diverging nozzle by the effect of vena contracta.

This is a good point.

I still don't understand why line "b" from the original post can't become choked if the diameter at the receiver was increased sufficiently. My understanding is the velocity at the throat is calculated from the pressure ratio between initial stagnation and exit, and the ratio between the throat and exit area. Just like in the case with the problem and solution above, the divergent part of the nozzle increased the throat velocity over a simple converging nozzle with the same minimum area. If the diverging section was made smaller in a c-d nozzle, the flow would decrease. If the diverging section was made larger, the flow would increase, yet you say it would not increase until it chokes. Why not? The first pressure ratio at which the throat can choke with a given geometry I believe is called the first critical pressure ratio.

 

[jack action]

I still don't understand why line "b" from the original post can't become choked if the diameter at the receiver was increased sufficiently.

That is why I changed my answer to "yes" in post #4. Previously, I was just looking at the variables in the mass flow rate equation, not realizing how the Mach number in it was influenced by the exit pressure. I was imagining the inlet flow velocity was part of the initial condition. Sorry for the confusion.

 

[MysticDream]

That is why I changed my answer to "yes" in post #4.

No problem. Thanks for clarifying.

It seems to me there would still be a limit to this pressure ratio and geometry of the diverging section. For example (referring again to the original post), the critical pressure ratio for a converging only nozzle is .528p1, but for line "c" it seems to be around .85p1, which is the first critical pressure ratio. Line "b" seems to be around .92p1, which could only be made the first critical pressure ratio if the divergent section diameter was increased (apparently). Is there a ratio at which it would not be possible to choke the flow? What about .95p1 or .99p1? At some point it seems the flow would become unsteady and the divergent section would not be effective. Perhaps the angle makes a difference. I haven't been able to find anything that addresses this.

 

[jack action]

Any exit pressure can create a choked flow given the proper exit area. The equation for the area ratio is:

The exit Mach number $M$ is found with the exit pressure $p$ from the stagnation pressure definition:

So for every $\frac{p}{p_t}$ there is a corresponding exit area $A$ that will choke our flow at the throat. (Remember, the throat area $A^*$ is fixed in our scenario.) Here is how it turns out:

If the exit pressure is $0.528\ p_t$ (dashed line), the area ratio is 1 because the exit Mach number is 1 and thus you have a choking condition at the exit; you thus have no diverging part in your nozzle. On the right side of the graph, the exit Mach number is subsonic, and on the left side, it is supersonic.

The angle doesn't matter, it is just a question of area ratio.

 

[MysticDream]

Any exit pressure can create a choked flow given the proper exit area.

This clarifies one of my questions. I appreciate it.

The angle doesn't matter, it is just a question of area ratio.

Does anything matter? What about the length of the diverging section? What if it was a 90 degree angle and you had a vena contracta effect and you increased the diameter beyond what is required for choking at the current pressure ratio. Would the velocity at the throat then begin to decrease and start behaving like a converging only nozzle? I'm trying to understand the limits.

 

[jack action]

What about the length of the diverging section?

Nope.

What if it was a 90 degree angle and you had a vena contracta effect and you increased the diameter beyond what is required for choking at the current pressure ratio. Would the velocity at the throat then begin to decrease and start behaving like a converging only nozzle?

Here it is different as the exit is also the throat. At the extreme case of 90°, it is a simple converging nozzle by definition. Therefore, if it is not choked at a certain diameter, it won't be choked increasing it further because you are also increasing the throat diameter (from the vena contracta) at the same time which, in our initial problem, we assumed was fixed.

No flow will ever turn a 90° sharp angle. If you don't build the walls to follow the streamlines, fluid will form the proper shape and energy will be lost doing so. In practice, no matter the angle, flow separation will always be there; it is just insignificant when the flow is not turbulent, thus why this theory can be applied.

I'm trying to understand the limits.

From memory, the limits for minimal losses is a 60° converging part with a 7-10° diverging part.

(source)​

Note that these angles are also true for exterior flow. To minimize flow separation, you will have a 60° windshield and 7-10° back window and trunk on a car with the smallest coefficient of drag.

 

[MysticDream]

From memory, the limits for minimal losses is a 60° converging part with a 7-10° diverging part...

That is all good information, including the images. Thanks a lot.