How do I apply a boundary condition to an external boundary of a solid domain to account for heat loss through thermal radiation, without modeling the external fluid field? I can account for convection heat losses by setting a Heat Transfer Coefficient and an Outside Temperature, but there's no option for radiation unless the solid domain is connected to a fluid domain.
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You can account for radiative heat losses by modifying the Heat Transfer Coefficient at the boundary. Heat loss through convection only is given by: q = h ( Tout -Twall ) where h and Tout are specified, and Twall is the calculated temperature at the boundary. To account for thermal radiaiton we would like to model: q = h ( Tout - Twall ) + sigma*emissivity*(Tout^4 - Twall^4) where sigma = 5.6703e-8 [W m^-2 K^-4] and emissivity is the wall emissivity. The above equation can be re-written as: q = hstar ( Tout - Twall ) where: hstar = h + sigma*emissivity*(Tout + Twall)*(Tout^2 + Twall^2) You can now write a CEL expression for hstar and set this as the Heat Transfer Coefficient. Below we are assuming that the Heat Transfer Coefficient due to convection only is 10 [W m^-2 K^-1] and Tout = 300 [K]. LIBRARY: CEL: EXPRESSIONS: emiss = 1 sigma = 5.6703e-8 [W m^-2 K^-4] hstar = 10 [W m^-2 K^-1] + sigma * emiss * (300[K] + T) * (300[K]^2 + T^2) END END END Adding this extra non-linearity to the Heat Transfer Coefficient can cause slower convergence or robustness problems. To avoid these problems you should set a Source Term Coefficient at the boundary that is equal to the derivative of sigma*emissivity*(Tout^4 - Twall^4) with respect to Twall, as shown below: LIBRARY: CEL: EXPRESSIONS: SourceTermCoefficient = -sigma*emiss*(4*T^3) END END END FLOW: DOMAIN: SolidDomain BOUNDARY: ExternalWall Boundary Type = WALL BOUNDARY CONDITIONS: HEAT TRANSFER: Heat Transfer Coefficient = hstar Option = Heat Transfer Coefficient Outside Temperature = 300 [K] END END BOUNDARY SOURCE: SOURCES: EQUATION SOURCE: energy Flux = 0 [W m^-2] Flux Coefficient = SourceTermCoefficient Option = Flux END END END END END END Make sure the Source Term Coefficient is negative and the actual source is zero! Your boundary condition will now account for both convection and radiative heat transfer. Note that in this example we have assumed that the Outside Temperature is the same for both the convection and radiation terms. In general this may not be true. In this case the general equation for heat flux can be written as: q = h ( Tout_c - Twall ) + sigma*emissivity*(Tout_r^4 - Twall^4)( Tout_c - Twall )/( Tout_c - Twall ) = hstar ( Tout_c - Twall ) where: hstar = h + sigma*emissivity*(Tout_r^4 - Twall^4)/( Tout_c - Twall ) where Tout_c is the Outisde Temperature for convection and Tout_r is the Outside Temperature for radiation. |
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