Deriving the Radiative Heat Flux

In Figure 4-5, consider a point P located on a surface that has an emissivity ε, reflectivity ρ, absorptivity α, refractive index n, and temperature T. The body is assumed opaque, which means that no radiation is transmitted through the body. This is true for most solid bodies.

Figure 4-5: Incoming irradiation (left), outgoing radiosity (right).

The total incoming radiative flux at P is called irradiation and denoted G. The total outgoing radiative flux at P is called radiosity and denoted J. This radiosity is the sum of reflected and emitted radiation:

(4-67)

According to the Stefan-Boltzmann law, eb(T) is the power radiated across all wavelengths and depends on the forth power of the temperature:

The net inward radiative heat flux, q, is then given by the difference between the irradiation and the radiosity:

(4-68)

Using Equation 4-67 and Equation 4-68, J can be eliminated and a general expression is obtained for the net inward heat flux into the opaque body based on G and T.

(4-69)

Most opaque bodies also behave as ideal gray bodies, meaning that the absorptivity and emissivity are equal, and the reflectivity is therefore obtained from the following relation:

(4-70)

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

(4-71)

This is the expression used for the radiative boundary condition.