I recently played "Crysis", a game where the protagonist wears a suit that allows the player to hide both himself and his heat signature. Then I watched Iron Man 3, where a kid suggests that Tony Stark should have implemented retro reflection panels in his suit.
So I'm thinking, well, as is the nature of things, people are going to be pursuing this sort of thing in real life too, pretty soon. But I'm trying to figure out whether a suit can contain a person's heat signature without emitting the heat somewhere. Is such a thing fundamentally possible to do, without over-heating the person within?
Answer
There are some nice ideas in other answers, but they are overseeing some basics. Let's do some thermodynamics. The efficiency of a thermal engine is bounded by the Carnot efficiency:
$$\eta \le 1 - \frac{T_c}{T_h} $$
Where $T_c$ is the temperature of the cold end and $T_h$ the heat source. Assuming we are in a cool environment, $T_c=0 C$, $T_h=37 C$, so:
$$\eta_C \simeq 12\%$$
Whatever you do to convert your body heat into something useful, you will still have to pump out 88% of what you produce. And this is a very optimistic estimation, given that you have to make it portable. Way to go.
Let's look at the heat sink option. A human body has a surface of roughly $2 \mathrm{m}^2$, so it radiates about $10^3 \mathrm{W}$, but we are getting in around $600 \mathrm{W}$ from our surroundings (again, assuming a temperature of 0 C). If we leave base with a heatsink of iron cooled with liquid nitrogen (76 K) we have:
$$\frac{c_e \Delta T}{P}=\frac{449\frac{J}{K kg} 234 K}{400 W} = 262 \mathrm{\frac{s}{kg}} \simeq 4 \mathrm{\frac{min}{kg}}$$
So if we manage to redirect all our excess heat to the sink we will finish it in a few minutes.* A full hour of invisibility would require loading an extra 15 kg of weight. Other metals have a better specific heat ratio, but expect no wonders here.
How to actually become IR invisible
In order to become truly invisible, what we need is to dissipate all our heat via conduction. The way I would do it (but I am not an expert), is a multi layered system, where the inner layers are designed to give away heat, while the outer layers have to allow for ventilation while shielding the IR emited by the dissipation layer. The peak emission for a human body is in the range of $ 9 \mu m$, so a metallic mesh of that width should be a pretty good shield.
Now, you "just" have to figure out how to build a cloth with that metal embedded, while being able to endure hard conditions. But that is mere Engineering, not Physics.
In this scenario, all our heat is being carried away by the air around us, but that will not create a strong signal, as it is not dense. Also, turbulence will quickly mix it with the surrounding air, cooling it instantly.
*- The specific heat varies with temperature, but as a ballpark it works.
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