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Consider a sufficiently large reservoir of compressed air at a pressure of $7 \text{ bar}$ (abs.). A tapping of $25 \text{mm}$ diameter of pipe length $1 \text { m}$. The pipe is open to the atmosphere on one end.

Applying Bernoulli's theorem between point 1 (inside the reservoir) and point $2$ (just at the outlet of the pipe), we get,

$$P_1 = P_2 +\frac12\rho v^2$$

Where $v$ is the velocity at the exit of the pipe.

Taking $P_1$ as $700000 \text{ Pa}$ and $P_2$ as $100000 \text{ Pa}$, density of compressed air as $8.4$ $\text{ kg/}$$m^3$, we get $v = 377 \text{ m/s}$.

(I am neglecting the friction losses, entry and exit losses as I want a rough estimate)

Is this calculation correct? Where am I making a mistake? Can the velocity reach supersonic without the use of a converging-diverging duct?

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  • $\begingroup$ So not worth accepting the answer? $\endgroup$
    – Solar Mike
    Commented Jul 10 at 4:40
  • $\begingroup$ @Solar Mike Done $\endgroup$
    – Utility ZC
    Commented Jul 10 at 10:00

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  1. It is a converging-diverging duct. It's just that the converging happens suddenly as the air goes from the reservoir into the pipe, and the diverging happens in the wall-less jet beyond the pipe outlet.
  2. Bernoulli's equation in that form is not going to be applicable once compressibility effects become significant. You're going to need some variant of the steady flow energy equation including internal energy and flow work, and some sort of assumption about the ratio of heat transfer to work transfer (probably either assuming the process is adiabatic and using the Poisson adiabat to relate pressure to temperature or assuming the process is isothermal and using Boyle's law).
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  • $\begingroup$ Thank you...Will the velocity not even be close to what I have calculated? $\endgroup$
    – Utility ZC
    Commented May 28 at 5:24
  • $\begingroup$ @UtilityZC Since you don't tell us the initial temperature, there's no telling what the velocity will be. $\endgroup$
    – Daniel Hatton
    Commented May 28 at 10:25
  • $\begingroup$ The temperature of the compressed air in the reservoir can be taken as 27 degree Celsius. $\endgroup$
    – Utility ZC
    Commented May 28 at 12:12
  • $\begingroup$ @UtilityZC Right: then you can apply the Poisson adiabat to find the temperature at the outlet (remembering to use absolute temperatures not Celsius temperatures), and apply the steady flow energy equation $c_pT_1+v_1^2/2 = c_pT_2+v_2^2/2$, assuming $v_1 = 0$ as you did before to find the velocity. Check at the end that friction and conduction really are negligible. $\endgroup$
    – Daniel Hatton
    Commented May 28 at 14:10
  • $\begingroup$ Thank you again...How do I calculate the temperature T2? Do I use P1/T1=P2/T2 relation? $\endgroup$
    – Utility ZC
    Commented May 29 at 4:13

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