Compressible choked gas flow through an orifice -- Excel formula

[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

 

[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 [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