Properties of participating media

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Figure 1: Attenuation of intensity in element of thickness dS

Absorption is the conversion of the radiation into internal energy of the medium; scattering is the change in direction of the radiation away from the direction of propagation without loss of energy (radiation scattering in cases of engineering interest are elastic scattering events); and emission is radiation originating from the medium that adds to the intensity.

Consider intensity $I λ$ that is propagating through a medium in a particular direction (Fig. 1). The spectral dependence is maintained in the relations being developed, because the properties of many media, particularly gases, are highly wavelength dependent. As the intensity moves through an element of thickness dS, it is attenuated by absorption into the medium, and by scattering from the medium.

The change in intensity is given by

${I_\lambda }(S + dS) = {I_\lambda }\left( S \right) + d{I_{\lambda ,attenuation}} = {I_\lambda }\left( S \right) - {\beta _\lambda }{I_\lambda }(S)dS\qquad \qquad(1)$

so that

$d{I_{\lambda ,attenuation}} = - {\beta _\lambda }{I_{\lambda ,attenuation}}(S)dS\qquad \qquad(2)$

The loss in intensity $d I λ, a t t e n u a t i o n$ in the element dS is assumed to be proportional to the local intensity $I λ S$ , the thickness of the element dS, and a property of the element called the attenuation coefficient, $β λ$ . The attenuation coefficient can also be shown to be the inverse of the mean free path for radiation through the medium; i.e., a large value of $β λ$ indicates large attenuation and a thus short mean free path. This derivation is based on an intuitive argument, but the results are confirmed by experiment, and can be rigorously derived using the electromagnetic theory relations for attenuating materials. Examination of eq. (2) shows that $β λ$ must have units of inverse length, and is usually in (1/m). Because attenuation is caused by both absorption in the element dS and scattering from that element, the attenuation coefficient is composed of an absorption coefficient, $κ λ$ , and a scattering coefficient, $σ s,λ$ (The $s$ subscript on $σ s λ.$ is used to avoid possible confusion with the Stefan-Boltzmann constant.) Both have units of inverse length, and

${\beta _\lambda } = {\kappa _\lambda } + {\sigma _{s\lambda }}\qquad \qquad(3)$

If the intensity change due to attenuation over a finite distance is needed, then eq. (2) can be rearranged and integrated over a path from S = 0 to a final distance $S$ :

$\int_{{I_\lambda }(S = 0)}^{{I_\lambda }(s)} {\frac{{d{I_{\lambda ,attenuation}}}}{{{I_\lambda }}}} = - \int_{S = 0}^S {{\beta _\lambda }(S)} dS\qquad \qquad(4)$

The result is

$\begin{array}{l} \int_{{I_\lambda }(S = 0)}^{{I_\lambda }(s)} {\frac{{d{I_{\lambda ,attenuation}}}}{{{I_\lambda }}}} = \ln {I_\lambda }(S) - \ln {I_\lambda }(S = 0) \\ = \ln \left[ {\frac{{{I_\lambda }(S)}}{{{I_\lambda }(S = 0)}}} \right] = - \int_{S = 0}^S {{\beta _\lambda }(S)} dS \\ \end{array}\qquad \qquad(5)$

Raising both sides to the exponential power gives

${I_\lambda }(S) = {I_\lambda }(S = 0)\exp [ - \int_{S = 0}^S {{\beta _\lambda }(S)} dS]\qquad \qquad(6)$

Transmittance: If the attenuation coefficient is uniform along S (which is assumed to be the case in this chapter; if it's not, then calculation of radiative transfer in other than the simplest geometries becomes quite computationally intense), then eq. (6) reduces to

${I_\lambda }(S) = {I_\lambda }(S = 0)\exp [ - {\beta _\lambda }S]\qquad \qquad(7)$

The intensity is seen to decrease exponentially as it travels along the path. We can define the transmittance of the medium $π λ (S)$ as

${\tau _\lambda }(S) = \exp [ - \int_{S = 0}^S {{\beta _\lambda }(S)} dS] = \exp \left( { - {\beta _\lambda }S} \right){\rm{ [if }}{\beta _\lambda } \ne F(S)]\qquad \qquad(8)$

The transmittance depends on the thickness of the medium and the value of the attenuation coefficient, and gives the ratio of the transmitted to the incident intensity. It should not be confused with the transmissivity, which is a property of an interface between two media, and is independent of the thickness.

Absorptance: If the attenuation is due only to absorption with no scattering, then the fraction of intensity absorbed plus that transmitted must sum to unity. The absorptance of the medium is then defined as

${\alpha _\lambda }(S) = 1 - {\tau _\lambda }(S) = 1 - \exp \left( { - {\kappa _\lambda }S} \right){\rm{ [if }}{\kappa _\lambda } \ne F(S)]\qquad \qquad(9)$

Again, this should not be confused with the surface property absorptivity, $α λ$ , (Section 9.3.1) which is independent of the thickness $S$ .

Emittance: Using a similar argument to that for Kirchhoff's Law for surface properties, the ability of an absorbing medium to emit radiation can be used to relate the absorptance to the property emittance $ε λ (S)$ :

${\varepsilon _\lambda }(S) = {\alpha _\lambda }(S) = 1 - {\tau _\lambda }(S) = 1 - \exp \left( { - {\kappa _\lambda }S} \right){\rm{ [if }}{\kappa _\lambda } \ne F(S)]\qquad \qquad(10)$

The emittance is used to predict the intensity traveling in a given direction due to emission from an isothermal medium along a path of length $S$ :

${I_\lambda }(S) = {\varepsilon _\lambda }(S){I_{\lambda b}}\left( T \right) = \left[ {1 - \exp ( - {\kappa _\lambda }S)} \right]{I_{\lambda b}}\left( T \right)\qquad \qquad(11)$

and for a differential element of medium $d S$ ,

$d{I_{\lambda ,emitted}} = {I_{\lambda b}}\left( T \right)\frac{d}{{dS}}\left[ {1 - \exp ( - {\kappa _\lambda }S)} \right] = {\kappa _\lambda }{I_{\lambda b}}\left( T \right)dS\qquad \qquad(12)$

Now consider an isothermal volume element $d V$ (Fig. 1). The intensity leaving the small element of length $d S$ and face area $d A s$ is given by eq. (12). Adding the intensity emitted by all elements parallel to the one shown in Fig. 1 gives the average intensity emitted in that direction by dV, or

$d{I_{\lambda ,emitted}}d{A_p} = \int_{d{A_p}} {{\kappa _\lambda }{I_{\lambda b}}\left( T \right)} dSd{A_s} = {\kappa _\lambda }{I_{\lambda b}}\left( T \right)\int_{d{A_p}} {dSd{A_s}} = {\kappa _\lambda }{I_{\lambda b}}\left( T \right)dV\qquad \qquad(13)$

where the final integral over all parallel volume elements is evaluated by noting that adding the volumes of all of the small elements of length dS and face area d $A s$ (as shown in Fig. 2) gives the total volume dV. The factor d $A p$ is the projected area of the entire volume element in the direction of the intensity. To get the total energy emitted by the volume element, we now use the definition of the intensity

${I_{\lambda b}} = \frac{{d{e_\lambda }}}{{dA\cos \theta d\omega d\lambda }}$

to find the energy emitted in one direction as

${d^3}{e_\lambda } = d{I_{\lambda ,emitted}}\left( T \right)d{A_p}d\Omega d\lambda = {\kappa _\lambda }{I_{\lambda b}}dVd\Omega d\lambda \qquad \qquad(14)$

Figure 2: Isothermal emitting volume element

where the cos $θ$ term in the definition of intensity is omitted because d $A p$ is already the projected area. To get the emission into all directions, integrate over the 4 $π$ of solid angles:

${d^2}{e_\lambda } = {\kappa _\lambda }{I_{\lambda b}}\left( T \right)dVd\lambda \int_{4\pi } {d\Omega = 4\pi {\kappa _\lambda }{I_{\lambda b}}\left( T \right)dVd\lambda } = 4{\kappa _\lambda }{E_{\lambda b}}dVd\lambda \qquad \qquad(15)$

Finally, we can integrate over all wavelengths to get the total emission from the volume element:

$de = 4dV\int_{\lambda = 0}^\infty {{\kappa _\lambda }{E_{\lambda b}}} d\lambda \qquad \qquad(16)$

If the spectral dependence of $κ λ$ is known, then the integral in eq. (16) can be evaluated, and we can define a mean absorption coefficient (the Planck mean) as

${\kappa _P} = \frac{{\int_{\lambda = 0}^\infty {{\kappa _\lambda }{E_{\lambda b}}} d\lambda }}{{\sigma {T^4}}}\qquad \qquad(17)$

and eq. (16) reduces to an equation for total emission into all directions from d $V$ :

$de = 4{\kappa _P}\sigma {T^4}dV\qquad \qquad(18)$

Scattering: If radiation is scattered from the direction $S$ into another direction ( $θ$ $φ$ ) defined relative to the direction of $S$ (Fig. 3) information is needed on how much of the scattered intensity goes into a particular direction. A function giving this information is defined as the scattering phase function, $Φ$

${\Phi _\lambda }(\theta ,\phi ) = \frac{{4\pi d{I_{\lambda ,s}}\left( {\theta ,\phi } \right)}}{{{\sigma _{s\lambda }}I(S)dS}}\qquad \qquad(19)$

Figure 3: Scattering from a volume element

The numerator in eq. (19) is 4 $π$ times the intensity scattered into a given direction, and the denominator is the scattered energy (into all directions) from the intensity passing through d $S$ . The intensity scattered into direction ( $θ$ $φ$ ) is

$d{I_{\lambda ,s}}\left( {\theta ,\phi } \right) = \frac{{{\Phi _\lambda }(\theta ,\phi ){\sigma _{s\lambda }}I(S)dS}}{{4\pi }}\qquad \qquad(20)$

Most scattering systems have randomly oriented particles such as soot or ash, and the scattering coefficient $σ s λ.$ is assumed to be independent of direction through the medium. There are exceptions such as radiation incident on the carbon filaments held in a transparent epoxy matrix in a filament wound structure such as a golf club shaft; in that case both $σ s λ.$ and $Φ$ depend on the angle of incidence onto the filaments. The shape of the distribution of scattered energy is usually quite complex, consisting of strongly peaked lobes when the particles and wavelength are close in magnitude. For spherical particles, electromagnetic theory was used by Gustav Mie (pronounced "me") (Mie, 1908) to predict the shape of $Φ$ , which is strongly dependent on both the refractive index of the particles and the parameter $\varsigma = \pi D/\lambda$ , where $D$ is the spherical particle diameter. Many approximations are used for $Φ$ to avoid the complication of following the path of multiply scattered radiation in radiative transfer calculations. The simplest assumption is that the scattered intensity is isotropic, in which case $Φ$ = 1.

We have now defined many of the important properties that must be considered in treating radiative transfer in the presence of a medium that affects radiative transfer; a participating medium. Now we can examine the calculation of radiative transfer considering the participating medium effects.

References

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

Mie, G., 1908, “Beiträge zur Optik trüber Medien speziell kolloidaller Metallösungen,” Ann. Phys., Vol. 25, pp. 377-445.

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