- #1

FranzDiCoccio

- 342

- 41

## Homework Statement

- A sphere of radius [itex]r_s[/itex] is at the center of a spherical shell of inner radius [itex]r_i=10\, r_s[/itex] and thickness [itex]s = 10\, {\rm cm}\ll r_i[/itex].

- The sphere has a temperature [itex]T_s=1073\, {\rm K}[/itex] and and an emissivity [itex]e=0.90[/itex].
- The inner surface of the shell has a temperature [itex]T_i = 873 \,{\rm K}[/itex] and it is made of asbestos . The heat transfer coefficient of asbestos is [itex]k_a = 0.090 J/(s K m)[/itex].
- The text says that the temperatures of the sphere and of the inner surface of the shell are "kept constant".

- The problem asks to find the temperature [itex]T_e[/itex] of the outer radius of the spherical shell.

## Homework Equations

- Stefan-Boltzmann law (radiated heat power)

- Heat transfer through a rod (should be ok for the spherical shell on account of its thinness)

## The Attempt at a Solution

Of course the problem is solved if one can estimate P in the second equation.

I guess it could be a matter of balance of heat radiated by the sphere and (possibly) by the inner surface of the shell. The sphere radiates heat, and, if the shell is not totally reflecting, it is going to absorb part of it.

If that is the case, it should also radiate some heat back.

I'm not sure of how to take into account that "the temperatures of the sphere and of the inner surface of the shell are kept constant".

The power absorbed by the inner shell could be

[tex]P_i = \sigma e_i A_i (T_s^4-T_i^4)[/tex]

But I'm not sure about at least a couple of things

- the S-B law applies to a concave surface. Perhaps it works because the shell radius is "large". Anyway, I do not know [itex]e_i[/itex]. The problem does not give it.

- it seems to me that the shell absorbs more energy than the sphere radiates (because of the larger surface).

[tex]P_i = \sigma e_s A_s (T_s^4-T_i^4)[/tex]

and all this power for some reason is trasmitted through the shell.

This does not convince me either.

I think that this problem should be solved using the two equations I wrote (these are the only equations about heat transfer that could be found in the book where I found the problem).