The Steel Greenhouse
A heated sphere in vacuum inside a thin steel shell. This is Postma's model, rebuilt from the equations and parameters printed on his figures (sphere 1.00 m, 2953 W; steel shell 1.50–1.52 m, starting at 0 K) and solved in your browser: heat conduction through the steel wall, radiation at both of its faces. Pick one of his runs, then the one with the shell's inner face emitting, and watch the energy ledger.
Run
Energy ledger since start
| Produced by the core | |
| Radiated to space | |
| Stored (sphere + shell warming up) | |
| Created from nothing |
created = to space + stored − produced, the same bookkeeping as "energy created = P_out + dU/dt − P_int" on Postma's figures. Energy conservation means it stays at zero.
Power right now
| Core | |
| Shell to space | |
| Shell radiation reaching the sphere | |
| …and lost by the shell? |
What changes between the runs
All three runs use the same equations for the steel wall (radial heat conduction) and its outer face (radiation to space at 0 K), with the same parameters.
Sphere held at 253.7 K: Postma's original run. The sphere's temperature is fixed by hand ("held constant" in his caption), and the shell's inner face absorbs the sphere's radiation but emits nothing back. The sphere cannot warm, because it is not allowed to.
Back-radiation added to the sphere: Postma's second run. The sphere now receives A_sp·σ·T_in⁴ from the shell,
but the shell's inner boundary is unchanged: the shell never loses what it sends. The same energy is received by the sphere and kept
by the shell, so the ledger fills with energy created from nothing. His own figure labels it: "energy created = P_back".
Shell emits from both faces: the sphere's equation is exactly the same as in his back-radiation run. The only change is one
term in the shell's inner boundary, −σT_in⁴: what the sphere receives from the shell, the shell loses. A surface that
absorbs infrared also emits it (Kirchhoff's law, which follows from the second law of thermodynamics). The ledger stays at zero,
space receives exactly the core's 2953 W once the system settles, and the sphere still ends up warmer than without the shell:
277.6 K instead of 253.7 K.
Why the sphere receives exactly A_sp·σ·T_in⁴: the inner face emits A_in·σ·T_in⁴ inward, but only the
fraction A_sp/A_in hits the sphere (everything the sphere emits hits the shell, and view factors obey reciprocity);
the rest crosses the cavity and lands back on the shell. That is also where the (r_sp/r_in)² in the boundary condition comes from.
Net heat always flows from the warmer sphere to the cooler shell. The shell does not heat the sphere; the core does. The shell reduces how fast the sphere loses heat, so the sphere must get warmer to get rid of the same 2953 W. The same thing keeps spacecraft instruments warm: a radioisotope heater of about 1 W, wrapped in multilayer insulation, in vacuum.
Not to be confused with the original: Willis Eschenbach's 2009 steel greenhouse already has the shell emitting from both faces ("a shell has two sides"). The absorb-only inner face comes from Postma's 2013 critique and his later simulations.