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What are the calculations to determine the level of cabin pressure at a given altitude once the cabin is punctured?

I was unable to determine the window size and the flow rate of the compression system on the recent Southwest accident aircraft, for example.

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  • $\begingroup$ Cabins are typically pressurized to approx. 8,000 feet of pressure-altitude. That is 75kPa, or 22 in Hg. Why do you want specific calculations? $\endgroup$ – abelenky Apr 18 '18 at 14:08
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    $\begingroup$ I've edited the question to show that I want to find the cabin pressure with the window out. Does it reach equilibrium or does it maintain a higher pressure. In order for that question to be answered, someone will need to calculate it, they may as well share their work. $\endgroup$ – jCisco Apr 18 '18 at 14:24
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It stabilizes at external pressure, in 40sec or less.

Air pressure at 30kft = 4.37psi, cabin pressure at 8kft = 10.92psi, from a calculator the flow rate through a 12in orifice at this differential = 475cuft/sec, volume of 737-700 cabin = 9000cuft, time to expel cabin volume holding all this constant = 19sec. Double this time for a linearly decreasing pressure differential.

Commercial aircraft exchange their cabin air 10-15 times per hour, or once in 4-6min, so the pressurization system will not be able to come anywhere close to keeping up. Without descent the cabin reaches external pressure in about 40sec. With descent it is even faster as the differential is lower, and therefore reached sooner.

There was a question about the source of the calculator flowrate. I looked at three:

http://www.efunda.com/formulae/fluids/calc_orifice_flowmeter.cfm#calc

http://www.pipeflowcalculations.net/orifice.xhtml

https://www.engineeringtoolbox.com/orifice-nozzle-venturi-d_590.html

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  • $\begingroup$ It would be good to provide references, especially for how you calculated the flow rates. $\endgroup$ – fooot Apr 20 '18 at 14:34
  • $\begingroup$ Flow rates are available from any of several calculators that deal with piping of gases. You have to find one that can deal with the structure sizes and pressures involved. At best I would consider this a rough approximation, but I think that is all that was needed. $\endgroup$ – Pilothead Apr 20 '18 at 15:25
  • $\begingroup$ Thanks for the references they were just what i was looking for. $\endgroup$ – jCisco Apr 26 '18 at 10:47
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You should read through this paper it outlines a lot of great info on calculating valve size for cabin outflow valves in pressurized aircraft it also addresses your second point (to an extent) on the flow rate of compression systems. The issue you will encounter is that the calculations will only hold for a controlled situation i.e. a situation where you have precise control over the valve (hole) size and can account for small leaks.

In the most generic terms you can find the calculation for flow rate through a hole in a pressure vessel here.

Your specific example is going to be a bit more complex since after a cabin pressure failure the plane will begin a descent rapidly to 10,000ft. so your exterior pressure will be rising as your interior pressure drops (you will need to factor that in to make the above solution a complete result). You also need to consider the airflow over the airframe its self since you are talking about a situation where the aircraft is in flight you have some fluid mechanics to play in as well.

Its called explosive decompression for a reason, its going to happen very very fast.

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  • $\begingroup$ Also, trying not to be grim, but a passenger apparently plugged the hole temporarily until other passengers could pull her back in. Unfortunately she did not survive it. I'm sure it would have been quite violent. $\endgroup$ – TomMcW Apr 18 '18 at 17:27
  • $\begingroup$ @TomMcW there was a similar situation worth reading up on here Im sure both incidents were horrific to live through. $\endgroup$ – Dave Apr 18 '18 at 17:31
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    $\begingroup$ @Dave National Airlines 27 was a very similar situation. $\endgroup$ – user71659 Apr 19 '18 at 1:03
  • $\begingroup$ The forces involved are astonishing. Air velocity at initial decompression would be nearly sonic, and the pressure if the window sealed rapidly and completely would be around 700lb. She died of blunt force trauma. $\endgroup$ – Pilothead Apr 19 '18 at 5:40
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    $\begingroup$ @Dave: At least the British Airways pilot did live through it. $\endgroup$ – Sean Jan 16 at 22:23

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