Choked Flow in a Converging Duct: Volume vs. Mass Flow Rate

[sid_galt]

If subsonic flow is accelerated in a converging duct to mach 1 and then the duct is further converged, choked flow will occur. Will flow at the exit of the duct be compressed as the volume flow rate is decreasing but the mass flow rate has to remain the same.

 

[sid_galt]

Can anyone help please?

 

[Q_Goest]

Hey Sid. This is a bit outside my area of expertise, but I believe supersonic wind tunnels don't operate quite like you're suggesting.

Supersonic tunnels have a convergent-divergent nozzle ahead of the test section, in which the flow is accelerated to sonic speed at the "throat" and reaches the required supersonic speed at the end of the diverging portion.

Ref: http://vonkarman.stanford.edu/tsd/pbstuff/tunnel/tunnelconfig.html
So to answer your question, the gas does not compress further after the point at which sonic velocity is reached because after this point, the duct diverges.

In fact, I don't think it will compress as it enters the convergent section either. The first law of thermo is relavent here, just not sure how to apply it for this type of flow.

 

[FredGarvin]

If subsonic flow is accelerated in a converging duct to mach 1 and then the duct is further converged, choked flow will occur. Will flow at the exit of the duct be compressed as the volume flow rate is decreasing but the mass flow rate has to remain the same.

By stipulating that the mass flow rate remains the same then I am gathering that you are not changing the upstream pressure. Once you reach M=1.0 in the duct that is the maximum it will go without an increase in area. The thing is, I don't see how you can have a continuing decrease in the nozzle and have sonic velocity upstream of that reduced area. The smallest area will always be the first to go sonic. If you decrease the area after that, you are simply moving the location of the sonic flow to another location and making it happen sooner. The only way I know of to further increase the density is to increase the upstream pressure.

 

[Clausius2]

By stipulating that the mass flow rate remains the same then I am gathering that you are not changing the upstream pressure. Once you reach M=1.0 in the duct that is the maximum it will go without an increase in area. The thing is, I don't see how you can have a continuing decrease in the nozzle and have sonic velocity upstream of that reduced area. The smallest area will always be the first to go sonic. If you decrease the area after that, you are simply moving the location of the sonic flow to another location and making it happen sooner. The only way I know of to further increase the density is to increase the upstream pressure.

I agree. Once a geometry of the duct is established, the sonic flow is reached at the section of minimum transversal area, for an upstream stagnant pressure $$P_o$$. So an steady regimen is achieved. If you try to enlarge the duct making it more longer such us the section keeps on converging more (imagine a telescopic duct), then there will be an unsteady regimen during some time. At the throat the pressure was minimum, but due to the new geometry, the sonic flow will find something like a wall which enhances a first increasing in pressure due to early shocks against it. It will cause a pressure wave which will travel towards upstream, forcing the upstream stagnation pressure to be changed to another $$P_1$$. If the duct is connected to a large gas vessel, this wave probably will dissipate, and a new steady regimen will be reached again with an upstream stagnation pressure $$P_o$$ but different conditions at the new throat (although they will be sonic too). This is as I see the movie.

 

[sid_galt]

Thank you for the replies.

nce a geometry of the duct is established, the sonic flow is reached at the section of minimum transversal area, for an upstream stagnant pressure . So an steady regimen is achieved. If you try to enlarge the duct making it more longer such us the section keeps on converging more (imagine a telescopic duct), then there will be an unsteady regimen during some time. At the throat the pressure was minimum, but due to the new geometry, the sonic flow will find something like a wall which enhances a first increasing in pressure due to early shocks against it. It will cause a pressure wave which will travel towards upstream, forcing the upstream stagnation pressure to be changed to another . If the duct is connected to a large gas vessel, this wave probably will dissipate, and a new steady regimen will be reached again with an upstream stagnation pressure but different conditions at the new throat (although they will be sonic too).

How can the pressure wave travel upstream when the flow has already gone sonic.

If the gas was say being pumped by a piston, then will the piston feel the pressure wave? I'm guessing that since it is traveling upstream, the piston would feel the pressure wave.

Are there any good books which deal with such flows? I don't mean a book on aerodynamics but a book which solely deals with airflows, jets, diffusion, etc.

 

[russ_watters]

The subject you're looking for is called "gas dynamics" and it deals with sonic/supersonic flow through nozzles.

 

[Clausius2]

How can the pressure wave travel upstream when the flow has already gone sonic.
.

Hey man!, I propose you to receive a medal for the most inquirer PF member. Your questions are very spicy, and it denotes your brain works very well and you have a great curiosity.

How can the pressure wave travel upstream?. Well, the sonic flow is an unstable one. I mean, I will bet sonic conditions doesn't exist in reality. The same happens with critical flows in open channel hydraulics. How do you define "sonic flow"? A flow such as M====1.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.?

Around sonic conditions there will be always fluctuations. To say the truth I don't dare to speak more about this stuff, because I really have reached a point here I cannot answer you with total assurement. This question could be dealed by means of gasdynamics steady equations everybody knows here, but I am not going to do that, because I would be lying you. Your question has a trick, because it implies an unsteady regimen, and given a geometry variation as you mentioned, only upstream fluid can feel it by means of a pressure wave, reshaping so all the fluid field. It must be on that way. During the unsteady process the sonic conditions could be destroyed momentarily, in the former sonic section, to be re-established in the new one downstream.

You know sidgalt, I really know one can sometimes be confused imagining flow configurations and trying to explain it with the steady one dimensional gas dynamics equations that every us learn in an undergraduate course. One must look farther in order to explain some other difficult flows.

Anyway your questions are always welcome.

EDIT: I have a question for you. What are you studying and where?. It is only curiosity. Please tell us.

 

[sid_galt]

Hey man!, I propose you to receive a medal for the most inquirer PF member. Your questions are very spicy, and it denotes your brain works very well and you have a great curiosity.

Thanks

EDIT: I have a question for you. What are you studying and where?. It is only curiosity. Please tell us.

I am in grade 11 currently in India. Recently bought a bunch of books on aerodynamics and gas turbines and went through them and can't stop thinking about them. As for my interests, basically everything in physics and engineering though currently it is more focused on mechanical engineering.

 

[Clausius2]

Thanks

You're welcome

I am in grade 11 currently in India. Recently bought a bunch of books on aerodynamics and gas turbines and went through them and can't stop thinking about them. As for my interests, basically everything in physics and engineering though currently it is more focused on mechanical engineering.

What is grade 11? Sorry I am not too much familiar with foreign educational systems. How old are you?.

 

[sid_galt]

What is grade 11? Sorry I am not too much familiar with foreign educational systems. How old are you?.

Grade 11 - 11th Class equivalent to the junior year in US school systems - that's two years before college. I am 15.5 years currently.

 

[Clausius2]

Grade 11 - 11th Class equivalent to the junior year in US school systems - that's two years before college. I am 15.5 years currently.

WOW!

When I was 15 years I didn't know anything of what you currently ask or talk about in this forum. I was only concerned of being a famous football player. If you keep on being interested on this stuff at such early ages, surely you will do something important in the future with your life.

 

[sid_galt]

Thanks!


BTW, I was thinking about this a bit more and realized, wouldn't decreasing the duct diameter more result in Prandt-Meyer compression waves which will increase the static pressure but decrease the total pressure?