400 m Ball Cooling in Vacuum

[nehorlavazapal]

Hi,

If we took the blood of all people here on Earth and made a ball of it (r=200, T = 310 K) how long would it take for it to cool down to ~ 285 K in vacuum (absent of sunglight)? How long would it take to cool down in air, with reasonable conduction speeds?

 

[mikeph]

empirical measurements can be used to approximate this to something like 10 W/m^2*K in air, where the K comes from the temperature difference between the ball and the air.

So for example your initial heat loss would be (4/3)*pi*200^2*10*(310 - 285), surface area times 10 times temperature difference. As the body cools this heat loss will drop as the surface temperature (310) will not be a constant, so you're solving a differential equation here.

Conductive effects are usually a lot higher, radiation will be a lot lower.

The link between heat flux and time to steady-state temperature is entirely governed by the heat capacity of the object.

 

[nehorlavazapal]

The total heat released will be 4180 J per kg × 25 × 35 billion = 3,65 × 10^15 J.

Wow, that's a lot of heat stored in human blood! S = 502 000 m^2. So in total each m^2 would need to radiate/conduct 7,2×10^9 J. That's a lot. Can somebody offer a model in some kind of computer program? This is basic stuff, so many programs should do the trick!

 

[mfb]

Using your values, the initial energy loss is 10W/(m^2*K) * 25K * 500 000m^2 = 125MW.
Assuming the interior conducts much better than the air/bubble surface, this gives a timescale of 7.2GJ/(125MW) = 338 days.

Assuming the air around the ball won't heat too much, this leads to an exponential decay of the temperature with a timescale of roughly one year.
WolframAlpha Plot (x in days, y in K)

Why human blood?

 

[SteamKing]

Must be a vampire or represent vampire related interests. After all, it's only a fortnight to Halloween!