[chris (OP)]

I received this question from Donald Piniach:

Thought experiment. I read it took a million years of bouncing around for photons to reach the surface of the sun and escape on average.

Ignoring all inconvient real experimental limitations, if we could keep a magical, immortal indestructible 100kg man in a vacuum bottle without any escape of heat energy, how hot would the temperature inside rise after a day, a year, and a million years? Assuming 2000 kcal a day, for numerical purposes, and no heat escaping, no solid food intake, just energy.  I assume the vacuum bottle might be a little heavier from the added energy, how much?

What do you think?

[evan_au]

I assume that once the temperature exceeded about 40C with 100% humidity, our 100kg man would be 100% dead.

His microbiome might turn him into a compost heap that gets a little hotter, so I am guessing that the limit might be around 45C.

Assuming the bottle is not much larger than the man, and doesn't have a supply of liquid oxygen as coolant and supply of breathable air, I expect that this temperature would be achieved well within 1 day.

Of course, an immortal and indestructible man might live a little longer, even without food or oxygen...

[Bored chemist]

Since an immortal and indestructible man wouldn't obey the laws of thermodynamics, what could we use to calculate an answer?
The question's meaningless.

[Kryptid]

I'll give this a shot, since this is exactly the kind of eccentric physics question that I would find myself thinking about.

2,000 kilocalories a day is 8,368,000 joules a day, 348,667 joules per hour, 5,811 joules per minute or 97 joules per second. 97 joules per second is, of course, the same as 97 watts. Since I've read (a long time ago) that mammals require about 10 times as much energy as reptiles of similar size thanks to the waste heat produced by their metabolism, I'll assume that 90% of that energy is turned into waste heat. This results in our human releasing about 87 watts of heat energy (given that I've seen other estimates between 80 and 100 watts for human waste heat, this seems to be a reasonable estimate).

So now you can easily calculate how many joules this "man-in-a-bottle" would accumulate over different time spans. After a day, it is 7,531,200 joules, after a year, it is 2,750,770,800 joules, and after a million years, it is 2,750,770,800,000,000 joules.

Converting that into a temperature measure is trickier, since human beings have a complex composition. However, I'll simplify it by assuming that our human has a heat capacity equal to their own mass in water (humans are mostly water anyway). Although heat capacity changes with temperature, I think we can still get some meaningful results here. Thankfully, I saved a calculation that I did a while ago where I found that the average heat capacity of water over its entire liquid range is 4.19555 joules per gram times Kelvins.

Our "man-in-a-bottle" has a starting body temperature of 310.15 Kelvins and a mass of 100,000 grams. It therefore takes about 419,555 joules to heat him up by 1 Kelvin. On day one, 7,531,200 joules of heat energy are added, resulting in an increase in temperature of 17.95 Kelvins (a total temperature of 328.1 Kelvins, 54.95 degrees Celsius or 130.91 degrees Fahrenheit). If this was a normal person, they would be quite dead on day 1. However, our super-man can keep going, so now on to a full year...

Before we can add a year's worth of energy, we have a new problem to face: we need to find out how much energy it will take to get our man to the boiling point. Why? Because the heat capacity of water vapor is different than that of liquid water. Also, energy will be consumed in the very act of boiling water (heat of vaporization). Without taking these into consideration, we can't expect our estimate to be accurate. Take note that this assumes that our man has enough room inside the bottle to vaporize into steam. If not, the water he is composed of will remain liquid for longer (due to the pressure increase) and probably throw off the calculations.

The boiling point of water is 373.15 Kelvins, so that's 63 Kelvins that would have to be added to our human from an unheated start. This results in a requirement of 26,431,965 joules, which will be subtracted from the total energy input over a year's time (2,750,770,800 joules - 26,431,965 joules) to give 2,724,338,835 joules left over after reaching the boiling point. The heat of vaporization of water is 2,256 joules per gram, so now we need 225,600,000 joules to take the water from liquid to gas. The remainder, 2,498,738,835 joules, will go into heating up the resulting water vapor.

The heat capacity of water vapor varies quite a bit depending upon the temperature. Right at water's boiling point, it's about 1,890 joules per gram times Kelvins, or 189,000,000 joules to heat up our steam man by one Kelvin. This would increase his temperature by 13.22 Kelvins for a final temperature at the end of one year of 386.37 Kelvins, 113.22 degrees Celsius or 235.8 degrees Fahrenheit. Since this is not significantly above water's boiling point, the given heat capacity of water vapor is probably accurate enough for the sake of this calculation.

A million years is going to be a doozy and I probably cannot calculate it accurately. I can, however, place somewhat of an upper limit on it (since heat capacity tends to increase for water vapor as its temperature goes up). If we assume (wrongly) that the water vapor's heat capacity remains at 1,890 joules per gram times Kelvins, we can calculate the resulting temperature after adding 2,750,770,547,968,035 joules (the energy remaining after boiling the water at its boiling point). The result is about 14.55 million Kelvins after a million years. This is almost certainly wrong. The thermal behavior of such a high temperature plasma would no doubt be very, very different from that of water vapor. I suspect our immortal man would be nowhere near that hot in actuality. I would need some kind of heat capacity for plasma at high temperatures to get a good ballpark estimate.

[yor_on]

Heh :)
Loved this one Kryptid
=

btw, I'm waiting for the book on it

[PmbPhy]

I received this question from Donald Piniach:

Thought experiment. I read it took a million years of bouncing around for photons to reach the surface of the sun and escape on average.

Ignoring all inconvient real experimental limitations, if we could keep a magical, immortal indestructible 100kg man in a vacuum bottle without any escape of heat energy, how hot would the temperature inside rise after a day, a year, and a million years? Assuming 2000 kcal a day, for numerical purposes, and no heat escaping, no solid food intake, just energy.  I assume the vacuum bottle might be a little heavier from the added energy, how much?

What do you think?

Since the problem is stated as a man inside a sealed, heat-proof bottle then there is flow of energy into or out of the bottle. This means that the temperature of the man would remain constant.

However it can be much more complicated if the statement of the problem was much more complete such as stating the initial conditions such as the atmosphere inside the bottle. If there is a vacuum and one must take into account the water evaporating off the man and filling the bottle with an atmosphere of water vapor. Then one must take into account the water vapor filling the bottle and coming to a constant pressure and then take into account the heat flow out of the man into the bottles internal atmosphere. It'd the be a much more complicated problem to solve since you'd have to solve the corresponding differential equation relating to the heat flow. Something that I'm not interested in doing myself.