Physics of a convergent nozzle

[Sunfire]

Hello,

Imagine a convergent nozzle; static pressure at exit is atmospheric. The fluid is air. The pressure on the pressurized side is P.

goal #1: achieve nozzle exit velocity somewhat below sonic
goal #2: have P as high as possible.

Is this possible to achieve through the geometry of the nozzle? What is the highest possible P?

Thank you

 

[Engineer_Phil]

All you need is Bernoulli's equation and the right set of assumptions.

 

[Sunfire]

What about the so-called `critical pressure ratio`` in subsonic nozzles - it is defined as

x = $(\frac{2}{\gamma+1})$$^{\frac{\gamma}{\gamma-1}}$ = 0.528 for air

Does this mean that for ANY subsonic nozzle, this is the highest pressure ratio before the exit flow becomes sonic

 

[Engineer_Phil]

I'm not formiliar with critical pressure ratio.

However bernoulli's equation is valid for incompressible fluids only. Air is considered incompressible as long as the velocity does not exceed 0.3 Mach.

 

[Sunfire]

It seems to me that no matter what convergent nozzle is used, it will get choked (exit air will be sonic) if the inlet pressure is more than about 2 times the outlet pressure

 

[Engineer_Phil]

If you assume gravity has no effect, and the inlet velocity is very small then Bernoulli's equations says:

V_exit=(2*(P_inlet-P_exit)/ρ)^1/2

provided V₂ is not more than 0.3M.

 

[Sunfire]

In this specific case, we are looking only at subsonic outlet velocities (e.g. nozzle is not choked)

 

[Engineer_Phil]

OK. I'm not sure if I helped you any. So are you saying that you are dealing with velocities (0.3 Mach) < V_exit < (Mach)?

 

[Sunfire]

Yes, I need to achieve fairly high velocities at outlet, say about (but below) M=1; also, need to find out what is the highest inlet nozzle pressure in this scenario. It seems the answer is 1/0.528 = 1.89 times the outlet pressure

 

[boneh3ad]

All you need is Bernoulli's equation and the right set of assumptions.

If you assume gravity has no effect, and the inlet velocity is very small then Bernoulli's equations says:

V_exit=(2*(P_inlet-P_exit)/ρ)^1/2

provided V₂ is not more than 0.3M.

By definition in the OP's question, the Mach number will be greater than 0.3. Bernoulli's equation does not apply here.

Hello,

Imagine a convergent nozzle; static pressure at exit is atmospheric. The fluid is air. The pressure on the pressurized side is P.

goal #1: achieve nozzle exit velocity somewhat below sonic
goal #2: have P as high as possible.

Is this possible to achieve through the geometry of the nozzle? What is the highest possible P?

Thank you

What about the so-called `critical pressure ratio`` in subsonic nozzles - it is defined as

x = $(\frac{2}{\gamma+1})$$^{\frac{\gamma}{\gamma-1}}$ = 0.528 for air

Does this mean that for ANY subsonic nozzle, this is the highest pressure ratio before the exit flow becomes sonic

It seems to me that no matter what convergent nozzle is used, it will get choked (exit air will be sonic) if the inlet pressure is more than about 2 times the outlet pressure

So, for a converging-only nozzle (or a straight tube with no area change), the critical pressure ratio of 0.528 represents the ratio of back pressure to total pressure where the nozzle is choked, i.e. the Mach number is unity. If you lower the back pressure, the Mach number doesn't change, nor does the total mass flow through your orifice. If you raise your reservoir pressure (P in your example), the Mach number will stay at 1 but the mass flow through the orifice will increase. You could raise P to any number you want so long as the pressure vessel serving as your reservoir can handle it. So, if you want to remain slightly below sonic conditions, just set it up such that your pressure ratio $p_b/p_t$ is ever so slightly greater than 0.528.

Keep in mind, the critical pressure ratio changes if you add a divergent duct of any sort to the end of your system.

 

[Sunfire]

Thank you for your reply. In my case, I am limited by the back pressure. It is 1 atmosphere. Also, I need to avoid sonic flows, e.g. have to keep the flow subsonic. This would mean I cannot increase P too much, no matter what kind of divergent nozzle I use

 

[boneh3ad]

Right, so if you are using simply a convergent nozzle and do not want to choke it and your back pressure is fixed at atmospheric pressure, then anything under 1.894 atm would be a subsonic outlet. How close you get to sonic is just a matter of how close you get to that pressure.

 

[Sunfire]

Thank you, this makes a lot of sense. I was unsure whether nozzle geometry would allow me to increase the reservoir pressure P... but it seems this cannot be done with nozzle modifications

 

[boneh3ad]

Not as long as you are using only a convergent nozzle. Using a convergent-divergent nozzle you could change the pressure ratios but the whole point of that is to go supersonic, so that's not really what you are lookin for.

You CAN play with the mass flow rate by changing your geometry and holding your pressures constant, but Mach number and pressure ratio will be dictated by the fact that you have only a converging nozzle.

 

[Sunfire]

boneh3ad, would you know some pressure ratios achievable for supersonic flows; after all supersonic might be a way for us to go...

 

[boneh3ad]

You can calculate various pressure ratios for whatever Mach number you happen to be interested in without much trouble. If you have a particular Mach number in mind then the rest is pretty easy. You could also work in reverse if you wanted and start with a pressure ratio and see what Mach number you can get out of that as well. It all depends on your application.

 

[pranj5]

Do convergent-divergent nozzles choke when the velocity at the throat exceeds Mach?

 

[boneh3ad]

Do convergent-divergent nozzles choke when the velocity at the throat exceeds Mach?

Exceed Mach? What do you mean by this? If you mean "exceed Mach 1" then the question doesn't have an answer, as the Mach number at the throat cannot exceed 1.

 

[pranj5]

I want to mean exceed Mach 1. Do you want to mean that even in convergent-divergent nozzles, the speed can't exceed Mach 1?

 

[boneh3ad]

It cannot exceed Mach 1 at the throat.

 

[pranj5]

OK. But, kindly tell me whether c/d nozzles will choke or not if the velocity at the throat attain Mach 1.

 

[boneh3ad]

Are your familiar with the definition of choked flow in a nozzle?

 

[pranj5]

Choking means there will be no change in downstream despite flow/pressure increase at the base, right?

 

[boneh3ad]

No. The downstream flow can always react to changes to the flow upstream. For example, mass flow rate will continue to change. Pressure can continue to change if the upstream pressure changes since a nozzle operating at its design Mach number features a set pressure ration from the inlet to the exit. Changing the downstream conditions further won't change the mass flow flow through the nozzle if the flow is choked, but do you know why? To me it just seems like you are reading a textbook definition of choked flow without actually understanding what it means. What is your background here? Have you taken a compressible flow course?

 

[pranj5]

Then kindly explain. I want to know.

 

[boneh3ad]

This is precisely why I asked if you've ever taken a course on fluid dynamics and/or compressible flow. I am trying to figure out where your base-level knowledge is and where your misconceptions arise. It will be much better for you if you go on a guided exploration of this topic than if I just blurt out the answer here.

So, I ask again:

1. Have you taken a fluid dynamics course before? What about compressible flow?
2. Why do you think that, in a choked nozzle, changing upstream conditions will not affect the downstream flow?
3. What exactly do you think choked means? If you have a converging-diverging nozzle, can you give a rough sketch of the Mach number along the length of the nozzle, starting from the reservoir and going all the way through to the exit plane (assuming the nozzle is operating ideally)?

 

[pranj5]

As far as I know, choking means no change in downstream while the condition at the upstream may increase because the velocity has attained its maximum speed at the throat. That's all I know.
And I must admit that I haven't taken any course on fluid dynamics ever separately. I am guessing based on what I have studied during my college years in aerodynamics.

 

[boneh3ad]

Aerodynamics is fluid dynamics. Air is a fluid. You still haven't answered all of my question, though.

1. Have you taken a compressible aerodynamics course (since it appears you have some sort of aerodynamics background)?
2. What makes you think that changing the upstream conditions results in no change in downstream conditions in choked flow?
3. Can you give me a rough sketch of what you think the Mach number, pressure, and temperature are doing in a converging-diverging nozzle?

If you really do have an aerdynamics background, you ought to at least be able to discuss this a little in a way that I can help you discover why your definition of choked flow is wrong, but you have to meet me halfway here.

 

[pranj5]

I have background in physics. No specialisation in fluid dynamics.

 

[boneh3ad]

You don't seem to want to answer my questions, and it seems like you are new to the topic, so I think you would be best served by doing some reading on the topic of compressible flows. Really, the phenomenon of choked flow is rooted in basic thermodynamics and fluid mechanics, so some background reading would be beneficial for you, I think. I'd suggest a few sources:
https://www.amazon.com/dp/0486419630/?tag=pfamazon01-20
https://www.amazon.com/dp/0072424435/?tag=pfamazon01-20

 

[pranj5]

Kindly supply some net links that I can download directly.

 

[boneh3ad]

I am unaware of any online resources that are as good as those books.

 

[pranj5]

At least give me some that can help me to get primary information.

 

[boneh3ad]

It used to be that potto.org had some incomplete, free, online textbooks available on these topics. They were okay... not great, but better than random stuff on the internet. It looks like all of their PDFs are missing but you could still browse the HTML versions. I can't guarantee that the texts are complete, and they aren't as good as the resources I already suggested.