... The formula for radiant power is (e * s) * area * T^4. Period. This is according to the Stefan-Boltzmann law, and no other variables are required at steady-state. The initial temperature of the heat source was 150F, or 338.71K. So we agreed that the input power to the heat source is sufficient for the equation (e * s) * (heat source area) * 338.71^4. The power input doesn't change. ... the total power output (and therefore power input) at the heat source, in initial conditions, was (we agreed on this) 82.12 W/m^2 * 510.065 m^2 = 41886.54 Watts. Power in = power out. ... [Jane Q. Public, 2014-09-11]
No. We've never agreed that the unchanging power input (my "constant electrical heating power") is "82 W/m^2". I've repeatedly failed to explain that the constant electrical heating power would only be "82 W/m^2" if the chamber walls were 0K blackbodies.
In this experiment there is a "... constant flow of energy into the plate from the electric heater... flowing in at a constant rate... the electric heater pumps in energy at a constant rate. ..."
Note that the constant rate of Dr. Spencer's electric heater would equal zero if the chamber walls were also at 150F. So any calculation of this crucial constant rate would also need to be zero in the case of chamber walls at 150F.
Since Jane's "82 W/m^2" value isn't the constant electrical heating power needed to keep the source at 150F inside 0F chamber walls, it isn't held constant. Here's where Jane actually calculated the constant electrical power heating the source inside 0F chamber walls:
... Calculate initial (denoted by "i") heat transfer from heat source to chamber wall. We are doing this only to check our work later. ... = 55.5913 [W/m^2]... [Jane Q. Public, 2014-09-10]
So Jane's source needs 55.6 W/m^2 of constant electrical heating power to stay at 150F inside 0F chamber walls. This value is held constant. After the enclosing shell is added and temperatures stabilize, conservation of energy demands that net heat transfer out equals Jane's 55.6 W/m^2. Does it?
... you should at least have tried drawing your boundary around your own goddamned heat source, both for initial conditions and your final result, to check your work. But you didn't. What you got was a universe-busting violation of conservation of energy. ... [Jane Q. Public, 2014-09-11]
No, I drew that boundary for both initial and final conditions to guarantee conservation of energy. In fact, I repeatedly suggested that you check your work by drawing a boundary between the source and the enclosing shell at your proposed steady-state temperatures, then calculating power in = power out using the original constant electrical power you calculated before the source was enclosed.
Let's do that:
Jane's constant electrical power of 55.6 W/m^2 flows into that boundary. At steady-state, power in = power out. But power out through that boundary is the net heat transfer from the source to the shell, and Jane calculates that as 27.8 W/m^2.
Since power in > power out, energy isn't conserved between the source and the enclosing shell at Jane's proposed "steady-state" temperatures.
... the total heat transfer now from heat source to the chamber wall is equal to: (heat transfer from heat source to the inside of the enclosing plate) PLUS (heat transfer from the outside of the enclosing plate to the wall). ... [Jane Q. Public, 2014-09-10]
Once again, conservation of energy means that power in = power out through any boundary where nothing inside that boundary is changing with time. Any heat transfer which doesn't cross the boundary can't be included because it can't change the total amount of energy inside the boundary.
... The "enclosing shell" (if by that you mean the passive plate that was inserted) is acted upon only by radiation. You should have drawn your shell around THAT, and that alone. ... [Jane Q. Public, 2014-09-11]
Let's draw a boundary around the enclosing shell to check Jane's work:
Jane's constant electrical power of 55.6 W/m^2 flows into that boundary. At steady-state, power in = power out. But power out through that boundary is the net heat transfer from the shell to the chamber walls, and Jane calculates that as 27.8 W/m^2.
Since power in > power out, energy isn't conserved between the source and the enclosing shell at Jane's proposed "steady-state" temperatures.
... the total heat transfer now from heat source to the chamber wall is equal to: (heat transfer from heat source to the inside of the enclosing plate) PLUS (heat transfer from the outside of the enclosing plate to the wall). ... Add them together for the total heat transfer: 27.7832 + 27.7813 = 55.5645 total heat transfer. ... [Jane Q. Public, 2014-09-10]
No. Since heat transfer from heat source to the inside of the enclosing plate never crosses a boundary drawn outside the enclosing plate, it can't affect energy conservation for that boundary. At Jane's temperatures, total heat transfer out through that boundary is actually just 27.8 W/m^2, while Jane's constant 55.6 W/m^2 electrical heating power still flows in.
Because power in = power out through any boundary where nothing inside that boundary is changing with time, Jane's "steady state" solution violates conservation of energy.