The Infinite Sleeping Bag

I've recently found myself pondering whether there is a limit to how hot you can make an object that gives off a constant amount of heat power by adding thicker and thicker layers of insulation around it.  This question could also take the following forms:

• Is there a limit to how cold an environment I can survive just by making my sleeping bag thicker and thicker?

• Is there a limit to how little power I can use to keep my pottery at an elevated temperature in a kiln if I make the walls thicker and thicker?

• Is there a limit to how long I can store ice in a cooler if I thicken the insulation?

I'm going to approximate the problem as a spherical shell of insulation (with non-zero thickness), fix the outside temperature, and set an incoming power distributed over the internal face.

Intuitively, as we add more insulation there’s clearly more distance through an insulating medium through which the heat must travel. However, as the insulation thickness increase with r, the surface area exposed to the outside world increases with r^2. So which wins?

Starting with Fourier’s law of thermal conduction, arranged for spherical shells, where k is thermal conductivity, dT/dr is the thermal gradient, Ar is the surface area at radius r, and Qdot is the heat power:

Rearranging and setting up the integrals:

Integrating, replacing T1-T2 with ∆T, and replacing some constants with C:

Doing some sneaky rearranging to prettify:

Finally, taking the limit as r2 approaches infinity (aka the insulation gets thick) using l’Hopital’s rule.

… and there we go! There does exist a limit, and it equals the heat power divided by 4 * π * thermal conductivity * internal radius.

This means that you cannot make yourself infinitely warm just by making your sleeping bag thicker. At some point you need to either make yourself smaller, or more conveniently decrease the thermal conductivity of the insulation. Try switching from down to a vacuum jacket.

If we say that a person’s ‘radius’ is 0.35 m, heat power is 100 W, and thermal conductivity of down is 0.025 W/m˚C, then the maximum possible ∆T is 909˚C. Turns out humans are in luck, and we could survive in an environment at absolute zero in a down sleeping bag. By my calculations the thickness would only need to be around 17 cm. Of course humans are not spheres, but this shows that the thickness needed would not be ridiculous.