Inconsistent Radiation Questions

[davidwinth]

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 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?

 

[hutchphd]

[

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 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?

I have no idea idea about the source you quote.
For a gray body with emissivity $\epsilon$ your result follows immediately from detailed balance$$ \epsilon\sigma{T_s}^4-\epsilon\sigma{T_{amb}}^4=\frac Q A_s$$
A more likely complication is that the body will not be gray in which case this would be true only for a given frequency and one would need to integrate over the spectrum at the two temperatures giving an effective value for each $\epsilon$ at Temp.

 

[davidwinth]

[
I have no idea idea about the source you quote.
For a gray body with emissivity $\epsilon$ your result follows immediately from detailed balance$$ \epsilon\sigma{T_s}^4-\epsilon\sigma{T_{amb}}^4=\frac Q A_s$$
A more likely complication is that the body will not be gray in which case this would be true only for a given frequency and one would need to integrate over the spectrum at the two temperatures giving an effective value for each $\epsilon$ at Temp.

Right, I know how to get my result for the case without an external sphere. I am wondering how to get the same result if I come at it by assuming the ambient is a sphere. I am quoting the textbook Heat Transfer by Nellis and Klein, which is a widely used textbook.

 

[hutchphd]

If the first sphere is encased in an identical larger sphere, then you just have a bigger sphere. If that is not the setup, a drawing is necessary. ( sorry I don't have Nellis at al )

 

[hutchphd]

N.B. It appears to me that your assumptions require $F_{21}=0$ for consistency. That will eliminate the pesky equation I think. Otherwise I'm shooting in the dark here!

 

[davidwinth]

If the first sphere is encased in an identical larger sphere, then you just have a bigger sphere. If that is not the setup, a drawing is necessary. ( sorry I don't have Nellis at al )

That is a problem with my description. The outer sphere is just a surface, not a solid. There is vacuum between the outer surface of the inner sphere and the larger shell.

 

[256bits]

N.B. It appears to me that your assumptions require $F_{21}=0$ for consistency. That will eliminate the pesky equation I think. Otherwise I'm shooting in the dark here!

I would think that would be the asymptote for the outer sphere approaching infinite distance, so it does make sense to me.

 

[davidwinth]

I discovered the problem. It was the assumption of the larger sphere being adiabatic. When that assumption is dropped, both equations agree.