Purdue ME 606: Radiative Heat Flux (Lecture 12)

Radiative Heat Flux

Consider a ray in direction $\hat s$ through a plane within a participating medium with normal direction $\hat n$

$\vec q''_\eta \cdot \hat n = \int_{4\pi}^{} I_\eta ({\hat s}) \left( {\hat n \cdot \hat s} \right) d\omega$

Remember that $\hat n \cdot \hat s = \cos \theta$

Of typically more interest is the divergence of the heat flux vector:

$\nabla \cdot \vec q''_\eta = \int_{4\pi} \frac{{dI_\eta}}{{ds}} d\omega = 4 \pi \kappa_\eta I_{b\eta} - \beta_\eta \int_{4\pi} I_\eta d\omega + \sigma_\eta \int_{4\pi} I_\eta d\omega_i$ $= \kappa_\eta \left( {4 \pi I_{b\eta} - \int_{4\pi} I_\eta d\omega} \right) = \kappa_\eta \left( {4 \pi I_{b\eta} - G_\eta} \right)$

where the incident radiation, $G_\eta = \int_{4\pi}^{} {I_\eta d\omega}$

Note: $G η = c u η$ $G = c u$ , where $u = \frac{{energy}}{{volume}}$ of radiation field.

FIGURE: FLUID ELEMENT IN SURROUNDINGS

Combined with the overall fluid energy equation, we have:

$\rho c_p \frac{{DT}}{{Dt}} = \nabla \cdot \left({k \nabla T} \right) + \beta_t T \frac{{Dp}}{{Dt}} + \mu \Phi_t - \nabla \cdot \vec q''_{rad}$

where:

$\frac{{D}}{{Dt}} = \frac{{\partial}}{{\partial t}} + \vec v \cdot \nabla$

$\nabla \cdot \left({k \nabla T} \right) = - \nabla \cdot \vec q''_{rad}$

$\beta_t T \frac{{Dp}}{{Dt}} =$ (compressible) flow work

$β t =$ thermal expansion coefficient

$μΦ t =$ viscous dissipation

1-D Transport

FIGURE: 1-D BOUNDED DOMAIN

Note: We assume that all quantities are spectral; therefore subscripts $λ$ are implied in all terms.

The optical lengths in directions normal to the interface and along the wave are:

$\tau_z = \int_0^z \beta dz$
$\tau_s = \int_0^s \beta ds$

Multiply RTE by $e^{\tau_s}$ :

$\frac{{d \left( {I e^{\tau_s}} \right)}}{{d \tau_s}} = \left ( {source} \right) e^{\tau_s}$

$I \left( {\tau_s,\hat s} \right) = I_{wall} \left( \hat s \right) e^{-\tau_s} + \int_0^{\tau_s} S \left( {\tau'_s , \hat s} \right) e^{- \left( {\tau_s - \tau'_s} \right) } d\tau'_s$

where $S \left( {\tau'_s , \hat s} \right) = \left( {1-\omega} \right) I_b \left( \tau'_s \right) + \frac{{\omega}}{{4 \pi}} \int_{4\pi} I \left( {\tau'_s,\hat s} \right) \Phi \left( {\hat s, \hat s_i} \right) d\omega_i$ ,