# Compressible choked gas flow through an orifice -- Excel formula

• I
stuartsjg
Hello,

I normally get these things working but I am a bit stuck as i don't feel I am getting sensible answers...

The problem is simple (!):
Q: Model the flow through an orifice with an upstream/supply pressure of 301BarA where the downstream pressure is in the range 1BarA to 301BarA. The orifice is 0.5mm in diameter and gas is Helium at 0C​

I am doing this in excel so formula are what I've copied out of that. I have:
P0 as 301BarA​
P1 as 1 to 301BarA​
T1 as 0C​
>> Determined density as p0 =((4*P0)/(0.0821*(T1+273)))/1000/1000 to get kG/cu.m so i get 5.35396E-05 which i believe to be correct.​
(checked at 1BarA aligns with textbook values)​
cp as 5.193​
cv as 3.116​
>> Determined specific heat ratio as y = 1.6667 which aligns with textbook values​
>> Determined p* as =(2/(y+1))^(y/(y-1)) which for helium is 0.487 which i think is OK too​
Critical pressure CP* = P1 / p*​
>> For example, with a P1 of 100BarA i get the critical pressure as CP* = 205.28BarA​
Determining sonic flow by p1/p2 and where p1 is greater than the critical pressure (else subsonic)​
>> eg 200BarA into 96BarA is supersonic flow as 200Bar is greater than (P2/p*) 96/0.487 = 197BarA​
but 200BarA into 104BarA is subsonic flow as 200Bar is less than 213.49BarA​
From this, i have a column which is intended for a sonic mass flow calculation and another for subsonic, and just an IF statement to use the appropriate calculation for the final result.​
Finally, the orifice is defined as​
Cd = 0.6 (arbitrary typical number)​
SupplyDia = 5mm (upstream pipe diameter)​
OrificeDia = 0.5mm >> calculated as 1.9635E-07 sq.m​

So, i think I've been successful in doing the subsonic flow, engineering toolbox has a calculator for this and my numbers align.
For reference, I've worked this out by:
mass flow kG/S mdot = Cd*(PI()/4)*((OrificeDia/1000)^2)*p0*(2*(((P0-P1)*100000))/(p0*(1-(OrificeDia/SupplyDia)^4)))^0.5​

As an example:
P0 = 200BarA, P1 = 160BarA​
>> mass flow mdot = 1.99086E-06 kG/s which i convert to volumetric by =(mdot/p0)*1000*60 or 3346.6 L/min​

Putting a practical head on, i can picture that sort of flow rate for that pressure and nozzle etc.

Going for a pressure closer to the critical point, just as we fall out of sonic flow, i get we should be looking at not too much over
3.16458E-06 kG/sec 5319.65L/min - however any way i try to implement the sonic calculations, i get wildly different answers!

What I've tried(!):
From engineering tool box (no calc for this, just eqn)​
mdot = (Cd*OrificeArea)*(SQRT(y*p0*P0*((2/y+1)^((y+1)/(y-1)))))​
>> Given the same just at the critical point entering into sonic flow, this gives mdot = 6.21982E-08 kG/se or 104.55L/min which i feel too low​
from chemeurope, a version which didnt need the density​
mdot = =(Cd*OrificeArea*P0*100000)*SQRT(((y*4.003)/((1+0.0045*0.1*P0)*8314.5*(T0+273)))*(2/(y+1))^((y+1)/(y-1)))​
>> this gives a result of 0.00218 kG/s or 3,658,503.09 L/min which is certainly too high!​
Ive done a bit of a cheat in excel which is to determine the highest mass flow rate (using the sub-sonic calculation) before we enter sonic flow, on the basis just before sonic flow and just after sonic flow are not going to be a million miles different.

This is a bit of a cheat as i would really like to get my sonic flow calculation correct and working.

One other odd effect i had not expected...

Using the sub-sonic flow method as described, it doesn't matter what input pressure P0 i have, whether its 10bar or 300bar, i always get the same volumetric flow rate, but a different mass flow rate, for example:

 P0 P1 m (total) volumetric P0 P1 m (total) volumetric P0 P1 m (total) volumetric P0 P1 m (total) volumetric BarA BarA kg/sec L/min BarA BarA kg/sec L/min BarA BarA kg/sec L/min BarA BarA kg/sec L/min 300​ 126​ 4.7469E-06​ 5319.650936​ 200​ 84​ 3.16E-06​ 5319.651​ 100​ 42​ 1.58E-06​ 5319.651​ 50​ 21​ 7.91E-07​ 5319.651​ 300​ 132​ 4.7469E-06​ 5319.650936​ 200​ 88​ 3.16E-06​ 5319.651​ 100​ 44​ 1.58E-06​ 5319.651​ 50​ 22​ 7.91E-07​ 5319.651​ 300​ 138​ 4.7469E-06​ 5319.650936​ 200​ 92​ 3.16E-06​ 5319.651​ 100​ 46​ 1.58E-06​ 5319.651​ 50​ 23​ 7.91E-07​ 5319.651​ 300​ 144​ 4.7469E-06​ 5319.650936​ 200​ 96​ 3.16E-06​ 5319.651​ 100​ 48​ 1.58E-06​ 5319.651​ 50​ 24​ 7.91E-07​ 5319.651​ 300​ 150​ 4.7217E-06​ 5291.478222​ 200​ 100​ 3.15E-06​ 5291.478​ 100​ 50​ 1.57E-06​ 5291.478​ 50​ 25​ 7.87E-07​ 5291.478​ 300​ 156​ 4.6263E-06​ 5184.568651​ 200​ 104​ 3.08E-06​ 5184.569​ 100​ 52​ 1.54E-06​ 5184.569​ 50​ 26​ 7.71E-07​ 5184.569​ 300​ 162​ 4.5289E-06​ 5075.407612​ 200​ 108​ 3.02E-06​ 5075.408​ 100​ 54​ 1.51E-06​ 5075.408​ 50​ 27​ 7.55E-07​ 5075.408​ 300​ 168​ 4.4294E-06​ 4963.846569​ 200​ 112​ 2.95E-06​ 4963.847​ 100​ 56​ 1.48E-06​ 4963.847​ 50​ 28​ 7.38E-07​ 4963.847​ 300​ 174​ 4.3275E-06​ 4849.719898​ 200​ 116​ 2.89E-06​ 4849.72​ 100​ 58​ 1.44E-06​ 4849.72​ 50​ 29​ 7.21E-07​ 4849.72​ 300​ 180​ 4.2232E-06​ 4732.842002​ 200​ 120​ 2.82E-06​ 4732.842​ 100​ 60​ 1.41E-06​ 4732.842​ 50​ 30​ 7.04E-07​ 4732.842​ 300​ 186​ 4.1163E-06​ 4613.003766​ 200​ 124​ 2.74E-06​ 4613.004​ 100​ 62​ 1.37E-06​ 4613.004​ 50​ 31​ 6.86E-07​ 4613.004​ 300​ 192​ 4.0065E-06​ 4489.96816​ 200​ 128​ 2.67E-06​ 4489.968​ 100​ 64​ 1.34E-06​ 4489.968​ 50​ 32​ 6.68E-07​ 4489.968​

I think the effect of the decreasing pressure P0 reducing the density is cancelling out the conversion from gravimetric to volumetric, although i would have thought less pressure = less density (OK) = less mass (OK) = less flow (not OK)... So that's probably an issue!

Anyway, I've attached the spreadsheet I am working on, sorry its a bit of a work in progress!

Sorry for the long question, i am hoping there's just a few simple mistakes in there... :)

Any help would be appreciated - its consumed an evening with much head scratching, i even asked my 5 year old Daughter and told me "stop being silly" and walked away...

Thanks,
Stuart G

#### Attachments

• OrificeFlow.xlsx
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• FluidDroog

FluidDroog
Hi Stuart,

Let's get to the bottom of this.
P0 as 301BarA
P1 as 1 to 301BarA
T1 as 0C
>> Determined density as p0 =((4*P0)/(0.0821*(T1+273)))/1000/1000 to get kG/cu.m so i get 5.35396E-05 which i believe to be correct.
(checked at 1BarA aligns with textbook values)
Perhaps a unit conversion problem here. At a P of 301 Bar (30.1 MPaA), rho = 46.01 kg/m^3, engineering toolbox has a good write up on this .

Using the correct density values in your spreadsheet, I compare the sonic output to Lenox Laser's orifice calculator . To do so, I need to convert mass flow and 'standard' liters/min from .

standard Liter/min = mdot * 60e3 * 22.414/MW * 1000

With P0 = 200 barA and T=0 C input into both sonic calculators the outputs are within 20% of each other. This validates your sonic formulas, the primary issue is the density calculation (check your units).

Why the 20% difference?
Finally, the orifice is defined as
Cd = 0.6 (arbitrary typical number)
I estimate the ideal Cd to be 0.72 following equation 2 from  which is about 20% different from 0.6.

References:
 Helium gas Density https://www.engineeringtoolbox.com/...html?msclkid=5aa74f92ae2d11ec8afe2093b604c2d8
 Lenox Laser Orifice Calculator https://lenoxlaser.com/resources/ca...tor/?msclkid=69c42910ae2011ec82ce61f227854fdd
 Converting mass flow to sccm https://www.physicsforums.com/threa...ic-centimeters-per-minute.917608/post-6614235
 NIST. Comparison of CFV Theoretical Models to Experimental https://tsapps.nist.gov/publication...0961&msclkid=3570c7ddae2711ecba4468e745bee897

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