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In many natural phenomena, materials processing, and manufactural situations, gas bubbles can form in liquid and solid phases. Their presence affects the thermophysical and radiative properties of the two-phase system and, hence, the transport phenomena. The glass melting process in industrial furnaces where bubbles are generated by chemical reactions is a typical example. In the last decade, several studies have focused on the radiative properties of typical glass foams. Fedorov and Viskanta (2000), Fedorov and Pilon (2002), and Pilon and Viskanta (2003) have been interested in low-density glass foam (about 10-30%). Rousseau et al. (2007a,b), Baillis et al. (2004), Dombrovsky et al. (2005), and Randrianalisoa et al. (2006a,b) have studied high-density foams (87-95%) when the volume fraction of gas bubbles is small.

These porous media are often treated as continuous media that refract, absorb, and scatter radiation. To analyze the combined heat transfer in a semi-transparent material of this type, one should determine the spectral radiative characteristics of the material. This can be done using different approaches:

  1. One can use the Mie theory or some particular approximation (like anomalous diffraction) to calculate the medium radiative characteristics at the known porosity and size distribution of spherical bubbles. In many cases, it is sufficient to use the hypothesis of independent scattering. For large bubbles, one can also employ a Monte Carlo simulation to take into account some effects of dependent scattering (Randrianalisoa and Baillis, 2010a,b). The theoretical basis of this approach was considered in some detail recently by Dombrovsky and Baillis (2010). The following Thermopedia articles are also recommended: Radiative properties of single particles and fibers: The hypothesis of independent scattering and the Mie theory, Geometrical optics approximation, Anomalous diffraction, Radiative properties of gas bubbles in semi-transparent medium, Radiative properties of polydisperse systems of independent particles.
  2. One can use the measurements of bi-directional or directional-hemispherical characteristics (both reflectance and transmittance) of glass foam samples in combination with a subsequent identification of the radiative properties using a mathematical procedure for the inverse problem solving (Baillis-Doermann and Sacadura, 2000; Baillis et al., 2004; Randrianalisoa et al., 2006b).
  3. A combined approach based on a combination of the Mie theory predictions and a simplified experimental/identification procedure appears to be very efficient in the case of a low volume fraction of gas bubbles (Dombrovsky et al., 2005). This method is reported in the article Semi-transparent media containing bubbles (for more detail, see Dombrovsky and Baillis, 2010). Note that the results of directional-hemispherical measurements allow employing the modified two-flux approximation developed by Dombrovsky et al. (2005, 2006) to identify the main spectral radiative properties of the medium.

Here, we will not reproduce the known relations for radiative properties of a refracting and weakly absorbing host medium containing polydisperse gas bubbles. These relations can be found elsewhere. A complete set of these relations, and also a comprehensive overview of the literature on this subject, have been given recently by Dombrovsky and Baillis (2010). However, a brief overview of the identification methods is given below.

The radiative characteristics, such as the extinction coefficient, the scattering albedo, and the scattering phase function of fused quartz containing bubbles, are determined by using an inverse method based on theoretical and experimental bi-directional transmittances (Baillis et al., 2004; Randrianalisoa et al., 2006,b). The experimental spectral bi-directional transmittance data were obtained from an experimental setup that includes a Fourier-transform infrared spectrometer (FTS 60 A, Bio-Rad, Inc.) (Baillis et al., 2004). The theoretical transmittances were obtained by solving the radiative transfer equation using the discrete ordinate method. Randrianalisoa et al. (2006b) improved the first inverse method of Baillis et al. (2004) by using a more accurate phase function and an adaptive composite quadrature to compute more precisely the intensities in the measurement direction. This adaptive quadrature enables accounting for the abrupt change of radiation direction at sample interfaces due to Fresnel refraction. In addition, a two-step inverse method was proposed to compute accurately and simultaneously the radiative parameters.

The improvements proposed by Randrianalisoa et al. (2006b), especially the scattering phase function, enable appropriately reproducing the scattering pattern of the samples. This issue results in a more realistic absorption coefficient. A comparison between the radiative properties in terms of the transport quantities obtained from the inverse method (see the symbols in Figs. 1 and 2) and the Mie theory approach (see the straight curves in Figs. 1 and 2) shows good agreement. Moreover, an exhaustive comparison between the experimental measurements of hemispherical transmittance and reflectance and the computational results (using the identified radiative characteristics) confirms the validity of the improved identification approach. Figure 3 depicts this comparison study for the case of a 9.9-mm-thick sample.

Figure 1. Comparison between the prediction and experimental data of the transport extinction coefficient of porous fused quartz.

Figure 2. Comparison between the prediction and experimental data of the transport scattering albedo of porous fused quartz.

Figure 3. Comparison between the computed and measured hemispherical transmittance for a 9.9-mm-thick sample.

It should be noted that a low-density glass foam containing spherical gas bubbles (see Fig. 4) is an excellent example, which seems to very interesting from the educational point of view. The reader can use a complete set of experimental and theoretical tools to study the visible and near-infrared properties of this semi-transparent material. It was useful for us to compare various approaches, both theoretical and experimental. No doubt, it would be also an interesting experience for potential readers of this article.

Figure 4. Close-up photograph of a fused quartz sample containing gas bubbles (the background is blue sky).

REFERENCES

Baillis, D., Pilon, L., Randrianalisoa, H., Gomez, R., and Viskanta, R., Measurements of radiation characteristics of fused quartz containing bubbles, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 149-159, 2004.

Baillis-Doermann, D. and Sacadura, J.-F., Thermal radiation properties of dispersed media: Theoretical prediction and experimental characterization, J. Quant. Spectrosc. Radiat. Transf., vol. 67, no. 5, pp. 327-363, 2000.

Dombrovsky L. A., and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Redding, CT: Begell House, 2010.

Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.

Dombrovsky, L., Randrianalisoa, J., and Baillis, D., Modified two-flux approximation for identification of radiative properties of absorbing and scattering media from directional-hemispherical measurements, J. Opt. Soc. Am. A, vol. 23, no. 1, pp. 91-98, 2006.

Fedorov, A. G. and Viskanta, L., Radiation characteristics of glass foam, J. Am. Ceram. Soc., vol. 83, no. 11, pp. 2769-76, 2000.

Fedorov, A. G. and Pilon, L., Glass foam: Formation, transport properties, and heat, mass, and radiation transfer, J. Non-Cryst. Solids, vol. 311, no. 2, pp. 154-173, 2002.

Pilon, L. and Viskanta, R., Radiation characteristics of glass containing bubbles, J. Am. Ceram. Soc., vol. 86, no. 8, pp. 1313-1320, 2003.

Randrianalisoa, J. and Baillis, D., Radiative transfer in dispersed media: Comparison between homogeneous phase and multiphase approaches, J. Heat Transfer, vol. 132, no. 2, pp. 023405.1-023405.11, 2010a.

Randrianalisoa, J. and Baillis, D., Radiative properties of densely packed spheres in semitransparent media: A new geometric optics approach, J. Quant. Spectrosc. Radiat. Transf., vol. 111, no. 10, pp. 1372-1388, 2010b.

Randrianalisoa, J., Baillis, D., and Pilon, L., Modeling radiation characteristics of semitransparent media containing bubbles or particles, J. Opt. Soc. Am. A, vol. 23, no. 7, pp. 1645-1656, 2006a.

Randrianalisoa, J., Baillis, D., and Pilon, L., Improved inverse method for radiative characteristics of closed-cell absorbing porous media, J. Thermophys. Heat Transfer, vol. 20, no. 4, pp. 871-883, 2006b.

Rousseau, B., Canizares, A., De Sousa Meneses, D., Matzen, G., Echegut, P., Di Michiel, M., and Thovert, J. F., Direct simulation of the high temperature optical behaviour of a porous medium based on a CT image, Colloids Surf., A, vol. 300, no. 1-2, pp. 162-168, 2007a.

Rousseau, B., De Sousa Meneses, D., Echegut, P., Di Michiel, M., and Thovert, J. F., Prediction of the thermal radiative properties of an X-ray μ-tomographied porous silica glass, Appl. Opt., vol. 46, no. 20, pp. 4266-4276, 2007b.

References

  1. Baillis, D., Pilon, L., Randrianalisoa, H., Gomez, R., and Viskanta, R., Measurements of radiation characteristics of fused quartz containing bubbles, J. Opt. Soc. Am. A, vol. 21, no. 1, pp. 149-159, 2004.
  2. Baillis-Doermann, D. and Sacadura, J.-F., Thermal radiation properties of dispersed media: Theoretical prediction and experimental characterization, J. Quant. Spectrosc. Radiat. Transf., vol. 67, no. 5, pp. 327-363, 2000.
  3. Dombrovsky L. A., and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Redding, CT: Begell House, 2010.
  4. Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie theory to analyze experimental data to identify infrared properties of fused quartz containing bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.
  5. Dombrovsky, L., Randrianalisoa, J., and Baillis, D., Modified two-flux approximation for identification of radiative properties of absorbing and scattering media from directional-hemispherical measurements, J. Opt. Soc. Am. A, vol. 23, no. 1, pp. 91-98, 2006.
  6. Fedorov, A. G. and Viskanta, L., Radiation characteristics of glass foam, J. Am. Ceram. Soc., vol. 83, no. 11, pp. 2769-76, 2000.
  7. Fedorov, A. G. and Pilon, L., Glass foam: Formation, transport properties, and heat, mass, and radiation transfer, J. Non-Cryst. Solids, vol. 311, no. 2, pp. 154-173, 2002.
  8. Pilon, L. and Viskanta, R., Radiation characteristics of glass containing bubbles, J. Am. Ceram. Soc., vol. 86, no. 8, pp. 1313-1320, 2003.
  9. Randrianalisoa, J. and Baillis, D., Radiative transfer in dispersed media: Comparison between homogeneous phase and multiphase approaches, J. Heat Transfer, vol. 132, no. 2, pp. 023405.1-023405.11, 2010a.
  10. Randrianalisoa, J. and Baillis, D., Radiative properties of densely packed spheres in semitransparent media: A new geometric optics approach, J. Quant. Spectrosc. Radiat. Transf., vol. 111, no. 10, pp. 1372-1388, 2010b.
  11. Randrianalisoa, J., Baillis, D., and Pilon, L., Modeling radiation characteristics of semitransparent media containing bubbles or particles, J. Opt. Soc. Am. A, vol. 23, no. 7, pp. 1645-1656, 2006a.
  12. Randrianalisoa, J., Baillis, D., and Pilon, L., Improved inverse method for radiative characteristics of closed-cell absorbing porous media, J. Thermophys. Heat Transfer, vol. 20, no. 4, pp. 871-883, 2006b.
  13. Rousseau, B., Canizares, A., De Sousa Meneses, D., Matzen, G., Echegut, P., Di Michiel, M., and Thovert, J. F., Direct simulation of the high temperature optical behaviour of a porous medium based on a CT image, Colloids Surf., A, vol. 300, no. 1-2, pp. 162-168, 2007a.
  14. Rousseau, B., De Sousa Meneses, D., Echegut, P., Di Michiel, M., and Thovert, J. F., Prediction of the thermal radiative properties of an X-ray μ-tomographied porous silica glass, Appl. Opt., vol. 46, no. 20, pp. 4266-4276, 2007b.
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