Bottles on a plane

[Editorial: The title just doesn't have the ring to it that Snakes on a Plane does... Maybe if I wrote it as m*therf*cking bottles on a m*otherf*cking plane...]

14 months ago, I went to Poland with friends of mine for a birthday. On the way over, I decided to do something nerdy...

I wanted to measure the pressure difference between the cabin at flight altitude and ground level.^1,2 On the way back, I bought two bottles of water; one I drank on the plane and closed it. The other one, I kept as a reference. As the plane descended, the air pressure gradually increased (my ears popped, yo!)... and the closed bottle got squashed. So... if I have the volume of something at a known pressure and the volume at another one, I can estimate the pressure at the second one... Thank you, Ideal Gas Law :)

Choosing the method
So I have my bottle of air and I have the other bottle... how to get my answer?

1. Fill the reference bottle (weigh it); Squeeze it out until it looks similar to crumpled bottle (weigh it again).
2. Submerge the bottles in water and compare the two volumes.
3. Put the bottle in a vacuum oven³; adjust the pressure in the oven until the bottle is back to its original shape. (One of the guys suggested that) 

I decided to go with 2 because it's better than eyeballing bottles to make them look the same and 3 was a bit of a beroepsmisvorming.⁴ What is nice about the vacuum oven, is that it's a direct measurement of the pressure. I'm sure there are other options - I'd like to hear them.

Procedure

I filled the reference bottle with water so that it would sink, although I was still stuck holding the crumpled bottle down. Also, for the crumpled bottle, I held it under the cold tap for a couple of minutes to chill the air, plus I took the temperature of the water. Then, if needs be, I can adjust the volume for temperature effects as well.⁵

I marked the water levels off on the lunch box (with - c - and without - a - the reference bottle and the crumpled reference bottle - b). Then I weighed how much water I had to add to go from a to b and to c:

a-b = 472g

a-c = 547g

Weight and volume are interchangeable here, so the ratio between the crumpled and reference bottles is the volume ratio, which in turn should tell me the pressure of the cabin: 0.86 atm. The height equivalent for my calculated pressure is: ~1250m.

For switching between altitudes and pressure, I used this handy correlation that I found on engineeringtoolbox.com for air pressure as a function of altitude:

p = (1 - 2.25577 10^-5 h)^5.25588

Where p is pressure in atmospheres (101,325 Pa) and h is height in m.

Sanity check

According to wikipedia,⁶ cabin pressures vary between an equivalent of sea level (0 m) and 2.4 km, which is a range of 1 atm to 0.75 atm. It also mentions that the actual cabin pressure depends on the type of plane; I flew with Wizzair and according to their Wikipedia page, they use Airbus A320. A quick google turned up this document that cites design specifications for the A320 as:

The cabin pressurisation system of the Airbus A320 is designed to maintain a maximum cabin altitude of 8,000ft (approximately 11.03psi) when the aircraft is cruising at high altitude by controlling the opening and closing of an outflow valve. Further, it is designed to limit the differential pressure between the cabin pressure and the ambient pressure to a maximum of 8.06psi. 

[Note: 8,000 feet = 2.4 km, 11.03 psi = .75 atm and a maximum differential of 0.55 atm. But no mention of what the pressure usually is...]

The cruising altitude for the A320 is about 11.7 km according to the first result on Google, which means that the calculated pressure differential is: 0.66 atm. This is within the safety criteria for the Airbus (it is about 88% of the max. pressure differential cited above).

Coming from a position of ignorance, that 12% lee-way makes me feel somewhat squeemish, but there are valve systems in place to balance out that the differential when it gets dangerous (it's mentioned in the above link from which I quoted). Thus in terms of safety, my numbers check out.

However harkening back to that wiki page on cabin pressurisation, this altitude is rather low for most airplanes - which doesn't assuage my doubts.

Errors

• My gut tells me that the biggest source for error is the plastic bottle; It's likely that the rigidity of the bottle means the deformation is less than it should be - hence the pressure is overestimated.
• I think my next biggest source of error is: using a wide lunch box instead of something narrow where changes in height are clearer.

Other its-bitsy sources of possible errors (in order of ascending unlikelihood/insignificance/scraping-the-barrel):

• My fat fingers holding the bottle under the surface
• The temperature difference between the water and room temperature is less than 20°C - on the Kelvin scale that's up to a 7% reduction, which corresponds to a 7% error in the pressure. Plus, it's an error in the other direction - if the the air contracts in the bottle the estimated pressure difference climbs.
• Diffusion of air back into the bottle from outside - since the difference is max. 0.25 atm, I don't see it have a big impact... I found a paper that gives the diffusivity values, but there was a server problem. :(

Next experiment:
I think the simplest thing to do is bring a balloon with me the next time:

• Inflate it in a bottle and fill the bottle
• Tie it off
• Fill the gap between the bottle and the balloon when I get down on the ground

*******

Reference bottle

Bottle in lunchbox

Difference in marks is the volume of the reference bottle

Holding the crumpled bottle under

Water temperature

This time in focus...

******

1. I like being able to measure and calculate things - I get quite a kick out of it.
2. I couldn't calculate the absolute pressure difference - not independently at least
3. We've got them at work.
4. I think it means to use the expertise/equipment/techniques of one's profession unnecessarily or excessively. There isn't a single word or idiomatic translation that I can think of that catches this well in English. The direct translation (professional deformation) doesn't sound quite right.
5. To be honest this is overkill This method has bigger issues.
6. I don't like the fact that the wikipedia page has the metric units in parentheses after the Imperial units.