Some time ago, Eli and his merry elves put together a lengthy comment on an even more lengthy paper (aka piece of trash) by Gerhard Gerlich and Ralf Tscheuschner, that being a paper so bad that it really was not worth the work, except the work the merry elves did was a piece of play.
Now the Rabett is quite happy with the project. It was maybe the first published blog generated reply to such nonsense (thus the grandfather of the 97% paper), and even happier about those who took part, some of whom blog, some of whom tweet and blog to this day, Chris Ho-Stuart, Chris Colose, Joel Shore, Arthur Smith and Joerg Zimmerman.
A major part of the comment was showing that absorbing layer models of the atmosphere lead to a warmer surface, in perfect agreement with the second law of thermodynamics. What happens, of course, is that each absorbing layer re-emits IR radiation, a part of which is absorbed by the layer below. This slows the rate at which the lower level cools by radiation. If the lowest level is heated by an outside source (such as the sun) and an equilibrium is established so that the energy into the system matches that of radiation from the system, then the temperature of the lowest level at equilibrium is higher than it would be in the absence of absorbing layers.
Of course, this did not meet with understanding amongst the lard heads, and Eli ran into it again recently on Bishop Hill. Curiously Chris Colose has been thinking about the problem too and has a couple of recent posts on the subject.
Eli's introduction to thermal radiation shielding was building very high temperature ovens (> 1200K) with multiple levels of radiation shielding during his graduate research, so, on an experimental level the answer was clear, but today while searching the net he came across a book on radiative transfer by Robert Siegel which considers the problem in detail starting with parallel piles of heat shielding layers which emit diffusely (e.g. the same in all directions)
in really complete detail. The model includes different emissivities for the inside and outside walls of each shielding level. Eli is not going to go full SoD on the bunnies, but those interested can find a detailed derivation of the heat flow per unit area between two parallel plates in just about any book on thermal transfer, or you can corner John Abraham at the next AGU. When a steady state is established the amount of heat flowing per unit area through each level q must be the same
Following Siegel, if we add these equations up, the right hand side is σ(T14-T24). Dividing by the co-factor of q on the left hand side yields
Heat transfer books usually stop there, because the MEs are interested in how to design shielding for thermal or cryogenic applications.
OTOH, Rabett and friends were looking at the case of a planet where the amount of incoming energy from the Sun or the star of your choice is q. The emissivity of the surface is going to be something like 0.95, that of the atmosphere at different levels, well that depends on the pressure, concentration and spectra of greenhouse gases, and, of course the specific humidity and where the clouds are. For CO2 the contribution is going to be between 0.19 and 0.12. For water vapor higher, as high as water vapor goes before condensing out
However, we can gain insight by setting ε1 equal to 1 and letting all of the other levels have the same emissivity, both inside and outside each shielding level. In that case
At a steady state, the same amount of energy has to be radiated to space. If there are no shielding levels, the amount of heat radiated per unit time is σT1o4. Consider the case where there is only the outermost heat shield (N=0) then
Canceling σ, multiplying both sides by 1/ε2 and bringing T1o4 to left hand side we get
All terms on the left hand side are positive, ε2 is less than or equal to 1, therefore T1, the temperature where there is one heat shielding level is greater than T1o, the temperature of the surface if there is no blocking.
If there are N equivalent heat shielding layers between the innermost and outmost layers, then similarly
The added term on the left hand side is again positive (if εs =1 then it is simply equal to N. If εs < 1 then (2/εs -1) > 1. In either case, especially the latter, T1 > T1o . The same can be done for spherical geometries, but one has to consider geometric factors, the ratios of the areas of the various shells to each other.
Siegel and other heat transfer books do the derivation.
How important are the geometric factors? They scale as An/Ao where A=4πR2 so at the risk of offending the punctilious the ratio is (Rn/Ro)2 The radius of the earth is 6371 km. Using a 10 km high atmosphere basically the troposphere, or at least the effective level which radiates to space in the CO2 bands, (Rn/Ro)2 = (6381/6371)2 = 1.003, so there will be a .3% difference from treating the system as a nest of sphere's or a series of parallel plates. Close enough.