Sizing valve for pressurized vessel to relieve under SUBcritical flow

[erk1313]

I need to size a valve to relieve air from a pressurized tank to drop pressure from 16psia to 14.76psia within 5 seconds. As the outlet pressure is greater than .5283 of the inlet pressure, this will occur as non-choked (i.e. sub-critical) flow. (Note that specification is not critical to safety). Is the following equation the correct approach? If so, how can I estimate the Expansion Factor (Y) if I don't know the Pressure Drop Ratio (Xt)?

From ISA specification for Compressible Gas Flow (EQ 11a) and summarized in this manufacturer brief:

Cv: flow coefficient
q: Volumetric Flow (SCFH)
N9 = 7320; conversion constant assumes units SCFH, PSIA, and R temperature
Fp = 1; pipe geometry factor without reducers
p1: inlet pressure
Y = 1 - x/(3*Fk*Xt); expansion factor
.....Fk = 1; Ratio of Specific Heats (assume air)
.....Xt: Pressure drop ratio factor (huh?)
M = 28.97; dry air
T1: inlet temperature (R)
Z = 1: Compressibility factor (assume air at standard T and P)
x = (P1-P2)/P1; ratio of pressure drop

I assume I can iterate through the equation to find the inlet pressure drop over time.
Thanks in advance for your help.

 

[ione]

That parameter, Xt, is also know as critical pressure drop ratio, and it is the pressure drop ratio corresponding to
choked flow condition

Xt = (P1 - Pchoke)/P1

 

[erk1313]

Would that equation for Xt still apply to non-choked flow?

So using the equation to find the critical pressure: Pchoke = 27.83 psia (air exhausting to atmosphere)

Then Pressure Drop Ratio: Xt = (16psia - 27psia) / 16psia = -0.688

 

[ione]

Upstream pressure in your case is P1 = 16 psia, so Pchoke = 0.5283*P1

 

[erk1313]

Duh! Thank you.

So finishing up, for the initial condition:
Pchoke = 8.5psia
Xt = 0.472
Y = 0.942
Flow = 8.56 SCFM (converted from SCFH)

Iterating every 0.2 sec... I get the following graph of inlet pressure for a valve of Cv = 2.0

Is this a close approximation I can trust?

 

[ione]

I think you'd need an Equation of State as well and not just what you've reported in your OP