Now that we've agreed on the inner shell temperature of ~149.9F, let's take the last step. Calculate the enclosed source temperature.

Draw a boundary just inside the inner surface of the enclosing shell. Because nothing in the boundary is changing with time, power in = power out. The same constant electrical power flows in as before the shell was added. Net radiative power flows out from the source to the enclosing shell's inner surface.

As before, that net radiative power is described by Wikipedia’s equation which accounts for areas and view factors.

#Completely surrounded by shell with finite conductivity.
var('sigma T_c T_h A_c A_h F_hc power epsilon_h epsilon_c')
eq1 = power == sigma*(T_h^4 - T_c^4)/((1-epsilon_h)/(epsilon_h*A_h) + 1/(A_h*F_hc) + (1-epsilon_c)/(epsilon_c*A_c))
soln4 = solve(eq1.subs(T_c=338.629929346551,power=15028.4258648090,sigma=5.670373e-8,epsilon_h=0.11,epsilon_c=0.11, F_hc=1, A_h=510.064471909788, A_c=511.185932522526),T_h)
soln4[0].rhs().n()

... Please explain what calculations you are using where, because I find it hard to tell the Sage-formatted calculations apart. [Jane Q. Public, 2014-09-08]

The first line "var('sigma..." declares my variables.
The line "eq1 = power == sigma..." is my "power in = power out" equation using Wikipedia's equation for net radiative power.
The next line plugs in all the relevant variables and solves it for the enclosed source temperature T_h.
The last line displays the answer.

So I've described my method for calculating the enclosed source temperature from start to finish. Before I post that final answer, can we agree with my method? If not, could you please describe your method?

On general principle, yes. When all factors are considered, this is true. I haven't disagreed with this general principle, and at this point I'm only really interested in seeing the rest of your calculations. Please explain what calculations you are using where, because I find it hard to tell the Sage-formatted calculations apart. [Jane Q. Public, 2014-09-08]

In order to explain what calculations I'm using, we have to first agree on the fundamental principle all my calculations are based on.

I'm glad we agree that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes.

Notice that this general principle applies to all systems, even if they're at different temperatures or out of (thermal/radiative) equilibrium.

Now suppose that nothing inside that boundary is changing with time. Since this includes the energy inside that boundary, the rate at which energy inside the boundary changes is zero. This means power in = power out through any boundary where nothing inside that boundary is changing with time.

If we can agree so far, just say "yes" and ignore the rest of this comment. Then we can move on to the final step, which is calculating the enclosed source temperature.

If we can't agree, here's why we first need to agree that power in = power out through any boundary where nothing inside that boundary is changing with time.

... a simple power-in = power-out view is not always the right answer. ... it shows how power-in = power-out calculations can easily mislead. [Jane Q. Public, 2014-09-07]

... your "energy conservation just inside the surface" won't work. ... [Jane Q. Public, 2014-09-07]

How could it mislead? Why won't it work? As long as nothing inside the boundary is changing, a simple power in = power out view is always the right answer.

... The problem is that an analysis of this kind, based on the assumption that power-in = power-out, is doomed to fail except in coincidental cases. Even conservation of energy can give very misleading results. ... power-in = power-out is not necessarily true, and in fact that is probably a very rare exception. Therefore, you aren't going to prove anything with this approach. I wanted to stop you before you wasted more of your time. [Jane Q. Public, 2014-09-07]

How is it doomed to fail? How could it give very misleading results? As long as nothing inside the boundary is changing, power in = power out is necessarily true.

... it does not translate directly into power in = power out at a boundary just inside the cavity surface. It most certainly does not if the bodies are not in thermal equilibrium, which again I must point out this system is not in. ... [Jane Q. Public, 2014-09-07]

No, energy is conserved even when the bodies aren't in thermal equilibrium. As long as nothing inside the boundary is changing, power in = power out.

... energy does not have to be conserved between two bodies at different temperatures. That was what Incorpora was saying in his book. ... [Jane Q. Public, 2014-09-07]

No, energy is conserved even between two bodies at different temperatures. As long as nothing inside the boundary is changing, power in = power out.

Can we agree that power in = power out through any boundary where nothing inside that boundary is changing with time? If so, then let's move on to the final step. Calculate the enclosed source temperature.

Since we've had to agree to disagree about the definition of the term "equilibrium" (whether radiative or thermal), it's necessary to agree on the fundamental principle of energy conservation using a simple statement that doesn't use the term "equilibrium" (of any kind).

Wait. Are you claiming that the enclosing hollow sphere is NOT at radiative equilibrium with its surroundings? [Jane Q. Public, 2014-09-08]

No. I'm saying that since we've had to agree to disagree about the definition of the term "equilibrium" (whether radiative or thermal), it's necessary to agree on the fundamental principle of energy conservation using a simple statement that doesn't use the term "equilibrium" (of any kind).

Once again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

... there is definitely no thermal equilibrium, and without at least radiative equilibrium, there is no equilibrium at all and we might as well just stop again right here. ... no thermal equilibrium ... [Jane Q. Public, 2014-09-08]

That's why I'm trying to see if we can agree on a general principle that applies even to systems that aren't in thermal equilibrium.

... why the hell are you trying to blame me for being confused? The condition you described is impossible, so how do you expect me to know what "equilibrium" you mean? ... [Jane Q. Public, 2014-09-04]

Since we've had to agree to disagree about the definition of the term "equilibrium" (whether radiative or thermal), it's necessary to agree on the fundamental principle of energy conservation using a simple statement that doesn't use the term "equilibrium" (of any kind).

... I very definitely did NOT mean net power at radiative steady-state represents zero energy flow. ... [Jane Q. Public, 2014-09-08]

Then our statements aren't equivalent, which means there's an innocent misunderstanding here. To help resolve this miscommunication, could we please agree on a general principle that applies to all systems, even if they're not in thermal or radiative equilibrium?

Once again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

... there is definitely no thermal equilibrium, and without at least radiative equilibrium, there is no equilibrium at all and we might as well just stop again right here. ... no thermal equilibrium ... [Jane Q. Public, 2014-09-08]

That's why I'm trying to see if we can agree on a general principle that applies even to systems that aren't in thermal equilibrium.

... why the hell are you trying to blame me for being confused? The condition you described is impossible, so how do you expect me to know what "equilibrium" you mean? ... [Jane Q. Public, 2014-09-04]

Since we've had to agree to disagree about the definition of the term "equilibrium" (whether radiative or thermal), it's necessary to agree on the fundamental principle of energy conservation using a simple statement that doesn't use the term "equilibrium" (of any kind).

... I very definitely did NOT mean net power at radiative steady-state represents zero energy flow. ... [Jane Q. Public, 2014-09-08]

Then our statements aren't equivalent, which means there's an innocent misunderstanding here. To help resolve this miscommunication, could we please agree on a general principle that applies to all systems, even if they're not in thermal or radiative equilibrium?

Once again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

Since you keep place qualifiers on energy conservation, your wording isn't equivalent to mine because my statement applies even for systems that aren't in radiative equilibrium.

But that should not matter because we are discussing a system in radiative equilibrium. ... [Jane Q. Public, 2014-09-08]

Really? Since when?

... There is no thermal equilibrium. Period. None. There MAY (and eventually would) arise a condition of radiative equilibrium for the (enclosing, passive, however you want to describe it) plate. But the other objects (heat source and chamber walls) do not meet this criteria because they are heated/cooled by means that may be other than radiative. "The system" is not in radiative equilibrium. ... [Jane Q. Public, 2014-09-03]

... I don't necessarily have a problem with a broader definition, but I prefer to stick to things that are pertinent to this discussion. So can we move on? [Jane Q. Public, 2014-09-08]

Since Jane's insisted that the system is not in radiative equilibrium, it's necessary to agree on a general principle that applies even for systems that aren't in radiative equilibrium. Then we can move on.

Can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

I'm not sure I agree with your wording. It could easily be misinterpreted to mean something it does not. I agree that power in minus power out of your boundary equals power through that boundary, which at radiative steady-state represents a constant rate of energy flow through that boundary. [Jane Q. Public, 2014-09-08]

I prefer my wording, which I think most people would agree is an equivalent statement regarding your drawn boundary, but (in my opinion) is less open to misunderstanding. I agree that power into your boundary minus power out of your boundary equals the power through the boundary, which at radiative equilibrium is equivalent to a constant rate of energy flow through that boundary. ... [Jane Q. Public, 2014-09-08]

... it is radiative equilibrium that forces it to be constant. ... [Jane Q. Public, 2014-09-08]

Your wording could easily be misinterpreted to mean a constant other than zero. Didn't you mean that net power through that boundary at radiative steady-state represents zero energy flow through that boundary? If not, our misunderstanding is much more fundamental than I first thought.

This principle applies even for systems that are changing, and even for systems that aren't in radiative equilibrium. Again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

How could my wording be easily misinterpreted? Once again, this fundamental principle applies even for systems that are changing, and even for systems that aren't at radiative steady-state.

I prefer my wording, which I think most people would agree is an equivalent statement regarding your drawn boundary, but (in my opinion) is less open to misunderstanding. I agree that power into your boundary minus power out of your boundary equals the power through the boundary, which at radiative equilibrium is equivalent to a constant rate of energy flow through that boundary. Were you trying to say something else? If not, let's please move on. [Jane Q. Public, 2014-09-08]

Once again, this principle applies even for systems that are changing, and even for systems that aren't in radiative equilibrium. Again, that's why I disagree with your claim that:

... energy does not have to be conserved between two bodies at different temperatures. That was what Incorpora was saying in his book. ... [Jane Q. Public, 2014-09-07]

Since you keep place qualifiers on energy conservation, your wording isn't equivalent to mine because my statement applies even for systems that aren't in radiative equilibrium.

Once again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes? Even for systems that are changing? Even for systems that aren't in radiative equilibrium?

Given your assumptions so far, I will not dispute your calculation of the temperature of the inner surface of the enclosing plate. Please continue your calculations, as a reply to my other comment, so we can continue this exchange in a linear fashion. [Jane Q. Public, 2014-09-08]

I'm glad you don't dispute the enclosing shell's inner temperature of ~149.9F, but we should agree on my assumption that energy is conserved before proceeding.

Can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

I'm not sure I agree with your wording. It could easily be misinterpreted to mean something it does not. I agree that power in minus power out of your boundary equals power through that boundary, which at radiative steady-state represents a constant rate of energy flow through that boundary. [Jane Q. Public, 2014-09-08]

How could my wording be easily misinterpreted? Once again, this fundamental principle applies even for systems that are changing, and even for systems that aren't at radiative steady-state.

Again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

Maybe an analogy would help. The rate at which water flows into a bathtub minus the water flowing out equals the rate at which water in the bathtub changes. No qualifications needed.

If we can't agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes, could you please explain exactly why we can't agree on this?

Obviously at radiative equilibrium energy between objects in the system is being transferred at a constant rate. [Jane Q. Public, 2014-09-07]

This principle applies even for systems that are changing, and even for systems that aren't in radiative equilibrium.

Again, can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

... Energy of an entire system is conserved. It need not be conserved between individual elements of that system. That's what I've been saying. ... Heat transfer between two bodies that are not at thermal equilibrium does not conserve energy between those two bodies. ... [Jane Q. Public, 2014-09-07]

Can we agree that energy conservation means that power going in minus power going out through some boundary equals the rate at which energy inside that boundary changes?

The reason my "dirt simple" calculation was wrong, as any reader of this exchange should be able to tell (and so should you have), that I misunderstood what your power figure represented. [Jane Q. Public, 2014-09-07]

I'm very sorry for not being more clear. I take full responsibility.

It absolutely does translate directly into power in = power out at a boundary just inside the cavity surface when everything inside that boundary isn't changing. In that case, the rate at which energy changes inside the boundary equals zero, which means power in = power out.

Are you also then presuming that power transferred from the outer surface of the enclosing plate to the chamber walls is the same as the power transferred from the heat source to that plate? [Jane Q. Public, 2014-09-07]

Anything else would violate conservation of energy. But we still have one more step before the net power transferred from the heat source to that enclosing plate becomes relevant.

No, of course I got the same answer, given your assumption that power-in = power-out: 149.59F. [Jane Q. Public, 2014-09-07]

Excellent. And can we also agree about the enclosing aluminum shell's final inner steady-state temperature?

Now to calculate the enclosing shell's inner temperature. At steady-state, power in = power out through some boundary. This time, draw the boundary within the enclosing shell. Again, constant electrical power flows in. But all the other boundaries we drew were in vacuum, so heat transfer was by radiation. This time the boundary is inside aluminum, so heat transfer out is by thermal conduction.

electricity = k*(T_h - T_c)/x (Eq. 4)

The shell's thickness "x" is 1mm, and the thermal conductivity "k" of aluminum is 215 W/(m*K). We just found that:

Outer shell temperature: 338.629792627809 K (149.864 F).

So:

Inner shell temperature: 338.629929668632 K (149.864 F).

Of course, that's a flat plate approximation of heat conduction through a spherical shell, which is derived here. That more accurate equation yields:

#Calculate enclosing shell's inner temperature.
var('T_c T_h power k r_c1 r_c2')
eq2 = power == 4*pi*k*r_c1*r_c2*(T_h - T_c)/(r_c2 - r_c1)
soln3 = solve(eq2.subs(T_c=338.629792627809,power=15028.4258648090,k=215,r_c1=6.378,r_c2=6.379),T_h)
soln3[0].rhs().n()

Inner shell temperature: 338.629929346551 K (149.864 F).

Now for the final step. Calculate the steady-state temperature of the enclosed heated plate (Jane's "source").

... My point was that it does not translate directly into power in = power out at a boundary just inside the cavity surface. It most certainly does not if the bodies are not in thermal equilibrium, which again I must point out this system is not in. ... [Jane Q. Public, 2014-09-07]

It absolutely does translate directly into power in = power out at a boundary just inside the cavity surface when everything inside that boundary isn't changing. In that case, the rate at which energy changes inside the boundary equals zero, which means power in = power out.

... energy does not have to be conserved between two bodies at different temperatures. That was what Incorpora was saying in his book. ... [Jane Q. Public, 2014-09-07]

No. Energy is always conserved. Always.

Once again, the next step is calculating the enclosing shell's final outer steady-state temperature once it's added. This should have only taken you a few minutes to calculate. Did you get a different answer than me?

Once again, energy is conserved, which means that if you draw a boundary around some system (like the heated plate), power going in minus power going out equals the rate at which energy inside that boundary changes. At steady-state, that rate is zero because the system doesn't change. So at steady-state, power in = power out.

Perhaps it would be more informative if you calculate ENERGY in and ENERGY out, since that is what is actually conserved. You seem to keep forgetting that (A) power is a RATE, not a unit of energy, and (B) we are not at thermal equilibrium. ... [Jane Q. Public, 2014-09-07]

No. Once again, I said that power going in minus power going out equals the rate at which energy inside that boundary changes. Once again, that rate is zero if the system doesn't change.

... are you suggesting that if I hollowed out enough of a mountain to make a hollow rock sphere (assume the rock is diffuse gray body) 1000 m diameter, suspended a 1m dia. sphere of the same rock in the center, and evacuated the cavity: the inner sphere is going to get much hotter than the surrounding rock? Power in = power out would seem to demand that very thing. [Jane Q. Public, 2014-09-07]

No. I've repeatedly told you that power in = power out demands that an unheated inner sphere will be at exactly the same temperature as the chamber walls.

... I am aware that view factor has to be taken into account. ... [Jane Q. Public, 2014-09-07]

Using what equation? A month ago I said we could use Wikipedia’s equation which includes areas, and later mentioned view factors. I've been using this equation to calculate the net heat transfer between the heated plate (Jane's "source") and the chamber walls.

If that's the equation Jane is thinking about using to take account of the view factor, Jane should ponder what happens in that equation when the two temperatures in that equation are equal. As I've repeatedly said, the net heat transfer goes to zero when the two temperatures are equal. Regardless of their areas.

... 788.01 W != 721.44 W (!!!) Power is not conserved. ... Power is not conserved. [Jane Q. Public, 2014-09-07]

... a simple power-in = power-out view is not always the right answer. ... it shows how power-in = power-out calculations can easily mislead. [Jane Q. Public, 2014-09-07]

... The black body example I gave shows why your "energy conservation just inside the surface" won't work. Aside from just "view factor" and a few other things, a certain amount of the power in (often a very significant amount) just ends up going right back out, but you often don't see that in the formulas. ... [Jane Q. Public, 2014-09-07]

No, it shows that Jane's equation doesn't correctly describe net radiative transfer between two surfaces. Once again, Wikipedia’s equation correctly takes into account the areas and view factor.

... The problem is that an analysis of this kind, based on the assumption that power-in = power-out, is doomed to fail except in coincidental cases. Even conservation of energy can give very misleading results. ... power-in = power-out is not necessarily true, and in fact that is probably a very rare exception. Therefore, you aren't going to prove anything with this approach. I wanted to stop you before you wasted more of your time. [Jane Q. Public, 2014-09-07]

No. Energy is always conserved. A boundary drawn around a system that isn't changing always has power in = power out. Always. Because energy is always conserved.

If power flowing in through a boundary weren't equal to power flowing out of that boundary, the system is either changing or energy isn't conserved. But energy is always conserved. So a boundary drawn around a system that isn't changing always has power in = power out. Always. Because energy is always conserved.

Once again, the next step is calculating the enclosing shell's final outer steady-state temperature once it's added. Did you get a different answer than me?

ACK! SORRY! Just to be clear, power in = power flowing into the boundary in question, and power out = power flowing out of that boundary.

A boundary drawn around a system that isn't changing always has power in = power out. Always. Because energy is always conserved.

... Let me rephrase what I was saying: at least theoretically, the power at the chamber wall is allowed to vary, in order to keep the temperature at 0 degrees F. But, if we draw a boundary around the system, and assume that the ONLY power in is what we put in, and the ONLY power out is what is removed, then of course it must be conserved. I was simply expressing my concern that your electricity figure may not be properly observing those boundaries. If your electricity figure is simply power in - power out... [Jane Q. Public, 2014-09-07]

Maybe this will help. It seems like Jane might think I meant power in = electrical heating power, and power out = cooling power of the chamber walls.

If so, that's not what I meant, and I'm sorry for not being more clear. I take full responsibility.

Just to be clear, power in = power flowing into the boundary in question, and power in = power flowing out of that boundary.

In my opinion, solving thermodynamics problems is mostly about choosing the most informative boundaries, then calculating steady-state solutions by setting power in = power out through that boundary.

From the start, the largest boundary I drew was "just inside the chamber walls" so the chamber walls and the cooler have always been outside all the boundaries. That means any power used by the cooler is simply being moved from some point outside the boundary to another point which is also outside the boundary. Because that power never crosses the boundary, it's irrelevant.