3.2.3. Radiation Transport Equation

For applications involving PMR, both the radiative heat flux and the divergence of the radiative heat flux are needed. The radiative heat flux vector provides the radiative flux to the boundary of the heat conduction region. The flux divergence provides one of the principal volumetric heat sources in the turbulent combustion region for fire applications.

3.2.3.1. Boltzmann Transport Equation

The spatial variation of the radiative intensity corresponding to a given direction and at a given wavelength within a radiatively participating material, $I(s)$ , is governed by the Boltzmann transport equation. In general, the Boltzmann equation represents a balance between absorption, emission, out-scattering, and in-scattering of radiation at a point. For combustion applications, however, the steady form of the Boltzmann equation is appropriate since the transient term only becomes important on nanosecond time scales which is orders of magnitude shorter than the fastest chemical reaction [7].

Experimental data shows that the radiative properties for heavily sooting, fuel-rich hydrocarbon diffusion flames ( $10^{-4}$ % to $10^{-6}$ % soot by volume) are dominated by the soot phase and to a lesser extent by the gas phase (Modest [8], pg. 425). Since soot emits and absorbs radiation in a relatively constant spectrum, it is common to ignore wavelength effects when modeling radiative transport in these environments. Additionally, scattering from soot particles commonly generated by hydrocarbon flames is several orders of magnitude smaller that the absorption effect and may be neglected [7]. With these assumptions in mind, the appropriate form of the Boltzmann radiative transport equation for heavily sooting hydrocarbon diffusion flames is

(3.62) $s_i {{\partial} \over {\partial x_i}} I\left(s\right) + \mu_a I\left(s\right) = {{\mu_a \sigma T^4} \over {\pi}} ,$

where $\mu_a$ is the absorption coefficient, $I(s)$ is the intensity along the direction $s_i$ , and $T$ is the temperature.

The flux divergence (on the right hand side of (3.33)) may be written as a difference between the radiative emission and mean incident radiation at a point,

(3.63) ${{\partial q_i^r} \over {\partial x_i}} = \mu_a \left[ 4 \sigma T^4 - G \right] ,$

where $G$ is the scalar flux. The quantity, $G/4\pi$ , is often referred to as the mean incident intensity [9].

The scalar flux and radiative flux vector represent angular moments of the directional radiative intensity at a point [8],

(3.64) $G = \int_{0}^{2\pi}\!\int_{0}^{\pi}\! I\left(s\right) \sin \theta_{zn} d \theta_{zn} d \theta_{az} ,$

(3.65) $q^{r}_{i} = \int_{0}^{2\pi}\!\int_{0}^{\pi}\! I\left(s\right) s_i \sin \theta_{zn} d \theta_{zn} d \theta_{az} ,$

where $\theta_{zn}$ and $\theta_{az}$ are the zenith and azimuthal angles respectively as shown in Ordinate Direction Definition, with ${\bf s} = \sin \theta_{zn} \sin \theta_{az} {\bf i} + \cos \theta_{zn} {\bf j} + \sin \theta_{zn} \cos \theta_{az} {\bf k}$ .

3.2.3.2. Radiation Intensity Boundary Condition

The radiation intensity must be defined at all portions of the boundary along which $s_i n_i < 0$ , where $n_i$ is the outward directed unit normal vector at the surface. The intensity is applied as a Dirichlet condition which must be determined from the surface properties and temperature. The diffuse surface assumption provides reasonable accuracy for many engineering combustion applications. The intensity leaving a diffuse surface in all directions is given by

(3.66) $I\left(s\right) = {1 \over \pi} \left[ \tau \sigma T_\infty^4 + \epsilon \sigma T_w^4 + \left(1 - \epsilon - \tau \right) q^{r,inc}_j n_j \right] ,$

where $\epsilon$ is the total normal emissivity of the surface, $\tau$ is the transmissivity of the surface, $T_w$ is the temperature of the boundary, $T_\infty$ is the environmental temperature and $q^{r,inc}_j$ is the incident radiation, or irradiation for direction $j$ . Recall that the relationship given by Kirchhoff’s Law that relates emissivity, transmissivity and reflectivity, $\rho$ , is

(3.67) $\rho + \tau + \epsilon = 1.$

where it is implied that $\alpha = \epsilon$ .