## Radiative Properties of Gas Bubbles in Semi-Transparent Medium

* Following from: *The Mie solution for spherical particles, Radiative properties of semi-transparent spherical particles

* Leading to: *Radiative effects in a semi-transparent liquid containing gas bubbles, Thermal radiation modeling in melt-coolant interaction

In many natural phenomena and materials processing, the presence of bubbles affects the thermophysical and radiative properties of the two-phase system and, hence, the transport phenomena. It is well known that radiation scattering by bubbles in the visible and infrared spectral ranges affects the optical properties of semi-transparent substances. One can remember the influence of bubbles on scattering of light in the near-surface layer of the ocean (Zhang et al., 1998; Thomas and Samnes, 1999; Soloviev and Lukas, 2006) and the effect of bubbles generated by chemical reactions on the glass melting process in industrial furnaces (Kim and Hrma, 1991; Fedorov and Pilon, 2002; Shelby, 2005; van der Schaaf and Beerkens, 2006). Similar structures with numerous bubbles or hollow microspheres in a semi-transparent host medium are considered advanced thermal insulation materials (Fricke and Tillotson, 1997; Baillis and Sacadura, 2000; Bynum, 2001; German and Grinchuk, 2002; Papadopoulos, 2005; Dombrovsky, 2005; Dombrovsky et al., 2007). In some cases, the pores in low-porous ceramics can be modeled as spherical gas bubbles (Manara et al., 1999). The role of steam bubbles may be considerable in high-temperature radiative heating of boiling water (Dombrovsky, 2004). The problem of the fuel-coolant interaction in a hypothetical nuclear reactor severe accident also involves near-infrared radiation transfer in water containing numerous steam bubbles (Kolev, 2007; Dombrovsky, 2007).

The general scattering problem for particles in a refracting and absorbing medium is considerably more complicated than the classic Mie problem. In several studies that have considered spherical particles in absorbing medium (Mundy et al., 1974; Chýlek, 1977; Quinten and Rostalski, 1996; Lebedev, 1999; Fu and Sun, 2001; Sudiarta and Chýlek, 2001a,b; Yang et al., 2002; Sun et al., 2004), two different kinds of optical characteristics of particles have been analyzed: the inherent properties calculated near the particle surface (in the near field) and the so-called apparent properties calculated at large distances from the particle (in the far field). Yang et al. (2002) showed that the apparent properties (and the corresponding efficiency factors of absorption and scattering) should be used when calculating the coefficients of the radiative transfer equation. This conclusion confirms the common practice in radiation transfer calculations in disperse systems (Dombrovsky, 1996).

The classical Mie solution for the absorption and scattering of radiation by a spherical particle relates to the case when the particle is in vacuum. According to the Mie theory, the dimensionless characteristics of absorption and scattering depend on the diffraction parameter, *x* = 2π*a* / λ, and the complex index of refraction of the particle material, *m* = *n* - *i*κ. Mundy et al. (1974) demonstrated that the formulas of the Mie theory are also valid for particles immersed in a refracting and absorbing medium with an arbitrary complex index of refraction, *m*_{e} = *n*_{e} - *i*κ_{e}. In so doing, the complex quantities = *m* / *m*_{e} (for cavities or gas bubbles, = 1 / *m*_{e}) and = *m*_{e}*x* should be substituted for *m* and *x* as independent variables in calculating the Mie coefficients, and the coefficient 2 / *x*^{2} in Eqs. (2) and (8) from the article The Mie solution for spherical particles should be replaced by

(1) |

where *r* ≥ *a* is the distance from the particle center. For the semi-transparent host medium treated in this study, κ_{e}*x* << 1.The latter condition can be treated as a small optical thickness of the host medium at distances comparable with the particle or bubble size. In this case, coefficient is independent of distance *r* and is determined by the following simple formula:

(2) |

Obviously, in the case of particles, cavities, or gas bubbles that do not absorb the radiation, the efficiency factor of absorption *Q*_{a} in an absorbing medium is negative, and the transport efficiency factor of scattering *Q*_{s}^{tr} is positive. Yang et al. (2002) confirmed the general conclusion by Mundy et al. (1974) for the limiting case of a weakly absorbing host medium for both homogeneous (uncoated) and coated spheres embedded in an absorbing host medium, but they derived another equation instead of Eq. (1) for considerable values of κ_{e}*x*:

(3) |

We do not consider the case of a strongly absorbing host medium. Therefore, the simple formula (2) will be used in subsequent calculations.

Fedorov and Viskanta (2000a,b) have reported one of the first theoretical analyses of the radiative properties of glass with bubbles. They considered large gas bubbles (compared with the wavelength of radiation) and used approximate analytical relations for the absorption and extinction efficiency factors derived by van de Hulst (1957, 1981) for the anomalous diffraction regime (see the article Anomalous diffraction). The scattering characteristics, including the scattering function of bubbles, were assumed to be the same as the corresponding characteristics of glass particles. The same approximation was used in the calculations performed by Pilon and Viskanta (2003). The following formulas are given by Fedorov and Viskanta (2000a,b) and Pilon and Viskanta (2003) for gas bubbles in glass:

(4) |

where, according to Fedorov and Viskanta (2000a,b), *Q*_{a}^{b} and *Q*_{s}^{b} were calculated by the complex index of refraction for gas *m*_{b}; *Q*_{a}^{g} was calculated by the complex index of refraction for glass *m*_{g}; and the diffraction parameter was taken to be the same in both cases, *x* = 2π*a* / λ. It can demonstrated both theoretically and through the use of direct calculations that the first equation In Eq. (4) is applicable only in the limiting case of |*m _{g}* - 1| << 1, κ

_{g}

*x*<< 1, when the Rayleigh-Gans approximation is valid (see the article Rayleigh-Gans scattering). The second equation in Eq. (4) is erroneous because the scattering depends on the ratio

*m*

_{b}/

*m*

_{g}, not on the value of

*m*

_{b}.

Following the studies done by Dombrovsky (2004) and Dombrovsky et al. (2005), consider the effect of bubbles on the radiative properties of a semi-transparent host medium, when all the bubbles are assumed to be of the same size. The absorption coefficient and the transport scattering coefficient of an absorbing medium containing bubbles can be calculated as follows:

(5a) |

(5b) |

where *f*_{v} is the volume fraction of bubbles, and α_{e} is the absorption coefficient of the host medium. We will examine the effect of bubbles on the radiative characteristics of the medium. For clarity, we rewrite Eq. (5a) as

(6) |

The results of calculations of the ξ(*x*) dependence for the most interesting range of the medium optical constants are given in Fig. 1. One can see that parameter ξ approaches the asymptotic value ξ = 1 for large bubbles in a weakly absorbing medium when *x* > 10 and κ*x* < 0.01.

**Figure 1. Relative absorption efficiency for gas bubbles in refracting and absorbing medium.**

In the infrared spectral range, the condition *x* > 10 is valid for all bubbles of radius *a >* 4μm. This means that the absorption coefficient of the medium containing large bubbles does not depend on the bubble size. In the case of *f*_{v} << 1, we have ξ *f*_{v} << 1; therefore, according to Eq. (6), the effect of bubbles on absorption in the range of semi-transparency is negligible and one can use the value for the medium without bubbles: α = α_{e}. It should be emphasized that the above analysis is based on the independent scattering model and the far-field approach when single bubbles are assumed to be located far from each other and can be treated as independent point scatterers. This means that there is no sense of formal substitution of relatively large values of the bubble volume fraction *f*_{v} > 0.05 in Eq. (6), as was recently done by Yin and Pilon (2006). The other physical limitation of the theoretical model discussed is the condition of small optical thickness of an elementary volume containing the representative number of bubbles *N*. In the opposite case, one cannot use the ordinary radiation transfer theory. This limitation leads to inequality

(7) |

Obviously, one should not employ the above model for bubbles of diffraction parameter *x* > 10 when the index of absorption of the host medium does not satisfy the inequality, κ_{e} << 0.005. The latter condition determines the boundaries of the spectral region of the medium semi-transparency, where the theoretical model is applicable.

A series of calculations using the Mie theory have demonstrated that, in a weakly absorbing medium, the absorption has almost no effect on the scattering of radiation by gas bubbles, and it is sufficient to treat the *Q*_{s}^{tr}(*x*) dependences for κ = 0 given in Fig. 2. The results of the calculations demonstrate that *Q*_{s}^{tr} is approximately constant when *x* > 10, and the limiting value of *Q*_{s}^{tr} corresponding to the range of geometrical optics can be estimated as follows:

(8) |

**Figure 2. Transport efficiency factor of scattering for gas bubbles in refracting and nonabsorbing medium.**

It is evident that *Q*_{s}^{tr} is independent of the bubble radius. Having substituted Eq. (8) into Eq. (5b) we obtain

(9) |

Let us compare the transport scattering coefficient with the absorption coefficient using the ratio between them

(10) |

It follows from Eq. (10) that scattering is predominant for the bubble volume fraction, *f*_{v} > 10κ_{e}*x*. With an invariable spectral index of absorption of the medium, the importance of scattering is defined by the ratio of the volume fraction of the bubbles to their radius: *f*_{v} / *a*. Note that the asymmetry factor of scattering is the same for all large bubbles and can be approximated as follows:

(11) |

The above results for radiation scattering by bubbles and the effect of the bubbles on the radiation absorption have been used in characterization of fused quartz containing gas bubbles (Dombrovsky et al., 2005), in radiation heat transfer analysis for water containing steam bubbles (Dombrovsky, 2004, 2007), and as a limiting solution for thin-wall glass microspheres in a polymer matrix (Dombrovsky, 2005; Dombrovsky et al., 2007). As was noted above, this solution is applicable only in the spectral ranges of host medium semi-transparency. In the opacity range, where the host medium is strongly absorbing, the radiation heat transfer problem totally degenerates and there is no sense of the medium containing bubbles in the volumetric radiative characteristics. This means that realistic heat transfer problems should be considered by a separate analysis of the spectral windows of the medium semi-transparency and for the bands of the medium opacity. This approach is realized in the applied problems considered in the articles Radiative effects in a semi-transparent liquid containing gas bubbles and Thermal radiation modeling in melt-coolant interaction. One can find more details on this subject in the study done by Dombrovsky and Baillis (2010).

At the end of this article, one should refer to the recent studies done by Durant et al. (2007a,b), which present a new approach to the long-standing problem of radiation extinction by an absorbing medium containing numerous randomly located particles of another substance. The extinction coefficient of the disperse system was obtained by Durant et al. (2007a,b) from an ensemble of the numerical solutions of Maxwell’s equations for many different realizations of the system by Monte Carlo simulation. The complex index of refraction of the host medium was either *m*_{e} = 1.33 - 0.02*i* or 1.33 - 0.03*i*. The index of refraction of nonabsorbing particles was either *n* = 1.55 or 2.8. The particle diffraction parameter was varied in the range from about 1 to 25. It was found that the independent scattering theory is in good agreement with the exact numerical solution at the volume fraction of particles less than about 5-10%. It was observed that the absorption of the host medium reduces the dependent scattering effects and makes the independent-scattering approximation applicable for larger volume fractions of particles.

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