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

• I
• stuartsjg
In summary, the conversation discusses the problem of modeling the flow through an orifice with specific parameters such as upstream and downstream pressure, diameter of the orifice, and type of gas. The speaker explains their attempts at calculating the flow rate using equations and formulas, but encounters difficulties with the sonic flow calculation. They also mention the unexpected result of decreasing pressure resulting in the same volumetric flow rate. The speaker hopes for assistance in finding and correcting any mistakes in their calculations.

#### 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
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

Last edited:

## 1. What is compressible choked gas flow through an orifice?

Compressible choked gas flow through an orifice refers to the flow of a gas, such as air or steam, through a small opening or orifice that is smaller than the diameter of the pipe it is flowing through. This type of flow is characterized by high velocities and a decrease in pressure as the gas passes through the orifice.

## 2. How is compressible choked gas flow through an orifice calculated?

The flow of a compressible gas through an orifice can be calculated using the Excel formula =SQRT(((2*gamma)/(gamma+1))^((gamma+1)/(gamma-1))*((P1-P2)/P1)*((A2/A1)^2)*((1-(P2/P1)^((gamma-1)/gamma)))), where gamma is the specific heat ratio, P1 is the upstream pressure, P2 is the downstream pressure, and A1 and A2 are the cross-sectional areas of the pipe and orifice, respectively.

## 3. What is the significance of "choked" flow in this context?

In compressible choked gas flow through an orifice, "choked" refers to the point at which the gas flow reaches its maximum velocity. This occurs when the gas velocity is equal to the local speed of sound, and any further decrease in downstream pressure does not result in an increase in flow rate. At this point, the flow is said to be choked and the flow rate is limited by the size of the orifice rather than the downstream pressure.

## 4. What are some applications of compressible choked gas flow through an orifice?

One common application of compressible choked gas flow through an orifice is in the design and operation of gas flow meters, such as orifice plates and venturi meters, which are used to measure the flow rate of gases in industrial and scientific settings. This type of flow is also important in aerospace engineering, where it is used in the design of rocket nozzles and other propulsion systems.

## 5. Are there any limitations to using the Excel formula for calculating compressible choked gas flow through an orifice?

While the Excel formula provides a useful approximation for calculating compressible choked gas flow through an orifice, it does have some limitations. For example, it assumes that the gas is ideal and that there are no heat losses or energy dissipation during the flow. In reality, these assumptions may not hold true, and the results may differ from actual flow rates. Additionally, the formula may not be suitable for highly complex or non-uniform flow systems.