Theoretical pressure decay of a pressurised vessel

[DSOTM]

I am looking to derive a method of plotting the theoretical pressure decay of a pressurised vessel. I would like to end up with a graph that plots internal vessel pressure against time.

Is this possible?

What assumptions would I need to make?

The following inputs will be known.

• Gas: air
• Vessel volume is constant
• Starting internal vessel pressure
• Internal vessel temperature remains constant
• External temperature remains constant
• External pressure remains constant
• Size of the leak aperture
• Test duration

The internal pressure difference will be small, in the region of a few hundred Pascals.

 

[Andrew Mason]

Too many variables and you may not have enough information. If the external temperature and pressure are changing you need to know the rate of change over time (unless we are to assume it is linear). But the real problem is that the interest level in solving it is too low for the amount of work involved in finding a solution.

 

[DSOTM]

Thanks. We would assume internal temperature, external temperature and external pressure to remain constant. That's not clear in my post so will update.

 

[Andrew Mason]

Welcome to PF, by the way!

 

[anorlunda]

You can make a transient simulation from the differential equations. Include

mass balance
energy balance
volume balance
perfect gas law
orifice flow equations
include all the constraints and assumptions you mentioned.

I don't see an assumption about moisture in your list.

 

[DSOTM]

Yikes, i wouldn't even know where to start with that.

What do you need to know about moisture?

We're looking for a somewhat simplistic approximation, if that's even possible.

 

[Vanadium 50]

What do you need to know about moisture?

When your gas leaks out, it cools. Moisture condenses around the aperture, shrinking it. This decreases the volume of gas leaving, but may increase its speed, which would further reduce the temperature.

In extreme cases, you can actually freeze the hole.

 

[Baluncore]

What is the type of gas and the average molecular weight of the gas ?
What is the viscosity of the gas ?

The first approximation will be an exponential decay.
The second approximation will be a couple of exponential decays due to switching between turbulent and laminar flow.

 

[Dullard]

This is a handy reference for the occasional plumber:

https://files.valinonline.com/userfiles/documents/instrvalvetechguide.pdf

See 'Gas Flow Calculations'- approx middle the document. You should be able to find published data giving you Cv values for different orifice sizes - you can get a reasonably good estimate of the flow with that and the equations in the document. Once you have that, it's pretty simple to use excel to calculate the 'new' (reduced) tank pressure and orifice flow at the interval of your choice.

This isn't perfect - you may need to 'adjust' your Cv based on experimental results. There are more 'precise' ways to do this, but I'm not sure that they're much more accurate - assumptions are required no matter what approach you use.

 

[FranzS]

I had quite good results with this semi-empirical formula developed by Lenox Laser (they also have a calculator on their website):
https://b6x8x4c7.rocketcdn.me/wp-content/uploads/2015/10/Fluid-flow-through-calibrated-orifices.pdf

 

[DSOTM]

When your gas leaks out, it cools. Moisture condenses around the aperture, shrinking it. This decreases the volume of gas leaving, but may increase its speed, which would further reduce the temperature.

In extreme cases, you can actually freeze the hole.

Ok with you now. So for a 50L vessel pressurised to 1000Pa we're looking at a leak of 0.5L/hr. So the flow rate is incredibly small. I would assume that the effects of moisture condensation are negligible.

 

[DSOTM]

What is the type of gas and the average molecular weight of the gas ?
What is the viscosity of the gas ?

The first approximation will be an exponential decay.
The second approximation will be a couple of exponential decays due to switching between turbulent and laminar flow.

Gas is air. Temperature 15 deg C. Google tells me that corresponds to 1.81x10-5 kg/(m.s).
Molecular weight of 28.97gram/mol.

The flow rate is extremely small, i would assume the flow is not turbulent.