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

AI Thread Summary
The discussion centers on modeling the flow through an orifice with helium gas under varying pressures using Excel. Key calculations include determining density, specific heat ratios, and critical pressures, with results aligning with textbook values for subsonic flow. However, discrepancies arise in sonic flow calculations, with results varying significantly based on different formulas. The primary issue identified is the density calculation, which may not be using the correct units, leading to inconsistent mass flow rates. The conversation emphasizes the importance of accurate density values and the potential impact of the discharge coefficient on flow calculations.
stuartsjg
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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
 

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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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