Question

A small sphere (emissivity = .9, radius = r1) is located at the center of a spherical asbestos shell (thickness = 1cm, outer radius = r2). The thickness of the shell is small compared to the inner and outer radii of the shell. The temperature of the small sphere is 800 C, while the temperature of the inner surface of the shell is 600 C, both temperatures remaining constant. Assuming that r2/r1 = 10 and ignoring any air inside the shell, find the temperature of the outer surface of the shell.

Answer 1

Temperatures:
T1: inner sphere
T2: inner surface of spherical shell
T3: outer surface of spherical shell

Radii (RE-DEFINED as per my convenience):
r1: inner sphere
r2: inner surface of spherical shell
r3: outer surface of spherical shell

Areas:
A1: associated with r1. A1 = 4*Pi*r1^2
A2: associated with r2. A2 = 4*Pi*r2^2

Because this is a series circuit, the Q_dot is the same for all resistances.

Equation for radiation heat transfer (assuming inner surface of shell behaves as a black body):
Q_dot = epsilon*sigma*A1*(T1^4 - T2^4)

Equation for conduction heat transfer in terms of thermal resistance R:
Q_dot = (T2 - T3)/R

Thermal resistance of a spherical shell:
R = (r3 - r2)/(4*Pi*k*r2*r3)

Thus:
epsilon*sigma*A1*(T1^4 - T2^4) = (T2 - T3)*(4*Pi*k*r2*r3)/(r3 - r2)

Substitute A1:
epsilon*sigma*4*Pi*r1^2*(T1^4 - T2^4) = (T2 - T3)*(4*Pi*k*r2*r3)/(r3 - r2)

Solve for T3:
T3 = T2 - epsilon*sigma*r1^2*(T1^4-T2^4)*(r3-r2)/(...

In terms of the radius ratio I choose to name j for no good reason, and thickness of spherical shell t,
r3 = r1*j
r2 = r3 - t
r2 = r1*j - t


T3 = T2 - epsilon*sigma*r1^2*(T1^4-T2^4)*t/(k*(r1*... - t)*r1*j)

And it simplifies:
T3 = T2 - epsilon*sigma*r1*(T1^4-T2^4)*t/(k*j*(r1*... - t))

Assuming that r1*j is MUCH larger than t, our equation simplifies:
T3 = T2 - epsilon*sigma*(T1^4 - T2^4)*t/(k*j^2)

And the result doesn't depend on r1.

Data: (MUST use Kelvin for temperatures)
T1:=1073.15 K; T2:=873.15 K; epsilon:=0.9; sigma:=5.67e-8 W/m^2-K^4; k:=0.09 W/m-K; j:=10; t:=0.01 m;

Result:
T3 = 830.9 Kelvin

Which translates back to:
T3 = 557.8 Kelvin