Let's say a person's hair was on fire and suddenly we (instantaneously) pumped all the air out of the (sealed) room, then after time period X we put all the air back. Is there an X such that we will put out the fire (and it won't start again), but we won't kill the person? What's the approximate value of X?
[Physics] If your hair was on fire and you entered a vacuum, could you outlast the fire
thermodynamics
Related Solutions
As everyone else is saying, if you assume Newton's law of cooling:
$$ \dot Q = m c_p \dot T = h A \Delta T $$
The equation for how you heat or cool is an exponential
$$ T(t) = T_\infty + \Delta T e^{ -\frac{hA}{mc_p} t } $$
The rate constant for growth (or dying) of temperature is the same (assuming other details of the material don't change much), so the half-life is the same regardless of the differences in temperatures or goal temperatures, but this does not mean you have to wait the same amount of time. As is the nature of half lives.
I think the best explanation would be a visual one:
Here I'm showing how the temperature would evolve with some made up, yet practical chosen values. If you like, ignore the actual numbers on the scales as they are not that important. What's important is you'll notice that for the red curve I "took it out of the fridge" for 1 hour, and then put it back and it isn't until another 6 hours later that is 41 degrees again. But, for the purple curve, which was out of the fridge long enough to nearly come to room temperature (8 hrs), it takes just as much time to cool down as it took to heat up.
In real life situations like this there tend to be lots of variables, so it's dangerous to make predictions based on (over?) simplified physical models.
Having said this, the article makes a good argument. Newton's law of cooling tells us that the rate of heat loss per unit area from an object is roughly proportional to the temperature difference between the object and it's surroundings. For a large and a small pan, both containing boiling water and both in the same kitchen, the larger pan has the larger surface area and it will lose more heat per second than the small pan.
You ned to distinguish carefully between heat loss and temperature fall. The temperature of the big pan will fall more slowly than the temperature of the small pan because its surface area to volume ratio is smaller. However it will lose heat faster.
The point the article is making is that the burners on your stove supply heat at a constant rate that is independant of the pan size. Because the larger pan loses more heat per second the net heat flow from the burner is lower, and it will take longer to get it back to the boil.
Of course this assumes that the efficiency of heat transfer from the burners to the pan is independant of pan size, but then as I mentioned at the outset in real life there are lots of such variables.
Best Answer
@AMCDawes gives a well reasoned explaination of how the physics would play out.
However in the scenario depicted by Dawes, he leaves out the part whereby we reintroduce the poor hapless subject to air (I assume 20% oxygen @ STP).
If we duty cycle this quickly enough the hair would definitely reignite [citation needed]*.
The issue is that flammable substances have a temperature called Autoignition temperature. At this temperature, as long as there is oxygen, the substance will spontaneously combust.
So if the hair was above the autoignition temperature, then by reintroducing the subject to air, his hair will reignite.
So to determine how long is needed, we would need to calculate how long it would take to significantly cool the hair below the autoignition temperature.
Hair is long and thin, therefore it can conduct heat away from itself relatively easily. Assuming a vacuum it would need to radiate the heat away. This process is going to take a LONG time, so no....the subject is likely to expire before that happens.
HOWEVER. Final point. During the initial decompression event, the air around the hair must be moved, to be REMOVED. Assuming a explosive decompression event, winds should be generated ~300m/s.
Citing birthday celebrations as empirical evidence and that 7 year olds do not have transonic breath, the wind generated by the decompression event should be enough to extinguish said flames by thermal conduction (and not by removal of the oxidiser).
*Please do not attempt this on live humans, Nazi Germany once did this please use that data instead.