Air flow through a hole in a submerged pressure tank

[Gustav Molin]

Hello! I will try to formulate myself as clearly as possible, but english isn't my main language. I'm having trouble solving this problem that should be relatively easy to solve.

Let's say we have a submerged pressure tank, constantly fed with air through a compressor, so it's maintained at a certain pressure P_tank. The pressure tank has a hole in it at a known height from the surface, and with a known diameter, so that air will leak out. I want to calculate the volume air flow through the hole [m³/s].

Can I use the formula

to conclude if the flow is choked by using
P* = P_atm + rho*g*h
P₀ = the pressure in the tube (between 1 and 4 bar or something)
gamma = 1,4 for air

If the flow is proven to be choked, does this mean the air speed is 340 m/s through the hole? Can I then just multiply the speed with the area [m²] of the hole to get the volume air flow [m³/s]?I have tried using this formula but I can't seem to figure out C_d.

Thanks in advance!

 

[Chestermiller]

Is this a homework problem?

 

[Gustav Molin]

No it's not a homework problem. I'm building a test setup for a company, that consists of a tank with multiple holes and want to ensure that the mass flow through every hole is equal.

 

[Edward Camic]

I believe in this case the discharge coefficient has to be determined via testing. There are published values for this coefficient based on the orifice type and the reference I typically use is table 8-2 in the latest edition of Sutton's Rocket Propulsion Elements. It highlights several liquid orifice types, diameters and the respective discharge coefficient for that kind of orifice.

Here personally, and I may be wrong so if someone thinks I am please let me know - I would love to hear you out, I would start with the volumetric flow rate Q = Cd * A * sqrt(2*ΔP/ρ). If you are feeding the tube with a compressor, the flow rate is that of the compressor output to the tube in total. If each orifice is the same, you can estimate the individual flow rate as the total flow of the compressor / the number of (equal size) outlets.

 

[Gustav Molin]

Thank you very much! I realized that all holes will have the same Cd so if I just guess what the Cd is, I can compare the different holes and calculate the estimated flow rate from there. You say that I can estimate the flow rate by dividing the total flow rate with the number of outlets. Is that true? I am thinking if the holes are placed at different depth, ΔP will differ from hole to hole and therefore the flowrate will differ. Is this correct?

Gustav

 

[Edward Camic]

Well yes there will be a hydrostatic pressure increase but its not unreasonable to start with assuming a consistent flow rate across all. Is the change in height so significant that the pressure difference will be considerable?

 

[Gustav Molin]

The change in depth will only be maximum 2 meters so maybe the pressure difference isn't significant. I'll try the test setup in a few days, then we'll see if the flow rate really differs at the different depths.

Thank you for being helpful! I really appreciate it

Gustav

 

[Gustav Molin]

I noticed the equation you provided is only applicable for incompressible fluids. This leads me to believe I have to find another way of solving this. Any ideas?

 

[Edward Camic]

Do you have reason to believe your fluid will exceed Mach 0.3?

 

[Gustav Molin]

I found a way of calculating the air flow rate. Thanks for the help.

 

[ahiddenvariable]

No it's not a homework problem. I'm building a test setup for a company, that consists of a tank with multiple holes and want to ensure that the mass flow through every hole is equal.

Are the holes below the liquid surface and you want the same mass flow of liquid through each hole? What is the hole above the liquid surface for? Please try to state exactly what you want to do, without expecting us to guess.

 

[Gustav Molin]

Are the holes below the liquid surface and you want the same mass flow of liquid through each hole? What is the hole above the liquid surface for? Please try to state exactly what you want to do, without expecting us to guess.

Clearly, the holes are below the liquid surface and I want the same air mass flow. Air is not a liquid. There is no hole above the liquid surface. Where did you get that from? Please read instead of guessing. Lastly, I already solved the problem, which I stated.

 

[nicknicknick]

Hi Gustav,
Would be grateful to know your solution please, I am looking at a similar problem.
Regards
Nick

 

[Gustav Molin]

Hi Nick!

Check out section 6-22, at the title "Compressible flow".
I used equations 6-118 and 6-122
https://chembugs.files.wordpress.com/2015/12/perrys-chemical-engineering-handbook1.pdf