The Radiosity Method for Diffuse-Gray Surfaces
The heat transfer by radiation is combined with convective and conductive heat transfer through a source term added to the heat equation along with the other contributions from the heat flux and boundary heat source boundary conditions. Recalling Equation 4-68, this source account for the difference between incident radiation, or irradiation, G, and radiation leaving the surface, or radiosity, J:
The radiosity, J, is given in Equation 4-67. It is the sum of reflected and emitted radiation. For diffuse-gray surfaces, J is defined by:
Here
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  G is the incoming radiative heat flux, or irradiation (SI unit: W/m2)
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ε is the surface emissivity (SI unit: 1), a dimensionless number in the range 0 ≤ ε ≤ 1. The diffuse-gray surface hypothesis corresponds to surfaces where ε is independent of the radiation wavelength.
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eb(T) is the blackbody hemispherical total emissive power (SI unit: W/m2).
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T is the surface temperature (SI unit: K).
The irradiation, G, at a given point is split into three contributions according to:
(4-82)
where:
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Gm is the mutual irradiation, coming from other boundaries in the model (SI unit: W/m2).
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Gext is the irradiation from external radiation sources (SI unit: W/m2). It is the sum of the products, for each external source, of the external heat sources view factor Fext by the corresponding source radiosity:
The first term of the sum gathers radiation sources located on a point. The second term stands for directional radiative sources.
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Gamb is the ambient irradiation (SI unit: W/m2), defined as:
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Famb is an ambient view factor; its value is equal to the fraction of the field of view that is not covered by other boundaries. Therefore, by definition, 0 ≤ Famb ≤ 1 at all points.
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Tamb is the assumed far-away temperature (SI unit: K) in the directions included in Famb.
The Surface-to-Surface Radiation Interface includes the following radiation types:
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Diffuse Surface (Surface-to-Surface Radiation interface) is the default radiation type. It requires accurate evaluation of the mutual irradiation, Gm. The incident radiation at one point on the boundary is a function of the radiosity, J, at every other point in view. The radiosity, in turn, is a function of Gm, which leads to an implicit radiation balance:
(4-83)
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Diffuse Mirror (Surface-to-Surface Radiation interface) is a variant of the Diffuse Surface radiation type with ε = 0. Reradiation surfaces are common as an approximation of a surface that is well insulated on one side and for which convection effects can be neglected on the opposite (radiating) side (see Ref. 18). It resembles a mirror that absorbs all irradiation and then radiates it back in all directions.
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Prescribed Radiosity (Surface-to-Surface Radiation interface) makes it possible to specify graybody radiation. The radiosity expression is then εeb(T). A user-defined surface radiosity expression can also be defined.
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Opaque Surface (Surface-to-Surface Radiation interface) is available when the surface-to-surface radiation method is Ray shooting. It accounts for specular reflection. The conservation equation reads
,
and the radiosity reads as in Equation 4-83.
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Semi-Transparent Surface (Surface-to-Surface Radiation interface) is available when the surface-to-surface radiation method is Ray shooting. It accounts for reflection, transmission and the conservation equation reads
,
and the radiosities read
(4-84)
(4-85)
The Surface-to-Surface Radiation interface handles the radiosity J as a shape function unless J is prescribed.
Whereas diffuse and specular reflectivities are handled by the different features of the Surface-to-Surface Radiation interface, only diffuse emissivity is considered.