I'm actually working on a cosmology paper, but it takes me into troubled waters!
For simplicity, let's suppose we have a spherical vessel with a vacuum inside. The walls of the vessel are heated such that they have a fixed temperature, T. Radiation moves inwardly as blackbody radiation, but energy is steadily lost from inside the vessel through a nonstandard (i.e., cosmological) mechanism. The energy is removed at all points within the sphere. My general understanding is that there should then be a gradient in the temperature measured at each point within the sphere, with the highest temperature at the walls and the minimum at the centre. Is this correct? How would one quantify this? What would be the minimum temperature at the centre? Lastly, I would be very grateful if you had references that could cover this.
Thanks for any help!
Matt Edwards
You are treating the sphere s a solid (no convection flows) So, ``Conduction of heat in solids" by Carslaw & Jeager. Oxford Science publication. P. 230 examines the sphere and cone.
Preston Guynn Thanks for your input. I can see my query sure looks strange! I'm being admittedly too cryptic here, but I'm concerned that is we talked cosmology, we could quickly lose track of the problem. Current cosmology is a mess and even to discuss it invites more confusion.
So, getting back to the problem, the shell is extremely thin relative to the interior, by many orders of magnitude. The interior is not a complete vacuum actually. Yes, a vacuum does not have a temperature, as you say. Now the time frame and distance scale is enormous. And the rate of heating of the shell is equal to the rate at which heat is lost from the interior. This will sound unbelievable, but this is the the setup. Since the time frame is so large, the system is in equilibrium and a Planck curve can be established. But what is its shape?
Forgot to mention, the shell is composed of a cold plasma at about 30 K.