# Physics of a convergent nozzle

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

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

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

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.

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

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

Vexit=(2*(Pinlet-Pexit)/ρ)1/2

provided V2 is not more than 0.3M.

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

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

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

Gold Member
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:

Vexit=(2*(Pinlet-Pexit)/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.

Sunfire said:
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.

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

Gold Member
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.

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

Gold Member
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.

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

Gold Member
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.

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

Gold Member
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.

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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?

Gold Member
It cannot exceed Mach 1 at the throat.

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

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

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