How Do You Model Radiative Heat Transfer Between Spheres in Thermofluids?

[Xaspire88]

Have this HW for a Thermofluids class and I'm a little confused.

Problem
A) Use small Biot number approximation to derive the energy equation for a small solid sphere of radius "a" which exchanges radiative energy with a large sphere of radius "b".

B) If B -> infinity, show the energy equation and find the temperature of the small sphere.

We are given:

Small Sphere (1)
Initial Temperature = T1
Emissivity = e

Large Sphere(2)
Black Body
Temperature = T2

I solved for the shape factors first since they were relatively easy to determine.
F12= 1
F21+F22=1
A1*F12=A2*F21
F21=$$\frac{A1}{A2}$$
F22=1-$$\frac{A1}{A2}$$

Next I tried solving for the Q1->2

Q=(Eb1-Eb2)/((1-e)/(A1*e)+1/(A1*F12))

But I'm not sure if this is correct. I don't think I have ever done a radiation problem between a Black and Gray body.

For the Energy Equation for part (A) Would I need to subtract Q1->2 from Q2->1? They should be equal and opposite I thought since there are only two surfaces exchanging energy, but maybe that's where I am confusing myself.

And for Part (B) I am really at a loss. It would seem as the radius of the large sphere goes to infinity that the heat exchange Q2->1 (heat exchange from the large sphere to the small one) would go to zero, but beyond that I have no idea how to solve for the equilibrium temperature of the small sphere using the initial temp.

Any help is appreciated. Thank You in advance.

 

[Nino MMP]

For part A, you are correct that Q1->2 should be subtracted from Q2->1. You can use the equations for Q1->2 and Q2->1 to derive the energy equation. The equation will be of the form:Q1->2 - Q2->1 = (T1 - T2)/tauwhere tau is the thermal time constant. The equation can then be rearranged to solve for the equilibrium temperature of the small sphere.For part B, as the radius of the large sphere goes to infinity, the heat exchange from the large sphere to the small one (Q2->1) will go to zero. This means that the energy equation reduces to Q1->2 = (T1 - T2)/tauSolving this equation for the equilibrium temperature of the small sphere gives the result T2 = T1. This means that the equilibrium temperature of the small sphere is equal to its initial temperature.