LOW POROSITY FOAMS
Following from: Spectral radiative properties of some important materials: Experimental data and theoretical models
Leading to: Radiative properties of gas bubbles in semi-transparent medium, Typical glass foams
For low-porosity closed cell foams, bubbles or cavities can be treated as a dispersed phase, while the surrounding host medium is a continuous phase. In various situations, the presence of bubbles affects the thermophysical and radiative properties of the two-phase system. Earlier investigations pointed out the effects of bubbles on the scattering characteristics of semitransparent systems, particularly in the visible and infrared spectral ranges. In natural phenomena and the manufacturing process, the influence of bubbles on scattering of light in the ocean (Zhang et al., 1998) and the glass melting process in the industrial furnaces (Fedorov and Pilon, 2002) have been demonstrated. For thermal insulating materials, bubbles or hollow spheres are created (or placed) in a host medium in order to increase the radiation extinction by scattering (Dombrovsky et al., 2007).
It goes without saying that the spectral radiative properties of a porous medium depend on both the refraction and absorption of the radiation in a host medium. It is known that the general expressions for the spectral radiative properties of a polydisperse system of independently scattering and absorbing particles or bubbles can be written as follows (Bohren and Huffman, 1983; Dombrovsky, 1996):
(1) |
In Eq. (1), α_{0,λ} is the absorption coefficient of the homogeneous matrix, and N_{t} denotes the total number of particle per unit volume. The single particle radiative characteristics (C_{a,λ}, C_{s,λ}, and C_{e,λ} = C_{a,λ} + C_{s,λ}) and the scattering phase function (ϕ_{λ}) of the particle are obtained by analyzing the interaction with a plane incident wave. The first difficulty in these equations comes from the determination of the term, including the absorption cross-section C_{a,λ}. For the case of low-porosity, optically large, and non-absorbing bubbles in weakly absorbing matrix, Dombrovsky (2004) showed that this term reduces to:
(2) |
The case of bubbles of arbitrary size is considered in more detail in a recent study done by Dombrovsky and Baillis (2010). The reader is also referred to the study done by Randrianalisoa et al. (2006).
Both the absorption and scattering characteristics of single spherical particles or bubbles surrounded by an absorbing and refracting host medium can be calculated on the basis of a generalized Mie theory, as discussed in the article Radiative properties of gas bubbles in semi-transparent medium. Of course, this approach is applicable in the case of a weakly absorbing medium only when the main assumptions of the radiation transfer theory are true (Dombrovsky and Baillis, 2010)
In the limit of optically large particles of arbitrary shape, the geometric optic theory can be used. In this case, Randrianalisoa and Baillis (2010a,b, 2011) developed a new ray-tracing method to determine the effective radiative properties of particles (or bubbles) embedded in a semi-transparent matrix. Compared with previous modeling, it avoids the use of the independent scattering hypothesis (assuming that the bubbles are far from each other and they scatter as a point). Indeed, when the bubbles are close to each other (especially for moderate and high-porosity material), the bubbles cannot be viewed as point scatterers: there are shadowing effects (the totally bubble surface cannot receive the incoming radiation due to the presence of close neighboring bubbles) and transportation effects (when the radiation crosses a bubble, the distance between the entering and exiting points is not negligible compared with the interparticle distance). For optically large bubbles in a weakly and moderately absorbing host medium, the limit of validity of the independent scattering theory [i.e., Eq. (1)] was pointed out by determining the critical porosity (see the article Typical glass foams; Randrianalisoa and Baillis, 2010b).
It is interesting to note that the relatively simple approach based on the Mie theory can be used to analyze the radiative properties of low-porosity semi-transparent materials (Manara et al., 1999, 2007; Dombrovsky et al., 2005).
In most of the previous studies, low-porosity foams have been investigated where independent scattering theory and smooth medium boundaries have been assumed. Although the recent studies done by Randrianalisoa and Baillis (2010a,b) investigated the limit of validity of the independent scattering theory and determined the critical porosity, more complete investigation (beyond the optically large bubble limit, weakly absorbing host medium, and optically smooth interfaces) should be conducted. Among the potential approaches, the T-matrix method (Mishchenko et al., 2000) and the full-wave Monte Carlo method (Durand et al., 2007) should be considered.
REFERENCES
Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
Dombrovsky, L. A., The propagation of infrared radiation in a semitransparent liquid containing gas bubbles, High Temp., vol. 42, no. 1, pp. 133-139, 2004.
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., Infrared radiative properties of polymer coatings containing hollow microspheres. Int. J. Heat Mass Transfer, vol. 50, no. 7-8, pp. 1516-1527, 2007.
Durant, S., Calvo-Perez, O., Vukadinovich N., and Greffet J.-J. (2007) Light scattering by a random distribution of particles embedded in absorbing media: full wave Monte Carlo solutions of the extinction coefficient,J. Opt. Soc. Am. A, vol. 24, no. 9, pp. 2953-2962, 2007.
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.
Manara, J., Caps, R., Raether, F., and Fricke, J., Characterization of the pore structure of alumina ceramics by diffuse radiation propagation in the near infrared, Opt. Commun., vol. 168, no. 1-4, pp. 237-250, 1999.
Manara, J., Reidinger, M., Korder, S., Arduini-Schuster, M., and Fricke, J., Development and characterization of low-emitting ceramics, Int. J. Thermophys., vol. 28, no. 5, pp. 1628-1645, 2007.
Mishchenko, M. I., Travis, L. D., and Macke, A., T-matrix method and its applications, in Light scattering by nonspherical particles: theory, measurements, and applications, eds. Mishchenko, M. I., Travis, L. D., Hovenier, J. W., San Diego: Academic, pp. 147-172, 2000.
Randrianalisoa, J. and Baillis, D., Radiative transfer in dispersed media: Comparison between homogeneous phase and multiphase approaches, ASME J. Heat Transfer, vol. 132, no. 2, pp. 023405.1-023405.11, 2010a.
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, 2006.
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., Coquard, R., and Baillis, D., Radiative transfer in two-phase material, in Heat Transfer in Multi-Phase Materials, eds. Öchsner, A. and Murch, G. E., Springer, Heidelberg, Berlin, 2011.
Zhang, X., Lewis, M., and Johnson, B., Influence of bubbles on scattering of light in the ocean, Appl. Opt., vol. 37, no. 27, pp. 6525-6536. 1998.
References
- Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, New York: Wiley, 1983.
- Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, New York: Begell House, 1996.
- Dombrovsky, L. A., The propagation of infrared radiation in a semitransparent liquid containing gas bubbles, High Temp., vol. 42, no. 1, pp. 133-139, 2004.
- 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., Infrared radiative properties of polymer coatings containing hollow microspheres. Int. J. Heat Mass Transfer, vol. 50, no. 7-8, pp. 1516-1527, 2007.
- Durant, S., Calvo-Perez, O., Vukadinovich N., and Greffet J.-J. (2007) Light scattering by a random distribution of particles embedded in absorbing media: full wave Monte Carlo solutions of the extinction coefficient,J. Opt. Soc. Am. A, vol. 24, no. 9, pp. 2953-2962, 2007.
- 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.
- Manara, J., Caps, R., Raether, F., and Fricke, J., Characterization of the pore structure of alumina ceramics by diffuse radiation propagation in the near infrared, Opt. Commun., vol. 168, no. 1-4, pp. 237-250, 1999.
- Manara, J., Reidinger, M., Korder, S., Arduini-Schuster, M., and Fricke, J., Development and characterization of low-emitting ceramics, Int. J. Thermophys., vol. 28, no. 5, pp. 1628-1645, 2007.
- Mishchenko, M. I., Travis, L. D., and Macke, A., T-matrix method and its applications, in Light scattering by nonspherical particles: theory, measurements, and applications, eds. Mishchenko, M. I., Travis, L. D., Hovenier, J. W., San Diego: Academic, pp. 147-172, 2000.
- Randrianalisoa, J. and Baillis, D., Radiative transfer in dispersed media: Comparison between homogeneous phase and multiphase approaches, ASME J. Heat Transfer, vol. 132, no. 2, pp. 023405.1-023405.11, 2010a.
- 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, 2006.
- 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., Coquard, R., and Baillis, D., Radiative transfer in two-phase material, in Heat Transfer in Multi-Phase Materials, eds. Öchsner, A. and Murch, G. E., Springer, Heidelberg, Berlin, 2011.
- Zhang, X., Lewis, M., and Johnson, B., Influence of bubbles on scattering of light in the ocean, Appl. Opt., vol. 37, no. 27, pp. 6525-6536. 1998.