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 This problem demonstrates the use of FlexPDE

 in the solution of problems in radiative transfer.

 Briefly summarized, we solve a Poisson equation

 for the radiation energy density, assuming that

 at every point in the domain the local

 temperature has come into equilibrium with the

 impinging radiation field.

 We further assume that the spectral character-

 istics of the radiation field are adequately

 described by three average cross-sections:

 the emission average, or "Planck Mean", sigmap;

 the absorption average, sigmaa; and the transport

 average, or "Rosseland Mean-Free-Path", lambda.

 These averages may, of course, differ in various

 regions, but they must be estimated by facilities

 outside the scope of FlexPDE.


 And finally, we assume that the radiation field

 is sufficiently isotropic that Fick's Law, that

 the flux is proportional to the gradient of the

 energy density, is valid.

 The problems shows a hot slab radiating across an

 air gap and heating a distant dense slab.



title 'Radiative Transfer'



   erad   { Radiation Energy Density }



   source         { declare the parameters, values will follow }

   lambdar         { Rosseland Mean Free Path }

   sigmap         { Planck Mean Emission cross-section }

   sigmaa         { absorption average cross-section }

   beta = 1/3     { Fick's Law proportionality factor }



  'air'        : source=0    sigmap=2    sigmaa=1    lambdar=10

  'hot slab'   : source=100  sigmap=10   sigmaa=10   lambdar=1

  'dense slab' : source=0    sigmap=10   sigmaa=10   lambdar=1


equations   { The radiation flow equation: }

   erad : div(beta*lambdar*grad(erad)) + source = 0




  region 1   { the bounding region is tenuous }

    use material 'air'


    natural(erad)=0       { along the bottom, a zero-flux symmetry plane }

    line to (1,0)

    natural(erad)=-erad   { right and top, radiation flows out }

    line to (1,1) to (0,1)

    natural(erad)=0       { Symmetry plane on left }

    line to close


  region 2   { this region has a source and large cross-section }

    use material 'hot slab'


    line to (0.1,0) to (0.1,0.5) to (0,0.5) to close


  region 3   { this opaque region is driven by radiation }

    use material 'dense slab'


    line to (0.8,0) to (0.8,0.3) to (0.7,0.3) to close






  contour(erad) as 'Radiation Energy'

  surface(erad) as 'Radiation Energy'

  vector(-beta*lambdar*grad(erad)) as 'Radiation Flux'


  { the temperature can be calculated from the assumption of equilibrium: }

  contour(sqrt(sqrt(erad*sigmaa/sigmap))) as 'Temperature'

  surface(sqrt(sqrt(erad*sigmaa/sigmap))) as 'Temperature'