Radiative Heat Transfer Models

This section details the mathematical formulations and API structures of the radiative transfer equation (RTE) solvers.

1. Governing Radiative Transfer Equation (RTE)

Radiative energy transport through an absorbing, emitting, and scattering medium along a direction vector $\mathbf{s}$ is governed by: $$\frac{d I_\nu(\mathbf{x}, \mathbf{s})}{d s} = -\left(\kappa_\nu + \sigma_s\right) I_\nu(\mathbf{x}, \mathbf{s}) + \kappa_\nu I_{b,\nu}(T) + \frac{\sigma_s}{4\pi} \int_{4\pi} I_\nu(\mathbf{x}, \mathbf{s}') \Phi(\mathbf{s}', \mathbf{s}) d\Omega'$$ where $I_\nu$ is the spectral radiative intensity, $\kappa_\nu$ is the spectral absorption coefficient, $\sigma_s$ is the scattering coefficient, and $I_{b,\nu}(T)$ is the blackbody emission.

2. Spherical Harmonics (P1) Radiation Model

The P1 model simplifies the RTE to a system of second-order diffusion equations for the incident radiation $G_\nu$:

$$\nabla \cdot \left( \Gamma_\nu \nabla G_\nu \right) - \kappa_\nu G_\nu = -4 \pi \kappa_\nu I_{b,\nu}(T)$$

where the diffusion coefficient is: $$\Gamma_\nu = \frac{1}{3 \kappa_\nu + \sigma_s (1 - g)}$$

P1 Boundary Conditions

At solid boundaries, a partial-flux balance yields: $$-\Gamma_\nu \frac{\partial G_\nu}{\partial n} = \frac{\epsilon_w}{2 (2 - \epsilon_w)} \left( 4 \pi I_{b,\nu}(T_w) - G_\nu \right)$$ where $\epsilon_w$ is the wall emissivity and $T_w$ is the wall temperature.

3. Discrete Ordinates Method (DOM)

The DOM discretizes the angular space into $M$ discrete directions $\mathbf{s}_m$ with weights $w_m$. The RTE is solved independently along each ordinate direction:

$$\mathbf{s}_m \cdot \nabla I_m = -\left(\kappa + \sigma_s\right) I_m + \kappa I_b(T) + \frac{\sigma_s}{4\pi} \sum_{k=1}^M w_k I_k \Phi(\mathbf{s}_k, \mathbf{s}_m)$$

3.1 Unstructured Angular Sweeps

On unstructured hexahedral grids, intensities are swept cell-by-cell downstream along the direction vectors. Cell-face intensities are calculated using spatial differencing schemes: $$I_{f,m} = \alpha_f I_{C,m} + (1 - \alpha_f) I_{\text{upwind},m}$$

3.2 DOM Boundary Conditions

For incoming directions ($\mathbf{s}_m \cdot \mathbf{n} < 0$), intensity is defined by wall reflection and emission: $$I_m(\mathbf{x}_w) = \epsilon_w I_b(T_w) + \frac{1 - \epsilon_w}{\pi} \sum_{\mathbf{s}_k \cdot \mathbf{n} > 0} w_k I_k(\mathbf{x}_w) |\mathbf{s}_k \cdot \mathbf{n}|$$

4. Multi-Physics Coupling API

• Coupling Source (qrad): The net radiative heat source added to the energy equation is: $$-\nabla \cdot \mathbf{q}_{\text{rad}} = \kappa \left( G - 4 \sigma_{\text{SB}} T^4 \right)$$ where $\sigma_{\text{SB}}$ is the Stefan-Boltzmann constant.
• Spectral Channel Bands: Supports splitting the spectrum into discrete absorption bands, with spectral properties computed from temperature and species fields using gray-gas or WSGGM models.