Participating Media Spherical Harmonics

The spherical harmonics method converts the RTE Eqn. (1721) to a set of partial differential equations in 3-D space by transforming the continuous directional dependence of radiative intensity into a series of orthogonal functions known as spherical harmonics. The lowest order of spherical harmonics, P1, can be written as one scalar equation and one vector equation [398]. Assuming linear isotropic scattering, the P1 equations become:

• ${\mathbf{q}}_{\lambda }$ is the radiative heat flux at wavelength $\lambda$ .
• ${\kappa }_{a,\lambda }$ is the absorption coefficient at $\lambda$ .
• $I_{b,\lambda }$ is the blackbody intensity at $\lambda$ .
• $I_{0,\lambda }$ is the initial intensity at $\lambda$ .
• ${\beta }_{\lambda }$ is the absorption coefficient plus the scattering coefficient at $\lambda$ .
• $I_0$ is the total initial intensity.

Let $G_{\lambda }=4\pi I_{0,\lambda }$ . Marshak's conditions are applied at the boundaries:

where:

• ${ϵ}_{\lambda }$ is emissivity at the wall for $\lambda$ .
• ${\tau }_{\lambda }$ is transmissivity at the wall for $\lambda$ .
• $\tau$ is total transmissivity.
• $q_{1\lambda }$ is the heat flux from the opposite boundary.
• $E_{bw\lambda }$ is the blackbody emissivity for $\lambda$ .
• $q_w$ is the heat flux at the wall.