Compressible choked gas flow through an orifice -- Excel formula

  • #1
stuartsjg
1
1
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:

P0P1m (total)volumetricP0P1m (total)volumetricP0P1m (total)volumetricP0P1m (total)volumetric
BarABarAkg/secL/minBarABarAkg/secL/minBarABarAkg/secL/minBarABarAkg/secL/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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Answers and Replies

  • #2
FluidDroog
7
6
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 [1].

Using the correct density values in your spreadsheet, I compare the sonic output to Lenox Laser's orifice calculator [2]. To do so, I need to convert mass flow and 'standard' liters/min from [3].

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 [4] which is about 20% different from 0.6.

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