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Physics of a convergent nozzle

  1. Apr 4, 2013 #1
    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
     
  2. jcsd
  3. Apr 4, 2013 #2
    All you need is Bernoulli's equation and the right set of assumptions.
     
  4. Apr 4, 2013 #3
    What about the so-called `critical pressure ratio`` in subsonic nozzles - it is defined as

    x = [itex](\frac{2}{\gamma+1})[/itex][itex]^{\frac{\gamma}{\gamma-1}}[/itex] = 0.528 for air

    Does this mean that for ANY subsonic nozzle, this is the highest pressure ratio before the exit flow becomes sonic
     
  5. Apr 4, 2013 #4
    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.
     
  6. Apr 4, 2013 #5
    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
     
  7. Apr 4, 2013 #6
    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.
     
  8. Apr 4, 2013 #7
    In this specific case, we are looking only at subsonic outlet velocities (e.g. nozzle is not choked)
     
  9. Apr 4, 2013 #8
    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)?
     
  10. Apr 4, 2013 #9
    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
     
  11. Apr 4, 2013 #10

    boneh3ad

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

    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 [itex]p_b/p_t[/itex] 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.
     
  12. Apr 4, 2013 #11
    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
     
  13. Apr 4, 2013 #12

    boneh3ad

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    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.
     
  14. Apr 4, 2013 #13
    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
     
  15. Apr 4, 2013 #14

    boneh3ad

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    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.
     
  16. Apr 4, 2013 #15
    boneh3ad, would you know some pressure ratios achievable for supersonic flows; after all supersonic might be a way for us to go...
     
  17. Apr 5, 2013 #16

    boneh3ad

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    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.
     
  18. Jul 16, 2016 #17
    Do convergent-divergent nozzles choke when the velocity at the throat exceeds Mach?
     
  19. Jul 17, 2016 #18

    boneh3ad

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    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.
     
    Last edited: Jul 17, 2016
  20. Jul 17, 2016 #19
    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?
     
  21. Jul 17, 2016 #20

    boneh3ad

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    It cannot exceed Mach 1 at the throat.
     
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