Calculating Choke and Mach Number for Nozzle/ Restrictor

[bwood02]

Hi,

I've run into a problem calculating flow velocity at a nozzle/restrictor using carbon dioxide at ambient temps. I'm attempting to calculate this value via direct measurement of a few critical values. Specifically the supply stagnation pressure and the down stream pressure are directly measured.

I've come upon two equations for relating the stagnation and supply pressures and mach value-but they dont seem to correlate with each other. maybe someone has encountered this in their own time.

Knowing isentropic coeff for c02 (k) is 1.29--> flow is choked for p*/Po<= .546 ...easy enough

I've determined from my system measurement and my ratio is well below the choke ratio, so i expect to have choked flow with M~1.0 (no diverging section of nozzle to further accelerate gas)

Now consider two specified eqns:

when M=1: 1) P0/P* = ((k+1)/2)^((k/(k-1)) relating P0 and P specified for conditions M=1
and when M is not 1 2) P0/P = (1+((k-1)/2)*M^2)^(k/(k-1)) relating P0 and P (in my case M<=1)

calculating P0 for p*= 85PSI in eq 1) you will find P0~ 155 PSI when M=1 per the equation assumption
but..
calculating P0 for p = 85PSI in eg 2) you will find P0~290 PSI when M=1 is used.

why don't these values converge about M=1 if eq1) specifically addresses m=1? I also know my supply pressure is well beyond the 290PSI stagnation pressure calculated.

I've run this a few different ways but i believe this to be the root of my issues to this point.

Thanks in advance for any insight..

 

[casflo]

You talk about nozzle/restictor as if they were similar. Take into account that the nozzles have a contoured throat and the restriction plates usually has a sharp-edge orifice.
This is a big difference because the nozzles have critical pressure in the throat, calculated by this equation:
Pc/P1 = (2/(k+1))**(k/(k - 1)). For k = 1.3 (typical fo CO2) is Pc/P1 = 0.546
Note: P1 is the nozzle upstream pressure and Pc the critical pressure.
If you have a square edge orifice instead a nozzle, take into account that this type of orifice has not critical pressure and assuming that the orifice discharges directly to the atmosphere its pressure drop will be P1 - 14.7 psia.
If there is a pipe downstream the orifice, its pressure drop will be P1 - P2, being P2 the back pressure in the pipe.
See the Crane Technical Paper No 410 where you can find in the Appendix A the differences between the nozzles and the square-edge orifices.

 

[georgeverghese]

The equation in (1) is correct

The equation in (2) is confusing - when M<1, the value of P, which is the pressure downstream of the nozzle, will only be influenced by the backpressure induced on this nozzle as a result of friction losses in the downstream system. And this backpressure is affected by the mach number of the fluid as it traverses the downstream system and the value of 4fL/D for the downstream. Your equation in (2) does not have this 4fL/D term in it.

See equation 6-123 and 6-124 in Perry Chem Engg Handbook 7th edition for this case when M<1 for the more exact adiabatic compressible flow. For a simpler expression ( approximated by isothermal compressible flow), use equation 6-114.