What Is the Best Method to Calculate Gas Flow Through an Orifice?

[FranzS]

Summary:: How to find gas flow through an orifice that links two vessels at different pressures.

Two vessels containing the same gas are linked through an orifice with diameter $d$. The pressure inside vessel 1 is $P_1$ and the pressure inside vessel 2 is $P_2$, and let's have $P_1>P_2$ (both are known values). These pressures never change despite the gas flow from vessel 1 to vessel 2 (I guess it's like assuming the vessels have infinite volumes).
Which formula should I use to find out the gas flow (for example, number of moles per second) from vessel 1 to vessel 2, as a function of the pressures $P_1$, $P_2$ and the orifice diameter $d$?

 

[BvU]

What did you find in your 'toolbox' (textbook, notes, ...) ?
Or on the net ?

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[FranzS]

What did you find in your 'toolbox' (textbook, notes, ...) ?
Or on the net ?

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I came across some "orifice equations" and "discharge coefficient", but these formulae take into account pipe diameters and/or potential energy of the fluid (gravity acceleration and height)... I honestly can't find what I'm searching for.
By the way, I'm assuming an ideally isothermal process.
I found this online calculator which provides what I'm looking for, but I do not know which formula it uses:
https://lenoxlaser.com/resources/calculators/orifice-calculator/

By the way, I see it was moved to Homework help, which actuaslly isn't.

 

[BvU]

We had a comparable thread here. Calculator there any use ?
At least is's orifice flow and it shows the equations.

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[haruspex]

Dimensional analysis reveals there is not enough info.
You are given only a pressure difference, $ML^{-1}T^{-2}$, and a linear distance, $L$.
It's unclear whether you are to find the volumetric flow or the mass flow. Since there's an irremovable M in the inputs, suppose it is mass flow. But that is $MT^{-1}$. There is no way to construct that. Combining the inputs to remove the L leaves $MT^{-2}$.

 

[FluidDroog]

Hi,

I use the LenoxLaser calculator for orifice flow approximations when I need something fast. I agree with you that their explanation is bit underwhelming [1]. From that I assume you are interested in details and accuracy.

The ISO standard for flow rate through orifices is ISO 5167-2. The equations and uncertainty analysis is there for you. As well as best practices. Note that this standard has a series for different orifice shapes.

If you need something simpler than the orifice equation. The Critical Flow Venturi formula is as simple as it gets (because it is an ideal case). For that, the relevant standard is ISO 9300.

Equations 1 and 2 of [4] is what I use for your problem. References:
[1] lenox laser orifice calculator explanation https://lenoxlaser.com/publications/fluid-flow-through-calibrated-orifices/
[2] ISO 5167-2 Orifice plates https://www.iso.org/standard/30190.html
[3]ISO 9300 CFV https://www.iso.org/standard/34272.html
[4] NIST comparison of theoretical CFV models to experiment. https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=830961