Bryan needs no introduction on this blog, but if we were to introduce him it would be as the fearless champion of Gerlich and Tscheuschner.
Bryan has been trying to teach me some basics on heat transfer from the Ladybird Book of Thermodynamics. In hilarious fashion we both already agree on that particular point.
So now here is a problem for Bryan to solve.
Of course, in Game of Thrones fashion, Bryan can nominate his own champion to solve the problem.
Spherical body, A, of radius ra, with an emissivity, εa =1. The sphere is in the vacuum of space.
It is internally heated by a mystery power source (let’s say nuclear, but it doesn’t matter), with power input = P.
The sphere radiates into deep space, let’s say the temperature of deep space = 0K to make the maths simpler.
1. What is the equation for the equilibrium surface temperature of the sphere, Ta?
The condition of case A, but now body A is surrounded by a slightly larger spherical shell, B, which of course is itself now surrounded by deep space at 0K.
B has a radius rb, with an emissivity, εb =1. This shell is highly conductive and very thin.
2a. What is the equation for the new equilibrium surface temperature, Ta’?
2b. What is the equation for the equilibrium temperature, Tb, of shell B?
The reason for the “slightly larger shell” is to avoid “complex” view factor issues. Of course, I’m happy to relax the requirement for “slightly larger” and let Bryan provide the more general answer.
The reason for the “highly conductive” and “thin” outer shell, B, is to avoid any temperature difference between the inside and the outside surfaces of the shell. That is, we can assume the outside surface is at the same temperature as the inside surface – both at temperature, Tb.
For anyone who wants to visualize some numbers: ra=1m, P=1000W, rb=1.01m
This problem takes a couple of minutes to solve on a piece of paper. I suspect we will wait a decade for Bryan’s answer. But I love to be proved wrong!