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).
We do not consider the case of a strongly absorbing host medium. Therefore, the simple formula (2) will be used in subsequent calculations.
where, according to Fedorov and Viskanta (2000a,b), Qab and Qsb were calculated by the complex index of refraction for gas mb; Qag was calculated by the complex index of refraction for glass mg; and the diffraction parameter was taken to be the same in both cases, x = 2πa / 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 |mg - 1| << 1, κgx << 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 mb / mg, not on the value of mb.
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 κx < 0.01.
Figure 1. Relative absorption efficiency for gas bubbles in refracting and absorbing medium.
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, κe << 0.005. The latter condition determines the boundaries of the spectral region of the medium semi-transparency, where the theoretical model is applicable.
Figure 2. Transport efficiency factor of scattering for gas bubbles in refracting and nonabsorbing medium.
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 me = 1.33 - 0.02i or 1.33 - 0.03i. 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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