About a dozen people who have read my article, the case of two different fluxes, have dismissed my central argument by invoking a silly theory. The most famous critic, Willis Eschenbach (of WUWT fame), thus writes:
Zoe, I just took a look at your page. I fear that you’ve made a mathematical mistake. The problem is that you have over-specified the equation. Let me explain by a parallel example:
It is a physical impossibility for there to be more water flowing out of the end of a hose than there is flowing through the hose. Can’t happen. The flow through the hose must be equal to the flow out the end.
In the same way, It is a physical impossibility for there to be more energy flowing out of the end of a block of concrete than there is flowing through the block. It is logically impossible. The flow through the block must be equal to the flow out the end.— Willis Eschenbach
Willis then went on to resolve my equations using his key “insight” that the radiation emerging out of an object “must” equal its conductive heat flux. In the language of my article, the assertion is: CSR = CHF (Conductive Heat Flux = Cold Side Radiation [radiation at interested end] ).
The emission is at any moment εσT⁴. If the emission is not balanced by absorption or heat flux the temperature and consequently the emission will drop.— Dirk Visser
This is essentially the same as Willis’ argument.
Other critics write:
If the heat flux is only 92 mW/m², then obviously geothermal can only make the surface about 36 kelvin.— Unnamed
Again we see the CSR = CHF assumption, then evaluated with Stefan-Boltzmann’s Law.
Geothermal is negligible.— Joseph Postma
Sun is more than 500 times as powerful as geothermal.— Unnamed
Both of these comments implicitly assume CSR = CHF.
All other critiques are just variations on the same theme. Only difference is how many implicit logical leaps they are from the core assumption that CSR “must equal” CHF.
In my article I clearly explained that there is a difference between conductive heat flux within a medium and the emergent electromagnetic radiation out of the medium, but it’s been lost on deaf ears for some people. I don’t know why (their denial), but I feel the need to shame them a little.
Let’s see what wikipedia says about a black body:
A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck’s law, meaning that it has a spectrum that is determined by the temperature alone…— Wikipedia
What is the conductive heat flux (CHF) of an object at thermal equilibrium (a uniform temperature)?
The conduction formula is:
CHF = Q/(A*Δt)
Obviously with a uniform temperature, ΔT equals 0, and thus CHF is also ZERO!
And what did Ludwig Boltzmann and Max Planck discover emitted from their radiation cavities which had a CHF of zero? Was it also zero as my critics assert with their CHF=CSR theory? No, of course not! What comes out of an object with CHS=0 is CSR=εσT⁴ , and not CHF=εσT⁴ [ as my 2nd critic evaluated ]. Nor is this CSR transient and headed for zero, as Willis and Dirk would have you believe.
Just as the wikipedia snippet above implies: ONLY the TEMPERATURE on the edge matters.
Now wikipedia is not always right about everything, but this is so commonly well known that I don’t need any other source. You can find essentially the same thing in every high school or college textbook. Every experiment since Gustav Kirchoff  has invalidated the CHF=CSR hypothesis, and reaffirmed my hypothesis: CHF and CSR are completely different and their relationship is inverse:
CSR = εσ(T-CHF*L/k)⁴
The greatest external emission is achieved at the lowest internal heat flux, assuming the hot side temperature is the same.
At thermal equilibrium (CHF=0), this formula drops to:
CSR = εσT⁴
Yes, just Stefan-Boltzmann’s Law
If my critics were correct, then all (even one!) experiments since 1859 would show their claim to be true. Yet none of them do, because my critics are … merely engaging in ideological mathematics and not real physics.
|CHF = CSR||CSR = εσ(T-CHF*L/k)⁴|
Geothermal is more than capable of delivering 0°C (CSR=~315 W/m²), despite the fact that its near surface CHF is ~92 mW/m². In fact, assuming same temperature at same depth, a smaller CHF yields a higher CSR. The CHF (~92 mW/m²) alone is not even enough information to determine the final temperature, and hence radiation out of the medium. Quoting CHF and comparing it to insolation is nothing but junk science.
Sincerely Yours, -Zoe
This video shows CHF through the water approaching zero. Gets to ~0.01 W/m² at the end.
This video shows CHF through the pan get to zero. See time 01:53.