How Long to Suffocate in Space

I previously looked at how long it would take to freeze in space.  In this post, I'm going to look at the other side of the losing life support coin: how long would it take to run out of air in space?

Since the heat being lost is dependent on the temperature, which is changing, we needed a differential equation before (well actually since it's all been worked out before we didn't, but where's the fun in that?).  This time, oxygen use is constant, so the math is much simpler.  On the other hand, there are still plenty of variables to introduce uncertainty.

To begin, if you are sealed in a room with a normal mix of air you will not run out of oxygen.  Rather CO₂ will build up to toxic levels, and you will die.  It is possible to scrub CO₂ from the air.  In fact, we currently do it on our spaceships.  A major issue Apollo 13 faced was getting circular CO₂ filters to work with a square hole.

What this means for us is we need to look at time for CO₂ to build up to lethal levels, and for how long it would take to run out of O₂ if the CO₂ is being removed.

We also need to consider that people consume O₂ at different rates, and even at different rates at different times.  VO₂ is a measure of what rate a person is consuming O₂.  Unfortunately, it is almost always used in the context of measuring peak VO₂ during exercise (as a measure of fitness).  It was hard to get good numbers for a resting person, but I settled on about 0.018 cubic meters per hour.  For the active rate, I still couldn't just use typical VO₂ numbers because they are for the max consumption rate during bursts of exercise.  A person moving around attempting to repair a ship wouldn't be using as much air as a person sprinting.  I found some good numbers from scuba diving forums and settled on 0.1 cubic meters per hour.

On the subject of the variation of people, different people will be able to tolerate different concentrations of CO₂ or levels of O₂.  It was hard finding a good number for lethal CO₂ concentration.  Most sites were concerned with long term exposure at a work environment (years), or short term accidental exposure (minutes).  I settled on 5% which is probably a bit low.

As for minimum O₂ concentration, similar problems apply.  Here I settled on 11%, compared to about 21% normally.  Ships could use slightly higher O₂ concentrations to begin with to help with loss of life support situations, but high O₂ levels have their own problems.

Additional issues are things like fires or venting atmosphere reducing the time.  Also, the number of people on a ship is harder to estimate.  On larger ships it is probably more constant.  But on small shuttles it could vary quite a bit.

Working in our favor is the fact that the respiration equation:
`"C"_6"H"_12"O"_6 + 6"O"_2 to 6"CO"_2 + 6"H"_2"O"`
Has a one to one mole ratio between O₂ and CO₂.  Additonaly, a mole of any gas takes up about 24 liters at normal temperature and pressure.  This means we can use the same formula for both O₂ consumption and CO₂ build up:
`t={V cdot Delta r}/{n cdot R}` Where: V is volume, `Delta r` is the change in the ratio of the gas, n is the number of people, and R is the rate that gas is changed.  As an example:
`t = 36.1 " hours" = {26 "m"^3 cdot 0.05}/{2 cdot 0.018 "m"^3/"hour"}`
This is the formula for a 26 cubic meter shuttle craft, with 2 people.  The change in concentration is 0.05 because CO₂ is effectively 0 normally.  0.018 cubic meters/hour is the resting CO₂ production rate.

I decided to give a range with worst case and best case scenarios.  The best case is resting, and not worrying about CO₂ (because it's being scrubbed).  The worst case is CO₂ build up will working to fix the ship.

As you can see there were some cases where heat loss was faster than the best case scenario.  I didn't expect it to even be close.

Name	Volume (`"m"^3`)	Crew	Heat (days)	Oxygen, Resting (days)	CO2, Doing Work (days)
Death Star II	2,144,000,000,000,000	2,500,000	437,837	218,370,370	17,866,667
Super Star Destroyer	12,645,900,000	300,000	967	10,733	878
Borg Cube	28,000,000,000	130,000	25,585	54,843	4,487
Enterprise-D	5,820,983	1,200	456	1,235	101
Enterprise	211,248	430	161.0	125.1	10.2
Runabout	569	3	11.3	48.3	4.0
Type 6 Shuttlecraft	26	2	12.8	3.3	0.3
TIE Fighter	8	1	0.7	2.0	0.2