Radiative transfer equation

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The fundamental equation that describes the change in intensity at a local position $S$ is found by combining the relations found in Section 10.5.2. The change in intensity is given by

Change in intensity = - loss by attenuation (absorption + scattering) + gain by emission + gain by inscattering into direction $S$ from other directions

Using the mathematical forms for each term,

$\begin{array}{l} d{I_\lambda }(S) = \\ {\rm{ }} - {\beta _\lambda }{I_\lambda }(S)dS \\ {\rm{ }} + {\kappa _\lambda }{I_{\lambda b}}(S)dS \\ {\rm{ }} + \frac{{{\sigma _{s\lambda }}dS}}{{4\pi }}\int_{{\Omega _i} = 0}^{4\pi } {{I_\lambda }\left( {\theta ,\phi } \right)} \Phi \left( {\theta ,\phi } \right)d{\Omega _i} \\ \end{array}\qquad \qquad(1)$

For isoptropic scattering ( $Φ$ = 1) and using d $Ω$ = sin $θ$ d $θ$ d $φ$ ,

$\begin{array}{l} d{I_\lambda }(S) = - {\beta _\lambda }{I_\lambda }(S)dS \\ {\rm{ }} + {\kappa _\lambda }{I_{\lambda b}}(S)dS + \frac{{{\sigma _{s\lambda }}dS}}{{4\pi }}\int_{{\Omega _i} = 0}^{4\pi } {{I_\lambda }\left( {\theta ,\phi } \right)} d{\Omega _i} \\ = - {\beta _\lambda }{I_\lambda }(S)dS \\ {\rm{ }} + {\kappa _\lambda }{I_{\lambda b}}(S)dS + \frac{{{\sigma _{s\lambda }}dS}}{{4\pi }}\int_{\theta = 0}^\pi {\int_{\phi = 0}^{2\pi } {{I_\lambda }\left( {\theta ,\phi } \right)} } \sin \theta d\theta d\phi \\ \end{array}\qquad \qquad(2)$

Dividing through by ${\beta _\lambda }dS \equiv d{\tau _\lambda }$ and defining $ω λ = σ s,λ / β λ$ , eq. (2) becomes

$\frac{{d{I_\lambda }({\tau _\lambda })}}{{d{\kappa _\lambda }}} = - {I_\lambda }({\tau _\lambda }) + (1 - {\omega _\lambda }){I_{\lambda b}}({\tau _\lambda }) + \frac{{{\omega _\lambda }}}{{4\pi }}\int_{\theta = 0}^\pi {\int_{\phi = 0}^{2\pi } {{I_\lambda }\left( {\theta ,\phi } \right)} } \sin \theta d\theta d\phi \qquad \qquad(3)$

The $μ λ$ is called the single scattering albedo, and is a measure of the importance of scattering in the overall attenuation in the medium, and d $τ λ$ is the differential optical thickness or opacity, a dimensionless path length that is a measure of the attenuation along a path. (This is not the same quantity as the transmittance, $τ$ (S), although the same symbol is commonly used for both.) Often, the form of the RTE is further simplified by defining the source function $i λ (τ λ)$ , which is the sum of the sources of intensity due to emission and in-scattering,

${i_\lambda }\left( {{\tau _\lambda }} \right) \equiv (1 - {\omega _\lambda }){I_{\lambda b}}({\tau _\lambda }) + \frac{{{\omega _\lambda }}}{{4\pi }}\int_{\theta = 0}^\pi {\int_{\phi = 0}^{2\pi } {{I_\lambda }\left( {\theta ,\phi } \right)} } \sin \theta d\theta d\phi \qquad \qquad(4)$

so that eq. (3) becomes

$\frac{{d{I_\lambda }({\tau _\lambda })}}{{d{\tau _\lambda }}} = {i_\lambda }({\tau _\lambda }) - {I_\lambda }({\tau _\lambda })\qquad \qquad(5)$

This is called the differential form of the radiative transfer equation, or RTE. Interpreting eq. (5) shows that the change in local intensity with optical thickness is due to the gain from the source function less the loss by attenuation. The RTE is a first-order integro-differential equation (with the integral buried in the source function). It can be formally integrated by multiplying through by the integrating factor exp( $τ λ$ ), giving

${I_\lambda }({\tau _\lambda })\exp ({\tau _\lambda }) + \frac{{d{I_\lambda }({\tau _\lambda })}}{{d{\tau _\lambda }}}\exp ({\tau _\lambda }) = {i_\lambda }({\tau _\lambda })\exp ({\tau _\lambda })\qquad \qquad(6)$

The left-hand side is equal to $\frac{d}{{d{\tau _\lambda }}}\left[ {{I_\lambda }({\tau _\lambda })\exp \left( {{\tau _\lambda }} \right)} \right]$ , and integrating with respect to optical thickness from $τ λ$ = 0 to $τ λ$ (S) results in

${I_\lambda }({\tau _\lambda }) = {I_\lambda }(0)\exp \left( { - {\tau _\lambda }} \right) + \int_{\tau {*_\lambda } = 0}^{{\tau _\lambda }} {{i_\lambda }} \left( {\tau {*_\lambda }} \right)\exp \left[ { - \left( {{\tau _\lambda } - \tau {*_\lambda }} \right)} \right]d\tau {*_\lambda }\qquad \qquad(7)$

Figure 1: Propagation of intensity

where ${\tau _\lambda }\left( S \right) = \int_{S* = 0}^S {{\beta _\lambda }} \left( {S*} \right)dS*$ . If the properties are invariant across the medium, ${\tau _\lambda }\left( S \right) = {\beta _\lambda }S$ .

Equation (7) is the integrated form of the RTE. It shows (Fig. 1) that the intensity at a position $τ λ$ along a path is due to intensity leaving the boundary at $τ λ$ =0 and exponentially attenuated, plus added (but attenuated) intensity from the source function integrated along the entire path.

Equation (7) is the fundamental relation for treating radiative heat transfer in a participating medium. If the medium is non-scattering and nonabsorbing, then examination of eq. (5) shows that d $I λ$ = 0, and the intensity is constant as expected.

Figure 2: Finding the radiative flux from the intensity.

Because this is a heat transfer book, the next step is to relate the intensity to the radiative heat flux. This is done by integrating the intensity passing through a plane over all incident angles as shown in Fig. 2. The flux in the n direction is

${{q''}_{rad, \lambda}} = \int_{\Omega = 0}^4 \pi {I_{\lambda}} s X n d \Omega = \int_{\Omega = 0}^4 \pi {I_{\lambda}}(\theta)cos \theta d \Omega \qquad \qquad (8)$

The integration over 4 $π$ solid angles accounts for radiation crossing d $A$ in both negative and positive directions, and $q'' r a d$ is the net positive radiative heat flux crossing dA in the positive $x$ direction in the wavelength range d $λ$ at $λ$ . The total radiative heat flux is then found by integrating eq. (8) over all wavelengths.

For use in an energy equation written on a volume element of the participating medium is needed. This is found by a radiative balance similar to that used in derivations of the mass, energy, or momentum equation. Considering the differences in radiative flux $q'' r a d,λ$ crossing each face of a volume element in a Cartesian coordinate system gives

$\nabla {{q''}_{rad}} = ({{q''}_{rad, x}}/x) + ({{q''}_{rad, y}}/y) + ({{q''}_{rad, z}}/Z) \qquad \qquad (9)$

Reflecting now on what is needed to solve a complete radiative transfer problem, then, at every volume element in the medium, the total radiative flux divergence $\nabla {{q’’}_{rad}}$ is needed. To find this, the total flux in each coordinate direction $q'' r a d, i$ is needed, and this requires knowledge of the spectral intensity in each direction passing through the local volume element. That in turn requires evaluation of the RTE at the location of the element at every wavelength and every direction. But solution of the RTE depends on knowledge of the entire temperature field throughout the participating medium. It is perhaps clear that a full solution of radiative transfer is computationally expensive, and often is the most time-consuming part of solving a mixed-mode heat transfer problem in a participating medium. Approaches in general use for solution of the integral RTE include the zonal method (subdividing the medium into isothermal volumes and boundary surfaces, and pre-calculating factors for the exchange among them that are similar to configuration factors, but that account for attenuation and emission by the medium), discrete ordinates (subdividing the 4 $π$ of solid angles into a limited number of discrete solid angle increments, and writing the RTE for each of these directions for each volume element), discrete transfer (somewhat similar to the discrete ordinates method, but integrating along a path through each intervening volume element), Monte Carlo (treating radiation as composed of energy "packets" that each follow the laws of emission, absorption, scattering, and surface interactions, and then following a large number of these packets through individual histories to find their ultimate fate), and direct FEM solution methods. For the differential form of the RTE, the $P n$ method (using spherical harmonics expansion of the intensity) has proven useful. For details on these and other general solution methods, refer to Siegel and Howell (2002) or Modest (2003). Because the complexity of the general case places these more complete solution methods outside the scope of this book, some useful limiting cases will be examined.

References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Modest, M.M., 2003, Radiation Heat Transfer, 2^nd ed., Academic Press, New York, NY.

Siegel, R. and Howell, J.R., 2002, Thermal Radiation Heat Transfer, 4^th ed., Taylor and Francis, New York, NY.

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Radiative transfer equation

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