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Heat transfer by conduction and convection is accounted for by Eq. (8) (or C in Table 2), which is also compatible with radiative heat transfer applied as a boundary condition on an opaque surface. However, glass and other media (e.g. combustion gases) are commonly semitransparent in glass processes, that is, they emit, absorb, and scatter IR radiation volumetrically.

Radiative heat transfer significantly differs from transport by advection and diffusion so that it cannot be mathematically described by an equation of the form of Eq. (9). It is instead governed by an integrodifferential equation, known as the radiative transfer equation (RTE) [3],

(14)

where represents the radiation intensity at a position and in the beam direction ; α and α_s are absorption and scattering coefficients, respectively; n is the index of refraction; the scattering phase function quantifying the portion of incident radiation from any direction redirected into direction ; and dΩ is a differential solid angle. The absorption coefficient and index of refraction are material properties, whereas the scattering coefficient and phase function Φ depend on physical conditions (e.g. size of soot particles), as well as on material characteristics. Note that Eq. (14) reduces to a first‐order differential equation when scattering is not considered.

As for the first term on the left side of Eq. (14), it represents the change in beam intensity per unit length in beam direction , whereas the second accounts for the decrease in beam intensity caused by the combined actions of absorption and scattering. While absorption has the effect of increasing local temperature, scattering only redirects a portion of the beam without absorbing energy. Finally, on the right side of Eq. (14), the first term accounts for emission, which tends to lower local temperatures, and the last term for scattering of IR from all directions into the beam path .

Radiation is a directional phenomenon and is in addition spectral in nature in that its intensity in principle depends on the wavelength of the IR beam. When spectral variations can be assumed to have negligible effects, Eq. (14) is written for a medium that is said to be gray. Extending the RTE to include spectral effects is straightforward [3], but not presented here.

Note that, Eq. (14) only accounts for IR intensity without determining directly temperatures within a material. Integrated over all directions, however, the net effects of absorption and emission are added to the source term S_T in the energy Eq. (8), thereby affecting local temperatures. There are several methods to account for the directional nature of the RTE. Referring to texts on radiation heat transfer for the details of their derivation [3], we will discuss some of them in Section 4.