Deriving the Radiative Heat Flux for Semitransparent Surfaces

In Figure 4-13, consider a point P located on a semitransparent surface that has an emissivity εu, diffuse reflectivity ρd,u, specular reflectivity ρs,u, refractive index nu, and temperature Tu on the upside, and an emissivity εd, diffuse reflectivity ρd,d, specular reflectivity ρs,d, refractive index nd, and temperature Td on the downside. As the surface is assumed semitransparent, some radiation is transmitted through the body.

Figure 4-14: Upside and downside incoming irradiation (left), upside outgoing radiosity (right). The downside outgoing radiosity is defined in a similar way.

The total incoming radiative flux at P is called irradiation, and is denoted Gu on the upside and Gd on the downside. The total diffuse outgoing radiative flux at P is called radiosity and denoted Ju on the upside and Jd on the downside. This radiosity is the sum of diffusively reflected radiation, emitted radiation and transmitted radiation coming from the other side of the semitransparent layer:

(4-97)

(4-98)

The net inward radiative heat fluxes on the upside and downside, qu and qd, are then given by the difference between the irradiation and the radiosity:

(4-99)

(4-100)

Bodies are considered to behave as ideal gray bodies, meaning that the absorptivity and emissivity are equal, and the reflectivity ρs is therefore obtained from the following relation:

(4-101)

(4-102)

Using Equation 4-97 to Equation 4-102, Ju and Jd can be eliminated and a general expression is obtained for the net inward heat fluxes into the semitransparent body based on Gu, Gd, Tu and Td:

(4-103)

(4-104)

Thus, for ideal gray bodies, q is given by:

(4-105)

This is the expression used for the radiative boundary condition.

Incident rays which angle of incidence (measured between the ray and the normal to the surface) is higher than the critical angle are not transmitted, regardless the transmittance of the surface. They contribute to total reflection instead. Hence the directional transmissivity coefficient can be defined as

where

is the critical angle. Using the following relation

we can establish