Thoughts on Heat Equilibria thru Spherical Shells
prompted by post on ScienceOfDoom.com
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  •   Bob Armstrong




    Sorry been busy & didn't get back to your question on the previous Brian post .



    I am surprised that , while several people commented that your question was treated in any elementary heat transfer physics course I believe only Ebel , other than myself , responded with an equation . Mine was the only equation I saw for a sphere as you specified .   Your question is exactly the sort that must be quantitatively understood -- as judged by how well they correspond to quantitative experiment , if planetary temperature is to be understood at the 3rd and 4th decimal place range of the data .   I've pointed out before the limitations of the internally heated ball analogy . At the Heartland conference a couple of speakers gave a value of about 0.02% ( Wikipedia , 0.03% )  of our external heating from the Sun . And external heating imposes the constraint that divergence = 0 over shells within . 

    I wish someone would do a "SteveGodard" on the experimental support for the mathematical abstractions , the crucial portions of which are 19th century . I'd really like to work with some others putting together an experimentally based , but with the theoretical understanding expressed in executable algorithms  implementable , and therefore experimentally testable by  the students , at the level of the PSSC course I had a half century ago .

    This return to the discipline of the classic experiment><algorithmatise physics with the ability of succinct array programming languages as an aid to the process is the thrust of my Heartland presentation : http://cosy.com/Science/HeartlandBasicBasics.html . These Gedankenexperiment undoubtedly were approximated by the realeExperimente upon which the equations were abstracted .  My standard reply in any assertion of qualitative effects is : " show me the equations " . In discussions like this , I want to ask " show me the experiments " .

    You thought experiment raises several interesting points .
    • Computations in energy are linear .  ( Otherwise talking in terms of  { % m ^ 2 3 }   would make no sense . ) 
      ( Sorry , I'm going to indulge in expressing computations in notation very close to executable APL . )

      Once you convert temperatures to energy via  { t ^ 4 }  ,  you can do simple arithmetic with the flows .  Thus you have the basic  { -/ ( t0 ; t1 ) ^ 4 } for net energy flow . And they are conserved . ( Not as Willis Eschenbach misspoke in his Steel Greenhouse post ,   w % m ^ 2  , rather , over closed surfaces .  Thus , the divergence theorem says that the sum over spherical shells around a source will be equal . Therefore the energy density must decrease by the inverse square of the radius . )
    • A energy flow isn't a temperature without a volume , and therefore a surface .  So the equations for spheres can always be expressed in terms of shells . ( In the limit , a delta function ) . I found this useful link I really need to study further with the equations for steady state conduction :  http://www.cdeep.iitb.ac.in/nptel/Mechanical/Heat%20and%20Mass%20Transfer/Conduction/Module%202/main/2.6.5.html. I was  looking for the equations for conduction because in these "thermostatics" , for a sphere ,  temperature follows either the inverse square-root law of radiation or the inverse law given in the table .

    • If a body is truly black , it by definition absorbs all the energy impinging on it . Thus there is no reflected energy to



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