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I have a formula for a sphere radiating to the environment where the environment is not at zero temperature. The sphere has a constant heat generated within itself. The formula for the temperature of the sphere under these conditions is:

##T_s = \left( \frac{Q_{gen}}{A_s \sigma \epsilon} + {T_{amb}^4} \right)^{\frac{1}{4}}##

Where ##Q_{gen}## is in Watts and the subscript

My starting point are the 4 radiosity equation in 4 unknowns given by Nellis & Klein, i.e.,

##Q_{gen1} = \frac{\epsilon_1 A_1(E_{b1} - J_1)}{1-\epsilon_1}##

##Q_{gen2} = \frac{\epsilon_2 A_2(E_{b2} - J_2)}{1-\epsilon_2}##

and

##Q_{gen1} = A_1 F_{12}(J_1- J_2)##

##Q_{gen2} = A_2 F_{21}(J_2- J_1)##

Solving the first set of equations for the radiosity gives:

##J_1 = E_{b1} - \frac{Q_{gen1} (1 - \epsilon_1)}{\epsilon_1 A_1}##

##J_2 = E_{b2} - \frac{Q_{gen2} (1 - \epsilon_2)}{\epsilon_2 A_2}##

Now plugging these into the second set of equations gives (after some algebra):

##Q_{gen1} = A_1 F_{12} \left( E_{b1} - \frac{Q_{gen1} (1 - \epsilon_1)}{\epsilon_1 A_1} - E_{b2} + \frac{Q_{gen2} (1 - \epsilon_2)}{\epsilon_2 A_2}\right)##

##Q_{gen2} = A_2 F_{21} \left( E_{b2} - \frac{Q_{gen2} (1 - \epsilon_2)}{\epsilon_2 A_2} - E_{b1} + \frac{Q_{gen1} (1 - \epsilon_1)}{\epsilon_1 A_1}\right)##

So these are general, but now I introduce the assumptions above, namely

##Q_{gen2} = 0##

##F_{12} = 1##

##F_{21} {A_2} = A_1##

##A_2 \rightarrow \infty##

The first equation results in:

##Q_{gen1}\left( \frac{1}{\epsilon_1}\right) = A_1(E_{b1} - E_{b2})##

Which is

##Q_{gen1} \left( \frac{1}{\epsilon_1} -1 \right)= A_1(E_{b1} - E_{b2})##

This is not the same result. My question, what happened? Shouldn't both equations give the same values?

##T_s = \left( \frac{Q_{gen}}{A_s \sigma \epsilon} + {T_{amb}^4} \right)^{\frac{1}{4}}##

Where ##Q_{gen}## is in Watts and the subscript

*s*refers to the sphere while the subscript*amb*refers to ambient. Fine and good, but I wanted to see if I could derive this equation from a circumstance where there are two concentric spheres, the inside sphere is identical to the above sphere and the outside sphere is super large, adiabatic, and at the temperature of the ambient.My starting point are the 4 radiosity equation in 4 unknowns given by Nellis & Klein, i.e.,

##Q_{gen1} = \frac{\epsilon_1 A_1(E_{b1} - J_1)}{1-\epsilon_1}##

##Q_{gen2} = \frac{\epsilon_2 A_2(E_{b2} - J_2)}{1-\epsilon_2}##

and

##Q_{gen1} = A_1 F_{12}(J_1- J_2)##

##Q_{gen2} = A_2 F_{21}(J_2- J_1)##

Solving the first set of equations for the radiosity gives:

##J_1 = E_{b1} - \frac{Q_{gen1} (1 - \epsilon_1)}{\epsilon_1 A_1}##

##J_2 = E_{b2} - \frac{Q_{gen2} (1 - \epsilon_2)}{\epsilon_2 A_2}##

Now plugging these into the second set of equations gives (after some algebra):

##Q_{gen1} = A_1 F_{12} \left( E_{b1} - \frac{Q_{gen1} (1 - \epsilon_1)}{\epsilon_1 A_1} - E_{b2} + \frac{Q_{gen2} (1 - \epsilon_2)}{\epsilon_2 A_2}\right)##

##Q_{gen2} = A_2 F_{21} \left( E_{b2} - \frac{Q_{gen2} (1 - \epsilon_2)}{\epsilon_2 A_2} - E_{b1} + \frac{Q_{gen1} (1 - \epsilon_1)}{\epsilon_1 A_1}\right)##

So these are general, but now I introduce the assumptions above, namely

##Q_{gen2} = 0##

##F_{12} = 1##

##F_{21} {A_2} = A_1##

##A_2 \rightarrow \infty##

The first equation results in:

##Q_{gen1}\left( \frac{1}{\epsilon_1}\right) = A_1(E_{b1} - E_{b2})##

Which is

**identical to my formula above**. Great, that is what I was hoping would happen! But the second equation gives:##Q_{gen1} \left( \frac{1}{\epsilon_1} -1 \right)= A_1(E_{b1} - E_{b2})##

This is not the same result. My question, what happened? Shouldn't both equations give the same values?

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